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