A Stability theory for stochastic processes

Stabilization of a delayed quantum system: the photon box case-study1

Abstract

We study a feedback scheme to stabilize an arbitrary photon number state in a microwave cavity. The quantum non-demolition measurement of the cavity state allows a non-deterministic preparation of Fock states. Here, by the mean of a controlled field injection, we desire to make this preparation process deterministic. The system evolves through a discrete-time Markov process and we design the feedback law applying Lyapunov techniques. Also, in our feedback design we take into account an unavoidable pure delay and we compensate it by a stochastic version of a Smith predictor. After illustrating the efficiency of the proposed feedback law through simulations, we provide a rigorous proof of the global stability of the closed-loop system based on tools from stochastic stability analysis. A brief study of the Lyapunov exponents of the linearized system around the target state gives a strong indication of the robustness of the method.

1 Introduction

In the aim of achieving a robust processing of quantum information, one of the main tasks is to prepare and to protect various quantum states. Through the last 15 years, the application of quantum feedback paradigms has been investigated by many physicists [21, 19, 5, 10, 16] as a possible solution for this robust preparation. However, most (if not all) of these efforts have remained at a theoretical level and have not been able to be give rise to successful experiments. This is essentially due to the necessity of simulating, in parallel to the system, a quantum filter [1] providing an estimate of the state of the system based on the historic of quantum jumps induced by the measurement process. Indeed, it is, in general, difficult to perform such simulations in real time. In this paper, we consider a prototype of physical systems, the photon-box, where we actually have the time to perform these computations in real time (see [6] for a detailed description of this cavity quantum electrodynamics system).

Taking into account the measurement-induced quantum projection postulate, the most practical measurement protocols in the aim of feedback control are the quantum non-demolition (QND) measurements [2, 18, 20]. These are the measurements which preserve the value of the measured observable. Indeed, by considering a well-designed QND measurement process where the quantum state to be prepared is an eigenstate of the measurement operator, the measurement process, not only, is not an obstacle for the state preparation but can even help by adding some controllability.

In [4, 9, 8] QND measures are exploited to detect and/or produce highly non-classical states of light trapped in a super-conducting cavity (see [11, chapter 5] for a description of such QED systems and [3] for detailed physical models with QND measures of light using atoms). For such experimental setups, we detail and analyze here a feedback scheme that stabilizes the cavity field towards any photon-number states (Fock states). Such states are strongly non-classical since their photon numbers are perfectly defined. The control corresponds to a coherent light-pulse injected inside the cavity between atom passages. The overall structure of the proposed feedback scheme is inspired by [7] using a quantum adaptation of the observer/controller structure widely used for classical systems (see, e.g., [12, chapter 4]). As the measurement-induced quantum jumps and the controlled field injection happen in a discrete-in-time manner, the observer part of the proposed feedback scheme consists in a discrete-time quantum filter. Indeed, the discreteness of the measurement process provides us a first prototype of quantum systems where we, actually, have enough time to perform the quantum filtering and to compute the measurement-based feedback law to be applied as the controller.

From a mathematical modeling point of view, the quantum filter evolves through a discrete-time Markov chain. The estimated state is used in a state-feedback, based on a Lyapunov design. Indeed, by considering a natural candidate for the Lyapunov function, we propose a feedback law which ensures the decrease of its expectation over the Markov process. Therefore, the value of the considered Lyapunov function over the Markov chain defines a super-martingale. The convergence analysis of the closed-loop system is, therefore, based on some rather classical tools from stochastic stability analysis [13].

One of the particular features of the system considered in this paper corresponds to a non-negligible delay in the feedback process. In fact, in the experimental setup considered through this paper, we have to take into account a delay of steps between the measurement process and the feedback injection. Indeed, there are, constantly, atoms flying between the photon box (the cavity) to be controlled and the atom-detector (typically ). Therefore, in our feedback design, we do not have access to the measurement results for the last atoms. Through this paper, we propose an adaptation of the quantum filter, based on a stochastic version of the Smith predictor [17], which takes into account this delay by predicting the actual state of the system without having access to the result of last detections.

In the next section, we describe briefly the physical system and the associated quantum Monte-Carlo model. In Section 3, we consider the dynamics of the open-loop system. We will prove, through theorem 1 that the QND measurement process, without any additional controlled injection, allows a non-deterministic preparation of the Fock states. Indeed, we will see that the associated Markov chain converges, necessarily, towards a Fock state and that the probability of converging towards a fixed Fock state is given by its population over the initial state. Also, through proposition 1, we will show that the linearized open-loop system around a fixed Fock state admits strictly negative Lyapunov exponents (see Appendix B for a definition of the Lyapunov exponent).

In Section 4, we propose a Lyapunov-based feedback design allowing to stabilize globally the delayed closed-loop system around a desired Fock state. The theorem 2 proves the almost sure convergence of the trajectories of the closed-loop system towards the target Fock state. Also, through proposition 2, we will prove that the linearized closed-loop system around the target Fock state admits strictly negative Lyapunov exponents.

Finally in Section 5, we propose a brief discussion on the considered quantum filter and by proving a rather general separation principle (theorem 3), we will show a semi-global robustness with respect to the knowledge of the initial state of the system. Also, through a brief analysis of the linearized system-observer around the target Fock state and applying the propositions 1 and 2, we show that its largest Lyapunov exponent is also strictly negative (proposition 3).

A preliminary version of this paper without delay has appeared as a conference paper [15]. The delay compensation scheme is borrowed from [6]. The authors thank M. Brune, I. Dotsenko, S. Haroche and J.M. Raimond from ENS for many interesting discussions and advices.

2 A discrete-time Markov process

As illustrated by Figure 1, the system consists in a high-Q microwave cavity, a box producing Rydberg atoms, and two low-Q Ramsey cavities, an atom detector and a microwave source. The dynamics model is discret in time and relies on quantum Monte-Carlo trajectories (see [11, chapter 4]). Each time-step indexed by the integer corresponds to atom number coming from , submitted then to a first Ramsey -pulse in , crossing the cavity and being entangled with it, submitted to a second -pulse in and finally being measured in . The state of the cavity is associated to a quantized mode. The control corresponds to a coherent displacement of amplitude that is applied via the micro-wave source between two atom passages.

In this paper we consider a finite dimensional approximation of this quantized mode and take a truncation to photons. Thus the cavity space is approximated by the Hilbert space . It admits as ortho-normal basis. Each basis vector corresponds to a pure state, called Fock state, where the cavity has exactly  photons, . In this Fock-states basis the number operator N corresponds to the diagonal matrix

 \bf N=diag(0,1,…,n\tiny max).

The annihilation operator truncated to photons is denoted by a. It corresponds to the upper -diagonal matrix filled with :

 Missing or unrecognized delimiter for \left

The truncated creation operator denoted by is the Hermitian conjugate of a. Notice that we still have , but truncation does not preserve the usual commutation that is only valid when .

Just after the measurement of the atom number , the state of the cavity is described by the density matrix belonging to the following set of well-defined density matrices:

 X={ρ∈C(n\tiny max+1)2|ρ=ρ†,\leavevmode\nobreak Tr(ρ)=1,\leavevmode\nobreak ρ≥0}. (1)

The random evolution of this state can be modeled through a discrete-time Markov process that will be described bellow (see [6] and the references therein explaining the physical modeling assumptions).

Let us denote by the control at step . Then , the cavity state after measurement of atom is given by

 ρk+1=Msk(ρk+\scriptsize12),ρk+\scriptsize12=Dαk−d(ρk) (2)

where,

• , , with operators and ( constant parameters). For any we set .

• where the unitary displacement operator is given by In open-loop, , (identity operator) and . Notice that .

• is a random variable taking the value when the atom is detected in (resp. when the atom is detected in ) with probability

 Pg,k=Tr(Mgρk+\scriptsize12M†g)(resp.\leavevmode\nobreak Pe,k=Tr(Meρk+\scriptsize12M†e)). (3)
• The control elaborated at step , , is subject to a delay of steps, being the number of flying atoms between the cavity and the detector .

We will assume through out the paper that the parameters , are chosen in order to have , invertible and such that the spectrum of and are not degenerate. This implies that the nonlinear operators and are well defined for all and that and belongs also to the state space defined by (1). Notice that and commute, are diagonal in the Fock basis and satisfy . The Kraus map associated to this Markov process is given by:

 Kα(ρ)=MgDαρD†αM†g+MeDαρD†αM†e. (4)

It corresponds to the expectation value of knowing and :

 \largeE(ρk+1\leavevmode\nobreak |\leavevmode\nobreak ρk,αk−d)=Kαk−d(ρk). (5)

3 Open loop dynamics

3.1 Simulations

We consider in this section the following dynamics

 ρk+1=Msk(ρk), (6)

obtained from (2) when . Figure 2 corresponds to 100 realizations of this Markov process with photons, and . For each realization, is initialized to the the same coherent state with as mean photon number. We observe that either tends to or . Since the ensemble average curve is almost constant, the proportion of trajectories for which tends to is given approximatively by .

3.2 Global convergence analysis

The following theorem underlies the observations made for simulations of Figure 2

Theorem 1.

Consider the Markov process obeying (6) with an initial condition defined by (1). Then

• for any , is a martingale

• converges with probability to one of the Fock state with .

• the probability to converge towards the Fock state is given by .

Proof.

Let us prove that is a martingale. Set . We have

 \largeE(Tr(ξρk+1)\leavevmode\nobreak |\leavevmode\nobreak ρk)=Pg,k%Tr(ξMgρkM†gPg,k)+Pe,k%Tr(ξMeρkM†ePe,k)=Tr(ξMgρkM†g)+Tr(ξMeρkM†e)=Tr(ρk(M†gξMg+M†eξMe)).

Since commutes with and and we have .

Considering the following function:

 Vn(ρ)=f(⟨n|ρ|n⟩),

where Notice that is -convexe, on and satisfies

 ∀(x,y,θ)∈[0,1],θf(x)+(1−θ)f(y)=θ(1−θ)2(x−y)2+f(θx+(1−θ)y). (7)

The function is increasing and convex and is a martingale. Thus is sub-martingale. Since

 ⟨n|Mg(ρ)|n⟩=cos2φnTr% (MgρM†g)⟨n|ρ|n⟩,⟨n|Me(ρ)|n⟩=sin2φnTr(MeρM†e)⟨n|ρ|n⟩

we have

 \largeE(Vn(ρk+1)\leavevmode\nobreak |\leavevmode\nobreak ρk)=Tr(MgρkM†g)f(cos2φnTr(MgρkM†g)⟨n|ρk|n⟩)+Tr(MeρkM†e)f(sin2φnTr(MeρkM†e)⟨n|ρk|n⟩)

Then (7), together with

 θ=Tr(MgρkM†g),\leavevmode\nobreak x=cos2φnTr(MgρkM†g)⟨n|ρk|n⟩,\leavevmode\nobreak y=sin2φnTr(MeρkM†e)⟨n|ρk|n⟩

yields to

 \largeE(Vn(ρk+1)\leavevmode\nobreak |\leavevmode\nobreak ρk)−Vn(ρk)=Tr(MgρkM†g)Tr% (MeρkM†e)(⟨n|ρk|n⟩)22(cos2φnTr(MgρkM†g)−sin2φnTr(MeρkM†e))2.

Thus we recover that is a sub-martingale, . We have also shown that implies that either or (assumption and invertible is used here).

We apply now the invariance theorem established by Kushner [13] (recalled in the Appendix A) for the Markov process and the sub-martingale . This theorem implies that the Markov process converges in probability to the largest invariant subset of

 {ρ∈X\leavevmode\nobreak |\leavevmode\nobreak Tr% (MgρM†g)=cos2φn or ⟨n|ρ|n⟩=0}.

But the set is invariant. It remains thus to characterized the largest invariant subset denoted by and included in .

Take . Invariance means that and belong to (the fact that and are invertible ensures that probabilities to jump with or are strictly positive for any ). Consequently . This means that . By Cauchy-Schwartz inequality,

 Tr(M4gρ)=Tr(M4gρ)% Tr(ρ)≥Tr2(M2gρ)

with equality if, and only if, and are co-linear. being non-degenerate, is necessarily a projector over an eigenstate of , i.e., for some . Since , and thus is reduced to . Therefore the only possibilities for the -limit set are or and

 Wn(ρk)=Tr(ρk|n⟩⟨n|)(1−Tr(ρk|n⟩⟨n|)\lx@stackrelk→∞⟶0in % probability.

The convergence in probability together with the fact that is a positive bounded () random process implies the convergence in expectation. Indeed

 limsupk→∞\largeE(Wn(ρk)) ≤ϵlimsupk→∞\largeP% (Wn(ρk)≤ϵ)+limsupk→∞% \largeP(Wn(ρk)>ϵ) ≤ϵ+limsupk→∞\largeP(Wn(ρk)>ϵ)≤ϵ,

where for the last inequality, we have applied the convergence in probability of towards . As the above inequality is valid for any , we have

 limk→∞\largeE(Wn(ρk))=0.

Furthermore, by the first part of the Theorem, we know that is a bounded martingale and therefore by the Doob’s first martingale convergence theorem (see the Theorem 4 of the Appendix A), converges almost surely towards a random variable . This implies that converges almost surely towards the random variable . We apply now the dominated convergence theorem

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This implies that vanishes almost surely and therefore

 Wn(ρk)=Tr(ρk|n⟩⟨n|)(1−Tr(ρk|n⟩⟨n|))\lx@stackrelk→∞⟶0almost surely.

As we can repeat this same analysis for any choice of , converges almost surely to the set of of Fock states

 {|n⟩⟨n|\leavevmode\nobreak |\leavevmode\nobreak n=0,1,…,n\tiny max},

which ends the proof of the second part.

We have shown that the probability measure associated to the random variable converges to the probability measure

 n\tiny max∑n=0pnδ(|n⟩⟨n|),

where denotes the Dirac distribution at and is the probability of convergence towards . In particular, we have

 \largeE(Tr(|n⟩⟨n|ρk))\lx@stackrelk→∞⟶pn.

But is a martingale and . Thus

 pn=⟨n|ρ0|n⟩,

which ends the proof of the third and last part.

3.3 Local convergence rate

According to theorem 1, the -limit set of the Markov process (6) is the discrete set of Fock states . We investigate here the local convergence rate around one of these Fock states denoted by for some .

Since , we can develop the dynamics (6) around the fixed point . We write with small, Hermitian and with zero trace. Keeping only the first order terms in (6), we have

 δρk+1=MskδρkM†skTr(Msk¯ρM†sk)−Tr(MskδρkM†sk)Tr(Msk¯ρM†sk)¯ρ.

Thus the linearized Markov process around the fixed point reads

where the random matrices are given by :

• with probability

• with probability .

The following proposition shows that the convergence of the linearized dynamics is exponential (a crucial robustness indication).

Proposition 1.

Consider the linear Markov chain (8) of state belonging to the set of Hermitian matrices with zero trace. Then the largest Lyapunov exponent is given by ()

 Λ=max\scriptsizen∈{0,…,n% \tiny max}n≠¯n(cos2φ¯nlog(|cosφn||cosφ¯n|)+sin2φ¯nlog(|sinφn||sinφ¯n|))

and is strictly negative: .

Proof.

Set for any . Since , we exclude here the case because . Since and are diagonal matrices, we have

 δρn1,n2k+1=an1,n2skδρn1,n2k (9)

where (resp. ) with probability (resp. ) and where and .

Denote by the Lyapunov exponent of (9) for . By the law of large numbers, we know that converges almost surely towards

 cos2φ¯n\leavevmode\nobreak log(|an1,n2g|)+sin2φ¯n\leavevmode\nobreak log(|an1,n2e|).

Thus, we have

 Λn1,n2=cos2φ¯n(log(|cosφn1||cosφ¯n|)+log(|cosφn2||cosφ¯n|))+sin2φ¯n(log(|sinφn1||sinφ¯n|)+log(|sinφn2||sinφ¯n|))

The function

 [0,π2]∋φ↦(cosφ|cosφ¯n|)cos2φ¯n\leavevmode\nobreak (sinφ|sinφ¯n|)sin2φ¯n

increases strictly from to when goes from to and decreases strictly from to when goes from to . Since , . Denote by for :

 Λn=cos2φ¯nlog(|cosφn||cosφ¯n|)+sin2φ¯nlog(|sinφn||sinφ¯n|).

Since , we have and is strictly negative. ∎

4 Feedback stabilization with delays

4.1 Feedback scheme and closed-loop simulations

Through out this section we assume that we have access at each step to the cavity state . The goal is to design a causal feedback law that stabilizes globally the Markov chain (2) towards a goal Fock state with photon(s), . To be consistent with truncation to photons, has to be far from (typically with in the simulations below).

The feedback is based on the fact that, in open-loop when , is a martingale. When [15] proves global almost sure convergence of the following feedback law

 αk=⎧⎪⎨⎪⎩ϵTr(¯ρ\leavevmode\nobreak [ρk,\bf a]) if Tr(¯ρρk)≥ηargmax|α|≤¯α(Tr(¯ρ\leavevmode\nobreak Dα(ρk)))% if Tr(¯ρρk)<η (10)

for any when are small enough. This feedback law ensures that is a sub-martingale.

When , we cannot set since will depend on and the feedback law is not causal. In [6], this feedback law is made causal by replacing by its expectation value (average prediction) knowing and the past controls , …, :

 ρ\tiny predk=Kαk−1∘…∘Kαk−d(ρk)

where the Kraus map is defined by (4).

We will thus consider here the following causal feedback based on an average compensation of the delay

 αk=⎧⎪ ⎪⎨⎪ ⎪⎩ϵTr(¯ρ\leavevmode\nobreak [ρ\tiny predk,\bf a]) if Tr(¯ρρ\tiny predk)≥ηargmax|α|≤¯α(Tr(¯ρ\leavevmode\nobreak Dα(ρ\tiny predg,k))Tr(¯ρ\leavevmode\nobreak Dα(ρ\tiny prede,k))) if Tr(¯ρρ\tiny predk)<η (11)

with

 ⎧⎨⎩ρ\tiny predg,k=Kαk−1∘…∘Kαk−d+1(MgDαk−dρkD†αk−dM†g)ρ\tiny predg,k=Kαk−1∘…∘Kαk−d+1(MeDαk−dρkD†αk−dM†e)

The closed-loop system, i.e. Markov chain (2) with the causal feedback (11) is still a Markov chain but with as state at step . More precisely, denote by this state where stands for the control delayed steps. Then the state form of the closed-loop dynamics reads

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ρk+1=Msk(Dβd,k(ρk))β1,k+1=αkβ2,k+1=β1,k⋮βd,k+1=βd−1,k. (12)

where the control law defined by (11) corresponds to a static state feedback since

 Extra open brace or missing close brace (13)

Notice that .

Simulations displayed on Figures 3 and 4 correspond to 100 realizations of the above closed-loop systems with and . The goal state contains photons and , and are those used for the open-loop simulations of Figure 2. Each realization starts with the same coherent state and . The feedback parameters appearing in (11) are as follows:

 ϵ=12¯n+1=17,η=110,¯α=1.

This simulations illustrate the influence of the delay on the average convergence speed: the longer the delay is the slower convergence speed becomes.

Remark 1.

The choice of the feedback law whenever might seem complicated for real-time simulation issues. However, this choice is only technical. Actually, any non-zero constant feedback law will seems to achieve the task here (see for instance the simulations of [6]). However, the convergence proof for such simplified control scheme is more complicated and not considered in this paper.

4.2 Global convergence in closed-loop

Theorem 2.

Take the Markov chain (12) with the feedback (11) where , and are given by (13) with . Then, for small enough and , the state converges almost surely towards whatever the initial condition is (the compact set is defined by (1)).

Proof.

It is based on the Lyapunov-type function

 V(χ)=f(Tr(¯ρρ\tiny pred))withρ\tiny pred=Kβ1∘…∘Kβd(ρ) (14)

where has already been used during the proof of theorem 1. The proof relies in 4 lemmas:

• in lemma 1, we prove an inequality showing that, for small enough , and are sub-martingales within .

• in lemma 2, we show that for small enough , the trajectories starting within the set