The paper is devoted to the study of nonlinear stochastic Schrödinger equations driven by standard cylindrical Brownian motions (NSSEs) arising from the unraveling of quantum master equations. Under the Born-Markov approximations, this class of stochastic evolutions equations on Hilbert spaces provides characterizations of both continuous quantum measurement processes and the evolution of quantum systems. First, we deal with the existence and uniqueness of regular solutions to NSSEs. Second, we provide two general criteria for the existence of regular invariant measures for NSSEs. We apply our results to a forced and damped quantum oscillator. . This reprint differs from the original in pagination and typographic detail. 1The interaction of the small system with the reservoir is simulated by L 1 = α 1 a, L 2 = α 2 a † , L 3 = α 3 N , L 4 = α 4 a 2 , L 5 = α 5 (a † ) 2 , and L 6 = α 6 N 2 , where β 1 , β 2 , β 3 are real numbers and α 1 , . . . , α 6 are complex numbers. Consider L k = 0 for all k ≥ 7. NONLINEAR STOCHASTIC SCHRÖDINGER EQUATIONS 3In Example 1, h represents the state space of a single mode of a quantized electromagnetic field and the vectors e n , with n ∈ Z + , provide the energy levels of the system. Because a destroys a photon, L 1 , L 4 model photon emissions. The operator a † describes the creation of a photon (see, e.g., [16,56]).Using (1), we can obtain a probabilistic representation of ρ t . Indeed, it is to be expected that ρ t = E|X t X t | (see, e.g., [6,31,51]). In Dirac notation, |x x|, with x ∈ h, stands for the linear operator defined by |x x|(z) = x, z x for any z ∈ h. Therefore, the probability that a measurement finds the system in the pure state x at time t ≥ 0 is E| x, X t | 2 assuming that E|X 0 X 0 | is the initial density operator and x is a vector of h of norm 1. On the other hand, the value of the observable A at time t T t (A) satisfies E X t , AX t = E X 0 , T t (A)X 0 (see, e.g., [3,34,35]). Recall that quantum observables are represented by self-adjoint operators in h.Second, X t is interpreted as the evolution of the state of a quantum system conditioned on continuous measurement (see, e.g., [4,8,19,30,51,59]). For instance, the following example describes the simultaneous monitoring of position and momentum of a quantum system whose evolution is governed by the Hamiltonian H (see, e.g., [32,57]).

We develop linear stochastic Schrödinger equations driven by standard cylindrical Brownian motions (LSSs) that unravel quantum master equations in Lindblad form into quantum trajectories. More precisely, this paper establishes the existence and uniqueness of the smooth strong solution Xt to a LSS with regular initial condition. Moreover, we obtain that the mean value of the square norm of Xt is constant. We also treat the approximation of LSSs by ordinary stochastic differential equations. We apply our results to: (i) models of quantum measurements of position and momentum; and (ii) a system formed by fermions.

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Applying probabilistic techniques we study regularity properties of quantum master equations (QMEs) in the Lindblad form with unbounded coefficients; a density operator is regular if, roughly speaking, it describes a quantum state with finite energy. Using the linear stochastic Schrödinger equation we deduce that solutions of QMEs preserve the regularity of the initial states under a general nonexplosion condition. To this end, we develop the probabilistic representation of QMEs, and we prove the uniqueness of solutions for adjoint quantum master equations. By means of the nonlinear stochastic Schrödinger equation, we obtain the existence of regular stationary solutions for QMEs, under a Lyapunov-type condition.

This paper develops weak exponential schemes for the numerical solution of stochastic differential equations (SDEs) with additive noise. In particular, this work provides first and second-order methods which use at each iteration the product of the exponential of the Jacobian of the drift term with a vector. The article also addresses the rate of convergence of the new schemes. Moreover, numerical experiments illustrate that the numerical methods introduced here are a good alternative to the standard integrators for the long time integration of SDEs whose solutions by the common explicit schemes exhibit instabilities.

The paper deals with the numerical solution of the nonlinear Itô stochastic differential equations (SDEs) appearing in the unravelling of quantum master equations. We first develop an exponential scheme of weak order 1 for general globally Lipschitz SDEs governed by Brownian motions. Then, we proceed to study the numerical integration of a class of locally Lipschitz SDEs. More precisely, we adapt the exponential scheme obtained in the first part of the work to the characteristics of certain finite-dimensional nonlinear stochastic Schrödinger equations. This yields a numerical method for the simulation of the mean value of quantum observables. We address the rate of convergence arising in this computation. Finally, an experiment with a representative quantum master equation illustrates the good performance of the new scheme.

Abstract. We address the problem of approximating numerically the solutions (Xt : t ∈ [0, T ]) of stochastic evolution equations on Hilbert spaces (h, ·, · ), with respect to Brownian motions, arising in the unraveling of backward quantum master equations. In particular, we study the computation of mean values of Xt, AXt , where A is a linear operator. First, we introduce estimates on the behavior of Xt. Then we characterize the error induced by the substitution of Xt with the solution Xt,n of a convenient stochastic ordinary differential equation. It allows us to establish the rate of convergence of E X t,n, AXt,n to E Xt, AXt , whereXt,n denotes the explicit Euler method. Finally, we consider an extrapolation method based on the Euler scheme. An application to the quantum harmonic oscillator system is included.

We study the non-linear quantum master equation describing a laser under the mean field approximation. The quantum system is formed by a single mode optical cavity and two level atoms, which interact with reservoirs. Namely, we establish the existence and uniqueness of the regular solution to the non-linear operator equation under consideration, as well as we get a probabilistic representation for this solution in terms of a mean field stochastic Schröndiger equation. To this end, we find a regular solution for the non-autonomous linear quantum master equation in Gorini-Kossakowski-Sudarshan-Lindblad form, and we prove the uniqueness of the solution to the non-autonomous linear adjoint quantum master equation in Gorini-Kossakowski-Sudarshan-Lindblad form. Moreover, we obtain rigorously the Maxwell-Bloch equations from the mean field laser equation.Keywords: Open quantum system, nonlinear quantum master equation, Maxwell-Bloch equations, quantum master equation in the Gorini-Kossakowski-Sudarshan-Lindblad form, existence and uniqueness, regular solution, Ehrenfest-type theorem, stochastic Schrödinger equation.