Stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays

Transcription

1 Stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays Pierre Rouchon, Mines-ParisTech, Centre Automatique et Systèmes, Delayed Complex Systems June 2012, Palma de Majorque Joint wor with Hadis Amini (Mines-ParisTech), Mazyar Mirrahimi (INRIA), Ram Somaraju (INRIA/Vrije Universiteit Brussel) Igor Dotseno, Serge Haroche (ENS, Collège de France) Michel Brune, Jean-Michel Raimond, Clément Sayrin (ENS).

2 Outline Feedbac and quantum systems Measurement-based feedbac of photons The LKB experiment The ideal Marov model Open-loop convergence Closed-loop experimental data Observer/controller design for the photon box A strict control Lyapunov control Quantum filter, separation principle and delay Conclusion: measurement-based versus coherent feedbacs Generalization to any discrete-time QND systems 1 Controlled QND Marov chains Open-loop convergence Feedbac, delay and closed-loop convergence Imperfect measurements 1 H. Amini et al. Preprint arxiv: , 2012.

3 Model of classical systems perturbation control system measure For the harmonic oscillator of pulsation ω with measured position y, controlled by the force u and subject to an additional unnown force w. x = (x 1, x 2 ) R 2, y = x 1 d dt x 1 = x 2, d dt x 2 = ω 2 x 1 + u + w

4 Feedbac for classical systems perturbation observer/controller system measure set point control feedbac Proportional Integral Derivative (PID) for d 2 dt 2 y = ω 2 y + u + w with the set point v = y sp ( ) ( ) u = K p y ysp d (y ) Kd dt y ysp Kint ysp with the positive gains (K p, K d, K int ) tuned as follows (0 < Ω 0 ω, 0 < ξ 1, 0 < ɛ 1: K p = Ω 2 0, K d = 2ξΩ 0,, K int = ɛω 3 0.

5 Feedbac for the quantum system S Key issue: bac-action due to the measurement process. Measurement-based feedbac: measurement bac-action on S is stochastic (collapse of the wave-pacet); controller is classical; the control input u is a classical variable appearing in some controlled Schrödinger equation; u depends on the past measures. Coherent feedbac: the system S is coupled to another quantum system (the controller); the composite system, S controller, is an open-quantum system relaxing to some target (separable) state (related to reservoir engineering). This tal is devoted to the first experimental realization of a measurement-based state feedbac. It has been done at Laboratoire Kastler Brossel of Ecole Normale Supérieure by the Cavity Quantum ElectroDynamics (CQED) group of Serge Haroche. 2 2 C. Sayrin et al.: Real-time quantum feedbac prepares and stabilizes photon number states. Nature, 477:73 77, 2011.

6 The closed-loop CQED experiment 3 Control input u = Ae ıφ ; measure output y {g, e}. Sampling time 80 µs long enough for numerical computations. 3 Courtesy of Igor Dotseno

7 The ideal Marov chain for the wave function ψ 4 g U R 1 U R 2 y g e U C D u 1 g g M g e M e M y M y Input u, state ψ = n 0 ψn n, output y : ψ+1/2 = M y ψ M y ψ, ψ +1 = D u ψ+1/2 with y = g (resp. e) with probability M g ψ 2 (resp. M e ψ 2 ); ( ) measurement Kraus operators M g = cos φ0 N+φ R 2 and ( ) M e = sin φ0 N+φ R 2 : M gm g + M em e = 1 with N = a a = diag(0, 1, 2,...) the photon number operator; displacement unitary operator (u R): D u = e ua ua with a = upper diag( 1, 2,...) the photon annihilation operator. 4 S. Haroche and J.M. Raimond. Exploring the Quantum: Atoms, Cavities and Photons. Oxford Graduate Texts, 2006.

8 The ideal Marov chain for the density operator ρ = ψ ψ Diagonal elements of ρ, ρ nn = n ρ n = ψ n 2, form the photon number distribution. D u M g ρ M gd u ( ) y = g with probability p g, = Tr Tr M g ρ M g ρ +1 = D u M e ρ M ed u ( ) y = e with probability p e, = Tr Tr M e ρ M e ( ) M g ρ M g ( ) M e ρ M e Displacement unitary operator (u R): D u = e ua ua with a = upper diag( 1, 2,...) the photon annihilation operator. ( ) Measurement Kraus operators M g = cos φ0 N+φ R 2 and ( ) M e = sin φ0 N+φ R 2 : M gm g + M em e = 1 with N = a a = diag(0, 1, 2,...) the photon number operator.

9 Open-loop behavior (u = 0) An experimental open-loop trajectory starting from coherent state ρ 0 = ψ 0 ψ 0 with n = 3 photons: ψ 0 = e n/2 n n n 0 n! n. A fast convergence towards n n for some n, followed by a slow relaxation towards vacuum 0 0 : decoherence due to finite photon life time around 70 ms (not included into the ideal model). Open-loop stability of ρ +1 = My ρ M y Tr(M y ρ M y ) convergence when φ 0 /π is irrational 5 for any n, ρ nn explaining this fast = n ρ n is a martingale: E ( ρ nn +1 ρ ) = ρ nn ; almost all realizations starting from ρ 0 converge towards a photon number state n n ; the probability to converge towards n n is given by the initial population ρ nn 0. This convergence characterizes a Quantum Non Demolition (QND) measurement of photons (counting photons without destroying them). 5 H. Amini et al., IEEE Trans. Automatic Control, in press, 2012.

10 Closed-loop experimental data Stabilization around 3-photon state Initial state coherent state with n = 3 photons State estimation via a quantum filter of state ρ est. Lyapunov state feedbac u = f (ρ ) stabilizing towards n n ρ is replaced by its estimate ρ est in the feedbac (quantum separation principle) Sampling period 80 µs Experience imperfections: detection efficiency 40% detection error rate 10% delay: 4 steps truncation to 9 photons finite photon life time atom occupancy 30%

11 Fidelity as control Lyapunov function In 6 we propose the following stabilizing state feedbac law based on the fidelity towards the target state n, u = f (ρ) =: Argmin υ [ ū,ū] V (D υ ρd υ) where V (ρ) = 1 F( n n, ρ) = 1 ρ n n and ū > 0 is small. Two important issues. The state ρ is not directly measured; output delay is of 4 steps: it was solved by a quantum filter taing into account the delay. V is maximum and equal to 1 for any ρ = n n with n n: no distinction between n = n + 1 (close to the target) and n (far from the target). This issue has been solved by changing the Lyapunov function V. 6 I. Dotseno et al.: Quantum feedbac by discrete quantum non-demolition measurements: towards on-demand generation of photon-number states. Physical Review A80:013805, 2009.

12 Lyapunov-based feedbac (goal photon number n) 7 V (ρ) = ( ) n ɛ n ρ n 2 + σ n n ρ n is a strict control Lyapunov function with ɛ > 0 small enough, n ν=1 1 ν 1, if n = 0; ν n 2 σ n = ν=n+1 1 ν 1, if n [1, n 1]; ν 2 0, if n = n; n ν= n+1 1 ν + 1, if n [ n + 1, + [, ν 2 ) and the feedbac u = f (ρ) =: Argmin V ɛ (D υ ρd υ (ū > 0 small). υ [ ū,ū] In closed-loop, V (ρ) becomes a strict super-martingale: E (V (ρ +1 ρ ) = V (ρ ) Q(ρ ) with Q(ρ) continuous, positive and vanishing only when ρ = n n. This feedbac law yields global stabilization for any finite dimensional approximation consisting in truncation to n max < + photons. global approximate stabilization for n max = +. 7 H. Amini et al.: CDC-2011.

13 The control Lyapunov function used for the photon box n max = 9. Coefficients σ n of the control Lyapunov function photon number n V (ρ) = ( 9 n=0 ɛ n ρ n 2 + σ n n ρ n )

14 Global approximate stabilization (n max = + ) 8 ) The feedbac u = Argmin V ɛ (D υ ρd υ ensures a strict υ [ ū,ū] closed-loop Lyapunov function V ɛ (ρ) = ( ) ɛ n ρ n 2 + σ n n ρ n n 0 with σ n log n, for n large (high photon-number cut-off). For any η > 0 and C > 0, exist ɛ > 0 and ū > 0 (small), such that, for any initial value ρ 0 with V ɛ (ρ 0 ) C, ρ n n converges almost surely towards a number inside [1 η, 1]. With Tr (ρ ) = 1, and ρ = ρ 0, this means, that almost surely, for large enough, ρ is close (wea-* topology) to the goal Foc state ρ = n n. 8 R. Somaraju, M. Mirrahimi, P.R.: CDC

15 Design of the strict control Lyapunov function 9 Exploit open-loop stability: for each n, n ρ n is a martingale; V (ρ) = 1 2 n n ρ n 2 is a super-martingale with E (V (ρ +1 ) / ρ ) = V (ρ ) Q(ρ ) where Q(ρ) 0 and Q(ρ) = 0 iff, ρ is a Foc state. For closing the loop tae σ n such that u σ n n D u ρd u n n 1. is strongly convex for ρ = n n 2. is strongly concave for ρ = n n, n n. This is achieved by inverting the Laplacian matrix associated to the control Hamiltionan H = ı(a a ). Remember that D u = e ıuh. 9 H. Amini et al., CDC 2011,

16 Estimation of ρ from the past measures y ν via a quantum filter ρ +1 = D u M y ρ M y D u ) Tr (M y ρ M y +1 = D u M y ρ est ( Tr M y ρ est ρ est M y D u ) M y Assume we now ρ and u. Outcome of measure no, y, defines the jump operator M y and we can compute ρ +1. Quantum filter and real-time estimation: initialize the estimation ρ est to some initial value ρ est 0 and update at step with measured jumps y and the nown controls u. Quantum separation principle for stabilization towards a pure state 10 : assume that the feedbac u = f (ρ) ensures global asymptotic convergence towards a pure state; then, if er(ρ est 0 ) er(ρ 0), the feedbac u = f (ρ est ) ensures also global asymptotic convergence towards the same pure state. 10 Bouten, van Handel, 2008.

17 A modified quantum filter with a measure delayed by one step Without delay the stabilizing feedbac reads M y ρ u = Argmin V (D est ) M y v Tr(M y ρ est M y ) D v With delay, we have only access to y 1 and the stabilizing feedbac uses the Kraus map K(ρ) = M g ρm g + M e ρm e: ( ) u = Argmin V D v K(ρ est )D v This is the same feedbac law but with another state estimation at step : K(ρ est M ) instead of y ρ est Tr(M y ρ est M y M y ). System theoretical interpretation: K(ρ est ) stands for the the prediction of cavity state at step. This prediction is in average (expectation value) since y {g, e} can tae two values. Quantum physics interpretation: K(ρ est ) corresponds to tracing over the atom that has already interacted with the cavity (entangled with cavity state) but that has not been measured at step. A delay of two steps involves two iterations of such Kraus maps,...

18 Conclusion: measurement-based versus coherent feedbac. Classical state-feedbac stabilization: continuous time systems with QND measurement (possible extension of M. Mirrahimi and R. van Handel, SIAM JOC, 2007), filtering stability (Belavin seminal contributions, see also van Handel,...). Stabilization by coherent feedbac: similarly to the Watt regulator where a mechanical system is controlled by another one, the controller is a quantum system coupled to the original one (Mabuchi, Nurdin, Gough, James, Petersen,...); related to quantum circuit theory (see last chapters of Gardiner-Zoller boo and the courses of Michel Devoret at Collège de France); Coherent feedbac is closely related to reservoir engineering: exploit and design the measurement process (here operators M µ ) and its intrinsic bac-action to ensure convergence of the ensemble-average dynamics towards a unique pure state (Ticozzi, Viola,... )

19 Watt regulator: a classical analogue of quantum coherent feedbac. 11 The first variations of speed δω and governor angle δθ obey to Third order system d dt δω = aδθ d 2 dt 2 δθ = Λ d dt δθ Ω2 (δθ bδω) with (a, b, Λ, Ω) positive parameters. d 3 δω = Λ d 2 δω Ω 2 d dt 3 dt 2 dt δω abω2 δω = 0 Characteristic polynomial P(s) = s 3 + Λs 2 + Ω 2 s + abω 2 with roots having negative real parts iff Λ > ab: governor damping must be strong enough to ensure asymptotic stability of the closed-loop system. 11 J.C. Maxwell: On governors. Proc. of the Royal Society, No.100, 1868.

20 Reservoir engineering stabilizing Schrödinger cats for the photon box 12 (a) Wigner functions of the various states that can be produced by such reservoir based on composite dispersive/resonant atom/cavity interaction. (b) (c) (d) 0.6 W(γ) (e) (f) (g) (h) 0 Im(γ) Re(γ) 3 12 Sarlette et al: PRL 107:010402,2011 and PRA to appear in 2012.

21 Control of a QND Marov chain with delay τ ρ +1 = M u τ µ (ρ ) =: M u τ µ ρ M u τ ( Tr M u τ µ ρ M u τ ) µ µ To each measurement outcome µ is attached the Kraus operator Mµ u C d d depending on µ and also on a scalar control input u R. For each u, m µ=1 Mu µ Mµ u = I, and we have the Kraus map K u (ρ) = m µ=1 Mu µρmµ u µ is a random variable ( taing values µ ) in {1,, m} with probability p u τ µ,ρ = Tr M u τ µ ρ M u τ µ. For u = 0, the measurement operators Mµ 0 are diagonal in the same orthonormal basis { n n {1,, d}}, therefore Mµ 0 = d n=1 c µ,n n n with c µ,n C. For all n 1 n 2 in {1,, d}, there exists µ {1,, m} such that c µ,n1 2 c µ,n2 2.

22 Open-loop convergence ρ +1 = M 0 µ (ρ ) For any initial condition ρ 0, with probability one, ρ converges to one of the d states n n with n {1,, d}. the probability of convergence towards the state n n depends only on ρ 0 and is given by n ρ 0 n. Proof based on the martingales n ρ n the super-martingale V (ρ) := n with Q(ρ) = 1 4 ( ) 2 n ρ n 2 satisfying E (V (ρ +1 ) ρ ) V (ρ ) = Q(ρ ) 0 ( n,µ,ν p0 µ,ρpν,ρ 0 cµ,n 2 n ρ n pµ,ρ 0 ) 2 cν,n 2 n ρ n p. ν,ρ 0 Q(ρ) = 0 iff exists n {1,..., d} such that ρ = n n.

23 Feedbac stabilization of ρ +1 = M u τ µ (ρ ) towards n n V 0 (ρ) = d n=1 σ n n ρ n with σ n 0 chosen such that σ n = 0 and for any n n, the second-order u-derivative of V 0 (K u ( n n )) at u = 0 is strictly negative (K u is the Kraus map): set of linear equations in σ n solved by inverting an irreducible M-matrix (Perron-Frobenius theorem). The function (ɛ > 0 small enough): V ɛ (ρ) = V 0 (ρ) ɛ 2 d n=1 ( n ρ n )2 still admits a unique global minimum at n n ; for u close to 0, u V ɛ (K u ( n n )) is strongly concave for any n n and strongly convexe for n = n. The delay of τ steps: stabilize the state χ = (ρ, β 1,, β τ ) (β l control input u delayed l steps) towards χ = ( n n, 0,..., 0) using the control-lyapunov function W ɛ (χ) = V ɛ (K β 1 (K β 2 ( K βτ (ρ)...))). For ū and ɛ small enough, the feedbac ( u = f (χ ) =: argmin E (Wɛ (χ +1 ) χ, u = ξ) ) ξ [ ū,ū] ensures global stabilization towards χ.

24 Quantum separation principle Estimate the hidden state ρ by ρ est satisfying ρ est +1 = Mu τ µ (ρ est ) where ρ obeys to ρ +1 = M u τ µ (ρ ) with the stabilizing feedbac u = f (ρ est, u 1,..., u τ ) computed using ρ est instead of ρ. If er(ρ est 0 ) er(ρ 0), ρ and ρ est converge almost surely towards the target state n n. Proof based on 13 : n ρ n [0, 1], linearity of E ( ) n ρ n ρ 0, ρ est 0 versus ρ0, decomposition ρ est 0 = γρ 0 + (1 γ)ρ c 0 with γ ]0, 1[. 13 Bouten, van Handel, 2008.

25 Imperfect measurements: the new observable state ρ The left stochastic matrix η: η µ,µ 0, 1] is the probability of having the imperfect outcome µ {1,..., m } nowing that the perfect one is µ {1,..., m}. ρ = E ( ρ ρ 0, µ 0,..., µ 1, u τ,..., u τ 1 ) obeys to 14 ρ +1 = L u τ µ ( ρ ), where L u µ ( ρ) = Lu µ ( ρ) ( ) with L u µ ( ρ) = m µ=1 η µ,µmµ u ρm µ u Tr L u µ ( ρ) µ is a random variable ( taing values ) µ in {1,, m } with probability p u τ µ, ρ = Tr L u τ µ ( ρ ). E ( ρ +1 ρ = ρ, u τ = u) = K u ( ρ) Assumption: for all n 1 ( n 2 in {1,, d}, ) there ( exists ) µ {1,, m }, s.t. Tr L 0 µ ( n 1 n 1 ) Tr L 0 µ ( n 2 n 2 ). Open-loop convergence of ρ towards n n with prob. n ρ 0 n. 14 R. Somaraju et al., ACC 2012 ( ;

26 Feedbac stabilization of ρ +1 = L u τ µ ( ρ ) towards n n With the previous function V ɛ (ρ) = V 0 (ρ) ɛ d 2 n=1 ( n ρ n )2 stabilize χ = ( ρ, β 1,, β τ ) towards χ = ( n n, 0,..., 0) using the control-lyapunov function W ɛ ( χ) = V ɛ (K β 1 (K β 2 ( K βτ ( ρ)...))). For ū and ɛ small enough, the feedbac ( u = f ( χ ) =: argmin E (Wɛ ( χ +1 ) χ, u = ξ) ) ξ [ ū,ū] ensures global stabilization of χ towards χ. Since ρ = E ( ρ ρ 0, µ 0,..., µ 1, u τ,..., u τ 1 ) convergences towards the pure state n n, ρ converges also towards the same pure state.

27 Quantum separation principle Estimate the hidden state ρ by ρ est satisfying where ρ obeys to ρ est +1 = Lu τ µ with the stabilizing feedbac ( ρ est ) ρ +1 = M u τ µ (ρ ) u = f ( ρ est, u 1,..., u τ ) computed using ρ est instead of ρ. µ = µ with probability η µ,µ. Filter stability: F ( ρ ( ) ( )) 2, ρ est Tr ρ ρ est ρ is always a sub-martingale 15. If er( ρ est 0 ) er(ρ 0), ρ and ρ est converge almost surely towards the target state n n. 15 P.R., IEEE Trans. Automatic Control, 2011.

28 Closed-loop experimental data Stabilization around 3-photon state Initial state coherent state with n = 3 photons State estimation via a quantum filter of state ρ est. Lyapunov state feedbac u = f (ρ ) stabilizing towards n n ρ is replaced by its estimate ρ est in the feedbac (quantum separation principle) Sampling period 80 µs Experience imperfections: detection efficiency 40% detection error rate 10% delay 4 sampling periods truncation to 9 photons finite photon life time atom occupancy 30%

29 The left stochastic matrix for the LKB photon box 16 For each control input u, we have a total of m = 3 7 = 21 Kraus operators. The jumps are labeled by µ = (µ a, µ c ) with µ a {no, g, e, gg, ge, eg, ee} labeling atom related jumps and µ c {o, +, } cavity decoherence jumps. we have only m = 6 real detection possibilities µ {no, g, e, gg, ge, ee} corresponding respectively to no detection, a single detection in g, a single detection in e, a double detection both in g, a double detection one in g and the other in e, and a double detection both in e. µ \ µ (no, µ c ) (g, µ c ) (e, µ c ) (gg, µ c ) (ee, µ c ) (ge, µ c ) or (eg, no 1 1-ɛ d 1-ɛ d (1-ɛ d ) 2 (1-ɛ d ) 2 (1-ɛ d ) 2 g 0 ɛ d (1-η g) ɛ d η e 2ɛ d (1-ɛ d )(1-η g) 2ɛ d (1-ɛ d )η e ɛ d (1-ɛ d )(1-η g + e 0 ɛ d η g ɛ d (1-η e) 2ɛ d (1-ɛ d )η g 2ɛ d (1-ɛ d )(1-η e) ɛ d (1-ɛ d )(1-η e + gg ɛ 2 d (1-ηg)2 ɛ 2 d η2 e ɛ 2 d ηe(1-ηg) ge ɛ 2 d ηg(1-ηg) 2ɛ2 d ηe(1-ηe) ɛ2 d ((1-ηg)(1-ηe) + ee ɛ 2 d η2 g ɛ 2 d (1-ηe)2 ɛ 2 d ηg(1-ηe) 16 R. Somaraju et al.: ACC 2012 (