Basic properties of nonlinear stochastic Schrödinger equations driven by Brownian motions

Abstract

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]).

Existence of invariant measures for the stochastic damped Schrödinger equation

“…on the space-time domain [0, ∞) × R d with an additive noise, by establishing the existence of an invariant measure and the asymptotic tightness of solutions of the equation. We emphasize that unlike in [MR,Assumption H1], our problem in the whole space R d does not allow any compact embeddings.…”

Regularity of Solutions to Linear Stochastic Schrödinger Equations

“…Notice that an unproved relation similar to (19) is used in the proof of Theorem 3 of [14] and is implicitly assumed in the sketchy proof of Theorem 2 of [15].…”

“…Since the backward Kolmogorov equation in infinite dimensions associated with (1) is the Heisenberg equation of motion, X t characterizes the time evolution of quantum observables (see, e.g., [1,14]). On the other hand, (1) plays a role in the development of the non-linear stochastic Schrödinger equations (see, e.g., [11,19]). Using the theory of globally Lipschitz stochastic evolution equation, Barchielli, Holevo, Paganoni and Zucca covered in [2,3] the existence and uniqueness of the strong solution of (1) when G and L k , with k ∈ N, are bounded operators.…”

Heisenberg evolution of quantum observables represented by unbounded operators

“…A slight change in the proof of Proposition 35 shows that (29) is still true if Hypothesis 2 holds and A * belongs to L((Dom(C), · C ), h).…”