Hamiltonian formulation of nonequilibrium quantum dynamics: geometric structure of the BBGKY hierarchy

TL;DR

This paper introduces a Hamiltonian framework for nonequilibrium quantum dynamics based on the BBGKY hierarchy, revealing its symplectic structure and enabling systematic approximations for complex many-body systems.

Contribution

It formulates the BBGKY hierarchy as Hamiltonian equations with geometric structure, allowing new approximation methods and insights into quantum many-body dynamics.

Findings

Systematic approximation scheme via canonical perturbation theory.

Application to Hubbard chain captures real-time dynamics.

Discovery of observable geometric phases in noncyclic evolutions.

Abstract

Time-resolved measurement techniques are opening a window on nonequilibrium quantum phenomena that is radically different from the traditional picture in the frequency domain. The simulation and interpretation of nonequilibrium dynamics is a conspicuous challenge for theory. This paper presents a novel approach to quantum many-body dynamics that is based on a Hamiltonian formulation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations of motion for reduced density matrices. These equations have an underlying symplectic structure, and we write them in the form of the classical Hamilton equations for canonically conjugate variables. Applying canonical perturbation theory or the Krylov-Bogoliubov averaging method to the resulting equations yields a systematic approximation scheme. The possibility of using memory-dependent functional approximations to close the Hamilton equations at a particular level of the hierarchy is discussed. The geometric structure of the equations gives rise to reduced geometric phases that are observable even for noncyclic evolutions of the many-body state. The formalism is applied to a finite Hubbard chain which undergoes a quench in on-site interaction energy U. Canonical perturbation theory, carried out to second order, fully captures the nontrivial real-time dynamics of the model, including resonance phenomena and the coupling of fast and slow variables.