REVIEW OF SOLID STATE QUANTUM BIT CIRCUITS Two strategies single particle states in semiconductor structures  global quantum states of superconducting Josephson circuits (A) Kane’s proposal : nuclear spins of P impurities in Si Chalmers NEC Quantronics U. Of New South Wales TU Delft (B) Electrons in quantum dots Schoelkopf et al, Yale SL SR B1) Charge: NTT B2) Spin: TU Delft, Harvard,… (C) Propagating states: flying qubits NIST e QHE edge states: LPA (ENS Paris) From charge states to phase states

First demonstration of coherent oscillations in a double dotd Coherent charge oscillations in a double dot (NTT, Hayashi et al. 2003) |R> |L> + |R> |L> - |R> |L> T2 (charge qubit) ~ 1 ns Result tells you two things: Coherent manipulation of electronic states in quantum dots is possible! So; quantum computing is feasible… Coherence time of charge states is very short (ns); spin states should be much better! But charge too much coupled to the environment ! spin expected better

Expts: TU Delft , Harvard B2) Spin qubits SL SR BZ Initial ideas: Loss & DiVincenzo (1998) 1-qubit control: magnetic (ESR) electric (modulate effective g-factor) 2-qubit coupling: exchange interaction between 2 dots Read-out through charge Expts: TU Delft , Harvard

GD GSR GSL b) 2e spin qubit in a double dot (NEW) Harvard U. C. Marcus team, GD GSR GSL Harvard Charge sensor Nature, june 2005, & Petta et al., in prep. Two electron spin qubit charge readout with QPC # e in dot

Bloch sphere in (1,1) S - T0 subspace

Measuring Spin Dephasing (T2) ε (electrostatic energy difference) Move from (0,2)S to (1,1) s & let evolve

dephasing causes failure to return to (0,2) Measuring Spin Dephasing (T2) (1,1)S (0,2)S ε 2t (1,1)T0 dephasing causes failure to return to (0,2) spin T2 ~ 10 ns

Short coherence time : 10 ns due to nuclei B Zeeman Nuclear Total

But coherence restored by spin echo e e tflip tflip model experiment tflip 40 ns 40 ns tflip pattern still observed at long times: coherence time TE =1.2 ms

team L. Kowenhouven& L. Vandersypen, TU Delft Electron spin resonance in a double dot B0 Gates ~ 30 nm gold Dielectric ~ 100nm calixerene Stripline ~ 400nm gold Expected AC current ~ 1mA Expected AC field ~ 1mT team L. Kowenhouven& L. Vandersypen, TU Delft

ESR detection via Pauli spin blockade Bext Energy 31jan06_20 hfac = g m B0 Flip right spin: ↑↑  ↑↓ = S11 + T0  S02 Flip left spin: ↑↑  ↓↑ = -S11 - T0  S02 Flip both spins: ↑↑  (T+ + 2 T0 + T- )/2  S02 Advantages: - low frequencies (B0 > 20mT, f > 100MHz. ) - not sensitive to electric fields (unlike single-dot ESR) - no confusion with ESR in leads

Pulsed ESR scheme

Rabi oscillations F. Koppens et al. pA

Solid State Qubits: way 2 single particle states in semiconductor structures  global quantum states of superconducting Josephson circuits A)Kane’s proposal : nuclear spins of P impurities in Si Phase qubit Flux qubit B) electrons in quantum dots (charge or spin) C P Box Charge-flux qubit SL SR C P Box C)Propagating states: flying qubits e Why superconductivity ? Why different flavors?

Josephson qubits come in different flavors Y ( N ) 1 2 >> 1 ~ 2 N phase charge-phase charge flux NIST TU Delft NEC, Chalmers Saclay, Yale (Devoret) Yale (Schoelkopf) Single Cooper pair boxes

N S Superconducting helps making qubits + Energy spectrum of an isolated electrode Superconducting state Non superconducting state S N singlet ground state + The Josephson junction

Building blocks for quantum bit circuits 1 0/1 U1 N ? ?

Basics of the Josephson junction single degree of freedom Josephson Hamiltonian :

V Phase qubits Ib tilted washboard potential ) ( d w10 w21 w32 d : extended phase conjugated of q on CJ d V 5 0.7 0.8 0.9 1 w10 w21 w32 ( ) p n E w h / - + Ib dc I 2 > I 1 > ac I 0 > tilted washboard potential

A flux quantum bit : the three junction loop  E Icirc 2t 0.5 F/Fo +Ip -Ip aEJ (0.5<a<1) g1 g2 EJ EJ  0.5 F0 Fx=F0/2 2D potential: Mooij et al. (TU Delft), (1999)

Single Cooper pair boxes The first ‘working’  qubit

energy spectrum Energy (EC) qubit Ng

Josephson Phase Qubits at UCSB UC Santa Barbara Team of John Martinis

Lifetime of state |1> Phase qubits = F0/2pI0cos d nonlinear inductor I d cos j = V ) (1/L J º sin LJ 2 p F I R I0 LJ C U(d) 5 0.7 0.8 0.9 1 w10 w21 w32 ( ) p n E w h / - + E0 E1 E2 G0 G1 G2 Gn Gn+1 1000 ~ <V> = 0 g10 wp DU <V> pulse (state measurement) Many ways to use non-linearity – for our qubit: 1) Non-linearity turns parabola to cubic -- Parameterized by dU & wp 2) QM : wavefunctions like H.O. – E levels now NOT equally spaced 3) Get tunneling, with ratio 200-1000 deep levels – no tunneling , Use top levels to tunnel, part of measuring process 4) Lifetime of deep levels given by dissipation, parameterized by R [ 2 3 ] / 1 4 I U - F = D p C [ ] 4 / 1 2 I p - ÷ ø ö ç è æ F = w 1: Tunable well (with I) 2: Transitions non-degenerate 3: Tunneling from top wells 4: Lifetime from R g10 @ 1 RC Lifetime of state |1>

better phase qubit : rf SQUID with dc SQUID readout U(d) ~5000 states “0” “1” … fast decay 1 F0 SQUID flux Switching current 10 mA Is If time Qubit Cycle Qubit Op Meas Amp Reset flux Measure p1 I 0 > I 1 > low noise bias qubit Is If "sample and hold" readout J. Martinis team (2003-2005) NIST & SB T2 : 10-50 ns (?)

Josephson-Junction Qubit State Preparation Wait t > 1/g10 for decay to |0> Qubit logic with current bias State Measurement: DU(Idc+dIp) Fast single shot – high fidelity potential |1> |0> I = Idc + dIdc(t) + Imwc(t)cosw10t + Imws(t)sinw10t phase H s · dIdc(t) + s · I + s · I ( 2 ) z x mwc y mws Apply ~3ns Gaussian pulse 1.0 0.8 0.6 0.4 0.2 0.0 0.7 0.5 0.3 |1> |0> Deep well Preparation Logic by DC and AC current Measurement by tunneling out of cubic well 2 ways to measure |1> : tunnel Prob. Tunnel 96% |0> : no tunnel pulse height of dIp

IC Fabrication Qubit Imwave Is If Al junction process X,Y readout Imwave Is Z 100mm (old design) If Made with standard fabrication techniques: metalization and etch, use insulator crossovers for complex circuits Point out structures via junction Al junction process & optical lithography Al Al SiNx Al Al2O3 substrate

Qubit Fidelity Tests (2006) ~90% visibility t Rabi: Ramsey: t (slightly detuned) Probability 1 state Echo: t (no detuning) t T1: Observe coherent oscillations by procedure… Uw produces transition 0 -> 1 -> 0 … Best data, with 70% visibility From this data we’ve now calibrated amplitude of drive, can define pi pulse Large Visibility! T1 = 110 ns, Tf ~ 85 ns

SUPERCONDUCTING FLUX QUBITS Group of Prof. Hans Mooij TU DELFT team Y. Nakamura NEC team K. Semba NTT

persistent-current quantum bit aEJ g1 (p) g2 (p) magnetic flux  0.5 Fo g1 g2 EJ EJ barrier scales with EJ, depends on a Tout a<1 to suppress influence of charge noise effective mass scales with junction capacitance C Tin tunneling: Tin = wm exp{ -0.64(EJ/EC)½} Tout = 1.6 wm exp{ -1.5(EJ/EC)½} for a=0.8

flux qubits : the three junction loop  E Icirc 2t 0.5 F/Fo +Ip -Ip aEJ (0.5<a<1) g1 g2 EJ EJ  0.5 F0 Fx=F0/2 2D potential: Mooij et al. (TU Delft), (1999)

SQUID readout of the flux qubit (readout # 1) Switching measurement (Ic 200 nA) mw on resonance

First spectroscopy of flux qubits switching current f GHz x10-3 F/Fo F/Fo-0.5 Van der Wal et al., Science 290, 773 (2000) also SUNY (Friedman, Lukens et al.)

Irinel Chiorescu and Yasu Nakamura (NEC, Delft) Science 299, 1869 (2003)

Dephasing: T2Ramsey, T2echo measurement (sample5) NEC Dephasing: T2Ramsey, T2echo measurement (sample5) correspond to detuning Ramsey interference p/2~2ns p/2 readout pulse t spin echo p/2~2ns p ~ 4ns p/2 readout pulse t/2 t/2

Observation of vacuum Rabi oscillations in a flux qubit coupled to a SQUID resonator K. Semba et al., NTT DISQUID~Mqub,SQIcirc,qubit SQUID qubit 6.7μm VSQUID Isw I |↓> |↑> Icirc,qubit M 6.3μm DISQUID Ithreshold DC SQUID detector: VSQUID = 0  qubit state |↑> ≠ 0  qubit state |↓>

Vacuum Rabi : measurement scheme |g1 |g0 p I qubit, LC-oscillator >　 |e0 |e1 |g1 |g0 |e1 |e0 |g1 2 → 3 |g0 excite qubit by a p-pulse shift qubit adiabatically shift qubit adiabatically readout qubit state 3 ⇔ 4 1 → 2 |e0 4 |g１ |g0

Vacuum Rabi oscillations Direct evidence of level quantization in a 0.1 mm large superconducting macroscopic ＬＣ-circuit J. Johansson et al., cond-mat/0510457 → to appear in Phys. Rev. Lett. (2006).

Quantization of Rabi period |g2 √ W ２ |e0 |g1 |g0 W |e1 J. Johansson et al., cond-mat/0510457 → to appear in Phys. Rev. Lett. (2006).

Readout of single Cooper pair boxes

Hamiltonian and energy spectrum 2 characteristic energies: 1 degree of freedom: 1 knob: or Hamiltonian:

energy spectrum Energy (EC) qubit Ng

Readout through the charge Energy (EC) Ng expectation value of the box charge: (measurement of the quantum state)

Capacitive coupling to a Single Electron Transistor EJ/EC=0.1 theory experiment Theory with no Josephson effect too slow ... V. Bouchiat et al. Quantronics (1996) V 1/2 1

Short coherence time : a few ns A box with a continuous readout Nakamura, Pashkin &Tsai (NEC,1999) Ng 0.5 V 2e- or 0 Continuous measurement by energy relaxation Pulse durationDt (ps) 200 400 600 5 First Rabi oscillations DC pulses from Ng=0.25 to 0.5 with duration Dt Short coherence time : a few ns 2000: WHY??

Open ports Decoherence The main difficulty a b + ? 1 1 U1 Decoherence Sources write Readout Open ports Decoherence

Readout port... lets noise in 1 Fluctuating environment A -meter detuning : Readout + environment DEPHASING signal (if A measured) : dephasing and readout closely related !

... and noise dephases... y = 2 1 + e ij

... and depolarises Fermi Golden Rule to summarize: dephasing dephasing excitation relaxation

connect only at readout time Solving the noise/readout dilemma: connect only at readout time Operate qubit at a stationary point: For readout: -Move away adiabatically at: better: stay there, and apply an ac drive (to be shown later on)

A general strategy now applied to different Josephson qubits A qubit ‘protected’ from decoherence: the quantronium gate 160 x160 nm A general strategy now applied to different Josephson qubits Vion et al, Science 2002; Esteve&Vion, cond-mat 2005

State dependent persistent currents The quantronium: 1) a split Cooper pair box 2 knobs : F U 1 d° of freedom 2 energies: State dependent persistent currents i

2) protected from dephasing hn01 energy (kBK) n01(GHz) d/2p d/2p Ng Ng EJ=0.86 kBK EC=0.68 kBK Optima working points exist in many qubits

1 3) with a readout junction discrimination Ng first readout of persistent currents with dc switching 1: switching 0: no switching 1 discrimination 3) with a readout junction d/2p Ng

c Rabi precession Rabi oscillations nµw readout rotation wRabi = aURF switching probability (%) nµw c X Y Effective field Note: visibility : <40%

Switching readout fidelity ? 40% contrast (only)

partly explains readout fidelity Relaxation during readout ramp ! T1=60ns T1=730ns partly explains readout fidelity

Optimal point for other qubits? the flux qubit Nakamura, Ciorescu, Bertet, Mooij et al. Delft, 2003-2004 shift in practice operating point readout point coupling to SQUID

Optimal point for other qubits? the flux qubit Chiorescu et al, TU Delft 2003 Nakamura et al., NEC 2005 Ib=0 Ib=-0.2 Ib=-0.4 2005: T2=400 ns

projection fidelity ? Fluctuating environment 1 ideal Quantum measurement: Readout: 1 Readout: 0 errors: wrong answer & projection error 1 Fluctuating environment A -meter

Switching readout fidelity ? 40% contrast (only)

partly explains readout fidelity Relaxation during readout ramp ! T1=60ns T1=730ns partly explains readout fidelity

Switching readout resets the qubit dc readout V d U resets the qubit dc pulse  switching simple, but: rep rate limited by quasiparticles fidelity <1 due to relaxation qubit reset : NOT QND

Towards QND readout ‘at’ optimal point flux qubit : charge qubit : charge-phase qubit : Quantum capacitance C/CJ j0 j1 1 SQUID inductance readout junction inductance quantum capacitance TU Delft Yale, Saclay Chalmers, Helsinki

dc versus ac readout in the quantronium PULSE IN PULSE OUT rf readout (M. Devoret, Yale) d U U d dc pulse “RF” pulse  switching  d dynamics in anharmonic potential simple, but: -fidelity 40% qubit reset : NOT QND more complex, but: -better fidelity ? -no reset: possibly QND

Towards non destructive readout at optimal point with an AC drive M. Devoret team at Yale I. Siddiqi et al., (2004) µW Pulse IN QuBit control d j0 j1 OUT 1 UJ optimal P Similar dispersive methods developed for other qubits

State dependent bifurcation The Josephson Bifurcation Amplifier M. Devoret team at Yale I. Siddiqi et al., (2004) µW Pulse IN QuBit control d j0 j1 OUT 180° -180° d amplitude µW drive amplitude µW phase j State dependent bifurcation 1 UJ optimal P Enhanced

Quantronium + JBA SETUP 50 W TN=2.5K G=40dB I LO demodulator 1.3-2 GHz M S -20dB -30dB NO bifurcation 300 K 4 K 0.6 K Quantronium from Yale 30 mK bifurcation

Rabi oscillations with the JBA p 45-50% 100ns 125ns JBA pulse Contrast : 50% (Saclay exprt)

Quantum Non Demolition ? read twice gate JBA readout partially QND 1 34% 100% 66% 0% 25% 9% 30% 36% 17% 83% Notice: relaxation again partly avoidable by tuning the qubit p 100ns 125ns 5ns 20ns 10ns 40ns 100ns A B 1 p & correlations 1 Note: results for flux-qubit now available

Dispersive readout of the flux qubit A. Lupascu et al. TU DELFT

Activation rates for different detuning values F = 775 MHz Fres=822 MHz Iac,bifurcation2 slope=udyn/(kT) Thy: M. Dykman

Activation rates for different detuning values

Optimal qubit manipulation and readout 87%

Rabi oscillations with optimal settings Dt = length of MW pulse

Ramsey oscillations with optimal settings Rabi oscillation Ramsey: wge-wmw= 69 MHz Ramsey frequency vs detuning Relatively strong low frequency fluctuations visible in the drift of the Ramsey frequency. QND data : analysis in progress

Experiments in Cavity QED with Superconducting Circuits Rob Schoelkopf Depts. of Applied Physics & Physics Yale University expt. Andreas Wallraff David Schuster Luigi Frunzio theory Steve Girvin Alexandre Blais Ren-Shou Huang And discussions w/ J. Zmuidzinas & M. Devoret Merci to D. Esteve & co. for assistance! Packard Foundation Keck Foundation Funding:

A Circuit Analog for Cavity QED 2g = vacuum Rabi freq. k = cavity decay rate g = “transverse” decay rate out L = l ~ 2.5 cm transmission line “cavity” 10 mm Cooper-pair box “atom” 10 GHz in Blais, Huang, Wallraff, Girvin & RS, cond-mat/0402216; to appear in PRA

Cavity QED with a Cooper pair box: first dispersive readout R. Schoelkopf, A. Wallraff, S. Girvin et al., Yale (2004) Dispersive readout with out of resonance photons

Dressed Artificial Atom: Resonant Case 2g T 1 “vacuum Rabi splitting”

Rabi Oscillations of Qubit Prf = +6 dB Prf = 0 dB Prf = 18 dB

Coherence time measurements with 2 pulse Ramsey sequence

Solid state qubits at work: CONCLUSION Solid state qubits at work: Semiconductor qubits recently demonstrated Superconducting qubits: qubit control single-shot readout Decoherence, QND readout, coupling in next lectures

2004 Fluctuating environment 1 A -meter The work on the Quantronium Quantronium Fluctuating environment dc gate dc gate µw qp trap box A -meter readout junction 1µm 2004 G. ITHIER E. COLLIN N. BOULANT D. VION P. ORFILA P. SENAT P. JOYEZ P. MEESON D. ESTEVE SPEC A. SHNIRMAN G. SCHOEN Y. MAKHLIN F. CHIARELLO

Thanks to NEC