HYBRID SUPERCONDUCTING QUANTUM COMPUTINGARCHITECTURES

by

Matthew A. Beck

A dissertation submitted in partial fulfillment ofthe requirements for the degree of

Doctor of Philosophy

(Physics)

at the

UNIVERSITY OF WISCONSIN–MADISON

2018

Date of final oral examination: 6/21/18

The dissertation is approved by the following members of the Final Oral Committee:Robert F. McDermott, Professor, PhysicsMark Saffman, Professor, PhysicsMark A. Eriksson, Professor, PhysicsMaxim G. Vavilov, Professor, PhysicsMikhail Kats, Assistant Professor, Electrical and Computer Engineering

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i

Abstract

Quantum computing holds the promise to address and solve computational problems that

are otherwise intractable on a classical, transistor based machine. While much progress has

been made in the last decade towards the realization of a scalable superconducting quantum

processor, many questions remain unanswered. The work contained in this thesis addresses

two equally important concerns; These are specifically that of quantum information storage

and transfer and the scaling of current qubit control and readout methods.

Superconducting quantum processors are exactly what their name implies: processors.

While the goal is to eventually build a universal quantum computer, it is not unreasonable to

envision near term quantum processors hard wired to perform specific computational tasks.

This idea of compartmentalized quantum processing necessitates that the quantum results

of a computation either be stored and/or transferred for latter / further use. A natural

candidate to realize such a quantum memory is the neutral Rydberg atom. The hyperfine

states of cesium atoms exhibit coherence times greater than 1 second while adjacent Rydberg

energy levels have electric dipole transitions in the gigahertz regime; These properties make

it a suitable candidate to realize a quantum memory and information bus between adjacent

superconducting processors yielding an unprecedented ratio of coherence time to gate time.

To realize such a computing architecture, the coherent coupling between a single Rydberg

atom and superconducting bus resonator must first be demonstrated. This first half of this

thesis details the development of a superconducting interface meant to realize strong coupling

to a single Rydberg atom.

To date, the experimental liquid Helium 4 K UHV cryostat has been built, characterized,

ii

and installed. Superconducting niobium coplanar waveguide (CPW) resonators have been

designed and fabricated to facilitate strong coupling to the Rydberg atom through on-chip

microwave field engineering. Additionally, the CPW resonators have been tailored to achieve

quality factors above 104 at 4 K. The project is currently still on-going with single-atom

trapping and state characterization near the 4 K chip surface under investigation.

The second portion of this thesis details the development of a superconducting single flux

quantum (SFQ) pulse generator for transmon qubit control. As the size of superconducting

quantum processors scales beyond the level of a few tens of qubits, the control hardware

overhead becomes untenable. For current technology based on microwave control pulses

generated at room temperature followed by amplification and heterodyne detection, the

heat load and physical footprint of the required classical hardware preclude brute force

scaling to qubit arrays more than ∼ 100. The work contained herein details the development,

fabrication, characterization and finally integration of a dc/SFQ driver with a transmon

qubit on a single chip as a first step towards an all superconducting digital control scheme of

quantum processors.

Details of the multi-additive layer processing and fabrication required to realize these

devices are discussed in the context of maintaining high (> 10 us) qubit coherent times and

small superconducting resonator loss. To date, coherent qubit rotations have been achieved

via application of SFQ pulses with pulse to pulse spacing aligned with subharmonics of the

qubit frequency. Interleaved randomized benchmarking (RB) of SFQ driven single qubit gates

realized are currently at 90% level. Future plans regarding a flip chip / multi-chip module

approach to increasing gate fidelities will also be discussed.

iii

Acknowledgments

I would like to take just a few lines to give thanks to some notable people without whom this

thesis and the work contained herein would not be possible.

First and foremost, I would like to extend my deepest gratitude to my advisor Dr. Robert

F. McDermott. It has been with Robert’s guidance that I have learned how to be a better

scientist.

To my graduate student forefathers in the group, Dr. David Hover, Dr. Steven Sendelbach,

Dr. Yuanfeng Gao and Dr. Guilhem Ribeill, I would like to take a sentence or two and

extend many thanks to you all for your patience and guidance during my first few years in

the group. The debt I owe you I hopefully can repay through forwarding these lessons to

younger students.

With regards to the hybrid atom–CPW project, I would like to thank the tireless efforts

of Jonathan Pritchard, Joshua Isaacs and P.I. Mark Saffman of the atomic team. You three

have been a pleasure to work with.

This acknowledgment section (nor this thesis) could not have been completed without a

mention of the Edward M. Leonard, Jr. The past few years working side by side in the lab

and clean room have been the most enjoyable professional experience of my life. Your tireless

efforts in always striving for cleaner samples, cleaner measurements and cleaner code have

been an inspiration. Your leadership in the lab has be a shining example to all in just how

much one person can achieve. I, as does the entire lab, owe you a great deal of gratitude.

I also would like to take a moment and thank the cowboys of the McDermott Family

Research Ranch, Alex, J.P., Nathan, Naveen, Ted, Brad, Ryan, and Joey. Thank you all for

iv

the many riveting discussions and helping to maintain a scalable lab.

Raising a child through undergraduate and graduate study could not have been possible

without the endearing love and support of my parents. I wish to give many thanks to my

mother who has been a bedrock of patience and understanding and to my father who showed

me what it means (and takes) to be a good man and father.

And finally, Alyssa and Zoe. Alyssa, you have been the love of my life since we were kids.

Thank you for everything you do everyday and showing me what happiness truly is. Zoe,

watching you grow up into the wonderful young adult you are has been a privilege. I can only

hope to teach you the number of lessons you have taught me. With that, this is for you, kid.

v

To my family.

vi

Contents

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Physical Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Quantum Bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Qubit Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Physical Qubit Realizations . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Neutral Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 Superconducting Qubits . . . . . . . . . . . . . . . . . . . . . . 91.3.3 Current Hybrid Quantum Computing Approaches . . . . . . . . 12

2 Josephson Physics and Superconducting Circuit Fundamentals . . . 142.1 Basic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 RCSJ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 Superconducting Quantum Interference Devices . . . . . . . . . 20

2.2 Quantization Of Electrical Circuits . . . . . . . . . . . . . . . . . . . . 222.3 Superconducting Qubit Zoo . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1 Phase qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.2 Flux Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.3 Transmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Circuit Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . 293.1 Cavity–Qubit Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1.1 Resonant Jaynes–Cummings Hamiltonian . . . . . . . . . . . . . 333.1.2 Dispersive Jaynes-Cummings Hamiltonian . . . . . . . . . . . . 34

vii

3.2 Qubit-Cavity Bringup . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.1 Coupling The Qubit–Cavity System to the Outside World . . . 373.2.2 Qubit Signs of Life . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.3 Finding the Qubit Frequency . . . . . . . . . . . . . . . . . . . 453.2.4 Orthogonal Qubit Axis Control . . . . . . . . . . . . . . . . . . 45

3.3 Qubit and Qubit Gate Benchmarking . . . . . . . . . . . . . . . . . . . 493.3.1 Characterizing and Calibrating the Qubit States . . . . . . . . . 513.3.2 Randomized Benchmarking of Qubit Gates . . . . . . . . . . . . 53

4 Hybrid Atom–CPW cQED Experiment . . . . . . . . . . . . . . . . . 574.1 By The Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.1 Cryostat Performance . . . . . . . . . . . . . . . . . . . . . . . 674.3 Superconducting CPW Resonator Design . . . . . . . . . . . . . . . . . 68

4.3.1 Engineering the Quality Factor . . . . . . . . . . . . . . . . . . 714.3.2 Measuring the Quality Factor . . . . . . . . . . . . . . . . . . . 784.3.3 Engineering the atom-CPW coupling strength . . . . . . . . . . 814.3.4 Stray Electric Field Compensation . . . . . . . . . . . . . . . . 91

4.4 Current Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 Single Flux Quantum Qubit Control . . . . . . . . . . . . . . . . . . . 945.1 Superconducting Supercomputing with Single Flux Quanta . . . . . . . 95

5.1.1 Rapid Single Flux Quantum . . . . . . . . . . . . . . . . . . . . 985.1.2 Energy Efficient SFQ . . . . . . . . . . . . . . . . . . . . . . . . 985.1.3 Reciprocal Quantum Logic . . . . . . . . . . . . . . . . . . . . . 98

5.2 SFQ Control of a Transmon Qubit . . . . . . . . . . . . . . . . . . . . 995.3 Bridging the Quantum–Classical Divide . . . . . . . . . . . . . . . . . . 103

5.3.1 SFQ Driver Design . . . . . . . . . . . . . . . . . . . . . . . . . 1045.3.2 SFQ–Qubit Circuit Layout . . . . . . . . . . . . . . . . . . . . . 1115.3.3 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.4 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.4.1 LHe SFQ Circuit Pre-screening . . . . . . . . . . . . . . . . . . 1215.4.2 100 mK Quantum Circuit Pre-screening . . . . . . . . . . . . . 1215.4.3 Initial SFQ–Qubit Bringup . . . . . . . . . . . . . . . . . . . . . 1245.4.4 Establishing Orthogonal Axis Control . . . . . . . . . . . . . . . 1335.4.5 Randomized Benchmarking of SFQ–Qubit Gates . . . . . . . . . 1375.4.6 SFQ Generated Quasiparticle Studies . . . . . . . . . . . . . . . 140

viii

5.5 Next Generation Design and Outlook . . . . . . . . . . . . . . . . . . . 1485.5.1 Flip Chip Design . . . . . . . . . . . . . . . . . . . . . . . . . . 148

A Josephson Energy of a dc SQUID . . . . . . . . . . . . . . . . . . . . . 151

B Preserving High Quality Nb . . . . . . . . . . . . . . . . . . . . . . . . . 154

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

ix

List of Tables

3.1 Single Qubit Cliffords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1 Bell State density matrix Lindblad superoperators . . . . . . . . . . . . . . . . . 614.2 Cryostat material leak-up rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3 Nb Plasma Etch Recipe Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 784.4 COMSOL Simulated Electric Field Compensation Parameters . . . . . . . . . . 92

5.1 dc/SFQ Circuit Inductances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.2 Chip Fabrication Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.3 SFQ gate fidelities determined with interleave randomized benchmarking for the

n = 3 and n = 41 subharmonic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

x

List of Figures

1-1 Bloch Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61-2 Qubit T1 / T ∗

2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71-3 Cesium Level Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101-4 Superconducting Z-Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2-1 Josephson Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162-2 Tilted Washboard Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182-3 Josephson IV Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3-1 cQED Circuit Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313-2 Qubit-Cavity Avoided Level Crossing . . . . . . . . . . . . . . . . . . . . . . . . 343-3 Inductive / Capactive Resonator Coupling . . . . . . . . . . . . . . . . . . . . . 383-4 Measured Resonator–Qubit Spectrum vs. Readout Power . . . . . . . . . . . . . 423-5 Resonator Spectroscopy Vs. Qubit Flux-Bias Current . . . . . . . . . . . . . . . 433-6 Qubit Spectroscopy Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463-7 Dilution Refrigerator Wiring Schematic . . . . . . . . . . . . . . . . . . . . . . . 503-8 Qubit I/π IQ Blob Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 513-9 Randomized Benchmarking of a Microwave X/2 Gate . . . . . . . . . . . . . . . 55

4-1 Controlled Phase Gate for Atom-CPW . . . . . . . . . . . . . . . . . . . . . . . 584-2 Bell state preparation fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624-3 Custom UHV Cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644-4 Cryostat Cold Finger Vibration Measurement . . . . . . . . . . . . . . . . . . . 694-5 Bell State Preparation Fidelity Vs. Temperature . . . . . . . . . . . . . . . . . . 704-6 Complex Conductivity Vs. Temperature . . . . . . . . . . . . . . . . . . . . . . 744-7 CPW Kinetic Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774-8 Predicted CPW Quality Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 794-9 Multiplexed Quarter-wave Resonators . . . . . . . . . . . . . . . . . . . . . . . . 804-10 Measured CPW Quality Factor Vs. CPW Geometry . . . . . . . . . . . . . . . . 82

xi

4-11 CPW �E Field Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834-12 Plated CPW Voltage Antinode Field Profile . . . . . . . . . . . . . . . . . . . . 854-13 Process for electroplating Cu on Nb . . . . . . . . . . . . . . . . . . . . . . . . . 874-14 Atom–Resonator micrograph and data . . . . . . . . . . . . . . . . . . . . . . . 904-15 Sample Mount / DC Compensation Pins . . . . . . . . . . . . . . . . . . . . . . 924-16 Image of Cs MOT trapped directly beneath superconducting sample . . . . . . . 93

5-1 JJ SFQ Phase Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965-2 Basic SFQ circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975-3 SFQ Driven Qubit Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015-4 Qubit State Leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025-5 dc/SFQ Driver Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055-6 dc/SFQ Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075-7 dc/SFQ driver WRSpice simulations . . . . . . . . . . . . . . . . . . . . . . . . 1095-8 dc/SFQ driver WRSpice simulations . . . . . . . . . . . . . . . . . . . . . . . . 1125-9 Fully Fabricated SFQ–Qubit Sample . . . . . . . . . . . . . . . . . . . . . . . . 1205-10 4.2 K Measurement IV Measurement . . . . . . . . . . . . . . . . . . . . . . . . 1225-11 Bonded Sample / ADR Prescreen Wiring Diagram . . . . . . . . . . . . . . . . 1235-12 Hysteretic dc/SFQ Driver at T = 100 mK . . . . . . . . . . . . . . . . . . . . . 1255-13 Thermally Cycled Shunt Resistor . . . . . . . . . . . . . . . . . . . . . . . . . . 1255-14 Initial SFQ–Qubit DR Wiring Diagram . . . . . . . . . . . . . . . . . . . . . . . 1275-15 Heavily processed qubit lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . 1285-16 Heavily processed qubit lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . 1305-17 Qubit Subharmonic Forest / Digital Rabi Oscillation . . . . . . . . . . . . . . . 1325-18 Wiring diagram with mixing on SFQ drive line . . . . . . . . . . . . . . . . . . 1345-19 Orthogonal SFQ Axis Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355-20 Orthogonal SFQ Control Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365-21 Application and Inverse of single qubit SFQ–Cliffords . . . . . . . . . . . . . . . 1385-22 Qubit T1 with and without background SFQ driving . . . . . . . . . . . . . . . 1395-23 Randomized benchmarking of SFQ driven gates . . . . . . . . . . . . . . . . . . 1405-24 Quasiparticle T1 poisoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435-25 Quasiparticle Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445-26 Quasiparticle induced qubit frequency shifts . . . . . . . . . . . . . . . . . . . . 1455-27 dωq Vs. Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475-28 dωqb Vs. Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

B-1 Al Fabrication Surface and x–ray Spectrogram . . . . . . . . . . . . . . . . . . . 155

xii

B-2 No Protection Layer Surface and x–ray Spectrogram . . . . . . . . . . . . . . . 157

1

Chapter 1

Introduction

There’s plenty of room at the bottom.

— R. P. Feynman (1959)

The meteoric rise in computing power seen in the last four decades has been accompanied

by an ever shrinking physical footprint in the size of the transistor. Moore’s law, named for

Intel founder Robert Moore, has accurately predicted roughly a doubling in the transistor

count per integrated circuit every 2 years since 1965. However, as the size of the transistor

becomes of order the electron wave function, the technological hurtles associated with

increasing the transistor count by even a few percent become seriously daunting. This

physical challenge is not alone in limiting the computing power of a classical transistor based

machine. There exists classes of computational problems whereby the solutions cannot be

calculated either efficiently (i.e. in polynomial time) by a classical machine or at all. To this

end, the control and readout of individual quantum systems has risen from scientific curiosity

to full scale research endeavors conducted by multiple universities and private companies in

an effort to build a quantum computer.

The work contained herein focuses on two different quantum computing architectures

utilizing superconducting circuits. The development and employment of a superconducting

CPW resonator photon bus between a neutral Rydberg atom and superconducting quantum

bit (qubit) is discussed. Additionally, the theory and realization of coherent qubit control via

2

Single Flux Quantum (SFQ) pulses is addressed.

1.1 Physical Information

The fundamental function of science is the extraction of information from an unknown

system. Systems may be left devoid of external perturbations and monitored in effort to

gain information of the so called “steady state.” Perturbations may also be applied in an

effort to gain insight concerning dynamical behavior subject to external stimuli. Whatever

the experiment may be, thermal dynamics, and more fundamentally, quantum mechanics, set

bounds on the amount of information one may ultimately extract [1].

The link between information and its physicality becomes apparent if one deals in system

entropies. In statistical thermodynamics, a system comprised of a multitude of particles can

be described in macroscopic ensembles of microscopic states (“microstates”) [2]. In 1902,

Gibbs generalized the work of Boltzmann in deriving a general form for the entropy that does

not require thermodynamic isolation. Specifically, the Gibbs ensemble

Ω ≡ NA!ΠNS!

, (1.1)

where NS is the number degeneracy of microstates with energy ES and NA = ∑SNS. Substi-

tuting Eq. (1.1) into the Boltzmann entropy equation yields

S ≡ kB log(Ω) = kB [log(NA!) − log(ΠNS!)] = −kB∑S

PS log(PS) , (1.2)

where PS = NS/NA is the probability of observing the system in state S. Equation Eq.(1.2) is

formally known as the Gibbs entropy. In the limit that all k possible microstates exist with

equal probability PS = 1/k, then Eq. (1.2) reduces to that derived by Boltzmann

S = kB log(k) (1.3)

3

The extension of Eq. (1.2) to the quantum mechanical case is straightforward. We wish

to describe the amount of information available to the experimenter for a given quantum

mechanical system. Most generally, we can describe any quantum state via its density matrix

ρ =∑

i

pi |ψi〉 〈ψi| , (1.4)

where pi is the probability of measuring the system ensemble in state |ψi〉. Physicality forcesnormalization upon the ensemble such that the sum of the probabilities equates to one

∑i

pi = Tr[ρ] = 1 . (1.5)

Additionally, one may make the distinction of pure states. Pure quantum mechanical states

have exactly one non-zero term in the density matrix.† Physically, this means that the state

can be completely described by one wave function |ψ〉 = ∑i ai |φi〉 with the condition that

∑ |ai|2 = 1. Mathematically, state purity may be defined as

Tr[ρ2] ≤ 1 (1.6)

where the equality holds for a pure state, ρ2 = ρ. Utilizing the definition of Eq. (1.4), one

can immediately write down the entropy for a general quantum state, first defined by Von

Neumann [1]

SVN = −∑i

pi log(pi) = −Tr [ρ log(ρ)] . (1.7)

The Von Neumann entropy takes on values 0 ≤ SVN ≤ log(k) where the lower bound is

saturated for pure states (log(1) = 0) and the upper bound is achieved for maximally mixed

states (pi = p = 1/k). When the upper bound is saturated, the probability of measuring any

microstate is completely random and thus any information concerning the macrostate is lost.†Also known as the "Schmidt Number" [3]

4

If we define the amount of information to be measured in terms of the maximum amount of

information to be lost, we can define a physical systems “free information”

B = Smax − SVN = log(k) + Tr [ρlog(ρ)] . (1.8)

While somewhat odd looking, Eq. (1.8) can be thought of as the informational equivalent to

the Helmholtz free energy in thermodynamics. The extension of this formalism to computing

and specifically the binary representation of information in what is now called “information

theory” was pioneered by Shannon in 1948 after discussions with Von Neumann [4].

1.2 Quantum Bits

With the ability to harness quantum superposition, to describe qubits one must change from

a discrete set of either logical 0 or logical 1 to a more continuous representation. A qubit

state can be generalized as a superposition of |0〉 and |1〉 up to a global phase

|ψ〉 = cos(θ/2) |0〉 + sin(θ/2)eiφ |1〉 . (1.9)

Borrowed from the nuclear magnetic resonance field, the Bloch sphere [5] provides a graphical

representation of a qubit state (see Fig. 1-1). A Bloch vector of unit length corresponds to

a pure quantum mechanical state while lengths less than unity describe “mixed” states. A

Bloch vector length of 0 is said to be maximally mixed. Single qubit logic gates are enacted

via rotations about the axes. Rotations taking one logical state to another are oft referred to

5

as π-gates and are represented by the X- and Y-Pauli matrices

X |0〉 =

⎡⎢⎢⎣0 1

1 0

⎤⎥⎥⎦⎡⎢⎢⎣10

⎤⎥⎥⎦ =

⎡⎢⎢⎣01

⎤⎥⎥⎦ = |1〉 (1.10)

Y |0〉 =

⎡⎢⎢⎣0 −i

i 0

⎤⎥⎥⎦⎡⎢⎢⎣10

⎤⎥⎥⎦ =

⎡⎢⎢⎣0

i

⎤⎥⎥⎦ = i |1〉 . (1.11)

The X and Y gates are the quantum mechanical equivalent of NOT gates for quantum bits

with a 90 degree phase between their axes of rotation. One may also rotate the qubit vector

onto the equator of the Bloch sphere, creating a superposition of |0〉 and |1〉 via a Hadamardgate,

H |0〉 = 1√2

⎡⎢⎢⎣1 1

1 −1

⎤⎥⎥⎦⎡⎢⎢⎣10

⎤⎥⎥⎦ = 1√

2

⎡⎢⎢⎣11

⎤⎥⎥⎦ = 1√

2(|0〉 + |1〉) . (1.12)

It is important to note that, by definition, any valid qubit gate A be unitary and thus preserve

the qubit vector length [6]

A†A = 1 . (1.13)

Not only are the relative amplitudes between the qubit states crucially important but also

the relative phase between them. Quantification of a qubits performance is discussed in the

next section.

1.2.1 Qubit Metrics

Perfect qubits would allow for the application of quantum gates such that arbitrary superpo-

sitions of |0〉 and |1〉 would remain in the predefined state until further acted upon and/or

measured. However, qubits are not immune from environmental perturbations [7], thus qubit

states are not infinitely long lived. Qubit performance is characterized via two physical

processes giving rise to two relevant timescales. The first, denoted as T1, encompasses the

rate at which the qubit relaxes from the excited state to the ground state, irreversibly losing

6

Figure 1-1: Bloch sphere representation of a qubit. For pure states, the vector length is 1and lives on the surface of the sphere.

energy to the surrounding environment. The rate, as described by Fermi’s golden rule, is

constant in time. The second, commonly denoted to as T ∗2 or Tφ, describes the process at

which the phase between qubit states becomes no longer well defined and coherence is lost.

The measurement of either time involves inducing coherent rotations about the Bloch

sphere followed by an adjustable idle time before finally concluding with qubit measurement.

The general procedures for measuring T1 and T ∗2 are displayed in Figures 1-2(a) and 1-2(b),

respectively. In a T1 measurement, the qubit is first coherently driven into the |1〉 statevia a π–rotation. It is then subsequently allowed to relax for increasing intervals of time

before a measurement of the state is made. The probability of measuring the qubit in the

|1〉 state is then plotted against the interleaved idle time between the initial gate and the

measurement and fit to a decaying exponential. The dephasing time T ∗2 is measured in

very much a similar manner except that the idle gate is between two π/2-rotations. In the

rotating frame, when the qubit Bloch vector is placed on the equator, noise induces incoherent

rotations about the �z axis. These incoherent kicks of the vector randomize the phase between

7

00.10.20.30.40.50.60.70.80.91

00.10.20.30.40.50.60.70.80.91

(a)

(b)

Figure 1-2: Qubit lifetime measurement protocols. (a) Qubit relaxation is measured via aπ-gate followed by a variable delay τ prior to measurement. (b) Qubit dephasing is measuredby a pair of π/2-gates with a variable delay interceded between them. Both measurementsrequire many repetitions.

the two computational states until such a time that all phase coherence between the qubit

states is lost and the probability of measuring the |1〉 state decays exponentially to 0.5 wherethen the qubit is maximally mixed. In any physical implementation, qubit measurements

project the state of the qubit onto either |0〉 or |1〉, thus measurements must be repeatedmany times over in order to build up necessary statistics.

1.3 Physical Qubit Realizations

Any physical quantum computing implementation must have particular attributes in order to

be considered a viable pathway to a large scale quantum computing architecture [8]. Known

as the “DiVincenzo criteria”, five necessary conditions must be met:

1. Physically realizable qubits that can be integrated into a larger system.

2. Qubit initialization must be robust.

8

3. Decoherence times (T1, T ∗2 ) must be longer than the measurement times

4. A set of universal 1 and 2 qubit quantum gates.

5. The qubits must be able to be measured.

In the framework of these criteria, current efforts in neutral atoms and superconducting

qubits are briefly reviewed.

1.3.1 Neutral Atoms

The neutral atom approach to quantum information utilizes adjacent hyperfine ground state

energy levels to encode logical qubit states [9]. A thorough review on the state of the art

is provided in [10]. The inherent atomic species homogeneity and weak magnetic dipole

transition between qubit states provides a double-edged sword in the context of quantum

information processing and the DiVincenzo criterion. Considering criteria 2 and 3, qubit

state preparation via optical pumping is robust against atom exchange and decoherence

times are intimately related with how strongly the system interacts with it’s environment.

Atomic qubit state lifetimes have now reached the ∼ 10 ms range [11]. However, the same

qubit homogeneity afforded by using a single atomic species presents issues when trying to

address single atomic qubits in multi-qubit arrays. Imperfect spatial alignment of the qubit

laser probe can couple to neighboring qubits and produce errors [12]. Additionally, the weak

magnetic moment between adjacent hyperfine states necessitates large fields be utilized to

induce coherent qubit rotations at rates comparable to the relaxation and dephasing rates.

Atomic qubit–qubit coupling is also severely limited by this weak moment for realizable

inter–qubit spacings of a few microns. This had led to much research in dynamical coupling

methods such as atom–atom collisions [13] and real / virtual photon exchange [14, 15].

Neutral atom quantum computing via Rydberg atoms aims to tackle the issues previously

addressed. By exciting an electron far outside the computational subspace to a large principle

quantum number n, qubit–qubit interactions are mediated via an electric dipole coupling [16],

9

with dipole-dipole interaction strength scaling as n4 [17]. This greatly increases not just the

strength of the interaction for a fixed distance between atoms, but also the physical range in

which a strong interaction can be realized. In fact, the first atomic two-qubit entangling gates

were performed utilizing the n = 50 and n = 51 Rydberg states in two Rb atoms separated

of order 1 cm in space [18].

Current efforts in neutral Rydberg atom quantum computing have achieved 7×7 qubitarrays with single qubit gate fidelities approaching 99% as measured via randomized bench-

marking [19]. CNOT gates realized utilizing nearest neighbor atoms have achieved raw

fidelities of 77% [20]. These impressive achievements have allowed the neutral approach

to satisfy the remaining DiVincenzo criteria albeit with an important caveat. Specifically,

atomic state readout is not just quantum state destructive, but qubit destructive [21]. "Push

out" techniques employ directionally imbalanced laser light resonant with a particular atomic

state. This imbalance applies light pressure to the atom if it is in the particular state of

interest. Readout is then performed by integrating the atomic fluorescence signal, the level

of which allows for determination of the qubit state. Once a measurement is performed, a

new atomic qubit must be reloaded into the trap for further processing.

A particular asset that neutral atoms do posses however is a large range of energy

level transitions spanning from the microwave to optical regime. This expanse makes them

attractive candidates to facilitate a microwave to optical photon transition, porting quantum

information processed on one superconducting chip to another or even between different

dilution refrigerators via a photonic flying qubit. While outside the scope of the work

presented here, the idea is interesting and the reader is referred to recent theoretical work by

Gard, B. T., et al for specifics [22].

1.3.2 Superconducting Qubits

Superconducting qubits can be aptly described as anharmonic quantum harmonic oscillators

[23]. Two general properties of superconductors allow for such a description; First, the bosonic

10

F = 3

F = 4

F = 5F = 6

F = 7F = 8

Figure 1-3: Ground state energy level diagram for Cesium. Commonly known as “clockstates,” the hyperfine states of the 62S1/2 level have a precisely defined splitting and makefor a long lived two level subspace to encode logical “0” and “1.” With the next nearesttransition energy in the optical regime, neutral atom qubits are robust against qubit stateleakage out of the computational manifold.

11

nature of the superconducting Cooper pair condensate allows for a single wave function

description of the system dynamics. The superconducting condensate acts as a incompressible

fluid whereby the entirety of all the multi-particle dynamics can be be reduced to a single

complex wave function ψ = √nCPeiφ, the normalization condition of which gives the total

number of Cooper pairs, nCP [24]. When subjected to harmonic boundary conditions (as

in either a distributed or lumped element LC circuit), the supercurrent will “slosh” back

and forth between the inductor (magnetic field) and capacitor (electric field) at the resonant

frequency ω = 1/√

LC. The combination of the superfluid condensate with correct boundary

conditions realizes a true physical manifestation of the quantum harmonic oscillator (QHO);

The same system which every undergraduate physics student studies. However, as any

undergraduate student who has done their homework should be able to tell you, the energy

spacings between all adjacent levels E|n〉 and E|n+1〉 are degenerate. That is to say that

ω|n〉→|n+1〉 = ω for all QHO number states |n〉. To realize an addressable quantum two-level

system (TLS), this degeneracy must be broken. Cooper pair tunneling, as first theoretically

described by Josephson [25] when looking at weakly coupled superconductors, gives rise to a

nonlinear inductance, and provides such a degeneracy breaking mechanism. Considered in

detail in section 2.1, the nonlinearity of the Josephson inductance arises naturally from the

first and second Josephson relations. When embedded in superconducting circuits, Josephson

junctions break the degeneracy of QHO energy states, allowing for the realization of engineered

quantum TLSs on a chip.

What sets the superconducting quantum computing effort apart from all other physical

implementations is the ability to directly engineer parameters of the interacting quantum

elements. Utilizing standard lithographical techniques, superconducting quantum circuits

with a wide range of parameters can be realized for use in experiments exploring the strong

and ultra-strong coupling regime of circuit quantum electrodynamics (cQED) and quantum

processors. We will explore this technology in far greater depth in the following chapters.

12

1.3.3 Current Hybrid Quantum Computing Approaches

The work contained is this thesis is not the first attempt at combining disparate quantum

systems. To provide a back drop for the work contained herein, we briefly review other efforts.

1.3.3.1 Superconductors Coupled To Atom Clouds

Some of the foremost work done in coherently coupling superconductors to atomic clouds

has been spearheaded by the group of Joseph Fortágh in collaboration with the group of

Reinhold Kleiner at the University of Tüebingen. In these experiments, clouds of Rb atoms

are cooled and then trapped via an on-chip superconducting "z-trap" [26, 27, 28] The z–trap

is a continuous wire that is physically laid out in the shape of the letter “z.” When current is

passed through the wire, the magnetic field (and hence the trapping potential) produced has

a shape similar to a cigar centered over the middle electrode. In 3 dimensions, the atoms are

trapped in this cigar shaped potential.

The z–trap is in close proximity to the current antinode of a superconducting CPW

resonator. This setup has achieved a magnetic dipole coupling between the hyperfine ground

state and the magnetic field of the CPW resonator. However, in the limit of single photon,

single atom coupling, the Rabi frequency g/2π < 1 Hz. The employment of Natom > 105

atoms is often utilized to provide a collective enhancement of the coupling. This, while also

driving the resonator with a classical RF field combats this low coupling strength with driven

Rabi oscillations of strength g/2π ≈ 2 kHz now having been observed in the lab.

1.3.3.2 Superconductors Coupled to Quantum Dots

Quantum dots (QDs) aim to achieve a scalable quantum computing architecture by harnessing

the power of trapping single electrons in harmonic potentials in the solid state. While

typically read out with either DC transport measurements or charge sensing, there has been

a growing interest in employing techniques developed in cQED to perform dispersive readout

of the QDs. Recent work in both the Petta group at Princeton [29, 30, 31] and Wallraff

13

(a) (b)

0

(nor

m.)

0

1.0

0

50

100

150

250-250 500-500

200

Figure 1-4: (a) Cartoon of a superconducting atomic z-trap. For a DC current, the vectorcomponents of the resulting magnetic field create a cigar shaped potential minimum centeredover the middle wire. (b) Numerical simulation of the z-trap magnetic field over the centerconductor for a bias current of 1 amp. The dark blue region in the center is the field minimumand provides the trapping potential. Larger bias currents will deepen the trap potential whilealso pulling the trap closer to the chip surface.

group at ETH-Zurich [32, 33, 34] have demonstrated coherent coupling between QDs and

superconducting CPW resonators. However, as explicitly mentioned in the Petta work, the

multilayer fabrication required to realize the QD system is detrimental to the performance of

the superconducting CPW resonator. Both groups report quality factors below 104. Realizing

coupling limited resonator performance in the face of the challenges presented by multilayer

fabrication is an active area of research.

14

Chapter 2

Josephson Physics and

Superconducting Circuit

Fundamentals

The career of a young theoretical physicist consists of treating the harmonic

oscillator in ever-increasing levels of abstraction

— Sidney Coleman (1975)

In 1911, Kamerling Onnes, while studying properties of metals at cryogenic temperatures,

came across a puzzling result. For a particular set of elements, all traces of electrical

resistance vanished below a certain critical temperature, Tc. This discovery, now known as

superconductivity, has to date spawned six Nobel prizes and a wealth of scientific endeavor

and physical insight spanning a large breadth of the physical sciences. A review of the

discovery and some controversy surrounding it can be found in [35]. In this chapter, we will

focus specifically on the applied aspects of these discoveries concentrating on the Josephson

effect and devices derived from it.

15

2.1 Basic Phenomena

BCS theory posits that the particles responsible for the super part of superconductivity

are pairs of bound electrons, formally referred to as Cooper pairs. Bosonic in nature, these

particles condense into an energy state below the Fermi energy whereby the collective lot of

all the particles can be described by one encompassing wave function

ψ = ψ0eiφ , (2.1)

where the normalization condition |ψ|2 = |ψ0|2 = nCP, the number of Cooper pairs.

The Josephson effect extends BCS theory by looking into the tunneling nature of Cooper

pairs across barriers, colloquially referred to as "weak links." Generally speaking, weak links

encompass any barrier between two superconducting electrodes where the critical current, the

maximum electrical current that be sustained prior to loss of superconductivity, is suppressed

[36]. These barriers can be made of insulators (such as an oxide) forming Superconductor–

Insulator–Superconductor (SIS) junctions or even normal non–superconducting metals forming

a Superconductor–Normal Metal–Superconductor (SNS) junction.

This first fundamental discovery about such systems is that, even in the absence of an

applied potential, they support a finite supercurrent across the electrode barrier. Know as a

the DC Josephson effect, the magnitude of the tunneling current is proportional to the sin of

the phase difference between the two superconducting wave functions of either side of the

barrier [37].

I(δ) = Ic sin(δ) . (2.2)

where Ic is the critical (maximum) supercurrent the junction can support and δ = φ2 − φ1 is

the phase difference between electrode wave functions.

The second relation derived by Josephson relates the rate of change of the phase difference

to the potential difference V placed across the electrodes,

16

(b)(a)

C LJ R

Figure 2-1: (a) Cartoon depiction of a Josephson junction. The phase difference δ = φ2 − φ1between the wave functions on either side of the junction is what allows for supercurrent to flowacross the boundary. (b) Resistive and Capacitively Shunted Junction circuit approximationfor a Josephson junction. When current biased in the superconducting state (Ib < Ic), theJosephson junction is modeled as a non–linear LC network with zero Ohm shunt resistance.When the the current bias Ib exceeds the critical current, the junction becomes resistive.

V = �

2edδ

dt. (2.3)

Through relating Eq. (2.2) and Eq. (2.3), an inherent inductance of the Josephson element

takes form

LJ = V

dI/dt= �/2e × dδ/dt

Ic cos(δ) × dδ/dt= Φ0

2πIc cos(δ)= L0

cos(δ) , (2.4)

where Φ0 = h/2e is the superconducting magnetic flux quantum. The nonlinearity of the

Josephson inductance defined in Eq. (2.4) will be a crucial point in the discussion of solid

state superconducting qubit functionality.

When no external bias current is applied, the potential energy of the junction is

U =∫

IV dt =∫

Ic sin(δ)Φ0

2πdδ

dtdt = IcΦ0

2π

∫sin(δ)dδ = EJ(1 − cos(δ)) (2.5)

where the Josephson energy EJ = IcΦ0/2π. We can then directly write down the Hamiltonian

17

for the JJ (ignoring constant offsets)

H = q2

2C − EJ cos(δ) , (2.6)

where q is the charge on the junction capacitor plates. The addition of an explicit external

bias current, Ib, modifies Eq (2.2) and consequently Eq. (2.5) in the following ways:

I(δ) = Ic sin(δ) − Ib (2.7a)

U =∫

IV dt = Ej(1 − cos(δ) − iδ) (2.7b)

where we have introduced the reduced bias current i = Ib/Ic. This new potential modifies the

Hamiltonian in Eq. (2.6) and produces what is commonly referred to as the “tilted washboard

potential”

H = q2

2C − Ej(cos(δ) − iδ) . (2.8)

A plot of the Josephson potential as a function of reduced bias current i is shown in Figure

2-2. For i = 0, we recover the potential in Eq. (2.6). The phase particle relaxes to the

minimum of the potential at phase values of δ = 2nπ. As the bias current is increased, the

term linear in the phase in Eq. (2.8) begins the dominate and tilts the potential. When the

bias current surpasses the critical current of the junction, the phase particle freely rolls down

the potential and a voltage V = Φ0/2π × dδ/dt develops across the Josephson junction.

2.1.1 RCSJ Model

A helpful model in exploring the dynamics of Josephson junctions is the resistive and

capacitvely shunted junction (RCSJ) model. In modeling the Josephson dynamics via

splitting the junction into constituent elements, the RCSJ model allows for direct calculation

of the dynamics through straight forward circuit analysis. Figure 2-1(b) displays a circuit

schematic of a Josephson junction in the RCSJ approximation. Writing down the total

18

Figure 2-2: Potential energy landscape for a current biased Josephson junction for differentvalues of reduced bias current i = Ib/Ic. For no bias current (blue curve), the potential ispurely (co)sinusoidal and the junction is in the supercurrent state with the phase particleliving in a potential minimum at δ = 2nπ and dδ/dt = 0. As the external bias current isincreased to appreciable fractions of the critical current (red curve), the term linear in δbegins to dominate and tilts the cosine potential. At values of i ≥ 1, the phase particle freelyrolls down the potential landscape. With the surpassing of the critical current, the junctionenters the voltage state with V = Φ0/2π × dδ/dt.

current in the RCSJ model of a Josephson junction, one must account for the current in the

junction, resistor, and capacitor,

I = Ic sin(δ) +V

R+ C

dV

dt. (2.9)

Making use of both the second Josephson relation and definition of the Josephson inductance,

Eq. (2.9) can be recast as

I

Ic

= sin(δ) + τJdδ

dt+ τJτRC

d2δ

dt2 , (2.10)

19

with the Josephson time constant τJ = LJ/R and the RC time constant τRC = RC. Making

the substitution dt = τJdε finally yields

I

Ic

= sin(δ) + dδ

dε+ βC

d2δ

dε2 , (2.11)

with βC = τRC/τJ = 2πIcR2C/Φ0. Formally known as the Stewart-McCumber parameter

[38, 39], βC is the ratio of the RC time constant to the Josephson time constant. Junction

dynamics in two limits of βC is worth further discussion.

Over damped Regime

When the RC time constant is much shorter than the Josephson time constant, βC < 1

and the junction is said to be over damped. For values of βC 1, the last term in Eq. (2.9)

can be ignored and the equation solved in both the steady state (i < 1 , dδ/dt = 0) and

dynamical phase (i > 1) limits.

In the steady (superconducting) state, the total current I < Ic and no voltage develops

across the junction. Solving for the phase in Eq. (2.11) yields

δ = arcsin(

I

Ic

). (2.12)

When the bias current surpasses the critical current value, a voltage develops across the

junction and the phase evolves in time. After rearrangement of terms in Eq. (2.11), we arrive

at the integral equation

δ =∫ I/Ic − sin(δ)

τJ

dt = 2arctan⎡⎣√1 − (Ic/I)2 tan

⎛⎝

√(I/Ic)2 − 1

τJ

t

⎞⎠+ Ic/I

⎤⎦ (2.13)

The junction phase can be seen to be evolving at a frequency ω =√(I/Ic)2 − 1/τJ . The time

averaged voltage then across the junction is then

〈V (t)〉 = Φ0ω

2π = IR√1 − (Ic/I)2 (2.14)

20

for I > Ic. Equation Eq. (2.14) is single valued and junctions in this regime do not exhibit

hysteresis.

Underdamped Regime

In the under damped regime, βC ≥ 1 and the second order derivative in time of δ in

Eq. (2.11) can not be ignored. For currents below the critical current, the results are the

same as in Eq. (2.12) again with no voltage developing. Upon return from the finite voltage

regime, under damped junctions do not immediately reenter the superconducting state with

the condition Ib < Ic being satisfied. Instead, the phase particle continues rolling down the

washboard potential until a finite “retrapping” current Ir is reached. This retrapping current

is inversely proportional to√

βC and the overall IV curve is hysteretic.

Representative IV curves can be found for the over damped and under damped case

in Figures 2-3(a) and 2-3(b), respectively. Data taken at T = 4.2 K on an unshunted 4

μm2 Nb-AlOx-Nb trilayer Josephson junction is shown in Figure 2-3(c). Above the critical

current of approximately 40 μA, the junction switches to the voltage state. Above the

superconducting gap voltage Vg = 2Δ/e, where Δ is the superconducting gap energy and e is

the fundamental electronic charge, the junction is completely normal and exhibits a linear

IV dependence that, when extrapolated to zero bias current, passes directly through zero

voltage as Ohm’s law demands.

2.1.2 Superconducting Quantum Interference Devices

When a loop of superconducting metal is interrupted by a JJ, interference of the supercon-

ducting wave function phase φ becomes directly measurable. Any superconducting loop

comprised of one or more JJ is referred to as a superconducting quantum interference device

(SQUID). Because all elements of a SQUID are superconducting, the single–valuedness of the

BCS wave function imposes a constraint on the phase of junctions within loops

∑i

δi +∑

j

2πΦj

Φ0= 2πn , (2.15)

21

0 1 2 3 4 5 6

V/IcR

0

0.5

1

1.5

2

I/Ic

0 1 2 3 4 5 6

V/IcR

0

0.5

1

1.5

2

I/Ic

Ir

40

80

120

160

0

-40

-80

-120

-160

0 5-5 -4 -3 -2 -1 1 2 3 4

Bia

s C

urre

nt (u

A)

Voltage (mV)

(a) (b)

(c)

Figure 2-3: (a) Representative nonhysteretic IV curve of a overdamped junction. (b)Representative hysteretic IV curve of an underdamped junction. The retrapping current Ir

for an unshunted junction should be very near 0 A. (c) IV curve of a 4 μm2 area Nb-Altrilayer junction measured at T = 4.2 K. The critical current of this particular junction wasapproximately 40 μA. The retrapping current is ∼ 1μA. The dashed line is drawn to showthe linear dependence on bias current for voltages above the superconducting gap voltage.

22

where δi is the phase of the ith junction and Φj the flux of the jth inductor.

In the case of a DC–SQUID, a loop interrupted by two parallel JJs, the total current

through the loop is simply sum of the current in both branches

IT = Ic1 sin(δ1) + Ic2 sin(δ2) , (2.16)

Taking |Ic1| = |Ic2| = Ic and expanding Eq. (2.15), the relation between junction phases is

δ1 − δ2 + 2π ΦΦ0

= δ1 − δ2 + 2πΦA

Φ0+ βL = 2πn . (2.17)

with the total flux Φ = ΦA + ΦLoop the sum of an external applied flux ΦA and the loop flux

ΦLoop = LIcirc and βL = 2πLIcirc/Φ0. The SQUID circulating current is then

Icirc =12Ic [sin(δ1) − sin(δ2)] . (2.18)

Combining Eq. (2.16) and Eq. (2.17) for n = 0 yields

IT = 2Ic cos(

πΦA

Φ0

)sin(δ) = IDC–SQUID

c sin(δ) (2.19)

where we have made the substitution δ ≡ (δ1 + δ2)/2. Equation (2.19) allows for the

description of a symmetric DC-SQUID as simply a single JJ with an external flux dependent

critical current IDC–SQUIDc = 2Ic cos (πΦA/Φ0). Further discussions of SQUIDs utilized in

superconducting classical and quantum computing can be found in Chapter 5 and Appendix

A.

2.2 Quantization Of Electrical Circuits

The ability to perform quantum algorithms with superconducting circuits means that the

constituent components comprising the processor must be behaving quantum mechanically.

The previous section discussed basic Josephson physics beginning with the two Josephson

23

relations. The RCSJ model and Stewart-McCumber parameter are very important is discussing

and understanding junction behavior even when the system temperature is an appreciable

percentage of the superconducting critical temperature, Tc. However, when the relevant

energy scales of the circuit Hamiltonian are much larger than the bath temperature, new

phenomena arise that can only be explained through quantum mechanics.

The energy stored in an unshunted, unbiased Josephson junction was derived in Eq. (2.6)

and is reprinted here for convenience,

H = q2

2C − EJ cos(δ) . (2.20)

The relevant temperature scale for a junction with area A = 0.1μm2 , critical current

density J = 10 A/cm2, capacitance per unit area C = 10 fF/μm2 are EJ/kb ≈ 230 mK

and Ec/kb ≈ 900 mK, where the junction charging energy Ec = e2/2C and C = C × A.For temperatures T � EJ/kb , Ec/kb, any quantum mechanical behavior is washed out by

incoherent processes due to the thermal bath. However, at temperatures achievable at the

cold stage of dilution refrigerators, the quantum mechanical behavior becomes manifestly

apparent.

The Hamiltonian for the Josephson element in Eq. (2.20) does not lend itself to direct

calculation. Approximation methods do however provide the necessary clarity to understand

how to make an artificial TLS and thus a qubit from a Josephson junction. We begin by

expanding the potential to second order in the phase and dropping constant terms which

yields

H = q2

2C + EJ

2 δ2 . (2.21)

While this expression may look straightforward, the choice of variables is a bit awkward.

Expressing δ is terms of the circuit node flux φ is a more natural choice. The relationship

between the junction phase and node flux is found by integrating Eq. (2.3) yielding

24

δ = 2πΦ0

∫V dt = 2π

Φ0φ . (2.22)

Substituting this back in to Eq. (2.21) yields

H = q2

2C + EJ

2

(2πΦ0

)2φ2 (2.23)

To this level of approximation, the Josephson junction is nothing more than a harmonic

oscillator with conjugate variables q and φ. In very much the same way as with the traditional

QHO [40], we can promote the conjugate variables to operators and provide them with the

appropriate commutation relation [φ, q] = i� having units of action.

Equating this Hamiltonian to that of the traditional QHO with q ↔ p, φ ↔ x, and

C ↔ m, we can find the equally spaced energy levels in this approximation.

12Cω2 = Ej

2

(2πΦ0

)2(2.24)

ω2 = Ej

C

(2π2eh

)2(2.25)

ω2 = EJe2

C

(2�

)2(2.26)

�ω =√8EcEJ (2.27)

In the language of 2nd quantization, the full Hamiltonian for this QHO is then

H = �ω(a†a + 1/2) (2.28)

with the raising (a†) and lowering (a) operators satisfying [a, a†] = i and

a =√

Cω

2�

(φ + i�

2 q

)(2.29a)

a† =√

Cω

2�

(φ − i�

2 q

). (2.29b)

25

While this illustrates how to go from the language of circuit parameters to that of

standard quantum mechanics, the results are lackluster. To this order of expansion, a

Josephson junction is a QHO with degenerate energy level spacings �ω. This would make a

very poor qubit indeed as addressing any two neighboring energy states with a resonant tone

would invariably lead to leakage out of the computational manifold. Thankfully however the

cos term is an infinite sum and we can expand to another order in φ. The quartic term in

the expansion is

V = −EJ

4!

(2e�

)4φ4 = −EJ

4!

(2e�

)4 (�

2Cω

)2

(a + a†)4 . (2.30)

The expansion of the last term is tedious as the order of the operators must be kept consistent.

However, only 6 terms in the expansion conserve excitation number n = a†a and yield

non-zero first order corrections. When added together, these 6 terms provide a correction to

the energy of the form

En = �ω(n + 1/2) − Ec(6n2 + 6n + 3)/3 , (2.31)

where now the energies between adjacent levels are

E10 = E1 − E0 = �ω − 4Ec (2.32a)

E21 = E2 − E1 = �ω − 8Ec (2.32b)

α = E21 − E10 = −4Ec . (2.32c)

We can see now that the degeneracy has been broken and an anharmonicity α exists of order

the charging energy Ec. With this anharmonicity now in place, we can truncate the Hilbert

space of the Josephson junction to the first two energy levels and adjust the Hamiltonian to

H = �ω(n + 1/2) → �ω

2 σz (2.33)

26

where σz is the Pauli operator. The exact same treatment can be applied to a true LC

oscillator circuit where the same QHO Hamiltonian can be arrived at albeit without the

higher order corrections.

The quantizing of a superconducting LC oscillator along with the justification of the

Josephson junction as a quasi TLS motivates the next chapter (and arguably the past 20

years worth of cQED and superconducting quantum computing) and it’s discussions of

Jaynes-Cummings physics on a superconducting circuit chip.

2.3 Superconducting Qubit Zoo

Before moving on to describing just how Jaynes-Cummings physics and QED style interactions

are realized on a chip, it is worth covering a few physical implementations of Josephson based

superconducting qubits and reviewing the differences between them.

2.3.1 Phase qubit

Initial demonstrations of quantum like behavior were first demonstrated in current biased

Josephson junctions [41]. Seminal work from the lab of John Clark at UC-Berkeley showed

that the Josephson phase particle, when subjected to the tilted washboard potential of a

current biased Josephson junction, exhibited quantized energy levels [42, 43]. It was these

levels that were used to make some of the very first superconducting qubits. Large capacitively

shunted Josephson junctions were current biased such that the ω10 transition was in the

microwave regime. Microwaves were applied directly through the DC bias line to excite

superpositions of the |0〉 and |1〉 state. Readout was performed via driving the |1〉 → |2〉transition. If the qubit was in the |1〉 state, it’s probability of tunneling through the barrier

would increase via the transition to the |2〉 state and a measurable voltage would develop. If

the qubit was in the ground state, the drive would be off resonant and the qubit would not

be excited. This is what is meant by "phase" in the qubit description; Is the qubit in the

27

superconducting or resistive state?

This style of qubit suffers from a multitude of issues. The current bias used to tilt the

potential is inherently noisy and limits the qubit phase performance as the energy spectrum is

directly dependent upon the bias stability. Additionally, the large capacitors used to shunt the

qubits have traditionally been made from amorphous dielectric materials that exhibit densely

populated energy gap spectrums that can absorb the qubit excitation [44]. Furthermore, the

readout is qubit state destructive with only the tunneling event (or lack there of) being the

indicator of the qubit state.

2.3.2 Flux Qubit

Flux qubits consist of a superconducting loop interrupted by one or more Josephson junctions

[45, 46, 47]. As a consequence of the single valued-ness of the superconducting wave function

at any point in space, the flux through a superconducting loop is quantized in units of the

superconducting magnetic flux quantum Φ0 = h/2e [48]. These loops are then flux biased

via an external current inductively coupled to the loop to a point where the induced flux

is Φ0/2. This fractional flux quantum frustrates the circulating current in the loop into

superpositions of clockwise and counter-clockwise propagation around the ring. The counter

propagating currents are mapped onto the qubit subspace and readout is performed via an

external SQUID magnetometer. Historically, flux qubits have been plagued by fabrication

issues leading to junction assymetry [49] and ultimately poor performance. Recent work has

focused on capacitively shunted (colloquially referred to as C-Shunt) flux qubits [50, 51] that

have begun to show promise in achieving better relaxation and coherence times.

2.3.3 Transmon

The transmon qubit [52], first developed at Yale in the lab of Rob Schoelkopf, can be described

exactly in the same fashion as the unbiased Josephson junction. The main difference is that

the junction (or two arranged as a DC SQUID for frequency tunability, see Appendix A)

28

is capacitively shunted to ground through a large (∼ 100 fF) planar capacitor. This shunt

capacitance all but eliminates the qubits sensitivity to charge noise by effectively dividing out

the kinetic q dependant term in the Hamiltonian. The trade off, however, is that the large

shunt capacitance also lowers the charging energy Ec and consequently the anharmonicity

generally enjoyed by its phase qubit counterpart. This small anharmonicity α ∼ 100 MHz is

an affordable price to pay in gaining the charge insensitivity that plagued earlier charge qubit

designs and has allowed for the building and control of multi-qubit processors with individual

qubit T1 and T ∗2 times approaching 100 μs [53]. The work detailed in the latter half of this

thesis employs a “rectmon” transmon qubit [54, 55] with a 75 fF shunt capacitance and a

center frequency of 4.8 GHz.

29

Chapter 3

Circuit Quantum Electrodynamics

Circuit Quantum Electrodynamics (cQED) allows for the ground breaking, Nobel prize

wining work pioneered by Serge Haroche [56] and David Wineland [57, 58] in cavity quantum

electrodynamics to be faithfully emulated and extended utilizing superconducting circuit

elements. Playing the role of the three dimensional cavity is the superconducting coplanar

waveguide (CPW) resonator. It’s atomic counterpart is played by the transmon qubit. The

electric dipole interaction between the two is mediated via a coupling capacitor. The ability to

directly engineer the on-chip elements has allowed for the exploration of a plethora of regimes

from strong to more recently ultra–strong coupling [59, 60]. In the latter, the excitation

exchange g rate between the cavity and qubit is of order the resonant frequency of the cavity

and/or qubit. Moreover, it has allowed for the exploration and development of the dispersive

coupling regime in which the detuning Δ ≡ ωr − ωq � g between the resonator and qubit is

larger than the coupling strength. It is this regime that quantum processors are operated

in. We will briefly review the physics of the coupled cavity–qubit system. Afterwards, we

will elucidate how to bring up a single cavity–qubit system in the lab, focusing on tangible

quantities.

30

3.1 Cavity–Qubit Hamiltonian

The coupled cavity-qubit system circuit diagram is displayed in Figure 3-1. The resonator

and qubit terms are annotated with the subscripts “r” and “q”, respectively with the coupling

capacitor denoted by Cc. We can write the Lagrangian down for this circuit by inspection

utilizing the node fluxes

L = 12Crφ

2r − φ2

r

2Lr

+ 12Cqφ

2q + EJ cos

(2πφq

Φ0

)+ 12Cc

(φr − φq

)2. (3.1)

The conjugate momenta, in this case the charges qr and qq, are found utilizing Hamilton’s

equations and are

qr =dLdφr

= Crφr + Cc

(φr − φq

), (3.2a)

qq =dLdφq

= Cqφq − Cc

(φr − φq

). (3.2b)

This is conveniently expressed in matrix notation as

�q = C�φ (3.3)

with

�q =

⎡⎢⎢⎣qr

qq

⎤⎥⎥⎦ , �φ =

⎡⎢⎢⎣φr

φq

⎤⎥⎥⎦ , C =

⎡⎢⎢⎣Cr + Cc −Cc

−Cc Cq + Cc

⎤⎥⎥⎦ . (3.4)

The Hamiltonian now is

H = �φ T �q − L . (3.5)

To eliminate the flux derivative term, we utilize the relation

�φ T �q = �φ T CC−1�q = �q T C−1�q . (3.6)

Substituting Eq. (3.6) into Eq. (3.5) and expanding we arrive at

31

Figure 3-1: Lumped element schematic of a resonator coupled to a qubit via coupling capacitorCc.

H = q2r

2

(Cq + Cc

det |C|)+ φ2

r

2Lr

+q2

q

2

(Cr + Cc

det |C|)

− Ej cos(2πφq

Φ0

)+ Cc

det |C|qrqq , (3.7)

where the det notation mean the determinant of the argument. The first two terms in Eq.

(3.7) are the adjusted resonator terms which in the language of second quantization can be

expressed as

Hr =q2

r

2

(Cq + Cc

det |C|)+ φ2

r

2Lr

= �ωr(a†a + 1/2) (3.8)

The third and fourth terms are the re-expressed qubit Hamiltonian which can be truncated

into the TLS formalism

Hq =q2

q

2

(Cr + Cc

det |C|)

− Ej cos(2πφq

Φ0

)= �ωq

2 σz (3.9)

The final term is the coupling between the resonator and qubit. We can express this is the

same formalism by expanding the charges into the appropriate raising and lowering operators

for the resonator and qubit. Utilizing the raising and lowering operator relations derived

32

earlier in Eq. (2.29a) and Eq. (2.29b), qq and qr equate to

qq = −i

√√√√�ωq det |C|2(Cr + Cc)

(σ− − σ+) (3.10a)

qr = −i

√√√√�ωr det |C|2(Cq + Cc)

(a − a†) . (3.10b)

Substituting these two into the coupling Hamiltonian

Hc =Cc

det |C|qqqr (3.11)

= − Cc

det |C|

√√√√�ωq det |C|2(Cr + Cc)

√√√√�ωr det |C|2(Cq + Cc)

(a − a†)(σ− − σ+)

= − �Cc√

ωqωr

2√(Cr + Cc)(Cq + Cc)

(aσ− − aσ+ − a†σ− + a†σ+) .

The outer two terms do not conserve excitation number and can be ignored. Once done, the

final form of Hc is

Hc = �Cc

√ωqωr√

(Cq + Cc)(Cr + Cc)(aσ+ + a†σ−) , (3.12)

where the coupling between the qubit and resonator is

g ≡ Cc√

ωqωr√(Cq + Cc)(Cr + Cc)

. (3.13)

Combining the results of Eq. (3.9), Eq. (3.8), Eq. (3.12), and Eq. (3.13), the full Hamiltonian

takes the form

H = �ωr(a†a + 1/2) + �ωq

2 σz + �g(aσ+ + a†σ−) , (3.14)

33

which is exactly the Jaynes-Cummings Hamiltonian for an atom interacting with a cavity

field.

We will briefly touch on two different limits of this Hamiltonian. Specifically, we will look

at both the resonant (ωq = ωr) dispersive (|ωq − ωr| � g) coupling regimes.

3.1.1 Resonant Jaynes–Cummings Hamiltonian

When the cavity and qubit frequencies are on resonance, they swap an excitation back forth

at rate g. The interaction term couples neighboring resonator-qubit states, where now the

eigenstates of the systems are a linear superposition of |0, n〉 and |1, n − 1〉. Re-expressingthe Janes-Cummings Hamiltonian in terms of state kets provides valuable intuition for the

resonant interaction,

HJC = �ωr(n |n〉〈n| + 1/2)

+ �ωq

2 (|1〉〈1| − |0〉〈0|)

+ �g√

n (|0, n〉〈1, n−1| + |1, n−1〉〈0, n|) .

(3.15)

The Schrödinger equation for the wave function describing the superposition of an excitation

in either the qubit or the resonator |ψ〉± = (2)−1/2(|0, n〉 ± |1 , n−1〉) now reads

HJC |ψ〉± =(�ωr(n + 1/2) − �ωq

2

)|0, n〉 /

√2

±(�ωr(n − 1/2) + �ωq

2

)|1, n−1〉 /

√2

+ �g√

n (|1, n−1〉 ± |0, n〉) /√2 ,

(3.16)

where the difference in energy between adjacent hybridized levels is ΔE± = 2g√

n.

What the eigenenergies and wave functions say is that when strongly coupled and on

34

Qubit Frequency (GHz)

Res

onat

or F

requ

ency

(GH

z)

5.405.39 5.41 5.42 5.435.385.37

Nor

mal

ized

Tra

nsm

issi

on

5.400

5.401

5.402

5.403

5.439

5.438

5.437

1.0

0.8

0.6

0.4

0.2

0.0

Figure 3-2: Resultant qubit-cavity spectrum from master equation simulation of the Jaynes-Cummings Hamiltonian using QuTip. As the qubit is tuned into resonance with the cavity,the system hybridizes and the cavity mode splits with splitting 2g = 6 MHz.

resonance, the coupled components lose their individual nature and the system as a whole

can no longer be described as a mere sum of the components. Figure 3-2 displays the results

of a master equation simulation of the Jaynes–Cummings Hamiltonian performed utilizing

QuTip. In the simulation, the qubit frequency (X–axis) is gradually tuned into resonance

with the readout resonator frequency at ωr = 5.4 GHz. The coupling strength g for the

simulation was set to 3 MHz.

3.1.2 Dispersive Jaynes-Cummings Hamiltonian

In the opposite limit, when ωq − ωr ≡ Δ � g, the cavity-qubit system is said to be

“dispersively” coupled and no direct excitation exchange between the two can occur. In order

to gain insight to the dynamics of this regime, we treat the coupling term as a perturbation

to the independent eigenenergies / wave functions of the uncoupled cavity and qubit.

35

The perturbation term couples states of differing photon number and qubit state so, to

first order in perturbation theory, the effect is zero. To second order, we find

E ′|0,n〉 = E|0,n〉 +

| 〈1, n − 1| H ′ |0, n〉 |2E|0,n〉 − E|1,n−1〉

(3.17)

= E|0,n〉 − �ng2/Δ

E ′|1,n−1〉 = E|1,n−1〉 +

| 〈0, n| H ′ |1, n − 1〉 |2E|1,n−1〉 − E|0,n〉

(3.18)

= E|1,n−1〉 + �ng2/Δ ,

with the unperturbed energies

E|0,n〉 = 〈0, n| H0 |0, n〉 (3.19)

= �ωr(n + 1/2) − �ωq/2

= �(ωrn − Δ/2) ,

E|1,n−1〉 = 〈1, n − 1| H0 |1, n − 1〉 (3.20)

= �ωr(n − 1/2) + �ωq/2

= �(ωrn +Δ/2) .

What this tells us is that the interaction can be represented in the both the photon number

and spin basis allowing a rewriting of the perturbing Hamiltonian as

H ′ = � a†a σz g2/Δ . (3.21)

The full Hamiltonian now has the form

36

H = H0 + H ′ (3.22)

= �ωr(a†a + 1/2) + �ωqσz/2 + � a†a σz g2/Δ .

Eq. (3.22) can be looked at in two distinct ways. In the first, the interaction term imparts

a cavity photon number state dependent shift on the qubit frequency where now the qubit

Hamiltonian is rewritten as

Hq = �ωqσz/2 + � a†a σz g2/Δ → �

2(ωq + 2 a†a g2/Δ)σz (3.23)

This Hamiltonian allows for a direct calibration of the superconducting cavity photon number

by performing qubit spectroscopy. This is a very powerful tool as it allows one not just to

distinguish the photon number, but also the photon number state distribution. This allows

one to distinguish between different distributions be they Bose-Einstein, thermal, or Fock.

The other view point of the interaction term is its effect on the resonator where the center

frequency is now influenced by the state of the qubit.

Hr = �ωra†a + � a†a σz g2/Δ → �(ωr + σz g2/Δ)a†a (3.24)

Where we have dropped constant terms. Looking at this form of the interaction, we see

that the resonator frequency is qubit state dependent. This allows for what is referred to

as a “weak” measurement; The interaction Hamiltonian commutes with the unperturbed

Hamiltonian and through interrogating the state of the weakly coupled resonator, we can

determine the qubit state without directly probing it. This form of the Hamiltonian is the

basis for all qubit quantum non-demolition (QND) readout.

37

3.2 Qubit-Cavity Bringup

There have been a great many review articles and theses covering in great mathematical

detail the nuances of Eq. (3.23) and Eq. (3.24). Instead of covering and/or reviewing the

material again, the following sections outline in detail how one measures a superconducting

transmon qubit in the lab. It is the hope of the author that this overview will provide a

general guideline in experimental procedure.

3.2.1 Coupling The Qubit–Cavity System to the Outside World

Having analyzed the circuit in Figure 3-1, we now have to extract the resonator frequency

encoded qubit information from the system so that we can determine and manipulate the

qubit state. We have two choices as to how to hook up the readout resonator to the outside

world. We can do it by either capacitively or inductively coupling the readout resonator to a

microwave feed line. Figures 3-3(a) and 3-3(b) illustrate the inductive and capacitive coupling

schemes for an LCR tank resonator circuit (we will ignore the qubit–resonator coupling for

now and just concentrate on the resonator–feed line coupling). While shown as discrete

elements, readout resonators are almost always distributed elements such as CPW resonators

where the resonant frequency is set by geometrical standing wave boundary conditions. We

can treat these CPW resonators as open / shorted transmission lines and apply microwave

transmission line theory [61] to derive the scattering parameters. We follow the formalism

given in [62].

For small internal loss, the input impedance of a shorted transmission line resonator with

characteristic impedance Z0 is

Z = Z0

αl + iπΔω/2ω0(3.25)

where α is the attenuation constant and Δω = ω − ω0. Relating the internal quality factor

Qi = π/4αl and substituting, Eq. (3.25) takes the form

38

R C L

Cc

Z0 Z0

R C L

Z0 Z0

M

(a) (b)

Figure 3-3: Readout resonator can be coupled to a microwave feed line either capacitively orinductively. (a) Inductively coupled LCR tank circuit with mutual coupling inductance M .(b) Capacitively coupled LCR tank circuit with coupling capacitance Cc.

Z = 4Z0Qi/π

1 + i2Qidx, (3.26)

where dx = Δω/ω0. Coupling the resonator to a feed line renormalizes the impedance by the

external coupling quality factor Qe

Z = Z0Qe

2Qi

(1 + i2Qidx) . (3.27)

Figure 3-3(a) shows a resonant tank circuit coupled to a feed line via a mutual inductance

M . The external quality factor for a inductively coupled resonator is

QMe = 2ωL

R∗ (3.28)

where L is the equivalent inductance of the LRC circuit given by

L = 1ω2C

= 4Z0

πω, (3.29)

and R∗ is the Norton equivalent circuit shunt resistance with form

R∗ = ω2M2/Z0

(1 + ω2M2/Z20)

≈ ω2M2

Z0. (3.30)

39

Substituting Eq. (3.29) and Eq. (3.30) back into Eq. (3.28), we arrive at

QMe = 8Z2

0

π(ωM)2 (3.31)

Alternatively, the resonator may be coupled capacitively as in Figure 3-3(b). The external

quality factor for a capacitively coupled resonator is given by

QCe = ωR∗C

2 (3.32)

where C is the equivalent capacitance of the LRC circuit

C = π

4ωZ0. (3.33)

R∗ again is the Norton equivalent shunt impedance which in the capacitive coupling case

equals

R∗ = 1 + ω2C2c Z2

0

ω2C2c Z0

≈ 1ω2C2

c Z0. (3.34)

Substituting Equations (3.33) and (3.34) into Eq. (3.32), we arrive at

QCe = π

8Z20(ωCc)2 . (3.35)

The forward scattering matrix element S21 for a shunt impedance is

S−121 = 1 + Z0

2Z . (3.36)

By substituting in Eq. (3.27) for Z and reorganizing, we finally arrive at

S21 =Qe + i2QiQedx

(Qe + Qi) + 2iQiQedx= Smin

21 + i2QT dx

1 + i2QT dx, (3.37)

where Smin21 = Qe/(Qe+Qi) is the minimum of the matrix element when on resonance (dx = 0)

40

and QT is the total quality factor given by

1QT

= 1Qi

+ 1Qe

. (3.38)

For qubit experiments, the external quality factor sets the time scale for qubit cavity

photon loading and unloading which limits the cavity photon ring up time and ultimately

determines the amount of time needed for averaging during cavity readout. It is important to

keep the cavity photon leak out rate κe ≡ ω/Qe greater than the cavity photon dissipation rate

ki ≡ ω/Qi such that the measurement photons leak into the detector instead of dissipating in

the cavity. This reasoning demands large Qi ∼ 105 such that reasonable couplings can be set.

3.2.1.1 Cavity Photon Number

The previous analysis of the coupling dynamics of a microwave resonator also allows for a

direct calculation of the cavity photon number as a function of the internal and coupling

quality factor. The power leaking in to the resonator Pi from an applied RF tone of power

Pa is

Pi = PAκe

ω= PA

Qc

. (3.39)

Photons both leak out of and are dissipated in the cavity which yields an output rate Po

Po = Pcavκe + κi

ω= Pcav

QT

. (3.40)

At equilibrium, the power leaking in and out of the cavity will be equal which yields the

relation between the intracavity power and the applied power

Pcav = PAQT

Qe

. (3.41)

41

In the absence of an applied drive tone, the resonator itself can be treated as a power source

that delivers power to the surrounding environment at rate

Pcav = n�ωκT = n�ω2

QT

, (3.42)

where n is the average photon number inside the cavity. Equating Eq. (3.41) and Eq. (3.42),

the average photon number inside the cavity can be directly related to the applied drive

power

n = PA ×(

Q2T

Qe�ω2

). (3.43)

3.2.2 Qubit Signs of Life

A particularly nice feature of the transmon qubit is that devoid of any microwave excitation

or control, the mere presence of it’s coupling to the readout resonator imparts a dispersive

frequency shift on the resonator. When renormalizing the frequencies of both the resonator

and the qubit for the weak interaction with the transmon, levels beyond the [|0〉 , |1〉] subspacemust be accounted for due to the qubit’s weak anharmonicity. The renormalized resonator

frequency, when coupled to a transmon qubit, takes the form

ω′r = ωr − χ12

2 , (3.44)

where

χij =g2

ωij − ωr

≡ g2

Δij

. (3.45)

This renormalized resonator frequency tells the experimentalist two important pieces of

information: The first is whether or not the qubit is functioning. The second, depending on

the sign of χ12, tells the experimentalist whether the qubit frequency is above or below the

bare, high power resonator frequency and limits the range required for search in order to

42

-120 -115 -110 -105 -100 -95 -90

Power (dBm)

6.155

6.156

6.157

6.158

6.159

6.16

6.161

6.162

6.163

6.164

6.165

Freq

uenc

y (G

Hz)

-50

-45

-40

-35

-30

-25

-20

S 21 (d

B)

Figure 3-4: Qubit readout resonator spectroscopy vs. readout power. In the high powerregime (P > −99 dBm), the resonator frequency is simply the bare resonator frequency ωr.As the power of the spectroscopy tone is lowered, the resonator frequency becomes unstableand difficult to measure in the average mode of the vector network analyzer. Finally, below acertain applied power (in this case -108 dBm), the full dispersive shift of the qubit on theresonator can be clearly seen. The upward shift in resonator center frequency denotes thatthe qubit frequency is below the cavity frequency, yielding a negative χ12.

determine the qubit frequency.

Figure 3-4 displays a two-dimensional vector network analyzer (VNA) plot where the

frequency and power of the microwave readout tone are being swept. For explicit on-chip

microwave powers greater than -98 dBm (not accounting for microwave losses due to cabling,

reflection, etc.), the resonator is at it’s bare resonant frequency that is power independent.

Between -98 and -108 dBm, the system is in an intermediate regime where the applied power

to the resonator is loading the cavity with a photon occupancy of order the critical photon

number ncrit = Δ2/4g2. Below -108 dBm of explicit applied power, the cavity frequency

shifts, becoming fully dressed by the interaction with the qubit. The upward shift in the

43

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Qubit Flux Bias Current (mA)

6.158

6.16

6.162

6.164

6.166

6.168

6.17

6.172

6.174

6.176 F

requ

ency

(GH

z)

-25

-20

-15

-10

-5

S 21 (d

B)

Figure 3-5: Qubit readout resonator spectroscopy vs. qubit flux bias current. The sinusoidaldependence of the qubit and thus resonator frequency is clearly seen. The spectrum does notdiverge at Φ = Φ0/2 however do to the 2-to-1 junction size ratio for the qubit SQUID. Themutual inductance MFB between the bias line and the SQUID loop was modeled to be 2.2pH, thus the current to achieve a full flux quantum of evolution is IFB = Φ0/MFB ≈ 1 mA.

center frequency of the resonator as measured in Figure 3-4 indicates that the qubit |1〉 → |2〉transition (but more importantly, the |0〉 → |1〉 transition) is below the cavity. Additionally,

the shift magnitude (∼ 2 MHz) allows one to estimate the qubit |1〉 → |2〉 transition frequencyto be approximately 2.5 GHz below the cavity for a designed coupling g = 100 MHz.

If the transmon capacitor is shunted with a dc-SQUID as opposed to a single junction,

one may adjust the qubit transition frequency via the application of an external flux to the

SQUID loop. To understand this is more detail, we need only to look at the Josephson energy

EJ of the SQUID loop

44

HSQUID = −EJ1 cos(δ1) − EJ2 cos(δ2) . (3.46)

The condition of flux quantization enforces that the equation for the phase difference between

the two junctions take the form

η ≡ δ2 − δ1 = 2nπ + 2πΦ/Φ0 , (3.47)

where Φ is an externally applied flux. For identical junctions, EJ1 = EJ2 = EJ and

ESQUID = −2EJ cos(πΦ/Φ0) . (3.48)

In the case of an asymmetric SQUID comprised of junctions with differing critical currents,

we can define a percentage difference

d = EJ2 − EJ1

EJ1 + EJ2, (3.49)

where then the Hamiltonian in Eq. (3.46) takes the form

H = EJ cos(πΦ/Φ0)√1 + d2 tan2(πΦ/Φ0) , (3.50)

where EJ = EJ1 + EJ2. Figure 3-5 shows the readout resonator spectrum as a function of

qubit flux bias current. The non-divergent sinusoidal dependency of the frequency means

that the qubit has a roughly 2-to-1 junction size ratio with a d = 1/3. At the upper most

frequency, the χ12 shift is now 12 MHz, yielding an approximate upper qubit frequency of

5.25 GHz again assuming a coupling g ≈ 100 MHz.

45

3.2.3 Finding the Qubit Frequency

The previous section detailed how to initially confirm that the qubit is at a minimum coupling

to the readout resonator in the expected way and, if designed for it, properly flux tunes. The

next step in bringing up the cavity-qubit system is determining the qubit frequency. The

previous sections gave upper and lower bounds for where in frequency space the fundamental

qubit transition should lie. That band was between 3.6 and 5.2 GHz given an estimated

coupling of 100 MHz.

Experimentally, the way to determine a qubit’s frequency at a particular flux bias point

(should the qubit be tunable) is to apply a microwave drive at the resonator center frequency

and monitor the level of the output signal as a function of qubit microwave drive frequency.

The qubit drive can be applied directly to the qubit via an independent control line or through

the readout resonator. The qubit drive tone is applied for times T � Tπ such that the qubit

state saturates at some superposition of |0〉 and |1〉 ensuring a measurable readout resonatorfrequency shift. Spectroscopy data of a transmon qubit is shown in Figure 3-6 where both

the qubit drive tone frequency and signal level were swept. The important feature to note is

how the qubit frequency line width becomes smaller as a function of qubit drive attenuation.

This effect is due to stimulated emission. Qubit drive photons can drive not just the |0〉 → |1〉transition but also drive the |1〉 → |0〉 transition which in spectroscopy looks like a parallel

qubit loss channel yielding a larger qubit transition line width. As the drive power is reduced,

less stimulated emission events occur and the extraneous loss channel diminishes reducing

the qubit line width.

3.2.4 Orthogonal Qubit Axis Control

With the qubit |0〉 → |1〉 transition frequency characterized via spectroscopy, coherently

controlling the qubit state vector is the next task. This is done via the application of a

resonant microwave drive tone that, in the lab frame, couples to the qubit via the interaction

46

0 2 4 6 8 10 12 14 16Qubit Attenuation (dB)

4.8

4.85

4.9

4.95

5

5.05

5.1

5.15

5.2

5.25

Qu

bit

Fre

qu

ency

(G

Hz)

3

3.5

4

4.5

5

5.5

S21

Cav

ity

Tra

nsm

issi

on

(ar

b. u

nit

s)

Figure 3-6: Qubit Spectroscopy Data. At high drive powers, the line width is large due to anadditional stimulated emission loss channel. As the drive power is reduced, the resonance linewidth reduces to a constant value the width of which can provide an estimate of the qubitrelaxation lifetime T1.

HD = �Γ(t) cos(ωqbt + γ)σx (3.51)

To achieve both amplitude and phase modulation, the qubit drive tone is usually pro-

duced via single sideband (SSB) modulation achieved through the utilization of an IQ

(In-phase/Quadrature) mixer. A standard IQ mixer has 4 ports: The local oscillator “LO”

port, an in-phase “I” and quadrature “Q” port, and finally an output “RF” port. A constant

microwave tone applied to the LO port is multiplied by the signals AI(t) and AQ(t) applied

at the I and Q ports, respectively. The resulting signal at the RF port is

ARF (t) = AI(t) cos(ωLOt) + AQ(t) sin(ωLOt) . (3.52)

47

For SSB modulation where ωq = ωLO − ωSSB, the signals applied at the I and Q ports should

be

AI(t) = Γ(t) cos(ωSSB + γ) , (3.53a)

AQ(t) = −Γ(t) sin(ωSSB + γ) . (3.53b)

The output tone at the RF port now takes the form

ARF (t) = Γ(t) cos(ωqt + γ) . (3.54)

The Hamiltonian for the qubit plus the drive tone in the lab frame now takes the form

H = �Γ(t) cos(ωqt + γ)σx + �ωq/2σz . (3.55)

The natural question to ask when looking at this equation is how one achieves independent

qubit X/Y axis control given only a σx operator. The answer lies in considering how the

qubit “sees” the application of the drive Hamiltonian in it’s own frame.

We can transform the Hamiltonian given in Eq. (3.55) for an arbitrary drive frequency

ωd into the interaction picture via the following transformation

VI = eiH0t/� V e−iH0t/�

= eiωqtσz/2�Γ cos(ωdt + γ)σxe−iωqtσz/2 .

(3.56)

After putting terms into matrix form and multiplying,

VI = �Γ cos(ωdt + γ)

⎡⎢⎢⎣ 0 eiωqbt

e−iωqbt 0

⎤⎥⎥⎦ . (3.57)

We now expand the cos term into it’s exponential form and multiply it throughout resulting

in

48

VI =�Γ2

⎡⎢⎢⎣ 0 e(i{ωd+ωq}t+γ) + e(i{ωq−ωd}t−γ)

e(−i{ωd+ωq}t−γ) + e(i{ωd−ωq}t+γ) 0

⎤⎥⎥⎦ (3.58)

In the interaction picture, wavefunctions are constant with respect to time while the ob-

servables are not. The solution to the Schrödinger equation in the interaction picture given

Eq. (3.58) is trivial, yielding 4 terms

ψ(t) = −iΓ2 ψ0

∫VIdt , (3.59)

where

∫VIdt =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∫V 11

I dt =∫

V 22I dt = C ≡ 0∫

V 12I dt = e(i{ωd+ωq}t+γ)

i(ωd + ωq)+ e(i{ωq−ωd}t−γ)

i(ωq − ωd).

∫V 21

I dt = e(−i{ωd+ωq}t−γ)

−i(ωd + ωq)+ e(i{ωd−ωq}t+γ)

i(ωd − ωq)

(3.60)

For drive frequencies near resonance, the first term in the off-diagonal elements for ψ(t)

become vanishingly small. We can then neglect these terms, which in combination of driving

on resonance (ωd = ωq) gives the interaction terms in Eq. (3.58) the form

VI =�Γ2

⎡⎢⎢⎣ 0 e−γ

eγ 0

⎤⎥⎥⎦ (3.61)

= �Γ2

⎡⎢⎢⎣ 0 cos(γ) − i sin(γ)

cos(γ) + i sin(γ) 0

⎤⎥⎥⎦ (3.62)

= �Γ2 [cos(γ)σx + sin(γ)σy] . (3.63)

49

It can be seen now directly that the phase γ of the sideband tone applied to the I and Q

ports of the IQ mixer determines which axis the qubit is rotated about. While in this analysis

we have explicitly dropped the time dependence of Γ, it is important to note that qubit drive

amplitude modulation is a key component in achieving precise rotation angles about the

Bloch sphere. For the work in this thesis, Gaussian profiles were utilized.

3.3 Qubit and Qubit Gate Benchmarking

Figure 3-7 displays a sample dilution refrigerator (DR) wiring schematic for a single qubit

/ resonator pair. Independent RF sources are required for the qubit (red) and resonator

(blue). In this schematic, the qubit and readout tones are produced via SSB modulation and

tied together onto a single microwave line with a 3 dB microwave splitter / coupler. The

tying together of signals is not necessary if the qubit has an independent addressing line

(not shown). These signals are then attenuated through multiple microwave attenuators at

different temperature stages of the DR before finally being applied to the device under test

(DUT). The DUT is generally encased in a superconducting sample box that helps to reduce

the magnetic field in the nearby vicinity due to Meissner shielding. In addition, extra layers

of radiation and magnetic shielding are applied around the sample. This can be done by

painting the inside of a mu-metal or Cryoperm can with ECCOSORB. After either driving

the qubit or sampling the resonator, the tones are then passed out of the sample and through

a series of microwave isolators and amplifiers. The isolators serve to reduce the noise and

hence back action emanating backwards through the chain from the input of the microwave

amplifiers. Once properly amplified, the signal coming out of the fridge is mixed back down

(green) to the sideband frequency with the I and Q ports then being digitized for analysis.

50

ADC

I QRF

LO

RO DACQB DAC

I QRF

LOI Q

RF

LO

300 K

60 K

3 K

mK

HEMT

HEMT

JPA

DUT

Timing / Instructions / Data Acquisition

20 dB Attenuator

3 dB Splitter

20 GHz LPF

Circulator / Isolator

Radiation Shielding

Magnetic Shielding

Figure 3-7: Sample dilution refrigerator wiring schematic for a single qubit / resonator pairexperiment.

51

3.3.1 Characterizing and Calibrating the Qubit States

Figure 3-8(a) shows a sample IQ “blob” distribution for the application of an idle (blue) gate

and a π-gate (orange) to a transmon qubit.The π-gate time was determined with a Rabi

oscillation experiment where the qubit is driven on resonance for variable amounts of time

resulting in an oscillatory output signal. Utilizing a high power readout protocol [63], large

separation in IQ space between the two states can be achieved. To convert this raw signal

into proper qubit states, the centroids (yellow crosses) of the two distributions are found and

a bisecting (black, dashed) line drawn between them. The data are then rotated and centered

onto the origin and binned according their distance along the bisector line. The result of this

manipulation is a histogram with two Gaussian distributions as can be seen in Figure 3-8(b).

Two distinct measures of the qubit readout chain may be obtained from the process.

The first, commonly referred to as the separation fidelity, is a measure of how well resolved

-5 0 5 10 15 20 25In-Phase (mV)

30

35

40

45

50

55

60

Qua

drat

ure

(mV)

Idle GatePi Gate

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Occ

upat

ion

Separation Fidelity

Single Shot Fidelity

-25 -20 -15 -10 -5 0 5 10 15 20 25Separation (mV)

0

10

20

30

40

50

60

Cou

nts

Idle GatePi Gate

= 100%

= 85.8%

(a) (b)

Figure 3-8: (a) Representative IQ data set for applied idle gate and π-gate. The largeseparation between the blobs allows for accurate qubit state discrimination. (b) Centered,rotated, and binned data IQ blob data. Two distinct measures of fidelity may be determinedfrom the processing of IQ data. The first is the separation fidelity which is a measure of howwell separated the data are. The second is the single shot readout fidelity which is a measureof the max distance between the integrals (dashed orange / dashed blue) of the two signals.The integral of the signal is also utilized to normalize the data such that qubit states (andthe superposition of them) may be put in normalized units.

52

the |0〉 and |1〉 qubit states are. To determine the separation fidelity, each resulting qubit

gate histogram is fit to a Gaussian (solid orange and blue curves). These gaussian fits are

then integrated over the separation range resulting in error functions (erf). The resulting

maximum difference between the erf(|1〉) and erf(|0〉) is the separation fidelity. Figure 3-8(b)

displays a qubit measurement that yields a 100% separation fidelity.

The other measure, known as the single shot fidelity, is defined as

F = 1 − P (1|0) − P (0|1) (3.64)

where P (x|y) defines the probability of measuring state x when state y has been prepared.

Experimentally, this is determined via the difference between the integrated π-gate signal

(dashed orange line, Figure 3-8(b)) and the integrated idle gate signal (dashed blue line,

Figure 3-8(b)). The single shot fidelity can be thought of in terms of extractable information

from the system. For instance, in the extreme limit that the fidelity is 0, the probability

of measuring either qubit state |1〉 or qubit state |0〉 are equal after having prepared state

|0〉. The state then is maximally mixed with a saturated entropy and no information can be

faithfully measured.

Finally, the calibration of the measurement chain to obtain results in qubit state occu-

pation is achieved by normalizing the integrated qubit state histograms (right hand y-axis,

Figure 3-8(b)). Using this calibration we can see that even when the qubit is in the ground

state, there is measurable population of the |1〉 state. This ground state preparation error

permeates along for the qubit rotations as a non-negligible amount of |0〉 state populationcan be seen for an applied π-gate. The single shot fidelity previously discussed is not only

limited by imperfect gates, but also by imperfect state preparation.

53

3.3.2 Randomized Benchmarking of Qubit Gates

Now that we have discussed how to implement arbitrary rotations about the Bloch sphere

and how to properly calibrate and measure the state of the qubit, the natural question to ask

is how well can one perform precise rotations / qubit gates and how does one characterize said

performance? For a single qubit, there are 24 unique rotations that preserve the octahedral

symmetry. These rotations belong to a class known as the Clifford group. Each Clifford maps

one point on the Bloch sphere to another. An easy way to think about these rotations is by

considering mapping the positive Z axis of the Bloch sphere to any one of the other 5 vertices

of which, when holding the Z axis then stationary, there are 4 unique rotations. The number

of unique Cliffords C for a single qubits is then

C = number of axes × rotations per axis = 6 × 4 = 24 . (3.65)

Table 3.1 lists the 24 unique Cliffords for a single qubit. Randomized benchmarking (RB)

aims to characterize the fidelity of qubit gates utilizing the power of randomization. The

RB protocol to characterize average qubit gate fidelity is to: (I) Apply a random sequence

of m Clifford operations to the qubit, (II) calculate and apply the unique Clifford CI that

inverts the Clifford sequence and brings the qubit back to |0〉, and (III) measure the ground

state population averaged over k different trials of the m Cliffords. The randomness of the

protocol depolarizes the environmental noise and is resilient against state preparation and

measurement (SPAM) errors. The resulting sequence fidelity is fit to

F = Apm + B . (3.66)

The average error per Clifford, r, is related to the fit parameter p via

r = (1 − p)(d − 1)d

(3.67)

54

Single Qubit CliffordsPauli I

XYY, X

2π/3 ±X/2, ±Y/2±Y/2, ±X/2

π/2 ±X/2±Y/2-X/2, Y/2, X/2-X/2, -Y/2, X/2

Hadamard X, ±Y/2Y, ±X/2X/2, Y/2, X/2-X/2, Y/2, -X/2

Table 3.1: List of single qubit Cliffords.

with d = 2n where n is the number of qubits. SPAM errors are encoded in the coefficient

A and linear offset B. While this protocol provides insight in the average fidelity per gate,

it provides no insight as to which gates, if any, may be bad apples and limiting the overall

sequence fidelity.

A subtle but important change to the RB protocol is the insertion of a target gate into

the m random Clifford sequence that we wish to measure the fidelity of. The interleaving of

this target gate (hence the name interleaved randomized benchmarking), allows for direct

comparison against of sequence of m Cliffords. The interleaved gate sequence decay pIG

allows for the calculation of error per interleaved gate

rIG = (1 − pIG/p)(d − 1)d

(3.68)

Figure 3-9(a) displays the interleaved RB experiment protocol. A reference experiment is

run with only randomized Clifford sequences of length m followed by an inversion gate CI .

The experiment is then re-run only with an interleaved gate, denoted here by G . The data

are then compared to extract the gate fidelity rIG. Sample data for taken in the lab at 40

55

( )

( )

(a)Reference

Experiment

(b)

5 10 15 20 25 30 35 40 45 50 55

Number of Cliffords,

0.55

0.6

0.65

0.7

0.75

0.8

0.85

Sequ

ence

Fid

elity

Interleaved Gate: IInterleaved Gate: X/2

Figure 3-9: (a) Interleaved randomized benchmarking experiment sequence. A randomizedClifford gate array is applied m times followed by subsequent inversion to the qubit |0〉 stateand measurement. The sequence is then repeated except for the interleaving of a desired gate,G, in the Clifford sequence. (b) Randomized benchmarking data taken for an idle (reference)gate and an X/2 gate applied to a transmon qubit in the lab at 40 mK. The curves areindependently fit with Eq. (3.66) and the extracted p’s are utilized in the calculation of theerror per gate utilizing Eq. (3.68).

56

mK is displayed in Figure 3-9(b). The applied gate was an X/2 gate with extracted fidelity

F = 1 − rIG = 99.7(3)%.

57

Chapter 4

Hybrid Atom–CPW cQED

Experiment

Any one physical implementation of a qubit is bound to have specific strengths and weaknesses

associated with it. The scaling up of any one physical qubit system to realize a quantum

architecture comprised of the millions of qubits deemed necessary to achieve a fault tolerant

processor will have to deal with the scaling of these particular weaknesses. Hybrid quantum

architectures aim to to combine physically disparate quantum systems in effort to circumvent

any one physical implementations weaknesses while exploiting their individual strengths. The

following sections detail efforts undertaken towards the demonstration of coherent coupling

between a superconducting CPW resonator and a single Rydberg Cs atom; The realization

of which would be a major milestone towards designing and building a larger integrated

atom-superconductor quantum processor.

4.1 By The Numbers

The commonality shared between cavQED and cQED quantum computing architectures is

that of the superconducting microwave resonator. Serving the same purpose in both physical

systems, the microwave resonator allows one to weakly probe the qubit system, be it atomic

58

(a) (b)

Figure 4-1: (a) Diagram of a single Rydberg Cs atom coupled to a planar superconductingCPW resonator. For maximal coupling, the atom is placed at the voltage antinode. (b)Energy level diagram for an atom–CPW photon Cz conditional phase gate. The Cs atomis excited out of the hyperfine ground state manifold (which is utilized as the 2 state qubitmanifold) to a large principle quantum number Rydberg state |r〉. The CPW mode is resonantwith the |r〉 → |r′〉 transition which evolves the state picking up a factor of 2π in the phasedepending on whether or not a photon resides in the cavity. The atom is then de-excitedback down into the qubit state manifold.

or artificial. The first step in the integration of the two disparate technologies is to show that

the large three–dimensional cavities commonly used in atomic experiments can be replaced

with the planar distributed LC oscillators used in most cQED experiments. We explore

the possibility of such a replacement by looking at the relevant numbers and timescales to

perform a conditional phase gate Cz between a single Rydberg atom and a superconducting

CPW resonator [64].

A schematic of the proposed coupling scheme is shown in Figure 4-1(a). An atom is placed

at the voltage antinode of the resonator so as to maximize the electric dipole coupling. Figure

4-1(b) illustrates the pulse sequence required to realize a conditional phase gate between

the atom and CPW cavity photon. The qubit states are encoded in the hyperfine levels

of the ground state. The |0〉 → |1〉 transition frequency for Cs is ω01 = 2π × 9.19263 GHz

59

(see Figure 1-3). A π–pulse excites the atom out of the computational manifold out to a large

principle quantum number n Rydberg state |r〉 via the application of a classical laser pulse of

duration τ|r〉 = π/Ω. The transition frequency between |r〉 and a nearby Rydberg state |r′〉 isthen tuned into resonance with the cavity mode of the CPW resonator. If a photon is present

in the resonator, the Rydberg state is coherently Rabi driven between |r〉 and |r′〉 pickingup a 2π evolution of the phase in time τrr′ = π/g, where g is the vacuum Rabi frequency

between the atom and the CPW resonator. The resultant Rydberg state is then mapped

back down to the computational |1〉 state via the application of another classical laser pulse

of time τ|r〉. The total time for the completion of the pulse sequence is

τTotal = 2τ|r〉 + τrr′ = π(2/Ω + 1/g) . (4.1)

The resulting gate from this series of operations is the conditional phase gate

Cz = |00〉〈00| − |01〉〈01| + |10〉〈10| + |11〉〈11| , (4.2)

where the first state designation is the Fock state of the resonator and the second the state

of the atomic qubit. When combined with single qubit rotations, the controlled phase allows

for the entanglement of the atom–cavity system providing a path for direct microwave to

optical photon conversion.

Teasing out the coupling scheme further, we look to the preparation of a Bell state utilizing

the Cz gate. The Bell state we wish to prepare is

|Ψ+〉 = 1√2(|01〉 + |10〉) , (4.3)

where again the first designation in the ket notation refers to the photon number inside the

resonator and the second the state of the atomic qubit. The gate sequence to prepare the

Bell state is

60

B+ = Ha Cz Ha (4.4)

where Ha is the Hadamard gate applied to the atom. The initial states required to realize the

Bell state are |ψ〉a = |0〉 and |ψ〉r = 1/√2 (|0〉 + |1〉) . Preparation errors for the resonator

superposition state are accounted for via a simulated qubit - cavity coupling of gqc = 2π × 100

MHz and a qubit T1 = 2μs. However, perfect qubit state initialization and preparation are

assumed. The Bell state preparation fidelity F for a range of simulated couplings g and

internal cavity quality factors Q was calculated utilizing

F = Tr[√√

ρ+ρ√

ρ+

], (4.5)

where ρ+ = |Ψ+〉〈Ψ+| is the ideal density matrix for the Bell state. The Cz gate implementation

described above is considered explicitly for the electronic transition between neighboring

Rydberg states 90s1/2 and 90p3/2 with dipole moment

drr′ = 〈90s, m = 1/2| d |90p, m = 3/2〉 =√2/9 × 8360 ea0 , (4.6)

where e is the fundamental charge and a0 is the Bohr radius. The radiative decay rates

incorporated into the simulation were γr = 820μs−1 and γr′ = 2 ms−1 for the |90p, m = 1/2〉and |90p, m = 3/2〉 states, respectively. The time evolution of the density matrix ρ was

calculated via the Lindblad master equation

ρ = i

�

[ρ, H

]+ 12∑

i

(2ciρc†

i − c†iciρ − ρc†

ici

). (4.7)

The superoperators ci considered for the simulation are listed in Table 4.1. Operators c1 and

c2 describe the relaxation of the Rydberg states |r〉 and |r′〉, respectively. Operators c3 and

c4 describe the loss and acquisition of cavity photons as a function of the cavity line width

κr = ωr/QT and the thermal cavity photon number n =[e−�ωr/kbT − 1

]−1.

61

Superoperator Physicality Mathematical Form

c1 Rydberg state |r〉 relaxation √γrσ

−r

c2 Rydberg state |r′〉 relaxation √γ′

rσ−r′

c3 Cavity photon number lowering√

κ(n + 1)a

c4 Cavity photon number raising√

κna†

Table 4.1: The superoperators utilized in calculating the Bell state density matrix ρ. c1,2describe the relaxation of Rydberg states |r〉 / |r′〉 at rates γr and γr′ , respectively. c3,4describe the coupling of photons in and out of the cavity as a function of the thermal photonnumber n.

Figure 4-2 shows the resulting contour plot of the Bell state preparation fidelity as a

function of the atom - cavity coupling strength g and the quality factor Q at T = 0 K. In the

limit that T = 0 K, n = 0 and thus c4 → 0, the physics is dominated solely by the cavity

photon loss rate κ � γ and the atom–cavity coupling g. We see that for reasonable coupling

rates in the single MHz regime and a quality factor no less that Q ≥ 105, the state can be

prepared with a fidelity F ≥ 99%. We will return to the effects of accounting for thermal

photons in a following section.

4.2 Experimental Apparatus

Ideally, experiments to test the theory outlined in the preceding section would take place

at millikelvin temperatures on the cold stage of a dilution refrigerator (DR). However, the

engineering challenges associated with integrating the milliwatt power lasers required for

single atom trapping and manipulation with the modest cooling powers (∼ 10μW) at the

mK stage of modern dilution refrigerators was deemed as challenge too great to tackle for

a first iteration experiment. Instead, we chose to build a custom UHV LHe cryostat as a

jumping off point to demonstrate that the two disparate quantum systems be integrated at

the most basic of levels.

The design implemented tandem suspended LN2 / LHe reservoirs from a 16.5" conflat

flange (see Figure 4-3). The LN2 vessel (red) was connected to the conflat flange via G10

62

Figure 4-2: Contour plot showing the Bell state preparation fidelity as simulated for differentcoupling strengths g and cavity quality factors Q. Regularly achieved single photon regimeinternal quality factors of Qi ≈ 105 when combined with moderate coupling strengths in thesingles of MHz, provides preparation fidelities greater than 99%.

support struts (green) 2.065"×1.750"×0.060" in size. Made from 0.055" rolled stainless steel,

the outer diameter of the LN2 vessel was 10" with a total length of 10". The LHe vessel

(blue) was also made from 0.055" rolled stainless steel with an overall length of 20". The

liquid capacity of the LN2 and LHe tanks was 10 and 27 liters, respectively. Each reservoir

had a 3003 series Al alloy wiring feed through plate bolted to the bottom of it in order

to facilitate the necessary wiring and electronics for the resonator measurement while also

providing real estate for thermal heat sinking while also serving as a radiation baffle. Each

reservoir base was made from an explosively joined 304 stainless steel-OFHC copper bi-metal

plate. Surrounding the LHe tank was a 1100 series Al alloy radiation shield (orange). This

shield is thermally sunk to the base plate of the LN2 reservoir. Bolted to the bottom of the

LHe reservoir was an oxygen free high-purity Copper (OFHC) sample cold finger (brown)

that extends down below the outer vacuum shield into the science chamber. The entire

63

cold space was enclosed via two custom made 17" long by 14" OD stainless steel full nipples

connectorized with 16.5" conflat flanges on either side. Below these, a 16.5" → 6" reducing

full nipple conflat flange was employed such that the UHV sample chamber with optical

access could be bolted on.

Microwave wiring was introduced into the cryostat via UHV rated SMA feed throughs

(Kurt J. Lesker P/N IFDCG012011) welded into a 2.75" conflat flange. The interior microwave

coax had a copper-nickel (CuNi) outer conductor with a silver–plated CuNi inner conductor.

Each signal line was thermally sunk to each successive temperature stage so as to mitigate

the heat load seen by the base of the LHe reservoir. Copper foil straps for heat sinking were

soldered on to wires and bolted to the top of the liquid nitrogen reservoir, the 77 K stage

plate, the top of the LHe reservoir and finally to the 4 K stage plate. Two wires were utilized

in the S21 through microwave measurement of the resonator sample. Sixty dB of attenuation

was placed on the 4 K stage plate and another 10 dB at the sample mount allowed for

measurements in both the high and low power regime. The output signal was amplified at 4

K via a Caltech Microwave HEMT amplifier (BW = 4-12 GHz, T < 5 K noise temperature,

P/N CITCRYO4-12A) followed by a room temperature NARDA HEMT amplifier.

Design and material choices were made in effort to simultaneously satisfy two (and often

contradictory) criteria; Minimize the heat load to the LHe vessel while preserving a clean

ultra-high vacuum (UHV) environment. While an exhaustive list of design choices used in

effort to satisfy these conditions would approach tome levels in length, it is worthwhile to

discuss a few particulars and their relevancy towards a working UHV cryostat.

Conflat UHV Flanges

While perhaps an obvious choice when compared to viton compression seals, it is worth a short

discussion of why Conflat all-metal seals are important for this design. Conflat seals provide

a thermally bakeable vacuum shield. Temperatures greater than 200 C are generally employed

to fully outgas adsorbed water off of the stainless steel surface while the copper gaskets

64

Figure 4-3: Machine drawing of the custom build UHV cryostat (left) and a picture of theinstalled cryogen vessels (right). Custom pant leg seals allow for cryogen transfer into thereservoirs. Not shown are the top plate wiring feed throughs or the optical access samplechamber.

65

Mat. A (cm2) LUR (empty, Torr L/sec) LUR (Inserted, Torr L/sec) LUR (Torr L/(sec cm2))

G10 90 1.0 × 10−5 1.3 × 10−5 3.3 × 10−8

Rogers 4000 series 13 2.8 × 10−8 1.2 × 10−7 7.1 × 10−9

Table 4.2: Measured leak-up rates (LUR) for G10 support struts and Rogers 4000 seriesmicrowave circuit board.

provide a seal capable of withstanding the 12-14 orders of magnitude pressure difference

between the inside of the vacuum space and the surrounding atmosphere environment. While

the maximum bakeable temperature of this system was limited by indium seals utilized in

the packaging of the Caltech HEMT microwave amplifier to below 110 C, the conflat flanges

still provided the sufficient and necessary vacuum interface to achieve the required pressures

for the experiment.

G10 Support Struts

G10 support struts were utilized in the tandem hanging of the LN2 and LHe reservoirs. G10

provides structural integrity and rigidity while also possessing a low thermal conductivity of

0.1 W/mK constant over a temperature range of 4.2 K to 293 K. The major concern with

G10 in this implementation, however, was its outgassing properties under UHV conditions.

The vacuum leak-up rate (LUR) for the G10 struts at room temperature was measured by

monitoring the vacuum leak-up rate for a turbo pumped stainless steel conflat full nipple both

empty and with the G10 strut present in the vacuum chamber. The results are tabulated

in table 4.2. The room temperature LUR for one G10 strut was measured to be 1.3 × 10−5

Torr L/sec. At room temperature, a LUR of this magnitude would be a major hurdle to

achieving the necessary base pressures required for single atom experiments, but at cryogenic

temperatures, the benefits of the low thermal conductivity far outweighs whatever slight

LUR they may maintain when at 77 K or 4.2 K.

66

Al 1100 / 3003 77 Kelvin shield

Two different alloys of aluminum were utilized in the construction of the 77 K shield. Aluminum

alloy 3003 (Al3003) was utilized for the fabrication of any and all flanges where drilling and /

or tapping of holes was required. Al3003 has a good machine-ability while also possessing a

low emissivity (ε3003 ≈ 0.03) when compared to it’s more commonly found 6061 counterpart

(ε6061 ≈ 0.1). The tubing for the shield was fabricated out of near elemental 1100 series

aluminum (Al1100) purely based on it’s near elemental emissivity value 0.01 < ε1100 < 0.03.

Wiring and Wiring Materials

The choice of materials related to microwave wiring was based on microwave signal performance

alone, relying on cryogenic operation to usurp any deleterious outgassing effects from the

intrinsic materials. PTFE Teflon insulated 50 Ω 860/200 μm outer/inner conductor coaxial

wire (Coax Co. P/N SC-086/50-SCN-CN) was utilized to route microwave signals from 300

K to 4.2 K. The outer (inner) conductor was made of (silver–plated) cupronickel with a

manufacturer specified thermal conductivity at 4.2 K of 7.0×10−3 W/m K. Normally one

would choose the cupronickel inner and outer conductor option for thermal conductivity

concerns (the thermal conductivity of the all cupronickel wire is specified as 9.84×10−4

W/m K) when wiring from room temperature but in a trade off to maximize microwave

performance, a single length of the silver-plated option was chosen to go between 293 K and

the 4.2 K stage plate with copper strap heat sinks placed at the top of the LN2 vessel, the 77

K stage plate, and the top of the LHe vessel. Each wire prior to installation was cleaned in

subsequents baths of Alconox, acetone, and isopropyl alcohol (IPA).

Rogers 4000 series microwave circuit board was utilized to transition from the SMA

cabling to the sample mount and provide a wire bonding surface in which to route the

microwave signals to the resonator. Again chosen specifically based upon the materials

microwave performance, the outgassing properties required testing. The results of the

outgassing experiments are tabulated in Table 4.2. At room temperature, the measured

67

material LUR was 7.1×10−9 Torr L/(sec-cm2), which, for a board size of 1 cm2, would only

require a pumping speed of 1 L/sec to maintain pressures below 10−8 Torr. When cooled to

LHe temperatures, this rate becomes even more negligible.

Choices related to DC wiring (HEMT power, electric field compensation pins, etc) were

based on minimizing outgassing. All DC wires were Kapton clad solid–core copper wire. The

gauge of the wire was chosen based upon operating current / voltage considerations. Kapton

/ polyimide outgassing rates for similar cables have been measured for the LIGO experiment

with reported rates as low as 1.0 × 10−8 Torr L/s/cm2 at 286 C.

4.2.1 Cryostat Performance

The base temperature of the sample mount when fully assembled reached Tbase = 5.04 K. The

limiting factors in achieving a lower temperature at the sample mount were direct exposure

to 300 K radiation necessitated by the experimental design and the poor base temperature of

the LHe reservoir base plate of T = 4.6 K. While no explicit tests were done to investigate

the mechanism for such a poor base plate temperature, it is currently believed that the

explosively joined bi–metal reservoir plate is to blame.

The base pressure of the cryostat with no active pumping beyond that provided by the

vessel cryopumping was measured to be below 10−9 Torr with an ion gauge and further

confirmed with a measured Cs MOT lifetime of 1.5 seconds, corresponding to a background

pressure of 6×10−10 Torr.

The vibrational motion of the cryostat was also measured utilizing a pair of power balanced

852 nm laser beams. The experimental setup is displayed in Figure 4-4(a). A collimated 852

nm laser beam was split into two power balanced arms utilizing a 50/50 beam splitter. One

arm was used as a reference beam so as to calibrate out any time dependent drifts in the

beam intensity. The other arm was passed into the cryostat and aligned so as 1/2 half of the

beam illuminated a razors edge that was bolted onto the cryostat cold finger. The motion

calibrated difference spectrum of the measurement is displayed in Figure 4-4(b). Motion

68

calibration was performed by walking the measurement beam off the razors edge in predefined

increments and monitoring the beam difference level. Integrating the motion spectrum out

to the knee frequency fknee = 1 KHz and taking the square root yielded an RMS motion of

1.3 μm.

4.3 Superconducting CPW Resonator Design

With the new temperature constraints imposed by the LHe cryostat, it is worth revisiting

the fidelity calculations for the Bell state preparation but now taking into account a finite

temperature. The average photon number in the resonator as a function of temperature is

given by the Bose-Einstein distribution

n =[e−�ωr/kBT − 1

]−1. (4.8)

These photons incoherently drive the |r〉 → |r′〉 Rydberg transition in addition to limiting thecavity superposition state preparation. Figure 4-5 displays the calculated fidelity for the Bell

state preparation as a function of the cavity temperature. Above 50 mK, incoherent driving

from the now non–negligible thermal photon occupation begins to play an appreciable roll in

the dynamics and the fidelity begins to decrease quickly.

With this in mind, the new figure of merit chosen to deem the implementation of the

superconductor-atom interface a success is the number of Rabi oscillations nRabi or coherent

excitation swaps between the disparate systems before total photon loss. The number of

coherent excitation swaps between a resonator and a qubit is defined as

nRabi =2g

κ + γ(4.9)

where κ and γ and the loss rates of the resonator and qubit, respectively. The dipole coupling

strength between the two systems is g and Q is the total quality factor. The photon loss

69

um2 /H

z

852 nm

50/50 Beam Splitter

(a)

(b)

ReferenceBeam

MeasurementBeam

Cryostat / Sample

Figure 4-4: (a) Diagram depicting the experimental setup to measure the vibrational motionof the cryostat cold finger. A collimated beam of 852 nm laser light was split utilizing a50/50 beam splitter the arms power balanced referenced to each other. One arm was used asa reference beam so as to be able to calibrate against drifts in the overall beam power. Theother was passed into the cryostat, passing by a razor’s edge that was bolted onto the coldfinger. (b) Vibrational motion data of the cryostat cold finger taken at 4.2 K. Integratingthe spectrum gives an RMS motion of 1.3 μm.

70

0.0 0.1 0.2 0.3 0.4 0.565

70

75

80

85

90

95

100

Temperature (mK)

Fide

lity

(%)

Figure 4-5: Simulation of the Bell state preparation fidelity as a function of temperature.Above approximately 50 mK, the fidelity begins to roll off as the thermal photon number inthe cavity becomes appreciable and incoherent driving begins to play a dominant role in thedynamics.

rates γR for Rydberg atoms is of order ∼ 1 ms−1, which, when compared to the photon loss

rate in state of the art superconducting planar resonators κ = ωr/Q ≈ 50 ms−1 for a center

frequency ωr = 2π × 5 GHz and Q = 105, can be ignored. Eq. (4.9) now reduces to

nRabi =2gκ

→ 2gQ

ωr

. (4.10)

For a fixed frequency ωr, we have only the product gQ to maximize in order to achieve the

strong coupling limit defined as nRabi > 1.

At the base temperature of the cryostat, the primary microwave loss mechanism limiting

the Q factor is thermal quasiparticles. We will review the basics of CPW resonator geometry

followed by a discussion of thermal quasiparticles and the Mattis-Bardeen conductivity. We

71

will then combine these two to form a full description of thermal quasiparticle dominated loss

in superconducting CPW resonators and how engineering the quality factor can be achieved

through judicious choice of resonator geometry.

Engineering the coupling strength g will also be addressed. It will be shown that for

standard CPW geometries, the electric field and thus the coupling between the atom and

resonator is very small at realistic atomic laser trap distances from the CPW surface. We will

show how to properly engineer the CPW electric field to extend far beyond the chip surface,

greatly enhancing the spatial extent of the field while introducing no additional loss.

4.3.1 Engineering the Quality Factor

4.3.1.1 CPW Basics

The CPW geometry can be thought of as a two dimensional coaxial wire. It is comprised of a

center trace surrounded on each side by a ground plane. The relevant physical dimensions are

the width of the center trace, W , and the gap between the center trace and the ground plane,

S. Conformal mapping techniques [65] allow for closed form equations of the capacitance Cg

and inductance Lg per unit length the CPW cross sectional geometry, given by

Cg = 4ε0εeffK(k0)K(k′

0)(4.11)

Lg =μ0

4K(k′

0)K(k0)

, (4.12)

where ε0, μ0 are the permittivity and permeability of free space, respectively. The effective

dielectric constant εeff, to first order, is the average between the relative dielectric substrate

permittivity and the permittivity of free space

εeff =εr + 12 . (4.13)

72

The K terms are the complete elliptic integrals of the first kind. The arguments k0 and k′0

are dependent on the CPW geometry in the following manner

k0 =W

W + 2S (4.14)

k′0 =

√1 − k2

0 . (4.15)

An important warning to heed is that K ′(k0) and K(k′0) are often interchanged in the

literature concerning CPWs. The impedance of the line is defined in the normal way as

Z =√

Lg/Cg . (4.16)

Terminating a CPW at either end with a capacitive open or inductive short will impose

standing wave boundary conditions. If both ends of the CPW are either shorted or open, the

CPW will support resonant frequencies at half wave integer multiples of the fundamental.

Should the terminations of the CPW be mixed and matched with one end being capacitive

with the other inductive, the CPW will support resonant frequencies at quarter wave integer

multiples.

4.3.1.2 Mattis-Bardeen Conductivity

The notion of describing the flow of electrons in a solid as a two component fluid arises

naturally even when considering the most basic of models. The Drude model offers a classical

interpretation of electrical conductivity through the application of Newton’s second law. The

force exerted on a particle of charge q from an electric field �E is

md�v

dt= q �E − m�v/τ (4.17)

73

where �v is the velocity and τ the relaxation time. Utilizing the relationship between current

density and the velocity of charge carriers,

�J = nq�v (4.18)

where n is the charge carrier volumetric density, Eq. (4.17) becomes

m

nq

d �J

dt= q �E − m

nqτ�J . (4.19)

Substituting Ohm’s law for an alternating electric field �E ≡ E0eiωt, the force equation takes

the form

σiωτE0eiωt + σE0eiωt = nq2τ

mE0eiωt . (4.20)

Dropping common terms, one arrives at a complex conductivity

σ = σ0

1 + ω2τ 2 + iωτσ0

1 + ω2τ 2 (4.21)

where σ0 = nq2τ/m. Only in the limits of very large frequency or very long relaxation time

does the imaginary part of Eq. (4.21) begin to contribute to the overall conductivity.

This simple model of complex (normal) conductivity fails in the limit of low temperature

/ long relaxation time and high frequencies as the field does not exponentially fall to 0 inside

the metal as predicted by the classical skin depth. This phenomena, known as the anomalous

skin effect, was first derived by Chambers [66] and formally extended for superconductors by

D.C. Mattis and John Bardeen [67].

In their seminal paper (culminating in the majority of Mattis’ PhD thesis), Mattis and

Bardeen provide a full quantum mechanical treatment of conductivity for both normal and

superconducting metals. For the superconducting case, the real and complex contributions

to the conductivity σ = σ1 − iσ2 are given by

74

0

0.25

0.50

0.75

1.00

10

10

10

10

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4-6: Normalized Mattis-Bardeen conductivities σ1 and σ2 verse reduced temperatureT/Tc. As the temperature approaches 0, the real part of the conductivity (blue curve)responsible for DC resistance goes to 0. In the opposite limit (T → Tc), the real part of theconductivity approaches the normal state conductivity at Tc. Conversely, the imaginary partof the conductivity approaches 0 as the critical temperature is reached.

σ1

σn

= 2�ω

∫ ∞

Δ[f(E) − f(E + �ω)] (E2 +Δ2 + �ωE)√

(Δ2 − E2)√(E + �ω)2 − Δ2

dE (4.22)

+ 1�ω

∫ −Δ

Δ−�ω[1 − 2f(E + �ω)] (E2 +Δ2 + �ωE)√

(Δ2 − E2)√(E + �ω)2 − Δ2

dE

σ2

σn

= 1�ω

∫ Δ

Δ−�ω

[1 − 2f(E + �ω)](E2 +Δ2 + �ωE)√(Δ2 − E2)

√(E + �ω)2 − Δ2

dE (4.23)

where σn is the normal state conductivity at the critical temperature Tc, and Δ is the BCS

superconducting energy gap. The second term in Eq. (4.22) is zero unless �ω � 2Δ.

Figure 4-6 displays the normalized real and imaginary parts of the Mattis-Bardeen

conductivity verse the reduced temperature T/Tc. As T → Tc, the real part of the conductivity

approaches the normal state conductivity when evaluated at Tc. Conversely, the imaginary

part of the conductivity approaches 0. In this opposite limit, the real part goes to 0 while

75

the imaginary part saturates.

The Mattis-Bardeen complex conductance gives rise to a complex impedance of the form

Zs =1

σd= 1

d

(σ1

σ21 + σ2

2+ i

σ2

σ21 + σ2

2

)≡ Rs + iωLk (4.24)

where d is the thickness of the metal and we have equated the real and imaginary parts to an

effective surface resistance Rs and what is commonly referred to as a kinetic inductance Lk.

The ratio of the imaginary to real parts of the impedance define the quality factor Q of the

system, which, when combined with the geometric contribution Lg from a CPW yields the

relation

Q = ω(Lk + Lg)Rs

= ωLk

Rs

(1 + Lg

Lk

)= σ2

σ1

(1 + Lg

Lk

). (4.25)

The ramification of Eq. (4.25) is that at temperatures of an appreciable fraction of the

superconducting critical temperature, the quality factor of a CPW resonator isn’t only just

temperature dependent but also geometry dependent.

The kinetic contribution to the total inductance is defined by looking at the energy stored

in the supercurrent,

12I2Lk =

12μ0λ2

∫j2dS , (4.26)

where μ0 is the permeability of free space, λ is the superconducting penetration depth, j is

the supercurrent density and the integral is taken over the cross sectional area dS of the CPW.

Rearranging Eq. (4.26), one can define the kinetic inductance in terms of the supercurrent

density j,

Lk = μ0λ2∫

j2 dS

(∫

j dS)2 (4.27)

The problem of calculating the kinetic inductance has now been reduced to calculating the

76

cross sectional supercurrent density. The cross sectional supercurrent density is in general

highly non-uniform necessitating numerical evaluation. However, in the thin film limit where

λ2/d W , a closed form equation for the kinetic inductance can be derived and is given by

Clem [68] as

Lk =μ0λ

Wq(d/λ)g(k, ε) (4.28)

where q(d/λ) is a thin-film thickness correction given by

q(x) = (sinh(x) + x)/8 sinh2(x/2) (4.29)

and g(k, ε) is an of order unity unit-less geometric correction given by

g(k, ε) = 12(1 − k)K2(k) ln

[2(1 − k)ε(1 + k)

], (4.30)

where ε 1.

Figure 4-7 displays the closed form and numerically evaluated kinetic inductance per

unit length for different CPW geometries. The penetration depth λ and metal thickness d

simulated were λ = d = 100 nm. Very good agreement can be seen between the numerical

analysis and the closed form provided by Clem for most values of the center trace width and

the gap. For wider center traces, cutoff procedures utilized in [68] to achieve closed form

solutions begin to annex non-negligible contributions to the supercurrent.

With the kinetic and geometric inductances now fully in hand, the thermal quasiparticle

limited internal quality factor can be directly modeled. Figure 4-8 displays the numerically

calculated quality factor Q as given in Eq. (4.25) for niobium with a superconducting gap

Δ = 1 mV at a temperature of 4.2 K again with λ/d = 1.

In the limit of S W , The geometric inductance scales as Lg → μ0S/W leading to the

ratio Lg/Lk ∝ S/λ. Conversely, when S ≈ W , the geometric contribution Lg ∼ μ0 while the

kinetic contribution reduces to Lk = μ0λ/W . Combined, this gives the ratio Lg/Lk ∝ W/λ.

77

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250.5

1

1.5

2

2.5

3

3.5

4x 10

Gap Width, S (um)

Lk

(Hen

ry/M

eter

)

W = 5, NumericalW = 5, ClemW = 10, NumericalW = 10, ClemW = 15, NumericalW = 15, ClemW = 20, NumericalW = 20, Clem

(a)

(b)

Figure 4-7: (a) Normalized supercurrent density averaged over the thickness of the tracesfor CPW geometry W = 6μm, S = 3μm and thickness d = 100 nm. (b) Closed form andnumerical evaluation of the kinetic inductance per meter for CPWs of different geometries.For S/W ≈ 1, the kinetic inductance is primarily a function of W . Conversely, for small S,the kinetic inductance has a non-negligible gap dependence.

78

Factors of 5–6 improvement in the internal quality factor are predicted with modest increases

with both the CPW center trace width and CPW gap.

4.3.2 Measuring the Quality Factor

To test the geometric dependence of the quality factor, Nb CPW resonators of differing

geometries were fabricated at the Wisconsin Center for Applied Microelectronics (WCAM)

on the University of Wisconsin - Madison campus. Three inch (75 mm) diameter, 500 μm

thick sapphire wafers were loaded into a magnetron sputter system. Prior to sputtering, the

3–inch Nb target was cleaned with a 400 W argon plasma for 2 minutes. For deposition, the

power was increased to 500 W and the three inch wafers were rotated at ωsample ≈ 2π × 1

Hz. Sputtering for 2 minutes yielded a film of 90 nm, which was verified with profilometry.

Photolithography began by spinning SPR-955 0.9CM photoresist on the wafers at 4500 RPM

for 60s followed by a pre-exposure bake at 100 C for 90 seconds. Wafers were then loaded

into a Karl Suss MA6 contact aligner. Hard contact mode was utilized with an exposure

wavelength of 365 nm and exposure time of 3.6 seconds. Wafers were then removed and

post-exposure baked (PEB) on a hotplate at 110 C for 90 seconds. Development was done

via immersion in Microchem CD-26 TMAH developer for 60s followed immediately by a DI

H2O rinse for another 60s. The resulting pattern was then checked via optical microscopy.

Subsequent reactive ion etching (RIE) of the pattern was carried out in either a Unaxis

Plasmatherm 790 RIE tool utilizing a sulfer hexaflouride (SF6) etch or a Unaxis Plasmatherm

770 RIE tool utilizing a boron trichloride etch depending on the tool availability. The sapphire

substrate utilized for these experiments has a negligible etch rate for either etch recipe.

Figure 4-9(a) shows the resulting sample. Each chip contained six quarter-wave CPW

Tool Gases Flow (SCCM) RF Power RIE/ICP (W) Rate (nm/min)

790 SF6 / O2 15 / 20 150 / NA 100770 BCl3 / Cl2 / Ar 18 / 9 / 18 50 / 300 45

Table 4.3: Parameters used for etching Nb films in either a PT 790 or 770.

79

0 5 10 15 20 25 30CPW Gap, S ( m)

1000

2000

3000

4000

5000

6000

7000In

tern

al Q

ual

ity

Fac

tor

Qi

W = 5 mW = 10 mW = 15 mW = 20 m

Figure 4-8: Numerical evaluation of Eq. (4.25) for a Nb film of thickness d = 100 nm,penetration depth λ = 100 nm, and superconducting gap Δ/e = 1 mV simulated at T = 4.2K. Just as with the kinetic inductance, in the limit of small gap S, the quality factor ispredicted to be relatively width independent and scale predominantly with the gap. As theCPW gap is widened, this dependence lessons and is replaced with a strong dependence ofthe width W .

resonators multiplexed in frequency between 4.5 and 5.0 GHz. Each resonator had a capacitive

coupling to the common feedline of 5 fF resulting in a coupling quality factor Qc ≈ 104.

Samples were aluminum wire bonded inside an aluminum box. The sample box was then

wired to a LHe dip probe and surrounded with a high permeability metal shield. This was

then immersed in LHe and the forward scattering matrix element was measured utilizing a

vector network analyzer (VNA). The resulting spectrum for one of the samples is displayed in

Figure 4-9(b). Each transmission dip was fit to Eq. (3.37) with fitting parameters Qi, Qe, and

ω0. The extracted internal quality factors Qi as a function of sample geometry are plotted in

Figure 4-10. The solid lines are the predicted Mattis–Bardeen quality factors from Eq. (4.25)

80

Frequency (GHz)4.5 4.6 4.7 4.8 4.9

S 21 (d

B)

-16

-14

-12

-10

-8

-6

-4

(a)

(b)

Figure 4-9: (a) Optical micrograph of the multiplexed quarter-wave resonator chip. Brightregions are Nb while dark are silicon. Each chip was 6×6 mm2 containing 6 resonatorsmultiplexed in frequency from 4.5 - 5.0 GHz. (b) Representative S21 transmission spectrum.Each sample was aluminum wire bonded inside an aluminum sample box. The box was thenattached to the SMA leads of a custom LHe dip probe and encased in a mu-metal shield.Samples were submerged in LHe and the transmission spectra taken for each dip.

81

for Nb with a measured critical temperature Tc = 8.8 K, zero-temperature penetration depth

λ0 = 87 nm, and superconducting gap Δ/e = 1 mV. There is a non-negligible dependence

on the resonant frequency for the Mattis–Bardeen conductivities, thus in order to obtain

better agreement between the theory and the data, the extracted center frequencies from the

measured resonators were used in the calculation of σ1 and σ2. This is why discrete steps are

seen in the solid prediction lines as frequencies were recycled every 6 measured resonators.

As history as shown, anything bearing the name “Bardeen” tends to be correct and great

agreement can be seen between the extracted resonator quality factor and that predicted by

Eq. (4.25). The geometry dependence predicted and thusly measured means that the loss can

be directly engineered simply by changing the geometrical layout of the CPW resonator with

almost an order of magnitude increase in the internal quality factor from a [W, S] = [5, 1]μm

to a [W, S] = [50, 30]μm pairing.

4.3.3 Engineering the atom-CPW coupling strength

The second but equally important term to maximize in the quest to reach strong coupling

between the CPW resonator and a single Rydberg atom is the vacuum coupling strength or

Rabi frequency g, defined as

g =�E · �d

�(4.31)

where �E is the photonic electric field vector and �d is the dipole moment of the atom. The

electric field of a CPW resonator is highly concentrated in the gap between the center

electrode and corresponding ground planes. Figure 4-11(a) shows the numerically calculated

zero-point electric field profile for a CPW resonator with center trace W = 20μm and a gap

width S = 10μm. The largest magnitude field is in the CPW gap between the center trace

and ground planes and falls off exponentially near the surface of the chip. Figure 4-11(b)

plots the absolute magnitude of the electric field and the coupling strength g (assuming the

82

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30

Inte

rnal

Qu

alit

y F

acto

r (x

103 )

CPW Gap, S (μm)

W = 5 μm W = 10 μm W = 20 μm W = 30 μm W = 50 μm

Figure 4-10: Extracted internal quality factor Qi for different CPW geometries. Error barsfor the fits are smaller than the symbol size. The solid lines are the predicted Mattis-Bardeenthermal quasiparticle limited quality factors for a superconducting gap Δ/e = 1 mV and azero temperature penetration depth λ0 = 87 nm. The discrete steps in the predictions resultfrom utilizing the extracted resonant frequencies for the calculation of the Mattis–Bardeenconductivities.

field and dipole moment are aligned) as a function of height above the chip surface.

The dipole moment vector utilized in the calculation was again for the electronic transition

between neighboring Rydberg states 90s1/2 and 90p3/2 with magnitude

drr′ = 〈90s, m = 1/2| d |90p, m = 3/2〉 =√2/9 × 8360 ea0 (4.32)

where e is the fundamental charge and a0 is the Bohr radius. For physically realizable atomic

trapping heights (z � 25μm) above the chip, the coupling strength is below 1 MHz resulting

83

in a nRabi < 1 even for a quality factor in excess of 104.

In effort to combat the low coupling rate and provide a reasonable distance at which the

atom could be trapped from the chip surface while also minimizing the amount of stray laser

light scattered on the chip surface, a thick–film copper electroplating process was developed

such that the voltage antinode of the resonator could be physically extended far into the

z-axis. Figure 4-12(a) displays a CAD drawing of the “science” end of the resonator. To

facilitate strong coupling to the atom, 2× ∼ 50μm tall copper pillars would be plated at

the capacitively shunted end of a quarter-wave CPW resonator. The superconducting Nb is

colored blue while the sapphire substrate is colored yellow. The electroplated copper pillars

are colored pink and the trapping laser is colored green. The red sphere denotes the optimal

trap placement for the atom.

(a) (b)

Figure 4-11: (a) False color contour plot of the zero-point (ncav = 0) electric field profilefor a CPW resonator with center trace width W = 20μm and a gap width S = 10μm.(b) Absolute magntiude zero-point electric field (left y-axis) and coupling strength g (righty-axis) as a function of height z above the chip surface. The solid blue line is taken to bedirectly over the center trace while the dashed red line is calculated in the gap between thecenter trace and ground plane. The coupling rate g was calculated for the dipole momentbetween Rydberg states 90s1/2 and 90p3/2. Even for the large dipole moment between theseneighboring electronic Cs Rydberg states, for realizable trapping distances (z � 25μm) abovethe chip surface, the coupling strength g 1 MHz.

84

Figures 4-12(b-c) show COMSOL simulations for the electric field between the plated

structures for a zero point voltage

〈V 2〉1/2 =√�ωr/2CCPW = 2μV (4.33)

placed on the center trace of the CPW where ωr = 2π × 5.4 GHz is the center frequency

of the resonator and CCPW = 0.44 pF. Figure 4-12(b) is a side-on view looking down the

�x axis. The plated structures are outlined in black. The false color maps the electric field

strength while the white arrows denote the field vector orientation. For the majority of the

height of the copper electrodes, the electric field is uniform in both magnitude (0.06 V/m)

and orientation (y). Figure 4-12(c) is an overhead view of the modeled field looking in the

−z direction. The electric field again is uniform in magnitude and orientation this time

throughout the 30 μm width of the copper pillars. Given this field magnitude, the coupling

is calculated to be g/2π ≈ 3 MHz, a factor of ∼ 10 higher than without the posts.

The fabrication of the copper pillar structures is outlined pictorially in Figure 4-13. A

base layer of 180 nm thick Nb was sputtered on to a 3-in, 500 μm thick single-side polished

sapphire wafer. Positive photoresist SPR-955 0.9cm was manually dispensed over the surface

of the wafer spin-coated at 4000 RPM followed by a pre-exposure bake on a contact hotplate

at 100 C for 90s. The photoresist was then lithographically patterned utilizing a 5:1 projection

i-line (365 nm) stepper. The wafer was then post exposure baked on a contact hotplate at 110

C for 90s. Development was performed via submersion and agitation in Microchem CD-26

TMAH developer for 60s. The wafer was then submerged and agitated in a bath of deionized

H20 for another 60s. The pattern was visually inspected under a microscope prior to etching.

Etching of the Nb base layer was performed in a Unaxis 790 reactive ion etcher utilizing the

recipe outline in Table 4.3. The plasma etch time was 2 min. The purposeful over etching

time was set to ensure a complete etch. The low reactivity of the sapphire substrates allows

this to be possible without trenching. Once etched, the sample was left submerged in room

temperature acetone for 12+ hours to ensure proper removal of the photoresist and residual

85

(a)

(b) (c)

Figure 4-12: (a) 3D CAD drawing of proposed electroplated structures at the voltage antinodeof a CPW resonator. (b) Side profile detailing zero-point electric field strength and profile assimulated in COMSOL. The False color maps the field strength in V/m while the arrowsdenote the field vector orientation. (c) Overhead profile detailing zero-point electric fieldstrength and profiles as simulated in COMSOL. Notice that in both simulations the fieldbetween the plates is homogeneous in both magnitude and vector orientation.

86

organics.

Initial experiments in directly trying to electroplate Cu onto blanket wafers of sputtered

Nb showed little success. The Cu films had very large grain size and required large plating

potentials to achieve the manufacturer recommended plating current densities of 10 A / cm2.

It was decided that the poor conductivity of Nb at room temperature in addition to it’s native

amorphous oxide was to blame. To combat this, an interstitial layer comprised of electron

beam evaporated titanium and palladium was added. Poor adhesion of Pd necessitates the

Ti adhesion layer. Negative photoresist AZ-5214 was spin coated at 4000 RPM followed by

a pre-exposure bake at 95 C for 60s. The wafer was then loaded on to a Karl Suss MA6

contact aligner and put into hard contact with the mask. The streets between the dies and

the desired areas intended for electroplating were left unexposed while the rest of the wafer

is exposed to 365 nm , 10 mW / cm2 light for 30 s.

The wafer was then image reversal baked at 110 C for 60s followed by a UV flodd exposure

for 60s. The development protocol is the same as for SPR-955. Once fully developed, the

wafer was placed inside a custom 4-pocket hearth electron beam evaporator. The patterned

wafer was argon ion milled for 17 seconds at a backing pressure of 4×10−5 Torr prior to

deposition. After, a 3 nm Nb↔Pd adhesion layer of titanium was evaporated at 0.5 nm/s as

measured with a crystal thickness monitor adjacent to the sample stage. Thirty nanometers

of Pd was then evaporated at 0.5 nm/s so as to provide a base metal for the electro-deposition

of the Cu structures that does not natively oxidize. Once finished, the wafer was removed

from the evaporator and left to sit in a bath of room temperature acetone for 12+ hours so as

to lift off the evaporated metal on top of the exposed negative resist. Mild sonication could

be used to accelerate the lift off process. Once complete, the lift off left the streets between

sample dies along with the capacitor ends of the resonators covered with the Ti/Pd bilayer.

The electro-deposition of the Cu capacitor plates began with the patterning of a photoresist

hard mask. Intervia BPN-65A negative photoresist was spun first at 400 RPM for 1 minute

with a ramp up to 800 RPM for 2+ minutes. While not explicitly measured, according to

87

Cu+

Cu+

Cu+

Cu+

Cu+ Cu+ Cu+Cu+

- - - - - - - - - - - -

Al2O3 Substrate

Niobium

SPR-955

AZ-5214

Ti/Pd

Intervia BPN

Cu2SO4 Sol’n

EP’d Cu

(a) (b)

(c) (d)

(e) (f)

Figure 4-13: Process for electroplating Cu on thin film Nb. (a) The process began with a180 nm thick sputtered layer of Nb (pink) on a 500 μm thick sapphire substrate (silver). (b)The base layer Nb was optically patterned with SPR-955 positive photoresist (dark blue)followed by a SF6 plasma etch. (c) Deposition of the Ti/Pd (yellow) interstitial platingadhesion layer was done via a negative resist (AZ-5214, purple) and electron beam evaporationfollowed by a lift off via submersion in acetone. Note that the undercut in the photoresistwas important to ensure that contact is broken between metal deposited in the developedareas and that on top of the photoresist. (d) Intervia BPN–65A photoresist (green) wasspun on exposed and developed to expose Ti/Pd surface. (e) The wafer was submerged inEnthone copper electroplating solution (blue) with a negative voltage potential applied acrossthe face of the wafer that attracts the positively charged Cu ions. (f) Once full plated, thewafer was removed from the solution, rinsed, dried, and diced. It was only then that thethick photoresist is removed revealing the lone standing copper structures (brown).

88

the spin-speed curve provided by the manufacturer, this should provide a resist thickness of

60 um. The resist was then pre-exposure baked (no post-exposure bake is required) at 60 C

for 3 minutes followed by 120 C for 9 minutes. The second bake time can be adjusted if the

resist is still soft after completion of initial 9 minute time. The resist was then allowed to

cool to room temperature. Once cool, the resist was exposed in a Karl Suss MA6 contact

aligner with 365 nm, 10 mW/cm2 UV light for 60s. Development was performed via puddle

develop in Microchem CD-26 developer for a minimum of 3×1 min intervals, with fresh

developer added every 1 minute. If, upon inspection after this first 3 minutes, the resist was

not fully developed, the process was continued until complete. Post development, many of

the developed areas had organic residue left over. This was subsequently removed via an

oxygen plasma ash descum of 50 mT O2 × 250 W × 5 min.

The now fully patterned wafer was placed into a single-wafer electroplating holder from

Advanced Micromachining Tools (AMMT.com). DC electrical contact between the holder

and the wafer was made via gold plated copper springs pressed into the front side of the

wafer. These finger springs were screwed into a gold plated copper ring housed in the interior

of the Teflon holder. The interior ring further made electrical contact with copper wires that

ran up the interior of a Teflon rod, eventually terminating in a female 3.5 mm socket.

Prior to submersion in the copper sulfate plating bath, the bath had to first be heated to

25 C and stirred at 200 RPM. It was important to use a properly leached polypropylene tub

as the acidic plating solution would become contaminated from impurities in the plastic if

not done. The plating solution used was commercially available and produced by Enthone

(www.enthone.com). Once heated and stirred, the prepared sample and a plating cathode

were both submerged facing each other into the plating bath approximately 10 inches apart.

The cathode used for this process was a 3"×6" rectangular Ti mesh. Potential was appliedto the Ti cathode and the wafer anode via a Dynatronix Pulse Series (dynatronix.com)

plating supply. The average forward plating current was set to provide a current density of

10 A/cm2 across the exposed wafer surface with the forward–to–reverse plating current and

89

plating time ratios set to 2:1 and 20:1, respectively. The total charge transfer was monitored

during the plating process. A total charge transfer of 1500 A·s resulted in structures of

∼ 50μm in height. Once complete, the sample was removed from the sample holder and

both were rinsed thoroughly in DI water. Failure to do so would allow the plating solution

to evaporate depositing difficult to remove copper sulfate crystals on the hardware / wafer.

To ensure survival of the freshly plated structures, the wafer was diced prior to the Intervia

BPN photoresist removal. Finally, removal of the Intervia photoresist was done via overnight

submersion in 75 C Intervia BPR photostripper.

Figure 4-14(a) displays the final product of the fabrication. The CPW geometry chosen

for the sample was W = 50μm and S = 25 μm, yielding an impedance Z = 50Ω. Samples

were 16 mm long by 4 mm wide tapering down to a with of 150 μm at the science end (red

box) so as to minimize the amount of stray trapping laser light deposited on the chip. The

Rayleigh length for a focused laser beam is

xR = πω20

λ(4.34)

where ω0 is the radius of the beam at the focus and λ is the laser light wavelength. For a

trap laser light of λ = 780 nm and a focus ω0 = 3 μm, the Rayleigh length is xR = 36μm.

The beam radius as a function of distance from the focus is given by

w(x) = w0

√1 +

[x

xR

]2(4.35)

For a maximum allowable beam radius w(x) ≡ wmax = 25μm, half the max height for the

copper electroplated structures, the max width of the chip at the trapping sight was

xmax = xR

√[wmax

w0

]2− 1 = 215μm . (4.36)

The tapering of the chip profile was achieved via a dicing saw with the overall width at the

position of the copper structures brought down to 150 μm.

90

100 um

50 um

(a) (b)

(c) (d)

(e) (f)

Figure 4-14: (a) Micrograph of the fabricated superconducting quarter wave CPW resonator.The chip was 16 mm long by 4 mm wide tapering down to a width of 150 μm at the scienceend. The resonator was inductively coupled to a feed line and capacitively shunted to ground.Image courtesy of Edward M. Leonard. (b) Profile view of the electroplated structures. (c)SEM image of electroplated structures. (d) Head on cartoon profile illustrating the necessityof a tapered chip (gray) profile. The beam (red) waist w scales by

√2 every Rayleigh length

xR away from the focus. (e) Microwave transmission across an electroplated resonator takenat T = 4.2 K. The internal quality factor was fitted to be Q4K

i = 3.0 × 104. (f) Microwavetransmission data taken across another electroplated resonator at T= 100 mK. At powerscorresponding to single photon occupation, the fitted quality factor was Q100mk

i = 1.5 × 105.

91

Figures 4-14(e-f) show microwave transmission data taken for two different samples; One

sample was measured at T = 4.2 K (e) and the other T = 100 mK (f). The T = 4.2 K

quality factor was power independent with an extracted value of Q4Ki = 3.0× 104. This value

matches with the predicted Mattis–Bardeen quasiparticle limited CPW quality factor for a

film of thickness d = 180 nm. This is important point because it could then be said that the

judicious placement of the copper towers at the voltage antinode of the resonator induce no

additional microwave loss. The extracted quality factor at T = 100 mK was power dependent,

with a single–photon occupation quality factor Q100mKi = 1.5 × 105. This value is comparable

with state of the art planar Nb CPW resonators at millikelvin temperatures.

When measured at the base temperature of the custom UHV cryostat, the internal quality

factor Qi = 1.0 × 104, in agreement with predictions from Mattis-Bardeen for Nb at T = 5

K. This value for Qi, when combined with the electroplated structure enhanced anticipated

Rabi frequency of g/2π = 3 MHz, yields a predicted number of coherent excitation swaps

nRabi ≈ 12 � 1, placing the interaction securely in the strong coupling regime [69].

4.3.4 Stray Electric Field Compensation

Residual gas adsorbates that condense on cryogenic surfaces have been shown to produce

non-negligible time dependent stray DC electric fields [70]. When combined with the large

DC polarizability of Rydberg atoms, the dynamics of atoms near cryogenic surfaces becomes

unpredictable. In effort to counteract the chaotic and deleterious effects of these adsorbates,

DC electric field compensation pins were incorporated with the sample mount.

Figure 4-15(a-b) show a profile and head-on view of the sample, respectively. The sample

mount is bolted on to the cold finger with 2×8-32 socket head cap screws with no thermal

grease applied between them. The four DC electric field compensation pins were arranged

around the atom trapping region (green circle, Figure 4-15(b)). COMSOL simulations were

performed to model the field profiles generated with 1 volt applied independently to each of

the 4 electrodes with the sample mount held at ground. The resultant electric field vectors

92

(a) (b)

DC E-Fieldcompensation pins

RF

Sample

ColdFinger

Sample Mount

1

4

2

3

4K Shield

Bracket

Figure 4-15: (a) Profile photograph of 1st generation sample and sample mount bolted on tothe cold finger. (b) Head-on view of sample and sample mount. Four DC electrodes wereplaced symmetrically about the single atom trapping site (green circle). Each electrode wascapable of being independently controlled via a custom high voltage control module.

Pin Voltages (V) Electric Field (V/m)

Pin 1 Pin 2 Pin 3 Pin 4 Ex Ey Ez

1 0 0 0 -6.2 -4.2 -5.31 1 0 0 -16.6 -7.0 -5.71 0 1 0 -16.6 7.0 -5.71 0 0 1 -6.2 4.2 -5.3

Table 4.4: COMSOL simulation results of electric field vector at atomic trapping region whenplacing a potential of 1 V in four different configurations.

for the four different configurations are compiled in Table 4.4

It is important to note that the problem is over constrained with 4 voltages to vary but

only three spatial dimensions; Thus, any two of the pins can be tied together and held at the

same potential while maintaining full 3 dimensional field control.

93

Figure 4-16: First image of Cs MOT (bright point, mid-frame) trapped directly beneath thesuperconducting CPW chip. Image courtesy of Joshua Isaacs thesis.

A custom bi-polar high voltage controller was built in order to automate the control of

the pin voltages and thus the electric field environment. Output voltages between ±1 kVwere controlled via a the output of a 20 bit (1 ppm) DAC fed with a control voltage of 10.5

V.

4.4 Current Status

As of this writing, the sample has been installed in the cryostat and the cryostat has been

installed on the optical table. To date, a MOT has been formed just beneath the chip (see

Figure 4-16) and efforts are currently underway toward achieving single atom trap loading,

cooling, and transport.

94

Chapter 5

Single Flux Quantum Qubit Control

The marriage of superconducting classical and quantum processor technology, on it’s face,

is a very natural path forward toward scaling up to qubit numbers of order the 1 million

or so required for implementation of fault tolerant surface codes. Both are mature tech-

nologies sharing many similar traits in terms of materials, fabrication, and operation. In

this chapter we review modern classical superconducting computing based on single flux

quantum technologies. Afterwards, we describe experiments and data demonstrating the

first successful implementation of SFQ based control of a transmon qubit. We describe in

detail the fabrication of the SFQ–qubit sample. We illustrate how to obtain orthogonal

qubit axis control which allows for orthogonal SFQ based qubit gates. We show data on the

interleaved randomized benchmarking (RB) performance of these gates. Furthermore, we

investigate limitations to SFQ gate performance by looking at non-equilibrium quasiparticles

(QP) generated by the SFQ circuit. Finally, we will describe future experiments where the

classical SFQ control circuit is decoupled from the transmon qubit by employing a multi–chip

module (MCM)/flip-chip architecture.

95

5.1 Superconducting Supercomputing with Single

Flux Quanta

The basic tenant of classical computing with Josephson based superconductor technology

is the regulation, storage, and transmission of SFQ pulses. For junctions that are critically

to over damped (βc ≥ 1), the junction phase δ can evolve in steps of 2π between successive

minimums of the junction potential without free cascade down the washboard before removal

of the bias current. When combined with the second Josephson relation, these successive

steps in the phase produce a voltage pulse in time with quantized area

A =∫

V dt = �

2e

∫dδ = Φ0 . (5.1)

The energetics of a SFQ JJ phase slip are displayed in Figure 5-1. A necessary condition

for the generation of SFQ phase slips is that βC ≤ 1 which generally involves the use of

external shunt resistors. In further circuit diagrams, the use of shunt resistors is implied for

all junctions but not explicitly shown.

In SQUID loops with βL � 1, stable circulating current flux states differ by ±Φ0 which

then are used to encode and store the logical “0” and “1”. Figure 5-2(a) displays the circuit

diagram for a 1–bit D latch SFQ memory circuit [71]. The dc–SQUID is comprised of

junctions J1 and J2 and inductor LS. At the beginning of the clock cycle, the current in

the SQUID is propagating counter–clockwise corresponding the “0” state. Static bias Ib is

applied very close to the critical current of J1. When an SFQ pulse is applied to the input, it

drives the current of J1 above the critical current and changes the flux state of the SQUID

to “0”, lowering now the total current through J1 and biasing J2. A CLK pulse entering the

circuit now drives J2 to switch producing a SFQ pulse at the output and resetting the flux

state of the SQUID. If, however, no pulse arrives at the input during the clock cycle, the CLK

pulse is instead dropped across junction J3 and no output is produced. For proper operation

of the circuit, the circulating current Icirc = Φ0/LS produced by the input SFQ pulse must

96

Figure 5-1: Phase particle evolution for a SFQ phase slip. A static bias i = Ib/Ic is appliedtilting the washboard potential. An current pulse is then applied to force i > 1 allowing thephase particle to traverse 2π after which it is returned to its static value. The voltage pulsedeveloped across the junction is quantized in area.

not exceed the critical current of J2. Additionally, the critical current of J3 should be less

than J2 such that when no pulse is applied to the input, the CLK pulse is dropped across it.

The transmission of information from processor node to processor node is performed by

the ballistic propagation of SFQ voltage pulses. However, the picosecond timescale of the

pulses implies enormous frequency content giving rise to pulse dispersion during propagation.

To combat this dispersion, SFQ repeater circuits can be used to transfer pulses over long

distances. Figure 5-2(b) displays the circuit diagram of a 2–stage Josephson transmission line

(JTL). Independent biases (or a common bias rail) are used to bring junctions J1 and J2 close

to their respective critical currents. An incoming SFQ pulse drops across junction J1 driving

it normal and producing another SFQ pulse at L2. This SFQ pulse is then subsequently

dropped across junction J2 producing yet again another SFQ pulse at the output. The choice

of series inductors and hence βL for each single junction RF–SQUID is important as setting

the max loop flux IcL < Φ0/2 forces forward propagation of the SFQ pulse instead of storage

as was the case in the D latch circuit.

With picosecond pulses flying back and forth between circuits, propagation non–reciprocity

is crucial for downward streaming of information. Figure 5-2(c) displays a two junction buffer

stage. For proper operation, the critical current of J1 should be larger than J2. When input

97

0 / 1

(a) (b)

(c) (d)

Figure 5-2: Common SFQ circuits. (a) 1–bit SFQ D latch. (b) Two junction JTL. (c)Unidirectional SFQ buffer stage. (d) SFQ 1×2 pulse splitter.

SFQ pulse arrive at L1 it subsequently drives J1 normal producing a SFQ pulse which is

subsequently dropped again across J2 reproducing the pulse at the output. However, if a

pulse arrives from the output, it drives J2 normal first dropping the voltage again at the

output not allowing it to pass to the input.

The ability to multiplex multiple circuit inputs with a single SFQ pulse is necessary to

fan out information and increase computing complexity. Figure 5-2(d) displays an 1× 2 SFQ

fanout. In parallel, junctions J2 and J3, each with equal critical currents I2 = I3 = I = I1√2,

look like a single junction with total critical Ic=√2I1. When an SFQ pulse drives the input,

the resulting SFQ pulse drives both branches producing a complimentary SFQ pulse at each

output without loss of amplitude.

Many different inceptions of SFQ computing have come along in the past 30+ years. We

briefly review the major implementations here.

98

5.1.1 Rapid Single Flux Quantum

First developed in 1985 by Likharev, Mukhanov, and Semenov [72], Rapid Single Flux

Quantum (RSFQ) technology was the first fully theoretically investigated superconducting

computing paradigm that showed promise to compete with CMOS. Junctions in RSFQ are

both externally shunted and biased via on–chip thin film resistors. While initial tests and

operating speeds showed promise, the static power consumption from the bias rails has limited

the scaling of the processor size to a few kb.

5.1.2 Energy Efficient SFQ

A variant of RSFQ, energy efficient SFQ (ee–SFQ, ERSFQ) [73] aimes to tackle the quiescent

power consumption issues previously discussed by replacing the bias resistors with supercon-

ducting inductors and current limiting JJs. During standby (no junction switching), zero

power is dissipated in the circuit. Additionally, during bring–up, local phase imbalance across

the junctions in the circuit will self–correct as the junctions will continuously switch until an

approximately global phase balance is reached.

5.1.3 Reciprocal Quantum Logic

Developed at Northrop Gumman in 2011, reciprocal quantum logic (RQL) [74] isn’t an

extension of RSFQ but instead an entirely new form of superconducting classical computing.

Instead of static biases, circuit elements are inductively coupled to and powered by a

distributed common AC rail which also serves as a global clock, serving to reduce and/or

eliminate the accumulation of timing jitter between circuit stages. Additionally, logical 0 and

1 are encoded in pairs of opposite polarity SFQ pulses, providing a natural flux set / reset of

the destination Josephson circuit. For the work contained in this chapter, traditional RSFQ

circuitry is employed.

99

5.2 SFQ Control of a Transmon Qubit

A detailed theoretical analysis of SFQ based transmon qubit control can be found in [75]. Here

we outline the calculations in that paper with an eye towards circuit design and experimental

control. A circuit diagram of the SFQ control idea is given in Figure 5-3(a). A generalized

SFQ producing circuit is capacitively coupled to a transmon qubit. The Hamiltonian for this

circuit is

H =

(Q − CcV (t)

)2

2CT

+ Φ2

2Lq

, (5.2)

where CT = Cq + Cc is the total capacitance of the circuit. Expanding the first term in

Eq. (5.2) yields

H = Q2

2CT

+ Φ2

2Lq

− Cc

CT

V (t)Q (5.3)

where we have dropped constant, non-operator dependent terms. The first two terms in Eq.

(5.3) are just the qubit self energy. Utilizing the methodologies outlined in Chapter 3, the

qubit terms can be re-expressed again in the TLS formalism

Hq =Q2

2CT

+ Φ2

2Lq

→ �ωq

2 σz . (5.4)

The last term in Eq. (5.3) is the interaction of the qubit with a capacitively coupled voltage

pulse V (t). To better illustrate the effect of this coupling, we expand the charge operator

into it’s truncated qubit state raising and lowering operators

Q = −i

√�ωqCT

2 (σ− − σ+) . (5.5)

Substituting this back into the coupling Hamiltonian yields

HSFQ = iCcV (t)√

�ωq

2CT

(σ− − σ+) . (5.6)

100

The additive combination of the truncated raising and lowering operators equate exactly to

the Pauli Y matrix

σ− − σ+ = −iσy . (5.7)

Finally substituting this last relation back into Eq. (5.6) yields

HSFQ = CcV (t)√

�ωq

2CT

σy . (5.8)

We can calculate the effect of this Hamiltonian on a general state |ψ〉 via the Schrödingerequation

−i�d |ψ〉

dt= HSFQ |ψ〉 (5.9a)

∫ d |ψ〉|ψ〉 = i

∫ HSFQ

�dt (5.9b)

ln[ |ψ(t)〉

|ψ0〉]= Cc

√ωq

2�CT

σy

∫V (t)dt = iCcΦ0δ(t − Δt)

√ωq

2�CT

σy (5.9c)

|ψ(t)〉 = exp[iCcΦ0δ(t − Δt)

√ωq

2�CT

σy

]|ψ0〉 , (5.9d)

where we have used the relation∫

V (t)dt = Φ0δ(t − Δt) for a train of delta-like SFQ pulses

arriving at intervals of Δt = 2nπ/ωq. By relating Eq. (5.9d) to the quantum mechanical

rotation operator D,

D(δθ) ≡ exp [iδθσy/2] , (5.10)

101

SFQSource

(a) (b) (c)

Figure 5-3: (a) Circuit diagram depiction of a transmon qubit capacitively coupled to ageneralized SFQ source (b) Voltage timing diagram of the output of the SFQ driver. The areaunderneath each pulse is exactly one flux quantum. The pulse-to-pulse spacing to coherentlydrive a qubit is at integer multiples of the qubit Bloch vector rotation period. (c) Blochsphere illustration of the qubit state vector trajectory when irradiated by trains of SFQpulses. The vector is displaced an angle dθ for each pulse. Between subsequent pulses, thestate vector precesses about the �z axis at the qubit frequency ωq a number n times beforethe next pulse arrives where n is an integer.

we can estimate a displacement angle δθ per SFQ pulse of

δθ/2 = CcΦ0

√ωq

2�CT

(5.11a)

δθ = CcΦ0

√2ωq

�CT

. (5.11b)

The ramifications of Eq. (5.11b) is that the accuracy of an arbitrary qubit rotation when

performed via SFQ pulses is now determined by a fundamental constant of Nature and a

lithographically adjustable circuit parameter. The absolute error due to the discrete rotation

angle is

1 − FSFQ = 1 − cos(δθ/2) � δθ2

4 . (5.12)

for small δθ 1.

Figure 5-3(b) depicts a time trace of an SFQ pulse train used to coherently drive a

transmon qubit. The area underneath each pulse is exactly 1 superconducting magnetic flux

102

4 4.5 5 5.5 6Frequency (GHz)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nor

mal

ized

Am

plitu

denSFQ = 10nSFQ = 25nSFQ = 50nSFQ = 100nSFQ = 1000

0 50 100 150 200 250 300 350 40010-4

10-3

10-2

10-1

100

P

50100Coupling Capacitor (aF)

nSFQ

(a) (b) 200400

Figure 5-4: (a) Normalized frequency spectrum of SFQ pulse trains of different number ofSFQ pulses. The SFQ pulse–to–pulse spacing was 200 ps corresponding to a drive frequencyof 5 GHz. For a small number of pulses, significant spectral weight can be seen at frequenciesabove and below the central peak. Frequency content below the central peak can drive higherorder qubit transitions. (b) Maximum qubit |2〉 state leakage probability as a function ofthe number of SFQ pulses required to achieve a qubit π–rotation. For a π–rotation requiringnπ

SFQ ≈ 200 SFQ pulses, the leakage error is below 10−3. The corresponding couplingcapacitance Cc required to yield a set number of SFQ pulses is plotted on the top x–axis.

quantum Φ0 with the inter-pulse timing set to an integer multiple of the qubit period. Figure

5-3(c) depicts the corresponding trajectory of the qubit Bloch vector (red) when irradiated

by such a pulse train. The Bloch vector undergoes discrete rotations of magnitude δθ about

the y axis for every pulse. Between pulses, the Bloch vector precesses freely about the z (|0〉)axis at frequency ωq.

A dominant source of error to consider is qubit state leakage into the |2〉 state. A singular

SFQ pulse has an enormous bandwidth in frequency space, scaling as ∼ 1/σ, where σ is the

SFQ pulse width. The reduced anharmonicity of a transmon qubit combined with this large

bandwidth will couple higher order qubit states out of the computational manifold resulting in

a non-negligible occupation of the qubit |2〉 state. Figure 5-4(a) shows the Fourier spectrumof a sequence of SFQ pulses in time for differing number of SFQ pulses. The pulse–to–pulse

spacing was set to be 200 ps resulting in a central frequency component centered at 5 GHz.

For pulse sequence numbers below nSFQ ≈ 100, significant spectral weight resides in the

103

side lobes of the spectrum. These side lobes can drive qubit rotations out of the computational

manifold resulting in poor control and large error.

In order to model the leakage, we extend the Hamiltonian to include a third level which

now takes the form

H|2〉Free = �ωq

⎡⎢⎢⎢⎢⎢⎢⎣0 0 0

0 1 0

0 0 1 + ω′q/ωq

⎤⎥⎥⎥⎥⎥⎥⎦

(5.13)

where ω′q = (E|2〉−E|1〉)/� is the frequency for the qubit |1〉 → |2〉 transition. The Hamiltonian

describing leakage out to the qubit |2〉 state is given by

H|2〉SFQ = CcΦ0

√2ωq

�Cq

⎡⎢⎢⎢⎢⎢⎢⎣0 −1 0

1 0 −√2

0√2 0

⎤⎥⎥⎥⎥⎥⎥⎦

. (5.14)

We model the system for a qubit fundamental transition frequency ωq = 2π × 5 GHz,

anharmonicity ω′q/ωq = 0.95, and qubit capacitance Cq = 100 fF. Figure 5-4(b) displays the

results from the full master equation simulation of the qutrit system as modeled in QuTiP

[76]. The maximum |2〉 state population is plotted against the required number of SFQ pulses

to achieve a π–rotation (bottom x–axis) and the corresponding coupling capacitance Cc (top

x–axis). For SFQ π–pulse lengths nπSFQ � 200, the maximum |2〉 state leakage is found to be

below 10−3. The corresponding coupling capacitances required to achieve this low–level of

leakage are well below 1 fF.

5.3 Bridging the Quantum–Classical Divide

In order to demonstrate the coherent control of a transmon qubit with SFQ pulses, an

experiment had to be designed that not only was extensible enough to provide definitive

proof of principle but also fabricable with the tools available in the lab and the clean room.

104

In the following sections, we describe the design, modeling and fabrication of superconducting

Nb experiments that combined both classical SFQ circuitry with a transmon qubit–CPW

resonator quantum circuit.

5.3.1 SFQ Driver Design

One of the most basic SFQ circuits is the dc/SFQ converter. Despite its misleading name,

the circuit actually converts a continuous RF tone into a time series array of SFQ pulses.

The circuit diagram of a three Josephson junction dc/SFQ converter is displayed in Figure

5-5(a). A junction JJT and shunt inductor LS form a RF SQUID. These two components

represent the “set” portion of circuit. The reset portion is comprised of two junctions JJR1

and JJR2 in parallel. The operation of the circuit is depicted in Figures 5-5(b) and 5-5(c).

During the first π phase evolution of the trigger tone (blue, top trace), the DC bias current

(purple) and trigger current add together in the trigger junction (green, middle trace) arm of

the RF SQUID. The two currents together are enough the exceed the critical current of the

trigger junction, causing it to go normal, creating a fluxon-antiluxon voltage pulse pair. The

fluxon propagates to the output of the circuit while the antifluxon sets the circulating current

flux state of the RF SQUID (orange arrows, Figure 5-5(b)). This circulating current flows

opposite in direction to the RF and DC bias currents bringing the total current through the

trigger junction back down to below it’s critical current value and allowing the Josephson

phase to only progress 2π down the tilted washboard potential (see Figure 5-1). During the

second π phase evolution of the trigger tone, the RF and DC current add together in the

reset arm of the converter triggering a SFQ pulse (red trace) that resets the state of the RF

SQUID. Every 2π phase evolution of the trigger current, the dc/SFQ converter arms, sending

out one SFQ pulse to the output and is subsequently reset. For proper flux set and reset of

the RF SQUID, it is important that the RF SQUID can support such a propagating current

and thus puts limits on βL, defined as

105

JJT

JJR1

JJR2

LS

IRF

IDC

Out

Out

Out

IRF

VT

VR

IRF

VT

VR

t

t

t

t

t

t

(a)

(b)

(c)

Figure 5-5: (a) Circuit diagram of a dc/SFQ converter. The circuit is comprised of 3Josephson junctions. One junction (JJT) serves as the SFQ trigger while the other 2 (JJR1/R2)serve to reset the circuit. (b) During the first π phase evolution of the RF tone, the DC biascurrent (purple) and the trigger current (blue) add together exceeding the critical current ofthe trigger junction (green). The junction briefly goes into the normal state and expels 2fluxon pulses. One fluxon pulse (green) propagates to the output of the circuit while the othersets the flux state (orange arrows) of the RF SQUID formed by JJT and LS. The circulatingcurrent in the SQUID quickly reduces the total current through the trigger junction allowingfor only 1 SFQ pulse to be generated. (c) During the second π phase evolution of the triggertone, the DC and RF currents add in the reset arm (red) of the dc/SFQ converter, triggeringa reset of the flux state of the RF SQUID.

106

βL = 2πIc,TLs

Φ0. (5.15)

For proper operation, 2 < βL < 6 [77]. To better illustrate these limits, we solve for the

Josephson phase in the trigger junction via circuit analysis. We begin by combining the reset

junctions JJR1 and JJR2 into a single junction JJR. The sum of the bias currents running

through each of the 2 legs of the SQUID is equal the the applied bias current IB,

IDC = IJJT + IJJR = Ic,T sin(δT) + Ic, R sin(δR) . (5.16)

The current shunted through the inductor is the sum of the trigger current and the reset

current

IL = IRF + Ic,R sin(δR) . (5.17)

Finally, the sum of the phases of the circuit must equate to an integer multiple of 2π yielding

the equation

2πΦ0

ILL + δR − δT = 2nπ . (5.18)

Combining equations Eq. (5.16), Eq. (5.17), and Eq. (5.18), we can derive a transcendental

equation of the trigger junction phase δT,

2πΦ0

L(IRF + IDC) − βL sin(δT) + arcsin(

IDC − Ic,T sin(δT)Ic,R

)= δT + 2nπ (5.19)

Figures 5-6(a) and 5-6(b) show the simplified circuit diagram and plots of the LHS (color)

and RHS (dashed lines) of Eq. (5.19) as a function of δT, respectively. The LHS values have

been artificially displaced in the y–axis so as to display them one on top of one another. As

βL is increased, more solutions, and thus more stable RF SQUID flux states, can be found

107

JJT

JJR

LS

IRF

IDC

-4 -3 -2 -1 0 1 2 3 4-30

-20

-10

0

10

20

30(a) (b)

LHS

Figure 5-6: (a) Simplified dc/SFQ converter circuit diagram. (b) Plot of the LHS ofEq. (5.19) as a function of the trigger junction phase. Dashed lines plot the RHS sideof the transcendental equation for varying values of n. As βL is increased, more solutions(simultaneous overlap of colored and dashed lines) are found which physically equates tomore stable flux states in the RF SQUID.

for Eq. (5.19). For βL values below 1, the SQUID cannot support a single flux quantum

equivalent circulating current.

Due to it’s relative simplicity in terms of operation (1 DC bias current + 1 RF trigger

current) in addition to the low number of junctions, the dc/SFQ circuit was chosen as a

first pass circuit to demonstrate SFQ based qubit control. The step first in integrating the

converter with a qubit was to design it in such a way that a minimum amount of input RF

power was required to trigger SFQ pulses. The onset power of SFQ pulses can be measured

via the appearance of Shapiro steps [78] in the junction IV curve. When irradiated with RF

tones, at particular biases a junction will switch into the voltage state at a rate commensurate

with the driving frequency ω. So long as the junction is properly shunted so as to not be

hysteretic (βC 1), the junction will switch into and reset from the voltage state without a

reduction of the static bias down to the retrapping current. Equating the energy required to

break cooper pairs with the photon energy of the RF driving tone, a very simple relation

develops relating the magnitude of voltage steps seen in the Josephson IV curve to the

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Inductor Value (pH)

L1 3.35L2 1.27L3 0.69L4 1.29L5 0.21L6 0.18L7 1.14L8 0.08L9 1.74L10 0.13L11 2.11

Table 5.1: dc/SFQ wiring inductances used in WRSpice circuit model.

frequency of the RF tone,

2eV = �ω (5.20a)

V = �ω

2e . (5.20b)

Whiteley Research Spice (WRSpice) [79] was used to model and engineer the driver at the

circuit element level for low (−60 dBm) RF input power operation. The modeled circuit is

shown in Figure 5-7(a). The design is based off of work in [80, 81] and the RSFQ cell library

of Stony Brook University [82]. The circuit is exactly the same as discussed previously with

the only change being the addition of a single junction Josephson transmission line (JTL) to

the output. Values for the modeled wiring inductances can be found in Table 5.1.

The resulting simulated IV curve for the driver with a modeled critical current density of

Jc = 1 kA/cm2, shunt inductor LS = 5 pH, and no applied RF power is displayed in Figure

5-7(b). The total critical current of the driver is the sum of the individual critical currents

109

(e)

(g)(f)

(d)(c)

(b)(a)

10LS (pH)

403020

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PRF

IDC

L1 L2

L3

L4

L5

L6

L7L9

L8

L10

L11

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ge (u

V)D

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(uV)

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403020

Figure 5-7: (a) dc/SFQ circuit modeled in WRSpice. Values for the inductances are listedin table 5.1. (b) Simulated IV curve for the dc/SFQ circuit in (a) for no applied RF withJc = 1 kA/cm2 and LS = 5 pH. (c-d) False color dc/SFQ IV curves as a function of applied 5GHz RF power for shunt inductors LS = {10, 20} pH. The onset of a clearly defined Shapirostep isn’t until the power is above -40 dBm. (e) Constant power (-60 dBm) IV curve as afunction of shunt inductance LS. Shunt inductance values corresponding to a 2.5 < βL < 5provide the largest operating region in DC bias current. (f-g) IV curves as a function of LSfor Jc = {0.9, 1.1} kA/cm2. Changing the critical current density by as much as ±10% stillyields an operable driver for proper choice of LS.

110

of the constituent junctions totaling 165 μA. Figure 5-7(c) plots the driver voltage (false

color) as a function of DC bias current (y–axis) and applied RF power (x–axis) for a driving

frequency of 5 GHz and shunt inductor LS = 10 pH. The onset of the n = 1 first Shapiro step

voltage V = �× (2π ×5 GHz)/2e ≈ 11.2μV Shapiro step can be seen for biases IDC � 100μA

and powers PRF � −45 dBm. The same simulation is displayed in Figure 5-7(c) only for

a shunt inductor LS = 20 pH. The two–fold increase in βL allows for more stable trigger

junction phase states and a resultant lowering in the required RF power to trigger single

2π phase slips in the trigger junction. Figure 5-7(e) displays again the simulation results

for the dc/SFQ driver but now sweeping the LS shunt inductor value from 1–50 pH for a

fixed driving frequency of 5 GHz and power PRF = -60 dBm. For shunt values LS< 20 pH

with corresponding βL < 2.5, only reductions in the overall critical current of the driver are

seen with no clear Shapiro step arising in the IV curves. For values of shunt inductance

20 < LS < 40 with corresponding 2.5 < βL < 5.0, a clear n=1 Shapiro step develops for bias

currents 110μA < IDC < 170μA. For higher shunt inductances, more than one Shapiro step

arises, limiting the operating DC bias current range.

This βL “sweet spot” for the low power onset of Shapiro Steps echoes and confirms the

results of Eq. (5.19) / Figure 5-6(b). Namely that for too low of a shunt inductance (βL),

only 1-2 solutions of the transcendental equation exist requiring large amounts of RF power

to drive transitions between stable states. As βL is increased, the number of stable phase

points increase allowing for the driving of 2π phase slips at lower RF power. It is worth

mentioning that while the simulations and design mind–set were for engineering drivers that

minimized the required applied RF power, they did not explicitly try to minimize the on–chip

dc/SFQ power dissipation. The energy dissipated for one 2π junction phase slip is

E =∫

Pdt =∫

IcV dt = Ic

∫V dt = IcΦ0 . (5.21)

For a junction being triggered at a clock rate f , the power dissipated then is just

111

PSFQ = E × f = IcΦ0f . (5.22)

For Ic = 100μA and clock rate / drive frequency f = 5 GHz, PSFQ ≈ 1 nW.

A feature of the dc/SFQ circuit and RSFQ circuits in general is the robustness to processing

/ fabrication induced variation. Figures 5-7(e–f) display the resultant IV curves again for the

driver depicted in Figure 5-7(a) but for ±10% variation in the critical current density with (e)

showing the results for a critical current density Jc = 0.9 kA/cm2 and (f) showing results for

the a critical current density Jc = 1.1 kA/cm2. When Jc is lowered (increased), the required

shunt inductance to obtain to widest operating margin in DC bias current simply shifts up

(down) by roughly the same percentage. This η ≈ 20% operating margin for a proper design

allows for devices to survive and operate appropriately even when exposed to the chaotic,

fractal-like nature of microfabrication processes and tools generally seen at university level

general user clean room facilities.

5.3.2 SFQ–Qubit Circuit Layout

Figure 5-8(a) displays a circuit diagram of the coupled SFQ–Qubit system. The output of

the dc/SFQ driver (red) is coupled to the qubit (blue) via a capacitor Cc. The qubit is a flux–

tunable transmon with mutual inductance MΦ to an external bias line (yellow). Readout of

the qubit is performed via capacitively coupling Cg to a quarter wave CPW resonator. Qubit

state dependent dispersive readout of the resonator is performed in microwave transmission

measurements via an inductive coupling Mc to a common feed line (black).

Figure 5-8(b) shows an overhead micrograph of the completed circuit. The colored outlines

correspond to the colored circuit elements in 5-8(a). The SFQ–qubit coupling capacitor was

in–plane with the qubit capacitor pad (blue). The qubit design was modeled from the Syracuse

University style “rectmon” transmon. The qubit capacitor pad was 40×400 μm in footprint

with a COMSOL / HFSS modeled self capacitance Cq ≈ 76 fF. The qubit capacitor pad was

112

(a) (b)

(c)

Figure 5-8: (a) Circuit diagram layout of coupled dc/SFQ driver to a transmon qubit. Theoutput of the driver (red) is capacitively coupled to the qubit (blue) via a coupling capacitorCc. The qubit capacitor pad Cq is shunted to ground via an asymmetric 2:1 DC SQUIDthat is externally flux biased (yellow) via a mutual coupling MΦ. The qubit is capacitivelycoupled to a quarter wave readout resonator (blue) that in turn is inductively coupled toa microwave feed line (black). (b) Micrograph of fabricated integrated dc/SFQ driver andqubit. Colored boxes outline the corresponding colored circuit elements shown in (a). (c)Zoomed in micrograph of fabricated dc/SFQ converter. The Josephson junctions are 1–to–1placed with respect to the corresponding circuit diagram.

113

shunted with two Josephson junctions forming a dc SQUID with area/energy asymmetry

α = 2 (see Appendix A) providing a value for d = 1/3.

Large modularity in physical design was afforded by the use of optical projection lithogra-

phy. Lithography masks were made such that a multitude of SFQ–qubit couplings Cc could

be tested ranging from sub–attoFarad to the single femtoFarad range. In addition, the entire

driver placement on the chip was modular with three physical distances of 0.5, 1.5, and

3.5 mm between the output of the driver and the qubit able to be realized. Furthermore,

readout resonator frequency ωr, feed line coupling Mc, and qubit coupling Cg could also be

modulated.

5.3.3 Fabrication

For the past 12+ years, traditional wisdom [83] has been that incorporating amorphous

dielectrics into the design of superconducting quantum circuits is a fool’s errand and that

doing so only serves to reduce the performance of the individual quantum circuit elements

by increasing the microwave loss. However, all the circuit diagrams, Hamiltonians, and

numerical simulations in the world mean nothing if the above proposed experiment can’t be

physically realized on a chip. With the minimum dc/SFQ stack requiring 3 layers (ground

plane / JJ electrode + wiring dielectric + JJ counter electrode), traditional wisdom had

to be discarded. The following describes in detail the fabrication required to integrate the

necessary mutlti–layer stack of SFQ fabrication with the traditional monolayer fabrication

employed in making superconducting quantum circuits. An overview of the 8 additive layer

(6 metal + 2 dielectric) process is displayed in Table 5.2.

Layer M0 The fabrication process began with the submersion of a bare intrinsic Si〈100〉

three–inch wafer in room temperature hydrofluoric (HF) acid for 60 seconds to strip the

native SiOx oxide. The wafer was then loaded into a DC magnetron sputtering tool with base

pressure Pb < 5 × 10−9 T. Prior to deposition, the Nb target was plasma cleaned at argon

pressure PAr= 4.25 mT at 400 W × 2 min. The wafer was rotated at ∼1 Hz during deposition

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LayerID

Materials Thickness(nm)

SurfacePre-treatment

DepositionMethod

Patterning Etching Purpose

M0 Nb 180 HF DC Sputter Projectioni-line

ICPGround PlaneResonatorQubit

V1 SiO2 130 – PECVD Projectioni-line

RIE Ground Vias

M1 Nb 90 ion–mill DC Sputter Projectioni-line

RIE SFQ JJ Base Electrode

V2 SiO2 180 – PECVD Projectioni-line

RIE JJ Vias

M2 Nb/Al-AlOx-Al/Nb 100 ion–mill DC Sputter +in-situ oxidation

Projectioni-line

RIE SFQ Bias WiringJJ Barrier / Counter Electrode

R Ti/Pd 3 + 20 ion–mill E-beam Evap Projectioni-line

– SFQ Shunt Resistors

QP Ti/Cu/Pd 93 ion–mill E-Beam Evap Projectioni-line

– Quasiparticle traps

QB Al-AlOx-Al 100 ion–mill E-Beam Evap E-BeamLitho

– Qubit JJs

Table 5.2: Layer–by–layer overview of the dc/SFQ–Qubit chip fabrication. Overall 8 additivelayers (6 metal + 2 dielectric) were required.

in which Nb was sputtered at 2×500 W×2 min with 10 minutes between sputters for a total

film thickness of 180 nm as measured with profilometry. For patterning, SPR-955 0.9 cm

positive photoresist (PR) was spun at 4500 RPM for 60 seconds followed by a pre-exposure

bake at 100 C × 90 seconds. Pattern exposure was done with a 365 nm i-line 5:1 projection

stepper. The wafer was then post-exposure baked at 110 C × 60s. Development of the

pattern was done via submersion and agitation in Microposit CD26 TMAH developer for 60s

followed immediately by submersion in DI water for another 60s. The wafer was then actively

dried with dry nitrogen gas. Etching of the Nb was performed in a PlasmaTherm PT770 with

a Cl2/BCl3 inductively coupled plasma (ICP). The etch was inspected visually prior to PR

stripping. The PR was stripped overnight in room temperature acetone followed subsequently

by baths in IPA and deionized (DI) water. The depth of the etch was characterized with

surface profilometry with an average trench depth of 210 nm corresponding to an over etch

of 30 nm.

Layer V1 The first of two SiOx wiring dielectrics, layer V1 was deposited using a

115

PlasmaTherm PT-70 plasma enhanced chemical vapor deposition (PECVD) tool. Prior to

wafer insertion, the tool platen was heated to 250 C. The chamber was then cleaned in an

O2 plasma for 2×250s periods followed by a recipe pre-seed of 500s. Only afterwards was

the wafer loaded into the tool. SiOx was deposited for 250s with a target thickness of 130

nm. Lithographic patterning was performed in the same manner as described above for

layer M0 with one additional step. A PR reflow bake at 125 C×180 s was performed post

development to induce ∠45◦ sloped sidewalls into the developed PR edges. The dielectric

etch was performed in a PlasmaTherm 790 reactive ion etcher (RIE) with a carbon platen.

Prior to loading the sample, the empty chamber was cleaned with an oxygen plasma for 7

minutes. The wafer was then etched with a CHF3 based RIE where the balance between

physical and chemical etching transfered the sloped sidewalls of the resist into the dielectric.

The sloped dielectric sidewall was crucial in eliminating step coverage issues in subsequent

layers. Post etch, the PR was stripped again in subsequent baths of acetone, IPA and DI

water. Profilometry yielded an average layer thickness of 132 nm.

Layer M1 Filling the role of the JJ bottom electrode, layer M1 began with a Ar–ion mil

at PAr = 2×10−4 T for 45 s to promote good metal–to–metal contact between M1 and M0

by milling the native oxide on M0. Deposition of the Nb was carried out the same way as

described for M0 with an important caveat. Specifically, the sputter pressure was adjusted to

provide a slightly compressive (200 MPa) intrinsic Nb film stress. Achieving a compressive

film stress for the JJ electrode layer was crucial as JJs made with tensile stress films have

been shown to have large leakage currents [84]. The sputtering pressure used varied as a

function of the target life increasing from approximately 4 mT for a newly installed target

and ending at 6.5 mT after 500 minutes of target sputter time. Patterning of the layer was

done in the same way as M0 with an additional PR reflow step as described above. Etching

of the layer was performed with an SF6 RIE plasma.

Layer V2 Layer V2 defined the SFQ JJ area. It was deposited, patterned and etched in

the same way as V1 only with a deposition time now of 350 seconds for a target thickness of

116

180 nm.

Layer M2 This Nb layer defined the SFQ JJ counter electrode. Involving the co–

deposition of both Nb and Al for the trilayer junctions, particular care was taken to follow

the exact steps detailed in the following:

(i) The wafer was loaded in the sputter system load lock which was then pumped out via a

roughing and turbo pump for t ≥ 1 h.

(ii) Commensurate with the load lock pump–out, the main sputter chamber was seeded

with 100 mT of O2 gas. This seeding process conditions the chamber chemistry provided

better fabrication run–to–run process uniformity.

(iii) After the pre-seed, the wafer was transferred from the load lock to the main chamber.

The pressure was allowed to recover to P≤ 5 × 10−8 T.

(iv) The wafer was Ar–ion milled for 25 seconds to prepare the exposed Nb surface of layer

M1 by removing the native oxide.

(v) The Al sputter target was cleaned at 200 W×2 m. Al deposition was then performed

at 40 W×2 m for a target thickness of 8 nm.

(vi) O2 gas was flowed into the chamber achieving a background pressure of 1 mT. All

pumps were then shut off from the main chamber and the pressure was allowed to rise

to 100 mT. The wafer was allowed to oxidize for 10 min with a target critical current

density Jc = 1 kA/cm2 .

(vii) After oxidation, the O2 gas was pumped out via the load lock roughing pump, turbo

pump and main chamber cryopump.

(viii) Al was again sputtered following the procedure described in (v) only this time for 90 s

for a target thickness of 6 nm,

117

(ix) A final cap of Nb was deposited for 2 min following the cleaning/sputtering steps

outlined earlier. The target thickness was 90 nm.

This counter electrode layer was coated with SPR955 photoresist and patterned with the

stepper. Etching of the Nb was performed in the Unaxis 790 RIE tool utilizing the already

mentioned SF6 RIE plasma. Once completed and prior to venting, a second 50 W Ar plasma

was run for 80 seconds prior to venting so as to remove residual SF6 plasma residue from the

wafer. The now exposed Al was then etched in a room temperature bath of MF-24A TMAH

PR developer. Test JJ patterns incorporating JJs of areas A = 2, 4, 8, and 16 μm2 were

then 4–wire probed to determine their room-temperature normal state resistance Rn. The

Ambegaokar–Baratoff relation [85]

(Jc × A)Rn =π

4

(2Δe

)(5.23)

with A the junction area, Δ the SC gap energy and e the fundamental charge, allowed for

determination of the critical current density Jc.

Layer V1/V2 Removal Up until this point, much of the PECVD SiOx dielectric was

still covering an area over layer M0 where the qubit and readout resonators needed to be

patterned. The patterning and etching of these layers to reveal this area of M0 was carried

in similar fashion to either of the individual dielectric etching procedures.

Qubit and Resonator Patterning With V1 and V2 layers now removed, the qubit

and resonator were patterned and etched utilizing the same methods described for layer M0.

Layer R AZ–5214 PR was spun on the wafer and baked at 95 C for 60 s. Pattern

exposure was carried out in the stepper. An image reversal (IR) bake was then done at

110 C for 60 seconds followed by an 60 s, 365 nm flood IR exposure in a Karl Suss MA6

contact aligner. Development of the pattern was performed in room temperature CD-26 for

45 seconds followed by a 30 second DI water rinse and dry N2 gas. When properly processed,

the resist should has a ∠45◦ undercut. The wafer was then loaded into a custom electron

118

beam evaporator. After loading into and pumping out the load lock, the sample was opened

to the main chamber and the pressure allowed to recover to below 5× 10−7 T. An Ar–ion mill

was performed for 17 seconds at 2 × 10−4 T backing pressure. A Ti adhesion layer was then

evaporated onto the wafer at a rate of 0.5 Å/s for a total of 3 nm. Pd was then evaporated

at a rate of 0.5 Å/s for a total of 30 nm. Once the evaporations were completed, the wafer

was placed in a Teflon tripod holder submerged in room temperature acetone overnight to

complete the liftoff process. Once all the extraneous Ti/Pd was lifted off, test resistor features

of 1.1 and 2.2 squares were 4–wire probed. The residual resistance ratio (RRR) the Ti/Pd

resistors was measured to be approximately 2 between 293 K and 4 K.

Layer QP Fabricating Cu quasiparticle (QP) traps began with spinning, patterning,

exposing, and developing AZ–5214 PR in the exact same manner as was done for layer R.

Once complete, the wafer was loaded into the evaporator and allowed to pump down again

to below 5 × 10−7 T. Following the same ion mill procedure described for the resistor layer,

Ti was evaporated at a rate of 0.5 Å/s for a total of 3 nm. Copper was then evaporated

at a rate of 1 Å/s for a total of 80 nm. Another layer of Ti was then evaporated again at

0.5 Å/s for another 3 nm. Finally, the resistor stack was completed with an evaporation

of Pd at a rate of 0.5 Å/s for a total of 10 nm. The total trap area fill factor was ∼ 35%.

While only the Cu plays a role in the trapping of QPs, the exposed Cu surface was found

to react negatively (color change) with the subsequent processing required for the qubit JJ

fabrication. An additional protective layer stack of Ti/Pd on top of the copper was found to

counteract this adverse reaction.

Layer QB The final step in the process was the patterning and double angle evaporation

of the QB JJs. This step in the process was carried out by our Syracuse collaborators. The

process began with a 150 W×30 second O2 plasma descum of the wafer prior to spinning on

the MMA / PMMA e–beam resist stack. The MMA was spun on first at 2500 RPM for 60

seconds for a target coating of 600 nm. The wafer was then baked at 170 C for 10 minutes.

The PMMA resist was then spun on at 3000 RPM for 60 seconds targeting a thickness

119

of 70 nm. The wafer was then baked again at 170 C for another 10 minutes. The Dolan

bridges [86] were written with a typical e–beam dosage of ∼ 1100μC/cm2 . The exposed

resist was then developed in a 3:1 ratio IPA–to–methylisobutylketone (MIKB) mixture for

60 seconds with agitation. The Dolan bridge on average was 130 nm wide. The e–beam

written junction trace widths were 180 and 360 nm with a designed junction overlap of 150

nm yielding a SQUID asymmetry ratio α = 2 (see Appendix A, Eq. (A.14)). The wafer was

then loaded into a custom e–beam evaporator. An Ar–ion mill was performed for 8 seconds

prior to Al deposition. The two Al evaporations were carried out at ±∠11◦. The base and

counter electrodes were evaporated to thickness of 55 nm and 35 nm, respectively with an

intermediary P= 9 T, 5% O2 in Ar oxidation carried out for 9 minutes. The subsequent liftoff

was performed in room temperature dichloromethane.

A photo of an undiced fully fabricated sample is displayed in Figure 5-9(a). Each chip was

8×8 mm2 and incorporated two independent experiments. Figure 5-9(b) displays an overhead

false–color optical image of the driver with the 4 junctions comprising the dc/SFQ driver plus

the JTL colored yellow. Layers M1 and M2 are colored blue and red, respectively. Finally,

a cross–sectional false–colored SEM image is shown in Figure 5-9(c). A focused ion beam

was used to mill into the sample so that the image could be taken. Nb layers are colored

red while SiOx layers are colored blue. The substrate has been colored yellow. The trilayer

junction formed by layers M1 and M2 is indicated by the black ×.

5.4 Measurement

In the following, we describe the process of how fully processed samples were prescreened

at multiple temperatures / environments such that only viable candidates were measured

at millikelvin temperatures of the DR. This multi–temperature prescreen process became

known colloquially as “The great filter.”†

†A reference to Robert Hanson and Nick Bostrom’s work concerning the Fermi Paradox

120

(a)

(c)

(b)

M0

M1

M2

V1

V2

2 mm

Figure 5-9: (a) Glamor shot of an undiced full processed chip. Two experiments were ableto be fabricated per chip. (b) overhead false–color optical image of the dc/SFQ driver.M1, M2 and the junctions are colored blue, red, and yellow, respectively. (c) False colorcross-sectional SEM image of the dc/SFQ driver layer stack. Nb is colored red while SiOxis colored in blue. The chip substrate in colored yellow. The junction formed between M1and M2 is indicated by the black ×. Images (a) and (c) courtesy of Edward M. Leonard, Jr.Image (b) courtesy of JJ Nelson.

121

5.4.1 LHe SFQ Circuit Pre-screening

The all Nb process for the SFQ drivers allowed for samples to be prescreened at 4.2 K in LHe

as a first step in filtering out poorly performing chips. Samples were placed on and aluminum

wire bonded to a custom dip probe mount containing bias and filtering electronics allowing

for a three–wire measurement of the driver along with four–wire measurements of unshunted

test junctions. Once bonded, the mount+sample were placed at the end of a custom dip

probe and surrounded by a high permeability magnetic shield. This was then submerged

in a 100L LHe storage dewar which was then placed inside a custom built Faraday cage to

reduce environmental RF noise. The measurement setup is depicted in Figure 5-10(a). The

three–wire driver bias lines each incorporated low–pass T-filters. The RF drive line had 20

dB of attenuation placed on the sample mount. IV curves were taken by applying a 1 Hz

bias tone through the I port of the three–wire setup and monitoring the voltage. Baseline

curves with no applied RF power were taken along with data at different drive frequencies

and applied power. Figure 5-10(b) shows baseline data along with data for a 5 GHz drive and

a 1.25 GHz drive at an explicit applied on–chip power of –40 dBm (no calibration was done

for microwave reflections or line losses). The baseline critical current of 130 μA corresponded

to a junction critical current density of Jc = 0.73 kA/cm2 . Shapiro steps of 10.3 μV and

2.6 μV can clearly be seen for the respective drive frequencies at bias currents exceeding

|I| > 100μA. Only samples exhibiting Shapiro steps at explicit applied powers of PRF < –35

dBm were moved on to more extensive testing the lower temperatures afforded by an ADR.

5.4.2 100 mK Quantum Circuit Pre-screening

For samples containing dc/SFQ drivers that passed the initial 4.2 K screening, the next

step in discerning suitability for the millikelvin stage of the DR was a repackaging into a

custom superconducting Al sample box and brief stint on the 100 mK stage of an ADR. The

wire bonded sample and sample box are shown in Figure 5-11(a). An ADR wiring diagram

for sample characterization is shown in Figure 5-11(b). A 6:1 coaxial relay was utilized

122

Figure 5-10: (a) Circuit diagram showing 4.2 K measurement setup for characterizing thedc/SFQ driver. A three–wire measurement was done with all passive bias electronics at 4.2K. (b) IV curves for different RF drive frequencies. The baseline IV curve shows a totaldriver critical current of Ic ≈ 130μA. Shapiro steps of 10.3 μV and 2.6 μV in size can be seenfor 5 GHz and 1.25 GHz drives, respectively. The power listed is the explicit power appliedat the chip not accounting for microwave line losses or reflections.

to switch between testing the SFQ drivers (RF1 / RF2) and passing in microwaves to the

quantum circuit (RO). Each line passed through a custom high frequency Eccosorb filter at

the millikelvin stage. The readout output was passed through another Eccosorb filter at the

millikelvin stage followed subsequently by a microwave isolator, K&L low pass microwave

filter and a HEMT microwave amplifier all at 3 K. Qubit flux bias lines were biased through 1

kΩ resistors sunk on the 3 K stage and passed through low-pass (-60 dB at 3 GHz) Eccosorb

filters on the cold stage. Three–wire measurements for driver characterization were comprised

of 2 kΩ and 1 kΩ resistors for the voltage and current bias respectively tied together either

on the 3 K plate or at the millikelvin stage depending on whether the cool down was shared

with experiments or not. At this temperature only qubit signs on life (see section 3.2.2) could

be investigated. Simple measurements reading out qubit cavities in transmission using a

2–port VNA allowed for rapid conclusions to the questions (1) “is the qubit alive?” and (2)

“does it flux tune appropriately?” Characteristics curves displaying a positive response to

123

RF2

FB2

RORO

IV2

RF1 FB1

IV1

FB1IV1

RF1

RO RO

IV2 FB2RF2

40

3 KmK

6:1 RelayH

F ESRB

HF ESR

B

HF ESR

B

HF ESR

B

LF ESRB

LF ESRB

LF ESRB

LF ESRB

K&

L

K&

L2 k 2 k1 k 1 k 1 k 1 k

HEMT

MuMetal Shield

(a)

(b)

Figure 5-11: (a) Bonded sample in sample mount. Each chip contained 2 independentexperiments (labels 1 & 2). Readout resonators were inductively coupled to a common feedline (RO). (b) ADR Wiring diagram for T = 100 mK sample prescreening. 6:1 channel relayswere used to increase wiring flexibility. Each quantum and classical circuit pair could betested independently.

124

both these questions are displayed in Figures 3-4 and 3-5, respectively.

IV curves of the SFQ drivers at T = 100 mK often displayed a slight amount of hysteresis

near the critical current. This was due to either the critical current exceeding such a value

that βC > 1 or, as was not found until imaged under a SEM, the shunt resistors failing after

thermal cycling. Figure 5-12 displays IV curves taken for a driver at T = 100 mK for no

applied RF power (blue) and for a 5 GHz drive at -40 dBm (orange). With no applied RF, the

driver was slightly hysteretic despite βC < 1 even for a slightly larger than designed 180 μA

critical current. However, as can be seen by the orange curve, the deleterious hysteresis could

often be suppressed with the application of microwave power to the driver. A clear Shapiro

step can be seen at a bias current of approximately 110 μA. This is not in the hysteretic

regime of the driver and thus constituted a perfectly suitable operating point.

Upon discovering the thermal cycling issue with drivers becoming hysteretic at 100 mK

despite being not at 4.2 K and only a modest increases in the critical current and thusly

βC , prescreened drivers exhibiting this behavior were imaged by SEM. Figure 5-13 shows

a depth profile SEM image of a dc/SFQ driver shunt resistor. Again the Nb metalization

layers are in pink, SiOx wiring dielectric is in blue and the shunt resistor is in green. The

process parameters for achieving a sloped side wall etch in Nb detailed in the previous sections

drifted resulting in a near vertical side wall profile of M2. With a thickness of 100 nm, the

vertical side wall all but broke many shunt resistor connections between the junction and

ground. Multiple thermal cycles of the samples exasperated the issue as resistors that did

make galvanic connection during deposition would break from the thermal stress.

5.4.3 Initial SFQ–Qubit Bringup

Only a handful of samples across multitudes of wafers survived the multi–temperature

prescreen procedure described in the previous sections. The few precious samples that did

survive were further tested on the millikelvin stage of a DR. Figure 5-14 displays the initial

wiring diagram afforded by the space available in the DR. Microwaves utilized to control

125

-300 -200 -100 0 100 200 300-300

-200

-100

0

100

200

300

Current

Volta

geNo RF5 GHz, -40 dBm

Figure 5-12: dc/SFQ IV curve for both no applied microwave power (blue) and a -40 dBm, 5GHz drive (orange) measured at 100 mK. For no applied microwave power, the driver wasslightly hysteretic. The application of microwaves, however, lowered the critical current valueand thusly βC to such a value that the hysteresis was suppressed and a DC bias currentoperating point of I = 110 μA could be found.

M0

M1

M2 R

Figure 5-13: Depth profile of SFQ driver layer stack showing the Nb metalization layers(pink), the SiOx wiring dielectric (blue) and the JJ shunt resistor (green). It is plain to seefrom this image that the reflow bake plus etch of layer M2 did not result in a sloped sidewall profile of the Nb. This caused many shunt resistors to break upon deposition or duringthermal cycling resulting in hysteretic dc/SFQ drivers.

126

and readout the qubit were SSB mixed in IQ mixers with the output of 1 GS/s DACS.

These signals were then combined with a 3 dB splitter at room temperature and fed into the

DR. Once at the cryogenic plates, the signal passed through stages of attenuation (50 dB

in total) in addition to Eccosorb and K&L 12 GHz low–pass filters before finally reaching

the input of the sample. At the output side, 3 microwave isolators sunk to the cold stage

separated the chip from the 4 K HEMT amplifier to prevent deleterious effects stemming

from back-action amplifier noise. The HEMT amplified signal was further amplified by two

Narda 2–8 GHz microwave amplifiers before finally being heterodyne mixed back down to the

side band frequency and digitized with a 1 GS/s Alazar analog to digital converter (ADC).

The qubit flux bias line began with a Stanford Research Systems (SRS) SIM928 voltage

source passed through a 10 kΩ resistor sunk on the 4 K plate. The DC current was then

subsequently filtered with a 250 MHz low–pass and low–frequency Eccosorb filter. The final

step taken in signal preparation was to insert a superconducting Nb wire into the chain to

help thermalize any hot electrons. The dc/SFQ driver was biased with a SRS SIM928 voltage

source passed through a 10 kΩ/1 kΩ 3–wire measurement combined at the 4 K stage. The

resulting single wire was then low–pass filtered, connected to a superconducting Nb wire and

subsequently fed into the dc/SFQ bias line. Initial tests of the quantum/classical interface

involved triggering the dc/SFQ driver direct from a coherent room temperature microwave

source where the signal could be turned on and off with a fast (∼ 1 ns) microwave switch.

The signal was attenuated by 40 dB and low–pass filtered before being applied to the RF

input of the dc/SFQ driver. The results reported from here on are for sample 070717C-C4.

Prior to SFQ experimentation, baseline measurements of the qubit lifetime and dephasing

rate were performed with standard microwave manipulation. The T1 and T2* curves are

shown in Figure 5-15(a) and Figure 5-15(b) respectively, for the qubit tuned to its upper sweet

spot frequency of ωq = 4.85 GHz. The fitted relaxation curve yielded a qubit T1 = 23.6±0.6μs.

The fitted Ramsey decay of the |1〉 population yielded a T ∗2 = 24.4 ± 0.8μs. While these

times were impressive for such a heavily processed sample, they were not indicative of the

127

Cryogenic Isolator

12 GHz Low Pass Filter

20 GHz Low Pass Eccosorb

FilterHF Ecc

300 MHz Low Pass Eccosorb

Filter

- +IQ Mixer

High Electron Mobility

Transistor

Voltage Source

3 dB Combiner

10 dB Attenuator10 dB

20 dB Attenuator

Microwave Generator

1 GS/s Arbitrary

Waveform Generator

12 GHz

12 GHz

1 kOhm

10 kOhm

LF Ecc

LF Ecc

Qubit Flux Bias

HEMT

HEMT

DC/SFQ Bias

ADCQ

LO RI

Q

I

AWG

Cavity Readout

QLO R

I

Q

I

LF Ecc

HF Ecc 10 dB20 dB

20 dB

20 dB

AWG

Q

I

AWG

Qubit Control

QLO R

I

QLO R

I

Q

I

- +

- +

ADC

Q

I

1 GS/s Analog-to-Digital

Converter

4 K mK

Variable Attenuator

DC/SFQ Trigger

fabricated chip

classicalcircuit

quantumcircuit

Al box

12 GHz

10 kOhm

20 dB20 dB

250 MHz

Nb coaxialstubNb

Nb

12 GHz

2 GHz

Nb

ECL microwave

switch

ECL microwave

switch

Figure 5-14: Wiring diagram for initial testing. The control of the SFQ tone was done with asimple ECL triggered microwave switch. Standard single side band mixing was utilized toon-demand create and shape the readout and qubit drive tones. These tones were coupledtogether at room temperature via a 3 dB splitter and fed into the DR.

128

average qubit performance measured across many different samples. The average measured

relaxation and dephasing times across all samples were T1 = T ∗2 ≈ 5μs.

After standard qubit manipulation and characterization with microwaves had been estab-

lished, basic functionality of the SFQ–qubit driving scheme was tested. SFQ driven Rabi

oscillations were measured by applying qubit transition frequency subharmonic drives to the

dc/SFQ circuit and dispersively measuring the qubit state. Subharmonic drives were required

due to microwave bleed through of the dc/SFQ driver, preventing the turning off coherent

driving of the qubit even for no DC bias current applied to the dc/SFQ circuit.

The false color image in Figure 5-16(a) displays the qubit occupation (false color) as a

function of SFQ drive time (y–axis) and SFQ bias current (x–axis) for a drive frequency

ωD = ωq/3. Above approximately 90 μA of bias current, fast oscillations of the qubit

population can be seen for times T< 500 ns. The corresponding dc/SFQ drive IV curve is

displayed below the false color image. A clear Shapiro step can be see at the same bias current

indicating the onset of consistent in–phase phase slips of the driver circuit. Figure 5-16(b)

(a) (b)

20 60 1000

0.5

1

20 60 1000.5

0.75

1

P 1 P 1

Figure 5-15: (a) Measured (black diamonds) exponential decay of SFQ–qubit P|1〉 lifetime.The extracted T1 (red line) was 23.6±0.6 μs. (b) Measured (black diamonds) dephasingtime T2∗. The extracted coherence time (red line) for the qubit was 24.4±0.8 μs. Averagetimes for all measured qubits were T1 = T ∗

2 ≈ 5μs.

129

displays another scan focused on the driver bias turn–on region. The turn–on of coherent

driving of the qubit is seen to correspond 1–to–1 with the onset of a Shapiro step in the

driver IV curve. An important note is that the oscillations are stable with respect to the bias

current over a range of approximately 15 μA allowing for bias noise resistant performance.

With the data taken in Figures 5-16(a-b) indicating appropriate dc/SFQ bias and mi-

crowave drive parameters, more standard SFQ based qubit metrology could be performed.

Figure 5-16(c) displays the classic chevron pattern of Rabi oscillations as a function a SFQ

drive frequency again centered at ωq/3. Ramsey fringes as a function of SFQ drive frequency

are displayed in Figure 5-16(d). It is worth noting that the chevron pattern in Figure 5-16(c)

and the Ramsey fringes in Figure 5-16 denote different qubit transition frequencies with

the center of the chevron pattern suggesting ωq/3 = 1.650 GHz and the Ramsey fringes

ωq/3 = 1.653 GHz, the latter of which is the true third subharmonic of the qubit frequency.

While more thoroughly addressed in section 5.4.6, we briefly mention here that continuous

driving of the dc/SFQ circuit as is done in the Rabi chevron experiment consistently gen-

erates quasiparticles that add a complex dissipative loss channel [87] to the qubit shifting

it’s fundamental frequency. The asymmetry in the 2D Ramsey experiment in Figure 5-16

displays the qubit frequency shifting back to its natural resonance after an interleaving I gate

between SFQ driven X/2 gates of length T = 200 ns.

While phase control of the SFQ drive tone was not afforded by the initial wiring displayed

in Figure 5-14, the high bandwidth of the microwave switch allowed for exploration of deep

subharmonic qubit dynamics. Figure 5-17(a) displays the results of a 36 hour scan over which

the drive frequency to the dc/SFQ circuit was swept from 0.48–1.25 GHz and the qubit state

monitored for 500 ns. Coherent oscillations of the qubit state can be seen for a multitude of

frequencies ωd = ωq/n for subharmonic number n ≥ 4. With coherent subharmonic dynamics

seen down to below 1 GHz, direct digital synthesis (DDS) was utilized to look at the qubit

dynamics at deeper subharmonic frequencies. Figure 5-17(b) displays the coherent Rabi

oscillation of the qubit when irradiated with pulses from the driver driven at ωq/41 ≈ 125

130

Tim

e (n

s)

0

100

200

300

400

500

90 100 110 120024

90 100 110 12050 60 70 80

Tim

e (

s)

0

VD

river

(mV

)

0.51.01.52.0

2.53.03.54.0

-2.340-2.336-2.332-2.328-2.324

V (

V)

(a) (b)

(c) (d)

1.641.645

1.651.655

1.661.648

1.651.652

1.6541.656

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

Tim

e (n

s)

0

100

200

300

400

500

Tim

e (n

s)

0

100

200

300

400

500

0.2

0.4

0.6

0.8

Figure 5-16: (a) Qubit occupation as a function of time and dc/SFQ driver bias currentfor a SFQ drive frequency of ωd ≈ ωq/3. The corresponding uncorrected driver IV curve isdisplayed underneath the false color plot. For bias currents Ib > 90μA, coherent oscillationsof the qubit state can be seen. (b) Zoom-in on the driver bias region between 90 and 125μA. Again the turn on of fast Rabi oscillations can be seen to coincide with the onset of aShapiro step in the IV curve of the dc/SFQ driver displayed below the false color plot. Themagnitude of the step was measured to be 3.46 μV in direct relation to the drive frequency.(c) Rabi chevron oscillations as a function of the dc/SFQ drive frequency centered at ωq/3.(d) Ramsey oscillations as a function of the dc/SFQ drive frequency centered again at ωq/3.

131

MHz. Discrete steps in the occupation can be seen every 1/125×10−6 s = 8 ns corresponding

the application of a single SFQ pulse being applied to the qubit once a drive period. At

the time of this writing, to our knowledge, this is the first demonstration of truly digital

control of a qubit state where, as previously mentioned, the precision of control is now set

lithographically through the coupling capacitor between the digital circuit and the quantum

circuit; lower the capacitance, increase the precision (with the cost paid for in gate time).

Subharmonic control also opens the door to rapid fundamental transition frequency

determination. In contrast to the procedures outlined in section 3.2.3 where a saturating

microwave tone is swept in frequency looking for a change in the readout tone level, a swept

SFQ tone positioned in the deep subharmonic frequency band would produce a forest of peaks

(or dips, depending on the read out resonator scheme); These peaks provide a spectroscopic

signature of the qubit fundamental transition. To look at this more concretely, we define the

consecutive qubit subharmonics as

ωn =ωq

n(5.24a)

ωn+1 =ωq

n + 1 . (5.24b)

Dividing the above equations by one another

ωn

ωn+1= n + 1

n. (5.25)

Straightforward algebra finally yields

n = ωn+1

ωn − ωn+1. (5.26)

The ramification of Eq. (5.26) is that the measurement of any 2 consecutive qubit subharmonic

frequencies provides the subharmonic number relating back to ωq. Applying this to the data

132

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2SFQ Drive Frequency (GHz)

0

50

100

150

200

250

300

350

400

450

500

Tim

e (n

s)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 50 100 150 200 250 300 350Time (ns)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

P 1

P1

161 162 163 164 165 166 167 168 1690.29

0.3

0.31

0.32

0.33

0.34

Time (ns)

(a)

(b)

Figure 5-17: (a) Qubit oscillations as a function of SFQ drive frequency. Subharmonic Rabioscillations as low as ωq/10 = 498 MHz can be seen. (b) Quarter period Rabi oscillation foran SFQ drive frequency of ωq/41 ≈ 125 MHz. The qubit occupation measurement samplingfrequency was set to 1 GHz. (inset) Close–up view of quantized steps in the Rabi oscillation.For a 125 MHz SFQ drive, 8 points were recorded per step in the occupation correspondingto the application of 1 SFQ pulse every T = 1/125 × 10−6 = 8 ns.

133

displayed in Figure 5-17(a), we see coherent Rabi oscillations at roughly both 1.24 GHz and

0.98 GHz. Substituting these numbers into Eq. (5.26) yields a value for n of approximately

3.8. Rounding this up to the next whole integer and multiplying by 1.24 GHz yields a

fundamental qubit frequency of ωqb ≈ 4.96 GHz, directly in line with the qubit spectroscopy

data for this sample displayed in Figure 3-6.

5.4.4 Establishing Orthogonal Axis Control

While the simple turning on and off of the microwave drive to the dc/SFQ converter allowed

for coherent oscillations of the qubit and quantized steps in the Rabi oscillation to be realized,

it did not afford phase control over the SFQ drive tone. To realize phase control of the SFQ

drive tone, the microwave switch was removed in favor for a standard single side band mixing

protocol as outlined in blue in Figure 5-18. Revisiting traditional orthogonal axis control with

shaped microwaves, we reprint the equation detailing to on resonance interaction between a

single side band mixed tone and a qubit in the rotating frame

VI =�Γ(t)2 [cos(γ)σx + sin(γ)σy] , (5.27)

where Γ(t) is the time varying amplitude of the mixed side band tone and γ it’s phase.

Extending this treatment to driving with SFQ pulses as opposed to shaped microwaves is

straightforward if one considers the dynamics in the time domain. Figure 5-19 displays circuit

diagrams a SFQ driven X gate and a SFQ driven Y gate. By adjusting the phase of the mixed

side band tone, a variable timing offset develops between SFQ driven X and Y rotations.

To measure and confirm SFQ gate orthogonality in the lab, an SFQ pulse sequence was

applied to the qubit very similar to a T ∗2 measurement only instead of the intermittent period

between subsequent X/2 gates being filled with an idle gate, a variable length and phase

SFQ driven Y gate was applied. Figure 5-20(a) displays the SFQ pulse timing diagram of

the orthogonality experiment. A XSFQ/2 pulse is applied to the qubit to set the control axis.

134

Cryogenic Isolator

12 GHz Low Pass Filter

20 GHz Low Pass Eccosorb

FilterHF Ecc

300 MHz Low Pass Eccosorb

Filter

- +IQ Mixer

High Electron Mobility

Transistor

Voltage Source

3 dB Combiner

10 dB Attenuator10 dB

20 dB Attenuator

Microwave Generator

1 GS/s Arbitrary

Waveform Generator

12 GHz

12 GHz

1 kOhm

10 kOhm

LF Ecc

LF Ecc

Qubit Flux Bias

HEMT

HEMT

DC/SFQ Bias

ADCQ

LO RI

Q

I

AWG

Cavity Readout

QLO R

I

Q

I

LF Ecc

HF Ecc 10 dB20 dB

20 dB

20 dB

AWG

Q

I

AWG

Qubit Control

QLO R

I

QLO R

I

Q

I

- +

- +

ADC

Q

I

1 GS/s Analog-to-Digital

Converter

4 K mK

Variable Attenuator

AWG

DC/SFQ Trigger

QLO R

I

Q

I

fabricated chip

classicalcircuit

quantumcircuit

Al box

12 GHz

10 kOhm

20 dB20 dB

250 MHz

Nb coaxialstubNb

Nb

12 GHz

2 GHz

Nb

variable based on desired tone

Figure 5-18: Wiring diagram for advanced SFQ–qubit testing including qubit state rotationorthogonality and randomized benchmarking of SFQ gates. The difference between thisdiagram and that displayed in Figure 5-14 is highlighted in blue.

135

Afterwards, a YSFQ gate of variable time t and phase γ is applied followed by another XSFQ/2

gate. Figure 5-20(b) displays overdetermined tomograms for SFQ subharmonic drives at the

3textrd and 41st subharmonic. In both cases when t = 0, the pulse sequence equates to a single

XSFQ gate. For the n = 3 case, there are 3 axes / values of γ for which orthogonality can

be achieved. This over-determination of axis orthogonality is simple to see when plotting

the phase evolution of two sine waves with frequencies 3 times that of each other as done on

the left hand side of Figure 5-20(c). Over the first π–phase evolution of the each wave, the

phases of the two waves align 2 times (black dots) indicating parallelism which results in the

continuous coherent driving of the qubit (blue lobes in LHS tomogram, 5-20(b)). The right

hand side of Figures 5-20(b) and 5-20(c) show the same dynamics only at a subharmonic

drive of n = 41. Now, instead of 3 orthogonal axes, there are correspondingly 41 orthogonal

values of 0 ≤ γ < π in which YSFQ may be set to. The signal overlaps (black dots) shown on

the RHS of Figure 5-20(c) indicate the phase matching of the qubit and SFQ drive tones.

AWGQ

LO RI

Q

I

dc/SFQ

AWG

Q

dc/SFQ

Figure 5-19: Single side band mixing allows for direct phase control over the output tone fromthe IQ mixers. These out of phase tones (red / blue) induce timing shifts between pulses forwhat is defined as a SFQ driven X gate and a SFQ driven Y gate. Once a particular phase /clock is set for the SFQ X gate, setting the SFQ Y gate is simply a matter of changing thephase γ of the side band tone.

136

(a)

(b)

(c)

Am

plitu

de (a

rb. u

nits

)

Phase (rad)

0

Am

plitu

de (a

rb. u

nits

)

Phase (rad)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 5-20: (a) Gate sequence to test SFQ based qubit gate orthogonality. Two SFQbased XSFQ/2 rotations were separated by a variable length and phase YSFQ gate. (b)Overdetermined tomograms for driving the qubit at ωd = ωq/3 and ωd = ωq/41. Values ofγ = m × π/3 (m × π/41) were in phase with the surrounding XSFQ gates and thus continueddriving the qubit state along those axes. (c) diagrammatic representation of the phaseevolution of the qubit verses that of the drive frequency. The LHS plot shows 2 phase overlaps(black dots) of the qubit evolution (red) and the drive evolution (blue) between 0 ≤ γ < π.The RHS plot shows the 40 overlaps between the qubit evolution and the drive tone againfor 0 ≤ γ < π.

137

5.4.5 Randomized Benchmarking of SFQ–Qubit Gates

The combination of coherent qubit state driving with orthogonal axis control allowed for

the establishment of SFQ based single qubit gates. Prior to characterization with IRB (see

section 3.3.2), it was thought prudent to secure the fact that all 24 single qubit Cliffords

could be realized. To test this, each Clifford gate was applied and subsequently inversed. The

|0〉 state population was then measured and taken as a measure of the fidelity of applying

only the one gate. Mathematically, the experiment took the form

CC−1 ≤ 1 , (5.28)

where SPAM errors were not explicitly corrected for. Resulting fidelities for the application

of and subsequent inverse of the 24 single qubit Cliffords are displayed in Figure 5-21 for the

n = 3, n = 4, and n = 39 subharmonic. It is important to note that while traditionally the

difference between a π and a π/2 rotation when utilizing microwaves is in the amplitude of the

pulse, for SFQ based gates the difference is the length of the pulse sequence, i.e. τX = 2×τX/2.

The fidelities averaged over all 24 Cliffords were 74±8%, 65 ± 11%, and 71 ± 8% for the

n = 3, n = 4, and n = 39 subharmonic, respectively. While promising in it’s own right

that the entire single qubit Clifford gate series could be realized with SFQ pulses, more

investigation was required into why the overall fidelities for the application and inverse

of single gates were so low. To do this, T1 measurements of the qubit were performed

utilizing standard microwave rotations but with an off–resonant SFQ drive applied while the

measurement was in progress. Figure 5-22 displays the relaxation of the qubit from the |1〉state for both no SFQ signal applied during the experiment and for a detuned, 2 GHz SFQ

drive. For SFQ drive lengths of 250 ns, the occupation of the qubit reduces quickly to a value

of ∼ 0.7 before falling off exponentially towards a background level of ∼ 0.2. This initial fall

off of the occupation for the background SFQ signal explains not just the average fidelity

achieved for the Cliffords displayed in Figure 5-21 but also the gate length dependency of

138

I,I X/2,-X/2-X/2,X/2X,X

Y/2,-Y/2-Y/2,Y/2Y,Y

-X/2,Y/2,X/2,-X/2,-Y/2,X/2

-X/2,-Y/2,X/2,-X/2,Y/2,X/2

Y,X,Y,XY,X/2,Y,X/2Y,-X/2,Y,-X/2X,Y/2,X,Y/2X,-Y/2,X,-Y/2X/2,Y/2,X/2,X/2,Y/2,X/2

-X/2,Y/2,-X/2,-X/2,Y/2,-X/2

Y/2,X/2,-X/2,-Y/2Y/2,-X/2,X/2,-Y/2-Y/2,X/2,-X/2,Y/2-Y/2,-X/2,X/2,Y/2X/2,Y/2,-Y/2,-X/2X/2,-Y/2,Y/2,-X/2-X/2,Y/2,-Y/2,X/2-X/2,-Y/2,Y/2,X/2

Gate Sequence

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Se

quen

ce F

idel

ity

Figure 5-21: The raw fidelities of the 24 single qubit Cliffords as applied by driving the SFQdriver at the n = 3, n = 4, and n = 39 subharmonic. The average gate fidelity for the n = 3subharmonic was 74±8%. A trend between sequence fidelity and overall gate length in timecan be seen with gates requiring full rotations (and thus driven for longer) having loweroverall fidelity.

that fidelity. The inset in Figure 5-22 displays a high density time series plot of the qubit T1

probability for both no background SFQ signal and with it. With the background SFQ signal

applied, the max obtainable probability falls off linearly with time for the first 250 ns. For

n = 3, the average length of all the gates (including their inversions) shown in Figure 5-21 is

76 ns. The corresponding max obtainable fidelity for a off–resonant SFQ pulse at ∼ 75 ns is

80%. The average obtained fidelity for the ωd = ωq/3 driven Cliffords was 75%.

A natural explanation for the sudden loss of occupation is that the driver is pushing the

qubit to higher states. However, this is not seen as only two qubit state IQ distributions

are ever resolved during the T1 experiments. Instead, what is seen is a spurious increase in

the |0〉 state population over the first 250 ns of the SFQ drive, indicative of quasiparticle

generation and subsequent heating. This put an overall limit on the max obtainable sequence

139

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Time ( s)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 50 100 150 200 250 300 350

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Time (ns)

No BG SFQW/ BG SFQ

Figure 5-22: Qubit T1 experiment with either the SFQ driver off (blue curve) or driven duringthe entire sequence at ωSFQ = 2π × 2 GHz. For the driver on, P|1〉 falls off immediately tobelow 0.7 with the background level rising to above 0.2. (inset) First 350 ns of P|1〉 relaxationexperiment. A fast linear decrease in the occupation can be seen for the first 250 ns.

fidelity for the interleaved RB experiments to be discussed in the following sections.

As discussed in section 3.3.2, interleaved RB provides a protocol to actively test the

fidelity of single gate operations by looking at the overall change in the sequence fidelity

as a function of the number of Clifford gates executed m. A subtle but important change

to implementing interleaved RB for SFQ based gates was the elimination of the identity or

idle gate I. The dynamic qubit frequency apparent in the asymmetry of the 2–dimensional

Ramsey plot of Figure 5-16(d) but not in the Rabi chevron plot of Figure 5-16(c) meant that

in order to stay on resonance with the qubit throughout the m Clifford gate long sequence, no

idle time could be afforded allowing the qubit to relax back to it’s unperturbed fundamental

transition frequency.

Figure 5-23 displays interleaved RB data for both the n = 3 and n = 41 subharmonics.

Fidelities F ≥ 90% were achieved for all applied single qubit gates at both subharmonics (see

Table 5.3). As discussed above, the best obtainable sequence fidelity for low Clifford number

was limited by quasiparticle heating with a maximum value F ≈ 70%. The combination

140

2 4 6 8 10 12 14 16 18Number of Cliffords

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Sequ

ence

Fid

elity

Interleaved Gate: None

Interleaved Gate: X

Interleaved Gate: X/2

Interleaved Gate: -X/2

2 4 6 8 10 12 14 16 18

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Number of Cliffords

Sequ

ence

Fid

elity

Interleaved Gate: None

Interleaved Gate: Y

Interleaved Gate: Y/2

Interleaved Gate: -Y/2

1 2 3 4 5 6 7 8

0.4

0.45

0.5

0.55

0.6

0.65

Sequ

ence

Fid

elity

Number of Cliffords

Interleaved Gate: None

Interleaved Gate: X

Interleaved Gate: X/2

Interleaved Gate: -X/2

1 2 3 4 5 6 7 8Number of Cliffords

0.4

0.45

0.5

0.55

0.6

0.65

Sequ

ence

Fid

elity

Interleaved Gate: None

Interleaved Gate: Y

Interleaved Gate: Y/2

Interleaved Gate: -Y/2

(a)

(b)

Figure 5-23: (a) Interleaved randomized benchmarking of SFQ driven qubit gates at ωSFQ =ωq/3. X–rotations are displayed in the left–hand column and Y–rotations in the right–handcolumn. Gate fidelities greater than 90% were achieved for all 6 SFQ gates. (b) Interleavedrandomized benchmarking of SFQ driven qubit gates at ωSFQ = ωq/41. Again, gate fidelitiesabove 90% are achieved. The ∼ 70% max fidelity for low Clifford number is due to heatingissues stemming from running the SFQ driver discussed in the text.

of long gate times (τgate > 150 ns) and limited on–board DAC memory allowed for only 8

interleaved Cliffords to be applied at the n = 41 subharmonic necessitating less precision and

larger error in the fits.

5.4.6 SFQ Generated Quasiparticle Studies

As previously mentioned, the max gate sequence fidelity and invariably the gate fidelities

themselves were limited by quasiparticle generation due to running of the dc/SFQ driver.

But, how can we say this with such assuredness? To definitively prove that it was in fact

141

Gate Fn=3 Fn=41

X 94.0 ± 1.0% 91 ± 2%X/2 95.5 ± 0.8% 93 ± 3%

-X/2 95.7 ± 0.5% 95 ± 3%Y 93.9 ± 0.5% 94 ± 3%

Y/2 96.9 ± 0.6% 92 ± 3%-Y/2 94.6 ± 0.6% 95 ± 2%

Table 5.3: SFQ gate fidelities determined with interleave randomized benchmarking for then = 3 and n = 41 subharmonic.

QPs limiting the operation fidelity, further experiments had to be carried out. Quasiparticles

relax through two main physical channels; They either recombine or are trapped [88]. These

two physical mechanisms lead to two different rates at which the average QP density nqp in a

system decays

dnqp

dt= −an2

qp − bnqp . (5.29)

The quadratic and linear terms model recombination and trapping, respectively. Dropping

the recombination term yields a simple exponential decay of the quasiparticle density

nqp(t) = nqp(0)e−bt . (5.30)

The qubit energy relaxation rate Γ is directly proportional to the number of QPs in the

vicinity of the qubit with the form

Γ(t) = cnqp(t) + Γ0 . (5.31)

The rate of decay of the qubit excited state is then

dP|1〉dt

= Γ(t)P|1〉 , (5.32)

which, when integrated, yields a super–exponential form [89] for P|1〉

142

P|1〉 = e〈nqp〉(exp[−t/Tqp]−1)e−t/T1 , (5.33)

where 〈nqp〉 is the average number of QPs, Tqp is the relaxation induced by 1 QP, and T1 the

relaxation time associated with all other processes. With this direct dependency of P|1〉 on

the number of QPs, an experiment could be devised to extract the 〈nqp〉 as a direct result ofrunning the dc/SFQ driver.

Figure 5-24(a) displays the measurement protocol for extracting the average number of

QPs generated during a variable time, off–resonant SFQ drive applied prior to a microwave

π–rotation of the qubit. The protocol is simply a standard microwave T1 experiment run

after a SFQ poisoning pulse. Figure 5-24(b) displays the P|1〉 decay probability (false color) as

a function SFQ drive time. The effect of running the SFQ driver for only a few microseconds

is seen immediately in the increase in decay. These curves were subsequently fit with Eq.

(5.33). The extracted 〈nqp〉 for the varying SFQ drive time is displayed in Figure 5-24(c).

The extracted background QP number was 0.10 for no applied poisoning pulse saturating

up to 2.1 for poisoning times τSFQ > 45μs. The inset of Figure 5-24 displays the first 5 μs

where the time axis has been scaled by ωSFQ = 133 MHz so that 〈nqp〉 could be displayed

as a function of the number of SFQ phase slips. The background 〈nqp〉 = 0.1 obscures the

dynamics for the first 200 phase slips after which a linear rise in the extracted QP number is

seen with slope (1.6 ± 0.2) × 10−3 QPs coupling to the qubit per dc/SFQ driver phase slip.

With some light now shed on the generation of QPs from the running of the dc/SFQ

driver, measuring their relaxation properties was subsequently performed. Figure 5-25(a)

displays the QP relaxation / T1 recovery experimental protocol. A poisoning pulse of fixed

length was applied followed by variable recovery time after which a standard T1 measurement

protocol was performed. Figure 5-25(b) displays the P|1〉 decay (false color) as a function of

recovery time. The recovery saturates at approximately 70 μs. Figure 5-25(c) displays the

fitted 〈nqp〉 as a function of recovery time. The QP decay constant b from Eq. (5.30) was

fitted to a value of b−1 = 17.6 ± 0.3μs. The inset shown in Figure 5-25(c) shows the same

143

0 10 20 30 40 500

20

40

60

80

100

Rea

dout

Del

ay

(us)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 10 20 30 40 50

0.5

1

1.5

2

2.5

0

SFQ

SFQ Drive Time (us)

(a)

(b) (c)

SFQ Drive Time (us)

0 200 400 6000

0.2

0.4

0.61.6(2)e-3

Phase Slips

Figure 5-24: (a) Experimental protocol to extract 〈nqp〉 from running the dc/SFQ driver. Anoff–resonant SFQ drive at frequency ωSFQ = ωqb/41 + δ is performed for a variable lengthof time τSFQ immediately followed by a microwave π–pulse. A variable length of idle timeτRO is then between the π–pulse and measurement. (b) Data on a series of T1 experimentsas a function of off–resonant SFQ drive time. The P|1〉 probability decay (false color) wasmeasured out to 110 μs for SFQ drives out to 50 μs. (c) Extracted 〈nqp〉 from the fits tothe data in (b). The number of QP saturates at approximately 45 μs at 〈nqp〉 ≈ 2.1. (Inset)Zoom in view of the first 5 μs scaled by the SFQ drive frequency. The extracted linear fit tothe data, once out of the background, yielded a coupling of 1.6 × 10−3 QPs/phase slip to thequbit.

data as in (c) but plotted on a semi–log scale. The linearity of the data when plotted this

way confirms that the QP decay is due to recombination solely and the earlier dropping of

the quadratic term in Eq. (5.29) was justifiable.

Just as in the Mattis and Bardeen work concerning thermal QPs, non–equilibrium QPs

also give rise to a complex impedance [87]; The real part responsible for the additional loss

discussed and measured above with the complex contribution giving rise to a frequency shift

of the qubit [90, 91]. To study the frequency effects of the SFQ generated non-equilibrium

QPs, a ω = ωq − 5MHz detuned Ramsey fringe experiment was run preceded by an SFQ

poisoning pulse of variable length. The pulse timing sequence / experimental protocol is

outlined is Figure 5-26(a). A detuning of 5 MHz was chosen so that many Ramsey fringes

could be measured within the pulse sequence timing / memory limits of the DAC/ADC

144

20 40 60 80 100 1200

1

2

3

SFQ(c)

0 20 40 60 80 100Recovery Time (us)

0

20

40

60

80

100

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a)

(b)

Recovery Time (us)

Rea

dout

Del

ay

(u

s)

20 40 60 80 100

100

10-1

Figure 5-25: (a) Experimental protocol to examine the relaxation of QPs generated fromthe dc/SFQ driver. The driver was run for a 〈nqp〉 saturating length of 50 μs producing abackground QP level 〈nqp〉 ∼ 2.5 followed by a variable recovery period before a standard T1measurement was performed. (b) Data on a series of T1 experiments as a function of recoverytime after the QP saturating SFQ drive. (c) Extracted 〈nqp〉 from the fits to the data in(b) as a function of recovery time. Eq. (5.30) was fit to the data with a decay constantb−1 = 17.6 ± 0.3μs. (Inset) Decay of the average QP number plotted on semi–log axes. Thelinearity confirms that the dominant QP relaxation mechanism was trapping.

hardware allowing for low–error fits. The measured Ramsey fringes as a function of SFQ drive

time are displayed in Figure 5-26(b). As the SFQ drive time increased, the max visibility of

the fringes decreased and a slight broadening of the oscillations could be seen in addition

to slight frequency shifts. Figure 5-26(c) displays the fitted Ramsey fringe frequencies as a

function of SFQ drive time. After an initial turn–on period of approximately 50μs, the qubit

frequency and hence the detuning shifts down linearly with the SFQ drive time.

In the limit where qubit relaxation is solely dominated by non–equilibrium QPs, the latter

term in Eq. (5.31) becomes negligible and the relaxation rate reduces to

Γ(t) → cnqp(t) . (5.34)

145

0 100 200 300 400 500 600 7000

0.51.01.52.02.53.03.54.04.5

Del

ay T

ime

(us)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

SFQ(c)

0 100 200 300 400 500 600 7004.9454.9504.9554.9604.9654.9704.9754.9804.9854.9904.995

Ram

sey

Det

unin

g (M

Hz)

(a)

(b)

SFQ Drive Time (ns) SFQ Drive Time (ns)

Figure 5-26: (a) Experimental protocol to examine induced qubit frequency shifts fromnon–equilibrium QPs. Prior to a detuned Ramsey fringe experiment, a variable length SFQpoisoning pulse was run. (b) 5 MHz qubit Ramsey fringes as a function of off–resonant SFQdrive time. A decrease in the max visibility along with a slight broadening and expansion ofthe fringes are commensurate with longer SFQ drives. (c) Fitted detuning frequency as afunction of SFQ drive time. After approximately a 50 μs delay, a linear shift in the qubitfrequency can be seen as a function of SFQ drive time.

The prefactor, determined from Fermi’s golden rule and the complex QP impedance, is given

by

c = ωq

π

√√√√ 2Δ�ωq

(5.35)

Additionally, the frequency shift of the qubit δωq is calculated to be (see [91, 92]) directly

proportional to the QP density as well with form

δωq = −nqpωq

2

⎡⎣ 1

π

√√√√ 2Δ�ωq

+ 1⎤⎦ . (5.36)

Combined, Eq. (5.34) and Eq. (5.36) provide a fundamental, materials constant only relation

146

δωq

Γ = −12

⎡⎣1 + π

√�ωq

2Δ

⎤⎦ ≡ m (5.37)

Figures 5-27(a–b) display two different experimental protocols for testing the relations

in Eq. (5.37). In 5-27(a), an off–resonant (ωSFQ = ωq/3 − δ = 1.6 GHz) SFQ drive is

applied for a variable amount of time followed immediately by either a qubit dephasing

or relaxation experiment. For 5-27(b), the length of the off–resonant SFQ drive is held

constant followed by a variable recovery period before performing either the dephasing or

relaxation experiment. The results of the variable poisoning and recovery time experiments

are displayed in Figures 5-27(c) and 5-27(d), respectively. For the variable poisoning length, a

linear relationship between the relaxation rate Γ and number of SFQ phase slips was measured

with a commensurate linear decrease in the qubit frequency where the explicit detuning

for the T ∗2 measurement was ∼ 5 MHz. Data in Figure 5-27(d) displays an exponential

decrease in the QP induced relaxation as a function of recovery time. This is to be expected

as the contribution from the leading term in Eq. (5.33) is exponential in 〈nqp〉 which was

measured to decrease exponentially as a function of time (see Figure 5-25). The return

of the Ramsey frequency to the purposefully detuned 7.5 MHz can be seen to coincide as

well with the decrease in 〈nqp〉. Figures 5-27(e) and 5-27(f) plot the extracted Ramsey

detuning against the extracted relaxation rates for the poisoning and recovery experiments,

respectively. For both experiments, the predicted linear relationship between dωqb and Γ

can be seen albeit with different slopes. The fitted slope for the poisoning experiment was

mPoison = −5.4 while the fitted slope for the recovery experiment was mRecovery = −1.313 .

When substituted back into Eq. (5.37), neither of these values produce a reasonable estimate

to the Al gap energy of Δ ∼ 160μeV. The variable length poisoning experiment predicts a

gap energy of ΔPoison = 1μeV while the recovery experiment predicts a closer yet still low

ΔRecovery = 37μeV.

Exactly why the two rates differ from each other is currently not yet known. The data

however is not an outlier as experiments done at Yale also looking at QP poisoning also saw a

147

4.94

4.95

4.96

4.97

4.98

4.99

5.00

Ram

sey

Det

unin

g (M

Hz)

50

60

70

80

90

100

110

Relaxation R

ate (kHz)

0 50 100 150

Recovery Time (us)

7.35

7.37

7.39

7.41

7.43

7.45

7.47

7.49

7.51

7.537.53

50

100

150

200

250

300

350

400

450

500

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Number of Phase Slips

Relaxation R

ate (kHz)

Ram

sey

Det

unin

g (M

Hz)

0 50 100 150 200 250 300 350 400 450-800

-700

-600

-500

-400

-300

-200

-100

0

0

50

100

150

0 5 10 15 20 25 30 35 40 45 50 55-350

-300

-250

-200

-150

-100

-50

0

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Recovery Tim

e (us)

Num

ber of Phase Slips

Relaxation Rate (kHz)

Qub

it Sh

ift(r

ad. m

s-1)

Qub

it Sh

ift(r

ad. m

s-1)

Relaxation Rate (kHz)

poisonrecovery

poison

(a) (b)

(d)(c)

(f)(e)

Figure 5-27: (a) QP Poisoning experiment protocol to study the relationship between qubitstate relaxation Γ and frequency shift δωq. (b) Complimentary experiment to again studythe relationship between Γ and δωq by measuring the relaxation properties of each after afixed length QP poisoning pulse. (c) QP poisoning / shifting of the decay rate Γ and qubitfrequency. A linear increase (decrease) was measured in the qubit relaxation (detuning). (d)QP relaxation data. The qubit relaxation rate was seen to decrease exponentially in timein line with the exponential decay in average QP number measured earlier. (e) MeasuredQP induced qubit frequency shift δωq Vs. Γ for the poison experiment. The fitted slope wasmPoison = −5.4 ± 0.09. (f) Measured QP induced qubit frequency shift δωq vs. Γ for therecovery experiment. The fitted slope was mRecovery = −1.313 ± 0.006.

148

linear trend between δωq and Γ albeit with a slope corresponding to too large of a gap energy.

It is worth noting that plotting the absolute qubit frequency change δfq = δωq/2π against the

change in Γ for the poisoning experiment predicts a value of m = −0.891 corresponding to

an Al gap energy Δ = 165μeV; This value is much more in line with direct tunnel–junction

gap measurements [93].

5.5 Next Generation Design and Outlook

With the demonstration of coherent qubit control utilizing SFQ pulses now having been

demonstrated and the limits to the fidelity of that control being thoroughly investigated, the

hybrid SFQ–qubit experiment has been warmed up and put out to pasture in it’s current

incantation. We describe here briefly plans currently in action towards realizing the next

generation of SFQ–qubit control.

5.5.1 Flip Chip Design

As previously mentioned, one of if not the main limitation to the fidelity of the initial SFQ

control scheme was the on–chip generation of QPs from the Cooper pair breaking inherent to

the operation of the dc/SFQ converter circuit. In order to combat this limitation, the next

generation design will acoustically decouple the dc/SFQ driver from the quantum circuit via

a multi–chip module (MCM) flip–chip architecture. Recent work [94] has shown that QP

poisoning over large (several cm) distances can be mediated by long–range phonons produced

by QP recombination at one physical locale breaking Cooper pairs at another remote location.

The introduction of an acoustic mismatch / decoupling between the dc/SFQ driver and the

quantum circuit will serve to mediate if not wholly eliminate QP poisoning of the qubit.

Figure 5-28(a) displays the circuit layout design of the proposed and currently in fabrication

flip–chip experiment. An 8×8 mm2 control chip (green) will house all of the microwave

readout, qubit control, and SFQ electronics. Enough lines are afforded to have two redundant

149

experiments per MCM. Indium bump bonds (blue squares) provide galvanic connection

between the two ground planes of the control and “quantum” chip (orange). Beyond

connecting the ground planes of the two chips, no galvanic signal transmission is currently

planned. Figure 5-28(b) displays a close–up of the microwave feed line on the control chip and

the shorted inductive coupler of one of the quarter-wave CPW resonators on the quantum

chip. Mutual coupling M across the flip–chip gap can be tuned by simply adjusting the length

of the coupler. Figure 5-28(c) displays a zoomed–in view of the resonator–qubit coupling plus

the output of the dc/SFQ driver and flux bias line. Again, capacitive or inductive coupling

across the air gap between the chips will facilitate signal transfer. Figure 5-28(d) shows a

close–up view of the Hypres designed dc/SFQ driver. A single bias current rail Ib supplies

the necessary current to both the dc/SFQ driver and the output JTLs. The input microwave

current RFSFQ drives phase slips in the dc/SFQ converter which are then multiplexed into two

separate outputs by the pulse splitter (PS) circuit. The output SFQ pulses are relegated to

either qubit by switch currents IS1 and IS2 which can be turned on by a fast (∼ 1 ns) current

supply. The SFQ pulses are then sent through a multi–stage JTL feeding a transmission line

to the qubit.

An interesting mode operation afforded by driving the qubit with SFQ pulses is over

clocking of the microwave drive tone to harmonics of the qubit frequency ωSFQ = 4 × ωq.

Without any form of gating, in this mode of operation pulses from a dc/SFQ driver would

arrive when the qubit vector was aligned with the ±x and ±y axes of the Bloch sphere. This

by itself would produce a zero–sum rotation as two of the discrete rotations would be in

phase with the other two out. However, replacing the simple dc/SFQ driver instead with

a bit–shift register slave to a clock frequency ωCLK = 4 × ωq where subsequent SFQ bit

pairs were alternating high or low, qubit gate times could be halved as compared to their

on–resonance counterparts.

150

(b)

(c)

(d)

(a)

(b) (c)

(d)

8 mm5 mm

Figure 5-28: (a) Layout schematic showing the bump bonded flip chip architecture for theSFQ–qubit project. An 8×8 mm2 carrier chip (green) houses the microwave and SFQ controlhardware for two redundant experiments. Flip–chip bonded to it via In bump bonds (blue)will be a 5×5 mm2 “quantum” chip (orange) housing both the qubits and readout resonators.(b) Close–up of the proposed inductive coupling scheme between the common feed line andthe resonators. No galvanic connections to facilitate signal flow is planned. Instead, allcoupling between the two chips with either be capacitive or inductive. (c) Close–up ofthe readout resonator–qubit pair plus the output coupling capacitor of the dc/SFQ circuit(bottom, green) and the flux bias line (right, green). (d) Zoom in of the dc/SFQ drivercircuit.

151

Appendix A

Josephson Energy of a dc SQUID

In section 2.2, the Hamiltonian for a single unbiased Josepshon junction was derived and in

section 2.3.3 it was stated how the transmon qubit was nothing more than a capacitively

shunted junction, making the trade off between charge noise sensitivity an anharmonicity. If,

however, the transmon qubit capacitor is shunted by two junctions forming a dc SQUID loop,

the energy of the SQUID loop must be taken into account. Looking at just the potential

terms for the junctions, the Hamiltonian for a general dc SQUID is

H = −E1 cos(δ1) − E2 cos(δ2) , (A.1)

where the subscripts refer to either junction 1 or 2 of the dc SQUID. As a consequence of flux

quantization, the phase difference between the two junctions in constrained by the magnetic

flux quantum

δ1 − δ2 = 2nπ + 2πΦΦ0

, (A.2)

where Φ is an externally applied flux. We define the total Josephson energy EΣ as the sum

of the individual junction energies

152

EΣ = E1 + E2 (A.3)

Substituting (A.2) and (A.3) into the second term of (A.1) and collecting common terms

gives

H = −EΣ cos(δ2) − E1(cos(δ1) − cos(δ2)) (A.4)

= −EΣ cos(δ2) − E1

[−2 sin

(δ1 + δ2

2

)sin

(δ1 − δ2

2

)](A.5)

= −EΣ cos(δ2) + 2E1 sin(δ) sin(

nπ + πΦΦ0

)(A.6)

= −EΣ cos(δ2) + 2E1 sin(δ)[sin(nπ) cos

(πΦΦ0

)+ cos(nπ) sin

(πΦΦ0

)](A.7)

= −EΣ cos(δ2) + 2E1 sin(δ) sin(

πΦΦ0

), (A.8)

where we have defined δ ≡ (δ1 + δ2)/2. This relation, when combined with (A.2) allows for

expansion of the first term in (A.1)

H = −EΣ cos(

nπ − πΦΦ0

− δ

)+ 2E1 sin(δ) sin

(πΦΦ0

)(A.9)

= −EΣ cos(δ) cos(

nΦΦ0

)− EΣ sin(δ) sin

(πΦΦ0

)+ 2E1 sin(δ) sin

(πΦΦ0

)(A.10)

= −EΣ

[cos(δ) cos

(nΦΦ0

)+ E1 − E2

E1 + E2sin(δ) sin

(πΦ0

Φ

)](A.11)

= −EΣ

[cos(δ) cos

(nΦΦ0

)+ α − 1

α + 1 sin(δ) sin(

πΦ0

Φ

)](A.12)

= −EΣ

[cos(δ) cos

(nΦΦ0

)+ d sin(δ) sin

(πΦ0

Φ

)](A.13)

= −EΣ cos(

nΦΦ0

)[cos(δ) + d sin(δ) tan

(πΦ0

Φ

)], (A.14)

where we have defined the ratio between the two Josephson energies α ≡ E1/E2 for E1 > E2

and d = (α − 1)/(α + 1). The final step is to note that the bracketed terms in (A.14) are

153

a generalized trigonometric identity relating sin and cos, which, when exploited, yields the

final form for the energy a general flux biased dc SQUID to be

H = −EΣ cos(

πΦΦ0

)√√√√1 + d2 tan2

(πΦΦ0

). (A.15)

The square root term prevents the energy from diverging as the external flux takes on values

approaching Φ0. In the limit of symmetric junctions E1 = E2 = EJ , d = 0, EΣ = 2EJ and

(A.15) takes on the more familiar form

H = −2EJ cos(

πΦΦ0

). (A.16)

154

Appendix B

Preserving High Quality Nb

In order to preserve and protect the 180 nm Nb ground plane (layer M0) from 7 additional

layers of processing as described in Chapter 5.3.3, sacrificial protection layers were employed

to shield the base layer from the subsequent processing. Initially, an 8 nm cap of Al was

employed because it could be sputtered sequentially with the Nb base layer without breaking

vacuum and Al has a low reactivity with the different plasma etch processes used in the

processing of subsequent layers. Until the fabrication of the joint SFQ–qubit circuit, the

practice of employing Al as a sacrificial cap layer had been used to great success in the

fabrication of other multi–layer circuits such as the SLUG microwave amplifier. However,

during initial attempts to fabricate the SFQ–qubit circuit, the final base layer etch (“Qubit

and Resonator Patterning”, Ch.5.3.3) left behind a metallic scum in the pockets where Nb

should have been fully etched away.

Figure B-1 displays a dark field false colored optical microscopy image of the readout

resonator–qubit coupling area. The resonator, qubit, and flux bias line are colored yellow,

red, and purple, respectively with the surrounding ground plane in blue. The etched pockets

between all these features should have been fully devoid of metal leaving only the surface of

the Si substrate behind. However, as is evident in Figure B-1, a silvery heterogeneous residue

was left behind after the final Nb-Al etch.

155

Binding Energy (eV)020040060080010001200

Co

un

ts /

s

x 104

0

2

4

6

8

10

12

Binding Energy (eV)050100150200

Co

un

ts /

s

0

2000

4000

6000

8000

10000

12000

14000O1s

Nb3pNb3d

C1s

Si2s

Al2s

Si2p

Al2p

Nb4p

(b) (c)

(a)

Figure B-1: (a) Dark field false colored optical microscopy image of the resonator (yellow)and qubit (red) coupling area. Ground plane has been color blue. The black areas are wherethe base Nb+Al protection layer should have been fully removed. A thin metallic scumhad been left on the wafer after processing that inhibited the quality factor of the readoutresonators. (b) X–ray surface spectrogram of a fully processed SFQ–qubit sample utilizing anAl protection layer. (c) Zoom in on the 0–200 eV binding energy region in (b) (red box). Al2p peaks were observed on the surface of the sample despite a full Al etch prior to patterningM0.

156

Circuits with this residue had high–power resonator internal quality factors below 1× 103

preventing efficient readout of the coupled qubits. In order to determine what the residual

scum was comprised off, circuits were placed inside an x–ray photoelectron spectroscopy

(XPS) tool and the surface chemistry was probed with an 11 keV Al source x–ray beam. The

resulting x–ray surface spectrogram is displayed in Figures B-1(b–c). Figure B-1(b) displays

a wide energy scan of the M0 surface. The oxygen and carbon signatures are from handling

and atmosphere exposure. Between 0–200 eV binding energy (red box, Figure B-1(b), B-1(c)),

residual traces of Al are seen to remain on the sample despite a chemical wet etch of the

protection layer. The Al was either being etched and subsequently redeposited or resisting

the etch and subsequently acting as a non–reactive barrier for the final M0 plasma etch.

This problem was solved by removing the Al protection layer altogether and simply not

etching the area of layer V1 covering the resonator–qubit pocket until the junction processing

was complete. Figure B-2(a) shows another dark field false colored optical microscopy image

off a sample processed without the Al protection layer. The corresponding x–ray spectrogram

of a fully completed sample is displayed in Figure B-2(b). Resonators processed with this

stack showed low–power, single photon level internal quality factors above 105.

157

(a)

(b)

Figure B-2: (a) Dark field false colored optical microscopy image of the resonator (yellow)and qubit (red) coupling area processed without an Al protection layer. Ground plane hasbeen color blue. No more silvery metallic scum can be seen on the surface of the substrate oretched metal layers. (b) X–ray surface spectrogram of a fully processed SFQ–qubit sampleutilizing without an Al protection layer.

158

References

[1] John Von Neumann. Mathematical Foundations of Quantum Mechanics. PrincetonUniversity Press, 1955.

[2] J. W. Gibbs. Elementary Principles in Statistical Mechanics. University Press, March1902.

[3] Barbara M. Terhal and Paweł Horodecki. Schmidt number for density matrices. Phys.Rev. A, 61:040301, Mar 2000.

[4] C. E. Shannon. A mathematical theory of communication. The Bell System TechnicalJournal, 27(3):379–423, July 1948.

[5] F. Bloch. Nuclear Induction. Physical Review, 70(7-8):460–474, 10 1946.

[6] M. Nielsen and I. Chuang. Quantum Computation and Quantum Information. CambridgeUniversity Press, 10th anniversary edition edition, 2010.

[7] R. J. Schoelkopf, A. A. Clerk, S. M. Girvin, K. W. Lehnert, and M. H. Devoret. Qubitsas Spectrometers of Quantum Noise, pages 175–203. Springer Netherlands, Dordrecht,2003.

[8] D. P. DiVincenzo. The Physical Implementation of Quantum Computation. Fortschritteder Physik, 48(9-11):771–783, 2000.

[9] Deutsch Ivan H., Brennen Gavin K., and Jessen Poul S. Quantum Computing withNeutral Atoms in an Optical Lattice. Fortschritte der Physik, 48(9⣳11):925–943,2018/03/29 2000.

[10] David S. Weiss and Mark Saffman. Quantum computing with neutral atoms. PhysicsToday, 70(7):44–50, 2018/03/29 2017.

[11] Yang Wang, Xianli Zhang, Theodore A. Corcovilos, Aishwarya Kumar, and David S.Weiss. Coherent Addressing of Individual Neutral Atoms in a 3D Optical Lattice.Physical Review Letters, 115(4):043003–, 07 2015.

[12] Katharina Gillen-Christandl, Glen D. Gillen, M. J. Piotrowicz, and M. Saffman. Compar-ison of Gaussian and super Gaussian laser beams for addressing atomic qubits. AppliedPhysics B, 122(5):131, 2016.

159

[13] D. Jaksch, H. J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller. Entanglement ofAtoms via Cold Controlled Collisions. Physical Review Letters, 82(9):1975–1978, 031999.

[14] Guan-Yu Wang, Qian Liu, Hai-Rui Wei, Tao Li, Qing Ai, and Fu-Guo Deng. Universalquantum gates for photon-atom hybrid systems assisted by bad cavities. ScientificReports, 6:24183 EP –, 04 2016.

[15] Roberto Stassi, Vincenzo Macrì, Anton Frisk Kockum, Omar Di Stefano, Adam Mira-nowicz, Salvatore Savasta, and Franco Nori. Quantum nonlinear optics without photons.Physical Review A, 96(2):023818–, 08 2017.

[16] E Brion, L H Pedersen, and K Mølmer. Implementing a neutral atom Rydberg gatewithout populating the Rydberg state. Journal of Physics B: Atomic, Molecular andOptical Physics, 40(9):S159, 2007.

[17] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin. Fast QuantumGates for Neutral Atoms. Physical Review Letters, 85(10):2208–2211, 09 2000.

[18] E. Hagley, X. Maître, G. Nogues, C. Wunderlich, M. Brune, J. M. Raimond, andS. Haroche. Generation of Einstein-Podolsky-Rosen Pairs of Atoms. Physical ReviewLetters, 79(1):1–5, 07 1997.

[19] T. Xia, M. Lichtman, K. Maller, A. W. Carr, M. J. Piotrowicz, L. Isenhower, andM. Saffman. Randomized Benchmarking of Single-Qubit Gates in a 2D Array of Neutral-Atom Qubits. Physical Review Letters, 114(10):100503–, 03 2015.

[20] K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower,and M. Saffman. Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits. Physical Review A, 92(2):022336–, 08 2015.

[21] M. P. A. Jones, J. Beugnon, A. Gaëtan, J. Zhang, G. Messin, A. Browaeys, andP. Grangier. Fast quantum state control of a single trapped neutral atom. PhysicalReview A, 75(4):040301–, 04 2007.

[22] Bryan T. Gard, Kurt Jacobs, R. McDermott, and M. Saffman. Microwave-to-opticalfrequency conversion using a cesium atom coupled to a superconducting resonator.Physical Review A, 96(1):013833–, 07 2017.

[23] R. Vijay, C. Macklin, D. H. Slichter, S. J. Weber, K. W. Murch, R. Naik, A. N. Korotkov,and I. Siddiqi. Stabilizing Rabi oscillations in a superconducting qubit using quantumfeedback. Nature, 490:77 EP –, 10 2012.

[24] J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Theory of Superconductivity. PhysicalReview, 108(5):1175–1204, 12 1957.

[25] B. D. Josephson. Possible new effects in superconductive tunnelling. Physics Letters,1(7):251–253, 1962.

160

[26] Simon Bernon, Helge Hattermann, Daniel Bothner, Martin Knufinke, Patrizia Weiss,Florian Jessen, Daniel Cano, Matthias Kemmler, Reinhold Kleiner, Dieter Koelle, andJózsef Fortágh. Manipulation and coherence of ultra-cold atoms on a superconductingatom chip. Nature Communications, 4:2380 EP –, 08 2013.

[27] F. Jessen, M. Knufinke, S. C. Bell, P. Vergien, H. Hattermann, P. Weiss, M. Rudolph,M. Reinschmidt, K. Meyer, T. Gaber, D. Cano, A. Günther, S. Bernon, D. Koelle,R. Kleiner, and J. Fortágh. Trapping of ultracold atoms in a 3He/4He dilution refrigerator.Applied Physics B, 116(3):665–671, 2014.

[28] H. Hattermann, D. Bothner, L. Y. Ley, B. Ferdinand, D. Wiedmaier, L. Sárkány,R. Kleiner, D. Koelle, and J. Fortágh. Coupling ultracold atoms to a superconductingcoplanar waveguide resonator. Nature Communications, 8(1):2254, 2017.

[29] Guido Burkard and J. R. Petta. Dispersive readout of valley splittings in cavity-coupledsilicon quantum dots. Physical Review B, 94(19):195305–, 11 2016.

[30] X. Mi, J. V. Cady, D. M. Zajac, P. W. Deelman, and J. R. Petta. Strong coupling of asingle electron in silicon to a microwave photon. Science, 355(6321):156, 01 2017.

[31] X. Mi, J. V. Cady, D. M. Zajac, J. Stehlik, L. F. Edge, and J. R. Petta. Circuit quantumelectrodynamics architecture for gate-defined quantum dots in silicon. Applied PhysicsLetters, 110(4):043502, 2018/03/29 2017.

[32] A. Stockklauser, P. Scarlino, J. V. Koski, S. Gasparinetti, C. K. Andersen, C. Reichl,W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff. Strong Coupling Cavity QED withGate–Defined Double Quantum Dots Enabled by a High Impedance Resonator. PhysicalReview X, 7(1):011030–, 03 2017.

[33] P. Scarlino, D. J. van Woerkom, A. Stockklauser, J. V. Koski, M. C. Collodo, S. Gas-parinetti, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff. All-MicrowaveControl and Dispersive Readout of Gate-Defined Quantum Dot Qubits in Circuit Quan-tum Electrodynamics. ArXiv e-prints, November 2017.

[34] A. J. Landig, J. V. Koski, P. Scarlino, U. C. Mendes, A. Blais, C. Reichl, W. Wegscheider,A. Wallraff, K. Ensslin, and T. Ihn. Coherent spin-qubit photon coupling. ArXiv e-prints,November 2017.

[35] Dirk van Delft and Peter Kes. The discovery of superconductivity. Physics Today,63(9):38–43, 2018/03/29 2010.

[36] K. K. Likharev. Superconducting weak links. Reviews of Modern Physics, 51(1):101–159,01 1979.

[37] K.K. Likharev. Dynamics of Josephson Junctions and Circuits. CRC Press, August1986.

[38] W. C. Stewart. Current–voltage characteristics of Josephson junctions. Applied PhysicsLetters, 12(8):277–280, 2018/03/29 1968.

161

[39] D. E. McCumber. Effect of ac Impedance on dc Voltage⣳-Current Characteristics ofSuperconductor Weak⣳-Link Junctions. Journal of Applied Physics, 39(7):3113–3118,2018/03/29 1968.

[40] Alexandre Blais, Ren-Shou Huang, Andreas Wallraff, S. M. Girvin, and R. J. Schoelkopf.Cavity quantum electrodynamics for superconducting electrical circuits: An architecturefor quantum computation. Physical Review A, 69(6):062320–, 06 2004.

[41] Michel H. Devoret, John M. Martinis, and John Clarke. Measurements of MacroscopicQuantum Tunneling out of the Zero-Voltage State of a Current-Biased JosephsonJunction. Physical Review Letters, 55(18):1908–1911, 10 1985.

[42] John M. Martinis, Michel H. Devoret, and John Clarke. Energy-Level Quantization inthe Zero-Voltage State of a Current-Biased Josephson Junction. Physical Review Letters,55(15):1543–1546, 10 1985.

[43] John M. Martinis, Michel H. Devoret, and John Clarke. Experimental tests for thequantum behavior of a macroscopic degree of freedom: The phase difference across aJosephson junction. Physical Review B, 35(10):4682–4698, 04 1987.

[44] John M. Martinis, K. B. Cooper, R. McDermott, Matthias Steffen, Markus Ansmann,K. D. Osborn, K. Cicak, Seongshik Oh, D. P. Pappas, R. W. Simmonds, and Clare C.Yu. Decoherence in Josephson Qubits from Dielectric Loss. Physical Review Letters,95(21):210503–, 11 2005.

[45] J. E. Mooij, T. P. Orlando, L. Levitov, Lin Tian, Caspar H. van der Wal, and SethLloyd. Josephson Persistent-Current Qubit. Science, 285(5430):1036, 08 1999.

[46] I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij. Coherent QuantumDynamics of a Superconducting Flux Qubit. Science, 299(5614):1869, 03 2003.

[47] A. Lupascu, C. J. M. Verwijs, R. N. Schouten, C. J. P. M. Harmans, and J. E. Mooij.Nondestructive Readout for a Superconducting Flux Qubit. Physical Review Letters,93(17):177006–, 10 2004.

[48] Michael Tinkham. Introduction to Superconductivity. Dover Books on Physics, 2ndedition, June 2004.

[49] Guido Burkard, David P. DiVincenzo, P. Bertet, I. Chiorescu, and J. E. Mooij. Asym-metry and decoherence in a double-layer persistent-current qubit. Physical Review B,71(13):134504–, 04 2005.

[50] Matthias Steffen, Shwetank Kumar, David P. DiVincenzo, J. R. Rozen, George A. Keefe,Mary Beth Rothwell, and Mark B. Ketchen. High-Coherence Hybrid SuperconductingQubit. Physical Review Letters, 105(10):100502–, 09 2010.

[51] Fei Yan, Simon Gustavsson, Archana Kamal, Jeffrey Birenbaum, Adam P Sears, DavidHover, Ted J. Gudmundsen, Danna Rosenberg, Gabriel Samach, S Weber, Jonilyn L.Yoder, Terry P. Orlando, John Clarke, Andrew J. Kerman, and William D. Oliver. The

162

flux qubit revisited to enhance coherence and reproducibility. Nature Communications,7:12964 EP –, 11 2016.

[52] Jens Koch, Terri M. Yu, Jay Gambetta, A. A. Houck, D. I. Schuster, J. Majer, AlexandreBlais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. Charge-insensitive qubit designderived from the Cooper pair box. Physical Review A, 76(4):042319–, 10 2007.

[53] Chad Rigetti, Jay M. Gambetta, Stefano Poletto, B. L. T. Plourde, Jerry M. Chow,A. D. Córcoles, John A. Smolin, Seth T. Merkel, J. R. Rozen, George A. Keefe, Mary B.Rothwell, Mark B. Ketchen, and M. Steffen. Superconducting qubit in a waveguidecavity with a coherence time approaching 0.1 ms. Physical Review B, 86(10):100506–, 092012.

[54] Jerry M. Chow, Jay M. Gambetta, Easwar Magesan, David W. Abraham, Andrew W.Cross, B R Johnson, Nicholas A. Masluk, Colm A. Ryan, John A. Smolin, Srikanth J.Srinivasan, and M Steffen. Implementing a strand of a scalable fault-tolerant quantumcomputing fabric. Nature Communications, 5:4015 EP –, 06 2014.

[55] A. D. Córcoles, Easwar Magesan, Srikanth J. Srinivasan, Andrew W. Cross, M. Steffen,Jay M. Gambetta, and Jerry M. Chow. Demonstration of a quantum error detectioncode using a square lattice of four superconducting qubits. Nature Communications,6:6979 EP –, 04 2015.

[56] Serge Haroche and Jean-Michel Raimond. Cavity Quantum Electrodynamics. 268(4):54–62, 1993.

[57] D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland. Generation ofNonclassical Motional States of a Trapped Atom. Physical Review Letters, 76(11):1796–1799, 03 1996.

[58] C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland. A “Schrödinger Cat”Superposition State of an Atom. Science, 272(5265):1131, 05 1996.

[59] Sal J. Bosman, Mario F. Gely, Vibhor Singh, Alessandro Bruno, Daniel Bothner, andGary A. Steele. Multi-mode ultra-strong coupling in circuit quantum electrodynamics.npj Quantum Information, 3(1):46, 2017.

[60] F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba. Characteristicspectra of circuit quantum electrodynamics systems from the ultrastrong- to the deep-strong-coupling regime. Physical Review A, 95(5):053824–, 05 2017.

[61] David M Pozar. Microwave engineering. Fourth edition. Hoboken, NJ : Wiley, ©2012.

[62] Benjamin A. Mazin. Microwave Kinetic Inductance Detectors. PhD thesis, CaliforniaInstitute of Technology, 2005.

[63] M. D. Reed, L. DiCarlo, B. R. Johnson, L. Sun, D. I. Schuster, L. Frunzio, and R. J.Schoelkopf. High-Fidelity Readout in Circuit Quantum Electrodynamics Using theJaynes-Cummings Nonlinearity. Physical Review Letters, 105(17):173601–, 10 2010.

163

[64] J. D. Pritchard, J. A. Isaacs, M. A. Beck, R. McDermott, and M. Saffman. Hybridatom-photon quantum gate in a superconducting microwave resonator. Physical ReviewA, 89(1):010301–, 01 2014.

[65] Jiansong Gao. The Physics of Superconducting Microwave Resonators. PhD thesis,California Institute of Technology, May 2008.

[66] R. G. Chambers. The anomalous skin effect. Proceedings of the Royal Society of London.Series A, Mathematical and Physical Sciences, 215(1123):481–497, Dec 1952.

[67] D. C. Mattis and J. Bardeen. Theory of the anomalous skin effect in normal andsuperconducting metals. Physical Review, 111(2):412–417, 07 1958.

[68] John R. Clem. Inductances and attenuation constant for a thin-film superconductingcoplanar waveguide resonator. Journal of Applied Physics, 113(1):013910, 2018/04/302013.

[69] M. A. Beck, J. A. Isaacs, D. Booth, J. D. Pritchard, M. Saffman, and R. McDermott.Optimized coplanar waveguide resonators for a superconductor–atom interface. AppliedPhysics Letters, 109(9):092602, 2018/04/30 2016.

[70] T. Thiele, S. Filipp, J. A. Agner, H. Schmutz, J. Deiglmayr, M. Stammeier, P. All-mendinger, F. Merkt, and A. Wallraff. Manipulating rydberg atoms close to surfaces atcryogenic temperatures. Physical Review A, 90(1):013414–, 07 2014.

[71] John Francis Bulzacchelli. A superconducting bandpass delta-sigma modulator for directanalog-to-digitaal conversion of microwave radio. PhD thesis, Massachusetts Institute ofTechnology, 2003.

[72] K. K. Likharev, O. A. Mukhanov, and V. K. Semenov. Resistive single flux quantumlogic for the Josephson-junction digital technology. de Gruyter, Germany, 1985.

[73] O. A. Mukhanov. Energy-efficient single flux quantum technology. IEEE Transactionson Applied Superconductivity, 21(3):760–769, 2011.

[74] Quentin P. Herr, Anna Y. Herr, Oliver T. Oberg, and Alexander G. Ioannidis. Ultra-low-power superconductor logic. Journal of Applied Physics, 109(10):103903, 2018/04/302011.

[75] R. McDermott and M. G. Vavilov. Accurate Qubit Control with Single Flux QuantumPulses. Physical Review Applied, 2(1):014007–, 07 2014.

[76] J. R. Johansson, P. D. Nation, and Franco Nori. QuTiP 2: A Python framework for thedynamics of open quantum systems. Computer Physics Communications, 184(4):1234–1240, 2013.

[77] K. K. Likharev. The properties of a weakly coupled superconducting ring as an elementwith several stable states. Radiotekhnika i Elektronika, 19:1494–1502, July 1974.

164

[78] Sidney Shapiro, Andre R. Janus, and Sandor Holly. Effect of Microwaves on JosephsonCurrents in Superconducting Tunneling. Reviews of Modern Physics, 36(1):223–225, 011964.

[79] S. R. Whiteley. Josephson junctions in SPICE3. IEEE Transactions on Magnetics,27(2):2902–2905, 1991.

[80] V. K. Kaplunenko, M. I. Khabipov, V. P. Koshelets, K. K. Likharev, O. A. Mukhanov,V. K. Semenov, I. L. Serpuchenko, and A. N. Vystavkin. Experimental study of theRSFQ logic elements. IEEE Transactions on Magnetics, 25(2):861–864, 1989.

[81] K. K. Likharev and V. K. Semenov. RSFQ logic/memory family: a new Josephson-junction technology for sub-terahertz-clock-frequency digital systems. IEEE Transactionson Applied Superconductivity, 1(1):3–28, 1991.

[82] A. Rylyakov. DC to SFQ Converter. http://www.physics.sunysb.edu/ Physics/RS-FQ/Lib/AR/dcsfq.html.

[83] John M. Martinis, K. B. Cooper, R. McDermott, Matthias Steffen, Markus Ansmann,K. D. Osborn, K. Cicak, Seongshik Oh, D. P. Pappas, R. W. Simmonds, and Clare C.Yu. Decoherence in Josephson Qubits from Dielectric Loss. Physical Review Letters,95(21):210503–, 11 2005.

[84] Kenichi Kuroda and Masahiro Yuda. Niobium-stress influence on Nb/Al-oxide/NbJosephson junctions. Journal of Applied Physics, 63(7):2352–2357, 2018/04/11 1988.

[85] Vinay Ambegaokar and Alexis Baratoff. Tunneling Between Superconductors. PhysicalReview Letters, 10(11):486–489, 06 1963.

[86] G. J. Dolan. Offset masks for lift-off photoprocessing. Applied Physics Letters, 31(5):337–339, 2018/04/11 1977.

[87] M. Lenander, H. Wang, Radoslaw C. Bialczak, Erik Lucero, Matteo Mariantoni, M. Nee-ley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao,A. N. Cleland, and John M. Martinis. Measurement of energy decay in superconductingqubits from nonequilibrium quasiparticles. Physical Review B, 84(2):024501–, 07 2011.

[88] C. Wang, Y. Y. Gao, I. M. Pop, U. Vool, C. Axline, T. Brecht, R. W. Heeres, L. Frunzio,M. H. Devoret, G. Catelani, L. I. Glazman, and R. J. Schoelkopf. Measurement andcontrol of quasiparticle dynamics in a superconducting qubit. Nature Communications,5:5836 EP –, 12 2014.

[89] Simon Gustavsson, Fei Yan, Gianluigi Catelani, Jonas Bylander, Archana Kamal,Jeffrey Birenbaum, David Hover, Danna Rosenberg, Gabriel Samach, Adam P. Sears,Steven J. Weber, Jonilyn L. Yoder, John Clarke, Andrew J. Kerman, Fumiki Yoshihara,Yasunobu Nakamura, Terry P. Orlando, and William D. Oliver. Suppressing relaxationin superconducting qubits by quasiparticle pumping. Science, 354(6319):1573–1577,2016.

165

[90] G. Catelani, J. Koch, L. Frunzio, R. J. Schoelkopf, M. H. Devoret, and L. I. Glazman.Quasiparticle relaxation of superconducting qubits in the presence of flux. PhysicalReview Letters, 106(7):077002–, 02 2011.

[91] G. Catelani, R. J. Schoelkopf, M. H. Devoret, and L. I. Glazman. Relaxation andfrequency shifts induced by quasiparticles in superconducting qubits. Physical Review B,84(6):064517–, 08 2011.

[92] John M. Martinis, M. Ansmann, and J. Aumentado. Energy decay in superconductingjosephson-junction qubits from nonequilibrium quasiparticle excitations. Physical ReviewLetters, 103(9):097002–, 08 2009.

[93] D. H. Douglass and R. Meservey. Energy gap measurements by tunneling betweensuperconducting films. i. temperature dependence. Physical Review, 135(1A):A19–A23,07 1964.

[94] U. Patel, Ivan V. Pechenezhskiy, B. L. T. Plourde, M. G. Vavilov, and R. McDermott.Phonon-mediated quasiparticle poisoning of superconducting microwave resonators.Physical Review B, 96(22):220501–, 12 2017.