quantum optics with electrical circuits: strong coupling ......- strongly non-linear devices for...

Quantum Optics with Electrical Circuits:Strong Coupling Cavity QED

Ren-Shou Huang, Alexandre Blais, Andreas Wallraff, David Schuster, Sameer Kumar, Luigi Frunzio, Hannes Majer,

Steven Girvin, Robert Schoelkopf

Yale University

2

‘Circuit QED’

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

Wallraff et al. [cond-mat/0407325]Nature (in press)

3

Atoms Coupled to Photons

1s

2p2sIrreversible spontaneous decay into the photon continuum:

12 1 1 nsp s Tγ→ + ∼

Vacuum Fluctuations:(Virtual photon emission and reabsorption)Lamb shift lifts 1s 2p degeneracy

Cavity QED: What happens if we trap the photonsas discrete modes inside a cavity?

Outline

Cavity QED in the AMO CommunityOptical Microwave

Circuit QED: atoms with wires attachedWhat is the cavity?What is the ‘atom’?Practical advantages

Recent Experimental ResultsQuantum optics with an electrical circuit

Future Directions

4

Cavity Quantum Electrodynamics (cQED)

2g = vacuum Rabi freq.κ = cavity decay rateγ = “transverse” decay rate

Strong Coupling = g > κ , γ , 1/t

t = transit time

Jaynes-Cummings Hamiltonian

† †12 ˆ ˆ

2)ˆ )(

2(el J

x zr a a a aE gH Eω σ σ σ σ− += + − +−+

Quantized FieldElectric dipole

Interaction2-level system5

Cavity QED: Resonant Case

01rω ω=with interactioneigenstates are:

( )

( )

1,0 ,1 ,02

1,0 ,1 ,02

+ = ↑ + ↓

− = ↑ − ↓

vacuumRabi

oscillations

6“dressed state ladders”

Microwave cQED with Rydberg Atoms

beam of atoms;prepare in |e>

3-d super-conducting

cavity (50 GHz)observe dependence of atom final

state on time spent in cavity

vacuum Rabi oscillations

measure atomic state, or …7Review: S. Haroche et al., Rev. Mod. Phys. 73 565 (2001)

cQED at Optical Frequencies

8(Caltech group H. J. Kimble, H. Mabuchi)

State of photons is detected, not atoms.

… measure changes in transmission of optical cavity

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”

9Blais, Huang, Wallraff, SMG & RS, PRA 2004

Cross-sectionof mode:

E B

10 µm

+ + --Lumped elementequivalent circuit

Advantages of 1d Cavity and Artificial Atom/g d E= i

Transition dipole:Vacuum fields:zero-point energy confined in < 10-6 cubic wavelengths 0~ 40,000d ea

10 x larger than Rydberg atomE ~ 0.25 V/m vs. ~ 1 mV/m for 3-d

10 µm

L = λ ~ 2.5 cm

Cooper-pair box “atom” 10

Resonator as Harmonic Oscillator

2 21 1( )2 2

H LI CVL

= +Lr Cr

mome m ntuLIΦ ≡ =

coordi te naV =† 12

ˆ ( )cavity rH a aω= +†

RMS

2

RMS

( )1 1 1

1 2

0 02 2 2

2r V

V V a a

C V

VC

ω

µω

= +

⎛ ⎞= ⎜ ⎟⎝ ⎠

= −∼11

Implementation of Cavities for cQEDSuperconducting coplanar waveguide transmission line

Q > 600,000 @ 0.025 K

12

1 cmNiobium films

Opticallithography

at Yalegap = mirror

6 GHz:1 @20n mKγ300mKω =

• Internal losses negligible – Q dominated by coupling

The Chip for Circuit QED

Nb

Nb

Nb

the ‘atom’

no wires attached to qubit! 13

14

Superconducting Tunnel Junction as aCovalently Bonded Diatomic ‘Molecule’

(simplified view)

1 pairsN +

pairsN1 pairsN +

pairsN810N ∼ 1 mµ∼tunnel barrier

aluminum island

aluminum island

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

bondinganti-bonding

Bonding Anti-bonding Splitting

( )12

ψ ± = ±810 1+

810 1+

810

810

anti-bonding bonding J 7 GHz 0.3 KE E E− = ∼ ∼Josephson coupling

J

2zEH σ= −

bonding

anti-bonding

↑ =

↓ = 15

Dipole Moment of the Cooper-Pair Box(determines polarizability)

2

1 2 3

1/(2 )1/ 1/ 1/

Cd e LC C C

=+ +

Vg

1 nm

L = 10 µm

+ ++ +

+ +

- -- -

- -

/gE LV=

~ 2 - md e µ

Vg

0

1C

2C

3C

J

2z x

gd VL

EH σσ −= − bonding

anti-bonding

↑ =

↓ = 16

Energy, Charge, and Capacitance of the CPB

↑

↓J

2z x

gE dH V

Lσ σ= − −JE

Ene

rgy

17

dEQdV

=

Cha

rge

Cap

acita

nce

no chargesignal

↑

↑

↓

↓

charge

dQCdV

= polarizability is state dependent

/g gC V e0 1 2deg. pt. = coherence sweet spot

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

J

2z xE dH V

Lσ σ= − − †

dc RMS( )V V V a a= + +

2RMS

1 2 3

1/(2 )1/ 1/ 1/

Ceg VC C C

=+ +

18

V

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

19

Dispersive Regime

01 r gω ω∆ = −

Large Detuning ofAtom from

Cavity

( )† †01R2

zH a a g a aω σ ω σ σ− += − + + +Large

Detuning ofAtom from

Cavity

†exp gU a aσ σ⎧ ⎫+ −⎛ ⎞= −⎜ ⎟⎨ ⎬⎝ ⎠∆⎩ ⎭

01 r gω ω∆ = −

2 2†

eff 0112r z z

g gH a aω σ ω σ⎛ ⎞ ⎛ ⎞

≈ − − +⎜ ⎟ ⎜ ⎟∆ ∆⎝ ⎠ ⎝ ⎠cavity freq. shift Lamb shift

†effH UHU=

effQND: [ , ] 0zH σ = 20

↑↓

Cavity Transmission PhaseControlled by State of Atom

Nb resonator20 mK

21

νr = 6.04133 GHzQ = 2π νr/κ ~ 10,000

Linewidthκ=2π x 0.6MHzκ-1 = 250 ns

QND Measurement of Qubit: Dispersive case

~ 5δθ °

∆

6.04133 GHzrν =rν /JE h

0Pha

se S

hift

2min2 / ~ 5gδθ κ= ∆ °

M/ 5 Hzg π =vacuum Rabi

frequency

r

min ~ 300 MHz( 0.05 )!ν

∆∼

012 ( )rνπ ν∆ = −ν01

22

Gate Sweep with Qubit Crossing Resonator

0

Pha

se S

hift

(a.u

.)

tune qubit thruresonance w/

cavity

0∆ =

rν /JE h

phase shiftchanges signat resonance

23

Spectroscopy of Qubit in CavitySend in 2 frequencies

•Readout•Spectroscopy

24

Pha

segn

gn

Pha

se

Data 1

ν01

-50

-40

-30

-20

-10

0

6.46.26.05.85.6

Probe (CW)cavity at νr

Spectroscopy (CW)at 6.3 GHz

near ν01

Attn

(dB

)

νr νs

↑

↓

Spectrum of Qubit

J

2z x

gE dH V

Lσ σ= − −

Vg

25gg

gCn

eV

=

Spe

c Fr

eque

ncy

(GH

z)

Cavity Phase

/g g gn C V e=1

EJ

Ene

rgy

Using Cavity to Map Qubit Parameter Space

26

0/Φ Φ

gg

gCn

eV

=

0∆ <

0∆ >

ν 01(G

Hz)

0

1

2Transition frequency of qubit

01 rω ω∆ = −

max ~ 6.7 GHz ~ 5.25 GHzJ CE E

Cavity phase shift

gg

gCn

eV

=

0/Φ Φ0∆ >

0∆ =

0∆ <

2e

Slice at ∆=00 1 2 3 4

Φ0

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= ×

27

28

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

450kHz/photon

0 20 40 60 80 100RF Power ΜW

6.15

6.16

6.17

6.18

6.19

6.2

Ν0G

Hz

0 50 100navg photons

0

10

20

ΝΝ0L

inew

idth

s

29

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⎛ ⎞⎡ ⎤≈ − + +⎜ ⎟⎢ ⎥∆ ⎣ ⎦⎝ ⎠

30

Measurement Back Action:Quantum Noise in ac Stark Shift

2† †

eff r 011 122 2 z

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

( )01[ ( )]12

i t te ω ϕψ − += ↓ + ↑

2

0

2 ˆ( ) ( )tgt n d nϕ τ δ τ

⎡ ⎤= +⎢ ⎥∆ ⎣ ⎦

∫

light shift random dephasing

31


2

0

2 ˆ( ) ( )tgt d nδϕ τ δ τ=

∆ ∫21 ( )( ) 2

222

0 0

2 ˆ ˆ( ) ' ( ) ( ')

ti t

t t

e e

gt d d n n

δϕδϕ

δϕ τ τ δ τ δ τ

−≈

⎛ ⎞= ⎜ ⎟∆⎝ ⎠

∫ ∫

Assuming gaussian fluctuations

32


Coherent state in driven cavity with damping rate κ

| |2ˆ ˆ( ) (0)n n neκ τ

δ τ δ−

=

τ

33


22 | '|2 2

0 0

222

22

2( ) '

2 1

(Gaussian inhomogeneous broadening)

2 4 1

(phase random walks--phase diffusion)(Lorentzian homogen

t tgt d d ne

g n t t

g n t t

κ τ τδϕ τ τ

κ

κκ

− −⎛ ⎞= ⎜ ⎟∆⎝ ⎠

⎛ ⎞≈ ⎜ ⎟∆⎝ ⎠

⎛ ⎞≈ ⎜ ⎟∆⎝ ⎠

∫ ∫

eous broadening)

34

Qubit Phase Diffusion (weak measurement)

222 2 4( ) gt n tδϕ

κ⎛ ⎞

≈ ⎜ ⎟∆⎝ ⎠2

21 2( )( ) 2 2exp 2t ti t ge e n t e ϕ

ϕϕ κκ

− −Γ−⎡ ⎤⎛ ⎞

= = − =⎢ ⎥⎜ ⎟∆⎢ ⎥⎝ ⎠⎣ ⎦

i t2 2

00

1 1( ) Im -i dt e( )

teS ϕ ϕω

ϕ

ωπ π ω ω

∞−Γ Γ⎧ ⎫

= − =⎨ ⎬ − + Γ⎩ ⎭∫

nϕΓ ∝ valid for ϕ κΓ

Measurement induced dephasing rate

35

Qubit Inhomogeneous Broadening (strong measurement)

2 221 12( ) ( )( ) 22 21 2exp

2t ti t ge e n t e

n

ϕϕϕ

ϕ

− − Γ−⎡ ⎤⎛ ⎞

= = − =⎢ ⎥⎜ ⎟∆⎢ ⎥⎝ ⎠⎣ ⎦

Γ ∝

022

2( )2i t

0

1 ( )21 1( ) Im -i dt e

2t

eS e ϕϕ

ω ω

ω

ϕ

ωπ π

−∞Γ− Γ −⎧ ⎫

= − =⎨ ⎬Γ⎩ ⎭

∫

nϕΓ ∝ valid for ϕ κΓ

36

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⎛ ⎞⎡ ⎤≈ − + +⎜ ⎟⎢ ⎥∆ ⎣ ⎦⎝ ⎠

Lorentzian

GaussiannϕΓ ∝

nϕΓ ∝

Summary of Dispersive Regime Results

Every thing works as predicted except the cavity enhanced lifetime has not been observed.

Non-radiative decay channels?-glassy losses in oxide barriers

-electroacoustic coupling to phonons? (Ioffe, Blatter)

-42

1

loss tangent 10εε∼

37

Dressed Artificial Atom: Resonant Case

? T

38

01 Rω ω=

“vacuum Rabi splitting”

2g

/ Rω ω

T

2γ κ+

1Fourier transform of HarocheRabi flopping expt.

39

First Observation of Vacuum Rabi Splitting for a Single Atom

Thompson, Rempe, & Kimble 1992

Cs atom in an optical cavity(on average)

photons

First Observation of Vacuum Rabi Splitting in a Superconducting Circuit

qubit detuned from cavity

40

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

===

Observing the Avoided Crossing of “Atom” & “Photon”

J rE ω= J rE ω<

41

Quantum Computation and NMR of a Single ‘Spin’

Quantum MeasurementSingle Spin ½

42

Box

SET Vgb Vge

Cgb Cc Cge

Vds

(After Konrad Lehnert)

43

NMR language

free evolution (analogousto gyroscopic precession)

1Ω

Quantum control of qubits

x

y

z

microwave pulse1

πpulse

0

π/2pulse

NOT NOT

44

Rabi Flopping of Qubit Under Continuous Measurement

FUTURE DIRECTIONS

- strongly non-linear devices for microwave quantum optics- single atom optical bistability- photon `blockade’

- single photon microwave detectors- single photon microwave sources- quantum computation

- QND dispersive readout of qubit state via cavity- resonator as ‘bus’ coupling many qubits- cavity enhanced qubit lifetime

45

46

SUMMARY

Cavity Quantum Electrodynamics

cQED

“circuit QED”

Coupling a Superconducting Qubit to a Single Photon

-first observation of vacuum Rabi splitting-initial quantum control results

Coupling Qubits via Cavity Mode

multiple CPBqubits in a cavity

20 µm

Nb

Nb

Nb

can integrate multiple qubits in a single cavity,with no additional fabrication complexity

47

Entanglement via Resonator “Bus”

Qubit coupling via virtualphoton exchange:

212 ~ /J g ∆~1cm

Room for many qubits in single resonator

2~ /op gΓ ∆Operation rate: (top~10-100 ns)

48( )2

NRNumber of Ops ~ max , / 40 1200op gγ κ⎡ ⎤Γ ∆ −⎣ ⎦ ∼

Multi-qubit readout:multiple cavity pulls 2 2

1 2

1 2

g g± ±

∆ ∆Transmission

frequency↓↓ ↑↑↓↑ ↑↓

Single readout line, 2 bits of information:Two qubit readout without extra wires⇒

Permits selective projection of 2 bit states 49

50

Single Atom Optical Bistability

2

r rc

c-15

c

11 /

250 photons

P 10 W

gn n

n

n

ω ω

κ

= +∆ +

≈

= ∼

ω2g

∆driveω0

1

2

3

0

1

2

, n↑ , n↓

∆

Comparison of cQED with Atoms and CircuitsParameter Symbol Optical cQED

with Cs atomsMicrowave

cQED/ Rydbergatoms

Super-conducting

circuitQED

Dipole moment d/eao 1 1,000 20,000

Vacuum Rabi frequency

g/π 220 MHz 47 kHz 100 MHz

Cavity lifetime 1/κ; Q 1 ns; 3 x 107 1 ms; 3 x 108 160 ns; 104

Atom lifetime 1/γ 60 ns 30 ms > 2 µsAtom transit time ttransit > 50 µs 100 µs Infinite

Critical atom # N0=2γκ/g2 6 x 10-3 3 x 10-6 6 x 10-5

Critical photon # m0=γ2/2g2 3 x 10-4 3 x 10-8 1 x 10-6

# of vacuum Rabi oscillations

nRabi=2g/(κ+γ) 10 5 100

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quantum optics with electrical circuits: strong coupling ......- strongly non-linear devices for...

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