1. Depts. of Applied Physics & Physics Yale University expt. Andreas Wallraff David Schuster Luigi Frunzio Andrew Houck Joe Schreier Hannes Majer Blake Johnson Circuit QED: Atoms and Cavities in Superconducting Microwave Circuits theory Alexandre Blais Jay Gambetta PI’s Rob Schoelkopf Steve Girvin Michel Devoret www.eng.yale.edu/rslab

2. Overview Quantum optics and Cavity QED The AC Stark shift & backaction of QND measurement – towards splitting the “atom” to see single photons The future?: - “bus” coupling of qubits - other possible (microscopic?) circuit elements Circuit QED: - One-d microwave cavities and coupling to JJ qubits Experiments showing strong coupling – splitting the photon The beauty of being off-resonant: - lifetime enhancement/suppression by cavity

3. State of the Art in Superconducting Qubits NIST/UCSB NEC/Chalmers Charge TU Delft/SUNY Flux Saclay/Yale Charge/PhasePhase Nonlinearity from Josephson junctions (Al/AlO x /Al) 1 st qubit demonstrated in 1998 (NEC Labs, Japan) “Long” coherence shown 2002 (Saclay/Yale) Several experiments with two degrees of freedom One example of C-NOT gate (2003, NEC again) Junction size # of Cooper pairs E J = E C So far only classical E-M fields: atomic physics with circuits Our goal: interaction w/ quantized fieldsQuantum optics with circuits

4. Cavity Quantum Electrodynamics (cQED) 2g = vacuum Rabi freq.  = cavity decay rate  = “transverse” decay rate Quantized Field Electric dipole Interaction 2-level system Jaynes-Cummings Hamiltonian Strong Coupling = g  t t = transit time

5. Cavity QED: Resonant Case vacuum Rabi oscillations “dressed state ladders” (e.g. Haroche et al., Les Houches notes) # of photons qubit state

6. Microwave cQED with Rydberg Atoms Review: S. Haroche et al., Rev. Mod. Phys. 73 565 (2001) 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 … P excited time

7. Optical Cavity QED … measure changes in transmission of optical cavity e.g. Kimble and Mabuchi groups at Caltech

8. 2004: Year of Strong Coupling Cavity QED superconductor flux and charge qubits Nature (London) 431, 159 & 162 (Sept. 2004) alkali atomsRydberg atoms semiconductor quantum dots Nature (London) 432, 197 & 200 (Nov. 2004) single trapped atom PRL 93, 233603 (Dec. 2004)

9. A Circuit Implementation of Cavity QED 2g = vacuum Rabi freq.  = cavity decay rate  = “transverse” decay rate L = ~ 2.5 cm Cooper-pair box “atom” 10  m 10 GHz in out transmission line “cavity” Theory: Blais et al., Phys. Rev. A 69, 062320 (2004)

10. Advantages of 1d Cavity and Artificial Atom 10  m Vacuum fields: zero-point energy confined in < 10 -6 cubic wavelengths Transition dipole: E ~ 0.2 V/m vs. ~ 1 mV/m for 3-d x 10 larger than Rydberg atom L = ~ 2.5 cm Cross-section of mode (TEM!): ++ - - E B 10  m

11. Implementation of Cavity on a Chip Superconducting transmission line Niobium films gap = mirror 6 GHz: 2 cm Si RMS voltage: even when

12. Qubits: Why Superconductivity? ~ 1 eV E 2  ~ 1 meV ATOM SUPERCONDUCTING NANOELECTRODE few electrons N ~ 10 9 total number of electrons superconducting gap “forest” of states

13. The Single Cooper-pair Box: an Tunable Artificial Atom ECEC EJEJ N N+1 2   ~ 1 meV) I “Zeeman shift” V “Stark shift” tunnel junctions (1 nm)

14. Note scale Pseudo spin ½: Coulomb EnergyJosephson tunneling Bias Gate The Real Artificial Atom Island containing 10 8 or 10 8 +1 pairs Nb Si Al

15. Energy Levels of Cooper Pair Box Tune  x with voltage: (Stark) Tune  z with  (Zeeman)

16. Coupling to Cavity Photons A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, PRA 69, 062320 (2004) Jaynes-Cummings

17. The Dipole Moment of the Cooper-Pair Box V 1 nm L = 10  m ++++ ++ ---- -- V CjCj CgCg CgCg

18. How Big Can a Dipole Coupling Get, Anyway? for a half-wave resonator: the fine structure constant in circuit form! “The Fine Structure Limit on Coupling” or g ~ 200 MHz on a 5 GHz transition

19. Comparison of cQED with Atoms and Circuits ParameterSymbolOptical cQED with Cs atoms Microwave cQED/ Rydberg atoms Super- conducting circuit QED Dipole momentd/ea o 11,00020,000 Vacuum Rabi frequency g/  220 MHz47 kHz100 MHz Cavity lifetime 1/  Q 1 ns; 3 x 10 7 1 ms; 3 x 10 8 160 ns; 10 4 Atom lifetime 1/  60 ns30 ms > 2  s Atom transit timet transit > 50  s100  s Infinite Critical atom # N 0 =2  /g 2 6 x 10 -3 3 x 10 -6 6 x 10 -5 Critical photon # m 0 =  2 /2g 2 3 x 10 -4 3 x 10 -8 1 x 10 -6 # of vacuum Rabi oscillations n Rabi =2g/(  ) 105100

20. The Chip for Circuit QED No wires attached to qubit! Nb Si Al

21. Microwave Setup for cQED Experiment Transmit-sideReceive-side

22. Measuring the Cavity Use microwave powers ~ 1 photon = 10 -17 watts

23. Bare Resonator Transmission Spectrum

24. First Observation of Vacuum Rabi Splitting for a Single Real Atom Thompson, Rempe, & Kimble 1992 Cs atom in an optical cavity Transmission

25. Bare Resonator Transmission Spectrum Qubit strongly detuned from cavity tune into resonance with cavity and repeat

26. Vacuum Rabi Mode Splitting by an Artificial Atom 2g Critical atom (N 0 ) & photon #: (M 0 ) Our Records So Far:

27. Observing the Avoided Crossing of “Atom” & “Photon” qubit photon |qubit + photon> = |qubit-photon> = A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. Girvin, and R. J. Schoelkopf, Nature (London) 431, 162 (2004)

28. CgCg Box Spontaneous Emission into Continuum? Decay rate: Power lost in resistor: “Atom” quality factor:

29. CgCg Box Spontaneous Emission into Resonator? Decay rate: On resonance: “Atom” quality factor: the Purcell factor in circuit guise!

30. CgCg Box Spontaneous Emission into Resonator? Decay rate: Off resonance: “Atom” quality factor: cavity enhancement of lifetime! Dispersive limit:

31. Off-Resonant Case: Lifetime Enhancement { See e.g. Haroche, Les Houches 1990 Really, a way to measure non-EM part of  “photonic part” of atom

32. Non-Radiative Decays of Qubit? Predicted cavity-enhanced lifetime ~ 0.001 s! Mechanism of non-radiative losses? Observed lifetimes ~ 1 -10  s

33. How to Measure without Dissipation? Transmission Frequency dielectric changes “length” of cavity A dispersive measurement – measures susceptibility, not loss “leave no energy behind”! (c.f. “JBA amplifier,” measures mag. suscept., by Devoret et al.)

34. Dispersive Circuit QED Dispersive regime: Small “mixing” of qubit and photon, but still small frequency shift of cavity!

35. Dispersive QND Qubit Measurement A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and RS, PRA 69, 062320 (2004) reverse of Nogues et al., 1999 (Ecole Normale) QND of photon using atoms!

36. Controlling the Qubit in the Cavity Large detuning of qubit frequency from cavity Add second microwave pulse to excite qubit  qubit Operate at gate-insensitive “sweet spot” for long coherence - A “clock” transition for SC qubits! (after Vion et al. 2002)

37. State Control of Qubit w/ Continuous Msmt. 85% contrast

38. “Unitary” Rabi Oscillations A. Wallraff et al., PRL 95, 060501 (2005)

39. On QND Measurements is a constant of motion, measure w/out changing it a superposition is dephased Phase shift of photons transmitted measures qubit state Photons in cavity dephase qubit

40. cavity freq. shift Lamb shift Probe Beam at Cavity Frequency Induces ac Stark Shift of Atom Frequency atom ac Stark shiftvacuum ac Stark shift

41. cQED Measurement and Backaction - Predictions measurement rate: dephasing rate: phase shift on transmission: quantum limit?: 2x limit, since half of information wasted in reflected beam (expt. still ~ 40 times worse)

42. AC-Stark Effect & Photon Shot Noise D. I. Schuster, A. Wallraff, A. Blais, …, S. Girvin, and R. J. Schoelkopf, cond-mat/0408367 (2004) g = 5.8 MHz g 2 /  =0.6 MHz shift measures n

43. Explanation of Dephasing What if 2g 2 /  >  ? Measurement dephasing from Stark random shifts Gaussian lineshape is sum of Lorentzians Qubit Response Frequency, s Coherent state has shot noise Peaks are Poisson distributed

44. Possibility of Observing Number States of Cavity? g 2 /    = 100 kHz g 2 /  = 5 MHz n = 1 Simulation g2/g2/  theoretical predictions: J. Gambetta, A. Blais, D. Schuster, A. Wallraff, L. Frunzio, J. Majer, S.M. Girvin, and R.J. Schoelkopf, cond-mat/0602322 see expt. results reported later this week: D. Schuster G3.00003 Tues 9:12 AM

45. Future Prospects/Directions cavity QED = testbed system for quantum optics nonlinear quantum optics - single atom/photon bistability - squeezing quantum measurements cavity enhancement of qubit lifetime? - measuring internal dissipation of qubits quantum bus for entanglement (cQED = “circuit quantum electrodynamics”)

46. Coupling Two Qubits via a Photon “long” range and non nearest-neighbor interactions! ala’ Cirac-Zoller ion trap gates 2 cm Address with frequency-selective RF coupling pulses

47. Two Qubits in One Cavity

48. First Two Qubit Cavity Measurements Gate voltage Flux

49. Strong Cavity QED with Polar Molecules? Dispersive qubit interaction

50. The Yale Circuit QED Team Dave Schuster Alexandre Blais (-> Sherbrooke) Andreas Wallraff (-> ETH Zurich) Steve Girvin

51. Summary “Circuit QED”: 1-d resonators + JJ atoms for strong coupling cQEC in the microwave circuit domain First msmt. of vacuum Rabi splitting for a solid-state qubit Dispersive QND measurements and backaction no dissipation - don’t heat the dirt! Control of qubit in cavity: long coherence time and high fidelity Numerous advantages for quantum control and measurement Theory: Blais et al., Phys. Rev. A 69, 062320 (2004) Vac Rabi: Wallraff et al., Nature 431, 132 (2004) AC Stark: Schuster et al., PRL 94, 123602 (2005) Qubit Control: Wallraff et al., PRL 95, 060501 (2005)

52. Circuit QED Publications High visibility Rabi oscillations & coherence time measurements: A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. 95, 060501 (2005) Circuit QED device fabrication: L. Frunzio, A. Wallraff, D. I. Schuster, J. Majer, and R. J. Schoelkopf, IEEE Trans. on Appl. Supercond. 15, 860 (2005) AC Stark shift & measurement induced dephasing: D. I. Schuster, A. Wallraff, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. 94, 123062 (2005) Strong coupling & vacuum Rabi mode splitting: A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. Girvin, and R. J. Schoelkopf, Nature (London) 431, 162 (2004) Circuit QED proposal: A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, PRA 69, 062320 (2004) see: www.eng.yale.edu/rslab