Quantum feedback control of solid-state qubits

Transcription

1 Quantum feedback control of solid-state qubits

2 The system we consider: qubit + detector qubit H e Vg V I(t) detector I(t) I(t) 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 INT H QB = (ε/)(c 1 + c 1 c + c )+H(c 1 + c +c + c 1 ) ε asymmetry, H tunneling Ω = (4H +ε ) 1/ /Ñ frequency of quantum coherent (Rabi) oscillations Two levels of average detector current: I 1 for qubit state 1, I for Response: ΔI = I 1 I Detector noise: white, spectral density S I DQD and QPC (setup due to Gurvitz, 1997) DET = l l l r r r + l r l, r r l + l r INT = Δ (, 1 1 )( r l + l r) lr H Eaa Eaa Taa ( aa) + H T cc cc aa aa S I = ei

3 H H=0 e I(t) Bayes rule: Bayesian formalism for DQD-QPC (qubit-detector) system Qubit evolution due to continuous measurement: 1) Diagonal matrix elements of the qubit density matrix evolve as classical probabilities (i.e. according to the classical Bayes rule) ) Non-diagonal matrix elements evolve so that the degree of purity ρ ij /[ρ ii ρ jj ] 1/ is conserved P( A ) ( ) ( ) i P R A P A i i R = P( Ak) P( R Ak) k (A.K., 1998) So simple because: 1) QPC happens to be an ideal detector ) no Hamiltonian evolution of the qubit Similar formalisms developed earlier. Key words: Imprecise, weak, selective, or conditional measurements, POVM, Quantum trajectories, Quantum jumps, Restricted path integral, etc. Names: Davies, Kraus, Holevo, Belavkin, Mensky, Caves, Gardiner, Carmichael, Knight, Walls, Gisin, Percival, Milburn, Wiseman, Habib, etc. (very incomplete list)

4 Quantum Bayes theorem (ideal detector assumed) 1> H > P e I(t) H = ε = 0 ( frozen qubit) After the measurement during time τ, the probabilities should be updated using the standard Bayes formula: Quantum Bayes formulas: ρ _ Measurement (during time τ): I actual D 1/ D 1/ I 1 I _ I Initial state: τ ρ11(0) ρ1(0) ρ1(0) ρ(0) 1 I I() t dt τ 0 P( I, τ ) = ρ11(0) P1 ( I, τ) + ρ(0) P ( I, τ) 1 P i ( I, τ ) = exp[ ( I I ) / ], i D π D D= S I I I S I I / τ, 1 i, τ I / i P( B ) ( ) ( ) i P A B P B i i A = P( Bk) P( A Bk) ρ11(0) exp[ -( I -I1) / D] 11( τ) = ρ ρ - - (0) exp[ ( I I ) / D] (0) exp[ ( I I ) / D] ρ ( τ) ρ (0) =, ρ( τ) = 1 -ρ11( τ) [ ρ ( τ) ρ ( τ)] [ ρ (0) ρ (0)] 1 1 1/ 1/ 1 1 k

5 H Bayesian formalism for a single qubit e I(t) e Vg V I(t) ε Hˆ QB = ( c1c1 cc) + H( c1c + cc1) 1Ò ÆI 1, ÒÆI, ΔI=I 1 -I, I 0 =(I 1 +I )/ S I detector noise ρ =- ρ =- ( H / )Im ρ + ρ ρ ( ΔI / S )[ It ()-I ] 11 Averaging over result I(t) leads to conventional master equation: 1 11 I 0 ρ 1 = i( ε / ) ρ1 + i( H / )( ρ11 -ρ) + ρ1( ρ11 -ρ)( ΔI / SI )[ It ()-I0]-γρ1 I() t -I =( ρ - ρ ) ΔI / + ξ(), t S = S I 0 11 γ = Γ- Δ Γ ensemble decoherence ( I) /4 SI, η = 1 - γ / = ( I) /4SI η (A.K., 1998) Γ Δ Γ detector ideality (efficiency), 100% Ideal detector (η=1, as QPC) does not decohere a qubit, then random evolution of qubit wavefunction can be monitored ξ dρ11/ dt =- dρ/ dt =-( H / )Imρ1 dρ / dt = i( ε / ) ρ + i( H/ )( ρ -ρ )-Γρ

6 Assumptions needed for the Bayesian formalism Detector voltage is much larger than the qubit energies involved ev >> ÑΩ, ev >> ÑΓ (no coherence in the detector, Ñ/eV << (1/Ω,1/Γ); Markovian approximation) Small detector response, ΔI << I 0, ΔI=I 1 - I, I 0 =(I 1 + I )/ Many electrons pass through detector before qubit evolves noticeably. (Not a really important condition, but simplifies formalism.) Coupling C ~ Γ/Ω is arbitrary [we define C = Ñ(ΔI) /S I H ] d d H ΔI ρ = - ρ = - Im ρ + ρ ρ [ I ( t )- I ] dt dt S I d ε H ΔI ρ = i ρ + i ( ρ - ρ ) + ρ ( ρ -ρ ) [ I( t) -I ]-γρ dt SI

7 Derivations 1) logical : via correspondence principle and comparison with decoherence approach (A.K., 1998) ) microscopic : Schr. eq. + collapse of the detector (A.K., 000) n ρ ij () t nt ( qubit detector k ) classical pointer information quantum frequent n number of electrons interaction collapse passed through detector 3) from quantum trajectory formalism developed for quantum optics (Goan-Milburn, 001; also: Wiseman, Sun, Oxtoby, etc.) 4) from POVM formalism (Jordan-A.K., 006)

8 Informational derivation of the Bayesian formalism Step 1. Assume H = ε = 0, frozen qubit Since ρ 1 is not involved, evolution of ρ 11 and ρ should be the same as in the classical case, i.e. Bayes formula (correspondence principle). Step. Assume H = ε = 0 and pure initial state, ρ 1 (0) = [ρ 11 (0) ρ (0)] 1/ For any realization ρ 1 (t) [ρ 11 (t) ρ (t)] 1/. Hence, averaging over ensemble of realizations gives ρ 1 av (t) ρ 1 av (0) exp[æ(δi /4S I ) t] However, conventional (ensemble) result (Gurvitz-1997, Aleiner et al.-1997) for QPC is exactly the upper bound: ρ 1 av (t)=ρ 1 av (0) exp[æ(δi /4S I ) t]. Therefore, pure state remains pure: ρ 1 (t)=[ρ 11 (t) ρ (t)] 1/. Step 3. Account of a mixed initial state Result: the degree of purity ρ 1 (t)/[ρ 11 (t) ρ (t)] 1/ is conserved. Step 4. Add qubit evolution due to H and ε. Step 5. Add extra dephasing due to detector nonideality (i.e., for SET).

9 Microscopic derivation of the Bayesian formalism Schrödinger evolution of qubit + detector for a low-t QPC as a detector (Gurvitz, 1997) d I I H ρ ρ ρ dt e e ρ d I I H ρ ρ ρ dt e e ρ n 1 n 1 n 1 n 11 = Im 1 n n n 1 n = + + Im 1 d ε H I + I II ρ ρ ρ ρ ρ ρ dt e e n n n n 1 n 1 n 1 1 = i 1 + i ( 11 ) If H = ε =0, this leads to ρ ρ n ρ ij () t nt ( k ) qubit detector pointer quantum interaction ρ frequent collapse Detector collapse at t = t k Particular n k is chosen at t k n n Pn ( ) = ρ11( tk ) + ρ( t k ) n ij tk n, nk ij tk ρ ( + 0) = δ ρ ( + 0) ρ nk ρij ( tk ) ij ( tk + 0) = nk nk ρ11 tk + ρ (0) P ( n) ρ (0) P ( n) ( ) ( t ) () t =, ρ() t = ρ ρ ρ ρ 1/ n [ ρ11( t) ρ( t)] ( It/ ) 1() 1(0) 1/, ( ) i e t = ρ Pi n = exp( Iit/ e), [ ρ11(0) ρ(0)] n! (0) P ( n) (0) P ( n) (0) P ( n) (0) P ( n) which are exactly quantum Bayes formulas classical information n number of electrons passed through detector k

10 Quantum efficiency of 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) Averin, 000; Pilgram et al., 00, Clerk et al., 00 Experimentally, η > 80% I(t) (using Buks et al., 1998). SET-transistor I(t) Very non-ideal in usual operation regime, η 1 Shnirman-Schön, 1998; A.K., 000, Devoret-Schoelkopf, 000 However, reaches ideality, η = 1 if: - in deep cotunneling regime (Averin, van den Brink, 000) - S-SET, using supercurrent (Zorin, 1996) - S-SET, double-jqp peak (η ~ 1) (Clerk et al., 00) - resonant-tunneling SET, low bias (Averin, 000) 3. SQUID V(t) Can reach ideality, η = 1 (Danilov-Likharev-Zorin, 1983; Averin, 000) 4. FET?? HEMT?? ballistic FET/HEMT??

11 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 I(t) d i ˆ 1 I [, ] [( ( ) k + I ρ i ij = Hqb ρ ij+ ρij ρkk I t )( Ii Ik ) + dt S k Ik + Ij + ( I( t) )( Ij Ik) ] γ ijρij (Stratonovich form) 1 γ = ( η 1)( I I ) /4 S I( t) = ρ ( t) I + ξ( t) ij i j I ii i i No measurement-induced dephasing between states iò and jò if I i = I j! Averaging over ξ(t) î master equation A.K., PRA 65 (00), PRB 67 (003)

12 Bayesian quantum feedback control of a qubit Since qubit state can be monitored, the feedback is possible! desired evolution H qb = Hσ X qubit H e control stage (barrier height) C<<1 detector feedback signal I(t) comparison circuit Bayesian equations ρ ij (t) environment Ruskov & A.K., 001 Goal: maintain perfect Rabi oscillations forever Idea: monitor the Rabi phase φ by continuous measurement and apply feedback control of the qubit barrier height, ΔH FB /H = F Δφ To monitor phase φ we plug detector output I(t) into Bayesian equations

13 K K z (τ) z Performance of quantum feedback Qubit correlation function C=1, η=1, F=0, 0.05, τ Ω /π cosωt C FHτ / ( τ ) = exp ( e 1) 16F C = ( ΔI) / SI H coupling F feedback strength D= Trρρ 1 desir For ideal detector and wide bandwidth, fidelity can be arbitrarily close to 100% D = exp( C/3F) D (synchronization degree) Fidelity (synchronization degree) C=C det =1 τ a =0 C env /C det = F (feedback factor) Experimental difficulties: necessity of very fast real-time solution of Bayesian equations wide bandwidth (>>Ω, GHz-range) of the line delivering noisy signal I(t) to the processor Ruskov & A.K., PRB-00

14 Simple quantum feedback of a solid-state qubit H qb = Hσ X qubit C<<1 detector 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!) H =H 0 [1 F φ m (t)] t X ( t) = [ I( t') I ] cos( Ωt') exp[ ( t t') / τ ] dt t 0 0 Y( t) = [ I( t') I ] sin( Ωt') exp[ ( t t') / τ ] dt I(t) control cos(ω t), τ-average local oscillator sin(ω t), τ-average (A.K., 005) Goal: maintain coherent (Rabi) oscillations for arbitrarily long time φ m = arctan( Y / X) (similar formulas for a tank circuit instead of mixing with local oscillator) Advantage: simplicity and relatively narrow bandwidth (1/ τ ~ Γ d << Ω) X Y phase φ m Essentially classical feedback. Does it really work?

15 Δφ rms (phase inaccuracy) Accuracy of phase monitoring via quadratures π/3 1/ C dimensionless coupling uncorrelated noise C<<1 C = 0.1, 0.3, no noise C<< τ [(ΔI) /S I ] (no feedback yet) (averaging time) p (Δφ) 1 0 τ [(ΔI) /S I ] = C = 0.1 Noise improves the monitoring accuracy! (purely quantum effect, reality follows observations ) dφ / dt = [ I( t) I0]sin( Ω t+ φ )( ΔI / S I ) dφ / dt = [ I( t) I ]sin( Ω t+ φ )/( X + Y ) m 0 1/Γ d =4S I /(ΔI) m 1/ (non-gaussian distributions) weak coupling C<< Δφ =φ -φ 4 m 1 8 Δφ (actual phase shift, ideal detector) (observed phase shift) Noise enters the actual and observed phase evolution in a similar way Quite accurate monitoring! cos(0.44) 0.9

16 Simple quantum feedback (fidelity) D, <X>(4/τ ΔI) C = 0.1 η = F/C 0. τ[(δi) /S I ]= classical feedback fidelity for different averaging τ (feedback strength) weak coupling C D feedback efficiency D FQ 1 F Tr ρ( t) ρ () t Q D max 90% 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/ τδi) X =0 for any non-feedback Hamiltonian control of the qubit

17 - nonideal detectors (finite quantum efficiency η) and environment - qubit energy asymmetry ε - frequency mismatch ΔΩ Quantum feedback still works quite well Main features: Effect of nonidealities D (feedback efficiency) Fidelity F Q up to ~95% achievable (D~90%) Natural, practically classical feedback setup Averaging τ~1/γ>>1/ω (narrow bandwidth!) Detector efficiency (ideality) η~0.1 still OK Robust to asymmetry ε and frequency shift ΔΩ Simple verification: positive in-phase quadrature X η eff = ε/h 0 = ΔΩ/CΩ=0. C = 0.1 τ [(ΔI) /S I ] = F/C (feedback strength) Simple enough experiment?!

18 Bayesian quantum feedback in more detail Ideal case (ideal detector, no extra dephasing, infinite bandwidth and no delay) C = ( ΔI) / S H, Coupling D - Feedback efficiency Analytics: D=exp(-C/3F) I D(F/C>>1) 1 works very well in ideal case We have also analyzed the following effects: Non-ideal detector & extra dephasing Finite bandwidth Time delay in the feedback loop Qubit parameter deviations (ε & H) Feedback of a qubit with ε 0 (needs special controller)

19 Detector nonideality and dephasing Effect of imperfect detector and extra dephasing due to coupling to environment Imperfection & extra dephasing D max (η e )<1 D max : maximum feedback efficiency η e =[1+4γS I /(ΔI) ] quantum efficiency Analytical D max =< P > P - purity Dmax = 1 P G( P ) dp 0 1 PG( P ) dp 0 GP ( ) = (1 P) 5/A A = ( η 1 1)/(1 P e ) exp[ ] Simple D max = [(1 ) ] ηe + ηe

20 Effect of finite detector bandwidth Available detector output signal I a (t) differs form the true one A modified signal to compensate this implicit time delay, Δ φ = φ Ω( t κτ ), D = D( k = k ) 1 t a 1 t a τa t τ a a a ( t t')/ τa max I ( t) = τ I( t')exp dt' exponential window a I ( t) = I( t') dt' rectangular window opt Narrower bandwidth (τ a ) reduced efficiency (D max ), loss of information

21 Effect of time delay in the feedback loop Signal from past time t-τ d ΔH fb /H = -F Δφ(t-τ d ) D< D(τ d =0) Full model: d Δφ Δ sin I Δ = φ ( I cos φ+ ξ( t)) FHΔφ( t τ ) dt d SI dδ φ ( ΔI) Simple model ( C 1): () t FH ( t ), S = ξ Δφ τ dt d ξ = SI D=<cosΔφ> simple classical model, C 0 F (T/τ d ) Sharp drop at Fτ d /T>1/4 ---different from ideal case Fτ d /T=1/4 separates stable region & unstable (oversteering ) region Simple model results full model results; Scaling behavior for Cτ d /T= C τ d /T

22 Effect of ε & H deviation (from ε =0, H =H 0 ) mistake in qubit monitoring obviously worsens D Solid: small ε decreases D very little Dashed: rescaling x-axis by C 1/ Dotted: exact monitoring significantly improves D Effect of H deviation quite similar to ε Dashed : rescaling x-axis by C Feedback operation is robust against small unknown deviations of qubit parameters.

23 Feedback of an energy-asymmetric qubit (ε 0) Despite the phase space becomes essentially two-dimensional, we still want to control only one parameter (H) Special controller: ΔH fb /H = -FΔφ m -F r sinφ m Δr m Change only one parameter H to reduce both Δφ m and Δr m D max (F r /F) optimized on F/C max (F r /F): optimized on F/C α=atan(ε/h) ρ m & ρ d are in two parallel planes D max-max (ε/h, C=0.1)>>D max-max (ε/h, C=0.3) D max-max (ε/h, C<<1) can approach 1 This controller works very well!

24 Conclusions

25 Quantum feedback in optics First experiment: Science 304, 70 (004) First detailed theory: H.M. Wiseman and G. J. Milburn, Phys. Rev. Lett. 70, 548 (1993)