Limit cycles in periodically driven open quantum systems

Paul Menczel, Kay Brandner

Research output: Contribution to journalArticleScientificpeer-review

2 Citations (Scopus)


We investigate the long-time behavior of quantum N-level systems that are coupled to a Markovian environment and subject to periodic driving. As our main result, we obtain a general algebraic condition ensuring that all solutions of a periodic quantum master equation with Lindblad form approach a unique limit cycle. Quite intuitively, this criterion requires that the dissipative terms of the master equation connect all subspaces of the system Hilbert space during an arbitrarily small fraction of the cycle time. Our results provide a natural extension of Spohn's algebraic condition for the approach to equilibrium to systems with external driving. Moreover, our theory leads to a rigorous condition for the emergence of dissipative discrete time crystals and covers also classical, periodically modulated Markov jump processes.

Original languageEnglish
Article number43LT01
Pages (from-to)1-9
JournalJournal of Physics A: Mathematical and Theoretical
Issue number43
Publication statusPublished - 30 Sep 2019
MoE publication typeA1 Journal article-refereed


  • asymptotics
  • cyclic machines
  • functional analysis and operator theory
  • mathematical physics
  • nonequilibrium systems
  • open quantum systems

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