Hamiltonian formulation of nonequilibrium quantum dynamics: Geometric structure of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy

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 are conspicuous challenges for theory. This paper presents an approach to quantum many-body dynamics that is based on a Hamiltonian formulation of the Bogoliubov-Born-Green-Kirkwood-Yvon 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 given 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 approach 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.