# Invariant measure for stochastic Schr\"odinger equations

## TL;DR

This paper investigates the invariant measures of quantum trajectories described by stochastic Schrödinger equations, proving uniqueness under certain conditions and exponential convergence, with explicit examples provided.

## Contribution

It establishes conditions for the uniqueness and exponential convergence of invariant measures for quantum trajectories governed by stochastic Schrödinger equations.

## Key findings

- Invariant measure is unique under ergodicity and purification conditions.
- Quantum trajectories converge exponentially fast to the invariant measure.
- Explicit expressions for invariant measures are derived in specific examples.

## Abstract

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 Schr\"odinger 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.