1

Quantum Optics with Electrical Circuits:Strong Coupling Cavity QED

Yale University

Experiment

Rob Schoelkopf Michel DevoretAndreas WallraffDavid SchusterHannes MajerLuigi FrunzioR. VijayaraghavanIrfan SiddiqiMichael MetcalfeChad RigettiAndrew Houck

Theory

Steven GirvinAlexandre Blais (Sherbrooke)Jay GambettaAxel AndreK. Moon (Yonsei)Terri Yu

R-S Huang (Ames Lab)K. Sengupta (Toronto)Cliff Cheung (Harvard)Aash Clerk (McGill)

NSF/Keck Foundation Center

forQuantum Information Physics

A Circuit Analog for Cavity QED2g = vacuum Rabi freq.κ = cavity decay rateγ = “transverse” decay rate

L = λ ~

2.5 cm

5 µmDC +6 GHz in

out

transmissionline “cavity”

2

E B

+ + --

10 µmBlais et al., Phys. Rev. A 2004

The Chip for Circuit QED

3

No wiresattached to qubit!

Nb

Nb

SiAl

First coherent coupling of solid-state qubit to single photon:A. Wallraff, et al., Nature (London) 431, 162 (2004)

Theory: Blais et al., Phys. Rev. A 69, 062320 (2004)

Advantages of 1d Cavity and Artificial Atom

RMS /g d E= i

Transition dipole:

45 µm

Vacuum fields:mode volume

0

Rydberg n=50

~ 40,00010

d ead∼

ERMS ~ 0.25 V/m

L = λ ~

2.5 cm

coaxial c

able

wave gu

ide

R λ≥

R λ

6 310 λ−

Cooper-pair box “atom”

Resonator as Harmonic Oscillator

2 21 1( )2 2

H LI CVL

= +Lr Cr

fluxcoord inateLIΦ ≡ =voltagemom entumV =† 1

2ˆ ( )cavity rH a aω= +

†RMS

RMS

2

ˆ ( )1 1 1ˆ0 02 2 2

1 22

r

V V

V VC

a a

C V

ω µ

ω

=

= +

⎛ ⎞= ⎜ ⎟⎝ ⎠

−∼5

6

The Artificial Atom

non-dissipative superconducting circuit elementnon-linear Josephson tunnel junction

⇒⇒

+n(2e)

Anharmonic!

ATOM

-n(2e)

SUPERCONDUCTINGTUNNEL JUNCTION

Non-linear LC resonator

1nm

WHY SUPERCONDUCTIVITY?

7

E

~ 1 eV

ATOM

few electrons N ~ 109total numberof electrons

N even

superconducting gap

“forest” of states

2∆ ~ 1 meV

SUPERCONDUCTINGNANOELECTRODE

8

Superconducting Tunnel Junction (The only non-linear dissipationless circuit element.)

Al2Ox tunnel barrierAl superconductor

Al superconductor

pairsN1 pairsN +

pairsN1 mµ∼810N ∼

1 pairsN +

Josephson Tunneling Splits the Bonding and Anti-bonding ‘Molecular Orbitals’

Covalently Bonded Diatomic ‘Molecule’

bondinganti-bonding

Bonding Anti-bonding Splitting

( )12

ψ ± = ±810 1+

810 1+

810

810

9

anti-bonding bonding J 4 20 GHz 0.2 - 1.0 KE E E− = −∼ ∼Josephson coupling

J

2zEH σ= −

bonding

anti-bonding

↑ =

↓ =

Two-level approximation justified by large chargingenergy.

SPLIT COOPER PAIR BOX QUBIT:THE “ARTIFICIAL ATOM” with two control knobs

E1

E0

hν01

( )20

1 1ˆ ( ) cos2jgCn

n n n nH E n n n EN π

⎡ ⎤+ + +⎛ ⎞= − −⎢ ⎥⎜ ⎟Φ⎢ ⎥⎝ ⎠⎣ ⎦

Φ∑

THE Hamiltonian (We mean it!) 10[Devoret & Martinis, QIP, 3, 351-380(2004)]

BOX QUANTUM LEVELS ANDMEASUREABLE QUANTITIES

charge

Ene

rgy

(kBK

)

E1

CgU/2eΦ/Φ0

E0

hν01

kk

EQU

∂=

∂

current

kk

EI ∂=∂Φ

11

BOX QUANTUM LEVELS ANDMEASUREABLE QUANTITIES

12

Ene

rgy

(kBK

)

E1

CgU/2eΦ/Φ0

E0

hν01

optimum working pointfor decoherence (sweet spot)

charge

kk

EQU

∂=

∂= 0

current

kk

EI ∂=∂Φ = 0

BOX QUANTUM LEVELS ANDMEASUREABLE QUANTITIES

Walraff et al.Wilson et al.

13

Ene

rgy

(kBK

)

E1

CgU/2eΦ/Φ0

E0

hν01

charge capacitance

current inductance

kk

EQU

∂=

∂

2

2k

kEC

U∂

=∂

kk

EI ∂=∂Φ

12

2k

kEL

−⎛ ⎞∂

= ⎜ ⎟∂Φ⎝ ⎠

optimum working pointfor decoherence (sweet spot)

Siddiqi et al.

Using the cavity to measure the state of the ‘atom’

qubit

14

V

0State dependent polarizability of ‘atom’ pulls the cavity frequency

cQED Dispersive Measurement

15A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and RS, PRA 69, 062320 (2004)

Coherent Control of Qubit in Cavity

signal forgnd state

signal for pure excitedstate

Rabi oscillations

85 % contrast

consistent with 100%fidelity of qubit rotation

High Visibility Rabi Oscillations

(i.e. inferred fidelity of operation)

Indicates no undesired entanglement with environment during operations.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M.H. Devoret, SMG, and RJS, Phys. Rev. Letters 95, 060501 (2005).

17

Dephasing, decoherence, decay

150.658 10 eV-s−≈ ×

151 nV acting on charge for 1 s 10 eV-se µ −=

Spins very coherent but hard to couple and very hard to measure.

Avoiding Dephasing1

0 1

0Ebox

2ggC V

e

→

←

1/2

2'sweet spot' for T

ν

( ) ( )( )( ) cos / 2 sin / 21 0i tt e ϕψ θ θ= +0

( ) ( )t

t i dt tϕ ν′ ′= − ∫ g0Q Q

Vν∂

= → → − ← ← =∂

Vion et al. Science 2002

Long Coherence Time

3 pulseexperiment

2 3000Qϕ π≈ ×

Coherence time (T2) ~ 500 nsAllows 100’s of operations on one qubit 20

21

Coherently Entangling a Single Photon and a Single ‘Atom’

Tune qubit into resonance with the cavity

( )1 1 photon, atom in ground state 0 photon, atom in excited state2±

Ψ = ±

E E+ −− = ‘vacuum Rabi splitting’

Single photon binds to the atom forming a ‘molecule’

First Observation of Vacuum Rabi Splitting in a Superconducting Circuit

qubit detuned from cavity

22

qubit detuned from cavity140 dBmprobeP = −

1710 W −=/ 2rn ω κ=

1n ≤qubit

tuned intoresonance( )1 qubit photon

2+

2g

( )1 qubit photon2

−

/ 2 12 MHz/ 2 0.6 MHz/ 2 1 MHz

2g πππ

κγ

===

23

24

What is a Measurement?

S

NStern Gerlach measures position not spin

but the field gradient entangles spin with position

out

in ( ) (product of space and s

0( ) ( ) (entangle

)

d

0

p

0

in

)

f f

f

f r f r

r

αβ

αβ

↑

↑

↓

↓

⎛ ⎞ ⎛ ⎞Ψ =

⎛ ⎞Ψ = ⎜ ⎟

⎝ ⎠

+⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝

=⎠

Macroscopically distinct(and distinguishable)‘pointer’ states areorthogonal

cQED Dispersive Readoutanalogous to Stern Gerlach

Input microwave beam is in a coherent state:

25

†

0ae

a

ζζ

ζ ζ ζ=

∼

Re ζ

Im ζ

o

in

ut

(product state

0

0(entangled

)

)

i ie eθ θα

ζ ζ

αζ

β

β−⎛ ⎞ ⎛ ⎞

⎛ ⎞Ψ = ⎜ ⎟

Ψ = +⎜ ⎟ ⎜ ⎟

⎝

⎝

⎠

⎝ ⎠ ⎠Re ζ

Im ζ

θθ−

Output beam is entangled with qubit:

What is the measurement time?

26

Re ζ

Im ζ

θ

θ−

2

12

14

N

N N

N

θθ θ

θ

∆ ∆ =∆ =

∆ =

=

What is the back action of this measurement?

zAcquiring information about ,we must necessarily lose information aboutthe phase of a superposition

σ

( )01[ ( )]12

i t te ω ϕψ − += ↓ + ↑

x y ii e ϕσ σ σ+ = + = Result of back action

27

2 2†

eff 0112r z z

g gH a aω σ ω σ⎛ ⎞ ⎛ ⎞

≈ − − +⎜ ⎟ ⎜ ⎟∆ ∆⎝ ⎠ ⎝ ⎠

cavity freq. shift Lamb shift

Probe Beam at Cavity Frequency Induces ‘Light Shift’ of Atom Frequency

2† †

eff r 011 122 2 z

gH a a a aω ω σ⎛ ⎞⎡ ⎤≈ − + +⎜ ⎟⎢ ⎥∆ ⎣ ⎦⎝ ⎠

atom ac Stark shift(light shift) vacuum ac Stark shift

2 cavity pulln= ×

28

AC-Stark Effect & Measurement Back Action

29D. I. Schuster et al., Phys. Rev. Lett. 94, 123602 (2005).

Atom ac Stark Shift (Light Shift)Induced by Cavity Photons

0.5 M

Hz pe

r pho

ton

= lin

e widt

h

30

Measurement Induced Dephasing:back action = quantum noise in the light Shift

| |2ˆ ˆ( )n n neκ τ

δ τ δ−

=fluctuations

in photon numbern

2† †

eff r 011 122 2 z

gH a a a aω ω σ⎛ ⎞⎡ ⎤≈ − + +⎜ ⎟⎢ ⎥∆ ⎣ ⎦⎝ ⎠

Photon shot noisefiltered by the cavity:

Lorentzian

GaussiannϕΓ ∝

nϕΓ ∝

31

32

Lorentzian to Gaussian Crossover

measurement induced dephasing greater line width

Weak Measurement Reaches the quantum limit*

(*if cavity is asymmetric)

measurement12

TϕΓ ≥

Dispersive cQED readout is simple enough that we cananalyze it quantum mechanically

1. Has minimum back action consistent with Heisenberg2. Allows calibration of qubit Rabi and Ramsey visiblity

33

Off resonant (from cavity) AC Stark Effect

34

55 MHz T2~150 ns

•RF control of transition frequency

•No decoherence(no measurement!)

•Fast rise times (~ns)

35

FUTURE DIRECTIONS

- strongly non-linear devices for microwave photon manipulation- single atom optical bistability- strong squeezing (Devoret/Siddiqi JBA in progress)- photon ‘blockade’ (photon antibunching)- single photon microwave detectors- single photon microwave sources- transferring qubit states to cavity photon states

as quantum memory

- resonator as ‘bus’ coupling many qubits- multiqubit control and readout- materials loss tangent characterization using

cavity enhanced qubit lifetime

36

Two Qubit Cavity Measurements

@DGate voltage

Flux

6) Performed first experiments on two qubits in a cavity.

qubit 1

“Triple” degeneracy point:

1 2cav ωω ω= = qubit 2

37

‘Tagged’ Single Photons on Demand

- Cliff Cheung senior thesis -

qubit

resonator 1

resonator 2CW drive

freq

uenc

y

timeAdiabatic Fast Passage:

1. pi pulse2. deexcitationsingle photon out

Similar scheme for single photon detection

38

Single Photon Detection- Cliff Cheung senior thesis -

phase shifted output CW drive

measurement

1Tϕ

Γ ∼

Resonator 2

Qubit under continuousmeasurement

inωsingle photon in

39

‘Circuit QED’Strong Coupling of a Single

Photon to a Cooper Pair Box

http://pantheon.yale.edu/~smg47http://www.eng.yale.edu/rslab/cQED

• “Cavity Quantum Electrodynamics for Superconducting Electrical Circuits: an Architecture for Quantum Computation,” A. Blais et al., Phys. Rev. A 69, 062320 (2004).

• “Coherent Coupling of a Single Photon to a Superconducting QubitUsing Circuit Quantum Electrodynamics,” A. Wallraff et al., Nature 431, 162 (2004).

• “AC Stark Shift and Dephasing in a Superconducting Qubit Strongly Coupled to a Cavity Field,” D.I. Schuster, et al., Phys. Rev. Lett. 94, 123602 (2005).

• “Approaching Unit Visibility for Control of a Superconducting Qubit with Dispersive Readout,” A. Wallraff et al., Phys. Rev. Lett. 95, 060501 (2005).

40

Fidelity of Single-shot Cavity QED Readout

Histograms of single shot msmts. Integrated probabilities

41

↓

↑

Measurement with ~ 5 photons in cavity; SNR ~ 2-3 in one qubit lifetime (T1)

F ~ 30-40%

Signal

↓

↑

107 shots

Two Qubit Gates via a Cavity QED “Bus”

42

2 cm

( )2

1 2 1 2gV σ σ σ σ+ − − += +∆

- Demonstrated new type of dispersive qubit measurement with cavity QED

Quantum Nondemolition Measurement of Qubit Spectrum via Cavity QED

D. Schuster et al., submitted to PRL, 2004.

no “junction resonances”!

43

• Measure phase of transmitted RF – “leave no energy behind”• Sensitive at “sweet spot” – may be reducing 1/f noise levels?• Readout method allows improved qubit coherence!

Quantum Optics with Electrical Circuits:Strong Coupling Cavity QEDA Circuit Analog for Cavity QEDThe Chip for Circuit QEDAdvantages of 1d Cavity and Artificial AtomResonator as Harmonic OscillatorThe Artificial AtomSuperconducting Tunnel Junction (The only non-linear dissipationless circuit element.)Bonding Anti-bonding SplittingSPLIT COOPER PAIR BOX QUBIT: THE “ARTIFICIAL ATOM” with two control knobsBOX QUANTUM LEVELS ANDMEASUREABLE QUANTITIESBOX QUANTUM LEVELS ANDMEASUREABLE QUANTITIESBOX QUANTUM LEVELS ANDMEASUREABLE QUANTITIESUsing the cavity to measure the state of the ‘atom’cQED Dispersive MeasurementCoherent Control of Qubit in CavityHigh Visibility Rabi OscillationsDephasing, decoherence, decayAvoiding DephasingLong Coherence TimeCoherently Entangling a Single Photon and a Single ‘Atom’First Observation of Vacuum Rabi Splitting in a Superconducting CircuitWhat is a Measurement?cQED Dispersive Readoutanalogous to Stern GerlachWhat is the measurement time?What is the back action of this measurement?Probe Beam at Cavity Frequency Induces ‘Light Shift’ of Atom FrequencyAC-Stark Effect & Measurement Back ActionMeasurement Induced Dephasing:back action = quantum noise in the light ShiftLorentzian to Gaussian CrossoverWeak Measurement Reaches the quantum limit*(*if cavity is asymmetric)FUTURE DIRECTIONSTwo Qubit Cavity Measurements‘Tagged’ Single Photons on Demand- Cliff Cheung senior thesis -Single Photon Detection- Cliff Cheung senior thesis -‘Circuit QED’Strong Coupling of a Single Photon to a Cooper Pair BoxFidelity of Single-shot Cavity QED ReadoutTwo Qubit Gates via a Cavity QED “Bus”Quantum Nondemolition Measurement of Qubit Spectrum via Cavity QEDRF Frequency Shift GateSpectroscopic Power Broadening and Saturation