APPLICATIONS OF SUPERINDUCTORS IN

SUPERCONDUCTING QUANTUM CIRCUITS

by

WENYUAN ZHANG

A dissertation submitted to the

School of Graduate Studies

Rutgers, The State University of New Jersey

In partial fulllment of the requirements

For the degree of

Doctor of Philosophy

Graduate Program in Physics and Astronomy

Written under the direction of

Michael Gershenson

And approved by

New Brunswick, New Jersey

May, 2019

ABSTRACT OF THE DISSERTATION

Applications of Superinductors in Superconducting Quantum

Circuits

By WENYUAN ZHANG

Dissertation Director:

Michael Gershenson

Superinductors are inductors whose microwave characteristic impedances are greater than

the resistance quantum, RQ = h/(2e)2 ≈ 6.5kΩ. They can be implemented using Josephson

junction chains and high kinetic inductance nanowires. In this dissertation, we explore ap-

plications of superinductors in both implementations in superconducting quantum circuits.

The dissertation consists of three parts. In the rst part, we discuss the uxon-parity-

protected qubit consisting of a Cooper-pair box (CPB) shunted by a superinductor made of a

chain of coupled asymmetric Superconducting Quantum Interference Devices (CASQUIDs).

The spectroscopic measurement of a prototype of the uxon-parity-protected qubit was per-

formed. We observed almost complete suppression of the single uxon tunneling across the

CPB due to the destructive Aharonov-Casher interference when the oset charge on the

CPB island was set to e mod(2e). A uxon-parity-protected qubit with a higher superin-

ductance can potentially be used to perform fault-tolerant Cliord gates. In the second part,

we studied the microwave losses in high-kinetic-inductance granular Aluminum lms using

superconducting coplanar-waveguide (CPW) resonators made of the lms. We observed that

the intrinsic losses in these resonators at low temperatures were limited by resonator cou-

pling to the two-level systems (TLS) in the environment. The demonstrated internal quality

ii

factors are comparable with those for CPW resonators made of conventional superconduc-

tors. The characterized granular Aluminum lms can be used to fabricate superinductors

for a wide range of applications in quantum metrology and quantum information processing.

In the third part, we discuss the one-dimensional Josephson metamaterial made of a similar

structure as the superinductor used in the uxon-parity-protected qubit. The metamaterial

demonstrated strong Kerr nonlinearity with the Kerr constant tunable over a wide range

from positive to negative values by the magnetic eld. The metamaterial is promising for use

as an active medium for quantum-limited Josephson traveling-wave parametric ampliers.

iii

Acknowledgments

I would like to thank my advisor Michael Gershenson for his mentorship throughout this

work. I cannot express my gratitude enough for his motivation and support. I would also

like to thank Professor Lev Ioe, whose theoretical proposal of the uxon-parity protected

qubit is the motivation of this experimental work.

Dr. Matthew Bell was the rst to introduce me to fabrication and measurement in the

lab. Though he moved to UMass Boston to start his tenure-track position a year after I joined

the lab, his guidance and support continued. He has oered great help in the fabrication of

the uxon-parity protected qubit discussed in Chapter 4. Later on, we collaborated on the

work discussed in Chapter 6 about the nonlinear Kerr eect in a metamaterial transmission

line.

Wen-Sen had worked in the semiconductor industry before coming to Rutgers for his

PhD. He has helped me better understand and appreciate the art of sample fabrication. He

is also full of cool ideas of DIY lab gadgets, which made the fabrication process much easier

for us. I can't be more grateful to have a lab mate like him.

Dr. Konstantin Kalashnikov joined the lab as a postdoc when I was about to nish my

graduate study. I have beneted from discussions with him as he brought up questions to

concepts I took for granted and helped further my understanding of the work I have done.

We also had a productive collaboration in analyzing the data about the low temperature

losses in the lms of granular Aluminum discussed in Chapter 5.

I would like to thank my committee members, Professors Karin Rabe, Weida Wu,

Amitabh Lath and Vladimir Manucharyan, for their valuable time and suggestions which

helped me in completion of this work.

I would like to thank the graduate students Plamen Kamenov and Tom DiNapoli for

their help with fabrications of the samples discussed in Chapter 5. I would also like to

iv

thank Hypres, Inc. for fabrication of the samples discussed in Chapter 6.

I thank my friends for their support and encouragement. Zhuohui, Rene, Lin-Ing, Wen-

Qing, Qin Xiao and many others.

Last, I thank my family especially my maternal grandma. She had generously supported

my education nancially as I grew up. Without her support, this journey would not be

possible. She is also an inspiration to me as an accomplished hematologist who has saved

thousands of lives with her knowledge and wisdom.

v

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

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

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

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. Superconducting qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2. Superinductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3. A chain of coupled asymmetric SQUIDs (CASQUIDs) . . . . . . . . . . . . . 6

1.4. Fluxon-parity-protected qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5. Josephson traveling-wave parametric amplier based on CASQUIDs . . . . . 9

2. Fluxon-parity-protected qubit: theoretical background . . . . . . . . . . 13

2.1. Single-qubit state and coherence . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2. Parity-protected qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3. Fluxon-parity-protected qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4. Inductive coupling of uxon-parity-protected qubit and readout resonator . . 21

3. Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1. Sample fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1. Josephson junction fabrication technique used for uxon-parity-protected

qubit fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.2. Deposition of disordered granular Aluminum lms . . . . . . . . . . . 27

vi

3.2. Measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3. Sample holders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4. Fluxon-parity-protected qubit : a prototype device . . . . . . . . . . . . . 32

4.1. Sample design and measurement . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2. First-tone measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3. Two-tone measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5. Microresonators fabricated from high-kinetic-inductance Aluminum lms 39

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2. Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3. Microwave characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3.1. The resonance frequency analysis . . . . . . . . . . . . . . . . . . . . . 42

5.3.2. The quality factor analysis . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3.3. The two-tone and time-domain measurements . . . . . . . . . . . . . . 46

5.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6. Josephson metamaterial with a widely tunable positive or negative Kerr

constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2. Metamaterial Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.3. Microwave Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7. Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.2. Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

vii

Appendix A. Design of hybrid superinductor . . . . . . . . . . . . . . . . . 71

Appendix B. Increase in transmission power through metamaterial tranmis-

sion line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

viii

List of Tables

4.1. Parameters of Josephson junctions in the representative device. . . . . . . . . 34

5.1. Summary of the measured parameters of AlOx resonators. . . . . . . . . . . . 42

5.2. Summary of the tting parameters. . . . . . . . . . . . . . . . . . . . . . . . . 44

6.1. Parameters of two Josephson metamaterial devices. . . . . . . . . . . . . . . . 54

ix

List of Figures

1.1. (a) A Josephson junction consisting of two electrodes (in white and grey)

separated by the tunnel barrier; (b) Schematic representation of a Josephson

junction with zero losses; (c) Schematic of a superconducting qubit. . . . . . 4

1.2. Schematic of the tunable superinductor consisting of two large (blue) and two

small (yellow) Josephson junctions per unit cell. α and α′ correspond to the

superconducting phases across the large junctions, and β1 and β2 correspond

to those across the small junctions. . . . . . . . . . . . . . . . . . . . . . . . 6

2.1. Bloch sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2. Schematic of a uxon-parity-protected qubit. . . . . . . . . . . . . . . . . . . 16

2.3. Schematic of the potential wells and the wave functions of the uxon-parity

protected qubit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4. Circuit diagram of a uxon-parity-protected qubit coupled inductively to a

readout resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5. Schematic of the magnitudes of the transmissions through the microwave

transmission line corresponding to |0〉 (red) and |1〉 (blue) states of the qubit

and the bare LC resonator (black). |0〉 and |1〉 states shift the resonant

frequency dispersively by χ0 and χ1 respectively. . . . . . . . . . . . . . . . . 24

3.1. Schematic representation of Josephson junction fabrication using the Man-

hattan pattern technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2. SEM images for junctions with in-plane dimensions (a) 100 × 100nm2 and

(b) 300×300nm2. (c) Variations of normal state resistance among nominally

identical junctions with in-plane dimension 100× 100nm2. . . . . . . . . . . 27

3.3. Schematics of the microwave measurement setup. . . . . . . . . . . . . . . . 29

x

3.4. Photographs of the sample holders. (a) Sample holder for launching from

SMA to microwave stripline and the sample holder (b) Sample holder for

launching from SMA to coplanar waveguide (CPW). . . . . . . . . . . . . . . 31

4.1. Sample design. (a) The schematics of the circuit containing the device and

the read-out lumped-element resonator. The CPB Josephson junctions are

shown as crosses. (b) The layout of the device, the read-out resonator, and

the MW transmission line. The superinductor consists of 36 coupled cells,

each cell represents a small superconducting loop interrupted by three larger

and one smaller Josephson junctions . . . . . . . . . . . . . . . . . . . . . . . 33

4.2. Spectroscopy of the readout resonator around full-frustration of the superin-

ductor loop. At ΦL/Φ0 = 0.5, the superinductor reaches maximum induc-

tance, resulting in a minimum of the readout resonant frequency. . . . . . . . 34

4.3. Panel (a): The transmitted microwave power |S21|2 at the rst-tone frequency

f1 as a function of the second-tone frequency f2 and the gate voltage Vg

measured at a xed value of ΦL = 0.5Φ0. The power maxima correspond to

the resonance excitations of the device (f2 = f01), the superinductor (fL), and

the read-out resonator (fR). Note that the resonance measurements could not

be extended below ∼ 1 GHz because of a high-pass lter in the second-tone

feedline. Panel (b): The frequency dependence of the transmitted microwave

power measured at Vg = 0V and ΦSL = 0.5Φ0. . . . . . . . . . . . . . . . . . 36

xi

4.4. Panel (a): The ux dependence of the device energy levels obtained by nu-

merical diagonalization of the Hamiltonian. The solid curves correspond to

ng = 0.5, the dashed curves - to ng = 0 (the blue curves correspond to the

ground state |0〉, the yellow curves - to the state |1〉, and the green curves

- to the state |2〉). For comparison we also plotted the dotted curves that

correspond to the fully suppressed uxon tunneling; in this case there are

no avoided crossings between the parabolas that represent the superinduc-

tor energies EL(m,Φ) = 12EL(m − Φ

Φ0)2 plotted for dierent m. Panel (b):

The dependences of the resonance frequencies f01 (red dots - ng = 0, red

squares - ng = 0.5) and f02 (blue down-triangles - ng = 0, blue up-triangles -

ng = 0.5 ) on the ux in the device loop. The theoretical ts (solid curves -

ng = 0.5, dashed curves - ng = 0) were calculated with the following parame-

ters: EJ = 6.25 GHz, the asymmetry between the CPB junctions 4EJ = 0.5

GHz, EC = 6.7 GHz, EL = 0.4 GHz (L = (Φ02π )2/EL h 0.4µH), ECL = 5 GHz. 37

5.1. (a) Microphotograph of a portion of the half-wavelength resonator capaci-

tively coupled to the coplanar waveguide transmission line. Light green - Al

ground plane and the central conductor of the transmission line, green - silicon

substrate, black - the central strip of the resonator made of strongly disor-

dered Al. (b) Several resonators with dierent resonance frequencies coupled

to the transmission line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2. The temperature dependences of resonance frequency shift δfTLSr (T )/fr0. . . 43

5.3. The dependencesQi(n) at T ≈ 25 mK for the resonators with dierent widths.

Solid curves represent the theoretical ts of the quality factor governed by

TLS losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.4. a) The dependences of Qi for resonator #1 on the pump tone power Pp for

several values of detuning ∆f between resonance and pump frequencies. (b)

The values of Qi measured versus detuning ∆f at a xed number of the pump

tone photons in the resonator np ≈ 1000. The error bars are derived from

the covariance matrix obtained from nonlinear tting of the measurement of

S21(f). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

xii

5.5. The time dependence of Re[S21] measured at T = 25 mK at a xed frequency

on the slope of a resonance dip. The microwave power corresponds to 〈n〉 ∼

1000. Each point corresponds to the data averaging over 1 sec. . . . . . . . . 48

5.6. (a) The pulse sequence. (b) The time dependence of |S21| measured at f0 =

2.4258 GHz. The pump pulse at fp = f0 + 1 MHz was applied between

t = 0 s and t = 0.5 s. The pump tone power corresponds to np ≈ 1000. Each

data point was averaged over 4000 cycles with the same readout delay time.

The inset shows CW measurement of S21 versus f with (red) and without

(blue) the pump signal and indicates the position of f0 used in the relaxation

time measurement. The readout power was at the single photon level for all

measurements on this plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.1. Josephson metamaterial based on a chain of coupled asymmetric SQUIDs.

(a) Circuit schematic of the metamaterial. Each unit-cell of the metamaterial

consists of two asymmetric SQUIDs coupled with a shared junction and is of

length a. Each SQUID in the unit-cell is threaded with a magnetic ux Φ and

has a capacitance to ground Cgnd. (b) Illustration of the three-metal-layer

layout of the device. Metal layer M0 (gray) represents the ground plane, M1

and M2 are the two metal layers which form the electrodes of coupled asym-

metric SQUIDs, red and green vias between M1 and M2 represent Josephson

junctions and M1-to-M2 vias respectively. The purpose of the ngers on M0

in gray and M1 in green which extend into the foreground is to increase the

capacitance of the SQUID array to ground (M0). (c) Optical image of the

measured Josephson metamaterial. . . . . . . . . . . . . . . . . . . . . . . . . 55

6.2. Low-power transmission measurements of the phase shift across the Josephson

metamaterial as a function of the magnetic ux for device 1 (lower panel) and

device 2 (upper panel) at dierent measurement frequencies. . . . . . . . . . 56

6.3. Wavenumber as a function of frequency for devices 1 (blue circles) and 2 (red

squares). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

xiii

6.4. Measurements of the microwave phase shift as a function of signal power where

= -70 dBm, at dierent values of the magnetic ux in the metamaterial unit

cells for device 1 (upper panel) and device 2 (middle panel). . . . . . . . . . 57

A.1. Design of hybrid superinductor. . . . . . . . . . . . . . . . . . . . . . . . . . 71

B.1. Increase in transmission power through the metamaterial transmission line. . 72

xiv

1

Chapter 1

Introduction

Quantum computers have the capacity to outperform classical computers in solving hard

problems such as quantum simulations [1, 2], factoring large numbers [3] and indexing un-

structured database [4]. A quantum computer utilizes coherent quantum systems called

quantum bit (qubit) for computation. A qubit provides a quantum superposition of |0〉 and

|1〉 states in contrast to a classical bit which is in either 0 or 1 state. The power of quantum

computing relies on the ability to control the qubits while preserving their coherence. Fault-

tolerant quantum computing can be realized by encoding a logical qubit using many physical

qubits as long as the error rate per physical qubit is below a threshold value [58]. The error

rate is dened as the ratio between the single gate operation time and the qubit coherence

time. The estimates of the threshold for dierent error correction codes very between 10−6

and 10−2. Lower error rates require lower circuit redundancy for a given correction code.

Physical implementations of qubits include trapped ions [9], quantum dots [10, 11],

nuclear [1214] and electron spins [15], photons [16] and superconducting circuits [17] among

others. Trapped ions and superconducting qubits have demonstrated gates with high delity

necessary for the surface error-correction code [18]. Though the delity of single qubit gates

of trapped ions is higher than that of the superconducting qubits, the gate operation time

of superconducting qubits is ∼ 103 times less [19, 20]. Thus the superconducting qubits

compare favorably with the trapped ion qubits for fault-tolerant quantum computing. In

addition, the fabrication and operation of superconducting circuits use technologies already

available in the semiconductor and telecommunication industries.

The circuit redundancy required for universal quantum computing with superconducting

qubits so far remains very high. It is estimated that it would require 103− 104 the state-of-

the-art physical superconducting qubits to realize a single logical qubit. One way to reduce

2

the circuit redundancy is to implement the so-called parity-protected qubits, which have

degenerate ground states corresponding to even and odd parities of Cooper-pairs or ux-

ons [21]. Parity-protected qubits have built-in circuit symmetry to suppress the transition

between dierent parities. So it requires lower circuit redundancy to realize fault-tolerant

Cliord gates using parity-protected qubits than using non-protected qubits.

One of the essential elements for the realization of the uxon-parity-protected qubits is

the so-called superinductor - an inductor with the impedance greater than the resistance

quantum RQ = h/(2e)2 ≈ 6.5kΩ. Superinductors oer a abroad range of applications for

novel quantum circuits. For example, they enable realization of high-impedance environment

and are an important resource for amplication of the amplitude of quantum uctuations

in phase. In this dissertation, we explore two types of superinductors. The rst type of

superinductor is based on Josephson inductance. It consists of a chain of coupled asym-

metric Superconducting Quantum Interference Devices (CASQUIDs). The advantage of

this superinductor is the tunability of its inductance and its nonlinearity by the external

magnetic ux. These superinductors can be used in the circuits where linearity of the in-

ductance is desirable. We studied the prototype uxon-parity-protected qubit consisting of

a Cooper-pair box (CPB) shunted by a CASQUIDs-based superinductor. The spectroscopy

of a prototype device of the qubit demonstrated that the parity of uxons in the loop was

preserved when the oset charge on CPB island was set to e mod (2e). A qubit with similar

design and a even larger superinductance can potentially be used to implement fault-tolerant

Cliord gates. The second type of superinductors is made of the high-kinetic-inductance

superconducting nanowires. We studied superinductors based on high-kinetic-inductance

disordered Aluminum lms. To study the dissipation processes in these lms, we fabri-

cated the superconducting coplanar waveguide (CPW) resonators and measured microwave

losses at ultra-low temperatures. We observed that the intrinsic losses in these resonators

were limited by the resonator coupling to two-level systems (TLS) in the environment. The

demonstrated internal quality factors were comparable to those for CPW resonators made

of conventional superconductors. The disordered Aluminum nanowires are promising for

applications in quantum metrology and quantum information processing.

In addition to fault-tolerant gate operations, fault-tolerant state measurements are needed

3

when performing quantum error corrections to avoid the propagation of errors detrimental

to the quantum information stored in the logical qubit. Fault-tolerant measurements re-

quire amplication of signals with quantum-limited noise level. Since fault-tolerant quan-

tum computing involves measurements of multiple qubits, the amplication also needs to

be broadband. Kerr nonlinearity in metamaterial transmission lines based on Josephson

junctions have been previously utilized to realize Josephson traveling-wave parametric am-

pliers (JTWPA) with nearly quantum-limited noise [22]. In this dissertation, we present

a novel matematerial on the basis of CASQUIDs with strong Kerr nonlinearity and Kerr

constant tunable by the magnetic eld over a wide range from positive to negative values.

The metamaterial is promising as an active medium for quantum-limited JTWPA.

1.1 Superconducting qubit

A Josephson junction consists of two superconducting electrodes separated by an insulating

barrier as illustrated in Fig. 1.1(a). The Al/AlOx/Al Josephson junctions are the most

widely used for superconducting qubits. Cooper pairs can tunnel without dissipation across

the junction. The supercurrent depends nonlinearly on the phase φ dierence between the

electrodes

I = Icsin(φ), (1.1)

where Ic is the critical current. The change in φ with time is associated with a voltage V

across the junction

dφ

dt=

2eV

~. (1.2)

From the two Josephson equations above, we can obtain the inductance associated with the

Josephson junction

LJ(φ) =Φ0

2π(dI

dφ)−1

=Φ0

2πIc cos(φ)

, (1.3)

4

Figure 1.1: (a) A Josephson junction consisting of two electrodes (in white and grey) sepa-rated by the tunnel barrier; (b) Schematic representation of a Josephson junction with zerolosses; (c) Schematic of a superconducting qubit.

where Φ0 is the ux quantum. Thus, a Josephson junction can be viewed as a non-dissipative

element with nonlinear inductance.

A Josephson junction is characterized by two energies. The rst one is the Josephson

energy EJ = ∆RQ/8RN , where ∆ = 1.76kBTc is the superconducting gap, and RN is

the normal state resistance of the junction [23]. The second one is the charging energy

EC = 4e2/2C corresponding to the transfer of one Cooper pair across the junction with

capacitance C. The Hamiltonian of the Josephson junction is

HJ = 4EC n2 − EJcosφ, (1.4)

where n and φ are a conjugate pair of coordinates corresponding to the number of Cooper

pairs stored in the capacitor and the phase across the junction respectively. Figure. 1(b)

shows the schematic representation of a Josephson junction with zero losses.

Figure 1.1(c) shows the schematic of a superconducting qubit consisting of a Josephson

junction shunted by an inductor. The Hamiltonion of the qubit is

H = 4EC n2 − EJcosφ+ ELφ

2, (1.5)

where EL = (Φ0/2π)2/L corresponds to the inductive energy of the inductor. The energy

levels of the system are non-equidistant. The two lowest energy states correspond to the

states |0〉 and |1〉 of the qubit. Various qubit designs corresponding to dierent EJ/EC and

EL/EJ have been explored in the process of improving the qubit coherence time during the

past two decades. A Mendeleev-like table of superconducting qubits can be found in Ref.

5

[24].

Superconducting qubits operate at temperatures < 20mK. At those temperatures, the

numbers of thermally excited quasiparticle and phonon are suppressed, which increases the

coherence time of qubits. Typically the energy dierence between the |0〉 and |1〉 states of

the superconducting qubit are designed to correspond to microwave frequencies < 10GHz.

1.2 Superinductor

If a wire's inductance is geometrical, its impedance Z would be limited by the ne structure

constant (Z ≤ 1/137RQ) since the geometric inductance is associated with energy stored in

electromagnetic elds. The kinetic inductance is not limited the same way for it is associated

with the kinetic energy of the Cooper pairs . Though kinetic inductance exists in normal

metals, it becomes signicant only at very high (terahertz) frequencies since electrons are

scattered on a short time scale. Cooper pairs in the superconducting condensate ow fric-

tionlessly at microwave frequencies [25], thus superinductors realized using superconductors

can operate at microwave frequencies with low losses, making it suitable for applications in

superconducting qubits.

Quantum uctuations of charge and ux uctuations in a LC resonator are related to

the ratio between its characteristic impedance Z0 =√L/C and the resistance quantum RQ

as [26]

δΦ/Φ0 =

√1

4πZ0/RQ (1.6)

δQ/2e =

√1

4πRQ/Z0. (1.7)

Therefore, superinductors allow quantum uctuations in ux larger than Φ0 and quantum

uctuations in charge less than 2e.

The kinetic inductance per unit length of a superconducting nanowire is

6

LK =m

2e2Ans, (1.8)

where A is the area of the cross section of wire, ns is the density of Cooper pairs and m is

the mass of an electron. Conventional superconductors have large kinetic inductance when

the current approaches the critical value because the density of electrons decreases. But

they have large losses for the density of normal electrons increases at the same time.

Superinductors can be realized using nanowires of disordered superconducting materi-

als, such as TiNx, NbN, and granular Aluminum, and using Josephson junction arrays (see

Eq. 1.3 for Josephson inductance of a single junction). Each approach has its advantages

and disadvantages. Superinductors made of Josephson arrays typically have larger in-plane

dimensions than that of nanowire-based superinductors. This results in a greater stray ca-

pacitance of Josephson arrays and lower frequencies of the self-resonance modes. On the

other hand, fabrication of compact superinductors based on nanowires requires enhanc-

ing the strength of disorder by approaching the disorder-driven superconducting-insulating

transition (SIT). Soft modes that might appear near the SIT may aect detrimentally the

performance of these superinductors by increasing losses [27]. Also, it is challenging to

reproducibly fabricate strongly disordered lms near the SIT.

1.3 A chain of coupled asymmetric SQUIDs (CASQUIDs)

Figure 1.2: Schematic of the tunable superinductor consisting of two large (blue) and twosmall (yellow) Josephson junctions per unit cell. α and α′ correspond to the superconduct-ing phases across the large junctions, and β1 and β2 correspond to those across the smalljunctions.

7

Figure 1.2 shows the schematics of a chain of coupled asymmetric Superconducting Quan-

tum Interference Devices (CASQUIDs). Each unit cell consists of two large junctions with

Josephson energy EJL (blue) and one small junctions with Josephson energy EJS (yellow),

where EJL > EJS . Two adjacent unit cells are coupled through a common large junction.

The large junctions form the meandered backbone for the chain of junctions.

The CASQUIDs is translationally invariant. The structure stays the same when shifted

horizontally by one unit cell and then mirror-reected with respect to the horizontal axis. In

Figure. 1.2, α and α′ correspond to the phase across the two large junctions, and β1 and β2

correspond to that across the two small junctions. Due to the transnational symmetry, the

phase dierence across the two large junctions within one unit cell are the same (α = α′).

Assuming the positive current ows clockwise, the phase across the junctions satises the

following constraints

α+ α′ + β1 =2π

Φ0Φ, and (1.9)

α+ α′ + β2 = −2π

Φ0Φ, (1.10)

where Φ corresponds to the external ux through the loop. By dening α+α′ ≡ ϕ, we have

β1 =2π

Φ0Φ− ϕ, and (1.11)

β2 = −2π

Φ0Φ− ϕ. (1.12)

In the classical limit where the charging energy is ignored 1, the total energy of the CASQUIDs

is

EJ(α) = −2EJLcos(ϕ

2)− EJScos(ϕ−

2π

Φ0Φ)− EJScos(ϕ+

2π

Φ0Φ). (1.13)

1The discussion in this section does not include the quantum eect by taking into consideration thecharging energy. Interested readers can refer to Ref. [28] for the discussion with quantum eects considered.

8

The inductance per unit cell for this CASQUIDs is

LJ(ϕ,Φ) =

(∂E2

J(ϕ)

∂2ϕ

)−1

(1.14)

=Φ0

2πIJS

([r

2+ 2 cos

(2π

Φ

Φ0

)]−[r

16+ cos

(2π

Φ

Φ0

)]ϕ2

)−1

. (1.15)

where r = EJL/EJS and IJS is the critical current corresponding to the large junction. The

linear and nonlinear part of inductance depend on r and are tunable by the external ux Φ

through the unit cell. For a certain r, the inductance increases and reaches the maximum

as the external ux increases from zero to full frustration (Φ = Φ0/2). For an innitely long

chain of CASQUIDs, the maximal inductance of the chain at full frustration correspond

to r = 4. The tunable inductor made of CASQUIDs have demonstrated an increase in

inductance by 1-2 orders of magnitude with the ux increases from zero to full frustration

[28].

In the superinductors realized using the CASQUIDs, the junctions of the backbone is

designed phase-slip-free with Josephson energy much greater than the charging energy as

the phase slip rate across a Josephson junction is ∼ exp(−√

8EJ/EC). In the experiment

shown in Ref. [28], EJS/EJC ∼ 100. The backbone of CASQUIDs can be replaced by

a nanowire made of disordered superconducting materials. In this case, the rst term in

Eq. 1.13 is replaced by the inductive energy ELϕ2/2 associated with the inductance of the

segment of the nanowire within one unit cell. When EL = 2EJS , the inductance at full

frustration reaches the maximal value.

1.4 Fluxon-parity-protected qubit

A uxon-parity-protected qubit consists of a Cooper pair box shunted by a superinductor

made of a chain of CASQUIDs. The uxon-parity-protected qubit is an example of physical

systems with built-in error correction. The Josephson energy EJ and the charging energy

EC of the junctions in the CPB satisfy the condition EJ > EC . The dynamics of the low

energy states of the qubit correspond to uxons tunneling across the Josephson junctions

in the CPB in and out of the superconducting loop. Each uxon is associated with phase

9

slips that change the superconducting phase across the CPB by an integer number of 2π.

The uxon tunneling process is dual to the Josephson eect that corresponds to Cooper

pairs tunnel cross the insulating barrier separating the two electrodes. The uxons that

change the phase by even and odd numbers of 2π are of even and odd parities respectively.

When the oset charge on the island of the CPB is emod(2e), the phase slip process that

change the phase across the CPB by 2π is suppressed due to Aharonon-Casher eect. The

lowest two energy states |0〉 and |1〉 of the qubit are superpositions of states associated with

uxons of even and odd parities respectively. Since the 2π phase slips are suppressed, so

is the transition between the states of even and odd parities. The qubit is thus protected

from energy decay. Meanwhile, the double phase slips corresponding to simultaneous 2π

phase slips across each junction in the CPB are allowed. When the double phase slip rate

is much greater than the energy dierence between the uxons separated by 4π, each qubit

state corresponds to a superposition of multiple uxon states of the same parity. When

the condition is met, the energy dierence between the two qubit states becomes negligibly

small and insensitive to charge and ux noises in the environment to the rst order. The

qubit is thus protected from pure dephasing.

Fault-tolerant Cliord gates can be performed by using certain types of nonlinear cou-

pling between the uxon-parity-protected qubit and the control channels. For example,

the fault-tolerant π/4 phase gate can be realized through quadratic coupling by tuning the

magnetic eld through the loops in the superinductor based on CASQUIDs [21]. Because of

the inherent properties of the Hamiltonian of the protected-qubit, it requires less eort to

actively check and correct for errors than that involved in fault tolerant operations realized

using non-protected qubit [29].

1.5 Josephson traveling-wave parametric amplier based on CASQUIDs

A qubit state is usually probed by the dispersive measurement using a high-Q readout

resonator. The qubit coupling to the resonator introduces a shift in the resonant frequency.

The state of the qubit can be derived by measuring the transmission of the readout resonator

at a xed frequency close to its resonance. To minimize the perturbation of qubit due to

readout signal, the readout resonator is usually measured with power corresponding to

10

the single photon level. For a critically coupled resonator of frequency fr = 6 GHz and

Qi = 10, 000 , this corresponds to microwave power ∼ −135 dBm = 4πhf2r /Qi.

Multiple stages of amplication are required for processing such a low-power signal. It

is necessary to amplify the signals at low temperatures before it reaches the room tempera-

ture pre-amps. Fault-tolerant operations require low temperature ampliers with quantum-

limited noise level to perform operations such as high delity measurement of supercon-

ducting qubit state and qubit entanglement [30]. The commercially available semiconductor

based high electron mobility transistor (HEMT) ampliers are able to amplify the signals at

4K, but would introduce noise corresponding to 10-20 photons due to thermal noise in the

transistor. Parametric ampliers based on Josephson junctions, rst developed by Yurke et

al. at Bell labs [31], are able to meet the requirement for fault-tolerant operations.

In a parametric amplier, a weak input signal is amplied through interacting with a

strong pump signal that modulates certain parameters of the nonlinear medium, such as the

index of refraction in nonlinear optical systems. There are mainly two types of parametric

ampliers based on Josephson junctions: the Josephson parametric amplier (JPA) and the

Josephson traveling-wave parametric amplier (JTWPA). The JPA relies on high-quality

resonators to prolong the interaction time between signals in the nonlinear medium [32

35]. As a result it has a limited bandwidth and dynamic range. The narrow bandwidth is

a limitation for multiplexing when a large number of qubits are measured simultaneously.

The JTWPA overcomes the limit by allowing interacting signals to propagate for a distance

much longer than their wavelength.

In a JTWPA, parametric amplication is realized in a transmission line that couples a

strong pump tone (ωp) to a weak signal (ωi) and an idler (ωi) via a degenerate four-wave

mixing process such that 2ωp = ωi + ωs. When the phase matching condition between the

three waves

∆k = 2kp − ks − ki = 0 (1.16)

is met, where kp, ks, and ki are the wave vectors of the pump, signal and idler respectively,

the gain of the signal ωi grows exponentially with the length of the transmission line.

11

The wave vector k of any signal transmitted through Josephson-junction-based transmis-

sion lines consists of a power independent part kL and a power dependent part kNL. kNL

is proportional to the intensity of the electromagnetic eld ∝ |E|2 due to Kerr eect similar

to that in optic bers [36]. For example, in the case of two dierent signals E1e−iω1t and

E2e−iω2t traveling simultaneously in the nonlinear transmission line, the power dependent

part of the wave vector of the signal at ω1 is modulated by the intensities of signals at ω1

and ω2

kNL(ω1) ∝ K(|E1|2 + 2|E2|2), (1.17)

where K is the Kerr constant. The two modulations are called self-phase modulation (SPM)

and cross-phase modulation (XPM).

In a chain of Josephson junctions, ∆kL = 2kLp − kLs − kLi > 0 and ∆kNL = 2kNLp −

kNLs − kNLi > 0 [37, 38]. At low pump power, both terms are close to zero. As the pump

power increases, ∆k = ∆kL + ∆kNL < 0. One method to achieve the phase matching

condition called resonant phase matching (RPM) is to modify the dispersion of the pump

tone by periodically loading the chain with resonators with the resonant frequencies close

to the pump frequency [22, 38]. The nonlinear dispersion of the pump close to the resonant

frequency compensates for the phase mismatch. For the JTWPA based on RPM, the pump

frequency can only be varied over a small range (∼ 100MHz ) around the resonant frequency

of the loaded resonators, and the signals at frequencies within the range can not be amplied.

In the transmission line made of a chain of CASQUIDs, the Kerr constant is tunable with

magnetic eld and even changes its sign. ∆kNL, which is proportional to the Kerr constant,

can be tuned to positive to compensate ∆kL. The pump frequency can be varied over a

broad range in the JTWPA based on the transmission line. Numerical studies have shown

that amplication can be realized for dierent pump frequencies in the range of 1-9GHz in

the CASQUIDs based JTWPA [39]. In addition, the nonlinearity in a chain of CASQUIDs

facilitates stronger interaction between a pump (ωp) and a signal (ωi) than that in a chain

of Josephson junctions. A JTWPA based on CASQUIDs is able to achieve comparable gain

as that based on RPM at shorter length. It is estimated to achieve 20dB gain with a length

12

∼ 650µm [39] , while it requires ∼ 3.3cm using RPM for similar gain [22].

13

Chapter 2

Fluxon-parity-protected qubit: theoretical background

2.1 Single-qubit state and coherence

A qubit state can be expressed as

|ψ〉 = cos(θ

2)|0〉+ eiφ sin(

θ

2)|1〉. (2.1)

with θ ∈ [0, π] and φ ∈ [0, 2π]. It can also be represented using the density matrix ρ = |ψ〉〈ψ|

ρ =1

2(I + axσx + ayσy + azσz), (2.2)

where σx, σy and σz are the Pauli matrices and ax = sin θ cosϕ, ay = sin θ sinϕ and az =

cos θ are their expectation values respectively. The qubit state can be represented by the

vector (ax, ay, az) called the Bloch vector within the sphere of unit radius called the Bloch

sphere. Figure. 2.1 shows the schematic representation of a Bloch sphere (black) and a

Bloch vector (blue).

For a qubit initialized at the excited state |1〉, its Bloch vector is of unit length and

pointing vertically downward. Due to the coupling between the qubit and the noise in

the environment, the Bloch vector decay to the position corresponding to the qubit state

at thermal equilibrium. The longitudinal and transversal decays are characterized by T1

and T ∗2 , called energy relaxation time and pure dephasing time respectively. The energy

relaxation is related to the noise at qubit frequency that is able to induce the transition

between |0〉 and |1〉 state. And the pure dephasing is due to the low frequency noise that

induces random change in the energy dierence between |0〉 and |1〉 state. The overall decay

14

Figure 2.1: Bloch sphere.

process is characterized by the coherece time T2

1

T2=

1

2T1+

1

T ∗2. (2.3)

2.2 Parity-protected qubit 1

In the quantum error-correction code (QECC) [58], a single logical qubit is encoded in

multiple physical ones. The logical qubit reside in the subspace corresponds to mutually

commuting operators Oα. Oα have eigenvalues +1 and −1. Eigenvalues of Oα cor-

responding to the code space are +1. The operators Oα are chosen such that any error

would project the logical qubit from the code space to its orthogonal space where at least one

of the eigenvalues of Oα corresponding to the logical qubit becomes −1. Oα are called

error syndromes for its capability of diagnosing the errors. By performing error syndrome

measurement, we are able to check for errors before applying corrections.

The process of implementing the QECC can be simplifed by designing a physical qubit

corresponds to the Hamiltonian H = −∑

α ∆αOα with degenerate ground states. The

degenerate ground states correspond to the code space. In order for the ground state to be

degenerate, additional symmetries Pβ need to be satised by H, [H,Pβ] = 0, and Pβ

does not commute with Oα. In addition, the energy required to excite the qubit from

1This section summarizes the discussions in Ref. [21] relevant to the purpose of this work.

15

the degenerate ground states to the excited states is larger than the thermal noise in the

environment (∆α kBT ) so that the error rate is suppressed.

In the phase space corresponding to a pair of canonical coordinates φ and q,

[cos(2πq), cos(mφ)] = 0, (2.4)

where m is an integer. We can use the two operators as error syndromes. We detect any

errors in φ and q by measuring cos(mφ) and cos(2πq) respectively. For the Hamiltonian

H = −Eqcos(2πq)− Eφcos(mφ), (2.5)

the lowest energy states are m-fold degenerate. The wave functions presented in the coor-

dinate representation are

ψs(φ) =∑k

δ[φ− 2π(k +s

m)], (2.6)

where s are integers between 0 andm−1 and k is summed over all integers. In the momentum

representation, the wave functions are

ψs(p) = exp(−2πiqs/m)∑l

δ(q − l). (2.7)

By embedding the system in a harmonic oscillator, we are able to create a energy gap

that separates the degenerate ground states from all higher excitations. Such a system

correspond to the Hamiltonian

H = −Eq cos(2πq)− Eφ cos(mφ) + 1/2V (t)φ2 +1

2m∗q2. (2.8)

When V (t) 1 and Eφm∗ 1, the coordinate φ are conned to the discrete values

2πk/m, where k is integer, corresponding to the maximum of cos(mφ). The ground states

are separated from excited states by√Eφ/m∗. Denote the discrete states as |k〉. The low

16

energy states of the system expressed in this basis becomes

H = −1/2Eφ (|k +m〉〈k|+ |k〉〈k +m|) +1

2V (t)

(2πk

m

)2

. (2.9)

When m = 2, the ground states are two fold degenerate corresponding to k at even and

odd numbers. They form the code space for the parity-protected qubit. When V (t) 1, the

qubit is decoupled from noise as well as external control and becomes a quantum memory.

But certain fault tolerant Cliord gates, such as the π/4 gate, can be performed by varying

V (t) [21].

2.3 Fluxon-parity-protected qubit

Figure 2.2: Schematic of a uxon-parity-protected qubit.

Figure 2.2 shows the schematic of the uxon-parity-protected qubit. The qubit consists

of a Cooper-pair Box (CPB) shunted by a superinductor. The CPB consists of two nomially

identical Josephson junctions separated by a superconducting island. EJ and EC correspond

to their Josephson and charging energies of the junctions respectively. The phase φ across

the CPB and the oset charge q associated with the superinductor are a conjugate pair of

coordinates [q, φ] ∼ i. We choose the gauge so that the phase at the two ends of the CPB are

17

±φ/2. The number of the charge n and the phase ϕ on the CPB island are also a conjugate

pair of coordinates [n, ϕ] ∼ i. ng is the oset charge on the CPB island controlled by the

external DC voltage bias. In the following discussion, φ and ϕ are the phases normalized by

Φ0/2π and q, n and ng are the charges normalized by 2e.

The CPB Hamiltonian is expressed as

HCPB = −EJcos(ϕ+ φ/2)− EJcos(ϕ− φ/2) + 4Ec(n− ng)2, (2.10)

= −2EJcos(ϕ)cos(φ/2) + 4Ec(n− ng)2. (2.11)

In the presence of the external magnetic eld Φ, the phase cross the superinductor cor-

responds to φ − 2πm0, where m0 = Φ/Φ0. The Hamiltonian of the superinductor with

inductive energy EL and charging energy ECL is expressed as

HSI =1

2EL(φ− 2πm0)2 + 4ECLq

2. (2.12)

EL is dened as (Φ0/2π)2(1/LSI), where LSI is the inductance of the superinductor. Its

charging energy ECL is associated with its parasitic capacitance to the ground. In the basis

of charging states |n〉, the CPB Hamiltonian is expressed in matrix form as

HCPB =

. . . −EJcos(φ/2) · · · · · · · · ·

−EJcos(φ/2) 4EC(N − 1− ng)2 −EJcos(φ/2) · · · · · ·... −EJcos(φ/2) 4EC(N − ng)2 −EJcos(φ/2) · · ·...

... −EJcos(φ/2) 4EC(N + 1− ng)2 · · ·...

......

.... . .

,

(2.13)

where N is an integer. When ng = N + δN , where δN ∈ [0, 1), in the limit of EJ < EC , the

eect of the transition between |n〉 and |n + 1〉 is signicant when n ' N . The low energy

18

subspace of the CPB is spanned by |N〉 and |N + 1〉 and the corresponding Hamiltonian is

HCPB =

−4EC(0.5− δN)2 −EJcos(φ/2)

−EJcos(φ/2) 4EC(0.5− δN)2

. (2.14)

When δN = 0.5, the Hamiltonian of the CPB when represented in the basis of 1/√

2(|N〉+

|N + 1〉), 1/√

2 (|N〉 − |N + 1〉) becomes

HCPB =

−EJcos(φ/2) 0

0 +EJcos(φ/2)

. (2.15)

The total Hamiltonian H = HCPB +HSI becomes

H =

H− 0

0 H+

(2.16)

where

H− = 4ECLq2 − EJcos(φ/2) +

1

2EL(φ− 2πm0)2, and (2.17)

H+ = 4ECLq2 + EJcos(φ/2) +

1

2EL(φ− 2πm0)2. (2.18)

H± can be considered as a system where a ctitious particle of mass 1/4ECL is moving in

the potential wells

V±(φ) = ±EJ cos(φ/2) +1

2EL(φ− 2πm0)2. (2.19)

The top plot in Figure 2.3 shows the schematic for V±(φ) with EJ EL. When EJ ?

ECL EL, the ground states of H± are localized around the minimums of ±EJ cos(φ/2).

Phenomenological, the localized states correspond the instantons called uxons. A uxon

state |m〉 corresponds to the state with phase dierence φ = 2πm (m ∈ Z) across the CPB.

Even and odd m correspond to uxon states of even and odd parities respectively. The |0〉

and |1〉 states of the qubit correspond to superpositions of uxons of even and odd parities.

19

Figure 2.3: Schematic of the potential wells and the wave functions of the uxon-parityprotected qubit. (a) Schematic of the potential wells V (φ)± with with EJ EL. Theinset shows the condition required for a uxon-parity-protected qubit. The energies of theuxons are separated from the energies corresponding to higher excitations by the plasmafrequency ωp. The energy dierence between the uxon states conned in two adjacentpotential minimums is ∼ 8π2EL. The two uxon states are coupled through the doublephase-slip process at the rate Edps. The uxon-parity-protected qubit correspond to thecondition when ωp Edps 8π2EL. (b) The wave functions for |0〉 and |1〉 states of theuxon-parity-protected qubit at m0 = 0 and ng = emod(e). The wave functions of |0〉and |1〉 states are superpositions of wave packages localized at even and odd numbers of 2πrespectively along φ, corresponding to states of even and odd parities.

20

The energy of uxons is separated by the plasma frequency ωp =√

2EJECL from the higher

excitations, the plasmons. The transition between two uxons |m〉and |m′〉 corresponds to

the phase-slip that change φ by 2π(m−m′). The 2π phase slip across the CPB corresponding

to transition from |m〉 and |m± 1〉 is suppressed when the oset charge on the CPB island

is emod(2e) due to the Aharonov-Casher eect [40]. The double phase-slips that change the

phase across the CPB by 4π become the dominant transition process. We will see later that

the description using uxons is only valid when we shunt the CPB with a superinductor.

The dynamics of the uxon states correspond to even or odd parities can be expressed

using the Hamiltonian 2

H = −Edps|m〉〈m+ 2| − Edps|m+ 2〉〈m|+ 2π2EL(m−m0)2, (2.20)

where the double phase-slip rate expressed in terms of g = 4√

2EJ/ECL and ωp =√

2EJECL

is

Edps ∼ g1/2exp(−g)ωp. (2.21)

The double phase slip rate is equivalent to the energy splitting corresponding to the particle

with kinetic energy 4ECLq2 tunneling between two minimums of the potential well V (φ). We

see from Eq. 2.20 that in order for the description using uxons to be valid, EL needs to be

less than the resonant frequency of the inductor that shunts the CPB. This is equivalent to

that the impedance of the inductor is greater than resistance quantum. So the superinductor

is necessary for uxon states to exist in the qubit.

The dependence of the energy dierence E01 between even and odd parity on m0 is

E01(m0) ∝

(√8Edpsπ2EL

)1/2

exp

(−

√8Edpsπ2EL

)√2EdpsEL| cos(4πm0)|. (2.22)

At full frustrations, |0〉 and |1〉 qubit states are degenerate and their energy dierence is

2The description of the system based on Josephson junction chains by instantons is similar to that inRef. [41].

21

rst order sensitive to the ux noise. At zero frustration, the energy dierence is non-zero

but rst order insensitive to the ux noise. When the amplitude of the double phase slips is

much greater than the energy between two uxons separated by δm = 2, Edps/8π2EL 1,

the two qubit states become almost degenerate. The qubit is thus protected from pure

dephasing at zero frustration.

When the oset charge on the CPB island is emod(2e), the charge noise induces the

nite coupling between |0〉 and |1〉 at the rate

t01 ∼ g1/4EC(δN − 0.5)√Edps/ωp. (2.23)

The eect is small when Edps ωp. Under this condition, the qubit is protected from

energy relaxation.

The condition to protect the qubit from decoherene, which includes the energy relaxation

and the pure dephasing, is ωp Edps 8π2EL [schematically shown in the inset in Figure

2.3(a)]. Under this condition, the double-phase slip rate is high enough for the mixing of

uxons of the same parity, such as |m − 2〉 and |m〉. Meanwhile it is much lower than

the plasma frequency to avoid mixing uxons with plasmons. The bottom plot in Figure.

2.3 schematically shows the wave functions corresponding to the protected qubit states

represented in the basis of φ. The wave functions are superpositions of wave packages

localized at positions corresponding to even and odd numbers of 2π. They share the same

Gaussian envelope corresponding to the ground state of the harmonic oscillator with respect

to the superinductor. The fault tolerant π/4 gate can be realized with the protected qubit

by varying the inductance of the superinductor adiabatically over a certain time while the

qubit still satises the aforementioned condition [21].

2.4 Inductive coupling of uxon-parity-protected qubit and readout res-

onator

The state of the uxon-parity-protected qubit can be determined dispersively through the

inductive coupling to the readout LC resonator, which is coupled inductively to a microstrip

22

Figure 2.4: Circuit diagram of a uxon-parity-protected qubit coupled inductively to areadout resonator.

transmission line. Figure 2.4 shows the circuit diagram corresponding to the uxon-parity-

protected qubit coupled to the resonator through the inductor Lc. The Hamiltonian of the

system can be divided into three parts: the resonator, the qubit and their coupling

H = Hres +Hqubit +Hcouple. (2.24)

We express Hres in terms of the creation a† and annihilation a operators of photons as

Hres =1

2~ωra†a(n+ 1), (2.25)

where n is the number of photons and ωr is the angular frequency of the resonator. The

Hamiltonian of the qubit with transition energy ω01 between |0〉 and |1〉 can be expressed

as

Hqubit = ~ω01σz. (2.26)

Let's dene the current Iq ows counter-clockwise around the qubit loop and the cur-

rent Ir ows clockwise around the resonator loop as shown in Fig. 2.4. The Hamiltonian

corresponding to the coupling between the qubit and the readout resonator is

Hcouple = LcIrIq. (2.27)

The phase around the qubit loop satisfy the constraint imposed by the magnetic ux Φext

23

through the loop

Φ0

2πφ+ LcIr + (Lc + LSI)Iq = Φext, (2.28)

where Lc and LSI are the inductance of the coupler and superinductor respectively. Re-

placing Iq in Eq. 2.27 using Eq. 2.28 and keeping terms containing parameters of both the

qubit and the LC resonator, we obtain

Hcoupler = −Φ0

2π· LcLc + LSI

φIr. (2.29)

The current Ir can be expressed in terms of a† and a as

Ir = −√

~ωr2L

(a− a†). (2.30)

Combine Eq. 2.30 and 2.29, we have

Hcoupler = ~ωrLc

Lc + LSI

√RQ

4πZ0φ(a− a†), (2.31)

where Z0 =√L/C is the characteristic impedance of the resonator. Dene the coupling

constant

g = ~ωrLc

Lc + LSI

√RQ

4πZ0. (2.32)

Hcoupler is expressed as

Hcoupler = gφ(a− a†). (2.33)

Next we discuss the dispersive shift of the resonant frequency for dierent qubit states.

Dene |i, n〉 as the eigenstate for Hres +Hqubit with energy E0i,n = Ei + 1/2~ωr(n+ 1) when

the qubit is in state i with energy Ei and there is n number of photons in the resonator.

24

Figure 2.5: Schematic of the magnitudes of the transmissions through the microwave trans-mission line corresponding to |0〉 (red) and |1〉 (blue) states of the qubit and the bare LCresonator (black). |0〉 and |1〉 states shift the resonant frequency dispersively by χ0 and χ1

respectively.

When the coupling is weak (g |ωr−ω01|), Hcoupler perturbs E0i,n by a small amount δEi,n

resulting in the dispersive shift χi = Ei,n+1 − Ei,n (Ei,n = E0i,n + δEi,n) of the resonant

frequency of the LC resonator. Fig. 2.5 shows a schematic example of the transmission

magnitude through the transmission line corresponding to |0〉 (red) and |1〉 (blue) states of

the qubit and the bare LC resonator (black). The qubit states |0〉 and |1〉 shifts resonant

frequency by χ0 and χ1 respectively. We are able to monitor the qubit state by measuring

the transmission at a xed frequency that corresponds to dierent transmission magnitudes

when the qubit is at |0〉 and |1〉. Note that the ideally protected states of the uxon-

parity-protected qubit is decoupled from linear coupling (〈0|φ|1〉 = 0). Nonlinear coupling

is required to operate and monitor the qubit at the protected state. For the experiment

presented in Chapter 4, since the superinductance is small compared to that in a ideally

protected qubit, we are still able to detect the state through the coupling method discussed

here.

25

Chapter 3

Experimental techniques

3.1 Sample fabrication

3.1.1 Josephson junction fabrication technique used for uxon-parity-protected

qubit fabrication

Al/AlOx/Al Josephson junctions were used for the sample of the uxon-parity-protected

qubit discussed in Chapter 4. The junctions were prepared using the Manhattan pattern

technique. In this technique, the metal deposition mask is formed from a bilayer e-beam re-

sist with an undercut created during the lithography. It allows bridge-free e-gun depositions

of top and bottom electrodes from two directions separated by 90 degrees.

The layout of the junction was designed using computer-aided design (CAD) as two in-

tersecting lines [see Fig. 3.1(a)]. E-beam lithography was used to pattern the the lift-o

mask for metal deposition according to the designed layout on top of the substrate. The

lift-o mask is made of a bilayer e-beam resist consisting of a 350-nm-thick poly(methyl

methacrylate) (PMMA) layer on top of a 100-nm-thick copolymer Methyl methacrylate

(MMA) layer [see Fig 3.1 (b) ]. The lithography was performed using a scanning electron

microscope (SEM). In the SEM, electrons scissor molecular chains by interacting with poly-

meric molecules in the e-beam resist. The blue region in Fig. 3.1(c) indicates the exposed

volume in the resist. The copolymer MMA has shorter molecular chains than does the

PMMA and is thus more sensitive to the exposure of electrons. As a result, an undercut was

formed at the bottom of the trench after the development [see Fig. 3.1(d)]. Following the

development, the sample was put into an Ozone asher for 5 minutes to remove the e-beam

resist residue on the substrate surface before metal deposition. Afterwards, the sample was

loaded into an oil-free high vacuum chamber with a bass pressure 4e-8 mTorr for metal

26

Figure 3.1: Schematic representation of Josephson junction fabrication using the Manhattanpattern technique. (a) The layout of the Josephson junction. (b-d) The e-beam lithographyprocess using a bilayer e-beam resist. (e-g) Metal deposition. (h) The cross section of aJosephson junction after metal lift-o.

deposition. The Aluminum electrodes were deposited at 45 degrees with respect to the sur-

face at rate 1A/s. Figure 3.1(e) shows the schematic representation of the deposition for

horizontal and vertical trenches labeled A and B respectively. The two black arrows indicate

the deposition directions. When the Aluminum was deposited along trench A [see Fig. 3.1

(f)], an electrode would form on the substrate only in trench A as long as the trench width

is less than the PMMA thickness [see Fig. 3.1 (f) and (g)]. The deposition along trench

B works similarly. We can start the deposition along either trenches. After depositing the

bottom Aluminum electrode, it was oxidized with dry oxygen without breaking vacuum be-

fore depositing the top electrode. The oxygen pressure and oxidation time was set to meet

the requirement for the normal state resistance of the Josephson junction. The thicknesses

for the top and bottom electrodes were 20nm and 60nm respectively. After deposition, the

device was passivated with 100 torr dry oxygen for 5 minutes before exposed to atmosphere.

27

The e-beam resist was then lift-o by by soaking the sample in the solution of Microchem

PG remover for 20 minutes at 80C. Figure 3.1 (h) shows the cross section of the junction

after lift-o.

By using the Manhattan pattern technique, we are able to fabricate Josephson junctions

with in-plane areas of 100×100nm2−300×300nm2 [See SEM images for Josephson junctions

in Fig. 3.2(a-b)]. Standard deviation of normal state resistance is ∼ 2.4% for junctions of

area 100 × 100nm2 [See Fig. 3.2(c)]. The variations of the in-plane areas for nominally

identical junctions are < 20%. The small spreading of the normal state resistance and of

the in-plane areas in Josephson junctions results in small variations in the Josephson energy

EJ and the charging energy EC among the nominally identical junctions. This is important

for realizing the uxon-parity-protected state since the CASQUIDs based superinductor

achieves the maximal superinductance when the unit cells are simultaneously frustrated and

the complete suppression of the single uxon tunneling requires identical phase slip rate

∼ exp(−√

8EJ/EC

)across each junction in the Cooper-pair box (CPB).

Figure 3.2: SEM images for junctions with in-plane dimensions (a) 100 × 100nm2 and (b)300×300nm2. (c) Variations of normal state resistance among nominally identical junctionswith in-plane dimension 100× 100nm2.

3.1.2 Deposition of disordered granular Aluminum lms 1

The standard method for the fabrication of disordered Al lms is the deposition of Al at a

reduced oxygen pressure [42, 43]. Such lms consist of nanoscale grains (3−4 nm in diameter)

partially covered by AlOx. The disordered granular Aluminum lms in the sample discussed

1This section is based on the supplemental material published in W. Zhang et al., Phys. Rev. Applied11, 011003 (2019).

28

in Chapter 5 were fabricated by DC magnetron sputtering of a 6N-purity Al target in the

atmosphere of Ar and O2. The base pressure for the sputtering system is < 1× 10−6mbar.

Typically, the partial pressures of Ar and O2 were 5× 10−3 mbar and (3÷ 7)× 10−5 mbar,

respectively. In order to improve reproducibility, prior to the disordered Al deposition the

target was pre-cleaned in a pure Ar-plasma by sputtering at a rate of 0.6 nm/s for 5 minutes.

The reactive DC sputtering of disordered Al was then initiated by introducing 1 sccm O2

and 115 sccm Ar gas mixture from two independent feedback-controlled mass ow meters

(MicroTrakTM and SmartTrakTM). By controlling the deposition rate and O2 pressure, the

resistivity of the lms can be tuned between 10−4 Ω·cm and 10−1 Ω·cm.

3.2 Measurement setup2

The experiments discussed in this work were performed using a BlueFors dilution refrig-

erator (DR) rated 200 µW at 100mK with a base temperature of ∼25 mK. The DR was

outtted with microwave coaxial cables from room to base temperature to perform mi-

crowave transmission measurement. A µ − metal shield was installed outside the cryostat

to reduce background magnetic elds. Fig. 3.2(a) shows the measurement setup for the

experiment on the uxon-parity-protected qubit discussed in Chapter 4. Inside the cryo-

stat, coaxial cables with stainless steel inner and outer conductor were used for wiring

inside the DR to transmit signals between the stages with dierent temperatures. Atten-

uators and low-pass lters were installed in the microwave input line to prevent leakage

of thermal radiation into the sample. The signal was amplied by a high-electron mo-

bility transistor (HEMT) amplier (Caltech CITCRYO 1-12, 35 dB gain between 1 and

12 GHz) at 4K before reaching room temperature and further amplied by two ampli-

ers with a total of 60 dB gain. Two Pamtech isolators (each provides 18dB isolation

between 3 and 12 GHz) were anchored at base temperature immediately after the device

to reduce the noise from ampliers. At room temperature, the probe signal at f and the

pump signal at fp, generated by two microwave synthesizers, were coupled to the input

of the cryostat through a directional coupler. At the output of the cryostat, the probe

2This section is based on the supplemental material published in W. Zhang et al., Phys. Rev. Applied11, 011003 (2019).

29

Figure 3.3: Schematics of the microwave measurement setup.

30

signal was downconverted to the intermediate frequency (IF) fIF = |f − fLO| ≈ 30MHz

using mixer M1 with the local oscillator signal fLO. The IF signal was acquired using

AlazarTech ATS 9870 at 1GS/s sampling rate. The magnitude and phase of the signal S21

was obtained by digital demodulation as a =√

(〈a2(t) sin2(2πft)〉+ 〈a2(t) cos2(2πft)〉) and

φ = arctan(〈a2(t) sin2(2πft)〉/〈a2(t) cos2(2πft)〉) − φ0 (here 〈...〉 stands for the time aver-

aging over integer number of periods, typically 106). The reference phase φ0 was provided

by mixer M2. The setup enables two-tone (pump-probe) spectroscopy and time-domain

measurements with microwave powers corresponding to 1− 1000 photon levels. A DC-gate

line was used for measuring the uxon-parity-protected qubit to tune the oset charge on

the CPB island. The DC-gate line was ltered with RC lters at the room temperature

and with a stainless-steel powder lter at the base temperature. A Niobium wire was used

to connect between the 4 K and the 700 mK anges. The DC and microwave signals are

combined using a bias-Tee at base temperature. The experiments on the granular aluminum

lms discussed in Chapter 5 and on the one-dimensional metamaterial discussed in Chapter

6 used the same setup except for the removal of DC-gate line [see Fig. 3.2(b)].

3.3 Sample holders

The samples were connected using wirebonds to RF-tight sample holders made of oxygen-free

high thermal conductivity (OFHC) copper. The sample holders contain SMA feedthroughs

and printed circuit boards (PCB) to route the signals from SMA connectors to the chips.

Two sample holders were designed to house samples with dierent types of input and output

ports. The sample holder shown in Fig. 3.3(a) was used for the sample with ports made of

microstrip lines. A 7 × 7mm2 square recess (left) was used to house the chip so that the

surface of the chip ush with the surface of the CPB. A plate with a rectangular window

(middle) was screwed on top of it to press the chip against the ground. The plate was

designed to be thick enough to prevent the bonding wires from touching the lid (right) but

not too thick that would induce parasitic modes within the frequency range of interest.

The sample holder shown in Fig. 3.3(b) was used for sample with ports made of coplanar

waveguide. Similar to the sample holder described above, the left piece was used to house

the sample. The middle piece containing the PCB routes the excitation signals to the

31

chip. The ground plane of the sample was connected to the ground plane of the PCB using

as many wirebonds as possible with lengths as short as possible so that the frequencies

corresponding to parasitic modes due to the inductance of wirebonds will be beyond the

measurement bandwidth. Silver paste was used for the electrical connections between the

PCB ground to the sample holder and between the central conductors of the SMA connector

and the PCB to avoid non-uniform magnetic eld in the device loops due to magnetic uxes

trapped in the solder.

The sample holder was mounted to the mixing chamber stage through cold ngers also

made of OFHC. In the experiment on the uxon-parity-protected qubit (Chapter 4) and

one-dimensional metamaterial transmission line (Chapter 6), the sample holder was housed

in another RF-tight cylindrical copper box. A superconducting coil providing magnetic elds

was slided on to the cylindrical box. The sample holder and the superconducting coil were

housed in a µ−metal shield to avoid stray eld due to magnetic parts within the cryostat.

Figure 3.4: Photographs of the sample holders. (a) Sample holder for launching from SMA

to microwave stripline and the sample holder (b) Sample holder for launching from SMA to

coplanar waveguide (CPW).

32

Chapter 4

Fluxon-parity-protected qubit : a prototype device 1

In this chapter, we present the microwave spectroscopy experiments of the prototype device

for uxon-parity-protected qubit consisting of a Cooper pair box (CPB) shunted by a tunable

superinductor. We observed the eect of the Aharonov-Casher (AC) interference on the

spectrum. The device spectrum varies as we change the oset charge ng on the CPB island

with the period ∆ng = 2e. The periodic variations are attributed to the charge modulated

AC interference between the uxon tunneling processes in the CPB Josephson junctions.

The |0〉 → |1〉 transition energy depends linearly on external ux Φext as it varies between

integers and half integers of ux quantum. The linear dependence corresponds to the energy

dierence between two uxons that dier by a single ux quantum Φ0. The measured phase

and charge dependences of the frequencies of the |0〉 → |1〉 and |1〉 → |2〉 transitions are in

good agreement with our numerical simulations. Almost complete suppression of the single

uxon tunneling due to destructive interference is observed for the charge ng = emod(2e).

As a result, the |0〉 and |1〉 states correspond to even and odd uxon parity. By further

increasing of the superinductance, the uxon-parity-protected qubit can be used for fault-

tolerant π/4 gate.

4.1 Sample design and measurement

The studied device (Fig. 4.1) consists of a superconducting loop that includes a CPB and a

36 unit cell superinductor[28]. We refer to this loop as the device loop. The magnetic ux Φ

in this loop controls the phase dierence across the CPB. Each unit cell of the superinductor

in this work consists of three large and one small Josephson junctions. The inductance L

1This chapter is based on the work published in M. T. Bell, W. Zhang et al., Phys. Rev. Lett. 116,107002 (2016).

33

Figure 4.1: Sample design. (a) The schematics of the circuit containing the device and theread-out lumped-element resonator. The CPB Josephson junctions are shown as crosses.(b) The layout of the device, the read-out resonator, and the MW transmission line. Thesuperinductor consists of 36 coupled cells, each cell represents a small superconducting loopinterrupted by three larger and one smaller Josephson junctions [44].

reaches its maximum when the unit cell is threaded by half a ux quantum, ΦL = Φ0/2. In

this regime of full frustration, L exceeds the Josephson inductance of the CPB junctions by

2 orders of magnitude. The device loop area ( 1850µm2) was designed much greater than

the superinductor unit cell area (15µm2) .

For dispersive measurement of the device resonances, a narrow portion of the device

loop with the kinetic inductance Lsh was coupled to the read-out lumped element resonator

as shown in Fig.4.1(b). The LC resonator is then coupled to a transmission line with

50Ohm characteristic impedance. The global magnetic eld, which determines the device

loop Φ and the unit cell of the superinductor ΦL, has been generated by a superconducting

solenoid. The oset charge on the CPB island was varied by a gate voltage Vg applied to

the microstrip transmission line [Fig. 4.1(b)]. The device was measured using the setup

described in Chapter 3.

The sample were fabricated using multiangle electron-beam deposition of aluminum

through a PMMA/MMA bilayer lift-o mask as described in Chapter 3. Six devices were

fabricated on the same chip; they can be addressed individually due to dierent resonant fre-

quencies of the read-out resonators. The parameters of the CPB junctions were nominally

the same for all six devices, whereas the maximum inductance of the superinductor was

systematically varied across six devices by changing the in-plane dimensions of the small

34

Table 4.1: Parameters of Josephson junctions in the representative device. Parameters ofthe CPB junctions correspond to the tting parameters; parameters of the superinductorjunctions were estimated using the Ambegaokar-Barato relationship and the resistance ofthe test junctions fabricated on the same chip [44].

Junctions In-plane areas, µm2 EJ ,GHz EC ,GHz

CPB 0.11× 0.11 6 6.4Superinductor large 0.30× 0.30 94 3.3Superinductor small 0.16× 0.16 25 11

junctions in the superinductor. Below we discuss the data for one representative device;

Table I summarizes the parameters of junctions in the CPB junctions and superinductor

(throughout the chapter, all energies are given in the frequency units, 1K ≈ 20.8GHz). All

measurements were performed at the base temperature of a dilutional fridge ∼ 20mK.

Figure 4.2: Spectroscopy of the readout resonator around full-frustration of the superinduc-tor loop. At ΦL/Φ0 = 0.5, the superinductor reaches maximum inductance, resulting in aminimum of the readout resonant frequency.

4.2 First-tone measurement

We perform rst-tone spectroscopy by measuring the transmission for the readout resoanator

as we vary the global magnetic eld while keeping the external DC bias for CPB island xed.

At full frustration, the superinductance reaches the maximum corresponding to the minimum

resonant frequency. Fig. 4.2 shows the rst-tone spectroscopy when ΦL ∼ 0.5Φ0. Because

35

the area of the unit cell is much smaller than the device loop, the phase across the CPB can

change by many periods within the region corresponding to the maximal superinductance.

The periodic change in the phase across the CPB corresponds to the wiggles around the

minimum in Fig. 4.2.

4.3 Two-tone measurement

In the two-tone measurements, the microwaves at the second-tone frequency f2 excited the

transitions between the |0〉 and |1〉 quantum states of the device, which resulted in a change

of its impedance [45]. This change was registered as a shift of the resonance of the readout

resonator probed with microwaves at the frequency f1. The microwave set-up used for these

measurements has been described in Refs. [28, 46, 47]. The resonance frequency f01 of the

transition between the |0〉 and |1〉 states was measured as a function of the charge ng and

the ux in the device loop. The f01 measurements could not be extended below ∼ 1 GHz

because of a high-pass lter in the second-tone feedline.

4.4 Results

The resonances corresponding to the |0〉 → |1〉 transition are shown in Fig. 4.3a as a

function of the gate voltage Vg at a xed value of the magnetic eld that is close to full

frustration of the superinductor unit cells (ΦL ' 0.5Φ0). The dependence f01 (Vg) is periodic

in the charge on the CPB island, ng, with the period ∆ng = 1 (here and below the charge

is measured in units 2e (mod 2e)). The increase of temperature above 0.3K resulted in

reducing the period in half due to the thermally generated quasiparticles population. Figure

4.3 also shows the resonance of the read-out resonator at fR = 6.45 GHz and the self-

resonance of the superinductor fL ≈ 5.5GHz. All three resonances are shown in Fig. 2b for

ng ≈ 0.47(Vg = 0) and ΦL ≈ 0.5Φ0. Weaker resonances observed at f2 ≈ 3 GHz and 4.8

GHz at Vg = −30mV correspond to the multi-photon excitations of the higher modes of the

superinductor.

Note that no disruption of periodicity neither by the quasiparticle poisoning [48] nor

by long-term shifts of the oset charge was observed in the data in Fig. 4.3(a) that were

36

Figure 4.3: Panel (a): The transmitted microwave power |S21|2 at the rst-tone frequencyf1 as a function of the second-tone frequency f2 and the gate voltage Vg measured at axed value of ΦL = 0.5Φ0. The power maxima correspond to the resonance excitations ofthe device (f2 = f01), the superinductor (fL), and the read-out resonator (fR). Note thatthe resonance measurements could not be extended below ∼ 1 GHz because of a high-passlter in the second-tone feedline. Panel (b): The frequency dependence of the transmittedmicrowave power measured at Vg = 0V and ΦSL = 0.5Φ0 [44].

measured over 80 min. With respect to the quasiparticle poisoning, this suggests that on

average, the parity of quasiparticles on the CPB island remains the same on this time scale.

In the opposite case, the so-called eye patterns would be observed on the dependences of

the resonance frequency on the gate voltage [49]. Signicant suppression of quasiparticle

poisoning was achieved due to the gap engineering [48] (the superconducting gap in the thin

CPB island exceeded that of the thicker leads by ∼ 0.2K), as well as shielding of the device

from infrared photons [50].

The expected ux dependence of the energy levels of the device is shown in Fig.4.4. This

ux dependence can be understood by noting that in the absence of uxon tunneling (the

dotted curves in Fig. 4.4a corresponding to ng = 0.5 and identical CPB junctions) dierent

states are characterized by a dierent number m of uxons in the device loop. At EJ EL

the energies of these states are represented by crossing parabolas EL(m,Φ) = 12EL(m− Φ

Φ0)2.

The phase slip processes mix the states with dierent numbers of uxons and lead to the

37

Figure 4.4: Panel (a): The ux dependence of the device energy levels obtained by numericaldiagonalization of the Hamiltonian (see the Supplemental Material of Ref. [44] for details,the tting parameters are listed below). The solid curves correspond to ng = 0.5, thedashed curves - to ng = 0 (the blue curves correspond to the ground state |0〉, the yellowcurves - to the state |1〉, and the green curves - to the state |2〉). For comparison we alsoplotted the dotted curves that correspond to the fully suppressed uxon tunneling; in thiscase there are no avoided crossings between the parabolas that represent the superinductorenergies EL(m,Φ) = 1

2EL(m − ΦΦ0

)2 plotted for dierent m. Panel (b): The dependencesof the resonance frequencies f01 (red dots - ng = 0, red squares - ng = 0.5) and f02 (bluedown-triangles - ng = 0, blue up-triangles - ng = 0.5 ) on the ux in the device loop.The theoretical ts (solid curves - ng = 0.5, dashed curves - ng = 0) were calculated withthe following parameters: EJ = 6.25 GHz, the asymmetry between the CPB junctions4EJ = 0.5 GHz, EC = 6.7 GHz, EL = 0.4 GHz (L = (Φ0

2π )2/EL h 0.4µH), ECL = 5 GHz[44].

level repulsion. The qualitative picture of uxon tunneling and AC interference is in good

agreement with the observed level structure shown in Fig. 4.4b.

Figure 4.4(b) shows the dependences of the resonance frequencies of the |0〉 → |1〉 and

|0〉 → |2〉 transitions (f01 and f02, respectively) on the ux in the device loop for the charges

ng = 0 and 0.5. In line with the level modeling, at ng = 0 the frequency f01 periodically

varies as a function of phase, but never approaches zero. On the other hand, when ng =

0.5, the amplitudes of uxon tunneling across the CPB junctions acquire the Aharonov-

Casher phase dierence π. Provided that the CPB junctions are identical, the destructive

interference should completely suppress uxon tunneling, which results in vanishing coupling

between the states |m〉 and |m ± 1〉 and disappearance of the avoided crossing. Since the

38

dierence EL(m,Φ)−EL(m± 1,Φ) is linear in Φ, the spectrum at ng = 0.5 should acquire

the sawtooth shape. This is precisely what has been observed in our experiment. To better

t the experimental data, we have assumed that the Josephson energies are slightly dierent

for the CPB junctions (4EJ < 0.5 GHz); for this reason, the minima of the theoretical

sawtooth-shaped dependence f01(Φ) are slightly rounded. Fitting allowed us to extract all

relevant energies (see the caption to Fig. 4.4). The amplitude of the single phase slips does

not exceed 0.2 GHz, the amplitude of the double phase slips Edps is 0.4 GHz, which is ∼ EL.

A protected qubit suitable for fault-tolerant operation requires Edps EL. This condition

can be satised by decreasing EL and meanwhile increasing ECL. It requires superinductor

with higher inductance and smaller in-plane dimension per unit length. A superinductor

with the backbone made of high-kinetic inductance disordered superconducting material will

be able to meet the requirement.

4.5 Conclusion

We have observed the eect of the Aharonov-Casher interference on the spectrum of the

Cooper pair box (CPB) shunted by a tuanble superinductor. Large values of L (EL EJ)

are essential for the observation of the AC eect with the Cooper pair box. We have

demonstrated that the amplitudes of the uxon tunneling through each of the CPB junctions

acquire the relative phase that depends on the CPB island charge ng. In particular, the phase

is equal to 0 (mod 2π) at ng = 2ne and π (mod 2π) at ng = e (2n+ 1). The interference

between these tunneling processes results in periodic variations of the energy dierence

between the ground and rst excited states of the studied quantum circuit; the period of the

oscillations corresponds to ∆q = 2e. The phase slip approximation provides quantitative

description of the data and the observed interference pattern evidences the quantum coherent

dynamics of our large circuit. By further increase of the superinductance, the qubit is able

to perform fault-tolerant Cliord gate.

39

Chapter 5

Microresonators fabricated from high-kinetic-inductance

Aluminum lms1

5.1 Introduction

The development of novel quantum circuits for information processing requires the imple-

mentation of ultra-low-loss microwave resonators with small dimensions [24]. Superconduct-

ing resonators have become ubiquitous parts of high-performance superconducting qubits

[51, 52] and kinetic-inductance photon detectors [53]. An important resource for resonator

miniaturization is the kinetic inductance of superconductors, LK , which can exceed the

magnetic (geometrical") inductance by orders of magnitude in narrow and thin supercon-

ducting lms [25]. High kinetic inductance translates into a high characteristic impedance

Z of the microwave (MW) elements, slow propagation of electromagnetic waves, and small

dimensions of the MW resonators. Ultra-narrow wires and thin lms of Nb and NbN [53, 54],

TiN [55], InOx[56, 57], and granular Al [58] were studied recently as candidates for high-LK

applications.

Research in high-LK elements also has an important fundamental aspect. According

to the Mattis-Bardeen (MB) theory [59], the kinetic inductance of a thin superconducting

lm LK(T = 0) is proportional to the resistance of the lm in the normal state, RN , and

thus increases with disorder. This theory, however, cannot be directly applied to strongly

disordered superconductors near the disorder-driven superconductor-to-insulator transition

(SIT). Recent theories predict a rapid decrease of the superuid density near the SIT and

the emergence of sub-gap delocalized modes that would result in enhanced dissipation at

microwave frequencies [27, 60]. Thus, the study of the electrodynamics of strongly disordered

superconductors may also contribute to a better understanding of the disorder-driven SIT.

1This chapter is based on the work published in W. Zhang et al., Phys. Rev. Applied 11, 011003 (2019).

40

In this chapter, we present a detailed characterization of the half-wavelength microwave

resonators fabricated from disordered Aluminum lms. Our interest in high-LK lms was

stimulated by the possibility of fabrication of superinductors (dissipationless elements with

microwave impedance greatly exceeding the resistance quantum RQ = h/(2e)2 [28, 61, 62]),

and the development of uxon-parity-protected qubit [44]. We have fabricated resonators

with an impedance Z as high as 5 kΩ, ultra-small dimensions and relatively low losses. The

study of the temperature dependences of the resonance frequency fr and intrinsic quality

factor Qi at dierent MW excitation levels allowed us to identify resonator coupling to

TLS in the environment as the primary dissipation mechanism at T . 250 mK; at higher

temperatures the losses can be attributed to thermally excited quasiparticles.

5.2 Experimental details

All microwave (MW) resonators studied in this chapter consisted of two parts. First, the

50-Ohm coplanar MW transmission line (TL) was formed on an intrinsic Si substrate by

electron beam deposition of a 140-nm-thick lm of pure Al through a lift-o mask, which

comprised of a 300-nm-thick e-beam resist (the top layer) and a 150-nm-thick copolymer

(the bottom layer). After the deposition of the bilayer resist and its patterning with e-beam

lithography, the sample was placed in a reactive ion etching system and etched with 75

mbar O2 plasma at a power of 30 watts for 30 seconds to remove any resist residue from

the substrate surface. The use of this pure Al transmission line facilitated the impedance

matching with the MW tract and eliminated spurious resonances (a large number of these

resonances is observed if high-Lk lms are used for both the TL and resonator fabrication).

After the second e-beam lithography with alignment precision better than 0.5 µm, several

half-wavelength disordered Al resonators were fabricated on the same substrate by reactive

DC magnetron sputtering as described in Chapter 3 (Fig. 5.1). The parameters of several

representative samples are listed in Table 5.1.

For the resonator characterization at ultra-low temperatures, we used a microwave setup

developed for the study of superconducting qubits described in Chapter 3. The resonators

were designed with the resonance frequencies fr ≈ 2 − 4 GHz, which allowed us to probe

the rst three harmonics of the resonators within the setup frequency range (2÷ 12) GHz.

41

Figure 5.1: (a) Microphotograph of a portion of the half-wavelength resonator capacitivelycoupled to the coplanar waveguide transmission line. Light green - Al ground plane andthe central conductor of the transmission line, green - silicon substrate, black - the centralstrip of the resonator made of strongly disordered Al. (b) Several resonators with dierentresonance frequencies coupled to the transmission line [63].

Dierent resonance frequencies of the resonators enabled multiplexing in the transmission

measurements. In order to ensure accurate extraction of the internal quality factor Qi, the

resonators were designed with a coupling quality factor Qc of the same order of magnitude

as Qi.

5.3 Microwave characterization

The resonators were characterized using a wide range of MW power PMW , two-tone (pump-

probe) measurements, and time domain measurements. The resonator parameters fr, Qi,

and Qc were found from the simultaneous measurements of the amplitude and the phase of

the transmitted signal S21(f) using the procedure described in Refs. [64, 65]. The kinetic

inductance LK of the central conductor of the resonators, which exceeded the magnetic

inductance by several orders of magnitude, was calculated as LK = 1/4f2rC (the capacitance

C between the resonator strip and the ground was obtained in the Sonnet simulations). The

parameters of several representative resonators are listed in Table 5.1.

The measured sheet kinetic inductance LK2 ≈ 2 nH/2 is similar to that reported for

granular Al lms in Ref. [66] and TiN in Ref. [67], and exceeds by a factor-of-2 LK2

42

realized for ultra-thin disordered lms of InOx [56, 68]. For the disordered Al lms with

ρ < 10 mΩ·cm, LK2 is in good agreement with the result of the MB theory [59], LK2(T =

0) = ~R2/π∆(0), where ∆(0) is the BCS energy gap at T = 0 K. Very large values

of LK2 allowed us to realize the characteristic impedance Z =√LK/C as high as 5 kΩ

for the resonators with narrow (w = 0.7 µm) central strips. The speed of propagation of

the electromagnetic waves in such resonators does not exceed 1% of the speed of light in

free space; accordingly, their length is two orders of magnitude smaller than that for the

conventional CPW resonators with the impedance Z = 50 Ω.

To identify the physical mechanisms of losses in the resonators, we measured the depen-

dences of fr and Qi on the temperature (T = 25÷450 mK) and the microwave power PMW .

Below we show that in the case of moderately disordered lms (resonators #2 − 4), both

the dissipation and dispersion at T < 0.25 K can be attributed to the resonator coupling

to the TLS [69] in the environment, whereas at higher temperatures they are controlled by

the T dependence of the complex conductivity of superconductors, σ(T ) = σ1(T )− iσ2(T )

[59].

Table 5.1: Summary of the measured parameters of AlOx resonators [63].

#w, l, fr, ρ, Tc, LK , Z,

µm µm GHz mΩ·cm K nH/

2 kΩ

1 11.0 1090 2.42 19.2 1.4 2.0 0.6

2 7.4 765 4.05 4.2 1.7 1.2 1.1

3 1.4 445 3.69 4.2 1.7 1.2 2.9

4 0.7 265 3.88 9.9 1.75 2.0 5.0

5.3.1 The resonance frequency analysis

We start the data analysis with the non-monotonic temperature dependence of the rela-

tive shift of the resonance frequency δfr(T )/fr0 ≡ [fr(T ) − fr(25mK)]/fr(25mK). Figure

5.2(a) shows the dependences δfr(T )/fr0 measured for three resonators (#2 − 4) with dif-

ferent width w. The non-monotonic character of these dependences is due to competing

eects of TLS [70] and thermally-induced quasiparticles on fr. The low-temperature part of

δfr(T )/fr0 is governed by the T -dependent TLS contribution to the imaginary part of the

complex dielectric permittivity ε(T ) = ε1(T )+iε2(T ). It should be noted that, in contrast to

43

Figure 5.2: The temperature dependences of resonance frequency shift δfTLSr (T )/fr0 (a)and the internal quality factor Qi (b) for the resonators #2− 4 measured at n ≈ 1(5) andn 1(4). The tting curves correspond to Eq. (5.2) and Eq. (5.7), respectively [63].

the TLS-related losses, the frequency shift δfTLSr is expected to be weakly power-dependent

[71]. Indeed, the temperature dependences measured for the dierent values of PMW almost

coincide; this simplies the analysis and reduces the number of tting parameters. The

low-temperature part of δfTLSr (T ) is well described by the following equation [53]:

δfTLSr (T )

fr0=Vfδ0

π

[Ψ<

(1

2+

1

2πi

hfrkBT

)− ln

(hfrkBT

)]. (5.1)

Here Ψ<(x) is the real part of the complex digamma function, the TLS participation

ratio Vf is the energy stored in the TLS-occupied volume normalized by the total energy

in the resonator, and the loss tangent δ0 characterizes the TLS-induced microwave loss in

weak electric elds at low temperatures kBT hfr. The product Vfδ0 is the only tting

parameter, its values are listed in Table 5.2. The obtained values of Vfδ0 are close to

that found for Al-based [71] and AlOx-based resonators [67, 72]. Note that resonator #4

demonstrates the most pronounced increase of fr(T ) with temperature due to the stronger

electric elds and a larger participation ratio characteristic of the high-Z resonators [73].

At T > 0.25 K, fr rapidly drops due to the decrease of the superuid density. The

44

Table 5.2: Summary of the tting parameters [63].

# ∆(0)/kBTc β Vfδ0 · 10−4 nc(0) · 10−3

2 1.96 0.60 1.4 50

3 1.98 0.55 4.8 1.6

4 1.88 0.38 6.7 0.23

dependences δfr(T ) over the whole studied T range can be described as

δfr(T )/fr0 = δfTLSr (T )/fr0 + δfMBr (T )/fr0 (5.2)

where

δfMBr (T )

fr0=

1

2

[σ2(T )− σ2(25mK)

σ2(25mK)

](5.3)

is the resonance shift due to the T -induced break of Cooper pairs and subsequent increase

of the kinetic inductance, calculated in the thin lm limit [71]. The only free parameter in

δfMBr (T )/fr0 is the gap energy ∆(0), which can be found by tting of the high-T portion

of δfr(T )/fr0 [Eq. (5.2)]; the measured ratio ∆(0)/Tc is about 10% greater than the BCS

value of 1.76kB, which is consistent with previously reported data [74].

5.3.2 The quality factor analysis

We now proceed with the analyses of losses. We observed the enhancement of the internal

quality factor Qi with increasing the average number of photons in the resonators, n =

2PMWQ2l /(Qchf

2r ) [75], where Ql = (1/Qi + 1/QC)−1 is the loaded quality factor. The

dependences Qi(n) for three resonators with dierent w measured at the base temperature

≈ 25 mK are shown in Fig. 5.3. Similar behavior of Qi(n) have been observed for many types

of CPW superconducting resonators (see, e.g. [53, 76] and references therein), including the

resonators based on disordered Al lms [58, 66]. Note that the increase of Qi with the input

MW power PMW is limited by the resonance distortion by bifurcation at PMW > P∗. For

the resonators with Ql & 104 the onset of bifurcation is observed for the microwave currents

I∗ =√

2P∗/Z which scale approximately as Idp/√Ql [77], where Idp is the Ginzburg-Landau

depairing current in the central strip [25].

45

The power-dependent intrinsic losses can be attributed to the resonator coupling to the

TLS with the Lorentzian-shaped distribution

g(ETLS) ∼ 1

(ETLS − hfr)2 + (~/τ2)2, (5.4)

where ETLS is the energy of TLS and τ2 is its dephasing time [78]. Once the MW power PMW

reaches some characteristic level Pc and the Rabi frequency of the driven TLS ΩR ∼√PMW

exceeds the relaxation rate 1/√τ1τ2, the population of the excited TLS increases, and the

amount of energy that the TLS with fTLS ≈ fr can absorb from the resonator decreases.

Thus, the high PMW burns the hole" in the density of states (DoS) of dissipative TLS. The

width of the hole is κ/2πτ2, the power-dependent factor can be written as

κ =

√1 +

(n

nc

)β, (5.5)

where n and nc correspond to PMW and Pc, respectively. Note that the exponent β is known

to be dependent on the electric eld distribution in a resonator [79], and the characteristic

power nc increases with temperature by orders of magnitude due to a strong T -dependence

of τ1 and τ2 [80, 81]. Taking into account the TLS saturation at high temperature, the power

dependence of the TLS-related part of the loss tangent can be expressed as follows [73]:

δTLS(n, T ) =Vfδ0

κtanh

(hfr

2kBT

). (5.6)

By tting the experimental data with Eq. (5.6) we found β and nc, the obtained param-

eters are listed in Table . We found that larger values of β correspond to wide strips, and

the extracted nc(0) scales as the square of the electric eld on the surface of the resonator.

The experimental dependences Qi(T ) measured for resonators #2−4 at n w 1 and n 1

[Fig. 5.2(b)] are well described by the sum of the TLS contribution [Eq. (5.6)] and the MB

term δMB = σ1(T )/σ2(T ) [71]:

Qi(T ) = δTLS(T, β, nc, Vfδ0) + δMB[T,∆(0)]−1. (5.7)

46

Figure 5.3: The dependences Qi(n) at T ≈ 25 mK for the resonators with dierent widths.Solid curves represent the theoretical ts of the quality factor governed by TLS losses [Eq.(5.5), see the text for details] [63].

The agreement of measured Qi with the prediction of Eq. (5.7) over the whole measured

temperature range proves that the losses in the developed resonators are limited by the sum

of TLS and MB terms.

5.3.3 The two-tone and time-domain measurements

We obtained an additional information on the TLS-related dissipation by performing the

pump-probe experiments in which Qi was measured at a low-power (n w 1) probe signal

while the power Pp of the pump signal at the frequency fp was varied over a wide range.

Figure 5.4(a) shows the dependences Qi(Pp) measured at dierent detuning values ∆f =

fp−fr = 0, ±1 MHz, and ±10 MHz. Note that we have not observed any changes in Qi when

the pump signal was applied at the second and third harmonics of the resonator. Also, Qi

was Pp-independent when we monitored the second harmonic and applied the pump signal

at the rst harmonic.

47

Figure 5.4: (a) The dependences of Qi for resonator #1 on the pump tone power Pp forseveral values of detuning ∆f between resonance and pump frequencies. (b) The valuesof Qi measured versus detuning ∆f at a xed number of the pump tone photons in theresonator np ≈ 1000. The error bars are derived from the covariance matrix obtained fromnonlinear tting of the measurement of S21(f) [63].

Since the resonator coupling to the pump signal varies by several orders of magnitude

within the detuning range 0 ÷ 10 MHz, it is more informative to analyze Qi as a function

of the average number of the pump" photons in the resonator, np = Pp(1 − |S21(fp)|2 −

|S11(fp)|2)/hf2p , where S21 and S11 = 1−S21 are the transmission and reection amplitudes

at the pump frequency, respectively. The dependence Qi on the detuning ∆f for a xed

np ≈ 1000 is depicted in Fig. 5.4(b). The resonance behavior of Qi(∆f) is expected since

only a narrow TLS band [Eq. (5.4)] contributes to dissipation: the hole" extension in the

DoS is limited by ∼ κ/τ2 around the pump frequency. Indeed, using the approach developed

in [82], one can obtain the following expression:

Qi(∆) = Q0

[1 +

(κ/2πτ2)2

∆f2 + κ(1/2πτ2)2

], (5.8)

where Q0 is the o-resonance quality factor, and introduced by Eq. (5.5) factor κ might

be calculated as κ = Qmax/Q0. The dephasing time is the only tting parameter and it is

found to be τ2 ≈ 60 ns. This result agrees with the measurements of the dephasing time for

individual TLS in amorphous Al2O3 tunnel barrier in Josephson junctions [83].

Interactions between the high-frequency (coherent, E > kBT ) TLS with the low-frequency

(thermal, E < kBT ) uctuators result in the TLS spectral diusion as well as the icker

48

noise. The telegraph noise in the resonance frequency fr is expected if some of the TLS

with f ≈ fr are strongly coupled to a resonator. Typical TLS densities for Al/AlOx junc-

tions are ∼1 (GHz·µm2)−1 [69]. Interestingly, the number of strongly coupled TLS for our

resonators (assuming that the strongly coupled TLS are in the oxide layer of the resonator)

is of the order of unity [1 (GHz · µm2)−1 × 0.1MHz × 104µm2]. To study the telegraphy

noise, we repetitively measured S21 at a xed frequency on a slope of the resonance dip for

a few minutes. Figure 5.5 shows an example of the measured telegraph noise in Re[S21].

The characteristic time scale of random switching between two Re[S21] levels is 1-10 sec-

onds. This noise can be attributed to interactions of the resonators with a small number of

strongly coupled TLS.

Figure 5.5: The time dependence of Re[S21] measured at T = 25 mK at a xed frequencyon the slope of a resonance dip. The microwave power corresponds to 〈n〉 ∼ 1000. Eachpoint corresponds to the data averaging over 1 sec [63].

We have performed the time domain measurements of the TLS relaxation time for res-

onator #1 using the pulse sequence shown in Fig. 5.6(a). A 0.5 s-long pump pulse was

applied to the resonator at the beginning of each duty cycle. A readout pulse at the single-

photon power level lasting for 36 ms followed the pump pulse and was digitized to obtain

S21. The readout delay time was varied between 0 s and 1 s. Figure 5.6(b) shows the re-

sult of the experiment at the readout frequency f0 = 2.4258 GHz and the pump frequency

fp = f0 + 1 MHz. The change in |S21(f0)| at t = 0.5 s is consistent with CW measurements

49

at the same readout frequency and power level when a pump tone was turned on and o.

This indicates that an upper limit of the TLS relaxation time for our sample is much less

than 36 ms.

Figure 5.6: (a) The pulse sequence. (b) The time dependence of |S21| measured at f0 =2.4258 GHz. The pump pulse at fp = f0 +1 MHz was applied between t = 0 s and t = 0.5 s.The pump tone power corresponds to np ≈ 1000. Each data point was averaged over 4000cycles with the same readout delay time. The inset shows CW measurement of S21 versusf with (red) and without (blue) the pump signal and indicates the position of f0 used inthe relaxation time measurement. The readout power was at the single photon level for allmeasurements on this plot [63].

5.4 Summary

In conclusion, we have fabricated CPW half-wavelength resonators made of strongly disor-

dered Al lms. Because of the very high kinetic inductance of these lms, we were able to

signicantly reduce the length of these resonators, down to ∼ 1% of that of conventional

CPW resonators with a 50 Ω impedance. Due to ultra-small dimensions and relatively

low losses at mK temperatures, these resonators are promising for the use in quantum su-

perconducting circuits operating at ultra-low temperatures, especially for the applications

50

that require numerous resonators, such as multi-pixel MKIDs [53, 77]. The high impedance

Z =√LK/C of the narrow resonators can be used for eective coupling of spin qubits

[84, 85]. The high resonator impedance imposes limitations on the strength of resonator

coupling to the transmission line. For the studied CPW resonators with Z ∼ 5 kΩ, the

strongest realized coupling (when half of the resonator length was used as the element of

capacitive coupling to the transmission line) corresponded to Qc ∼ 104. On the other hand,

for many applications, such as large MKID arrays that require a high loaded Q factor, this

should not be a limitation.

We have shown that the main source of losses in these resonators at T Tc is the

coupling to the resonant TLS. A comparison of our results with those of the other groups

shows that the obtained Qi values, increasing from (1÷2)×104 in the single-photon regime

to 3×105 at high microwave power, are typical for the CPW superconducting resonators

with similar TLS participation ratios. This implies that the disorder in Al lms does not

introduce any additional, anomalous losses. Most likely, the relevant TLS are located near

the edges of the central resonator strip either in the native oxide on the Si substrate surface

or in the amorphous oxide covering the lms. Further increase of Qi can be achieved by the

methods aimed at the reduction of surface participation, such as substrate trenching (see

[86] and references within) and increasing the gap between the center conductor and the

ground plane [79]. The evidence for that was provided by the results of Ref. [66] obtained

for the modied three-dimensional microstrip structures based on disordered Al lms. It is

also worth mentioning that the losses can be reduced using TLS saturation by the microwave

signal outside of the resonator bandwidth but within the TLS spectral diusion range. A

fundamental issue pertinent to all strongly disordered superconductors is the development of

a better understanding of the impedance of superconductors near the disorder-driven SIT.

This issue requires further research, and the microwave experiments with the resonators

made of disordered Al and other disordered materials demonstrating the SIT may shed light

on the nature of this quantum phase transition.

51

Chapter 6

Josephson metamaterial with a widely tunable positive or

negative Kerr constant1

6.1 Introduction

In conventional optics, a material whose refractive index n is aected by the intensity of

an electric eld n ∝ |E|2 is known as a Kerr medium [87]. Analogous to nonlinear optics,

microwave superconducting circuits exhibit the Kerr eect due to a nonlinear response of

their kinetic Josephson inductance that determines the circuit impedance. In supercon-

ducting circuits based on Josephson junctions the Kerr eect originates from the φ2 term

in the expansion of the Josephson inductance L(φ) = Φ0/(2πIc cosφ), where φ is the su-

perconducting phase across the junction, Ic is the junction critical current, and Φ0 is the

magnetic ux quantum. The Kerr eect in superconducting circuits has been used to gen-

erate squeezed states of light [88], traveling-wave parametric ampliers [22, 39, 89, 90], and

superconducting quantum bits [91].

In this chapter, we demonstrate a novel Josephson metamaterial with a Kerr constant

tunable over a wide range that includes both positive and negative values. Such a nonlinear

medium can nd applications in wave-packet rectication [92], analogues of nematic optical

materials [93], superinductances [28], and in Josephson traveling-wave parametric ampliers

(JTWPA) [39], which was the motivation behind the present work. The metamaterial is

composed of a one-dimensional chain of asymmetric superconducting quantum interference

devices (SQUIDs) with nearest-neighbor coupling through common Josephson junctions Fig.

6.1(a). The same magnetic ux threads all SQUIDs to allow for tunability of the chromatic

and nonlinear dispersion. Tunable superconducting metamaterials [94] composed of passive

[95], and electrically active meta-atoms such as SQUIDs [9699] or qubits [100102] have

1This chapter is based on the work published in W. Zhang et al., Phys. Rev. Appl. 8, 051001 (2017).

52

been investigated. Below we discuss a novel topology with direct coupling between meta-

atoms in a structure with a tunable Kerr constant which can change sign. This design

oers signicant advantages for several applications, including parametric amplication in

a JTWPA. As the magnetic ux on the metamaterial is varied, we observe a monotonic

dependence of the chromatic dispersion and a Kerr constant which varies over a wide range

from positive to negative. This novel metamaterial compares favorably with the Josephson

circuits previously used for parametric amplication [103] in two important aspects. First,

the Kerr eect is much stronger and the magnitude of a Kerr constant can be easily tuned

by the magnetic ux Φ in the SQUID loops. Second, the sign of the Kerr constant is also

ux-dependent, which is an important resource for the development of quantum-limited

parametric ampliers and other superconducting circuits.

6.2 Metamaterial Design

The design of the proposed metamaterial is shown in Fig. 6.1(a); it resembles the design

of the Josephson superinductor introduced by us in Ref. [28] . Each unit-cell of length a

is composed of two coupled asymmetric SQUIDs with a single smaller Josephson junction

with critical current Ijs0 and capacitance Cjs in one arm and two larger Josephson junctions

with critical current Ijl0 = rIjs0 and capacitance Cjl = rCjs in the other arm. Here r is the

ratio between the areas of the larger and smaller junctions. The eld dependent Josephson

inductance of the metamaterial is

L(φ,Φ) = L0

([r2

+ 2 cos(

2πΦ

Φ0

)]−[ r

16+ cos

(2π

Φ

Φ0

)]φ2)−1

,

(6.1)

where L0 = ϕ0/Ijs0, ϕ0 = Φ0/(2π), and φ is the phase dierence across a unit-cell. At

a critical value r0 = 4 the rst term in L(φ) vanishes at Φ/Φ0 = 0.5 and the quadratic

term dominates [28, 39]. Propagation of electromagnetic waves with wavelengths λ a in

this metamaterial in the absence of dissipation is described by the following nonlinear wave

equation for superconducting phases on the nodes between unit-cells ϕ(z, t) [28, 37]

53

a2

L0

[r2

+ 2 cos(

2πΦ

Φ0

)]∂2ϕ

∂z2+ a2Cjs

(r2

+ 2) ∂4ϕ

∂t2∂z2−

Cgnd∂2ϕ

∂t2− γ ∂

∂z

[(∂ϕ∂z

)3]= 0,

(6.2)

where γ = a4/(3ϕ20L0)[r/16 + cos(2πΦ/Φ0)]. Cgnd is the distributed capacitance between

the metamaterial and the ground plane. The linear (low-power) dispersion relation and

solution to Eq. (6.2) is

k =ω√L0Cgnd

a

√[r2 + 2 cos

(2π Φ

Φ0

)]− ω2L0Cjs

(r2 + 2

) (6.3)

and A(z) = A0e−i(k+α)z respectively, where

α =3γk5|A0|2

8ω2Cgnd, (6.4)

and A0 is the superconducting phase amplitude, see Ref. [39]. Electromagnetic waves

which propagate in this metamaterial acquire a phase shift −αz, where z is the direction of

propagation along the metamaterial which depends on the intensity |A0|2 analogous to light

traveling in a Kerr medium [36, 87]. The Kerr constant γ and thus the intensity dependent

phase shift can vary over a wide range with magnetic ux tuning, and can even change sign

from positive to negative.

6.3 Microwave Characterization

To demonstrate the tunable properties of the Josephson metamaterial, several devices were

fabricated at Hypres Inc. using the standard Nb/AlOx/Nb trilayer process with a nominal

critical current density of 30 A/cm2. The devices are shown schematically in Fig. 6.1(b).

The perforated bottom metal layer M0 (gray) acted as the ground plane; it was separated

from the metamaterial structure by 150 nm of SiO2. Metal layers M1 (green) and M2

54

Table 6.1: Parameters of two Josephson metamaterial devices [104].

Device r Cjs(fF)

Cjl(fF)

Cgnd(fF)

Ijs0(µA)

Ijl0(µA)

RNs(kΩ)

RNl(kΩ)

1 5.9 50 300 75 0.25 1.5 8.44 1.42

2 7 50 350 75 0.25 1.75 8.43

(blue) form the coupled asymmetric SQUID structure of the metamaterial. The Josephson

junctions are shown in red, and the vias between M1 and M2. Fig. 6.1(c) shows an optical

image of the device. The design parameters of two representative devices are listed in Table

1. The junction critical currents were determined from the Ambegaokar-Barato formula

[25] using the normal state resistance RN(s,l) of the on-chip smaller and larger test junctions,

respectively, measured at room temperature. The variations in the normal state resistance

within the same batch of devices did not exceed 1%. Each SQUID in the unit-cell has a

loop area of 13 × 7µm2 and the unit-cell which is composed of two SQUIDs has a length

a = 14µm. Each device measured contains 125 unit-cells and have a physical length l = 125a

(1.75 mm).

Investigation of the chromatic and nonlinear dispersion in the Josephson metamaterial

was performed in the cryostat described in Chapter 3. The Josephson chain was included in

the microwave transmission line, and transmission measurements were performed with an

Anritsu 37369A vector network analyzer. A superconducting solenoid was used to provide

a uniform magnetic ux bias to all SQUID loops in the metamaterial.

The dispersion of the Josephson metamaterial in the linear (low-power) regime was in-

vestigated with the transmission measurements of the phase shift as a function of magnetic

ux (Fig. 6.2). The linear transmission measurements were performed at a signal power

of (−130 dBm) (−100 dBm), where S21 was independent of the signal intensity. Figure

6.2 illustrates the low-power phase shift across the metamaterial −lk(Φ, f) measured for

devices 1 and 2. Solid lines are ts to Eq. (6.3) utilizing the design parameters listed in

Table 6.1 and Ijs0 as the only tting parameter (for brevity only one measurement frequency

f = 4 GHz is shown for device 1). The values of Ijs0 = 0.21 ± 0.01µA, the same for both

devices, were slightly lower than Ijs0 = 0.25µA estimated using the Ambegaokar-Barato

formula (Table 6.1). The chromatic dispersion of the metamaterial at Φ/Φ0 = 0.5 is shown

55

Figure 6.1: Josephson metamaterial based on a chain of coupled asymmetric SQUIDs. (a)Circuit schematic of the metamaterial. Each unit-cell of the metamaterial consists of twoasymmetric SQUIDs coupled with a shared junction and is of length a. Each SQUID inthe unit-cell is threaded with a magnetic ux Φ and has a capacitance to ground Cgnd. (b)Illustration of the three-metal-layer layout of the device. Metal layer M0 (gray) representsthe ground plane, M1 and M2 are the two metal layers which form the electrodes of coupledasymmetric SQUIDs, red and green vias between M1 and M2 represent Josephson junctionsand M1-to-M2 vias respectively. The purpose of the ngers on M0 in gray and M1 in greenwhich extend into the foreground is to increase the capacitance of the SQUID array toground (M0). (c) Optical image of the measured Josephson metamaterial [104].

for both devices in Fig. 6.3. Solid lines are the expected k(f) dependence calculated with

the initial design parameters in Table 6.1. The eective plasma frequency at Φ/Φ0 = 0.5 of

the elementary unit-cell is fp = [L(Φ = 0.5Φ0)(r/2 + 2)Cjs]−1/2/2π which corresponds to 8

GHz and 12 GHz for device 1 and 2, respectively. The phase velocity υ = a/√L(Φ)Cgnd

for both devices varied between 3 × 106 m/s and 1.5 × 106 m/s for Φ = 0 and Φ = 0.5Φ0

, respectively. The characteristic impedance Z =√L(Φ)/Cgnd of the metamaterial varied

between 60 Ω and 145 Ω over the magnetic ux range from 0 to 0.5Φ0.

Figure 6.4 shows the main result of this work: the dependence of the microwave phase

shift at a signal frequency of 4 GHz as a function of signal power P . An estimated signal

56

Figure 6.2: Low-power transmission measurements of the phase shift across the Josephsonmetamaterial as a function of the magnetic ux for device 1 (lower panel) and device 2(upper panel) at dierent measurement frequencies. Solid lines are ts to Eq. (6.3) [104].

Figure 6.3: Wavenumber as a function of frequency for devices 1 (blue circles) and 2 (redsquares) extracted from the tting procedure of the data at Φ/Φ0 = 0.5 in Fig. 6.2. Solidlines are a plot of Eq. (6.3) with the design parameters listed in Table 6.1 [104].

57

Figure 6.4: Measurements of the microwave phase shift as a function of signal power Pwhere P0 = -70 dBm, at dierent values of the magnetic ux in the metamaterial unit cellsfor device 1 (upper panel) and device 2 (middle panel). Transmission measurements wereperformed at a signal frequency of 4 GHz. Solid lines are ts to Eq. (6.4). (lower panel)shows the Kerr constant of the metamaterial unit cell derived from the ts of the powerdependent transmission phase data in the upper two panels normalized to the Kerr constantof the small Josephson junction γ/γJJ as a function of magnetic ux. As magnetic ux istuned, γ/γJJ varies over a wide range and changes sign from positive to negative [104].

.

58

power at the mixing chamber of P0 = -70 dBm was attenuated at room temperature with

a programmable attenuator (Aeroex 8311) to vary the signal power P to port 1 of the

metamaterial. Microwave transmission measurements were performed over several xed

values of the magnetic ux 0 ≤ Φ ≤ Φ0/2. For each magnetic ux, the phase across the

metamaterial depends on the input power. Near zero eld, the phase across the metamaterial

decreases with signal power (i.e. a positive Kerr constant), similar to a linear chain of

Josephson junctions which would exhibit a non-tunable Kerr constant γJJ = a4/(6ϕ20L0)

[37] which is magnetic ux independent. Fig. 6.4 (lower panel) shows the Kerr constant of

the metamaterial unit-cell derived from the ts of the power dependent transmission phase

data in the upper two panels of Fig. 6.4 normalized to γJJ of the small junctions in the

metamaterial. In contrary to a linear chain of junctions, as the magnetic ux increases, the

magnitude and sign of the phase shift changes: γ becomes negative at 0.3 ≤ Φ/Φ0 ≤ 0.5 as

shown in Fig. 6.4 (lower panel). The magnitude of the Kerr constant in the metamaterial is

similar to γJJ for a linear chain of junctions with critical currents equal to that of the smaller

junctions in the metamaterial γ/γJJ = 2[r/16 + cos(2πΦ/Φ0)]. However, the metamaterial

can be driven with higher microwave power since the majority of the current ows through

the backbone formed by larger critical current junctions. This feature allows for an increase

in the dynamic range of JTWPAs composed of this metamaterial in comparison to linear

chains of junctions. According to Eq. (6.4) it was expected that the Kerr constant γ whould

change sign at a magnetic ux of Φ/Φ0 = cos−1(−r/16)/(2π), which is Φ/Φ0 = 0.3 and

Φ/Φ0 = 0.33 for device 1 and 2 respectively. Indeed, for both devices the sign change of the

Kerr constant was observed in the ux range 0.3 − 0.33(Φ/Φ0) as shown in the γ/γJJ vs.

Φ/Φ0 data in Fig. 6.4 (lower panel). In Fig. 6.4 the solid lines are ts to Eq. (6.4) calculated

with the design parameters in Table 6.1 and Ijs0 = 0.21µA. Best ts were obtained with

P0 = −70 dBm ±1.5 dB as a tting parameter which takes into account the uncertainty

of actual signal power level at Port 1 of the metameterial for dierent tunings of magnetic

ux. The nonlinear wave equation (Eq. (6.2)) which describes the behavior of the Josephson

metamaterial is in good agreement with the phase measurement data.

59

A nonlinear medium with a tunable Kerr constant, which can change sign, is very promis-

ing for the development of Josephson traveling-wave parametric ampliers [39]. State-of-

the-art JTWPAs rely on a four-wave-mixing process which require perfect phase matching

between signal, idler and pump waves propagating along a nonlinear transmission line. In

recent works [22, 38, 89, 105] the required relations between chromatic dispersion and self-

(SPM) and cross-phase (XPM) modulations were realized by tuning the pump frequency

near a pole or band-gap introduced into the nonlinear transmission line via sophisticated

dispersion engineering techniques. The unique feature of the studied metamaterial enables

phase matching due to compensation of the positive chromatic dispersion between the signal,

idler, and pump waves by the negative SPM and XPM . The theory of operation of such a

novel JTWPA was described in Ref. [39].

6.4 Summary

In conclusion, we have developed a unique one-dimensional Josephson metamaterial whose

Kerr constant is tunable over a wide range, and can change sign from positive to negative.

The metamaterial is composed of a chain of coupled asymmetric SQUIDs. The dispersion

properties of the metamaterial are varied with an external magnetic ux threading each

SQUID loop in the array. The transmission measurements of the phase of microwaves prop-

agating along the metamaterial at low and high signal powers veried predictions of a non-

linear wave equation governing the microwave response of the medium. Such a metamaterial

can be used as the nonlinear medium for parametric amplication and phase-matching in a

four-wave-mixing process in Josephson traveling-wave parametric ampliers, its use elimi-

nates the need for complex dispersion engineering techniques.

60

Chapter 7

Conclusions and future work

7.1 Conclusions

In this dissertation, we explored (a) superinductor-based parity-protected qubits, (b) novel

disordered granular Aluminum lms for the superinductor fabrications, and (c) superinductor-

inspired one-dimensional superconducting circuits for parametric amplication of microwave

signals at ultra-low temperatures.

In Chapter 2 and 4, we discussed the operation of the parity-protected qubits and our ex-

periments with a prototype uxon-parity-protected qubit formed by a superconducting loop

consisting of a Cooper-pair box (CPB) with EJ < EC and a superinductor. The superin-

ductor in the qubit was realized as a chain of coupled asymmetric SQUIDs (CAQUIDs)

frustrated by the external magnetic eld. The backbone of the superinductor consists of

junctions with EJ EC . The dynamics of the low energy states of the qubit correspond

to uxons tunneling across the Josephson junctions in the CPB in and out of the supercon-

ducting loop. The uxon tunneling process is dual to the Josephson eect where Cooper

pairs tunnel between two electrodes separated by an insulating barrier. The phase dierence

of the uxon tunneling amplitudes across the two junctions of a CPB depends on the oset

charge on the CPB island due to Aharonov-Casher eect. By measuring the qubit spec-

trum when biasing the oset charge on the CPB at emod(2e), we observed almost complete

suppression of the tunneling of a uxon that change the phase across the CPB by 2π. The

lowest two energy states are composed of uxons of even and odd parities carrying even and

odd numbers of ux quantum. Symmetry between EJ and EC of the two junctions forming

the CPB is essential as it allows the 2π-phase-slip process to happen across either one of

the CPB junctions with equal probability. We reduced the spread of the Josephson and

61

charging energies in the CPB junctions down to ~5% by using the Manhattan pattern fab-

rication technique. Further increase of the inductance of the superinductor will enhance the

quantum ucuations in phase across the CPB in order to perform fault tolerant operations

based on uxon-parity-protected qubits which require each qubit states to be superpositions

of multiple states corresponding to dierent uxons of the same parity.

In Chapter 5, we studied superconducting coplanar-waveguide (CPW) resonators fab-

ricated from disordered (granular) lms of Aluminum. The studied disordered supercon-

ducting lms have high kinetic inductance and low microwave losses. The intrinsic quality

factors of the resonators are limited at ultra-low temperatures by the resonator coupling to

two-level systems in the environment and are comparable with those of resonators made of

conventional superconductors. Nanowires of width 1µm made of the high kinetic inductance

material have characteristic impedance greater than resistance quantum. The wavelength for

microwaves traveling in the transmission lines made of the nanowires are extremely short

(∼ 200µm). The high kinetic inductance nanowires are easier to design and have more

compact in-plane dimensions and higher self-resonant frequencies compared to superinduc-

tors made of a chain of Josephson junctions of the same inductance. In addition to high

kinetic inductance and small dimension, the well-understood losses make these disordered

Aluminum resonators promising for a wide range of microwave applications which include

kinetic inductance photon detectors and superconducting quantum circuits.

In Chapter 6, we studied a noval superinductor-inspired metamaterial transmission lines

based on CAQUIDs. In a transmission line based on Josephson junctions, because of the

nonlinear Josephson inductance, the wave vectors for a transmitted signal composed of

a part linear with respect to its frequency and a nonlinear part due to the power of its

own or that of other signals. The nonlinear eect is called Kerr eect similar to that in

optic bers. The ratio between the nonlinear part versus the power is proportional to a

constant called Kerr constant. We observed that the Kerr constant in the transmission

line based on CASUIQDs is tunable over a wide range from positive to negative values

with magnetic elds. The tunable Kerr nonlinearity of these CAQUID-based transmission

lines facilitate the phase matching of the four-wave mixing process that couples a strong

pump tone to a weak microwave signal for the quantum-limited parametric amplication at

62

ultra-low temperatures.

7.2 Future work

One of the directions of future work on uxon-parity-protected qubits would be optimization

of CASQUIDs-based superinductors. In particular, replacement of Josephson junctions in

the superinductor backbone with a nanowire made of granular Aluminum could signi-

cantly increase the self-resonance frequencies of the superinductor. The optimal working

point for uxon-parity-protected qubits, as discussed in Chapter 2, is when the magnetic

eld through the qubit loop is at zero-frustration and the oset charge on the CPB island

is emod(2e). In the prototype qubit, we observed that the qubit frequency under the two

conditions are close to the resonant frequency of the readout resonator and the frequency

corresponding to self-resonant mode of the superinductor. In order to bring down the qubit

frequency and to raise the self-resonant frequency of the superinductor at the same time,

we need to increase the superinductance while reducing its parasitic capacitance. For the

superinductor used in the prototype qubit, the Josephson inductance corresponding to the

backbone per unit cell is 3nH. The same inductance can be realized using the a 1.5 μm long

and 0.1μm wide wire made of disordered granular Aluminum lms. The in-plane area of the

superinductor can thus be shrunk by at least ve times (see Appendix A for the design of

the "hybrid" superinductor made of granular-Aluminum and Al/AlOx/Al Josephson junc-

tions). The superinductor with smaller in-plane dimensions would have smaller parasitic

capacitance and thus higher self-resonant frequency.

The studied metamaterial transmission line have shown an average 10dB increase in

signal when the pump is applied (see Appendix B). Future studies on the transmission line

include characterization of the gain by comparing the transmission powers with respect to

the signals passing and bypassing the transmission line and characterization of the noise

associated with the amplication.

63

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Appendix A

Design of hybrid superinductor

Figure A.1: Design of hybrid superinductor. The backbone of the superinductor (black)consists of nanowires made lms of granular Aluminum. Josephson junctions are formed atthe crossing of two Aluminum nanowires (blue).

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Appendix B

Increase in transmission power through metamaterial

tranmission line

Figure B.1: Increase in transmission power through the metamaterial transmission linewith a strong pump applied at 3.5 GHz and Φ/Φ0 ≈ 0.5 . The ripples are due to impedancemismatch between the transmission line and the embedding environment.