Quantum Optics with Electrical Circuits: Strong Coupling Cavity QED

Experiment Theory

Rob Schoelkopf Michel Devoret Andreas Wallraff Steven Girvin Alexandre Blais (Sherbrooke) David Schuster NSF/Keck Foundation Hannes Majer Jay Gambetta Center Luigi Frunzio Axel Andre R. Vijayaraghavan for K. Moon (Yonsei) Irfan Siddiqi Quantum Information Physics Terri Yu Michael Metcalfe Chad Rigetti Yale University R-S Huang (Ames Lab) Andrew Houck K. Sengupta (Toronto) Cliff Cheung (Harvard) Aash Clerk (McGill)

1 A Circuit Analog for Cavity QED 2g = vacuum Rabi freq. κ = cavity decay rate γ = “transverse” decay rate out

cm 2.5 λ ~ transmissionL = line “cavity”

E B DC + 5 µm 6 GHz in - ++ -

2 Blais et al., Phys. Rev. A 2004 10 µm The Chip for Circuit QED

Nb No wires Si attached Al to qubit! Nb

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) 3 Advantages of 1d Cavity and Artificial Atom gd= iERMS / Vacuum fields: Transition dipole: mode volume10−63λ de~40,000 a0

∼ 10dRydberg n=50 ERMS ~ 0.25 V/m

cm ide .5 gu 2 ave λ ~ w L = R ≥ λ e abl l c xia coa

R λ 5 µm Cooper-pair box “atom” 4 Resonator as Harmonic Oscillator

1122 L r C H =+()LI CV r 22L

Φ ≡=LI coordinate flux

ˆ † 1 V = momentum voltage Hacavity =+ωr ()a2 ˆ † VV=+RMS ()aa

11ˆ 2 ⎛⎞1 CV00= ⎜⎟ω 22⎝⎠2 ω VV= r ∼ 12− µ RMS 2C 5 The Artificial Atom non-dissipative ⇒ superconducting circuit element non-linear ⇒ Josephson tunnel junction

1nm +n(2e) -n(2e)

SUPERCONDUCTING Anharmonic! TUNNEL JUNCTION

ATOM Non-linear LC resonator 6 WHY SUPERCONDUCTIVITY?

E few electrons N ~ 109 total number of electrons “forest” of states N even

~ 1eV 2∆ ~ 1meV

superconducting gap ATOM SUPERCONDUCTING NANOELECTRODE7 Superconducting Tunnel Junction (The only non-linear dissipationless circuit element.)

N +1 pairs Al superconductor N pairs 8 N ∼ 10 Al2Ox tunnel barrier ∼ 1mµ N pairs Al superconductor N +1 pairs

Josephson Tunneling Splits the Bonding and Anti-bonding ‘Molecular Orbitals’ Covalently Bonded Diatomic ‘Molecule’

anti-bonding bonding

8 Bonding Anti-bonding Splitting

8 1 10 + 1 108 ψ =± ()8 ± 8 2 10 10 + 1

EEanti-bonding −=bonding EJ ∼∼4 −20 GHz 0.2 - 1.0 K Josephson coupling

↑= bonding Two-level approximation E z H =− J σ justified by large charging ↓= anti-bonding 2 energy. 9 SPLIT COOPER PAIR BOX QUBIT: THE “ARTIFICIAL ATOM” with two control knobs

E1

hν01

E0

⎡⎤nn++11n+ n ˆ 2 ⎛⎞π Φ ( ) HE=−⎢⎥C ()nNg nn−Ej cos⎜⎟ ∑ Φ 2 n ⎣⎦⎢⎥⎝⎠0

THE Hamiltonian (We mean it!) [Devoret & Martinis, QIP, 3, 351-380(2004)]10 BOX QUANTUM LEVELS AND MEASUREABLE QUANTITIES

charge E ∂E 1 Q = k k ∂U

) hν K 01 B k (

Energy current

∂Ek E0 I = k ∂Φ Φ/Φ0

CgU/2e 11 BOX QUANTUM LEVELS AND MEASUREABLE QUANTITIES

optimum working point for decoherence (sweet spot) charge E ∂E 1 Q = k = 0 k ∂U

) hν K 01 B k (

Energy current

E ∂Ek 0 Ik = = 0 ∂Φ Φ/Φ0

CgU/2e 12 BOX QUANTUM LEVELS AND MEASUREABLE QUANTITIES Walraff et al. Wilson et al. optimum working point for decoherence (sweet spot) charge capacitance E ∂E 2 1 k ∂ Ek Qk = C = ∂U k ∂U 2

) hν K 01 B k (

Energy current inductance

∂E 2 −1 E k ⎛⎞∂ Ek 0 Ik = Lk = ⎜⎟2 ∂Φ ⎝⎠∂Φ Φ/Φ0

CgU/2e 13 Siddiqi et al. Using the cavity to measure the state of the ‘atom’

qubit

V

0 State dependent polarizability of ‘atom’ pulls the cavity frequency14 cQED Dispersive Measurement

15 A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and RS, PRA 69, 062320 (2004) Coherent Control of Qubit in Cavity

signal for pure excited state

85 % contrast signal for gnd state

Rabi oscillations

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

≈×0.658 10−15 eV-s

1 nV acting on charge e for 1 µs =10−15 eV-s

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

E box 1 ← 0

0 → 1 CgVg 1/2 2e

'sweet spot' for T2 Vion et al. Science 2002 ν ∂ν = →→QQ−←←=0 t ∂V ϕν()ti=−∫ dt′ (t′) g 0 ψθ()te=+cos()/2 10itϕ ()sin(θ/2) Long Coherence Time

3 pulse experiment

Qϕ ≈ 23π × 000

Coherence time (T2) ~ 500 ns

Allows 100’s of operations on one qubit 20 Coherently Entangling a Single Photon and a Single ‘Atom’

Tune qubit into resonance with the cavity

1 Ψ= ()1 photon, atom in ground state ±0 photon, atom in excited state ± 2

‘vacuum Rabi splitting’ EE+−− =

Single photon binds to the atom forming a ‘molecule’

21 First Observation of Vacuum Rabi Splitting in a Superconducting Circuit

qubitqubit detuned detuned Pprobe =−140 dBm fromfrom cavity cavity = 10−17 W

= n ωrκ /2 n ≤ 1 qubit tuned into 2g 2g / 2π = 12 MHz resonance 1 1 κ / 2π = 0.6 MHz ()qubit + photon ()qubit − photon 2 2 γ / 2π = 1 MHz

22 23 What is a Measurement?

Stern Gerlach measures position not spin N but the field gradient entangles spin S with position

⎛⎞α Ψ=in f (r)⎜⎟ (product of space and spin) ⎝⎠β ⎛⎞α ⎛0 ⎞ Ψ=out fr↑ ( )⎜⎟+ fr↓ ( )⎜⎟ (entangled) Macroscopically distinct 0 β (and distinguishable) ⎝⎠ ⎝⎠

‘pointer’ states are ff↑ ↓ = 0 orthogonal 24 cQED Dispersive Readout analogous to Stern Gerlach Input microwave beam is in a coherent state:

ζ a† ζ ∼ e 0 Im ζ a ζζ= ζ

⎛⎞α Re ζ Ψ=in ζ ⎜⎟ ⎝ β ⎠ (product state) Im ζ Output beam is entangled with qubit:

⎛⎞α ⎛0 ⎞ θ iiθθ− Re ζ Ψ=out ζζee⎜⎟+ ⎜⎟ ⎝⎠0 ⎝ β ⎠ −θ (entangled) 25 What is the measurement time?

Im ζ ∆θ ∆=N 1

θ ∆=θ 2θ Re ζ −θ ∆=NN 1 N = 4θ 2

26 What is the back action of this measurement?

Acquiring information about σ z , we must necessarily lose information about the phase of a superposition

+ xyiϕ σσ=+ieσ= Result of back action

1 −+it[(ωϕt)] ψ =↓( +e 01 ↑) 2 27 Probe Beam at Cavity Frequency Induces ‘Light Shift’ of Atom Frequency

22 ⎛⎞gg† 1⎛⎞ Haeff ≈−⎜⎟ωrzσωa−⎜01 +⎟σz ⎝⎠∆∆2⎝⎠

cavity freq. shift Lamb shift atom ac Stark shift vacuum ac Stark shift (light shift) =×2n cavity pull

11⎛⎞g2 ⎡⎤ Ha≈−ω ††aωσ+2 aa+ eff r ⎜⎟01 ⎢⎥z 22⎝⎠∆ ⎣⎦ 28 AC-Stark Effect & Measurement Back Action

D. I. Schuster et al., Phys. Rev. Lett. 94, 123602 (2005). 29 Atom ac Stark Shift (Light Shift) Induced by Cavity Photons

n to o h p r e p h z t d H i w M 5 e . in 0 l =

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

11⎛⎞g 2 ⎡⎤ Ha≈−ω ††aωσ+2 aa+ eff r ⎜⎟01 ⎢⎥z 22⎝⎠∆ ⎣⎦

κ Photon shot noise − ||τ n fluctuations 2 filtered by the cavity: δτnnˆˆ()δ = ne in photon number

Γ∝ϕ n Gaussian

Lorentzian Γ∝ϕ n

31 Lorentzian to Gaussian Crossover

measurement induced dephasing Æ greater line width

32 Weak Measurement Reaches the quantum limit* (*if cavity is asymmetric)

1 Γ T ≥ ϕ measurement 2

Dispersive cQED readout is simple enough that we can analyze it quantum mechanically 1. Has minimum back action consistent with Heisenberg 2. Allows calibration of qubit Rabi and Ramsey visiblity

33 Off resonant (from cavity) AC Stark Effect

•RF control of transition frequency

•No decoherence (no measurement!)

•Fast rise times (~ns)

55 MHz T2~150 ns

34 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

35 Two Qubit Cavity Measurements 6) Performed first experiments on two qubits in a cavity. Flux

Gate voltage@D “Triple” degeneracy point: ω ==ω ω cav 1 2 qubit 1 qubit 2 36 ‘Tagged’ Single Photons on Demand - Cliff Cheung senior thesis -

qubit resonator 1 CW drive resonator 2 frequency

time Adiabatic Fast Passage: 1. pi pulse single photon out 2. deexcitation

37 Similar scheme for single photon detection Single Photon Detection - Cliff Cheung senior thesis -

CW drive phase shifted output

Resonator 2

Qubit under continuous 1 measurement Γϕ ∼ Tmeasurement single photon in

ωin

38 ‘Circuit QED’ Strong Coupling of a Single Photon to a Cooper Pair Box

http://pantheon.yale.edu/~smg47 http://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 Qubit Using 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 39 (2005). 40 Fidelity of Single-shot Cavity QED Readout

Histograms of single shot msmts. Integrated probabilities

107 shots ↓ ↓ ↑ F ~ 30-40%

↑

Signal Measurement with ~ 5 photons in cavity;

SNR ~ 2-3 in one qubit lifetime (T1) 41 Two Qubit Gates via a Cavity QED “Bus”

m 2 c

g 2 V =+ ()σ +−σσ−σ+ ∆ 12 12 42 Quantum Nondemolition Measurement of Qubit Spectrum via Cavity QED - Demonstrated new type of dispersive qubit measurement with cavity QED

no “junction resonances”!

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

• 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! 43