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

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

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

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

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

The readout problem 1) Readout should be FAST : for high fidelity ( ) Ideally, 2) Readout should be NON-INVASIVE Unwanted transition caused by readout process errors (but full dephasing can’t be avoided !!!) 3) Readout should be COMPLETELY OFF during quantum state preparation (avoid backaction) II.1) Linear resonator

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) II.1) Linear resonator

Physical realization L=3.2cm, fn=n 1.8GHz 3mm Coupling capacitor Cc Typical lateral dimensions : 10mm - 1-dimensional mode - Very confined : - Large voltage quantum fluctuations - Quality factor easily tuned by designing Cc II.1) Linear resonator

CPB coupled to a CPW resonator A. Blais et al., PRA 69, 062320 (2004) Cg Vext 2-level approximation + Rotating Wave Approximation Jaynes-Cummings Hamiltonian II.1) Linear resonator

Strong coupling regime with superconducting qubits GEOMETRICAL dependence of g Easily tuned by circuit design Can be made very large ! (Typically : 0 – 200MHz) Strong coupling condition naturally fulfilled with superconducting circuits (Q=100 enough for strong coupling !!) II.1) Linear resonator

The Jaynes-Cummings model couples only level doublets |g,2> |e,1> |g,n+1> , |e,n> |g,1> |e,0> Exact diagonalization possible |g,0> Restriction of HJ-C to |g,n+1> , |e,n> Note : |g,0> state is left unchanged by HJ-C with Eg,0=-d/2 II.1) Linear resonator

The Jaynes-Cummings model couples only level doublets |g,2> |e,1> |g,n+1> , |e,n> |g,1> |e,0> Exact diagonalization possible |g,0> Coupled states II.1) Linear resonator

The Jaynes-Cummings model II.1) Linear resonator

The Jaynes-Cummings model II.1) Linear resonator

Two interesting limits RESONANT REGIME (d=0) II.1) Linear resonator

DISPERSIVE REGIME (|d|>>g) DISPERSIVE REGIME (|d|>>g) Two interesting limits RESONANT REGIME (d=0) DISPERSIVE REGIME (|d|>>g) DISPERSIVE REGIME (|d|>>g) II.1) Linear resonator

QUBIT STATE READOUT QUBIT STATE READOUT Two interesting limits QUANTUM STATE ENGINEERING QUBIT STATE READOUT QUBIT STATE READOUT RESONANT REGIME (d=0) DISPERSIVE REGIME (|d|>>g) DISPERSIVE REGIME (|d|>>g) II.1) Linear resonator

The Jaynes-Cummings model : dispersive interaction 2) Light shift of the qubit transition in the presence of n photons Field in the resonator causes qubit frequency shift and decoherence 1) Qubit state-dependent shift of the cavity frequency Cavity can probe the qubit state non-destructively with the dispersive coupling constant II.1) Linear resonator

Dispersive readout of a transmon: principle |0> or |1> ?? II.1) Linear resonator

Dispersive readout of a transmon: principle |0> or |1> ?? w=wc II.1) Linear resonator

Dispersive readout of a transmon: principle |0> or |1> ?? |a0> w=wc II.1) Linear resonator

Dispersive readout of a transmon: principle |0> or |1> ?? |a0> w=wc p |0> |1> -p wd/wc II.1) Linear resonator

Dispersive readout of a transmon: principle |0> or |1> ?? |a0> L.O w=wc or ??? p |0> |1> -p wd/wc II.1) Linear resonator

Typical implementation (Saclay) 5 mm (f0=6.5GHz) 80mm Q=700 40mm 2mm (optical+e-beam lithography) II.1) Linear resonator

Typical setup (Saclay) MW meas 20dB 50MHz Fast Digitizer A(t)‏ I MW drive LO (t)‏ Q COIL Vc G=56dB dB 300K G=40dB 20dB TN=2.5K 50 4K DC-8 GHz 30dB 600mK 1.4-20 GHz 4-8 GHz 18mK 20dB 50 II.1) Linear resonator

Signature for strong coupling: 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) 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) on resonance II.1) Linear resonator A. Wallraff et al., Nature 431, 162 (2004)

Qubit spectroscopy with dispersive readout MW drv Pump TLS MW meas Probe resonator phase   /c II.1) Linear resonator

Typical spectroscopy of a transmon + cavity circuit II.1) Linear resonator

Rabi oscillations measured with dispersive readout Variable-length drive MW drv x 10000  Ensemble averaging Projective measurement MW meas T2R=316 ns Y X II.1) Linear resonator

Dispersive readout : the signal-to-noise issue |0> or |1> ?? |a0> Ideal amplifier L.O w=wc or ??? p |0> |1> -p wd/wc II.1) Linear resonator

Dispersive readout : the signal-to-noise issue |0> or |1> ?? Real amplifier TN=5K L.O |a0> w=wc or ??? No discrimination in 1 shot p |0> |1> -p wd/wc II.1) Linear resonator

Dispersive readout : the signal-to-noise issue QUANTUM- LIMITED AMPLIFIER ?? |a1> |0> or |1> ?? |a0> Real amplifier TN=5K L.O w=wc or in one single-shot ?? p |0> |1> -p wd/wc II.1) Linear resonator

How to build an amplifier with minimal noise ??? signal out pump signal in Nonlinear resonator l/4 l/4 Junction causes Kerr non-linearity 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 max M. J. Hatridge, R. Vijay, D. H. Slichter, J. Clarke and I. Siddiqi, Phys. Rev. B 83, 134501 (2011) II.2) Nonlinear resonator (courtesy I. Siddiqi)

A nonlinear resonator as quantum-limited amplifier Small signal Saturated linecut. SINE WAVES FLIPPED OR UN FLIPPED TO GO UP OR DOWN -128 dB for 1 photon power - 1dB compression at -130 in other system, but here not a problem. want faithful representation of analog signal. animate to add qubit as digital (small signal digital). then run as saturated. binary signals? if not a two state signal, lose info in saturated regime. since qubit is binary, we don’t care (courtesy I. Siddiqi) II.2) Nonlinear resonator

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 INPUT DRIVE OUTPUT The experimental implementation of our circuit QED plus paramp architecture looks like this. Readout photons enter from the input and are directed through a circulator into the cavity, where they INTERACT with the qubit and acquire a phase shift that depends on the qubit state. The readout photons, now carrying info about the qubit state in their phase, then REFLECT out of the cavity and pass through several circulators, which isolate the qubit from the strong pump used to bias the parametric amplifier. The readout photons combine with this pump in a directional coupler and are amplified by the paramp. The amplified signal reflects out and is directed to the ouput port, where it is further amplified, mixed down to zero frequency, and digitized. We have done this with a variety of qubit and paramp samples. The data I will show in this talk come from two different qubit samples (and two different paramps). Our first sample didn’t have great coherence properties, but the latest sample has shown a factor of 4 improvement in T1. We operate where the dispersive shift is comparable to the cavity linewidth, so that the qubit-dependent phase shift is substantial. (courtesy I. Siddiqi) II.2) Nonlinear resonator

Individual measurement traces readout off readout on In our measurement protocol, we PREPARE the qubit state with a microwave pulse and then turn on the readout to measure the qubit. Let’s FOLLOW THREE different individual time traces to see what happens. Here the phase shift signal has been CONVERTED TO A VOLTAGE SIGNAL through HOMODYNE detection. The white region denotes that the readout is off, while the grey region shows when the readout is on. We prepare the qubit state here. Once we turn the readout on, the traces SEPARATE AND MOVE TO SOME NEW VALUE depending on the qubit state. In this case, the red and green traces were prepared in the excited state using a pi pulse, while the blue trace was prepared in the ground state with a 2 pi pulse. We can choose a DISCRIMINATION THRESHOLD and say that signals above are in the excited state, and signals below are in the ground state. Note that the our signal-to-noise ratio is good enough so that we can readily determine whether our signals is above or below this threshold. We know that the qubit has a finite lifetime, and so it will eventually decay back to the ground state. However, if we continue to measure the qubit, we force it to be in an eigenstate, either 0 or 1. This means as we monitor the qubit, it will decay to the ground state with a SUDDEN JUMP, and THIS IS in fact WHAT WE SEE. The jump is a STOCHASTIC process, so it happens at a different time on each trace. If we extract the jump times from many traces, we find they are exponentially distributed with a time constant that is equal to the independently measured T1 of the qubit, as we would expect. This is the first time such quantum jumps of a superconducting qubit have been observed. R. Vijay, D.H. Slichter, and I. Siddiqi, PRL 106, 110502 (2011) (courtesy I. Siddiqi) II.2) Nonlinear resonator

Bivalued histograms Single-shot discrimination of qubit state The dots in the middle that rise above the gaussian fits represent about 3% of the total number of traces; these are the ones in transition. Single-shot discrimination of qubit state (courtesy I. Siddiqi) II.2) Nonlinear resonator

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

Other strategy : sample-and-hold detector integrated with qubit Kerr-nonlinear resonator l/4 l/4 pump L H BISTABILITY FOR Pd /Pc = 0.2 0.5 1.0 1.8 - II.2) Nonlinear resonator

wd The Cavity Josephson Bifurcation Amplifier (CJBA) Pd H state M. Devoret group, Yale JBA: I. Siddiqi et al., PRL (2004) CJBA: M. Metcalfe et al, PRB (2007) MW drive : Pd(t) , wd jin Non linear resonator jout Pd Switching from L to H : BIFURCATION 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) H L Pd H state Bistable region wd L state wc II.2) Nonlinear resonator 44

wd The Cavity Josephson Bifurcation Amplifier (CJBA) Pd H state M. Devoret group, Yale JBA: I. Siddiqi et al., PRL (2004) CJBA: M. Metcalfe et al, PRB (2007) MW drive : Pd(t) , wd jin Non linear resonator jout H Pd Pd H state Bistable region L wd L state wc II.2) Nonlinear resonator 45

SINGLE-SHOT QUBIT READOUT Readout of transmon with CJBA MW drive : Pd(t) , wd jin Non linear resonator jout qubit in |0> or |1> Pd Pd H state SINGLE-SHOT QUBIT READOUT |1> wd L state wc|1> wc|0> |0> II.2) Nonlinear resonator 46

Rabi oscillations visibility h01 h12 tp,12 Dt Pswitch (%) TRabi=500ns Dt (µs) Mallet et al., Nature Physics (2009) Single-shot 93% contrast Rabi oscillations See also A. Lupascu et al., Nature Phys. (2007) II.2) Nonlinear resonator