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Invariant Measure for Stochastic Schrödinger Equations

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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 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 toward this invariant measure. We illustrate our results with examples where we can derive explicit expressions for the invariant measure.

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Acknowledgements

The research of T.B., Y.P. and C.P. has been supported by the ANR project StoQ ANR-14-CE25-0003-01. The research of T.B. has been supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02. The research of M.F. was supported in part by funding from the Simons Foundation and the Centre de Recherches Mathématiques, through the Simons-CRM scholar-in-residence program. Y.P. acknowledges the support of ANR project NonStops ANR-17-CE40-0006 and of the Cantab Capital Institute for the Mathematics of Information at the University of Cambridge.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, 31062, Toulouse Cedex 9, France

    T. Benoist & C. Pellegrini

  2. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    M. Fraas

  3. Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405, Orsay, France

    Y. Pautrat

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  1. T. Benoist
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  2. M. Fraas
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  3. Y. Pautrat
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  4. C. Pellegrini
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Correspondence to T. Benoist.

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Communicated by Alain Joye.

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Benoist, T., Fraas, M., Pautrat, Y. et al. Invariant Measure for Stochastic Schrödinger Equations. Ann. Henri Poincaré 22, 347–374 (2021). https://doi.org/10.1007/s00023-020-01001-4

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