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
Circuit QED Blais et al. Phys. Rev. A 69, 0630 (004) Wallraff et al. [cond-mat/040735] Nature (in press)
Atoms Coupled to Photons s 1s p Irreversible spontaneous decay into the photon continuum: p 1 s+ γ T 1 ns 1 Vacuum Fluctuations: (Virtual photon emission and reabsorption) Lamb shift lifts 1s p 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 Quantum optics with an electrical circuit Future Directions 4
Cavity Quantum Electrodynamics (cqed) g = vacuum Rabi freq. κ = cavity decay rate γ = transverse decay rate t = transit time Strong Coupling = g > κ, γ, 1/t Jaynes-Cummings Hamiltonian ˆ E 1 el EJ + H = ω ( ) ˆ ˆ r aa+ + σ x σ z g( aσ + σ a) Electric dipole Quantized Field -level system Interaction 5
Cavity QED: Resonant Case ωr = ω 01 with interaction eigenstates are: +,0 = 1,1 +,0,0 = 1,1,0 ( ) ( ) vacuum Rabi oscillations dressed state ladders 6
Microwave cqed with Rydberg Atoms beam of atoms; prepare in e> vacuum Rabi oscillations 3-d superconducting cavity (50 GHz) observe dependence of atom final state on time spent in cavity measure atomic state, or Review: S. Haroche et al., Rev. Mod. Phys. 73 565 (001) 7
cqed at Optical Frequencies State of photons is detected, not atoms. measure changes in transmission of optical cavity (Caltech group H. J. Kimble, H. Mabuchi) 8
transmission line cavity A Circuit Analog for Cavity QED L = λ ~.5 cm g = vacuum Rabi freq. κ = cavity decay rate γ = transverse decay rate out DC + 6 GHz in 5 µm Lumped element equivalent circuit Blais, Huang, Wallraff, SMG & RS, PRA 004 E - Cross-section of mode: B + + - 10 µm 9
Advantages of 1d Cavity and Artificial Atom g = d ie/ Vacuum fields: Transition dipole: zero-point energy confined d ~40,000ea in < 10-6 0 cubic wavelengths E ~ 0.5 V/m vs. ~ 1 mv/m for 3-d L = λ ~.5 cm 10 x larger than Rydberg atom 10 µm Cooper-pair box atom 10
Resonator as Harmonic Oscillator L r C r 1 1 H = ( LI) + CV L Hˆ = ω ( a a+ ) cavity V = V a+ a RMS r 1 RMS( ) 1 1 1 C 0 V 0 = ω V ωr = 1 µ V C Φ LI = V = momentum coordinate 11
Implementation of Cavities for cqed Superconducting coplanar waveguide transmission line Q > 600,000 @ 0.05 K Optical lithography at Yale Niobium films 1 cm gap = mirror 6 GHz: ω = 300mK n γ 1 @0mK Internal losses negligible Q dominated by coupling 1
The Chip for Circuit QED Nb Nb the atom Nb no wires attached to qubit! 13
Superconducting Tunnel Junction as a Covalently Bonded Diatomic Molecule (simplified view) 8 N 10 N +1 pairs N pairs aluminum island tunnel barrier aluminum island N pairs N +1 pairs 1µ m Cooper Pair Josephson Tunneling Splits the Bonding and Anti-bonding Molecular Orbitals anti-bonding bonding 14
Bonding Anti-bonding Splitting 1 ψ = ± ± ( ) 8 10 + 1 10 8 10 8 8 10 + 1 E E = E 7 GHz 0.3 K anti-bonding bonding J = bonding Josephson coupling H = = anti-bonding 15 E J σ z
Dipole Moment of the Cooper-Pair Box (determines polarizability) L = 10 µm 1 nm - - - - - - + + + + E = V g / L C 1 C V g V g + + C 3 0 = bonding H = E J z σ d Vg L x σ 1/ C d = ( e ) L 1/ C 1+ 1/ C + 1/ C 3 d ~e -µ m = anti-bonding 16
Energy, Charge, and Capacitance of the CPB Energy E J H E J z = σ d V L g x σ Charge no charge signal Q = de dv charge Capacitance C = dq dv polarizability is state dependent 0 1 CV g g / e deg. pt. = coherence sweet spot 17
Using the cavity to measure the state of the atom H E J z = σ g = d x Vσ L ( e) 1/ C 1/ C + 1/ C + 1/ C V = V + V a + a 1 3 ( ) dc RMS V RMS V 0 18 State dependent polarizability of atom pulls the cavity frequency
Dispersive Regime Large Detuning of Atom from Cavity = ω 01 ωr g 19
Large Detuning of Atom from Cavity ω σ ω σ σ ( + ) 01 z H = + Ra a+ g a + a U exp g = aσ + a σ Heff = UHU = ω 01 ωr g g 1 g Heff ωr σ z a a ω 01 + σz cavity freq. shift Lamb shift QND: [ H, σ ] = 0 0 eff z
Cavity Transmission Phase Controlled by State of Atom Nb resonator 0 mk Linewidth κ=π x 0.6MHz κ -1 = 50 ns ν r = 6.04133 GHz Q = π ν r /κ ~ 10,000 1
QND Measurement of Qubit: Dispersive case ν 01 = π ( ν ) 01 ν r ν r EJ / h ν r = 6.04133 GHz min ~ 300 MHz ( 0.05 ν!) r Phase Shift 0 δθ ~5 δθ = g / minκ ~5 g / π = 5MHz vacuum Rabi frequency
Gate Sweep with Qubit Crossing Resonator ν r = 0 EJ / h tune qubit thru resonance w/ cavity Phase Shift (a.u.) 0 phase shift changes sign at resonance 3
Spectroscopy of Qubit in Cavity Send in frequencies Readout Spectroscopy ν 01 Attn (db) 0-10 -0-30 -40-50 Probe (CW) cavity at ν r 5.6 5.8 6.0 Spectroscopy (CW) at 6.3 GHz near ν 01 6. 6.4 Phase Phase ν r Data 1 ν s n g n4 g
Spectrum of Qubit H E = J σ z d L V g σ x Energy V g E J Spec Frequency (GHz) Cavity Phase 1 n = C V / e g g g n g = C V g g 5 e
Φ / Φ ν 01 (GHz) Using Cavity to Map Qubit Parameter Space 0 1 0 Transition frequency of qubit Cavity phase shift > 0 < 0 0 1 3 4 C gvg ng = e Slice at =0 = ω E max ~ 6.7 GHz J ω 01 r Φ / Φ E C 0 Φ 0 > 0 = 0 < 0 e ~ 5.5 GHz n g = C 6 g V e g
Probe Beam at Cavity Frequency Induces Light Shift of Atom Frequency H g 1 g ω σ ω σ eff r z a a 01 + z cavity freq. shift atom ac Stark shift (light shift) = n cavity pull Lamb shift vacuum ac Stark shift g eff r 01 ω ω σ 1 1 H a a + a a+ z 7
Atom ac Stark Shift (Light Shift) Induced by Cavity Photons Ν0 GHz 6. 6.19 6.18 6.17 6.16 6.15 navg photons 0 50 100 450kHz/photon 0 0 0 40 60 80 100 RF Power ΜW 0 10 Ν Ν0 Linewidths 8
Measurement Induced Dephasing: back action = quantum noise in the light Shift 1 g 1 Heff ωra a ω01 a a σz + + δnˆ( τ) δnˆ = ne κ τ n fluctuations in photon number 9
Measurement Back Action: Quantum Noise in ac Stark Shift g eff r 01 ω ω σ 1 1 H a a+ + a a+ ψ 1 ( i [ t ( t )] ) e ω + ϕ = + 01 ϕ τδ τ g () t = n + d nˆ ( ) 0 t z light shift random dephasing 30
Measurement Back Action: Quantum Noise in ac Stark Shift t g () t = d nˆ ( ) δϕ τ δ τ 0 e 1 iδϕ () t e δϕ () t Assuming gaussian fluctuations t g () t = d d ' nˆ( ) nˆ( ') δϕ τ τ δ τ δ τ t 0 0 31
Measurement Back Action: Quantum Noise in ac Stark Shift Coherent state in driven cavity with damping rate κ δnˆ( τ) δnˆ(0) = ne κ τ τ 3
Measurement Back Action: Quantum Noise in ac Stark Shift t t κ ' g τ τ δϕ () t = dτ dτ ' ne 0 0 g nt κt 1 (Gaussian inhomogeneous broadening) g 4 n t κt 1 κ (phase random walks--phase diffusion) (Lorentzian homogeneous broadening) 33
Qubit Phase Diffusion (weak measurement) g 4 δϕ () t n t κ 1 ϕ () t iϕ () t g Γ t e = e = exp nκ t = e ϕ κ 1 iωt Γ 1 ϕt ϕ S( ω) = Im -i dt e e = π π ( ω ω 0 0) +Γϕ Γ n valid for Γϕ κ ϕ Γ Measurement induced dephasing rate 34
Qubit Inhomogeneous Broadening (strong measurement) e = e = exp nt = e ϕ 1 1 ϕ i () t 1 g Γ ϕ Γ n () t ( ϕt) ( ω ω0 ) 1 ( ) 1 Γ t i t ϕ ω 1 Γϕ S ω = e = e ( ) Im -i dt e π 0 π Γ ϕ Γ n valid for Γϕ κ ϕ 35
Measurement Induced Dephasing: back action = quantum noise in the light Shift 1 g 1 Heff ωra a ω01 a a σz + + δnˆ( τ) δnˆ = ne κ τ n fluctuations in photon number Γϕ Γϕ n n Gaussian Lorentzian 36
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 ε ε -4 loss tangent 10 -electroacoustic coupling to phonons? (Ioffe, Blatter) 1 37
Dressed Artificial Atom: Resonant Case ω = ω 01 R? T T g γ + κ Fourier transform of Haroche Rabi flopping expt. vacuum Rabi splitting 1 ω ω / 38 R
First Observation of Vacuum Rabi Splitting for a Single Atom Cs atom in an optical cavity (on average) photons 39 Thompson, Rempe, & Kimble 199
First Observation of Vacuum Rabi Splitting in a Superconducting Circuit P probe = 140 dbm 17 = 10 W = n ω κ / r qubit detuned from cavity n 1 g /π = 1 MHz κ /π = 0.6 MHz γ / π = 1 MHz qubit tuned into resonance 1 qubit photon ( + ) g 1 qubit photon ( ) 40
Observing the Avoided Crossing of Atom & Photon E J = ωr EJ < r ω 41
Quantum Computation and NMR of a Single Spin Single Spin ½ Quantum Measurement V gb C gb C c C ge V ds Box SET V ge (After Konrad Lehnert) 4
NMR language Quantum control of qubits z y microwave pulse 1 x Ω 1 0 π/ pulse π pulse ree evolution (analogous to gyroscopic precession) NOT NOT 43
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 -initial quantum control results 46
Coupling Qubits via Cavity Mode Nb multiple CPB qubits in a cavity Nb 0 µm Nb can integrate multiple qubits in a single cavity, with no additional fabrication complexity 47
Entanglement via Resonator Bus ~1cm Qubit coupling via virtual photon exchange: J 1 ~ g / Room for many qubits in single resonator Operation rate: Γ op ~ g / (t op ~10-100 ns) Number of Ops ~ Γ max ( ) op γ NR, κ g / 40 100 48
Multi-qubit readout: multiple cavity pulls Transmission g ± ± g 1 1 frequency Single readout line, bits of information: Two qubit readout without extra wires Permits selective projection of bit states 49
Single Atom Optical Bistability ω n r c g = ωr + + 50 photons = n -15 P κ c 10 W 1 1 n/ n c 3 1 0, n, n 1 0 ω drive g 50 ω
Comparison of cqed with Atoms and Circuits Parameter Symbol Optical cqed with Cs atoms Microwave cqed/ Rydberg atoms Superconducting circuit QED Dipole moment d/ea o 1 1,000 0,000 Vacuum Rabi frequency g/π 0 MHz 47 khz 100 MHz Cavity lifetime 1/κ; Q 1 ns; 3 x 10 7 1 ms; 3 x 10 8 160 ns; 10 4 Atom lifetime 1/γ 60 ns 30 ms > µs Atom transit time t transit > 50 µs 100 µs Infinite Critical atom # N 0 =γκ/g 6 x 10-3 3 x 10-6 6 x 10-5 Critical photon # m 0 =γ /g 3 x 10-4 3 x 10-8 1 x 10-6 # of vacuum Rabi oscillations n Rabi =g/(κ+γ) 10 5 100 51