Adiabatic theorem for a class of stochastic differential equations on a Hilbert space

Abstract

We derive an adiabatic theory for a stochastic differential equation,under a condition that instantaneous stationary states of L 1 (s) are also stationary states of L 2 (s). We use our results to derive the full statistics of tunneling for a driven stochastic Schrödinger equation describing a dephasing process.

Lindbladians for controlled stochastic Hamiltonians

“…So far, no approximation has been made. However, the time ordering remains a major complication 8 . For its precise meaning one can either go back to equation (2.3), or alternatively, see the discussion and graphical representation in appendix A.…”

“…7 We shall use script characters to denote super-operators. 8  may be viewed as the grand canonical partition function of a 1-D quantum gas with short range interaction.…”