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Theoretical and Mathematical Physics

Schrödinger-Belavkin equations and associated Kolmogorov and Lindblad equations

Oleg Georgievich Smolyanov, Aubrey Truman

Published Aug 1, 1999• 12 citations
Publisher
DOI
10.1007/bf02557405

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Representations of solutions of Lindblad equations by randomized Feynman integrals over trajectories are obtained by averaging similar representations for solutions of stochastic Schrödinger equations (Schrödinger–Belavkin equations). An approach based on the application of Chernoff’s theorem is applied. First, (randomized) Feynman formulas approximating Feynman path integrals are obtained; these formulas contain integrals over finite Cartesian powers of the space of values of the functions over which the Feynman integrals are taken.
Abstract We present an alternative derivation of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation based on complex Riccati equations and the associated dynamical invariants. Our formulation introduces time-dependent coefficients, which satisfy the constraints imposed by the Robertson-Schrödinger uncertainty relations, as previously discussed by Dekker et al. in the time-independent case. This time dependence may provide new insights and perspectives for extending the framework to non-Markovian systems.
The Schrödinger and Neumann equations for open quantum systems of the quantum gas type were obtained, these describing their temperature dependence. Then modified Lindblad equations were obtained, describing exponential relaxation and diffusion damping due to slow wandering in the configuration space. The solution of a single exact Neumann equation with terms from the Kolmogorov equation, which describes random motions of quantum particles in momentum space, leads to diffusion damping of spin magnetization signals in momentum space exponentially with time t3/3 and complex oscillations of products of momenta and coordinates.