Multi-Mode Quantum Synchronization

Abstract

The dynamics of interacting many-particle systems on quantum mechanical scale is a broad subject of research in modern physics. The understanding of the quantum correlations, or entanglement, between the particles in an isolated many-body quantum system may however be challenging, as the amount of interactions may be large. To be able to describe the system in a simpler way, mean field theory is often applied; the interactions of all individual particles are substituted by an averaged effect. First and second order mean field approximations are applied to the equations of motion of the amplitudes <ân> for each mode of the many-body system. In first order mean field, under the assumption that each mode has the same magnitude of the (constant) amplitude, the Kuramoto equation follows for the phases of the complex amplitudes. To study this first order approximation and to refine the mean field solution, the equations of motion have been extended to second order mean field. After that, numerical integration is used to solve the system of coupled differential equations for two modes. The solutions seem unstable, which is verified by defining quantum fluctuations beyond mean field. These are shown not to be negligible for a two mode system.It was discovered that the mean field equations of motion show divergent fluctuations for a system of two modes, which opens a whole set of questions on the validity of the first order mean field approximation and therefore the use of the Kuramoto model for quantum many-body dynamics. The extension to the second order mean field solution might be promising as this does include correlations between the operators.