Josephson qubits. P. Bertet. SPEC, CEA Saclay (France), Quantronics group

Josephson qubits P. Bertet SPEC, CEA Saclay (France), Quantronics group

Outline Lecture 1: Basics of superconducting qubits Lecture 2: Qubit readout and circuit quantum electrodynamics 1) 2) 3) Readout by a linear resonator Nonlinear resonators for high-fidelity readout Resonant qubit-resonator coupling: quantum state engineering and tomography Lecture 3: 2-qubit gates and quantum processor architectures

Fabrication techniques small junctions 1) e-beam patterning 2) development 3) first evaporation 4) oxidation 5) second evap. 6) lift-off 7) electrical test e-beam lithography e- Al/Al2O3/Al junctions O2 PMMA PMMA-MAA SiO2 I.3) Decoherence small junctions Multi angle shadow evaporation

QUANTRONIUM (Saclay group) gate I.3) Decoherence 160 x160 nm

FLUX-QUBIT (Delft group) I.3) Decoherence

TRANSMON QUBIT (Saclay group) 40µm 2µµ I.3) Decoherence

The ideal qubit readout relax. 2 β = p α 0>+β 1> 1 0 a + b 0? 1 1 0 a + b 0? 1 a + b 0? 1 1111100 tmeas << T1 p= α 2 1 0 Hi-Fi 0000000 Quantum Non Demolishing (QND) BUT.HOW??? SURPRISING DIFFICULT AND INTERESTING QUESTION FOR SUPERCONDUCTING QUBITS

The readout problem 1) Readout should be FAST : t meas << T1 : 1µ s for high fidelity ( F ﾣ 1 t meas / T1 ) Ideally, t meas : 10ns 2) Readout should be NON-INVASIVE Unwanted transition caused by readout process (but full dephasing can t be avoided!!!) errors 3) Readout should be COMPLETELY OFF during quantum state preparation (avoid backaction)

Readout by a linear resonator 1D CPW resonator Superconducting artificial atom R. Schoelkopf, 2004 A. Blais et al., Phys. Rev. A 69, 062320 (2004) A. Walraff et al., Nature 431, 162 (2004) I. Chiorescu et al., Nature 431, 159 (2004) Modern readout methods by coupling to a resonator (CIRCUIT QUANTUM ELECTRODYNAMICS)

Physical realization L=3.2cm, fn=n 1.8GHz 3mm 50µm Coupling capacitor Cc 10mm 20µm Typical lateral dimensions : 10µm - 1-dimensional mode - Very confined : Vcav ﾣ 10 5 λ3 - Large voltage quantum fluctuations δ V0 : µv - Quality factor easily tuned by designing Cc

CPB coupled to a CPW resonator A. Blais et al., PRA 69, 062320 (2004) ( ) Vˆg = δv0 aˆ + aˆ + + Vgext Cg Vext ωc θˆ n 2 ˆ ˆ ˆ ˆ Htot = EJ cosθ + 4EC ( n ng ) + hωc aˆ + aˆ 2 ˆ ˆ Htot = EJ cos θ + 4EC ( nˆ ngext ) + hωc aˆ +aˆ + 8(Cg δv0ec / 2e )nˆ(a + a + ) Hˆ q Hˆ cav H int 2-level approximation + Rotating Wave Approximation Htot ωge ; σ z + ωc (a + a + 1/ 2) + g (σ + a + σ a + ) 2 Jaynes-Cummings Hamiltonian g = 2eδV0 (Cg / C ) 0 nˆ 1

Strong coupling regime with superconducting qubits g = 2eδV0 (Cg / C ) 0 nˆ 1 GEOMETRICAL dependence of g Easily tuned by circuit design Can be made very large! (Typically : 0 200MHz) g ﾣ 200MHz >> γ,κ ﾣ 100 500kHz Strong coupling condition naturally fulfilled with superconducting circuits (Q=100 enough for strong coupling!!)

The Jaynes-Cummings model g,3> d e,3> d e,2> g,2> e,1> g,1> e,0> HJ C = HJ C ωge σ z + ωc (a + a + 1/ 2) + g (σ + a + σ a + ) 2 couples only level doublets g,n+1>, e,n> Exact diagonalization possible g,0> Restriction of HJ-C to g,n+1>, e,n> g, n + 1 g, n + 1 e, n e, n (δ =ω (n + 1)ωc δ / 2 g n +1 g n +1 ( n + 1) ω + δ / 2 c Note : g,0> state is left unchanged by HJ-C with Eg,0=-δ/2 ge ωc )

The Jaynes-Cummings model g,3> d e,3> d e,2> g,2> e,1> g,1> e,0> HJ C = ωge σ z + ωc (a + a + 1/ 2) + g (σ + a + σ a + ) 2 HJ C couples only level doublets g,n+1>, e,n> Exact diagonalization possible g,0> Coupled states +, n = cos θ n e, n + sinθ n g, n + 1 E +,n, n = sinθ n e, n + cosθ n g, n + 1 E,n θn = h = (n + 1)hωc + 4g 2 ( n + 1) + δ 2 2 h = (n + 1)hωc 4g 2 (n + 1) + δ 2 2 2g n + 1 1 tan 1 2 δ

The Jaynes-Cummings model 1.08 e, n E/(hν c) 1.04 1.00 0.96 0.92 g, n + 1 g, n + 1 e, n -5 0 δ/g 5

The Jaynes-Cummings model 1.08 +,n e, n E/(hν c) 1.04 g, n + 1 1.00 g, n + 1 2g n + 1,n 0.96 e, n 0.92-5 0 δ/g 5

Two interesting limits 1.08 +,n e, n E/(hν c) 1.04 g, n + 1 1.00 g, n + 1 2g n + 1,n 0.96 RESONANT REGIME (δ=0) e, n 0.92-5 0 δ/g 5

Two interesting limits 1.08 +,n e, n E/(hν c) 1.04 g, n + 1 1.00 g, n + 1 2g n + 1,n 0.96 RESONANT REGIME (δ=0) e, n 0.92-5 DISPERSIVE REGIME ( δ >>g) 0 δ/g 5 DISPERSIVE REGIME ( δ >>g)

Two interesting limits 1.08 +,n e, n E/(hν c) 1.04 g, n + 1 QUANTUM STATE ENGINEERING 1.00 2g n + 1 g, n + 1,n 0.96 QUBIT STATE e,readout n 0.92-5 DISPERSIVE REGIME ( δ >>g) RESONANT REGIME (δ=0) 0 δ/g QUBIT STATE READOUT 5 DISPERSIVE REGIME ( δ >>g)

The Jaynes-Cummings model : dispersive interaction δ >> g + ωge + χ ω + 2 χ ( a a + 1/ 2) ge + H J C / h ﾣ σ z + (ωc + χσ z )a a = σ z + ωc a +a 2 2 g2 with χ = the dispersive coupling constant δ 1) Qubit state-dependent shift of the cavity frequency ω% c = ωc + χσ z Cavity can probe the qubit state non-destructively 2) Light shift of the qubit transition in the presence of n photons δωge = 2 χ n Field in the resonator causes qubit frequency shift and decoherence

Dispersive readout of a transmon: principle ωc + χσ z 0> or 1>??

Dispersive readout of a transmon: principle Veiωc t Veiωc t ωc + χσ z 0> or 1>?? ω=ωc

Dispersive readout of a transmon: principle Veiωc t Veiωc t ωc + χσ z α1> α0> ω=ωc 0> or 1>??

Dispersive readout of a transmon: principle Veiωc t Veiωc t Ve iωc t + ωc + χσ z φ α1> α0> ω=ωc π 2χ 0> φ π 1> 0,96 1,00 ωd/ωc 1,04 0> or 1>??

Dispersive readout of a transmon: principle Veiωc t Veiωc t Ve iωc t + ωc + χσ z φ α1> α0> L.O ω=ωc φ0 π or φ1??? 2χ 0> φ π 1> 0,96 1,00 ωd/ωc 1,04 0> or 1>??

Typical implementation (Saclay) 5 mm (f0=6.5ghz) Q=700 80µµ g = 45MHz 40µµ 2µµ (optical+e-beam lithography)

Typical setup (Saclay) MW meas MW drive COIL Vc LO db 20dB 20dB 50MHz I Fast Digitizer Q G=56dB A(t) φ(t) 300K G=40dB TN=2.5K 50Ω 4K DC-8 GHz 30dB 600mK 1.4-20 GHz 20dB 4-8 GHz 50Ω 18mK

Observation of the vacuum Rabi splitting with electrical circuits (courtesy of S. Girvin) Signature for strong coupling: Placing a single resonant atom inside the cavity leads to splitting of transmission peak 2008 vacuum Rabi splitting atom off-resonance observed in: cavity QED R.J. Thompson et al., PRL 68, 1132 (1992) I. Schuster et al. Nature Physics 4, 382-385 (2008) on resonance circuit QED A. Wallraff et al., Nature 431, 162 (2004) quantum dot systems J.P. Reithmaier et al., Nature 432, 197 (2004) T. Yoshie et al., Nature 432, 200 (2004) 28 A. Wallraff et al., Nature 431, 162 (2004)

Qubit spectroscopy with dispersive readout -120 Probe resonator phase MW meas g Pump TLS -122 φ ( ) MW drv -124-126 Some e 5,25 Drive freq (GHz) π e ϕ g π ω/ωc 5,30 5,35

Typical spectroscopy of a transmon + cavity circuit ν01 ν12 νχ

Rabi oscillations measured with dispersive readout Δt MW drv Variable-length drive MW meas Projective measurement x 10000 Ensemble averaging -108 0-111 X ϕ ( ) Y T2R=316 ns -114 1-117 0 200 400 600 t (ns) 800 1000

Dispersive readout : the signal-to-noise issue Veiωc t Veiωc t Ve iωc t + ωc + χσ z φ α1> α0> Ideal amplifier L.O ω=ωc φ0 π or φ1??? 2χ 0> φ π 1> 0,96 1,00 ωd/ωc 1,04 0> or 1>??

Dispersive readout : the signal-to-noise issue Veiωc t Veiωc t Ve iωc t + Real amplifier TN=5K ωc + χσ z φ L.O ω=ωc φ0 π or α0> φ1??? 2χ 1> 0,96 No discrimination in 1 shot 0> φ π α1> 0> or 1>?? 1,00 ωd/ωc 1,04

Dispersive readout : the signal-to-noise issue Veiωc t Veiωc t QUANTUMLIMITED AMPLIFIER?? Ve iωc t + ωc + χσ z φ α1> 0> or 1>?? α0> Real amplifier TN=5K L.O ω=ωc φ0 π or φ1 2χ 0> φ π 1> 0,96 in one single-shot?? 1,00 ωd/ωc 1,04

How to build an amplifier with minimal noise??? pump signal in signal out Nonlinear resonator λ/4 λ/4 Junction causes Kerr non-linearity K + 2 2 Hc =ω h ac a + h (a ) a 2 + Resonator can behave as parametric amplifier K. Lehnert group M. Devoret group I. Siddiqi group II.2) Nonlinear resonator

A nonlinear resonator as quantum-limited amplifier δmax II.2) Nonlinear resonator M. J. Hatridge, R. Vijay, D. H. Slichter, J. Clarke and I. Siddiqi, Phys. Rev. B 83, 134501 (2011) (courtesy I. Siddiqi)

A nonlinear resonator as quantum-limited amplifier Small Saturated signal II.2) Nonlinear resonator (courtesy I. Siddiqi)

Signal-to-noise enhancement by a paramp M. Castellanos-Beltran, K. Lehnert, APL (2007) (quantum limit on how good an amplifier can be : Caves theorem) Actually reached in several experiments : quantum limited measurement II.2) Nonlinear resonator

Qubit and amplifier at 30 mk OUTPUT INPUT II.2) Nonlinear resonator DRIVE (courtesy I. Siddiqi)

Individual measurement traces readout off readout on R. Vijay, D.H. Slichter, and I. Siddiqi, PRL 106, 110502 (2011) II.2) Nonlinear resonator (courtesy I. Siddiqi)

Bivalued histograms Single-shot discrimination of qubit state II.2) Nonlinear resonator (courtesy I. Siddiqi)

Other strategy : sample-and-hold detector integrated with qubit pump λ/4 λ/4 Nonlinear resonator used as threshold detector II.2) Nonlinear resonator

Other strategy : sample-and-hold detector integrated with qubit Kerr-nonlinear resonator λ/4 λ/4 pump H Pd /Pc = 0.2 0.5 1.0 1.8 I Ic 0.2 0.1 0 L 2 0 II.2) Nonlinear resonator 2 - BISTABILITY FOR Ω > Ωc = 3 Ω

The Cavity Josephson Bifurcation Amplifier (CJBA) M. Devoret group, Yale MW drive : Pd(t), ω d ϕ in JBA: I. Siddiqi et al., PRL (2004) CJBA: M. Metcalfe et al, PRB (2007) Non linear resonator ϕout H Pd Pd H state Bistable region L Switching from L to H : BIFURCATION ωd L state ωc II.2) Nonlinear resonator Stochastic process governed by thermal or quantum noise. M.I. Dykman and M.A. Krivoglaz, JETP 77, 60 (1979) M.I. Dykman and V.N. Smelyanskiy, JETP 67, 1769 (1988)

The Cavity Josephson Bifurcation Amplifier (CJBA) M. Devoret group, Yale MW drive : Pd(t), ω d ϕ in JBA: I. Siddiqi et al., PRL (2004) CJBA: M. Metcalfe et al, PRB (2007) Non linear resonator ϕout H Pd Pd H state Bistable region ωd L state ωc II.2) Nonlinear resonator Switching probability L 1,0 0,8 0,6 0,4 0,2 0,0-36 -35-34 Power Pd (db) -33

Readout of transmon with CJBA MW drive : Pd(t), ω d ϕ in Non linear resonator ϕout qubit in 0> or 1> Pd H state L state 2χ ω c 1>ω c 0> II.2) Nonlinear resonator ωd Switching Porbability Pd 1,0 0,8 1> 0,6 0,4 0,2 0,0 0> -38-37 -36-35 -34-33 SINGLE-SHOT QUBIT READOUT Power Pd (db)

Rabi oscillations visibility hν12 0 t tπ,12 250ns 400ns Pswitch (%) hν01 2 TRabi=500ns Mallet et al., Nature Physics (2009) t (µs) Single-shot 93% contrast Rabi oscillations II.2) Nonlinear resonator See also A. Lupascu et al., Nature Phys. (2007)