Subsystem dynamics under random Hamiltonian evolution

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.

16 pages, 8 figures; v3: slightly re-structured and an additional appendix

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• Compelling Bounds on Equilibration Times -- the Issue with Fermi's Golden Rule
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• Entanglement Trajectory and its Boundary