Simple quantum feedback. of a solid-state qubit

Transcription

1 e Vg qubit (SCPB) V Outline: detector (SET) Simple quantum feedback of a solid-state qubit Alexander Korotkov Goal: keep coherent oscillations forever qubit detector controller fidelity APS 5, LA, 3// ntroduction (Bayesian formalism for continuous quantum measurement, Bayesian quantum feedback) Simple quantum feedback of a qubit C =. η = classical feedback. τ[ ( ) /S ]= feedback strength Support: cond-mat/44696

2 The system we consider: qubit + detector qubit H e Vg V detector Double-quantum-qot (DQD) and quantum point contact (QPC) e Cooper-pair box (CPB) and single-electron transistor (SET) H = H QB + H DET + H NT H QB = (ε/)(c + c c + c )+H(c + c +c + c ) ε asymmetry, H tunneling Ω = (4H +ε ) / /Ñ frequency of quantum coherent (Rabi) oscillations Two levels of average detector current: for qubit state, for Response: = Detector noise: white, spectral density S DQD and QPC (setup due to Gurvitz, 997) DET = l l l r r r + l r l, r r l + l r NT = (, )( r l + l r) lr H Eaa Eaa Taa ( aa) Alexander Korotkov + H T cc cc aa aa S = e

3 e Vg V Bayesian formalism for a single qubit ˆ ε HQB = ( cc cc ) + Hcc ( + cc ) Ò Æ, ÒÆ = -, =( + )/, S detector noise ρ =- ρ =- ( H / )m ρ + ρ ρ ( / S )[ t ()- ] ρ = i( ε / ) ρ + i( H / )( ρ -ρ) + ρ( ρ -ρ)( / S )[ t ()-]-γρ γ = Γ- Γ ensemble decoherence ( ) /4 S, (A.K., 998) η = - γ / Γ= ( ) /4S Γ detector ideality (efficiency), η % deal detector (η=) does not decohere a single qubit; then random evolution of qubit wavefunction can be monitored For simulations: () t - =( ρ- ρ) / + ξ(), t Sξ = S Averaging over ξ(t) ï conventional master equation Similar formalisms developed earlier. Key words: mprecise, weak, selective, or conditional measurements, POVM, Quantum trajectories, Quantum jumps, Restricted path integral, etc. Names: Davies, Kraus, Holevo, Mensky, Caves, Gardiner, Carmichael, Plenio, Knight, Walls, Gisin, Percival, Milburn, Wiseman, Onofrio, Habib, Doherty, etc. (incomplete list)

4 S (ω)/s S (ω)/s qubit Measured spectrum of qubit coherent oscillations α C=3 3.3 detector C = HS ( ) / ε /H= ω/ω α =. η = classical limit ω /Ω Alexander Korotkov S What is the spectral density S (ω) of detector current? Assume classical output, ev» Ω ε = Γ = ( ω ), η ( ) / 4 = S + S Ω ( ) Γ ( ω Ω ) +Γ ω Spectral peak can be seen, but peak-to-pedestal ratio 4η 4 (result can be obtained using various methods, not only Bayesian method) Weak coupling, α = C/8 «S ( ω ) S ηs ε / H = + + ω Ω H Γ ( /4 ) 4 ηs( + ε / H ) + + [( ω Ω ) Γ ( H / Ω )] A.K., LT 99 Averin-A.K., A.K., Averin, Goan-Milburn, Makhlin et al., Balatsky-Martin, Ruskov-A.K., Mozyrsky et al., Balatsky et al., Bulaevskii et al., Shnirman et al., Bulaevskii-Ortiz, 3 Shnirman et al., 3 Contrary: Stace-Barrett, 3 (PRL 4)

5 Quantum feedback control of a qubit Since qubit state can be monitored, the feedback is possible! desired evolution feedback signal H qb = Hσ X qubit H e control stage (barrier height) C<< detector comparison circuit Bayesian equations ρ ij (t) environment Ruskov-Korotkov, Goal: maintain desired phase of coherent (Rabi) oscillations in spite of environmental dephasing (keep qubit fresh ) dea: monitor the Rabi phase φ by continuous measurement and apply feedback control of the qubit barrier height, H FB /H = F φ To monitor phase φ we plug detector output into Bayesian equations

6 K z (τ) Performance of quantum feedback (no extra environment) Qubit correlation function C=, η=, F=,.5, τ Ω /π cosωt C FHτ / Kz( τ ) = exp ( e ) 6F (for weak coupling and good fidelity) Detector current correlation function ( ) cosωt FHτ / K ( τ ) = ( + e ) 4 C FH / S exp ( e τ ) + δτ ( ) 6F Alexander Korotkov D (synchronization degree) Fidelity (synchronization degree) C= η = d e = "direct" τ a / T=/ F (feedback factor) C = ( ) / S H coupling - τ a available bandwidth F feedback strength D= Trρρ desir /3 For ideal detector and wide bandwidth, fidelity can be arbitrarily close to % D = exp( C/3F) Ruskov & Korotkov, PRB 66, 44(R) ()

7 Quantum feedback in presence of decoherence by environment D (synchronization degree) C env /C det =..5 C=C det = τ a = F (feedback factor) D max (synchronization) D max C C env det C env /C det (relative coupling to environment) Big experimental problems: necessity of very fast real-time solution of the Bayesian equations wide bandwidth (>>Ω, GHz-range) of the line delivering noisy signal to the processor

8 Simple quantum feedback of a solid-state qubit H qb = Hσ X qubit C<< detector H =H [ F φ m (t)] Alexander Korotkov control cos(ω t), τ-average local oscillator sin(ω t), τ-average (A.K., cond-mat/44696) dea: use two quadrature components of the detector current to monitor approximately the phase of qubit oscillations (a very natural way for usual classical feedback!) t - t X ( t) = [ ( t') ] cos( Ωt') exp[ ( t t') / τ ] dt Goal: maintain coherent (Rabi) oscillations for arbitrary long time φ m = -arctan( Y / X) Y( t) = [ ( t') - ] sin( Ωt') exp[ -( t-t') / τ ] dt - (similar formulas for a tank circuit instead of mixing with local oscillator) Advantage: simplicity and relatively narrow bandwidth (/ τ ~ Γ d <<Ω) Anticipated problem: without feedback the spectral peak-to-pedestal ratio <4, therefore not much information in quadratures phase (surprisingly, situation is much better than anticipated!) X Y φ m

9 φ rms (phase inaccuracy) Accuracy of phase monitoring via quadratures π/3 / no noise C<< /Γ d =4S /( ) C dimensionless coupling Alexander Korotkov (no feedback yet) τ [( ) /S ] uncorrelated noise C<< C =.,.3, 3 3 (averaging time) p ( φ) τ [( ) /S ] = C =. Noise improves the monitoring accuracy! (purely quantum effect, reality follows observations ) dφ / dt = [ ( t) ]sin( Ω t+ φ)( / S ) dφ / dt = [ ( t) ]sin( Ω t+ φ )/( X + Y ) m m / (non-gaussian distributions) φ =φ -φ 4 m 8 φ (actual phase shift, ideal detector) (observed phase shift) Noise enters the actual and observed phase evolution in a similar way Quite accurate monitoring! cos(.44).9 weak coupling C<<

10 Simple quantum feedback (fidelity) D, <X>(4/τ ) C =. η = F/C. τ[( ) /S ]= classical feedback fidelity for different averaging τ (feedback strength) weak coupling C D feedback efficiency D FQ F Tr ρ( t) ρ () t Q D max 9% des (F Q 95%) How to verify feedback operation experimentally? Simple: just check that in-phase quadrature X of the detector current is positive D = X (4/ τ ) X = for any non-feedback Hamiltonian control of the qubit

11 - nonideal detectors (finite quantum efficiency η) and environment - qubit energy asymmetry ε - frequency mismatch Ω Quantum feedback still works quite well Main features: Effect of nonidealities D (feedback efficiency) Fidelity F Q up to ~95% achievable (D~9%) Natural, practically classical feedback setup Averaging τ~/γ>>/ω (narrow bandwidth!) Detector efficiency (ideality) η~. still OK Robust to asymmetry ε and frequency shift Ω Simple verification: positive in-phase quadrature X η eff = ε/h =. Ω/CΩ=. C =. τ [( ) /S ] = F/C (feedback strength) Simple enough experiment?!

12 Quantum feedback in optics Recent experiment: Science 34, 7 (4) First detailed theory: H.M. Wiseman and G. J. Milburn, Phys. Rev. Lett. 7, 548 (993)

13 Conclusion Very straightforward, practically classical feedback idea (monitoring the phase of oscillations via quadratures) works well for the qubit coherent oscillations Price for simplicity is a less-then-ideal operation (fidelity is limited by ~95%) Feedback operation is much better than expected Relatively simple experiment (simple setup, narrow bandwidth, inefficient detectors OK, simple verification)