Quantum feedback control of solid-state qubits and their entanglement by measurement

Transcription

1 NANO 4, St. Petersburg, June 4 Quantum feedback control of solid-state qubits and their entanglement by measurement Rusko Ruskov and Outline: Continuous measurement of a single qubit Bayesian formalism Experimental predictions and proposals Quantum feedback of a solid-state qubit Entanglement of two qubits by measurement Recent developments of the Bayesian theory PRB 66, 44(R) (), PRB 67, 335(R) (3), Review: cond-mat/969 Acknowledgement: Q. Zhang, D. Averin, W. Mao 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 SQUIDs H = H QB + H DET + H INT H QB = (ε/)(c + c -c + c )+H(c + c +c + c ) ε - asymmetry, H tunneling Ω = (4H +ε ) / / frequency of quantum coherent (Rabi) oscillations Two levels of average detector current: I for qubit state, I for Response: I = I -I Detector noise: white, spectral density S I

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) > 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 cc ) + Hcc ( + cc ) I, I I=I I, I =(I +I )/, S I detector noise d d I ρ = ρ = HIm ρ + ρρ [ I( t) I] dt dt SI d I ρ = iερ + ih( ρ ρ) + ρ( ρ ρ) [ I( t) I] γρ dt SI A.K., 998 γ =Γ ( I) /4 S, Γ ensemble decoherence I γ / Γ = ( I ) / 4SI detector ideality (ef ciency), η η = Γ fi % For simulations: I() t I ( ρ ρ) I / + ξ(), t Sξ = SI Averaging over ξ(t) master equation Ideal detector (η=) does not decohere a single qubit; then the random evolution of the qubit wavefunction can be monitored Similar formalisms developed earlier. Key words: Imprecise, weak, selective, or conditional measurements, POVM, Quantum trajectories, Quantum jumps, Restricted path integral, etc.

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, 997) collapse classical information Detector collapse at t = t k Particular n k is chosen at t k d I I H ρ ρ ρ dt e e ρ d I I H ρ ρ ρ dt e e ρ n n n n = + Im n n n n = + + Im d ε H I + I II ρ ρ ρ ρ ρ ρ dt e e n n n n n n = i + i ( ) + n n 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 ρ tk ρ ( ) ( ) + ( t ) k If 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 / n [ ρ( t) ρ( t)] ( It/ ) () () /, ( ) i e t = ρ Pi n = exp( Iit/ e), [ ρ() ρ()] n! which are exactly quantum Bayes formulas

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

7 Ideality of realistic solid-state detectors (ideal detector does not cause single qubit decoherence). Quantum point contact Theoretically, ideal quantum detector, η= A.K., 998 (Gurvitz, 997; Aleiner et al., 997) Experimentally, η > 8% (using Buks et al., 998). SET-transistor Very non-ideal in usual operation regime, η Shnirman-Schőn, 998; A.K.,, Devoret-Schoelkopf, However, reaches ideality, η = if: - in deep cotunneling regime (Averin,, vanden Brink, ) - S-SET, using supercurrent (Zorin, 996) - 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. SQUID V(t) Can reach ideality, η = (Danilov-Likharev-Zorin, 983; Averin, ) 4. FET?? HEMT?? ballistic FET/HEMT??

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

9 Some experimental predictions and proposals Direct Bayesian experiments (998) Measured spectral density of Rabi oscillations (999,, ) Bell-type correlation experiment () Quantum feedback control of a qubit (, 4) Entanglement by measurement () Measurement and entanglement by a quadratic detector (3) Squeezing of a nanomechanical resonator (4) A. Korotkov, R. Ruskov, D. Averin, W. Mao, Q. Zhang, K. Schwab

10 S I (ω)/s S I (ω)/s qubit α C=3 3.3 Measured spectrum of Rabi oscillations (or spin precession) detector C = I HS ( ) / I ε /H= ω/ω α =. η = classical limit ω /Ω S I What is the spectral density S I (ω) of detector current? ε = Γ = ( ω ), η ( I) / 4 = S + S Ω ( I ) 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 Assume classical output, ev» Ω I ( ω ) ηs ε / H = S + + Ω Γ ( ω /4 H ) 4 ηs( + ε / H ) + + [( ω Ω ) Γ ( H / Ω )] A.K., LT 99 A.K.-Averin, 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., Contrary: Stace-Barrett, 3

11 on off τ A Q A = I A dt Bell-type correlation experiment τ on off τ B Q B = I B dt detector A qubit detector B δ B = (<Q B > - Q B )/ Q B Q A is fixed (selected) (Q A /τ A - I A ) / I A = δq A > δq A = δq A > τ A ( I A ) /S A = 3 τ Ω /π.6,, -.3 conventional A.K., after π/ pulse 3 τ Ω /π Idea: 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=). Advantage: no need to record noisy detector output with GHz bandwidth; instead, we use two detectors and fast ON/OFF switching.

12 Quantum feedback control of a solid-state qubit qubit H e control stage (barrier height) C<< detector feedback signal desired evolution comparison circuit Bayesian equations ρ ij (t) Ruskov & A.K., environment Goal: maintain desired phase of Rabi oscillations in spite of environmental dephasing (keep qubit fresh ) Idea: 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 optics is discussed since 993 (Wiseman-Milburn), recently first experiments (Armen et al., ; Geremia et al., 4).

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

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

15 detector (phase inaccuracy) φ rms qubit H=H [ F φ m (t)] π/3 / no noise C<< Simple quantum feedback of a qubit cos(ωt), τ-average sin(ωt), τ-average p ( φ) uncorrel. noise τ [( I) /S I ] = C =. control X Y phase φ m φ C =.,.3, (averaging time) τ [( I) /S I ] A.K., 4 Idea: two quadrature components of the detector current carry information on the phase of qubit Rabi oscillations Quadratures are extracted by mixing with a local oscillator or using a tank circuit Surprisingly, the phase calculated from two quadratures is very close to actual phase of Rabi oscillations (noise improves the monitoring accuracy, real evolution follows the observed behavior) C- dimensionless coupling τ averaging time φ - inaccuracy of phase monitoring

16 Efficiency of the simple quantum feedback (fidelity) D, <X>(4/τ I) C =. η = classical feedback. τ[( I) /S I ]= F/C fidelity for several values of coupling (feedback strength) η eff = ε/h =. Ω/CΩ=. C =. τ [( I) /S I ] = F/C fidelity for nonideal detectors The price for simplicity is the limitation of fidelity by about 9% Experimental verification by positive average in-phase quadrature X <X>=Dτ I/4 Works well for nonideal detectors, D =.5 for η =. Relatively simple experiment!

17 Two-qubit entanglement by measurement entangled u= qubit 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 + HINT ˆ H = ε ( aa aa) + H( aa + aa) + ε ( bb bb) + H( bb + bb) QB a a b b I( )=I( ), states indistinguishable by measurement Bell = ( )/ does not evolve Ruskov & A.K., Collapse into Bell state (spontaneous entanglement) with probability /4 starting from fully mixed state Hb

18 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 = η = entangled, P=/4 oscillatory, P=3/4 Ω t S I (ω)/s 4 ω /Ω ω /Ω ) ρ in ρ Bell, probability ρ B () (/4 for fully mixed state) ) ρ in oscillatory state, probability ρ B () (3/4 for fully mixed state) spectral peak at Rabi frequency Ω, S peak /S 3/3 Entanglement due to common quantum noise; however, detector is needed Ruskov & A.K., PRB (3) S I (ω)/s Peak/noise = (3/3)η

19 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+ E+5 Ω τ E+5 6E+4 8E+4 E+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

20 V ga V Quadratic Quantum Detection V gb V(φ) Mao, Averin, Ruskov, A.K., 3 (PRL, 4) V, I quadratic H a Hb qubit a SET qubit b I bias Linear detector I( ) I( )=I( ) Noninear detector I( ) I( )=I( ) φ, q Quadratic detector I( )=I( ) I( ) I( ) I( )= I( ) Quadratic detection is useful for quantum error correction (Averin-Fazio, )

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

22 Two-qubit quadratic detection: scenarios and switching Three scenarios: (distinguishable by average current) ) collapse into - = B, current I, flat spectrum ) collapse into - = B, current I, flat spectrum 3) collapse into remaining subspace 34 B, current (I +I )/, 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 ) Slightly different Rabi frequencies, Ω=Ω -Ω Γ B B = Γ B B = ( Ω) / Γ, Γ= η ( I) /4S ( I ) Γ SI ( ω) = S + ( Ω ) + ω /( ) Γ Ω ) Slightly nonquadratic detector, I I 4 Γ B B = [( I I ) / I] Γ / 34 4 Mao, Averin, Ruskov, Korotkov, 3 4 Ω ( I ) Γ SI ( ω) = S + 3 ( ω 4 Ω ) +Γ ω 4 8( I ) + 7 Γ( I I ) + [4 ( I) / 3 Γ( I I ) ] 4 ω 4

23 Bayesian approach to continuous position measurement of a nanomechanical resonator ω GHz, T mk, quantum behavior ˆ ˆ pˆ mω x H = + m 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 QPC = +..) Detector noise lr, SX = S ei V(t) Coupling k T osc ( ) V C meas I X = π M + M x ρlρre = I + kx Sm ω τ Evolution equation (Stratonovich form) d ˆ ρ( xx, ) i [ H, ρ] ρ( xx, ) ( It ( ) I) kx ( x x) k x + = + + x x dt S Cooling by feedback Hˆ fb = Fxˆ, F = γ ( mω x + p) Ruskov- Korotkov, 4 resonator Formalism similar to A. Hopkins et al., 3 & Doherty-Jacobs, 999 x m, ω

24 QND squeezing of a nanoresonator by feedback squeezing Constant voltage no squeezing at C << (Hopkins, Jacobs, Habib, Schwab, 3) We consider periodic V(t) squeezing possible! (Ruskov-Korotkov, 4) Dψ, D c.m. ( x / x) sine modulation D ψ D cm.. ω = ω D = Dψ + D cm.. no modulation with modulation ω t /π C =. ω / ω η = C = η =.8 η = C =. γ ω = Dcm..<< D ψ Squeezing resonances ω at ω = n Dψ, D c.m. Squeezing pulse modulation D cm.. ( x / x) D ψ ω = ω ω t /π /5 / /3 C =.5 δt /T =.5 η = C = ω / ω

25 Conclusions Bayesian formalism for continuous quantum measurement is simple (almost trivial); but still a new interesting subject in solid-state mesoscopics Quantum feedback of a single qubit can keep the Rabi oscillations in a qubit for arbitrary long time; feedback can be realized in a simple way via quadratures Two qubits can be made fully entangled just by measuring them with an equally coupled detector Various experimental predictions have been made using the Bayesian formalism; the range of applications still expanding No experiments yet; hopefully, coming soon

26 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 (t) ideal detector ξ 3 (t) fully correlated Id (t) S classical current S ξ (t) + detector S+S A.K., AS+ θ S K =, SI = S + S S d d I ρ = ρ = HIm ρ + ρρ [ I( t) I] dt dt SI d I ρ = iερ + ih( ρ ρ ) + ρ ( ρ ρ ) [ I( t) I ] + ik[ I( t) I ] γ mρ dt SI I K correlation between output and backaction noises Fundamental limits for ensemble decoherence Γ = γ + ( I) /4S I, γ Γ ( I) /4S I Γ = γ + ( I) /4S I + K S I /4, γ Γ ( I) /4S I + K S I /4 Translated into energy sensitivity: (Є I Є BA ) / / or (Є I Є BA Є I,BA ) / /

27 Direct Bayesian experiments (A.K., 998) Idea: check the evolution of (almost pure!) qubit state given by Bayesian equations. ρ Reρ Evolution from /-alive to Density matrix purification /3-alive Schrödinger cat by measurement stop & check.5 ρ Re ρ Im ρ..5.. t. 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. Start with completely mixed state.. Measure and monitor the Rabi phase. 3. Stop evolution (make H=) at state. 4. Measure. Difficulty: need to record noisy detector current, typical required bandwidth ~ - GHz.

28 Possible experimental confirmation? (STM-ESR experiment similar to Manassen-989) peak noise 3.5 (Colm Durkan, private comm.)