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

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 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 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 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 . 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 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 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 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 .

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 $d$ steps between the measurement process and the feedback injection. Indeed, there are, constantly, $d$ atoms flying between the photon box (the cavity) to be controlled and the atom-detector (typically $d=5$). Therefore, in our feedback design, we do not have access to the measurement results for the $d$ last atoms. Through this paper, we propose an adaptation of the quantum filter, based on a stochastic version of the Smith predictor , which takes into account this delay by predicting the actual state of the system without having access to the result of $d$ 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 . The delay compensation scheme is borrowed from . The authors thank M. Brune, I. Dotsenko, S. Haroche and J.M. Raimond from ENS for many interesting discussions and advices.

A discrete-time Markov process

The annihilation operator truncated to $n^{\text{\tiny max}}$ photons is denoted by a. It corresponds to the upper $1$-diagonal matrix filled with $(\sqrt{1},\ldots,\sqrt{n^{\text{\tiny max}}})$:

The truncated creation operator denoted by ${\text{\bf a}}^{\dagger}$ is the Hermitian conjugate of a. Notice that we still have ${\text{\bf N}}={\text{\bf a}}^{\dagger}{\text{\bf a}}$, but truncation does not preserve the usual commutation $[{\text{\bf a}},{\text{\bf a}}^{\dagger}]=1$ that is only valid when $n^{\text{\tiny max}}=+\infty$.

Just after the measurement of the atom number $k-1$, the state of the cavity is described by the density matrix $\rho_{k}$ belonging to the following set of well-defined density matrices:

The random evolution of this state $\rho_{k}$ can be modeled through a discrete-time Markov process that will be described bellow (see and the references therein explaining the physical modeling assumptions).

$s_{k}$ is a random variable taking the value $g$ when the atom $k$ is detected in $g$ (resp. $e$ when the atom $k$ is detected in $e$) with probability

The control elaborated at step $k$, $\alpha_{k}$, is subject to a delay of $d$ steps, $d$ being the number of flying atoms between the cavity $C$ and the detector $D$.

It corresponds to the expectation value of $\rho_{k+1}$ knowing $\rho_{k}$ and $\alpha_{k-d}$:

Open loop dynamics

We consider in this section the following dynamics

2 Global convergence analysis

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

Consider the Markov process $\rho_{k}$ obeying (6) with an initial condition $\rho_{0}\in{\mathcal{X}}$ defined by (1). Then

for any $n\in\{0,\cdots,n^{\text{\tiny max}}\}$, $\text{Tr}\left(\rho_{k}\left|n\right>\left<n\right|\right)=\left<n\right|\rho_{k}\left|n\right>$ is a martingale

$\rho_{k}$ converges with probability $1$ to one of the $n^{\text{\tiny max}}+1$ Fock state $\left|n\right>\left<n\right|$ with $n\in\{0,\cdots,n^{\text{\tiny max}}\}$.

the probability to converge towards the Fock state $\left|n\right>\left<n\right|$ is given by $\text{Tr}\left(\rho_{0}\left|n\right>\left<n\right|\right)=\left<n\right|\rho_{0}\left|n\right>$.

Let us prove that $\text{Tr}\left(\rho_{k}\left|n\right>\left<n\right|\right)$ is a martingale. Set $\xi=\left|n\right>\left<n\right|$. We have

where $f(x)=\tfrac{x+x^{2}}{2}.$ Notice that $f$ is $1$-convexe, $f^{\prime}\geq\tfrac{1}{2}$ on $$ and satisfies

The function $f$ is increasing and convex and $\left<n|\rho_{k}|n\right>$ is a martingale. Thus $V_{n}(\rho_{k})$ is sub-martingale. Since

We apply now the invariance theorem established by Kushner (recalled in the Appendix A) for the Markov process $\rho_{k}$ and the sub-martingale $V_{n}(\rho_{k})$. This theorem implies that the Markov process $\rho_{k}$ converges in probability to the largest invariant subset of

with equality if, and only if, $M_{g}^{4}\rho$ and $\rho$ are co-linear. $M_{g}^{4}$ being non-degenerate, $\rho$ is necessarily a projector over an eigenstate of $M_{g}^{4}$, i.e., $\rho=\left|m\right>\left<m\right|$ for some $m\in\{0,\ldots,n^{\text{\tiny max}}\}$. Since $\text{Tr}\left(M_{g}\rho M_{g}^{{\dagger}}\right)=\cos^{2}\!\varphi_{n}>0$, $m=n$ and thus ${\mathcal{X}}_{n}$ is reduced to $\{\left|n\right>\left<n\right|\}$. Therefore the only possibilities for the $\omega$-limit set are $\text{Tr}\left(\rho\left|n\right>\left<n\right|\right)=0$ or $1$ and

The convergence in probability together with the fact that $W_{n}(\rho_{k})$ is a positive bounded ($W_{n}\in$) random process implies the convergence in expectation. Indeed

where for the last inequality, we have applied the convergence in probability of $W_{n}(\rho_{k})$ towards . As the above inequality is valid for any $\epsilon>0$, we have

Furthermore, by the first part of the Theorem, we know that $\text{Tr}\left(\rho_{k}\left|n\right>\left<n\right|\right)$ is a bounded martingale and therefore by the Doob’s first martingale convergence theorem (see the Theorem 4 of the Appendix A), $\text{Tr}\left(\rho_{k}\left|n\right>\left<n\right|\right)$ converges almost surely towards a random variable $l_{n}^{\infty}\in$. This implies that $W_{n}(\rho_{k})$ converges almost surely towards the random variable $l_{n}^{\infty}(1-l_{n}^{\infty})\in$. We apply now the dominated convergence theorem

This implies that $l_{n}^{\infty}(1-l_{n}^{\infty})$ vanishes almost surely and therefore

As we can repeat this same analysis for any choice of $n\in\{0,1,\ldots,n^{\text{\tiny max}}\}$, $\rho_{k}$ converges almost surely to the set of of Fock states

We have shown that the probability measure associated to the random variable $\rho_{k}$ converges to the probability measure

where $\delta(\left|n\right>\left<n\right|)$ denotes the Dirac distribution at $\left|n\right>\left<n\right|$ and $p_{n}$ is the probability of convergence towards $\left|n\right>\left<n\right|$. In particular, we have

which ends the proof of the third and last part.

3 Local convergence rate

According to theorem 1, the $\Omega$-limit set of the Markov process (6) is the discrete set of Fock states $\{\left|n\right>\left<n\right|\}_{n\in\{0,\ldots,n^{\text{\tiny max}}\}}$. We investigate here the local convergence rate around one of these Fock states denoted by $\bar{\rho}=\left|\bar{n}\right>\left<\bar{n}\right|$ for some $\bar{n}\in\{0,\ldots,n^{\text{\tiny max}}\}$.

Thus the linearized Markov process around the fixed point $\bar{\rho}$ reads

where the random matrices $A_{s_{k}}$ are given by :

$A_{g}=\frac{M_{g}}{\cos\varphi_{\bar{n}}}$ with probability $P_{g}=\cos^{2}\!\varphi_{\bar{n}}$

$A_{e}=\frac{M_{e}}{\sin\varphi_{\bar{n}}}$ with probability $P_{e}=\sin^{2}\!\varphi_{\bar{n}}$.

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

Consider the linear Markov chain (8) of state $\delta\rho$ belonging to the set of Hermitian matrices with zero trace. Then the largest Lyapunov exponent $\Lambda$ is given by ($\varphi_{n}=\varphi_{0}+n{\vartheta}$)

Set $\delta\rho^{n_{1},n_{2}}=\left<n_{1}|\delta\rho|n_{2}\right>$ for any $n_{1},n_{2}\in\{0,\ldots,n^{\text{\tiny max}}\}$. Since $\text{Tr}\left(\delta\rho_{k}\right)\equiv 0$, we exclude here the case $(n_{1},n_{2})=(\bar{n},\bar{n})$ because $\delta\rho_{k}^{\bar{n},\bar{n}}=-\sum_{n\neq\bar{n}}\delta\rho_{k}^{n,n}$. Since $A_{e}$ and $A_{g}$ are diagonal matrices, we have

where $s_{k}=g$ (resp. $s_{k}=e$) with probability $\cos^{2}\varphi_{\bar{n}}$ (resp. $\sin^{2}\varphi_{\bar{n}}$) and where $a_{g}^{n_{1},n_{2}}=\tfrac{\cos\varphi_{n_{1}}\cos\varphi_{n_{2}}}{\cos^{2}\varphi_{\bar{n}}}$ and $a_{e}^{n_{1},n_{2}}=\tfrac{\sin\varphi_{n_{1}}\sin\varphi_{n_{2}}}{\sin^{2}\varphi_{\bar{n}}}$.

Denote by $\Lambda^{n_{1},n_{2}}$ the Lyapunov exponent of (9) for $(n_{1},n_{2})\neq(\bar{n},\bar{n})$. By the law of large numbers, we know that $\frac{\log\left(\left|\prod_{l=0}^{l=k}a_{s_{k}}^{n_{1},n_{2}}\right|\right)}{k+1}$ converges almost surely towards

increases strictly from to $1$ when $\varphi$ goes from to $\arcsin(|\sin\varphi_{\bar{n}}|)$ and decreases strictly from $1$ to when $\varphi$ goes from $\arcsin(|\sin\varphi_{\bar{n}}|)$ to $\frac{\pi}{2}$. Since $(n_{1},n_{2})\neq(\bar{n},\bar{n})$, $\Lambda^{n_{1},n_{2}}<0$. Denote by $\Lambda^{n}=\Lambda^{n,\bar{n}}$ for $n\in\{0,\ldots,n^{\text{\tiny max}}\}$:

Since $(n_{1},n_{2})\neq(\bar{n},\bar{n})$, we have $\Lambda^{n_{1},n_{2}}\leq\max_{n\neq\bar{n}}\Lambda^{n}$ and $\Lambda=\max_{n\neq\bar{n}}\Lambda^{n}$ is strictly negative. ∎

Feedback stabilization with delays

Through out this section we assume that we have access at each step $k$ to the cavity state $\rho_{k}$. The goal is to design a causal feedback law that stabilizes globally the Markov chain (2) towards a goal Fock state $\bar{\rho}=\left|\bar{n}\right>\left<\bar{n}\right|$ with $\bar{n}$ photon(s), $\bar{n}\in\{0,\ldots,n^{\text{\tiny max}}\}$. To be consistent with truncation to $n^{\text{\tiny max}}$ photons, $\bar{n}$ has to be far from $n^{\text{\tiny max}}$ (typically $\bar{n}=3$ with $n^{\text{\tiny max}}=10$ in the simulations below).

The feedback is based on the fact that, in open-loop when $\alpha_{k}\equiv 0$, $\text{Tr}\left(\bar{\rho}\rho_{k}\right)=\left<\bar{n}|\rho_{k}|\bar{n}\right>$ is a martingale. When $d=0$, proves global almost sure convergence of the following feedback law

for any $\bar{\alpha}>0$ when $\epsilon,\eta>0$ are small enough. This feedback law ensures that $\text{Tr}\left(\bar{\rho}\rho_{k}\right)$ is a sub-martingale.

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

The closed-loop system, i.e. Markov chain (2) with the causal feedback (11) is still a Markov chain but with $(\rho_{k},\alpha_{k-1},\cdots,\alpha_{k-d})$ as state at step $k$. More precisely, denote by $\chi=(\rho,\beta_{1},\cdots,\beta_{d})$ this state where $\beta_{l}$ stands for the control $\alpha$ delayed $l$ steps. Then the state form of the closed-loop dynamics reads

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

Notice that $\rho^{\text{\tiny pred}}_{k}=\rho^{\text{\tiny pred}}_{g,k}+\rho^{\text{\tiny pred}}_{e,k}$.

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

The choice of the feedback law whenever $\text{Tr}\left(\bar{\rho}\rho^{\text{\tiny pred}}_{k}\right)<\eta$ 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 ). However, the convergence proof for such simplified control scheme is more complicated and not considered in this paper.

2 Global convergence in closed-loop

It is based on the Lyapunov-type function

where $f(x)=\frac{x+x^{2}}{2}$ has already been used during the proof of theorem 1. The proof relies in 4 lemmas:

The combination of lemmas 2, 3 and 4 shows then directly that $\chi_{k}$ converges almost surely towards $\bar{\chi}$. We detail now these 4 lemmas. ∎

For $\epsilon>0$ small enough and for $\chi_{k}$ satisfying $\text{Tr}\left(\bar{\rho}\rho^{\text{\tiny pred}}(\chi_{k})\right)\geq\eta$,

For $\alpha$ small the Baker-Campbell-Hausdorff formula yields

Since $\alpha_{k}=\epsilon\text{Tr}\left(\bar{\rho}[\rho^{\text{\tiny pred}}_{k},{\text{\bf a}}]\right)$, we get

Thus for $\epsilon>0$ small enough and uniformly in $\rho^{\text{\tiny pred}}_{k}\in{\mathcal{X}}$

Using the fact that $f$ is increasing and $f(x+y)\geq f(x)+y/2$ for any $x,y>0$, we get

When $\eta>0$ is small enough, any state $\chi_{k}$ satisfying the inequality $\text{Tr}\left(\bar{\rho}\rho^{\text{\tiny pred}}(\chi_{k})\right)<\eta$ yields a new state $\chi_{k+1}$ such that $\text{Tr}\left(\bar{\rho}\rho^{\text{\tiny pred}}(\chi_{k+1})\right)\geq 2\eta$.

Since $M_{g}$ and $M_{e}$ are invertible, there exists $\zeta\in]0,1[$ such that, for any $\chi$, $\text{Tr}\left(\rho^{\text{\tiny pred}}_{g}(\chi)\right)\geq\zeta$ and $\text{Tr}\left(\rho^{\text{\tiny pred}}_{e}(\chi)\right)\geq\zeta$ ($\rho^{\text{\tiny pred}}_{g}$ and $\rho^{\text{\tiny pred}}_{e}$ are defined in (13)). Denote by ${\mathcal{X}}_{\zeta}$ the compact set of Hermitian semi-definite positive matrices with trace in $[\zeta,1]$: for any $\chi$, $\rho^{\text{\tiny pred}}_{g}(\chi)$ and $\rho^{\text{\tiny pred}}_{e}(\chi)$ are in ${\mathcal{X}}_{\zeta}$. Let us prove first that, for any $\rho_{g},\rho_{e}\in{\mathcal{X}}_{\zeta}$

Take now $\eta<\tfrac{\delta}{2}$ and $\chi_{k}$ such that $\text{Tr}\left(\bar{\rho}\rho^{\text{\tiny pred}}(\chi_{k})\right)\leq\eta$. According to (11), $\alpha_{k}$ is chosen as an argument of

and thus $p>1-\frac{1-2\eta}{1-\eta}=\frac{\eta}{1-\eta}$. ∎

Sample paths $\chi_{k}$ remaining in the set $\{\text{Tr}\left(\bar{\rho}\rho^{\text{\tiny pred}}(\chi)\right)\geq\eta\}$ converges almost surely to $\bar{\chi}$ as $k\rightarrow\infty$.

We apply first the Kushner’s invariance theorem to the Markov process $\chi_{k}$ with the sub-martingale function $V(\chi_{k})$. It ensures convergence in probability towards ${\mathcal{I}}$ the largest invariant set attached to this sub-martingale (see appendix). Let us prove that ${\mathcal{I}}$ is reduced to $\{\bar{\chi}\}$.

Invariance associated $\alpha\equiv 0$ implies that $\beta_{1}=\ldots=\beta_{d}=0$. Thus the above equality reads

Since $\text{Tr}\left(M_{e}\rho M_{e}^{\dagger}\right)+\text{Tr}\left(M_{g}\rho M_{g}^{\dagger}\right)=1$, we recover $\text{Tr}\left(M_{g}\rho M_{g}^{\dagger}\right)=\cos^{2}\!\varphi_{\bar{n}}$ the same condition as the one appearing at the end of the proof of theorem 1. Similar invariance arguments combined with $\text{Tr}\left(\bar{\rho}\rho\right)>0$ imply then $\rho=\bar{\rho}$. Thus ${\mathcal{I}}$ is reduced to $\{\bar{\chi}\}$.

where $\|\cdot\|$ is any norm on the $\chi$-space. The continuity of $\chi\mapsto\text{Tr}\left(\bar{\rho}\rho^{\text{\tiny pred}}(\chi)\right)$ implies that, $\forall\delta>0$,

As $0\leq\text{Tr}\left(\bar{\rho}\rho^{\text{\tiny pred}}(\chi)\right)\leq 1$, we have

The process $\text{Tr}\left(\bar{\rho}\rho^{\text{\tiny pred}}(\chi_{k})\right)$ is a bounded sub-martingale and therefore, by Theorem 4 of the Appendix A, we know that it converges for almost all trajectories remaining in the set $\{\text{Tr}\left(\bar{\rho}\rho^{\text{\tiny pred}}(\chi)\right)\geq\eta\}$. Calling the limit random variable $\text{fid}_{\infty}$, we have by dominated convergence theorem

This trivially proves that $\text{fid}_{\infty}\equiv 1$ almost surely and finishes the proof of the Lemma. ∎

3 Convergence rate around the target state

Around the target state $\bar{\chi}=(\bar{\rho},0,\ldots,0)$ the closed-loop dynamics reads

Set $\chi=\bar{\chi}+\delta\chi$ with $\delta\chi=(\delta\rho,\delta\beta_{1},\ldots,\delta\beta_{d})$ small. Computations based on

yield the following linearized closed-loop system

where $s_{k}\in\{g,e\}$, the random matrices $A_{s_{k}}$ are given by $A_{g}=\frac{M_{g}}{\cos\varphi_{\bar{n}}}$ with probability $P_{g}=\cos^{2}\!\varphi_{\bar{n}}$ and $A_{e}=\frac{M_{e}}{\sin\varphi_{\bar{n}}}$ with probability $P_{e}=\sin^{2}\!\varphi_{\bar{n}}$.

Set $\delta\rho^{n_{1},n_{2}}_{k}=\left<n_{1}|\delta\rho_{k}|n_{2}\right>$ for any $n_{1},n_{2}\in\{0,\ldots,n^{\text{\tiny max}}\}$. Since $\text{Tr}\left(\delta\rho_{k}\right)\equiv 0$, we exclude here the case $(n_{1},n_{2})=(\bar{n},\bar{n})$ because $\delta\rho_{k}^{\bar{n},\bar{n}}=-\sum_{n\neq\bar{n}}\delta\rho_{k}^{n,n}$. When $(n_{1},n_{2})$ does not belong to $\{(\bar{n}-1,\bar{n}),(\bar{n}+1,\bar{n}),(\bar{n},\bar{n}-1),(\bar{n},\bar{n}+1)\}$, we recover the open-loop linearized dynamics (9):

where $s_{k}=g$ (resp. $s_{k}=e$) with probability $\cos^{2}\!\varphi_{\bar{n}}$ (resp. $\sin^{2}\!\varphi_{\bar{n}}$) and where $a_{g}^{n_{1},n_{2}}=\tfrac{\cos\varphi_{n_{1}}\cos\varphi_{n_{2}}}{\cos^{2}\!\varphi_{\bar{n}}}$ and $a_{e}^{n_{1},n_{2}}=\tfrac{\sin\varphi_{n_{1}}\sin\varphi_{n_{2}}}{\sin^{2}\!\varphi_{\bar{n}}}$. A direct adaptation of the proof of proposition 1 shows that the largest Lyapounov exponent $\Lambda_{0}$ of this dynamics is strictly negative and given by

For $(n_{1},n_{2})\in\{(\bar{n}-1,\bar{n}),(\bar{n}+1,\bar{n}),(\bar{n},\bar{n}-1),(\bar{n},\bar{n}+1)\}$, we just have to consider $x=\delta\rho^{\bar{n},\bar{n}-1}$ and $y=\delta\rho^{\bar{n}+1,\bar{n}}$ since $\delta\rho$ is Hermitian. Set $z_{j,k}=\delta\beta_{j,k}$. We deduce from (17) that the process $X_{k}=(x_{k},y_{k},z_{1,k},\ldots,z_{d,k})$ is governed by

where $s_{k}=g$ (resp. $s_{k}=e$) with probability $\cos^{2}\!\varphi_{\bar{n}}$ (resp. $\sin^{2}\!\varphi_{\bar{n}}$) and

Take $\mu>0$ to be defined later, set $\sigma=|\cos{\vartheta}|\in]0,1[$ and consider

A direct computation exploiting (18) yields

Take $\mu=\tfrac{\sqrt{\bar{n}}+\sqrt{\bar{n}+1}}{\sigma^{d-1}}$, then

Because $\sigma<1$, for $\epsilon>0$ small enough ($\epsilon<\tfrac{1-\sigma}{2(\bar{n}+1)}$) the norm $V(X_{k})$ is a super-martingale converging exponentially almost surely towards zero. Thus the largest Lyapunov exponent of the linear Markov chain (18) is strictly negative. To conclude, we have proved the following proposition:

Consider the linear Markov chain (17). For small enough $\epsilon>0$, its largest Lyapunov exponent is strictly negative.

Quantum filter and separation principle

The feedback law (11) requires the knowledge of $(\rho_{k},\beta_{1,k},\cdots,\beta_{d,k})$. When the measurement process is fully efficient and the jump model (2) admits no error, the Markov system (12) represents a natural choice for the quantum filter to estimate the value of $\rho$. Indeed, we define the estimator $\chi^{\text{\tiny est}}_{k}=(\rho^{\text{\tiny est}}_{k},\beta_{1,k},\cdots,\beta_{d,k})$ satisfying the dynamics

Note that, similarly to any observer-controller structure, the jump result, $s_{k}=g$ or $e$, is the output of the physical system (2) but the feedback control $\alpha_{k}$ is a function of the estimator $\rho^{\text{\tiny est}}$. Indeed, $\alpha_{k}$ is defined as in (11):

where the predictor’s state $\rho^{\text{\tiny pred,est}}_{k}$ is defined as follows:

We will see through this section that, even if do not have any a priori knowledge of the initial state of the physical system, the choice of the feedback law through the above quantum filter can ensure the convergence of the system towards the desired Fock state. Indeed, we prove a semi-global robustness of the feedback scheme with respect to the choice of the initial state of the quantum filter.

Through the next subsection, we establish a sort of separation principle implying this semi-global robustness of the closed-loop system with respect to the initial state of the filter equation. Also through the short Subsection 5.3 we provide a heuristic analysis of the local convergence rate of the filter equation around the target Fock state.

2 A quantum separation principle

We consider the joint system-observer dynamics defined for the state $\Xi_{k}=(\rho_{k},\rho^{\text{\tiny est}}_{k},\beta_{1,k},\ldots,\beta_{d,k})$:

We have the following result, a quantum version of the separation principle ensuring asymptotic stability of observer/controller from stability of the observer and of the controller separatly.

Consider any closed-loop system of the form (21), where the feedback law $\alpha_{k}$ is a function of the quantum filter: $\alpha_{k}=g(\rho^{\text{\tiny est}}_{k},\beta_{1,k},\ldots,\beta_{d,k})$. Assume moreover that, whenever $\rho^{\text{\tiny est}}_{0}=\rho_{0}$ (so that the quantum filter coincides with the closed-loop dynamics (12)), the closed-loop system converges almost surely towards a fixed pure state $\bar{\rho}$. Then, for any choice of the initial state $\rho^{\text{\tiny est}}_{0}$, such that $\text{ker}\rho^{\text{\tiny est}}_{0}\subset\text{ker}\rho_{0}$, the trajectories of the system converge almost surely towards the same pure state: $\rho_{k}\rightarrow\bar{\rho}$.

where $\{s_{j}\}_{j=0}^{k-1}$ denotes the sequence of $k$ first jumps. Finally, through simple computations, we have

At this point, we apply the assumption $\text{ker}\rho^{\text{\tiny est}}_{0}\subset\text{ker}\rho_{0}$ and therefore, one can find a constant $\gamma>0$ and a well-defined density matrix $\rho_{0}^{c}$ in ${\mathcal{X}}$, such that

Now, considering the system (21) initialized at the state $(\rho^{\text{\tiny est}}_{0},\rho^{\text{\tiny est}}_{0},0,\ldots,0)$, we have by the assumptions of the theorem and applying dominated convergence theorem:

This implies the almost sure convergence of the physical system towards the pure state $\bar{\rho}$. ∎

3 Local convergence rate for the quantum filter

Let us linearize the system-observer dynamics (21) around the equilibrium state $\bar{\Xi}=(\bar{\rho},\bar{\rho},0,\ldots,0)$. Set $\Xi=\bar{\Xi}+\delta\Xi$ with $\delta\Xi=(\delta\rho,\delta\rho^{\text{\tiny est}},\delta\beta_{1},\ldots,\delta\beta_{d})$ small, $\delta\rho$ and $\delta\rho^{\text{\tiny est}}$ Hermitian and of trace 0. We have the following dynamics for the linearized system (adaptation of (17)):

where $s_{k}\in\{g,e\}$, the random matrices $A_{s_{k}}$ are given by $A_{g}=\frac{M_{g}}{\cos\varphi_{\bar{n}}}$ with probability $P_{g}=\cos^{2}\!\varphi_{\bar{n}}$ and $A_{e}=\frac{M_{e}}{\sin\varphi_{\bar{n}}}$ with probability $P_{e}=\sin^{2}\!\varphi_{\bar{n}}$.

At this point, we note that by considering $\widetilde{\delta\rho}_{k}=\delta\rho^{\text{\tiny est}}_{k}-\delta\rho_{k}$, we have the following simple dynamics:

Indeed, as the same control laws are applied to the quantum filter and the physical system, the difference between $\delta\rho^{\text{\tiny est}}_{k}$ and $\delta\rho_{k}$ follows the same dynamics as the linearized open-loop system (8). But, we know by the proposition 1 that this linear system admits strictly negative Lyapunov exponents. This triangular structure, together with the convergence rate analysis of the closed-loop system in proposition 2, yields the following proposition whose detailed proof is left to the reader:

Consider the linear Markov chain (22). For small enough $\epsilon>0$, its largest Lyapunov exponent is strictly negative.

Conclusion

We have analyzed a measurement-based feedback control allowing to stabilize globally and deterministically a desired Fock state. In this feedback design, we have taken into account the important delay between the measurement process and the feedback injection. This delay has been compensated by a stochastic version of a Smith predictor in the quantum filtering equation.

In fact, the measurement process of the experimental setup admits some other imperfections. These imperfections can, essentially, be resumed to the following ones: 1- the atom-detector is not fully efficient and it can miss some of the atoms (about 20%); 2- the atom-detector is not fault-free and the result of the measurement (atom in the state $g$ or $e$) can be inter-changed (a fault rate of about 10%); 3- the atom preparation process is itself a stochastic process following a Poisson law and therefore the measurement pulses can be empty of atom (a pulse occupation rate of about 40%). The knowledge of all these rates can help us to adapt the quantum filter by taking into account these imperfections. This has been done in , by considering the Bayesian law and providing numerical evidence of the efficiency of such feedback algorithms assuming all these imperfections.

References

Appendix A Stability theory for stochastic processes

We recall here the Doob’s first martingale convergence theorem, the Doob’s inequality and the Kushner’s invariance theorem. For detailed discussions and proofs we refer to (Chapter 2) and (Sections 8.4 and 8.5).

The following theorem characterizes the convergence of bounded martingales:

Let $\{X_{n}\}$ be a Markov chain on state space ${\mathcal{X}}$ and suppose that

this is $X_{n}$ is a submartingale. Assume furthermore that ($x^{+}$ is the positive part of $x$)

Now, we recall two results that are often referred as the stochastic versions of the Lyapunov stability theory and the LaSalle’s invariance principle.

For the statement of the second theorem, we need to use the language of probability measures rather than the random processes. Therefore, we deal with the space $\mathcal{M}$ of probability measures on the state space ${\mathcal{X}}$. Let $\mu_{0}=\sigma$ be the initial probability distribution (everywhere through this paper we have dealt with the case where $\mu_{0}$ is a Dirac on a state $\rho_{0}$ of the state space of density matrices). Then, the probability distribution of $X_{n}$, given initial distribution $\sigma$, is to be denoted by $\mu_{n}(\sigma)$. Note that for $m\geq 0$, the Markov property implies: $\mu_{n+m}(\sigma)=\mu_{n}(\mu_{m}(\sigma)).$

Appendix B Lyapunov exponents of linear stochastic processes

where $A_{s_{k}}$ is a random matrix taking its values inside a finite set $\{A_{1},\ldots,A_{m}\}$ with a stationary probability distribution for $s_{k}$ over $\{1,\ldots,m\}$. Then