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Repeated Interaction Processes in the Continuous-Time Limit, Applied to Quadratic Fermionic Systems

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Abstract

We study a class of Lindblad equations on finite-dimensional Fermionic systems. The model is obtained as the continuous-time limit of a repeated interaction process between Fermionic systems with quadratic Hamiltonians, a setup already used by Platini and Karevski for the one-dimensional XY-model. We prove a necessary and sufficient condition for the convergence to a unique stationary state, which is similar to the Kalman criterion in control theory. Several examples are treated, including a spin chain with interactions at both ends.

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Acknowledgements

I thank my advisor Stéphan Attal for introducing this subject to me and helping me throughout the redaction of this article and my co-advisor Claude-Alain Pillet for interesting remarks and discussions about Fermionic systems. I also thank the two anonymous referees for their very detailed work and numerous suggestions.

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Correspondence to Simon Andréys.

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Work supported by ANR Project “StoQ” N\({}^\circ \)ANR-14-CE25-0003.

Communicated by Alain Joye.

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Andréys, S. Repeated Interaction Processes in the Continuous-Time Limit, Applied to Quadratic Fermionic Systems. Ann. Henri Poincaré 21, 115–154 (2020). https://doi.org/10.1007/s00023-019-00852-w

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