Synthesizing arbitrary photon states in a superconducting resonator

Transcription

1 Synthesizing arbitrary photon states in a superconducting resonator Max Hofheinz, Haohua Wang, Markus Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O Connell, D. Sank, M. Weides, J. Wenner, J.M. Martinis, A.N. Cleland Department of Physics, University of California, Santa Barbara Collège de France, March

2 Generation of quantum states in a resonator Classical drive on qubit and resonator

3 Generation of quantum states in a resonator Classical drive on qubit and resonator

4 Generation of quantum states in a resonator Classical drive on qubit and resonator

5 Generation of quantum states in a resonator Classical drive on qubit and resonator

6 Generation of quantum states in a resonator Classical drive on qubit and resonator

7 Generation of quantum states in a resonator Classical drive on qubit and resonator

8 Generation of quantum states in a resonator Classical drive on qubit and resonator

9 Generation of quantum states in a resonator Classical drive on qubit and resonator

10 Generation of quantum states in a resonator Classical drive on qubit and resonator arbitrary states only coherent states

11 Generation of quantum states in a resonator drive the qubit couple it to the harmonic oscillator arbitrary states arbitrary states

12 Device die size: 6 6mm

13 Coplanar waveguide resonator U V ν x very small capacitors are open ends each mode forms a quantum harmonic oscillator we use fundamental mode at frequency ν = c eff L make out of superconductor for high quality factor Q Performance: Q or T.5... µs

14 Superconducting qubits level system hard to make in a circuit very strong nonlinearity is enough allows to address transition exclusively we use Josephson junction as nonlinear circuit element

15 Josephson junction superconductor superconductor phase φ L tunnel barrier phase φ R Dynamics are determined by: I = I sinδ V = δ e t δ = φ L φ R phase difference I critical current of junction V DC Josephson relation AC Josephson relation

16 Josephson junction V Dynamics are determined by: I = I sinδ V = δ e t δ = φ L φ R phase difference I critical current of junction DC Josephson relation AC Josephson relation

17 Superconducting phase qubit U Φ bias k s δ Φ bias ( e H = δ Φ bias) L e I cos δ + Q }{{} C U [δ,q] = ei

18 Superconducting phase qubit U Φ bias k s e g δ Φ bias ( e H = δ Φ bias) L e I cos δ + Q }{{} C U [δ,q] = ei

19 Superconducting phase qubit U Φ bias k s e g δ Φ bias ( e H = δ Φ bias) L e I cos δ + Q }{{} C U [δ,q] = ei

20 Superconducting phase qubit U Φ bias k e g s δ Φ bias ( e H = δ Φ bias) L e I cos δ + Q }{{} C U [δ,q] = ei

21 Superconducting phase qubit U e δ Φ bias e g Γ e = 5Γ g Γ g readout squid δ Φ bias ( e H = δ Φ bias) L e I cos δ + Q }{{} C U [δ,q] = ei

22 UCSB phase qubit readout flux bias microwaves energy relaxation time T 6 ns quality factor Q 4 phase coherence time T 5 ns nonlinearity MHz limits gate time ns readout visibility P e g P g e.9

23 Coupling qubit and resonator H = hν q σ + σ + Ω ( ( ) aσ+ + a σ + hνr a a + )

24 Spectroscopy of qubit-resonator coupling Frequency (GHz) ν r Ω/π ν q Excited state probability Pe (%) high probability indicates eigenfrequencies of qubit-resonator system Qubit flux bias (mφ ) 5

25 Spectroscopy of qubit-resonator coupling coupling 6.6 on off 4 Frequency (GHz) ν r Ω/π ν q Excited state probability Pe (%) high probability indicates eigenfrequencies of qubit-resonator system Qubit and resonator only interact when ν q ν r Coupling can be turned on and off Qubit flux bias (mφ ) 5

26 Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

27 Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

28 Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

29 Generating Fock states: Pumping photons one by one Ω H int = Ω ( ) aσ+ + a σ total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

30 Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

31 Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

32 Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

33 Generating Fock states: Pumping photons one by one Ω H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

34 Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

35 Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

36 Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

37 Generating Fock states: Pumping photons one by one 3Ω H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

38 Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

39 Generating Fock states: Pumping photons one by one 3Ω H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

40 Generating Fock states: Pumping photons one by one H int = Ω ( aσ+ + a σ ) total photon number resonator qubit π Ω π Ω preparation measurement π 3Ω τ Only possible with good pulses: π-pulse: > 98 % fidelity E. Lucero et al. PRL, 47 (8) swap-pulse: > 98 % fidelity

41 Fock states: Clear Rabi oscillations resonator qubit preparation measurement τ Excited state probability Pe (%) Interaction time τ (ns) Prepared state

42 Fock states: Rabi frequencies scale as n Prepared state Rabi frequency (MHz) Normalized Fourier amplitude

43 Fock states: Photon number distribution for.8 probability photon number

44 Fock states: Photon number distribution for.8 probability photon number

45 Fock states: Photon number distribution for 3.8 probability photon number

46 Fock states: Photon number distribution for 4.8 probability photon number

47 Fock states: Photon number distribution for 5.8 probability photon number

48 Fock states: Photon number distribution for 6.8 probability photon number

49 Fock states: Photon number distribution for 7.8 probability photon number

50 Fock states: Photon number distribution for 8.8 probability photon number

51 Fock states: Photon number distribution for 9.8 probability photon number

52 Fock states: Photon number distribution for.8 probability photon number

53 Fock states: Photon number distribution for.8 probability photon number

54 Fock states: Photon number distribution for.8 probability photon number

55 Directly driving the resonator creates a coherent state Probabilities for Fock states are Poisson distributed P n (a = α ) = an e a n! resonator qubit τ

56 Directly driving the resonator creates a coherent state Probabilities for Fock states are Poisson distributed P n (a = α ) = an e a n! resonator qubit τ

57 Directly driving the resonator creates a coherent state Probabilities for Fock states are Poisson distributed P n (a = α ) = an e a n! resonator qubit τ

58 Directly driving the resonator creates a coherent state nω Probabilities for Fock states are Poisson distributed P n (a = α ) = an e a n! resonator qubit τ

59 Directly driving the resonator creates a coherent state Probabilities for Fock states are Poisson distributed P n (a = α ) = an e a n! resonator qubit τ

60 Coherent state: Non-periodic time traces resonator qubit τ Excited state probability (%) Interaction time τ (ns) Displacement α

61 Coherent state: Superposition of Fock states Displacement α Rabi frequency (MHz)

62 Coherent state: Superposition of Fock states Displacement α Probability Rabi frequency (MHz) Photon number

63 Generalizing the pump sequence to arbitrary states only perform partial state transfers leave some probability behind

64 Generalizing the pump sequence to arbitrary states only perform partial state transfers leave some probability behind

65 Generalizing the pump sequence to arbitrary states Ω only perform partial state transfers leave some probability behind

66 Generalizing the pump sequence to arbitrary states only perform partial state transfers leave some probability behind

67 Generalizing the pump sequence to arbitrary states Ω, Ω only perform partial state transfers leave some probability behind

68 Generalizing the pump sequence to arbitrary states qubit and resonator entangled during sequence phases matter swaps act on several photon numbers

69 Algorithm for arbitrary states: Reverse Engineering ψ = 3 ( + + i ) g g e g e g

70 Algorithm for arbitrary states: Reverse Engineering ψ = 3 ( + + i ) g g g g e e Time-reversed problem easier: start with the desired state eliminate states from top to bottom Law, Eberly, PRL 76, 55 (996) resonator qubit

71 Algorithm for arbitrary states: Reverse Engineering g e g e g resonator qubit

72 Algorithm for arbitrary states: Reverse Engineering ψ = ( ) g + (.57i.577 ) e g e g e g resonator qubit

73 Algorithm for arbitrary states: Reverse Engineering g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase g resonator qubit

74 Algorithm for arbitrary states: Reverse Engineering ψ = (( i) +.63 ) g + (.58 +.i) e g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase g resonator qubit

75 Algorithm for arbitrary states: Reverse Engineering g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase swap phase g resonator qubit

76 Algorithm for arbitrary states: Reverse Engineering ψ = (( i) +.63 ) g + (.568i ) e g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase swap phase g resonator qubit

77 Algorithm for arbitrary states: Reverse Engineering g g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase swap phase swap amplitude resonator qubit

78 Algorithm for arbitrary states: Reverse Engineering ψ = (( i) ) g + (.849i ) e g g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase swap phase swap amplitude resonator qubit

79 Algorithm for arbitrary states: Reverse Engineering g g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase swap phase swap amplitude resonator qubit

80 Algorithm for arbitrary states: Reverse Engineering ψ = ( i) g g g g e e parameters per photon step: qubit drive pulse amplitude qubit drive pulse phase swap phase swap amplitude resonator qubit

81 Calibrations impulse response (sampling scope) imperfections of IQ-mixer (spectrum analyzer) cable response (qubit as sampling scope) various parameters for readout SQUID π pulses (amplitude and frequency) measure pulse amplitude swap pulse amplitude swap pulse time for each photon number first order correction for finite rise time of swap pulses qubit/resonator dephasing rate when off resonance resonator displacement/drive amplitude ratio resonator drive phase readout visibility BUT: no per-state calibrations; just run the calculated sequence

82 A superposition of and 3 ψ = + 3 Pe Probability (a.u.) time (ns) Frequency (MHz) Probability Photon number

83 A superposition of and 3 ψ = + 3 and φ = + i 3 look the same! Pe Probability (a.u.) time (ns) Frequency (MHz) Probability Photon number

84 Full description of a resonator state Example: ψ = + 3 Density matrix: ρ = ψ ψ Wigner function: W (α) = π ψ D ( α)πd( α) ψ m 3 i ρ mn Im(α) + π + π π W (α) 3 n Re(α) π

85 Full description of a resonator state Example: ψ = + i 3 Density matrix: ρ = ψ ψ Wigner function: W (α) = π ψ D ( α)πd( α) ψ m 3 i ρ mn Im(α) + π + π π W (α) 3 n Re(α) π

86 Wigner tomography: measurement protocol resonator qubit Preparation of state ψ (Law & Eberly protocol) ρ = F ψ ψ +... τ State reconstruction displacement: ρ ρ α = D( α)ρd(α) qubit resonator interaction for time τ qubit measurement Pe interaction time τ Pn Photon number n W (α) = π Tr (ρ απ) = π Im(α) Re(α) ( ) n P n n= + π + π π π W (α)

87 Controlling photon number ψ = 7 + π + π Im(α) W (α) π Re(α) π 8 6 m 4 Fidelity: F = ψ ρ ψ =.75 ρmn i n 4 6 8

88 Controlling photon number ψ = 8 + π + π Im(α) W (α) π Re(α) π 8 6 m 4 Fidelity: F = ψ ρ ψ =.7 ρmn i n 4 6 8

89 Controlling superpositions ψ = + + π + π Im(α) W (α) π Re(α) π m ρmn Fidelity: F = ψ ρ ψ = n i

90 Controlling superpositions ψ = + + π + π Im(α) W (α) π Re(α) π m ρmn Fidelity: F = ψ ρ ψ = n i

91 Controlling superpositions ψ = π + π Im(α) W (α) π Re(α) π m ρmn Fidelity: F = ψ ρ ψ = n i

92 Controlling superpositions ψ = π + π Im(α) W (α) π Re(α) π m ρmn Fidelity: F = ψ ρ ψ = n i

93 Controlling superpositions ψ = π + π Im(α) W (α) π Re(α) π m ρmn Fidelity: F = ψ ρ ψ = n i

94 Controlling phase ψ = +e 8 iπ π + π Im(α) W (α) π Re(α) π m 3 ρmn Fidelity: F = ψ ρ ψ = n i

95 Controlling phase ψ = +e 8 iπ π + π Im(α) W (α) π Re(α) π m 3 ρmn Fidelity: F = ψ ρ ψ = n i

96 Controlling phase ψ = +e 8 iπ π + π Im(α) W (α) π Re(α) π m 3 ρmn Fidelity: F = ψ ρ ψ = n i

97 Controlling phase ψ = +e 3 8 iπ π + π Im(α) W (α) π Re(α) π m 3 ρmn Fidelity: F = ψ ρ ψ = n i

98 Controlling phase ψ = +e 4 8 iπ π + π Im(α) W (α) π Re(α) π m 3 ρmn Fidelity: F = ψ ρ ψ = n i

99 Voodoo cat ψ k = e i π 3 k 3 dead + π + π Im(α) alive W (α) zombie π Re(α) π 8 m 6 4 ρmn i n 4 6 8

100 Voodoo cat ψ = k=,, n=,3,6,9... e i π 3 k n n! n Im(α) Re(α) + π + π π π W (α) 8 6 m 4 ρmn Fidelity: F = ψ ρ ψ = n i

101 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.5 µs 3 4 n 3 4 i

102 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =. µs 3 4 n 3 4 i

103 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.35 µs 3 4 n 3 4 i

104 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.5 µs 3 4 n 3 4 i

105 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.65 µs 3 4 n 3 4 i

106 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.8 µs 3 4 n 3 4 i

107 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.95 µs 3 4 n 3 4 i

108 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =. µs 3 4 n 3 4 i

109 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.4 µs 3 4 n 3 4 i

110 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.7 µs 3 4 n 3 4 i

111 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =. µs 3 4 n 3 4 i

112 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.3 µs 3 4 n 3 4 i

113 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ =.9 µs 3 4 n 3 4 i

114 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ = 3.5 µs 3 4 n 3 4 i

115 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ = 4.4 µs 3 4 n 3 4 i

116 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ = 6. µs 3 4 n 3 4 i

117 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ = 9. µs 3 4 n 3 4 i

118 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ = µs 3 4 n 3 4 i

119 Decay of resonator states (Haohua Wang) ψ = π + π Im(α) W (α) π Re(α) π 4 3 m ρmn Delay: τ = 5 µs 3 4 n 3 4 i

120 Conclusions Full control over qubit state extended to resonator: arbitrary states (N < due to coherence) deterministic generation full characterization Max Hofheinz, H. Wang, M. Ansmann, R. Bialczak, E. Lucero, M. Neeley, A. O Connell, D. Sank, M. Weides, J. Wenner, J.M. Martinis, A.N. Cleland Fock states: Nature 454, 3 34 (8) decoherence: H. Wang, PRL, 44 (8) Arbitrary states: Nature 459, (9) decoherence: H. Wang, PRL 3, 44 (9) Funding: IARPA, ARO, NSF

121 Superconducting circuits: Microwave quantum optics atom Josephson qubit optical cavity LC or wave-guide resonator light microwaves (ν 7 GHz) Advantages: Qubits and cavities can be coupled at will Microwave signal waveforms can be fully controlled Drawbacks: Low photon energy requires low temperatures T < 5 mk Large solid state system limits coherence times

122 Calibrations impulse response (sampling scope) imperfections of IQ-mixer (spectrum analyzer) cable response (qubit as sampling scope) various parameters for readout SQUID π pulses (amplitude and frequency) measure pulse amplitude swap pulse amplitude swap pulse time for each photon number first order correction for finite rise time of swap pulses qubit/resonator dephasing rate when off resonance resonator displacement/drive amplitude ratio resonator drive phase readout visibility BUT: no per-state calibrations; just run the calculated sequence

123 Calibrating the swap pulse length Calibrate with the pulse lengths from the Fock state tune-up. 5 programmed pulse length (ns) / n

124 Detuning time calibration: Qubit resonator Ramsey fringes resonator qubit π π..8.6 Pe Time between pulses (ns)

125 Custom microwave electronics

126 Sub-ns timing resolution with a GHz DAC deconvolved signal DAC 4 GHz Gaussian lowpass MHz Signal (mv) Time (ns)

127 Sub-ns timing resolution with a GHz DAC deconvolved signal DAC 4 GHz Gaussian lowpass MHz Signal (mv) Time from programmed edge (ns)