Existence, uniqueness and approximation for stochastic Schrödinger equation: the Poisson case

In quantum physics, recent investigations deal with the so-called ”quantum trajectory” theory. Heuristic rules are usually used to give rise to “stochastic Schrödinger equations” which are stochastic differential equations of non-usual type describing the physical models. These equations pose tedious problems in terms of mathematical justification: notion of solution, existence, uniqueness, justification... In this article, we concentrate on a particular case: the Poisson case. Random measure theory is used in order to give rigorous sense to such equations. We prove existence and uniqueness of a solution for the associated stochastic equation. Furthermore, the stochastic model is physically justified by proving that the solution can be obtained as a limit of a concrete discrete time physical model. Introduction Many recent developments in quantum mechanics deal with “stochastic Schrödinger equations”. These equations are classical stochastic differential equations (also called Belavkin equations) which describe random phenomena in continuous measurement theory. The solutions of these equations are called “quantum trajectories”, they give account of the time evolution of an open quantum system undergoing a continuous measurement. In quantum mechanics, the result of a measurement is inherently random, as is namely expressed by the axioms of the theory. The setup is as follow. A quantum system is characterized by a Hilbert space H (with finite or infinite dimension) and an operator 1 ρ, self-adjoint, positive, trace class with Tr[ρ] = 1. This operator is called a “state” or a “density matrix”. The measurable quantities (energy, momentum, position...) are represented by the self-adjoint operators on H and are called “observable” of the system. The accessible data are the values of the spectrum of the observable. In finite dimension for example, if A = ∑p i=0 λiPi denotes an observable where λi are the eigenvalues of A and Pi the eigenprojectors, the observation of the λi is governed by the probability law: Pρ[to observe λi] = Tr[ρ Pi]. (1) As opposed to classical system (described by classical mechanics), a quantum system is disturbed by the result of the measurement. Conditionally to the result, the reference state of the system is modified. If we have observed the eigenvalue λi, the state ρ collapses to the new state: ρ −→ ρi = PiρPi Tr[ρ Pi] (2) This is the principle of the ”wave packet reduction”. Thus the quantum trajectory theory is the study of the evolution of the state of the system undergoing a sequence of measurement. The probability theory (1) and the wave packet reduction (2) give rise to a random variable ρ which is the new reference state. This random state describes the first evolution, the second evolution is then given by a second measurement. However according to the fact that PiPj = 0 if i 6= j and PiPi = Pi, if we have observed the result λi during the first measurement, a second measurement with the same observable gives us the following result: Pρ1i [to observe λi] = 1. The principle (2) imposed the second state to be ρ = ρ almost surely and so on. The probability law described by (1) and (2) is the description of what is called a direct measurement. Such procedure is not interesting in terms of dynamics because after one measurement the evolution is stopped. The physical procedure used in order to get around this obstacle is an interaction setup. Our system interacts with another system, after the interaction, a measurement is performed on the interacting system, so we get back a partial information of the evolution of our system without destroy it. A repeated scheme of interaction is used to obtain a significant evolution. The system is in contact with a chain of identical quantum system. Each system interacts one after the other with our system during a defined time. A measurement of the same observable is performed after each interaction on the interacting system. Each measurement gives us a random modification of the reference state of our system without destruction. So the probability theory (1) and the wave packet reduction (2) allow us to describe a sequence of random state called “discrete quantum trajectory”. The probabilistic framework of this discrete model is going to be deeply studied in the section (1). This model is called repeated quantum measurement. It will be shown that the evolution of the state can be described by a discrete stochastic equation and Markov property. A continuous model in a similar framework is considered in quantum optics. The system is in contact with a continuous field (a photon stream, a Bozon field, a laser...) and 2 a continuous time measurement is performed on the interacting system. In this situation, the first rigorous results are due to Davies which have described the time evolution of the state of the system from which we observe the photon emission. Using heuristic rules one can derive stochastic equations from this description. This is the setup of the Belavkin equations. Depending on the observable which is considered, there are two type of stochastic differential equations, one is diffusive and the other is driven by a counting process. If the continuous quantum trajectory is denoted by (ρt)t representing the state of the system at time t, it satisfies either a diffusive equation: dρt = L(ρt)dt+ [ρtC ⋆ + Cρt − Tr [ρt(C + C)] ρt] dWt (3) where Wt designs a one-dimensional Brownian motion, or a jump equation: dρt = L(ρt)dt+ [ J (ρt) Tr[J (ρt)] − ρt ] (dÑt − Tr[J (ρt)]dt) (4) where Ñt is assume to be a counting process with intensity ∫ t 0 Tr(J (ρs)]ds (the different operator are going to be described in the article). These equations pose tedious problem in terms of mathematical justification. In the literature the question of existence and uniqueness of a solution is not treated. Classical theorems can not be applied directly. Furthermore the way of writing the jump-equation is not clear. How can we consider a driving process which depends on the solution? There is no intrinsic existence for such process in this way. Even the notion of solution is then not clear and a clearly probability space must be defined to give sense for such equations. Regarding the physical justification of the use of the Belavkin equation model, the mathematical framework needs a heavy analytic machinery (Von-Neumann algebra, conditional expectation in operator algebra, Fock space, quantum filtering...). This high technology contrasts with the intuition of the heuristics rules. In this article for the very first time, the continuous model is obtained as a limit of the previous discrete model. The discrete evolution equation, describing the discrete procedure of measurement, appears as a discrete analog of the continuous time stochastic equation. The physical idea behind this convergence is the following. The continuous field is considered as a chain of quantum space which interacts one after the other during a time h and a measurement is performed after each interaction. The continuous limit (h goes to zero) gives rise then to the continuous model and the Belavkin equations. In this article we shall focus on the case of jump-equation (4), the diffusive case is treated in details in [22]. The problem of the right way of writing the equation and the right notion of solution is treated with the use of random Poisson measure. In order to prove the convergence theorem we use random variable coupling and a comparison between the discrete process and a Euler scheme of the continuous time equation . This article is structured as follow. The section (1) is devoted to present the discrete model of repeated quantum measurement. The mathematical model is namely defined as well as the discrete random variable 3 sequence which gives account of the modification of the state of the studied system. We present a natural probability space attached with this sequence. It is shown that this sequence or discrete quantum trajectory is governed by a Markov property. It is namely a Markov chain which satisfies a stochastic finite difference equation. We study the continuous model in the section (2). We deal with the jump-equation. By martingale problems theory, we define a rigorous probabilistic framework to deal with such equation. The random measure theory is namely used to define the driving process (Ñt). This allows us to give a clearly sense of the notion of solution in this non-usual situation. Next the question of existence and uniqueness of a solution is treated in details. Finally the section (3) is devoted to the link between the discrete and the continuous model. The solution of the jump-equation (continuous quantum trajectory) is going to be obtained as a continuous limit of the discrete quantum trajectory. With the discrete model, we define a process which depends on a time slice h. Random coupling theory is next used to realize and to compare this discrete process and the continuous quantum trajectory in the same space. Actually the discrete process is compared to a Euler scheme of the jump-Belavkin equation. The Euler approximation is treated in details in this non-usual context. 1 Quantum repeated interactions: A Markov chain 1.1 Quantum repeated measurement As it was described in the introduction the result of a quantum measurement is inherently random and a direct measurement destroy the evolution of a system. In order to get around the ”wave packet reduction” and to observe a significant dynamic, a principle of quantum interaction is necessary. This physical procedure is used experimentally (se Haroche [10]) in quantum optics or in quantum information. A little system is then in contact with a chain of other quantum system which interacts one after the other during a defined time h. Attal-Pautrat in [3] have rigorously shown that this model is an approximation of an equivalent continuous model when h goes to zero. This model of interaction is called quantum repeated interactions, the main goal of this section is to describe the quantum measurement principle in this framework. A measurement is performed at each interaction on the interacting system; the random character of the measurement gives rise to a random sequence of state describing the little system. This section is devoted to the setup of the mathematical description of this sequence which is our discrete quantum trajectory. Let us introduce the principle of quantum repeated interaction. We denote by H0 the Hilbert space representing the system from which we study the evolution. In repeated quantum interaction the interacting field is represented as a chain of identical Hilbert spaces H. Each copy describes a quantum system which is a piece of the environment. Each one interacts with the small system H0 one after the other during a time interval h. 4 The mathematical formalism of the interaction model is the following. We shall focus on the case H0 = H = C, the general case is treated in [22]. Let us describe the first interaction. The compound system describing the interaction is given by the tensor product H0⊗H. According to the principle of quantum mechanics the evolution of the coupling system is described by an unitary operator which acts on the states of the compound system. If ρ is any state on H0⊗H, following the so-called Schrödinger picture, the effect of the interaction is: ρ → U ρU. For instance we do not come into the question of approximation and the length time interaction is not specified. But we must keep in mind that the unitary-operator depends on the time interaction, that is U can be written as U = exp(ihH) where h is the time of interaction and H is a self-adjoint operator called Hamiltonian of the system (this will be precise in the section (3)). After the first interaction, a second interaction with a second copy is considered with the same fashion. And so on. The Hilbert space describing the repeated interaction is given by the countable tensor product: TΦ = H0 ⊗ ⊗ k≥1 Hk. (5) The countable tensor product ⊗ k≥1Hk mean the following. We consider a fixed orthonormal basis of H: (Ω, X), the projector P{Ω} is called the ground state (or vacuum state) of H and the tensor product is taken with respect to Ω. In order to have precision on countable tensor product one can see [2]. The k-th interaction is described by a unitary-operator denoted by Uk. It acts like U on H0 ⊗ Hk and like the identity operator on the rest of the tensor product. If ρ is any state on TΦ, the effect of this operator is: ρ → Uk ρU k . We then define a sequence of operators acting on TΦ by: { Vk+1 = Uk+1Vk V0 = I (6) This sequence clearly describes the effect of the successive interactions and the k first interactions are described by: ρ → Vk ρ V ⋆ k . The above description is the classical setup of repeated quantum interactions. We shall now explain the measurement principle. Let us describe the measurement on the k-th piece of environment. We use the projection principle (2) to describe the measurement at the k-th interaction. According to the fact that we work in 2-dimension we consider an observable on Hk of the form A = λoP0+λ1P1. The natural ampliation as an observable TΦ is: A := k ⊗ j=0 I ⊗ (λoP0 + λ1P1)⊗ ⊗