Large Time Behaviour and Convergence Rate for Non Demolition Quantum Trajectories

Introduction

In quantum optics, indirect measurements are often used . Usually a system is probed by light beams (direct photodetection, homodyne and heterodyne detection schemes) or conversely, atoms probe a photon field trapped in a cavity. Such experiments are promising towards the manipulation of quantum states . They are designed to extract information from a quantum system on site and without destroying it. The idea is to avoid direct interaction of the quantum system with a macroscopic apparatus (photo detector, screen …). Instead the physical setup is the following: a quantum system $\mathcal{S}$ (from which we want to extract information) is put in interaction with an auxiliary quantum system $\mathcal{E}$. After interaction, a measurement on $\mathcal{E}$ is performed. Due to the entanglement between $\mathcal{S}$ and $\mathcal{E}$, the result of the measurement holds some information on $\mathcal{S}$. Conditionally to this result one can compute the evolution of $\mathcal{S}$.

One of the best example of such setups is Serge Haroche’s group experiment at LKB. They have successfully implemented a scheme of repeated interactions and measurements which allows to measure the number of photons in a cavity (without destroying the photons). The principle consist in putting the system $\mathcal{S}$ (the cavity photon field) in contact with a sequence of auxiliary systems (Rydberg atoms) $\mathcal{E}_{k}$ which interact one after the other with $\mathcal{S}$. After each interaction a measurement on the atom $\mathcal{E}_{k}$ which has just finish to interact is performed. Such a procedure, called repeated quantum indirect measurements, allows to monitor the system $\mathcal{S}$ and to have an estimation of the number of photons inside the cavity.

A particular feature in the Serge Haroche’s group experiment is that only Quantum Non Demolition (QND) measurement are performed. Such a scheme is at the cornerstone of the mathematical study of the long time behavior of $\mathcal{S}$. In , the authors show that the state of the system $\mathcal{S}$ converges when the number of interactions tends to infinity. More precisely they show a convergence which is compatible with the wave function collapse postulate. In particular it is shown that the state of $\mathcal{S}$ behaves in infinite time as if a direct Von Neumann measurement on $\mathcal{S}$ would have been performed at time . Essentially these results concern discrete time model where the time of interaction $\tau$ between $\mathcal{S}$ and a piece $\mathcal{E}_{k}$ is fixed. They apply to general nondemolition measurement scheme of which Serge Haroche’s group experiment is an example (see and references therein).

When the time of interaction goes to zero, this yields to continuous time models. In it has been shown that quantum repeated interactions model are a powerful approximation of the so called Quantum Langevin equation. In , it is shown that the continuous time approximation ($\tau$ goes to zero) of repeated quantum indirect measurements lead to jump-diffusion stochastic differential equations (see also ). Such equations are namely the equations which describe the evolution of a quantum system undergoing indirect continuous measurements . They are called stochastic master equations and their solutions quantum trajectories.

In this article, we focus on the stochastic master equations describing general continuous time quantum nondemolition measurement. Our main purpose is to describe the long time behavior of the state of $\mathcal{S}$ when the time $t$ goes to infinity. In particular if $(\rho(t))$The process $(\rho(t))$ is actually the quantum trajectory describing the evolution of the state of $\mathcal{S}$ which undergoes indirect continuous measurement describes the stochastic evolution of $\mathcal{S}$ undergoing indirect QND measurement, we show that $(\rho(t))$ converges to a pure state $|\Upsilon\rangle\langle\Upsilon|$. This convergence is obtained by studying in detail the quantities $(q_{\alpha}(t))$ defined by

The article is structured as follows. In Section 1, we introduce the stochastic models describing the generic stochastic master equations. Next we present the particular case of nondemolition stochastic master equations. This allows us to define the processes $(q_{\alpha}(t))$. We then study the properties of these processes and show that they are bounded martingales. Section 2 is devoted to the main convergence theorem. From the martingale and boundedness property of $(q_{\alpha}(t))$, we conclude that these processes converge almost surely. This allows us to present the main convergence result and to define precisely the random variable $\Upsilon$. Next, using appropriate Girsanov change of measure, we show that this convergence is exponentially fast. Finally we investigate the problem of estimation.

Non destructive quantum trajectories

This section is devoted to present the continuous time stochastic processes which describe quantum trajectories. As announced these stochastic processes are solutions of particular type of jump-diffusion stochastic differential equations.

Before presenting the SDEs, let us introduce some notations. The quantum system is represented by a finite dimensional Hilbert space denoted by $\mathcal{H}$. We denote the set of density matrices by $\mathcal{S}(\mathcal{H})=\{\rho\in\mathcal{B}(\mathcal{H}),\rho\geq 0,\textrm{Tr}[\rho]=1\}$. A density matrix represents a general system mixed state. A system in a pure state $|\phi\rangle\in\mathcal{H}$ corresponds to a special case where the density matrix is the projector onto $|\phi\rangle$. In this situation the corresponding density matrix is $\rho_{|\phi\rangle}=|\phi\rangle\langle\phi|$. In the rest of the article, if not specified, the term state refers to a density matrix.

Let us consider a family $C_{i},i=0,\ldots,n$ of operators in $\mathcal{B}(\mathcal{H})$ and let $H\in\mathcal{B}(\mathcal{H})$ such that $H=H^{*}$ i.e. $H$ is a Hermitian operator. On $\mathcal{S}(\mathcal{H})$, we introduce the following functions:

for all states $\rho\in\mathcal{S}(\mathcal{H})$.

where $\rho_{0}\in\mathcal{S}(\mathcal{H})$.

The equation (2) is called a stochastic master equation and its solution is called a quantum trajectory.

Equation (2) is a ”generic”One can generalize these equations by introducing time dependent and random coefficients SDE describing the evolution of a system undergoing continuous indirect measurements. Results of existence and uniqueness of the solution of (2) can be found in .

In Eq. (2), the operator $L$ is a usual Lindblad operator . These operators appear in the definition of the master equation in the Markovian approach of Open Quantum Systems.

From Eq. (2), one can introduce the measurement record counting processes:

These processes are counting processes with stochastic intensity given by

During an experiment, these processes would correspond to the counting measurement records an experimenter would obtain. For example $\hat{N}_{i}(t)$ could correspond to the total number of photons arrived on a detector up to time $t$. In section 2.3 we discuss in more details what would be the equivalent for a continuous measurement record.

In terms of $\hat{N}_{i}(t)$, Eq. (2) can be written as

In the next section we introduce a nondemolition condition on this evolution and study the large time behaviour of $(\rho(t))$.

2 Non demolition condition

A measurement process is called nondemolition if one can find a basis $\mathcal{P}$ of $\mathcal{H}$ such that any element of $\mathcal{P}$ is unmodified by the measurement process. If, at a given time, a system is in one of the basis states, it will remain in it at any future time with probability one.

Let $\mathcal{P}$ be a basis of $\mathcal{H}$. A measurement process fulfills a nondemolition condition for $\mathcal{P}$ if any state of $\mathcal{P}$ is stable under the measurement process: for any $|\alpha\rangle\in\mathcal{P}$ if at time $s$, $\rho(s)=|\alpha\rangle\langle\alpha|$ then for any time $t>s$, $\rho(t)=\rho(s)$, almost surely.

The stable states $|\alpha\rangle\langle\alpha|,\alpha\in\mathcal{P}$, are called pointer states.

We assume from now on that $H$ and the $C_{i}$’s are diagonal in the basis $\mathcal{P}$. The main result we prove in this section is the equivalence between this diagonal assumption and a nondemolition condition for $\mathcal{P}$.

Attached to these decompositions we introduce the following quantities which will be used further

Here $\overline{z}$ is $z$ complex conjugate. In the basis $\mathcal{P}$, we denote a matrix $A=(A_{\alpha\beta})_{\alpha,\beta}$.

Our study of $(\rho(t))$ is mainly based on the study of the diagonal elements of $(\rho(t))$ in the basis $\mathcal{P}$. If a direct measurement identifying all the pointer states would have been performed at time $t=0$, then the system after this measurement would have been in the pointer state $|\alpha\rangle\langle\alpha|$ with probability $\rho_{\alpha\alpha}(0)={\rm Tr}[\rho(0)|\alpha\rangle\langle\alpha|]$. If the same direct measurement is performed at time $t>0$ the probability to obtain the same system pointer state is $\rho_{\alpha\alpha}(t)={\rm Tr}[\rho(t)|\alpha\rangle\langle\alpha|]$. So, the evolution of the diagonal elements of $(\rho(t))$ in $\mathcal{P}$ gives us information on the distribution of such direct measurement outcomes.

In the sequel, for all $\alpha\in\mathcal{P}$, we use the notations

As a preliminary, let us prove that the $(q_{\alpha}(t))$, $\alpha\in\mathcal{P}$ are $(\mathcal{F}_{t})$ martingales solutions of Dade-Doleans type of SDEs.

where $\displaystyle\langle r_{i}(t)\rangle=\sum_{\gamma}r(i|\gamma)q_{\gamma}(t)$ and $\displaystyle\langle\theta_{i}(t)\rangle=\sum_{\gamma}\theta(i|\gamma)q_{\gamma}(t)$, for all $t\geq 0$.

In particular, the stochastic processes $(q_{\alpha}(t)),\alpha\in\mathcal{P}$ are $(\mathcal{F}_{t})$ martingales. As solution of Dade-Doleans type of SDEs, they can be expressed in the following form

In the case where $\theta(i|\alpha)=0$, for some $i=p+1,\ldots,n$, if a jump of the corresponding $\hat{N}_{i}$ occurs at some time $t$, one can see that $q_{\alpha}(t)$ vanishes ($q_{\alpha}(u)=0,$ for all $u\geq t$). In this situation in order to give a sense to the second expression one can consider that $\ln(0)=-\infty$ and $\exp(-\infty)=0$. Nevertheless the second expression will be used only when $\theta(i|\alpha)>0$ for all $\alpha\in\mathcal{P}$ and for all $i=p+1,\ldots,n$. Let us stress that in this situation, if $q_{\alpha}(0)\neq 0$, we have $q_{\alpha}(t)>0,$ for all $t\geq 0$. Although, in section 2.2, we discuss some interesting properties of $(q_{\alpha}(t))$ when $\theta(i|\alpha)=0$ for some $i$.

In order to obtain the expression (5), we have to compute $d\rho_{\alpha\alpha}(t)$ by using (3). To this end we have to plug the diagonal condition into the expression of $L$, $H_{i}$, $J_{i}$ and $v_{i}$. This way we can compute the following expression.

From now on we only need the expression of the diagonal elements $\alpha=\beta$ in the pointer basis. In other words the stochastic differential equations for $(q_{\alpha}(t))$ do not depend on the off diagonal elements of the system state. This way remarking also that $(L(\rho))_{\alpha\alpha}=0$ for any $\alpha\in\mathcal{P}$, it is easy to derive Equation (5).

The second part follows from the fact that the processes $(W_{i}(t))$ and $(\hat{N}_{i}(t)-\int_{0}^{t}v_{i}(\rho(s-))ds)$ are $(\mathcal{F}_{t})$ martingales. This way the stochastic processes $(q_{\alpha}(t))$ are local martingales but since they are bounded they are true martingales. The solution (6) is the usual expression of the solution of a Dade-Doleans SDE. ∎

We are now equipped to prove the equivalence between the diagonal assumption and the nondemolition condition.

The quantum stochastic master equation (3) fulfills a nondemolition condition for $\mathcal{P}$ if and only if $H$ and all the operators $C_{i}$ are diagonal in the basis $\mathcal{P}$.

Let us first prove that the diagonal condition imply the nondemolition condition. We need to prove that if at time $s$, $\rho(s)=|\alpha\rangle\langle\alpha|$, then at any time $t>s$, $\rho(t)=|\alpha\rangle\langle\alpha|$ almost surely. Since $\rho(t)$ is a Markovian process, we can, without loss of generality, set $s=0$.

Put $\rho(0)=|\alpha\rangle\langle\alpha|$ for some $\alpha\in\mathcal{P}$. We have $q_{\beta}(0)=0$ for any $\beta\neq\alpha,\beta\in\mathcal{P}$ and ${\rm Tr}[\rho(t)]=1$ for any time $t$. Then, looking at (6), we have $q_{\beta}(t)=0$ and $q_{\alpha}(t)=1$ for all $t\geq 0$ almost surely. It implies

We now prove that the nondemolition condition implies the diagonal assumption. If at time $s$, $\rho(s)=|\alpha\rangle\langle\alpha|$, then for any time $t>s$, $\rho(t)=|\alpha\rangle\langle\alpha|,$ almost surely. The expectation of $\rho(t)$ conditioned on $\rho(s)=|\alpha\rangle\langle\alpha|$ with $\alpha\in\mathcal{P}$ is:

Since $(W_{i}(t))$ and $(\hat{N}_{i}(t)-\int_{0}^{t}v_{i}(\rho(u-))du)$ are martingales,

At this stage, since $\rho(u)=\rho(s)$ for all $u\geq s$ almost surely, we get

Let $\beta\in\mathcal{P},\beta\neq\alpha$. The condition $L(|\alpha\rangle\langle\alpha|)_{\beta\beta}=0$ implies $(C_{i})_{\beta\alpha}=0$ for all $i=0,\ldots,n$. Using this result, the condition $L(|\alpha\rangle\langle\alpha|)_{\beta\alpha}=0$ implies $H_{\beta\alpha}=0$. Hence, $H$ and all the $C_{i}$’s must be diagonal in the basis $\mathcal{P}$. ∎

In the next section we use the martingale property of $(q_{\alpha}(t))$ to study its long time behavior. Before let us prove that this martingale property is equivalent to the nondemolition condition.

The processes $(q_{\alpha}(t))$ are $(\mathcal{F}_{t})$ martingales if and only if (3) fulfills a nondemolition condition for $\mathcal{P}$.

We already proved that the nondemolition condition implies the martingale property of $(q_{\alpha}(t))$. Let us prove the converse.

We suppose that for any $\alpha\in\mathcal{P}$, the process $(\rho_{\alpha\alpha}(t))$ is a martingale. This assumption is true only if the drift part of (3) for $\rho_{\alpha\alpha}(t)$ is null whatever is the initial state. Hence, for any arbitrary $\rho\in\mathcal{S}(\mathcal{H})$, we must have $L(\rho)_{\alpha\alpha}=0$. Take $\rho=|\beta\rangle\langle\beta|$ with $\alpha\neq\beta$. As seen earlier, the condition $L(\rho)_{\alpha\alpha}=0$, implies that the $C_{i}$’s must be diagonal in the basis $\mathcal{P}$. Now put $|\alpha+\beta\rangle=\frac{1}{\sqrt{2}}(|\alpha\rangle+|\beta\rangle)$. Take $\rho=|\alpha+\beta\rangle\langle\alpha+\beta|$. The condition $L(\rho)_{\alpha\alpha}=0$ implies $H_{\alpha\beta}-H_{\beta\alpha}=0$. Put $|\alpha+i\beta\rangle=\frac{1}{\sqrt{2}}(|\alpha\rangle+i|\beta\rangle)$. Take $\rho=|\alpha+i\beta\rangle\langle\alpha+i\beta|$. The condition $L(\rho)_{\alpha\alpha}=0$ implies $H_{\alpha\beta}+H_{\beta\alpha}=0$. We can thus conclude that $H$ must also be diagonal in the basis $\mathcal{P}$.

As proved earlier this diagonal property is equivalent to the nondemolition condition. ∎

The next section is devoted to the large time behavior of $(q_{\alpha}(t))$ and to interpretations of the obtained convergence in terms of wave function collapse.

Convergence and wave function collapse

In this section we show the almost sure convergence of the processes $(q_{\alpha}(t))$ when $t$ goes to infinity. Under some non degeneracy conditions, we can identify the limit random variables $(q_{\alpha}(\infty))$. More precisely, in this context we show that $q_{\alpha}(\infty)$ is equal to $1$ for a pointer $\Upsilon\in\mathcal{P}$ and for the others. The pointer state $\Upsilon$ is a random variable and we find its distribution. We next show that this imply that $(\rho(t))$ converges almost surely to one of the pointer states $|\alpha\rangle\langle\alpha|$. In particular, we show that the probability for the limit pointer state to be $|\alpha\rangle\langle\alpha|$ is $q_{\alpha}(0)=\rho_{\alpha\alpha}(0)$. This is what is predicted by the von Neumann projection postulate if a direct measurement would have been performed at time . Thinking of the limit state in terms of random variable, in the limit $t\to\infty$, the system state is a random variable with the same law as the one predicted by von Neumann projection postulate for a direct measurement at time .

In the following subsections we present some useful properties implied by this convergence.

Let us express our non degeneracy condition

Assumption (ND): For any $(\alpha,\beta)$ with $\alpha\neq\beta$ there exists $i\in\{0,1,\ldots,n\}$ such that

either $r(i|\alpha)\neq r(i|\beta)$ if $i\leq p$

or $\theta(i|\alpha)\neq\theta(i|\beta)$ if $i>p.$

Under Assumption (ND), there exist random variables $q_{\alpha}(\infty),\alpha\in\mathcal{P}$ which take values in $\{0,1\}$ such that

The random variables $q_{\alpha}(\infty),\alpha\in\mathcal{P}$, satisfy

As a consequence there exists a random variable $\Upsilon$ with values in $\mathcal{P}$ such that

where $[q_{\alpha}(t),q_{\alpha}(t)]$ corresponds to the stochastic bracket of $(q_{\alpha}(t))$. We then have

Again by the dominated convergence Theorem, the quantities

converge when $t$ goes to infinity. Then from (2.1) it follows that necessarily

The above $L^{2}$ convergences imply the almost sure convergences up to an extraction. More precisely, there exist subsequences $(t^{i}_{n})$ such that for all $i=0,\ldots,n,\lim_{n\to\infty}t^{i}_{n}=\infty$ and such that almost surely

Since the processes $(q_{\alpha}(t))$ converge almost surely, by uniqueness of the almost sure limits and using the boundedness of $\langle\theta_{i}(t)\rangle$, we can conclude that almost surely, for all $\alpha\in\mathcal{P}$

Then it follows that, almost surely, for all $\alpha\in\mathcal{P}$

This way, almost surely, for all $\alpha\neq\beta$

It follows that, almost surely, for all $\alpha\neq\beta$

Finally using Assumption (ND) one can conclude that

For the second part, let us come back to the definition of $\Upsilon$. This defines a random variable taking values in the set of pointer states $\mathcal{P}$ (for $\omega\in\Omega\setminus\Omega^{\prime}$, we can put $\Upsilon(\omega)=\psi$, where $\psi\notin\mathcal{P}$, this will appear with probability 0). It is then clear that

This result is crucial in the following. In particular it will allow us to use a Girsanov transformation ”in infinite time horizon”.

2 Exponential rate of convergence

In this section, we study the convergence speed of the processes $(q_{\alpha}(t))$. In particular, we shall show an exponential convergence. To this end, we study the following quantities

Since $(q_{t}(\alpha))$ can vanish in the case where $\theta(i|\alpha)=0$ for some $i$, this quantity can be finite or infinite. Furthermore, we limit our study to pointer states $\alpha,\gamma\in\mathcal{P}$ such that $q_{\alpha}(0)\neq 0$ and $q_{\gamma}(0)\neq 0$. Remark, if $q_{\beta}(0)=0$ for some pointer state $\beta$ we have $q_{\beta}(t)=0$ for any time $t$.

First, let us start by studying the case where $q_{\alpha}(t)>0$ and $q_{\gamma}(t)>0$, for all $t\geq 0$, almost surely. As already discussed this is ensured when for any $i\in\{p+1,\ldots,n\}$, $\theta(i|\alpha)>0$ and $\theta(i|\gamma)>0$. In this case, using (5), we have almost surely, for all $t\geq 0$,

for all $\gamma\in\mathcal{P}$ such that $q_{\gamma}(0)\neq 0$.

and in terms of filtration we get the following Radon Nykodim formula

The following theorem expresses the exponential convergence speed towards $\Upsilon$.

Assume Assumption (ND) is satisfied. Assume that $\alpha,\gamma\in\mathcal{P}$ are such that $q_{\alpha}(0)\neq 0$, $q_{\gamma}(0)\neq 0$, and such that $\theta(i|\alpha)>0$, $\theta(i|\gamma)>0$ for all $i=p+1,\ldots,n.$ Then, we have

More generally, in terms of the random variable $\Upsilon$, we have

More precisely each term of the sums is nonnegative. Now from Assumption (ND), the quantity (37) is equal to zero if and only if $\alpha=\Upsilon$. This underlines the exponential rate convergence towards $\Upsilon$.

Let us assume the interaction of our measurement apparatus with the system involves only one hermitian operator $C_{i}=C$. In other words $n=0$. Let us also assume we can choose either to have a continuous process as our measurement record or a counting process. In other words, either the quantum stochastic master equation is

In the diffusive case (38), the convergence rate (37) is equal to $(c(\alpha)-c(\Upsilon))^{2}$. In the counting case (39) it is equal to $-c(\Upsilon)^{2}[\ln(c(\alpha)^{2}/c(\Upsilon)^{2})+1-c(\alpha)^{2}/c(\Upsilon)^{2}]$. A simple study shows that

So the choice of a counting process may lead to a higher convergence rate. But it comes at a price. Suppose $C$ has two different eigenvalues of equal norm: $c(\alpha)\neq c(\beta)$, $|c(\alpha)|=|c(\beta)|$. The non degeneracy assumption (ND) is not fulfilled for the jump equation (39) whereas it is fulfilled for the diffusive equation (38).

We now study the situation where it exists $i\in\{p+1,\ldots,n\}$ and $\alpha\in\mathcal{P}$, such that $\theta(i|\alpha)=0$. In this case we shall study the following stopping times

Assume that $q_{\alpha}(0)\neq 0$, we have $q_{\alpha}(T(\alpha))=0$ and $q_{\alpha}(t)>0$, for all $t<T(\alpha)$ as well as $q_{\alpha}(t)=0$, for all $t\geq T(\alpha)$. This way, if $T(\alpha)<\infty$, the process $(q_{\alpha}(t))$ converges to zero in finite time.

We have the following proposition which describes the distribution of $T(\alpha),\alpha\in\mathcal{P}$.

Let $\alpha\in\mathcal{P}$ such that there exists $i\in\{p+1,\ldots,n\}$ such that $\theta(i|\alpha)=0$. Then,

where $\displaystyle\lambda(\alpha|\gamma)=\sum_{i,\,\,s.t.\,\,\theta(i|\alpha)=0}\theta(i|\gamma)$.

Let us note that taking $t$ goes to infinity we get

In the next section, we address the problem of convergence when one does not have access to the process $(\rho(t))$ but only to the measurement records.

3 Stability

Usually, in experiments the initial state $\rho_{0}$ of the system $\mathcal{H}$ is unknown (this is sometimes that we want to estimate). This way we cannot have access to the quantum trajectory $(\rho(t))$. Nevertheless we have still access to the results given by the measurement apparatus. These results are directly connected to the quantum trajectory (2). In terms of processes, the results of the measurement are described by output processes in the following way. The observed processes are given by

for the diffusive part of the evolution and by

for the counting processes. The quantities $y_{i}(t)$ and $\hat{N}_{i}(t)$ are the measurements recorded by the apparatus. In an homodyne or heterodyne detection scheme, $y_{i}(t)$ would represent the detected photo current integrated up to time $t$ whereas, in a direct photodetection scheme, $\hat{N}_{i}(t)$ would be the number of photons detected up to time $t$. The quantum trajectory can be expressed as follows

Let us treat the case where $\theta(i|\alpha)\neq 0,$ for all $\alpha\in\mathcal{P}$ and for all $i=p+1,\ldots,n$. Eq. (46) are still Dade-Doleans exponential and the solution of (46) are given by

for all $\alpha\in\mathcal{P}$ and for all $t\geq 0$.

which shows the stability of the estimation and the convergence rate is the same.

Acknowledgments

C. P. acknowledges financial support from the ANR project HAM-MARK, N∘ ANR-09-BLAN-0098-01.

T. B. thanks Denis Bernard for helpful discussions and acknowledges financial support from ANR contracts ANR-2010-BLANC-0414.01 and ANR-2010-BLANC-0414.02.

References