Depts. of Applied Physics & PhysicsYale University
expt.Andreas WallraffDavid SchusterLuigi Frunzio
Andrew HouckJoe Schreier
Hannes MajerBlake Johnson
Circuit QED: Atoms and Cavities in Superconducting Microwave Circuits
theoryAlexandre BlaisJay Gambetta
PI’sRob Schoelkopf
Steve GirvinMichel Devoret
www.eng.yale.edu/rslab
Overview• Quantum optics and Cavity QED
• The AC Stark shift & backaction of QND measurement – towards splitting the “atom” to see single photons
• The future?:- “bus” coupling of qubits- other possible (microscopic?) circuit elements
• Circuit QED:- One-d microwave cavities and coupling to JJ qubits
• Experiments showing strong coupling – splitting the photon
• The beauty of being off-resonant:- lifetime enhancement/suppression by cavity
Cavity Quantum Electrodynamics (cQED)
2g = vacuum Rabi freq.
= cavity decay rate
= “transverse” decay rate
† †12 ˆ ˆ
2)ˆ )(
2(el J
x zr a a a aE
gHE
Quantized FieldElectric dipole
Interaction2-level system
Jaynes-Cummings Hamiltonian
Strong Coupling = g t
t = transit time
Cavity QED: Resonant Case
r a
vacuumRabi
oscillations
“dressed state ladders”(e.g. Haroche et al., Les Houches notes)
# ofphotons
qubit state
+ ,0 ,1
- ,0 ,1
Microwave cQED with Rydberg Atoms
Review: S. Haroche et al., Rev. Mod. Phys. 73 565 (2001)
beam of atoms;prepare in |e>
3-d super-conducting
cavity (50 GHz)
observe dependence of atom finalstate on time spent in cavity
vacuum Rabi oscillations
measure atomic state, or …
Pexcited
time
Optical Cavity QED
… measure changes in transmission of optical cavity
e.g. Kimble and Mabuchi groups at Caltech
2004: Year of Strong Coupling Cavity QED
superconductor flux and charge qubitsNature (London) 431, 159 & 162 (Sept. 2004)
alkali atoms Rydberg atoms
semiconductor quantum dotsNature (London) 432, 197 & 200 (Nov. 2004)
single trapped atomPRL 93, 233603 (Dec. 2004)
A Circuit Implementation of Cavity QED2g = vacuum Rabi freq.
= cavity decay rate
= “transverse” decay rate
L = ~ 2.5 cm
Cooper-pair box “atom”10 m10 GHz in
out
transmissionline “cavity”
Theory: Blais et al., Phys. Rev. A 69, 062320 (2004)
Advantages of 1d Cavity and Artificial Atom
10 m
Vacuum fields:zero-point energy confined in < 10-6 cubic wavelengths
Transition dipole:
/g d E
0~ 40,000d ea
E ~ 0.2 V/m vs. ~ 1 mV/m for 3-dx 10 larger than Rydberg atom
L = ~ 2.5 cm
Cross-sectionof mode (TEM!):
+ + --
E B
10 m
Implementation of Cavity on a ChipSuperconducting transmission line
Niobium filmsgap = mirror
300mK 1 @ 20 mKn 6 GHz:
2 cm
Si
RMS voltage: 0 2 V2
R
R
VC 0n even when
Qubits: Why Superconductivity?
~ 1 eV
E
2~ 1 meV
ATOM SUPERCONDUCTINGNANOELECTRODE
few electronsN ~ 109
total numberof electrons
superconducting gap
“forest” of states
The Single Cooper-pair Box:an Tunable Artificial Atom
EC
EJNN+1
2~ 1 meV)
I
“Zeeman shift”
V
“Stark shift”
tunnel junctions
(1 nm)
Note scale
Pseudo spin ½: 8 8; 10 10 1
JosephsonCoulombeff
2 2x zEE
H B ����������������������������
Coulomb Energy Josephson tunneling
Bias Gate
The Real Artificial Atom
Island containing108 or 108 +1
pairs
Nb
Nb
Si
Al d
Coupling to Cavity Photons
A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, PRA 69, 062320 (2004)
40~ m ~ 10 d e ea
0 0 0gCg E d eV eV
C
d
ˆ ˆ ˆ2 2el J
box x z
E EH
†ˆcavity RH a a
†int
ˆ ( )H a ag
Jaynes-Cummings
How Big Can a Dipole Coupling Get, Anyway?
20
0 2R R
R
ZV
C
20
1 12 4R RC V
0/ 2R RC Z
for a half-wave resonator: / 2
0 0 0 050 ~ /Z c
20 0 0~
2 K
g eV e Z ZR
0 02 2
/K
Z cR h e
the fine structure constantin circuit form!
2~ 4%
r
g “The Fine Structure
Limit on Coupling”
or g ~ 200 MHz on a 5 GHz transition
Comparison of cQED with Atoms and Circuits
Parameter Symbol Optical cQED with Cs atoms
Microwave cQED/
Rydberg atoms
Super-conducting
circuitQED
Dipole moment d/eao 1 1,000 20,000
Vacuum Rabi frequency
g/ 220 MHz 47 kHz 100 MHz
Cavity lifetime 1/Q 1 ns; 3 x 107 1 ms; 3 x 108 160 ns; 104
Atom lifetime 1/ 60 ns 30 ms > 2 s
Atom transit time ttransit > 50 s 100 s Infinite
Critical atom # N0=2/g2 6 x 10-3 3 x 10-6 6 x 10-5
Critical photon # m0=2/2g2 3 x 10-4 3 x 10-8 1 x 10-6
# of vacuum Rabi oscillations
nRabi=2g/() 10 5 100
The Chip for Circuit QED
No wiresattached to qubit!
Nb
Nb
SiAl
Microwave Setup for cQED Experiment
Transmit-side Receive-side
det ~ 40n
Measuring the Cavity
Use microwave powers ~ 1 photon = 10-17 watts
incident rP n
/ cycle /circulating incident r rP QP n
Bare Resonator Transmission Spectrum
First Observation of Vacuum Rabi Splitting for a Single Real Atom
Thompson, Rempe, & Kimble 1992
Cs atom in an optical cavity
Tra
nsm
issi
on
Bare Resonator Transmission Spectrum
Qubit strongly detuned from cavity
tune into resonance with cavity and repeat
Vacuum Rabi Mode Splitting by an Artificial Atom
2g
2 *0 2/ 2 0.003M g T
20 2 / 0.01N g
Critical atom (N0) & photon #: (M0)
2
2 * 50 2/ 2 10M g T
2 40 2 / 10N g
Our Records So Far:
phobit ,0 ,1
quton ,0 ,1
Cg Box
Spontaneous Emission into Continuum?
1
2 2 201
T
e ZP g
/gC C
0 50 R Z
Decay rate:
0 sinI I t
0I e
Power lost in resistor: 2
220 0
gCP I R e ZC
C
“Atom” quality factor:
2 2
2 20
1 /a
eQ
Z g
Cg Box
Spontaneous Emission into Resonator?
2
0
Re( )Tg Z
Z
/gC C
0R QZ
Decay rate:
0 sinI I t
0I e
On resonance: Res 0Re( )Z QZ
C
“Atom” quality factor:
2
2aQ Qg
2 2 20
0 0
Re( )T QZg Z g g
Z Z
the Purcell factorin circuit guise!
Cg Box
Spontaneous Emission into Resonator?
Decay rate:
Off resonance:
0
Res 2Re( )1 2 /
QZZ
C
“Atom” quality factor:
2
2aQ Qg
2 2 2 20
2 20 0
Re( )T QZg Z g g
Z Z
cavity enhancementof lifetime!
0
2
Res 0Re( ) ~ /Z QZ ,g Dispersive limit:
Off-Resonant Case: Lifetime Enhancement
,0
01 R
{
See e.g. Haroche, Les Houches 1990
,0,0
,0
,0 cos ,0 sin ,1
,0 sin ,0 cos ,1
For : 1g
g
2 2,0 cos sin
2 2,0 sin cos
Really, a way to measure non-EM part of
2g
“photonic part”of atom
Non-Radiative Decays of Qubit?
NR
0?
NR
Predicted cavity-enhanced
lifetime ~ 0.001 s!
Mechanism of non-radiative losses?
Observed lifetimes ~ 1 -10 s
How to Measure without Dissipation?
T
rans
mis
sion
Frequency
dielectricchanges“length” of cavity
A dispersive measurement – measures susceptibility, not loss
“leave no energy behind”!
(c.f. “JBA amplifier,” measures mag. suscept., by Devoret et al.)
Dispersive Circuit QED
g
Dispersive regime:
a r g
Small “mixing” of qubit and photon,
but still smallfrequency shiftof cavity!
Dispersive QND Qubit Measurement
A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and RS, PRA 69, 062320 (2004)
reverse of Nogues et al., 1999 (Ecole Normale)
QND of photon using atoms!
Controlling the Qubit in the Cavity
• Large detuning of qubit frequency from cavity• Add second microwave pulse to excite qubit
qubit
Operate at gate-insensitive“sweet spot” for long coherence -A “clock” transition for SC qubits!
(after Vion et al. 2002)
“Unitary” Rabi Oscillations
A. Wallraff et al., PRL 95, 060501 (2005)
On QND Measurements2 2
†eff 01
1
2r z z
g gH a a
, eff 0z H z is a constant of motion,measure w/out changing it
, eff 0x H a superposition is dephased
Phase shift of photons transmitted measures
qubit state
Photons in cavity dephase qubit
n
2 2†
eff 01
1
2r z z
g gH a a
cavity freq. shift Lamb shift
Probe Beam at Cavity Frequency Induces ac Stark Shift of Atom Frequency
2† †
eff r 01
1 12
2 2 z
gH a a a a
atom ac Stark shift vacuum ac Stark shift
2 cavity pulln
cQED Measurement and Backaction - Predictions
measurement rate:
dephasing rate:
phase shift on transmission:2
0
2g
2 20 0
12 2m
m r
Pn
T
2 20 02 2
r
Pn
1mT quantum
limit?:2x limit, since half of information
wasted in reflected beam
(expt. still ~ 40times worse)
AC-Stark Effect & Photon Shot Noise
D. I. Schuster, A. Wallraff, A. Blais, …, S. Girvin, and R. J. Schoelkopf, cond-mat/0408367 (2004)
• g = 5.8 MHz
• g2/=0.6 MHz
• shift measures n
Explanation of Dephasing
What if 2g2/ > ?
• Measurement dephasing from Stark random shifts
• Gaussian lineshape is sum of Lorentzians
22 n g
22 n g
Qub
it R
espo
nse
Frequency, s
( )( )
!
nnn
P n en
• Coherent state has shot noise
• Peaks are Poisson distributed
Possibility of Observing Number States of Cavity?
g2/ • = 100 kHz
• g2/= 5 MHz
• n = 1
Simulation
g2/
theoretical predictions: J. Gambetta, A. Blais, D. Schuster, A. Wallraff, L. Frunzio, J. Majer, S.M. Girvin, and R.J. Schoelkopf, cond-mat/0602322
see expt. results reported later this week: D. Schuster G3.00003 Tues 9:12 AM
Future Prospects/Directions
cavity QED = testbed system for quantum optics
• nonlinear quantum optics- single atom/photon bistability- squeezing
• quantum measurements
• cavity enhancement of qubit lifetime? - measuring internal dissipation of qubits
• quantum bus for entanglement
(cQED = “circuit quantum electrodynamics”)
Coupling Two Qubits via a Photon
“long” range and non nearest-neighbor
interactions!
ala’ Cirac-Zollerion trap gates
2 cm
Address with frequency-selective RF coupling pulses
† †1 2
1,22 2a a
r j j jj
H a a g a a
Two Qubits in One Cavity
First Two Qubit Cavity Measurements
0.3 0.2 0.1 0 0.1 0.2gate voltage, Vgarb.
4.6
4.8
5
5.2
xulfsaib,bra.
Gate voltage
Flu
x
Strong Cavity QED with Polar Molecules?
0/ 2 / ~ 100 kHz
/ 2 5 kHz
/ 2 ~ 2 Hz
g dE h
12
6
~ 5 GHz
~ 10
/ 2 5 MHz
5 Debyes
Q
d
2 2 100 / 2 10M g 2 6
0 2 / 10N g
2 ~ 1 ms
4swaptg
Dispersivequbit
interaction
The Yale Circuit QED Team
Dave Schuster
Alexandre Blais (-> Sherbrooke)
Andreas Wallraff(-> ETH Zurich)
Steve Girvin
Summary
• “Circuit QED”: 1-d resonators + JJ atoms for strong coupling cQEC in the microwave circuit domain
• First msmt. of vacuum Rabi splitting for a solid-state qubit • Dispersive QND measurements and backaction
no dissipation - don’t heat the dirt!
• Control of qubit in cavity: long coherence time and high fidelity
• Numerous advantages for quantum control and measurement
2* ~ 500 nsT
,g
Theory: Blais et al., Phys. Rev. A 69, 062320 (2004)Vac Rabi: Wallraff et al., Nature 431, 132 (2004)AC Stark: Schuster et al., PRL 94, 123602 (2005)Qubit Control: Wallraff et al., PRL 95, 060501 (2005)
Visibility 95%1~ 8 sT
Circuit QED Publications
High visibility Rabi oscillations & coherence time measurements:
A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, S. M. Girvin, and R. J. Schoelkopf,
Phys. Rev. Lett. 95, 060501 (2005)
Circuit QED device fabrication:
L. Frunzio, A. Wallraff, D. I. Schuster, J. Majer, and R. J. Schoelkopf,
IEEE Trans. on Appl. Supercond. 15, 860 (2005)
AC Stark shift & measurement induced dephasing:
D. I. Schuster, A. Wallraff, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Girvin, and
R. J. Schoelkopf, Phys. Rev. Lett. 94, 123062 (2005)
Strong coupling & vacuum Rabi mode splitting:
A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. Girvin,
and R. J. Schoelkopf, Nature (London) 431, 162 (2004)
Circuit QED proposal:
A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, PRA 69, 062320 (2004)
see: www.eng.yale.edu/rslab