Algebraic Theory of Quantum Synchronization and Limit Cycles under Dissipation

TL;DR

This paper develops a comprehensive algebraic framework for understanding quantum synchronization and limit cycles in dissipative quantum systems, providing analytical tools and criteria for their existence and absence.

Contribution

It introduces necessary and sufficient algebraic conditions for quantum limit cycles and synchronization, offering exact solutions and criteria applicable to various quantum systems.

Findings

Derived algebraic criteria for quantum limit cycles and synchronization.

Provided exact analytical solutions using dynamical symmetry algebra.

Demonstrated synchronization in fermionic cold atom systems.

Abstract

Synchronization is a phenomenon where interacting particles lock their motion and display non-trivial dynamics. Despite intense efforts studying synchronization in systems without clear classical limits, no comprehensive theory has been found. We develop such a general theory based on novel necessary and sufficient algebraic criteria for persistently oscillating eigenmodes (limit cycles) of time-independent quantum master equations. We show these eigenmodes must be quantum coherent and give an exact analytical solution for all such dynamics in terms of a dynamical symmetry algebra. Using our theory, we study both stable synchronization and metastable/transient synchronization. We use our theory to fully characterise spontaneous synchronization of autonomous systems. Moreover, we give compact algebraic criteria that may be used to prove absence of synchronization. We demonstrate synchronization in several systems relevant for various fermionic cold atom experiments.