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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
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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
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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
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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
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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
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Optical Cavity QED
… measure changes in transmission of optical cavity
e.g. Kimble and Mabuchi groups at Caltech
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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)
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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)
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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
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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
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Qubits: Why Superconductivity?
~ 1 eV
E
2~ 1 meV
ATOM SUPERCONDUCTINGNANOELECTRODE
few electronsN ~ 109
total numberof electrons
superconducting gap
“forest” of states
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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)
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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
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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
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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
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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
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The Chip for Circuit QED
No wiresattached to qubit!
Nb
Nb
SiAl
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Microwave Setup for cQED Experiment
Transmit-side Receive-side
det ~ 40n
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Measuring the Cavity
Use microwave powers ~ 1 photon = 10-17 watts
incident rP n
/ cycle /circulating incident r rP QP n
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Bare Resonator Transmission Spectrum
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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
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Bare Resonator Transmission Spectrum
Qubit strongly detuned from cavity
tune into resonance with cavity and repeat
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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
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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
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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!
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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:
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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
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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
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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.)
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Dispersive Circuit QED
g
Dispersive regime:
a r g
Small “mixing” of qubit and photon,
but still smallfrequency shiftof cavity!
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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!
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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)
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“Unitary” Rabi Oscillations
A. Wallraff et al., PRL 95, 060501 (2005)
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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
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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
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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)
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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
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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
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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
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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”)
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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
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Two Qubits in One Cavity
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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
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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
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The Yale Circuit QED Team
Dave Schuster
Alexandre Blais (-> Sherbrooke)
Andreas Wallraff(-> ETH Zurich)
Steve Girvin
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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
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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