Subsystem dynamics under random Hamiltonian evolution

TL;DR

This paper analyzes the evolution of a quantum subsystem's density matrix under complex Hamiltonian dynamics modeled by random matrices, providing exact calculations and insights into the transition to random states.

Contribution

It introduces an analytical framework using a noncentral correlated Wishart ensemble to describe subsystem dynamics under chaotic Hamiltonian evolution.

Findings

Exact eigenvalue density calculations for the reduced density matrix.

Identification of the convergence time to random states.

Observation of eigenvalue collisions and phase transitions in distribution.

Abstract

We study time evolution of a subsystem's density matrix under unitary evolution, generated by a sufficiently complex, say quantum chaotic, Hamiltonian, modeled by a random matrix. We exactly calculate all coherences, purity and fluctuations. We show that the reduced density matrix can be described in terms of a noncentral correlated Wishart ensemble for which we are able to perform analytical calculations of the eigenvalue density. Our description accounts for a transition from an arbitrary initial state towards a random state at large times, enabling us to determine the convergence time after which random states are reached. We identify and describe a number of other interesting features, like a series of collisions between the largest eigenvalue and the bulk, accompanied by a phase transition in its distribution function.