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, 062320 (2004) Wallraff et al. [cond-mat/0407325] Nature (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 Quantum optics 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 Rydberg atom 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 qubit! 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 10 +1 ± 2 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 ω01 z ††− + Ha=− σ +ωσR a+g()a+aσ 2 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[(ωϕ01 t)] ψ =↓( +e ↑) 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 qubits 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,