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

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 an 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 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.

Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time-Dependent RDMFT (TD-RDMFT)

“…All the off-diagonal elements of the 1-RDM are able to obtain a complex phase-factor, but because the diagonal is necessarily real, the occupation numbers do not have a quantum phase [117]. This lack of a corresponding quantum phase for the natural occupation numbers is not limited to the 1-RDM, but exists for the diagonal of any p-RDM if the BBGKY hierarchy is truncated at the pth order [151].…”

Bogoliubov-Born-Green-Kirkwood-Yvon chain and kinetic equations for the level dynamics in an externally perturbed quantum system

“…. N ] concerning positions and velocities [13][14][15]23,24,27 . We denote an empirical distribution function,…”

“…The final two terms associated with L I correspond to the perturbation contribution of the Hamiltonian as result of interactions [13][14][15]23,24,[28][29][30] . Although this hierarchy produces a scheme which determines the kinetic equations of motion, it does not accommodate for the nature of quantum systems with parametrically evolving Hamiltonians 15,25,26 . In the following we derive a generalised BBGKY hierarchy that describes the level dynamics associated to non-equilibrium systems which extends to parametrically evolving Hamiltonians, using the Pechukas model.…”

“…This stimulates the search for alternative theoretical methods, which could provide some useful figures-of-merit describing large quantum systems out of equilibrium. A common approach to achieve quantum coherence in non-equilibrium many body dynamics is Keldysh Green function theory 15 , however this approximation is limited to short time intervals where its errors grow as a power of time 15 . This paper aims at developing basic elements of such an approach, which would seem especially useful for, but not necessarily restricted to, modelling of adiabatic quantum computers.…”

“…This description is advantageous as it is expected to have significant developments in non-equilibrium processes such as decohenrence 1,3,15 . We, eventually, would like to address the question to what extent this approach can describe adiabatic quantum computing.…”