Diffusion and dissipation in complex quantum systems

The Hamiltonian of a quantum-mechanical system contains a time-dependent parameter X. The system is prepared in a highly excited eigenstate of the instantaneous Hamiltonian at t=0, and we compute the amplitudes 〈n(t)‖ψ(t)〉 to be in the eigenstates ‖n〉 of the instantaneous Hamiltonian H^(X(t)) at later times. It is found that for generic systems, with no symmetries yielding additional quantum numbers, the occupation probability spreads diffusively away from the initial state. This diffusion is an irreversible process, and can be related to a model for dissipation. The diffusion constant D is investigated numerically in a random matrix model as a function of Ẋ, the rate of change of the parameter. Good agreement with theoretical predictions is found in two limiting regimes. When Ẋ is large, D is proportional to Ẋ ${}_{}{}^2{}_{}{}^{}$, corresponding to the Ohmic dissipation predicted by the Kubo formula. When Ẋ is small, Landau-Zener transitions are the mechanism for diffusion, and D is proportional to Ẋ ${}_{}{}^{3/2}{}_{}{}^{}$.