Poisson and Diffusion Approximation of Stochastic Schrödinger Equations with Control

Quantum trajectories” are solutions of stochastic differential equations of nonusual type. Such equations are called “Belavkin” or “Stochastic Schrödinger Equations” and describe random phenomena in continuous measurement theory of Open Quantum System. Many recent investigations deal with the control theory in such model. In this article, stochastic models are mathematically and physically justified as limit of concrete discrete procedures called “Quantum Repeated Measurements”. In particular, this gives a rigorous justification of the Poisson and diffusion approximation in quantum measurement theory with control. Furthermore we investigate some examples using control in quantum mechanics. Introduction Recent developments and applications in quantum mechanics deal with “Stochastic Schrödinger Equations” (also called Belavkin Equations [12]). These equations are classical stochastic differential equations; they describe random phenomena in continuous measurement theory. The solutions of these equations are called “Quantum Trajectories”, they give account of the time evolution of reference states of open quantum system undergoing a continuous measurement. A classical physical model ([12]) used in quantum optics is the one of an interaction between a two-level atom and a continuous field which describes the environment. The evolution of the small system (the atom) is observed by performing a quantum measurement. Because of the ”Wave Packet Reduction”, an indirect continuous measurement is then performed on the field in order not to destroy the information contained in the atom; we get then partial data of this system. These partial information are rendered by a stochastic evolution of the reference state of the small system. Without control, one 1 consider essentially two types of stochastic models described by stochastic differential equations. They are called classical Belavkin Equations or Stochastic Schrödinger Equations and their solutions are called “classical quantum trajectories”. 1. The “diffusive equation” (Homodyne detection experiment) is given by dρt = L(ρt)dt+ [ρtC ⋆ + Cρt − Tr (ρt(C + C)) ρt]dWt (1) where Wt describes a one-dimensional Brownian motion. 2. The “jump equation” (Resonance fluorescence experiment) is dρt = L(ρt)dt+ [ J (ρt) Tr[J (ρt)] − ρt ] (dÑt − Tr[J (ρt)]dt) (2) where Ñt is a counting process with stochastic intensity ∫ t 0 Tr[J (ρs)]ds. First mathematical results concerning the evolution of an atom system undergoing a continuous measurement are due to Davies in [11]. He gives namely a description of the time evolution of the state of an atom system from which we study the detection of photon emission. With this description, heuristic rules can be used to derive stochastic Schrödinger equations. A way to obtain rigorous result is the use of Quantum Filtering Theory ([6],[7]). Such theory needs a high analytic machinery using Von Neumann algebra, conditional expectation in operator algebra and fine properties of the Non-commutative Probability Theory. Many recent applications, in quantum optics or modern engineering, needs an exterior control in the interaction and measurement experiences. Such investigations was motivated by precision and optimization constraints in order to obtain reliable performance in experimental physics. Control actions can be of very different types and can be resumed by a continuous modification of the parameters of experiences. For example in quantum optics, the modification of the intensity of a laser are used to monitor the evolution of atoms. Such procedures are called “open loop control” or deterministic control. Finer strategies needs the use of stochastic control. Following the evolution of the system and the different results of measurements, the interaction is modified in order to control the progress of experiences. As the evolution of a system undergoing a measurement is stochastic (cf equations (1) and (2)), control, in such situations, own a random character. This is called “closed loop control” or “feedback control”. Usually, the evolution of an open quantum system is described by a unitary-evolution described in continuous time by a process (Vt) which satisfies a quantum Langevin equations. Mathematically, the control effect is then rendered by the modification of this unitary-evolution. The technical difficulties of Quantum Filtering Theory are increased by the introduction of control. In [27],[8],[7], it was investigated how classical Belavkin equations (1) and (2) are modified in presence of control with such tools. A more intuitive approach in terms of physical and mathematical justification consists to use a discrete model called “Quantum Repeated Interactions”. The setup is as follows. 2 The field is represented as an infinite chain of identical small quantum system (a spin chain for example). Each pieces of environment interacts one after the other with the atom during a time interval of length h. Such discrete interaction model is shown in particular to converge (h → 0) to continuous time models described by Quantum Langevin Equations (cf [4]). At each interaction, a measurement on the environment is performed. The random results of observations give rise to discrete stochastic processes describing the procedure. As regards just the small system, its evolution during the successive measurements is described by classical Markov chains called “discrete quantum trajectories”. In [23] and [22] it is shown that these discrete trajectories (without control), converge in distribution to solutions of classical Belavkin Equations (1) and (2). In this article, we present a way to introduce control effects in the discrete model of quantum repeated interactions. Stochastic models of quantum measurement with control are then justified by convergence theorems in the same way of [23] and [22]. Next we investigate some applications. This article is structured as follows. The first section is devoted to the discrete model of quantum repeated interactions with control. We define the probabilistic framework which describe the random character of repeated quantum measurements. We show that the quantum trajectories which describe the evolution of the small system are classical controlled Markov chains. Next we focus on a particular case of a two-level atom in contact with a spin chain. We show that quantum trajectories describing this model satisfy finite difference stochastic equations which appears as approximations of continuous time stochastic differential equations. We present next asymptotic conditions to come into the problems of convergence. The second section is then devoted to continuous models. From the approximation model of a two level system of Section 1, we establish Belavkin equations with control. Depending on the observable which is measured, it gives rise of two different continuous model. Next we justify such models by proving that the solutions of Belavkin equations can be obtained as limit of discrete quantum trajectories. In the last section we present some applications of the continuous model. On the one hand, we study a concrete example of an atom monitored by a laser. By modelling a suitable interaction discrete model and by adapting the result of Section 2, we obtain a stochastic model for this concrete example. On the other hand we come into the problem of ”optimal control” which uses general stochastic control results. This is applied to our subject in the diffusive case. 1 Discrete Controlled Quantum Trajectories We make here precise the mathematical framework of quantum repeated measurements with control. It is shown that the principle of quantum repeated measurements gives rise to Markov chains which are called ”discrete controlled quantum trajectories”.