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arXiv:2510.21939 · Low Temperature Physics

The evolution of Liouville von Neumann master equations in the Pechukas–Yukawa framework

Mumnuna A. Qureshi

Published Dec 1, 2025• 0 citations
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Abstract

This paper presents a novel formalism for the out-of-equilibrium dynamics of the density matrix, capable of describing highly entangled many-body interactions. The evolution of quantum states is evaluated via eigenvalue dynamics of a general Hamiltonian system, perturbed by a parametrically evolving variable λ(t) that carries the time-dependence. This is achieved using the Pechukas–Yukawa mapping of the evolution of the energy levels governed by their initial conditions on a generalized Calogero–Sutherland model of a 1D classical gas. As such, quantum systems can be described exactly in their entirety from eigenvalue dynamics. Under this description, we provide an improved understanding of the relationship between nonequilibrium quantum phase transitions and decoherence, which has significant impacts on a wide range of applications.

DOI
10.1063/10.0039872

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This paper presents a novel formalism for the out-of-equilibrium dynamics of the density matrix, capable of describing highly entangled many-body interactions. The evolution of quantum states is evaluated via eigenvalue dynamics of a general Hamiltonian system, perturbed by a parametrically evolving variable λ(t) that carries the time-dependence. This is achieved using the Pechukas–Yukawa mapping of the evolution of the energy levels governed by their initial conditions on a generalized Calogero–Sutherland model of a 1D classical gas. As such, quantum systems can be described exactly in their entirety from eigenvalue dynamics. Under this description, we provide an improved understanding of the relationship between nonequilibrium quantum phase transitions and decoherence, which has significant impacts on a wide range of applications.
We consider the evolution of the quantum states of a Hamiltonian that is parametrically perturbed via a term proportional to the adiabatic parameter $\ensuremath{\lambda}(t)$. Starting with the Pechukas-Yukawa mapping of the energy eigenvalue evolution in a generalized Calogero-Sutherland model of a one-dimensional classical gas, we consider the adiabatic approximation with two different expansions of the quantum state in powers of $d\ensuremath{\lambda}/dt$ and compare them with a direct numerical simulation. We show that one of these expansions (Magnus series) is especially convenient for the description of nonadiabatic evolution of the system. Applying the expansion to the exact cover 3-satisfiability problem, we obtain the occupation dynamics, which provides insight into the population of states and sources of decoherence in a quantum system.
Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing a new approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in case of an adiabatically slow external perturbation. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of with cubic repulsion. Equilibrium statistical mechanics of this gas allows to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e. approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and inter-level transition amplitudes, they allow to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as 1/N, where N is the number of the Pechukas gas particles (i.e. energy levels in the system).
The primary goals of this dissertaion are to describe and elucidate a new formalism to study the out of equilibrium dynamics of integrable models, and to apply this formalism to some specific problems. In particular, I describe the “quench dynamics” of a gas of interacting bosons in one dimension, and provide details of some preliminary work on the isotropic Heisenberg chain. The formalism, based on earlier work by Yudson, provides a way of getting around some of the difficulties involved in calculating the time evolution of arbitrary initial states evolved with integrable Hamiltonians. In addition to this, I aim to discuss some of the general aspects of quench dynamics in quantum systems. The description of the nonequilibrium dynamics of a given system depends significantly on initial states, and the time scales at which the system is probed compared with the various inherent time scales in the system. I present the experimental context and motivation for these studies, and survey the existing techniques and efforts at understanding the relaxation of systems that are far from their equilibrium states or ground states.
The conditions under which quantum-classical Liouville dynamics may be reduced to a master equation are investigated. Systems that can be partitioned into a quantum-classical subsystem interacting with a classical bath are considered. Starting with an exact non-Markovian equation for the diagonal elements of the density matrix, an evolution equation for the subsystem density matrix is derived. One contribution to this equation contains the bath average of a memory kernel that accounts for all coherences in the system. It is shown to be a rapidly decaying function, motivating a Markovian approximation on this term in the evolution equation. The resulting subsystem density matrix equation is still non-Markovian due to the fact that bath degrees of freedom have been projected out of the dynamics. Provided the computation of nonequilibrium average values or correlation functions is considered, the non-Markovian character of this equation can be removed by lifting the equation into the full phase space of the system. This leads to a trajectory description of the dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees of freedom. The results are illustrated by computations of the rate constant of a model nonadiabatic chemical reaction.
The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.
Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing an approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in the case of an adiabatically slow external perturbation, though it is not restricted to adiabatic systems. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of particles with cubic repulsion. Equilibrium statistical mechanics of this Pechukas gas allows us to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the Pechukas gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e., approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and interlevel transition amplitudes, they allow us to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as 1/N1/N1/N, where NNN is the number of the ``Pechukas gas particles'' (i.e., energy levels in the system).