Noisy quantum measurement of solid-state qubits: Bayesian approach

Transcription

1 Noisy quantum measurement of solid-state qubits: Bayesian approach Outline: Bayesian formalism Continuous measurement of a single qubit deal and nonideal solid-state detectors Bayesian formalism for entangled qubits Experimental predictions and proposals Measured spectrum of Rabi oscillations Bell-type experiment Quantum feedback for a qubit Entanglement by measurement Quadratic quantum detection SPE, Santa Fe, June 3 Review: cond-mat/969, in Quantum Noise in Mesoscopic Physics (Kluwer, 3) Acknowledgement: Rusko Ruskov Support:

2 Examples of solid-state qubits and detectors qubit H e detector Double-quantum-qot and quantum point contact (QPC) Cooper-pair box and single-electron transistor (SET) Two SQUDs H = H QB + H DET + H NT H QB = (ε/)(c 1+ c 1 -c + c )+H(c 1+ c +c + c 1 ) ε - asymmetry, H tunneling Ω = (4H +ε ) 1/ frequency of quantum coherent (Rabi) oscillations Two levels of average detector current: 1 for qubit state 1, for Response: = 1 - Detector noise: white, spectral density S

3 H e What happens to a qubit state during measurement? For simplicity (for a moment) H=ε=, infinite barrier (frozen qubit), evolution due to measurement only Orthodox answer Conventional (decoherence) answer (Leggett, Zurek) exp( Γt) exp( Γt) 1 1 1> or >, depending on the result no measurement result! ensemble averaged Orthodox and decoherence answers contradict each other! applicable for: Orthodox Conventional (ensemble) Bayesian Single quantum systems yes no yes Continuous measurements no yes yes Bayesian formalism describes gradual collapse of single quantum systems Noisy detector output should be taken into account

4 H e Bayesian formalism for a single qubit ˆ ε HQB = ( cc 1 1 cc ) + Hcc ( 1 + cc 1) 1Ò Æ 1, ÒÆ = 1 -, =( 1 + )/, S detector noise d d ρ11 = ρ = Hm ρ1 + ρ11ρ [ ( t) ] dt dt S d ρ1 = iερ1 + ih( ρ11 ρ) + ρ1( ρ11 ρ) [ ( t) ] γρ1 dt S A.K., 1998 γ =Γ ( ) /4 S, Γ ensemble decoherence 1 γ / Γ = ( ) / 4S detector ideality (ef ciency), η 1 η = Γ fi % For simulations: () t ( ρ ρ11) / + ξ(), t Sξ = S Averaging over ξ(t) î master equation Similar formalisms developed earlier. Key words: mprecise, weak, selective, or conditional measurements, POVM, Quantum trajectories, Quantum jumps, Restricted path integral, etc. Names: E.B. Davies, K. Kraus, A.S. Holevo, C.W. Gardiner, H.J. Carmichael, C.M. Caves, M.B. Plenio, P.L. Knight, M.B. Mensky, D.F. Walls, N. Gisin,.C. Percival, G.J. Milburn, H.M. Wiseman, R. Onofrio, S. Habib, A. Doherty, etc. (very incomplete list)

5 Microscopic derivation of the Bayesian formalism ρ ij n (t) n(t k ) qubit detector pointer quantum interaction Schrödinger evolution of qubit + detector for a low-t QPC as a detector (Gurvitz, 1997) collapse classical information Detector collapse at t = t k Particular n k is chosen at t k d H ρ ρ ρ dt e e ρ d H ρ ρ ρ dt e e ρ n 1 n 1 n 1 n 11 = m 1 n n n 1 n = + + m 1 d ε H + ρ ρ ρ ρ ρ ρ dt e e n n n n 1 n 1 n 1 1 = i 1 + i ( 11 ) n n 11 k ρ P( n) = ρ ( t ) + ( t k ) n ij tk n, nk ij tk ρ ( + ) = δ ρ ( + ), ρ nk ρij tk ij ( tk + ) = nk nk ρ11 tk ρ ( ) ( ) + ( t ) k f H = ε =, it leads to ρ ρ ρ () P ( n) ρ () P ( n) () P ( n) () P ( n) () P ( n) () P ( n) () t =, ρ() t = ρ ρ ρ ρ 1/ n [ ρ11( t) ρ( t)] ( t/ ) 1() 1() 1/, ( ) i e t = ρ Pi n = exp( it/ e), [ ρ11() ρ()] n! which are exactly quantum Bayes formulas

6 P "Quantum Bayes theorem (ideal detector assumed) 1> H > e Measurement (during time τ): D 1/ _ actual 1 H = ε = (frozen qubit) After the measurement during time τ, the probabilities can be updated using the standard Bayes formula: Quantum Bayes formulas: _ nitial state: τ ρ11() ρ1() ρ1() ρ() 1 () t dt τ P(, τ ) = ρ11() P1 (, τ ) + ρ() P (, τ ) 1 P i (, τ ) = exp[ ( ) / ], i D π D D= S S / τ, 1 i, τ / i P( B ) ( ) ( ) i P A B P B i i A = P( Bk) P( A Bk) k ρ11() exp[ ( 1) / D] 11( τ) = ρ ρ ρ () exp[ ( ) / D] () exp[ ( ) / D] ρ ( τ) ρ () =, ρ( τ ) = 1 ρ11( τ ) [ ρ ( τ) ρ ( τ)] [ ρ () ρ ()] 1 1 1/ 1/ 1 1

7 qubit (ε, H) Nonideal detectors with input-output noise correlation classical noise affecting ε signal quantum backaction noise classical noise affecting ε ξ (t) = Aξ 1 (t) ideal detector ξ 3 (t) fully correlated d (t) S classical current S1 ξ1 (t) + detector S+S1 A.K., d d ρ = ρ = Hm ρ + ρ ρ [ ( t) ] dt dt S AS1+ θ S K =, S = S + S S 1 d ρ = iερ + ih( ρ ρ ) + ρ ( ρ ρ ) [ ( t) ] + ik[ ( t) ] γρ dt S K correlation between output and backaction noises Fundamental limits for ensemble decoherence Γ = γ + ( ) /4S, γ Γ ( ) /4S Γ = γ + ( ) /4S + K S /4, γ Γ ( ) /4S + K S /4 Translated into energy sensitivity: (Є Є BA ) 1/ / or (Є Є BA Є,BA ) 1/ /

8 deality of realistic solid-state detectors (ideal detector does not cause single qubit decoherence) 1. Quantum point contact Theoretically, ideal quantum detector, η=1 A.K., 1998 (Gurvitz, 1997; Aleiner et al., 1997) Experimentally, η > 8% (using Buks et al., 1998). SET-transistor Very non-ideal in usual operation regime, η 1 Shnirman-Schőn, 1998; A.K.,, Devoret-Schoelkopf, However, reaches ideality, η = 1 if: - in deep cotunneling regime (Averin,, vanden Brink, ) - S-SET, using supercurrent (Zorin, 1996) - S-SET, double-jqp peak (Clerk et al., )??? S-SET, usual JQP (Johansson et al.), onset of QP branch (?) - resonant-tunneling SET, low bias (Averin, ) 3. SQUD V(t) Can reach ideality, η = 1 (Danilov-Likharev-Zorin, 1983; Averin, ) 4. FET?? HEMT?? ballistic FET/HEMT??

9 Bayesian formalism for N entangled qubits measured by one detector qb 1 qb qb qb N ρ (t) Up to N levels of current detector d i ˆ 1 [, ] [( ( ) k + ρ i ij = Hqb ρ ij+ ρij ρkk t )( i k ) + dt S k k + j + ( ( t) )( j k) ] γijρij (Stratonovich form) 1 γ = ( η 1)( ) /4 S ( t) = ρ ( t) + ξ( t) ij i j ii i i Averaging over ξ(t) î master equation No measurement-induced dephasing between states iò and jò if i = j! A.K., PRA 65 (), PRB 67 (3)

10 Some experimental predictions and proposals Direct experiments (1998) Measured spectral density of Rabi oscillations (1999,, ) Bell-type correlation experiment () Quantum feedback control of a qubit (1) Entanglement by measurement () Measurement by a quadratic detector (3)

11 Direct Bayesian experiments (A.K., 1998) dea: check the evolution of (almost pure!) qubit state given by Bayesian equations 1. ρ 11 Reρ 1 Evolution from 1/-alive to Density matrix purification 1/3-alive Schrödinger cat by measurement stop & check.5 ρ 11 Re ρ 1 m ρ t 1. Prepare coherent state and make H=.. Measure for a finite time t. 3. Check the predicted wavefunction (using evolution with H to get the state time 1. Start with completely mixed state.. Measure and monitor the Rabi phase. 3. Stop evolution (make H=) at state Measure. Difficulty: need to record noisy detector current, typical required bandwidth ~ 1-1 GHz.

12 S (ω)/s S (ω)/s qubit α C= Measured spectrum of Rabi oscillations (or spin precession) detector C = HS ( ) / ε /H= 1 ω/ω α =.1 η = 1 classical limit ω /Ω S What is the spectral density S (ω) of detector current? ε = Γ = ( ω ) 1, 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 «1 S Assume classical output, ev» Ω ( ω ) ηs ε / H = S + + Ω Γ 1 ( ω /4 H ) 1 4 ηs(1 + ε / H ) [( ω Ω ) Γ (1 H / Ω )] A.K., LT 99 Averin-A.K., A.K., Averin, Goan-Milburn, 1 Makhlin et al., 1 Balatsky-Martin, 1 Ruskov-A.K., Mozyrsky et al., Balatsky et al, Bulaevskii et al., Shnirman et al., Bulaevskii-Ortiz, 3

13 Possible experimental confirmation? (STM-ESR experiment similar to Manassen-1989) peak noise 3.5 (Colm Durkan, private comm.)

14 on off τ A Q A = A dt Bell-type correlation experiment τ on off τ B Q B = B dt detector A qubit detector B δ B = (<Q B > - Q B )/ Q B Q A is fixed (selected) (Q A /τ A - A ) / A = τ A ( A ) /S A = τ Ω /π.6,, -.3 δq A > δq A = δq A > conventional A.K., after π/ pulse 1 3 τ Ω /π dea: two consecutive finite-time (imprecise) measurements of a qubit by two detectors; probability distribution P(Q A, Q B, τ) shows the effect of the first measurement on the qubit state. Proves that the qubit remains in a pure state during the measurement (for η=1). Advantage: no need to record noisy detector output with GHz bandwidth; instead, we use two detectors and fast ON/OFF switching.

15 Quantum feedback control of a solid-state qubit desired evolution Ruskov & A.K., 1 control stage (barrier height) feedback signal comparison circuit qubit H e C<<1 detector Bayesian equations ρ ij (t) environment Goal: maintain desired phase of 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 Quantum feedback in quantum optics is discussed since 1993 (Wiseman-Milburn), recently first successful experiment in Mabuchi s group (Armen et al., ).

16 K z (τ) Performance of quantum feedback (no extra environment) Qubit correlation function C=1, η=1, F=,.5, τ Ω /π cos Ωt C FHτ / Kz( τ ) = exp ( e 1) 16F (for weak coupling and good fidelity) Detector current correlation function ( ) cos Ωt FHτ / K ( τ ) = (1 + e ) 4 C FH / S exp ( e τ 1) + δτ ( ) 16F D (synchronization degree) Fidelity (synchronization degree) C=1 η =1 d e = "direct" τ a / T=/ F (feedback factor) C = ( ) / S H coupling -1 τ a available bandwidth F feedback strength D= Trρρ 1 desir 1/3 For ideal detector and wide bandwidth, fidelity can be arbitrary close to 1% D = exp(-c/3f) Ruskov & Korotkov, PRB 66, 4141(R) ()

17 Suppression of environment-induced decoherence by quantum feedback D (synchronization degree) C=C det =1 τ a = C env /C det = F (feedback factor) D max det D max (synchronization) Cenv 1 1 C 1 + C / C env det C env /C det (relative coupling to environment) f qubit coupling to the environment is 1 times weaker than to the detector, then D max = 99.5% and qubit fidelity 99.75%. (D = without feedback.)

18 Two-qubit entanglement by measurement entangled u= qubit 1 qubit detector ρ (t) Ha Hb DQDa QPC DQDb H a V ga qubit a V SET V gb qubit b Assume symmetric case: equal symmetric qubits, ε a =ε b =, H a =H b, Ω a =Ω b, equal coupling, C a =C b, no direct interaction, u= Hˆ = Hˆ ˆ ˆ QB + HDET + HNT ˆ H = ε ( aa aa) + H( aa + aa) + ε ( bb bb) + H( bb + bb) QB a a b b ( )=( ), states indistinguishable by measurement BellÚ = ( Ú- Ú)/ does not evolve Ruskov & A.K., Collapse into BellÚ state (spontaneous entanglement) with probability 1/4 starting from fully mixed state Hb

19 H a V ga qubit a V V gb Hb SET qubit b Continuous measurement (detector is ON all the time) Two scenarios of evolution from mixed state ρ Bell (t) C = 1 η = entangled, P=1/4 oscillatory, P=3/4 Ω t S (ω)/s 4 1 ω /Ω ω /Ω 1) ρ in ρ Bell, probability ρ 11 B () (1/4 for fully mixed state) ) ρ in oscillatory state, probability 1-ρ 11 B () (3/4 for fully mixed state) spectral peak at Rabi frequency Ω=H/Ñ, S peak /S 3/3 Entanglement due to common quantum noise; however, detector is needed Ruskov & A.K., PRB (3) S (ω)/s Peak/noise = (3/3)η

20 H a V ga qubit a ρ Bell (t) V SET V gb qubit b Hb Small imperfections: switching between entangled and oscillatory states Switching time distributions B _ >O O _ >B E+ 1E+5 Ω τ E+5 6E+4 8E+4 1E+5 Ω t Different Rabi frequencies: Γ B = ( Ω) /Γ Different coupling: Γ = Γ B ( C/ C) /8 Environmental dephasing: Γ = ( γa + γ )/ Using trivial feedback procedure (applying noise in undesirable state), we can keep two qubits entangled Detector may be nonideal (just 3η/3 peak), so SET is OK Γ B O B = ΓB O /3 b

21 Quadratic Quantum Detection Mao, Averin, Ruskov, A.K., 3 V ga V V gb V(f) V, quadratic H a Hb qubit a SET qubit b bias Linear detector ( ) ( )=( ) Noninear detector ( ) ( )=( ) φ, q Quadratic detector ( )=( ) ( ) ( ) ( )= ( ) Quadratic detection is useful for quantum error correction (Averin-Fazio, )

22 S (ω)/s S (ω)/s S (ω)/s Linear detector analytical 6 numerical 4 1 ω /Ω 3 Nonlinear detector 6 4 Quadratic detector ω /Ω 3 1 ω /Ω 3 S Two-qubit detection (oscillatory subspace) ( ω) = S + 1 η 8 Ω ( ) Γ 3 ( ω Ω ) +Γ ω Γ= ( ) /4 S, = = Spectral peak at Ω, peak/noise = (3/3)η Extra spectral peaks at Ω and S ( ω ) (Ω is the Rabi frequency) 4 Ω ( ) Γ = S + ( ω 4 Ω ) +Γ ω ( =, =, = ) Peak only at Ω, peak/noise = 4η Mao, Averin, Ruskov, A.K., 3

23 Two-qubit quadratic detection: scenarios and switching Three scenarios: (distinguishable by average current) 1) collapse into - Ú = 1Ú B, current Æ, flat spectrum ) collapse into - Ú = Ú B, current ÆÆ, flat spectrum 3) collapse into remaining subspace 34Ú B, current ( Æ + ÆÆ )/, spectral peak at Ω, peak/pedestal = 4η. B ρ (t) B 34 B ε Ωt 6E+4 8E+4 3) Slightly asymmetric qubits, ε B B ε Γ 34 = Γ/ Ω B Switching between states due to imperfections 1) Slightly different Rabi frequencies, Ω=Ω 1 -Ω 1 Γ 1B B = Γ B 1 B = ( Ω) / Γ, Γ= η ( ) /4S ( ) Γ 1 S ( ω) = S + ( Ω ) 1 + ω /( ) Γ Ω ) Slightly nonquadratic detector, 1 4 Γ B B = [( ) / ] Γ / Mao, Averin, Ruskov, Korotkov, 3 4 Ω ( ) Γ S ( ω) = S + 3 ( ω 4 Ω ) +Γ ω 4 8( ) Γ( ) 1 + [4 ( ) / 3 Γ( ) ] 1 4 ω 1 4

24 Conclusions Bayesian formalism for continuous quantum measurement is simple (almost trivial); but still a new interesting subject in solid-state mesoscopics Several experimental predictions have been already made; however, many problems not studied yet No experiments yet (except one); hopefully, coming soon