Simple quantum feedback of a solid-state qubit

Transcription

1 Simple quantum feedback of a solid-state qubit Alexander Korotkov H =H [ F φ m (t)] control qubit CÜ detector I(t) cos(ω t), τ-average sin(ω t), τ-average X Y phase φ m Feedback loop maintains Rabi oscillations for infinitely long time Advantage: simplicity and relatively narrow bandwidth Anticipated problem: not much information in quadratures (surprisingly, feedback loop works much better than anticipated!) Support: Alexander Korotkov cond-mat/696

2 Simple quantum feedback of a solid-state qubit Alexander Korotkov We propose an experiment on quantum feedback control of a solidstate qubit, which is almost within the reach of the present-day technology. Similar to the earlier proposal, the feedback loop is used to maintain the coherent oscillations in a qubit for an arbitrary long time; however, this is done in a significantly simpler way, which requires much smaller bandwidth of the control circuitry. The main idea is to use the quadrature components of the noisy detector current to monitor approximately the phase of qubit oscillations. The price for simplicity is a less-than-ideal operation: the fidelity is limited by about 95%. The feedback loop operation can be experimentally verified by appearance of a positive in-phase component of the detector current relative to an external oscillating signal used for synchronization. Support: Alexander Korotkov cond-mat/696

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

4 φ rms (phase inaccuracy) Accuracy of phase monitoring via quadratures π/3 / no noise C<< /Γ d =S I /( I) C dimensionless coupling Alexander Korotkov (no feedback yet) τ [( I) /S I ] uncorrelated noise C<< C =.,.3, 3 3 (averaging time) p ( φ) τ [( I) /S I ] = C =. Noise improves the monitoring accuracy! (purely quantum effect, reality follows observations ) dφ / dt = [ I( t) I]sin( Ω t+ φ)( I / S I ) dφ / dt = [ I( t) I ]sin( Ω t+ φ )/( X + Y ) m m / (non-gaussian distributions) weak coupling Cá φ =φ -φ m 8 φ Best approximation X +Y =(S I / I) (/5)( / -).6 (actual phase shift, ideal detector) (observed phase shift) Noise enters the actual and observed phase evolution in a similar way

5 (fidelity) D, <X>(/τ I) C =. η = Quantum feedback performance F/C Alexander Korotkov. τ[( I) /S I ]= classical feedback..5 8 fidelity for different averaging τ (feedback strength) Fidelity F 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 Very simple verification just positive in-phase quadrature X η eff =.5. (weak coupling C).5 ε/h =. Ω/CΩ=. C =. τ [( I) /S I ] =. nonideal detectors F/C D F F Tr ρ( t) ρ des() t D X (/ τ I) X in-phase quadrature of the detector current Simple experiment?!

6 Quantum feedback in optics Recent experiment: Science 3, 7 () First detailed theory: H.M. Wiseman and G. J. Milburn, Phys. Rev. Lett. 7, 58 (993) No experimental attempts of quantum feedback in solid-state yet (even theory is still considered controversial) Experiments soon?

7 Conclusions (simple quantum feedback of a solid-state qubit) Very straightforward, practically classical feedback idea (monitoring the phase of oscillations via quadratures) works well for the qubit coherent oscillations Much simpler realization than for quantum feedback of Ruskov-Korotkov, Phys. Rev. B 66, (R) () Price for simplicity is a less-then-ideal operation (fidelity is limited by ~95%) Feedback performance is much better than expected Relatively simple experiment (simple setup, narrow bandwidth, inefficient detectors OK, simple verification)

8 Quadratic quantum measurements W. Mao, D. Averin, R. Ruskov, and A. Korotkov Stony Brook University and UC Riverside Phys. Rev. Lett. 93, 5683 () Studied setup: two qubits and detector V ga V I(t) V gb V(f) V, I quadratic H a Hb qubit a SET qubit b I bias Qubits can be made % entangled by measurement This is done in an easier way than using a linear detector, as in Ruskov-Korotkov, PRB 67, 35(R) (3) Support: φ, q

9 Quadratic quantum measurements W. Mao, D. Averin, R. Ruskov, and A. Korotkov Stony Brook University and UC Riverside Phys. Rev. Lett. 93, 5683 () We develop a theory of quadratic quantum measurements by a mesoscopic detector. It is shown that the quadratic measurements should have non-trivial quantum information properties, providing, for instance, a simple way of entangling two non-interacting qubits. We also calculate output spectrum of a detector with both linear and quadratic response, continuously monitoring two qubits. Support:

10 Studied setup: two qubits and detector V ga V I(t) V gb V(f) V, I quadratic H a Hb qubit a SET qubit b I bias Setup is similar to Ruskov-Korotkov, PRB 67, 35(R) (3), but a nonlinear (instead of a linear) detector is considered φ, q Linear detector I( ) I( )=I( ) Noninear detector I( ) I( )=I( ) Quadratic detector I( )=I( ) I( ) I( ) I( )= I( ) Quadratic detection is useful for quantum error correction (Averin-Fazio, )

11 Bayesian formalism for a nonlinear detector j QBs DET σz j=, t δσ z + δσ z λσσ z z }) ξ + }) ξ j σz H = H + H + [({ t t ({ ] t( x) = + + Assumed: ) weak tunneling in the detector, ) large detector voltage (fast detector dynamics, and 3) weak response. The model describes an ideal detector (no extra noises). Recipe: Coupled detector-qubits evolution and frequent collapses of the number n of electrons passed through the detector Two-qubit evolution (Ito form): d ρ = ih [, ρ] + [ It ( ) I ][ ( I dt S + I I ) iϕ ] ρ γ ρ kl QBs kl k l kl kl kl kl * kl = (/ )( Γ + +Γ-)[( tk tl ) + kl t ], kl = arg( tktl ) γ ϕ ϕ δ = quadratic detector ρ jj j, k ( -) k, ( -) j + + I = I I = Γ Γ t S = Γ +Γ t j

12 S I (ω)/s S I (ω)/s S I (ω)/s Linear detector 8 analytical 6 numerical ω /Ω 3 Nonlinear detector 6 Quadratic detector 6 ω /Ω 3 ω /Ω 3 =, =, 3=, = Alexander Korotkov S I Two-qubit detection (oscillatory subspace) ( ω) = S + η 8 Ω ( I ) Γ 3 ( ω Ω ) +Γ ω Γ= ( I) / S, I = I I = I I 3 3 Spectral peak at Ω, peak/noise = (3/3)η (Ω is the Rabi frequency) Extra spectral peaks at Ω and S I ( ω ) Ω ( I ) Γ = S + ( ω Ω ) +Γ ω ( I = I I, I = I, I = I ) 3 3 (Ruskov-Korotkov, ) (analytical formula for weak coupling case) Peak only at Ω, peak/noise = η Mao, Averin, Ruskov, Korotkov,

13 Two-qubit quadratic detection: scenarios and switching B ρ (t) Three scenarios: (distinguishable by average current) B 3 B ε Ωt 6E+ 8E+ 3) Slightly asymmetric qubits, ε B B ε Γ 3 = Γ/ Ω ) collapse into - Ú = Ú B, current I Æ, flat spectrum ) collapse into - Ú = Ú B, current I ÆÆ, flat spectrum 3) collapse into remaining subspace 3Ú B, current (I Æ +I ÆÆ )/, spectral peak at Ω, peak/pedestal = η. B Switching between states due to imperfections ) Slightly different Rabi frequencies, Ω=Ω -Ω Γ B B = Γ B B = ( Ω) / Γ, Γ= η ( I) /S ( I ) Γ SI ( ω) = S + ( Ω ) + ω /( ) Γ Ω ) Slightly nonquadratic detector, I I Γ B B = [( I I ) / I] Γ / 3 Ω ( I ) Γ SI ( ω) = S + 3 ( ω Ω ) +Γ ω 8( I ) + 7 Γ( I I ) + [ ( I) / 3 Γ( I I ) ] ω

14 S I /S Effect of qubit-qubit interaction ω/ω Ω= H ( εσ + σ σσ j j QBs = j z j x)/+ z z j ν - interaction between two qubits First spectral peak splits (first order in ν), second peak shifts (second order in ν) ω - =[ +(ν/)] / - ν/ ω + =[ +(ν/)] / + ν/ ω =[ +(ν/)] / =ω - + ω + ν Conclusions (quadratic quantum measurements) Conditional (Bayesian) formalism for a nonlinear detector is developed Detector nonlinearity leads to the second peak in the spectrum (at Ω), in purely quadratic case there is no peak at Ω (very similar to classical nonlinear and quadratic detectors) Qubits become entangled (with some probability) due to measurement, detection of entanglement is easier than for a linear detector (current instead of spectrum)

15 Quantum nondemolition (QND) squeezing of a nanoresonator R. Ruskov, K. Schwab, and A. Korotkov UC, Riverside and LPS resonator x m, ω Squeezing S ( x / x) 8 6 /5 / /3 C =.5 δt /T =.5 η = QPC Support: V(t) I(t) ω/ω Application: ultrasensitive force detection; sensitivity beyond standard quantum limit cond-mat/66

16 Quantum nondemolition (QND) squeezing of a nanoresonator R. Ruskov, K. Schwab, and A. Korotkov UC, Riverside and LPS We show that the nanoresonator position an be squeezed significantly below the ground state level by measuring the nanoresonator with a quantum point contact or a single-electron transistor and applying a periodic voltage across the detector. The mechanism of squeezing is basically a generalization of quantum nondemolition measurement of an oscillator to the case of continuous measurement by a weakly coupled detector. The quantum feedback is necessary to prevent the heating due to measurement backaction. We also discuss a procedure of experimental verification of the squeezed state. Support: cond-mat/66

17 resonator QPC x QND squeezing of a nanoresonator V(t) m, ω I(t) Ruskov, Schwab, Korotkov, cond-mat/66 Hˆ pˆ / m m xˆ / = + ω ˆ DET = l l l + r r r + ( l r +..) l r l, r H Eaa Eaa Maa Hc H ˆ ( ˆ INT = Mxaa l r + Hc..) lr, ω GHz, T 5 mk, quantum behavior T< ω or Tτ obs /Q< / Model similar to Hopkins, Jacobs, Habib, Schwab, PRB 3 (continuous monitoring and quantum feedback to cool down) New feature: Braginsky s stroboscopic QND measurement using modulation of detector voltage squeezing becomes possible Potential application: ultrasensitive force measurements Other most important papers: Doherty, Jacobs, PRA 999 (formalism for Gaussian states) Mozyrsky, Martin, PRL (ensemble-averaged evolution)

18 Stroboscopic QND measurements Quantum nondemolition (QND) measurements (Braginsky-Khalili book) (a way to suppress measurement backaction and overcome standard quantum limit) Idea: to avoid measuring the magnitude conjugated to the magnitude of interest Standard quantum limit Example: measurement of x(t )-x(t ) p> xt ( ) xt ( ) / x First measurement: p(t )> / x(t ), then even for accurate second measurement inaccuracy of position difference is x(t )+(t -t ) /m x(t )> (t -t ) / / m Stroboscopic QND measurements (Braginsky et al., 978; Thorne et al., 978) oscillator Difference in our case: Idea: second measurement exactly one oscillation period later is insensitive to p (or t = nt/, T=π/ω ) continuous measurement weak coupling with detector quantum feedback to suppress heating

19 QPC x V(t) Bayesian formalism for continuous measurement of a nanoresonator m, ω I(t) ˆ ˆ ˆ H = p / m+ mωx / H Eaa Eaa Maa Hc ˆ DET = l l l + r r r + ( l r +..) l r l, r H ˆ ( ˆ INT = M xa l a r + H. c.) Current lr, ( ) ρ ρ Ix = π M + M x re V / = I + kx Detector noise Sx = S ei Recipe: quantum Bayes procedure l Nanoresonator evolution (Stratonovich form), same Eqn as for qubits: dρ( x, x ) i ˆ ρ( x, x ) = [ H, ρ ] + I( t)( Ix + Ix I ) ( Ix + Ix I ) dt S I = I ρ( x, x), I( t) = I + ξ( t), S = S Ito form (same as in many papers on conditional measurement of oscillators): dρ( x, x ) i ˆ k k = [ H, ρ] ( x x ) ρ( x, x ) + ( x+ x x ) ρ( x, x ) ξ( t) dt S η S x x ξ

20 ρ Evolution of Gaussian states Assume Gaussian states (following Doherty-Jacobs and Hopkins-Jacobs-Habib-Schwab), then ρ(x,x ) is described by only 5 magnitudes: x, p - average position and momentum (packet center), D x, D p, D xp variances (packet width) Assume large Q-factor (then no temperature) Voltage modulation f(t)v : k = f() t k, Ix = f()( t I + kx), SI = f() t S Then coupling (measurement strength) is also modulated in time: I C = f( t) C, C = k / S mω = / ω τmeas Packet center evolves randomly and needs feedback (force F) to cool down d x / dt = p / m+ ( k / S )sgn[ f( t)] D ξ ( t) d p / dt = mω x + ( k / S )sgn[ f ( t)] D ξ( t) + F( t) Packet width evolves deterministically and is QND squeezed by periodic f(t) d D / dt = (/ m) D ( k / S ) f( t) D x xp x p / ω xp ( / η) ( ) ( / ) ( ) xp xp / (/ ) p ω x ( / ) ( ) x xp d D dt = m D + k S f t k S f t D d D dt = m D m D k S f t D D x xp

21 Squeezing S D x, D x f(t) ( x / x) Squeezing by sine-modulation, V(t)=V sin(ωt) D x D x D = D x + D x no modulation with modulation C =. ω = ω t /π η = C = η =.8 η = ω / ω Alexander Korotkov ω Squeezing obviously oscillates in time, maximum squeezing at maximum voltage, momentum squeezing shifted in phase by π/. S max t ( x) / Dx Analytics (weak coupling): S( ω ) = 3 η, ω =.36 ω C / η η - detector efficiency, C coupling x = ( /mω ) / ground state width D x =( x), D x = x - x Quantum feedback: F = mω γ x γ p x p Ruskov-Schwab-Korotkov (same as in Hopkins et al.; without modulation it cools the state down to the ground state) Feedback is sufficiently efficient, D x ÜD x Squeezing up to.73 at ω =ω

22 Squeezing S f(t) D x, D x Squeezing by stroboscopic (pulse) modulation pulse modulation D x ω = Alexander Korotkov ω /n D x << D x 3 5 D using ( x / x) x 8 6 D x =( x) ω t /π /5 / /3 C =.5 δt /T =.5 η = ω / ω feedback ω t /π Momentum squeezing as well Dx Dp Dxp 3 5 ω t /π Sá Efficient squeezing at ω=ω /n Ruskov-Schwab-Korotkov Tr[ ρ ( x, x )] State purification (natural QND condition)

23 Squeezing by stroboscopic modulation f(t) Squeezing S Squeezing S =( x / x) ( x / x) 3 δ t =.5T C = 8 η = ω / ω τ wait /T = C =. ω = ω δt /T Analytics (weak coupling, short pulses) Maximum squeezing Linewidth S( ω / n) = 3η ωδt ω = 3 ω C( δt) πn 3η C dimensionless coupling with detector δt pulse duration, T =π/ω η quantum efficiency of detector (long formula for the line shape) Finite Q-factor limits the time we can afford to wait before squeezing develops, τ wait /T ~Q/π Squeezing saturates as ~exp(-n/n ) after n = 3 η / C ( ωδt) measurements Therefore, squeezing cannot exceed S C Q η

24 Observability of nanoresonator squeezing Procedure: ) prepare squeezed state by stroboscopic measurement, ) switch off quantum feedback N 3) measure in the stroboscopic way x X N = N j= For instantaneous measurements (δt ) the variance of X N is DXN, = + ( x ) at N mω S NCωδt S Then distinguishable from ground state (S=) in one run for Sà (error probability ~S -/ ) j S squeezing, x ground state width Not as easy for continuous measurements because of extra heating. D X,N has a minimum at some N and then increases. However, numerically it seems min D ( x ) / S (only twice worse) N X, N Example: min N DX, N /( x) =.78 for C =., η=, δt/t =., /S=.36 Squeezed state is distinguishable in one run (with small error probability), therefore suitable for ultrasensitive force measurement beyond standard quantum limit

25 Conclusions (QND squeezing of a nanoresonator) Periodic modulation of the detector voltage modulates measurement strength and periodically squeezes the width of the nanoresonator state ( breathing mode ) Packet center oscillates and is randomly heated by measurement; quantum feedback can cool it down (keep it near zero in both position and momentum) Sine-modulation leads to a small squeezing (<.73), stroboscopic (pulse) modulation can lead to a strong squeezing (>>) even for a weak coupling with detector Still to be done: correct account of Q-factor and temperature Potential application: ultrasensitive force measurement beyond standard quantum limit