On the reduced dynamics of a subset of interacting bosonic particles

TL;DR

This paper derives a hierarchical framework for the reduced quantum dynamics of interacting bosons, leading to a nonlinear mean-field equation and extensions for decoherence effects, generalizing the Gross-Pitaevskii equation.

Contribution

It introduces a hierarchical approach to bosonic subdynamics and derives a nonlinear mean-field equation, with extensions for incoherent effects, advancing the understanding of many-body quantum dynamics.

Findings

Derivation of a hierarchical expansion for bosonic subdynamics.

Recovery of the Gross-Pitaevskii equation for contact interactions.

Proposal of a nonlinear Lindblad-type equation for decoherence.

Abstract

The quantum dynamics of a subset of interacting bosons in a subspace of fixed particle number is described in terms of symmetrized many-particle states. A suitable partial trace operation over the von Neumann equation of an $N$-particle system produces a hierarchical expansion for the subdynamics of $M\leq N$ particles. Truncating this hierarchy with a pure product state ansatz yields the general, nonlinear coherent mean-field equation of motion. In the special case of a contact interaction potential, this reproduces the Gross-Pitaevskii equation. To account for incoherent effects on top of the mean-field evolution, we discuss possible extensions towards a second-order perturbation theory that accounts for interaction-induced decoherence in form of a nonlinear Lindblad-type master equation.