Electrical quantum engineering with superconducting circuits

Transcription

1 switching probability P. Bertet & R. Heeres SPEC, CEA Saclay (France), Quantronics group swap duration (ns) Electrical quantum engineering with superconducting circuits

2 Outline Lecture 1: Basics of superconducting qubits 1) Introduction: Hamiltonian of an electrical circuit 2) The Cooper-pair box 3) Decoherence of superconducting qubits Lecture 2: Qubit readout and circuit quantum electrodynamics 1) Readout using a resonator 2) Amplification & Feedback 3) Quantum state engineering Lecture 3: Multi-qubit gates and algorithms 1) Coupling schemes 2) Two-qubit gates and Grover algorithm 3) Elementary quantum error correction Lecture 4: Introduction to Hybrid Quantum Devices

3 Rationale for the hybrid way CIRCUIT QED

4 Rationale for the hybrid way CIRCUIT QED COMBINING QUANTUM SYSTEMS FOR NOVEL QUANTUM DEVICES

5 Rationale for the hybrid way G. Kurizki et al., PNAS (2015)

6 This lecture : spins / superconducting circuits SUPERCONDUCTING CIRCUITS +SPINS IN SOLIDS

7 These lectures : spins / superconducting circuits Superconducting qubits - Macroscopic circuits (100s µm) - Easily controlled, entangled, readout - Ultimate microwave detectors Spins in crystals - Long coherence times (1s 6hrs) - Optical transitions Superconducting circuits to improve spin detection Superconducting circuits to mediate interaction between spins Spins to store quantum information from superconducting qubits Spins to convert superconducting qubit state into optical photons

8 Outline Lecture 1: Basics of superconducting qubits 1) Introduction: Hamiltonian of an electrical circuit 2) The Cooper-pair box 3) Decoherence of superconducting qubits Lecture 2: Qubit readout and circuit quantum electrodynamics 1) Readout using a resonator 2) Amplification & Feedback 3) Quantum state engineering Lecture 3: Multi-qubit gates and algorithms 1) Coupling schemes 2) Two-qubit gates and Grover algorithm 3) Elementary quantum error correction Lecture 4: Introduction to Hybrid Quantum Devices 1) Spins for hybrid quantum devices 2) Circuit-QED-enabled high-sensitivity magnetic resonance 3) Spin-ensemble quantum memory for superconducting qubit

9 Desired features : Spins for hybrid quantum devices Which spin systems??? Long coherence times Nuclear-spin-free host crystal Nuclear spin bath Decoherence Electron spin Nuclear spins Carbon : 12 C has no nuclear spins, 13 C has spin ½ but 1.1% nat. Abundance Silicon : 28 Si has no nuclear spins, 29 Si has spin ½ but 4.7% nat. Abundance Both materials can be isotopically purified : magnetically silent crystals

10 Desired features : Spins for hybrid quantum devices Which spin systems??? Long coherence times Nuclear-spin-free host crystal Low dc magnetic field for compatibility with superconducting circuits Superconducting resonator / circuit ωω 0 Spins ωω ss (BB 0 ) Need ωω ss BB 0 = ωω 0. But large BB 0 causes vortices Microwave losses! (even w. parallel field) Aluminum : BB Gs Niobium : BB 0 1 T NbTiN : BB 0 5 T Circuits with Josephson junctions? Probably BB Gs

11 Desired features : Spins for hybrid quantum devices Which spin systems??? Long coherence times Nuclear-spin-free host crystal Low dc magnetic field for compatibility with superconducting circuits Nitrogen-vacancy centers in diamond Bismuth donors in silicon 209 Bi e -

12 Nitrogen-Vacancy (NV - ) centers in diamond Conduction V NV excited Excite in green NV ground 637nm Fluorescence in red (637nm) Valence Detection at the single emitter level at 300K using confocal microscopy Single NV 1 µm Gruber et al., Science 276, 2012 (1997)

13 Spin-dependent photoluminescence Ground state is spin triplet, solid-state spin-qubit 3 E Dm=0 1 A 3 A 2.88 GHz Optical pumping leads to strong polarization in m s =0 Spin-dependent photoluminescence : Optical detection of magnetic resonance (ODMR)

14 Spin Hamiltonian : Notations In all these lectures, we use dimensionless spin operators SS = SS/ħ SS zz = 1 2 SS xx = 1 2 SS yy = 1 2 Spin ½ = 1 2 σσ zz = 1 2 σσ xx 0 ii ii 0 = 1 2 σσ yy Spin SS zz = SS xx = SS yy = 1 0 ii 0 ii 0 ii 2 0 ii 0 Magnetization of a spin : MM = γγγ SS γγ ee 2ππ GYROMAGNETIC RATIO = 28 GHz/T for a free electron

15 Spin Hamiltonian S=1 HH ħ = DDSS zz 2 γγ ee BB 00 SS + EE SS xx 2 SS yy 2 + AASS II + QQII zz 2 ZERO-FIELD SPLITTING ZEEMAN SPLITTING STRAIN-INDUCED SPLITTING HYPERFINE INT. WITH 14 N Quadrupole 14 N D/2π=2.878 GHz ω - ω + ω - ω + D/2π = GHz B 0 (mt)

16 Spin Hamiltonian S=1 HH ħ = DDSS zz 2 γγ ee BB 00 SS + EE SS xx 2 SS yy 2 + AASS II + QQII zz 2 ZERO-FIELD SPLITTING ZEEMAN SPLITTING STRAIN-INDUCED SPLITTING HYPERFINE INT. WITH 14 N Quadrupole 14 N D/2π=2.878 GHz ω - ω + ω - ω + D/2π = GHz B 0 (mt)

17 Spin Hamiltonian S=1 HH ħ = DDSS zz 2 γγ ee BB 00 SS + EE SS xx 2 SS yy 2 + AASS II + QQII zz 2 ZERO-FIELD SPLITTING ZEEMAN SPLITTING STRAIN-INDUCED SPLITTING HYPERFINE INT. WITH 14 N I=1 Quadrupole 14 N D/2π=2.878 GHz ω - ω + ω - ω + D/2π = GHz B 0 (mt)

18 Hyperfine ODMR spectrum 14 N zoom 14 N + 13 C NC + 13 C

19 Decoherence mechanisms in spins (1) : energy relaxa Γ 1 = 1/TT 1 Γ 1 = Γ 1,rrrrrr + Γ 1,ppp Γ 1,ppp

20 Decoherence mechanisms in spins (1) : energy relaxa Γ 1 = 1/TT 1 Γ 1 = Γ 1,rrrrrr + Γ 1,ppp Γ 1,ppp In free space, and at X-band frequencies (7 9GHz), Γ 1,rrrrrr ss 1 For NV Γ 1,ppp 300ss 1 i.e. TT 1 = 3mmmm Γ 1,ppp 10 2 ss 1 i.e. TT 1 100ss AT LOW TEMPERATURES, ENERGY RELAXATION IS IN GENERAL NEGLIG

21 Decoherence mechanisms in spins (2) : dephasing SPIN-BATH : paramagnetic impurities or nuclear spins BB 0 BB llllll ωω ss = γγ ee (BB 0 +BB llllll ) ρρ(ωω) BB lllllll Due to spin bath, spins of same species have slightly different frequenc (inhomogeneous broadening) ωω

22 Decoherence mechanisms in spins (2) : dephasing SPIN-BATH : paramagnetic impurities or nuclear spins BB 0 BB llllll (tt) ωω ss (tt) = γγ ee (BB 0 +BB llllll (tt)) BB lllllll (t) ρρ(ωω) Due to spin bath, spins of same species have slightly different frequenc (inhomogeneous broadening) Dephasing is due to the slow evolution of the spin-bath under flip-flop events ωω

23 Various coherence times π/2 T π/2 MEASURE SS zz Ramsey pulse sequence Sensitive to inhomogeneous broadening + slow noise SS xx = ee TT TT 22 αα αα 2 π/2 T π T π/2 MEASURE SS zz Hahn-echo pulse sequence Insensitive to static noise SS xx = ee 2TT TT 22 ββ ββ 2 3 π/2 π π π/2 T T T T MEASURE SS zz Dynamical decoupling pulse sequenc Insensitive to low-frequency noise SS xx = ee NNNN TT γγ γγ 2 3 Because spin-bath is slow, in general TT 2 TT 2 TT 2DDDD

24 Hahn echo on NVs in isotopically purified diamond G. Balasubramyan et al., Nature Materials (2008) Ramsey fringe sequence Hahn echo TT 2 = 7μμμμ TT 2 = 2mmmm Typical : TT 2 TT 2 100

25 Summary : NV centers for hybrid quantum devices Single electron trapped in a diamond lattice Can be operated in BB Gs because of zero-field splitting Long coherence times possible in ultra-pure crystals Can be optically reset in its ground state Individual NVs / ensembles can be characterized at 300K with ODMR

26 Bismuth donors in silicon e Bi Bi +

27 Bismuth donors in silicon e - Bi Bi e - Same Hamiltonian as P:Si (cf M. Pioro lectures) HH ħ = BB 00 ( γγ ee SS γγ nn II) + AAII SS ZEEMAN EFFECTHYPERFINE Two differences : Nuclear spin I=9/2 Large hyperfine coupling AA 2ππ = GHz Useful to introduce FF = II + SS the total angular momentum Note : HH, FF zz = 0 so that energy eigenstates are always states with well-defined mm FF = mm SS + mm II

28 frequency GHz 10 Bi:Si energy levels HH ħ = BB 00 γγ ee SS γγ nn II + AAII SS B

29 frequency GHz 10 The low-field limit HH ħ = BB 00 γγ ee SS γγ nn II + AAII SS 11 states 5 F= B0 9 states 5 F= LOW-FIELD γγ ee BB 0 AA Eigenstates of FF, mm FF Hybridized eletro-nuclear spin states αα 1/2, mm II + ββ + 1 2, mm II 1

30 frequency GHz 10 The low-field limit HH ħ = BB 00 γγ ee SS γγ nn II + AAII SS 11 states 5 F= B0 9 states 5 F= «allowed» transitions at low field

31 Summary : Bismuth donors in Silicon for hybrid quantum device Bi e - Single electron trapped in a silicon lattice Can be operated in BB Gs because of large hyperfine interaction Long coherence times in isotopically purified silicon Rich level diagram (naturally occurring λ transitions)

32 Spin-LC resonator coupling B 0 BB 11 L SPIN C/2 C/2 ωω 0 = 1/ LLLL BB 11 = δδbb 11 ( aa + aa + )

33 Spin-LC resonator coupling Classical drive B 0 BB 11 L SPIN C/2 C/2 κκ ωω 0 = 1/ LLLL P in BB 11 (tt) = δδbb 11 (ααee iiωω 0tt + αα ee iiωω 0tt ) αα = nn = PHOTON NUMBER PP iiii κκκωω 0

34 Spin-LC resonator coupling BB 11 Interaction Hamiltonian : HH iiiiii = MM BB 1

35 Spin-LC resonator coupling BB 11 Interaction Hamiltonian : HH iiiiii = MM BB 1 = γγγ SS δδbb 11 ( aa + aa + )

36 Spin-LC resonator coupling SS z y δδbb 1, δδbb 11 BB 11 x δδbb 1, Interaction Hamiltonian : HH iiiiii = MM BB 1 = γγγ SS δδbb 11 ( aa + aa + ) HH iiiiii ħ = γγγγbb 1, SS zz aa + aa + γγγγbb 1, SS xx ( aa + aa + )

37 Spin-LC resonator coupling SS z y δδbb 1, δδbb 11 BB 11 x δδbb 1, Interaction Hamiltonian : HH iiiiii = MM BB 1 = γγγ SS δδbb 11 ( aa + aa + ) HH iiiiii ħ = γγγγbb 1, SS zz aa + aa + γγγγbb 1, SS xx ( aa + aa + ) Fast rotating term : neglected

38 Spin-LC resonator coupling SS z y δδbb 1, δδbb 11 BB 11 x δδbb 1, Interaction Hamiltonian : HH iiiiii = MM BB 1 = γγγ SS δδbb 11 ( aa + aa + ) 1 ωω 0 0 Projection on { 0, 1 } HH iiiiii ħ = γγγγbb 1, SS zz aa + aa + γγγγbb 1, SS xx ( aa + aa + ) HH iiiiii ħ = γγγγbb 1, 0 SS xx 1 (σσ + σσ + )( aa + aa + )

39 Spin-LC resonator coupling SS z y δδbb 1, δδbb 11 BB 11 x δδbb 1, Interaction Hamiltonian : HH iiiiii = MM BB 1 = γγγ SS δδbb 11 ( aa + aa + ) ωω Projection on { 0, 1 } HH iiiiii ħ = γγγγbb 1, SS zz aa + aa + γγγγbb 1, SS xx ( aa + aa + ) HH iiiiii ħ = γγγγbb 1, 0 SS xx 1 ( σσ + σσ + )( aa + aa + ) Rotating-Wave Approximation HH iiiiii ħ = gg ( σσ aa + + σσ + aa) gg = γγγγbb 1, 0 SS xx 1

40 Coupling constant estimate (1) : Magnetic field fluctuatio δδii 00 BB 11 δδbb 1, μμ 0 4ππππ δδii 0 with δδii 0 = ωω 0 ħ 2ZZ 0 and ZZ 0 = LL/CC CURRENT FLUCTUATIONS RESONATOR IMPEDANCE For large coupling need resonators with High frequency ωω 0 (but fixed by the spins!) Low impedance i.e. low L and high C In practice, for 2D resonators : 10Ω < ZZ 0 < 300Ω

41 Coupling constant estimate δδii 00 δδbb 11 r δδbb 1 μμ 0 4ππππ δδii 0 with δδii 0 = ωω 0 ħ 2ZZ 0 γγ ee 2ππ = 28GGGGGG/TT gg = γγ ee δδbb 1, 0 SS xx 1 Bi:Si (9-10) : 0 SS xx 1 = 0.47 NV centers : 0 SS xx 1 = 1/ 2 ZZ 0 = 50Ω, rr = 1μμμμ ZZ 0 = 15Ω, rr = 20nnnn NV centers ωω ss 2222 = gg 2ππ = 70Hz gg 2ππ = 6kHz ωω ss Bi:Si = gg = 120Hz 2ππ gg 2ππ = 11kHz

42 Coupling regimes Overall, spin-resonator coupling constant (up to 10kHz for extreme dimensions) gg 2ππ khz Comparison to resonator and spin damping rates? Resonators : Highest quality factor level is Q=10 6 i.e. energy damping rate κκ = ωω 0 QQ ss 1 gg Spins : in isotopically pure crystals, possible to obtain TT 2 = μμμμ i.e. dephasing rate or even lower than g gg < κκ or even gg κκ : «bad cavity» REGIME ( circuit QED)

43 Outline Lecture 1: Basics of superconducting qubits 1) Introduction: Hamiltonian of an electrical circuit 2) The Cooper-pair box 3) Decoherence of superconducting qubits Lecture 2: Qubit readout and circuit quantum electrodynamics 1) Readout using a resonator 2) Amplification & Feedback 3) Quantum state engineering Lecture 3: Multi-qubit gates and algorithms 1) Coupling schemes 2) Two-qubit gates and Grover algorithm 3) Elementary quantum error correction Lecture 4: Introduction to Hybrid Quantum Devices 1) Spins for hybrid quantum devices 2) Circuit-QED-enabled high-sensitivity magnetic resonance 3) Spin-ensemble quantum memory for superconducting qubit

44 A. Blais et al., PRA 69, (2004) J. Gambetta et al., PRA 77, (2008) Spins in a «bad cavity»: the model drive κ ω 0 aa oooooo g HH ħ = ωω ss 2 σσ zz + ωω 0 aa + aa + gg(aa + σσ + aaσσ + ) + drive at ω 0 HH tt = iii κκββ( ee iiωω 0tt aa + ee iiωω 0tt aa + ) In rotating frame at ω 0 : HH ħ = δδσσ zz + gg aa + σσ + aaσσ + + ββ( aa + aa + ) Damping terms (taken into account in Lindblad form) : - energy in cavity at rate κκ = ωω 0 /QQ - Spin dephasing at rate γγ 2

45 Spins in a «bad cavity»: the model aa = κκ 2 aa + κκββ iiii σσ σσ = (iiii + γγ 2 ) σσ + iiii σσ zz aa σσ + = ( iiii + γγ 2 ) σσ + iiii σσ zz aa σσ zz = 2iiii σσ + aa σσ aa + Approximations : «bad cavity limit» gg κκ Field-spin correlations are neglected σσ + aa = σσ + aa σσ zz aa = σσ zz aa σσ aa = σσ aa To find spin steady-state operators : 1) Solve for field <a> without spin 2) Take stationary values of spin operators for spin driven by cavity field (classical Rabi oscillation in field <a> in the cavity), with additional decay channel provided by the cavity γγ PP Adiabatic elimination of the cavity field, see B. Julsgaard et al., PRA 85, (2012) C. Hutchison et al., Canadian Journ of Phys. 87, 225 (2009)

46 The Purcell effect ħωω ss g ωω 0, QQ κκ = ωω 0 /QQ γγ PP γγ PP = 4gg2 κκ ωω ss ωω 0 κκ 2 B. Julsgaard et al., PRA 85, (2012) C. Hutchison et al., Canadian Journ of Phys. 87, 225 (2009) - New way to initialize spins in ground state? - Can be tuned by changing spin/resonator detuning ωω ss ωω 0

47 Field radiated by the spins Steady-state value of the cavity field aa = 2ββ κκ ii 2gg κκ Cavity field w/o spin Output signal from N identical spins σσ Field radiated by spin in cav Spin signal proportional to σσ aa oooooo = ii 2NNgg κκ σσ

48 Conventional Pulsed Inductive Detection Electron Spin Resonance (ESR) Excite spins Echo emission B 0 Sensitivity Minimal number of spins NN mmmmmm detected with a SNR = 1 in a single echo? τ τ π/2 π echo time π π/2

49 Sensitivity of an inductive detection spectrometer gg resonator ωω 0, QQ B 0 TT EE L.O. I Q SSSSSSSSSSSS = aa oooooo dddd TT 2 NNNN κκ NNNNNNNNNN = aa nnnnnnnnnn,ii 2 dddd = TT 2 nn II Number of noise photons in the detected quadrature bandwidth nn II = SS II(ωω) ħωω

50 Sensitivity of an inductive detection spectrometer gg resonator ωω 0, QQ B 0 TT EE L.O. I Q NN mmmmmm ~ nn II pp CC A.Bienfait et al., Nature Nano (2015) Spin polarization For spin ½ at T pp = tanh ħωω 0 2kkkk Number of noise photons in the detected quadrature bandwidth nn II = SS II(ωω) ħωω Single-spin signal Cooperativity CC = gg2 TT 2 κκ

51 Sensitivity of an inductive detection spectrometer nn eeee T nn aaaaaa B 0 nn II = nn eeee,ii + nn aaaaaa,ii nn eeee,ii = 1 2 coth ħωω 0 2kkkk Using a «noiseless» Josephson Parametric Amplifier nn aaaaaa,ii = 0 nn eeee,ii 0.5 QUANTUM LIMIT TT/(ħωω 0 /kk) Quantum limit for magnetic resonance nn II = 0.25

52 EPR spectroscopy : state-of-the-art 1000 Commercial ESR spectrometer (300K) 100 nn II 10 1 QUANTUM LIMIT Sigillito et al., APL 2014 (1.7K) 0.1 1E-12 1E-10 1E-8 1E-6 1E pp CC

53 Quantum limited ESR with Parametric Amplifier 1.4 mm 5µm 2D lumped element Superconducting Al resonator B 1 field gg 2ππ = 55HHHH

54 Quantum limited ESR with Parametric Amplifier 1.4 mm 2D lumped element Superconducting Al resonator Copper box ω 0 /2π = 7.24 GHz Q =

55 Quantum limited ESR with Parametric Amplifier Sig in Sig out pump Josephson Parametric Amplifier 23 GHz 3MHz BW (freq. tunable) Zhou et al. PRB 89, (2014).

56 Quantum limited ESR with Parametric Amplifier 20mK plate Attenuators Circulators JPA 2-axis coil w. sample

57 The Spins: Bi donors in 28 Si 10 allowed ESR-like low B Resonator Implanted Bi mf = 4 -> mf = G mf = 3 -> mf = Gs

58 Spin echo detection π/2 π echo 300 µs 300 µs 2.5µs 5µs A. Bienfait et al., Nature Nano (2015)

59 Coherence time T2e = 9.3 ms A π/2 A π T 2 =9ms : typical for Bi: 28 Si A. Bienfait et al., Nature Nano (2015)

60 Spectrometer single-shot sensitivity SNR=7 ΔSS zz = A. Bienfait et al., Nature Nano (2015) Sensitivity : NN 1111 = = spins per echo Gain comp. to state-of-the-art (Sigillito et al., APL, 2014) Consistent with expectations from formula NN mmmmmm ~ 1 pp nnωω 0 QQTT EE 1 gg

61 EPR sensitivity : summary 1000 NN mmmmmm Commercial ESR spectrometer (300K) B 0 nn II QUANTUM LIMIT NN mmmmmm 10 7 Sigillito et al., APL 2014 (1.7K) C. Eichler et al., arxiv(2016) A.Bienfait et al., Nature Nano (2015) NN mmmmmm = E-12 1E-10 1E-8 1E-6 1E pp CC 10 4 improvement over state-of-the-art NN mmmmmm = S. Probst (2017)

62 Absolute sensitivity and spin relaxation time TT 1 Repetition rate?? Limited by time TT 1 needed for spins to reach thermal equilibr TT = 0 TT T 1 = 0.35 s A Q Spectrometer absolute sensitivity : 1700 spin/ HHHH «Short» T 1 due to spontaneous emission in the cavity (Purcell effect)

63 Observing the Purcell effect for spins ωω 0, QQ gg γγ PP 11 TT 11 = γγ PP + ΓΓ NNNN γγ PP = 4QQgg2 ωω QQ 2 ωω ss ωω 0 ωω 0 2 A. Bienfait et al., Nature (2016)

64 Outline Lecture 1: Basics of superconducting qubits 1) Introduction: Hamiltonian of an electrical circuit 2) The Cooper-pair box 3) Decoherence of superconducting qubits Lecture 2: Qubit readout and circuit quantum electrodynamics 1) Readout using a resonator 2) Amplification & Feedback 3) Quantum state engineering Lecture 3: Multi-qubit gates and algorithms 1) Coupling schemes 2) Two-qubit gates and Grover algorithm 3) Elementary quantum error correction Lecture 4: Introduction to Hybrid Quantum Devices 1) Spins for hybrid quantum devices 2) Circuit-QED-enabled high-sensitivity magnetic resonance 3) Spin-ensemble quantum memory for superconducting qubit

65 Hybrid quantum processor 1) Long coherence time 2) Economical in processing qubits 3) Intrinsic low-crosstalk in gates and qubit readout Few-qubit quantum processor N-qubit memory QUANTUM MEMORY.. ψψ 1 ψψ ii ψψ NN Few-qubit processor + N-qubit memory : N-qubit computer Interest :

66 Memory operations QUANTUM MEMORY

67 Memory operations QUANTUM MEMORY ψψ 1 ψψ 2 ψψ NN 1 WRITE

68 Memory operations QUANTUM MEMORY.. ψψ 1 ψψ 2 ψψ ii ψψ NN 1 WRITE.

69 Memory operations QUANTUM MEMORY ψψ 1 ψψ ψψ NN ψψ ii 1 2 WRITE. READ

70 Memory operations QUANTUM MEMORY WRITE. READ RESET

71 Memory operations QUANTUM MEMORY.. Entangled states ψψ = ii 1 ii NN = 0,1 cc ii1 ii NN ii 1 ii NN WRITE. READ RESET

72 Sketch of hybrid quantum processor Long coherence time (>s) NV Bi:Si Active reset possible

73 Spin ensemble resonator system gg eeeeee = NN HH/h = gg kk aa σ,kk + hcc = gg eeeeee aa bb + aabb kk gg kk2 kk NN Bright mode bb = gg kk gg eeeeee σ +,kk (1 excitation shared by N spins) Coupling of the resonator to one collective spin mode

74 WRITE step : Single-photon transfer

75 WRITE step : Single-photon transfer DC Frequency-tuning by flux ω Spins ω Qubit

76 WRITE step : Single-photon transfer DC Frequency-tuning by flux ω Spins ω Qubit

77 WRITE step : Single-photon transfer Qubit drive DC Frequency-tuning by flux ω Spins ω Qubit

78 WRITE step : Single-photon transfer Qubit drive DC Frequency-tuning by flux Single-shot Qubit readout ω Spins ω Qubit

79 WRITE step : Single-photon transfer 30 mk refrigerator Transmon Bus SQUID readout junction

80 Spectroscopy 24/55 Resonator transmission Simulations

81 Qubit characterization Pulse sequences e> Duration t Readout Rabi ( g> + e>)/ 2 T1 π wait time t Readout T1 qubit = 1.75 µs π/2 with small detuning T2 qubit = 2.2 µs T2 evolution time t Readout

82 Transmon and quantum bus interaction : the SWAP gate Resonant SWAP gate Adiabatic SWAP gate bus qubit π τ R bus qubit π τ R 1,0 0,8 e,0> SWAP e,0> g,1> e,0> g,1> 0,6 Pe 0,4 0,2 g,1> 0, Interaction time, τ (ns) aswap Adiabatic ramp duration, τ (ns)

83 Transmon and quantum bus interaction : the SWAP gate Resonant SWAP gate Adiabatic SWAP gate bus qubit π τ R bus qubit π τ R 1,0 0,8 e,0> SWAP e,0> g,1> e,0> e,0> g,1> 0,6 Pe 0,4 0,2 g,1> 0, Interaction time, τ (ns) g,1> aswap Adiabatic ramp duration, τ (ns)

84 Prepare the Qubit in WRITE step : Single-photon transfer

85 WRITE step : Single-photon transfer DC Transfer from Qubit to resonator

86 WRITE step : Single-photon transfer DC Transfer from Qubit to resonator

87 WRITE step : Single-photon transfer DC Transfer from resonator to spins

88 WRITE step : Single-photon transfer Spins Resonator Vacuum Rabi oscillation

89 WRITE step : Single-photon transfer DC Transfer BACK from spins to Qubit

90 WRITE step : Single-photon transfer DC Transfer BACK from spins to Qubit

91 Qubit readout WRITE step : Single-photon transfer

92 WRITE: storage of 1 Spins Resonator Vacuum Rabi oscillation

93 WRITE: storage of 1 Spins Resonator Vacuum Rabi oscillation

94 WRITE: storage of 1 Spins Resonator Vacuum Rabi oscillation τ s

95 WRITE: storage of 1 Spins Resonator Vacuum Rabi oscillation τ s τ r

96 WRITE: storage of 1 Spins Resonator Vacuum Rabi oscillation Decay in T 2 100ns as expected from the linewidth Bright state TT 2 Dark state ω τ s τ r

97 WRITE efficiency Spins Resonator Vacuum Rabi oscillation E WRITE = 95 % E WRITE = f(τ s ) Spins damping Resonator damping τ s

98 WRITE: storage of / 2 Spins Resonator Vacuum Rabi oscillation State tomography by no pulse (I), π/2(x), π/2(y) 0 π/2(x) π/2(y) 1

99 WRITE: storage of / 2 Spins Resonator Vacuum Rabi oscillation State tomography by no pulse (I), π/2(x), π/2(y) 0 Coherence retrieved τ s τ r π/2(x) π/2(y) 1

100 Spin-photon entanglement spin qubit X(π) τ s /2 τ s /2 δ τ readout bus i ( B I + e δτ B I ) 1,0 0,1 / 2 Hyperfine structure! Oscillations in δτ : Evidence for spin-photon entanglement P e τ (ns) Y. Kubo et al., PRL (2011)

101 Lecture Conclusions Fruitful marriage of circuit Quantum Electrodynamics and Magnetic Resonance Magnetic resonance detection reaching the quantum limit of sensitivity Quantum fluctuations of the field affect spin dynamics (Purcell effect) Use squeezing as a resouce to improve sensitivity even further Quantum memory applications within reach Perspectives Reach single-spin detection sensitivity Build a platform for spin-based quantum computation