Quantum Optics with Electrical Circuits: Strong Coupling Cavity QED

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

Yale University ‘Circuit QED’

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

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

2 Atoms Coupled to Photons

2s 2p Irreversible spontaneous decay into the photon continuum:

2 ps→+1 γ T1 ∼ 1 ns 1s

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

Cavity QED: What happens if we trap the photons as discrete modes inside a cavity? 3 Outline

‰ Cavity QED in the AMO Community ‰ Optical ‰ Microwave

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

‰ Recent Experimental Results ‰ with an electrical circuit

‰ Future Directions

4 Cavity Quantum Electrodynamics (cQED)

2g = vacuum Rabi freq. κ = cavity decay rate γ = “transverse” decay rate t = transit time

Strong Coupling = g > κ , γ , 1/t

Jaynes-Cummings Hamiltonian

E E Hˆ =+ω (aa††+ 1 ) el σσˆˆ− J −g(aσ−++σa) r 2 2 xz2 Electric dipole Quantized Field 2-level system Interaction 5 Cavity QED: Resonant Case ω = ω r 01 with interaction eigenstates are: 1 +=,0 (↑,1 +↓,0 ) 2 1 −=,0 ()↑,1 −↓,0 2

vacuum Rabi oscillations

“dressed state ladders” 6 Microwave cQED with Rydberg Atoms

vacuum Rabi oscillations beam of atoms; prepare in |e>

3-d super- conducting observe dependence of atom final cavity (50 GHz) state on time spent in cavity measure atomic state, or …

Review: S. Haroche et al., Rev. Mod. Phys. 73 565 (2001) 7 cQED at Optical Frequencies

State of photons is detected, not atoms.

… measure changes in transmission of optical cavity 8 (Caltech group H. J. Kimble, H. Mabuchi) A Circuit Analog for Cavity QED 2g = vacuum Rabi freq. κ = cavity decay rate γ = “transverse” decay rate out

cm 2.5 transmission λ ~ line “cavity” L =

Cross-section of mode: DC + B 5 µm E 6 GHz in - ++ - Lumped element equivalent circuit 9 Blais, Huang, Wallraff, SMG & RS, PRA 2004 10 µm Advantages of 1d Cavity and Artificial Atom gd= iE/ Vacuum fields: Transition dipole: zero-point energy confined de~40,000 a in < 10-6 cubic wavelengths 0 10 x larger than E ~ 0.25 V/m vs. ~ 1 mV/m for 3-d cm 2.5 λ ~ L =

10 µm Cooper-pair box “atom” 10 Resonator as Harmonic Oscillator

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

Φ ≡ LI = momentum

ˆ † 1 V = coordinate Hacavity =+ωr ()a2

† VV=+RMS ()aa

112 ⎛⎞1 CV00= ⎜⎟ω 22⎝⎠2 ω V = r ∼ 12− µV RMS 2C 11 Implementation of Cavities for cQED Superconducting coplanar waveguide transmission line Q > 600,000 @ 0.025 K

Optical lithography 1 cm at Yale Niobium films gap = mirror

6 GHz:

ω = 300mK nmγ 1@20K

• Internal losses negligible – Q dominated by coupling 12 The Chip for Circuit QED

Nb

Nb the ‘atom’

Nb

no wires attached to ! 13 Superconducting Tunnel Junction as a Covalently Bonded Diatomic ‘Molecule’

(simplified view)

N +1 pairs aluminum island N pairs N ∼ 108 tunnel barrier ∼ 1mµ N pairs aluminum island N +1 pairs

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

anti-bonding bonding

14 Bonding Anti-bonding Splitting

8 1 10 +1 108

ψ =±8 8 ()10 +1 ± 2 10

EEanti-bonding −=bonding EJ ∼∼7 GHz 0.3 K Josephson coupling ↑= bonding E H =− J σ z ↓= anti-bonding 2 15 Dipole Moment of the Cooper-Pair Box (determines polarizability)

Vg -- L = 10 µm C1 ---- 1 nm EL=Vg / C ++++ 2 C Vg ++ 3 0 1/C de= (2 )L 2 1/CC12++1/ 1/C3

↑= bonding EJ z d x H =− σ − Vgσ de~2 -µm ↓= anti-bonding 2 L 16 Energy, Charge, and Capacitance of the CPB ↓ E E zxd J HV=− J σ − σ ↑ g

Energy 2 L

↓ no charge dE signal Q = charge Charge ↑ dV

↑ dQ polarizability is C = state dependent ↓ dV Capacitance CV / e 01 2 gg deg. pt. = coherence sweet spot 17 Using the cavity to measure the state of the ‘atom’

E d † HV=− J σ zx− σ VV=+V ()a+a 2 L dc RMS

(2e) 1/C2 gV= RMS 1/CC12++1/ 1/C3

V

0 State dependent polarizability of ‘atom’ pulls the cavity frequency18 Dispersive Regime

Large Detuning of Atom from Cavity

∆=ω01 −ωr g

19 ω Ha=− 01 σ z +ωσ††a+g()a− +aσ+ 2 R Large ⎧ g ⎛⎞+ † − ⎫ Detuning of Ua=−exp ⎨ ⎜⎟σσa⎬ Atom from ⎩⎭∆ ⎝⎠ Cavity † Heff = UHU

∆ =−ω01 ωr g

22 ⎛⎞gg† 1⎛⎞ Haeff ≈−⎜⎟ωrzσωa−⎜01 +⎟σz ⎝⎠∆∆2⎝⎠ cavity freq. shift Lamb shift 20 QND : [Heff ,σ z ] = 0 Cavity Transmission Phase Controlled by State of Atom

Nb resonator 20 mK ↑ ↓

Linewidth νr = 6.04133 GHz Q = 2πν/κ ~ 10,000 κ=2π x 0.6MHz r κ-1 = 250 ns

21 QND Measurement of Qubit: Dispersive case

ν01 ∆ =−2(π ν 01 ν r ) ∆

ν = 6.04133 GHz ν r r EhJ /

∆min ~ 300 MHz

(∼ 0.05ν r !)

δθκ= 2/g 2 ∆°~5 δθ ~5° min g /π = 5HM z 0 Phase Shift vacuum Rabi frequency 22 Gate Sweep with Qubit Crossing Resonator

∆ = 0 tune qubit thru resonance w/

ν r cavity EhJ /

phase shift 0 changes sign at resonance Phase Shift (a.u.)

23 Spectroscopy of Qubit in Cavity Send in 2 frequencies •Readout ν01 •Spectroscopy

νr νs Probe (CW) cavity at ν r ↑ 0

-10 Phase ↓ -20 -30 n -40 g

Attn (dB) Data -50 1

5.6 5.8 6.0 6.2 6.4

Spectroscopy (CW) Phase at 6.3 GHz

near ν01 24 ng Spectrum of Qubit E d HV=− J σ zx− σ 2 L g

Vg Cavity Phase

EJ Energy Spec Frequency (GHz)

1 nCgg= Vg / e CgVg n = 25 g e Using Cavity to Map Qubit Parameter Space

∆ =−ω01 ωr Transition frequency of qubit Cavity phase shift 2 Φ / Φ 0 ∆ > 0 Φ /Φ 0 ∆ > 0 1 ∆ = 0 Φ0 0 ∆ < 0 ∆ < 0

(GHz) 2e 01

ν 0 1 2 3 4 C V Slice at ∆=0 n = g g g C V e n = g g g e max EEJC~ 6.7 GHz ~ 5.25 GHz 26 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⎝⎠∆ ⎣⎦ 27 Atom ac Stark Shift (Light Shift) Induced by Cavity Photons

navg photons 0 50 100 6.2  6.19

 20 6.18 GHz  Linewidths  0 6.17

Ν 10 450kHz/photon 0 6.16 ΝΝ 6.15 0 0 20 40 60 80 100 RF Power ΜW 28 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 − ||τ n fluctuations 2 δτnnˆˆ()δ = ne in photon number

29 Measurement Back Action: Quantum Noise in ac Stark Shift

11⎛⎞g2 ⎡⎤ Ha≈+ω ††aωσ+2 aa+ eff r ⎜⎟01 ⎢⎥z 22⎝⎠∆ ⎣⎦

1 −+it[(ωϕt)] ψ =↓( +e 01 ↑) 2 2g 2 ⎡ t ⎤ ϕ()tn=+⎢ ∫ dτδnˆ(τ)⎥ ∆ ⎣ 0 ⎦

light shift random dephasing 30 Measurement Back Action: Quantum Noise in ac Stark Shift

2g 2 t δϕτ()td= ∫ δnˆ(τ) ∆ 0

1 − δϕ 2 ()t eeitδϕ () ≈ 2 Assuming gaussian fluctuations

2 2 tt 2 ⎛⎞2g δϕτ()td= ⎜⎟∫∫dτ'δnˆˆ(τ)δn(τ') ⎝⎠∆ 00

31 Measurement Back Action: Quantum Noise in ac Stark Shift

Coherent state in driven cavity with damping rate κ

κ − ||τ δτnnˆˆ()δ(0)= ne2

τ 32 Measurement Back Action: Quantum Noise in ac Stark Shift

2 2 tt κ −−|'ττ| 2 ⎛⎞2g δϕ ()td= ⎜⎟∫∫τ dτ 'ne2 ⎝⎠∆ 00

2 2 ⎛⎞2g 2 ≈ ⎜⎟nt κt1 ⎝⎠∆ (Gaussian inhomogeneous broadening) 2 ⎛⎞24g 2 ≈ ⎜⎟ntκt1 ⎝⎠∆ κ (phase random walks--phase diffusion) (Lorentzian homogeneous broadening)

33 Qubit Phase Diffusion (weak measurement)

2 2 2 ⎛⎞24g δϕ ()tn≈ ⎜⎟ t ⎝⎠∆ κ 1 2 2 − ϕ 2 ()t ⎡⎤ −itϕ () ⎛⎞2g −Γ t ee==2 exp ⎢⎥−2⎜⎟nκ t=eϕ κ∆ ⎣⎦⎢⎥⎝⎠ 11⎧⎫∞ Γ S(ω) =− Im -i dt eitω e−Γϕt = ϕ ⎨⎬∫ 22 ππ⎩⎭0 ()ω−ω0 +Γϕ

Γ∝ϕ n valid for Γϕ κ Measurement induced dephasing rate 34 Qubit Inhomogeneous Broadening (strong measurement)

112 2 −−ϕ 2 ()tt⎡⎤(Γ)2 −itϕ () 12⎛⎞g 2 ϕ ee==22exp ⎢⎥−⎜⎟nt=e 2 ∆ ⎣⎦⎢⎥⎝⎠

Γ∝ϕ n

2 ()ωω− 0 ∞ 1 2 − −Γ()t 2 11⎧⎫it ϕ 2Γ Se(ω) =− Im ⎨⎬-i∫ dt e ω e 2 = ϕ π ⎩⎭0 2π Γϕ

Γ∝ϕ n valid for Γϕ κ

35 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 − ||τ n fluctuations 2 δτnnˆˆ()δ = ne in photon number

Γ∝ϕ n Gaussian

Lorentzian Γ∝ϕ n

36 Summary of Dispersive Regime Results

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

Non-radiative decay channels? ε -glassy losses in oxide barriers loss tangent 2 ∼ 10-4 ε1 -electroacoustic coupling to phonons? (Ioffe, Blatter)

37 Dressed Artificial Atom: Resonant Case

? T ω01 = ωR

T 2g

γ +κ 2

Fourier transform of Haroche 1 ω /38ω Rabi flopping expt. “vacuum Rabi splitting” R First Observation of Vacuum Rabi Splitting for a Single Atom (on average) Cs atom in an optical cavity

photons

Thompson, Rempe, & Kimble 199239 First Observation of Vacuum Rabi Splitting in a Superconducting Circuit

qubitqubitdetuneddetuned 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

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

EJr= ω EJr< ω

41 Quantum Computation and NMR of a Single ‘Spin’

Single Spin ½ Quantum Measurement

Vds C C C gb Box c ge Vgb Vge SET

42 (After Konrad Lehnert) Quantum control of NMR language z

y x microwave pulse 1

Ω1

0

π/2 π pulse pulse free evolution (analogous to gyroscopic precession) NOT NOT43 Rabi Flopping of Qubit Under Continuous Measurement

44 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 SUMMARY

Cavity Quantum Electrodynamics

cQED

“circuit QED”

Coupling a Superconducting Qubit to a Single Photon

-first observation of vacuum Rabi splitting 46 -initial quantum control results Coupling Qubits via Cavity Mode

multiple CPB qubits in a cavity

Nb

Nb 20 µm

Nb can integrate multiple qubits in a single cavity, with no additional fabrication complexity 47 Entanglement via Resonator “Bus”

Qubit coupling via virtual photon exchange: ~1cm 2 Jg12 ~/∆

Room for many qubits in single resonator 2 (t ~10-100 ns) Operation rate: Γop ~/g ∆ op

⎡⎤2 Number of Ops ~ Γ∆op max γ NR ,κ ()g / ∼ 40 −1200 ⎣ ⎦ 48 Multi-qubit readout: multiple cavity pulls gg22 ±±12 ∆ ∆ Transmission 12

↓↓ ↑↑ frequency ↓↑ ↑↓

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

Permits selective projection of 2 bit states 49 Single Atom Optical Bistability

g 2 1 ωωrr=+ ∆ 1/+ nnc

nc ≈ 250 photons -15 P1= κnc ∼ 0W

2 3 1 ω 2 g 2 ∆ 0 1 ωdrive ∆ 0 ↑, n ↓, n 50 Comparison of cQED with Atoms and Circuits

Parameter Symbol Optical cQED Microwave Super- with Cs atoms cQED/ conducting Rydberg circuit atoms QED

Dipole moment d/eao 1 1,000 20,000 Vacuum Rabi g/π 220 MHz 47 kHz 100 MHz frequency 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 µs

Atom transit time ttransit > 50 µs 100 µs Infinite 2 -3 -6 -5 Critical atom # N0=2γκ/g 6 x 10 3 x 10 6 x 10 2 2 -4 -8 -6 Critical photon # m0=γ /2g 3 x 10 3 x 10 1 x 10

# of vacuum Rabi nRabi=2g/(κ+γ) 10 5 100 oscillations

51