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Article Dans Une Revue Annales Henri Poincaré Année : 2021

Invariant measure for stochastic Schrödinger equations

Résumé

Quantum trajectories are Markov processes that describe the time-evolution of a quantum system undergoing continuous indirect measurement. Mathematically, they are defined as solutions of the so-called "Stochastic Schrödinger Equations", which are nonlinear stochastic differential equations driven by Poisson and Wiener processes. This paper is devoted to the study of the invariant measures of quantum trajectories. Particularly, we prove that the invariant measure is unique under an ergodicity condition on the mean time evolution, and a "purification" condition on the generator of the evolution. We further show that quantum trajectories converge in law exponentially fast towards this invariant measure. We illustrate our results with examples where we can derive explicit expressions for the invariant measure.

Dates et versions

hal-02263935 , version 1 (06-08-2019)

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Tristan Benoist, Martin Fraas, Yan Pautrat, Clément Pellegrini. Invariant measure for stochastic Schrödinger equations. Annales Henri Poincaré, 2021. ⟨hal-02263935⟩
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