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arXiv:2004.12598 · Mathematical Notes

Exact Dynamics of Moments and Correlation Functions for GKSL Fermionic Equations of Poisson Type

Iu. A. Nosal’, Alexander Evgen’evich Teretenkov

Published Nov 1, 2020• 14 citations
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Abstract

Gorini-Kossakowski-Sudarshan-Lindblad equation of Poisson-type for the density matrix is considered. The Poisson jumps are assumed to be unitary operators with generators, which are quadratic in fermionic creation and annihilation operators. The explicit dynamics of the density matrix moments and Markovian multi-time ordered correlation functions is obtained.

DOI
10.1134/s0001434620110358

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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.
For a model of a multilevel system interacting with several baths at zero temperature, it is shown that its dynamics becomes Markovian after the bath correlation time. We take into account not only the contribution of the bath spectral density, which leads to a continuous correlation function, but also the ohmic contribution to the spectral density, which leads to a renormalization of both equations and initial conditions. An explicit Gorini–Kossakowski–Sudarshan–Lindblad equation describing the dynamics of the system after the bath correlation time is derived, and the form of initial conditions for this equation is obtained. They do not coincide with the exact initial conditions due to the renormalization associated with the ohmic contribution and due to the short initial non-Markovian time interval preceding the bath correlation time.
Open quantum systems are, in general, described by a density matrix that is evolving under transformations belonging to a dynamical semigroup. They can obey the Franke–Gorini–Kossakowski–Lindblad–Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space of dimension 2. First, we find final fixed states (called pointers) of an evolution of an open system, and we then obtain a general solution to the FGKLS equation and confirm that it converges to a pointer. After this, we check that the solution has physical meaning, i.e., it is Hermitian, positive and has trace equal to 1, and find a moment of time starting from which the FGKLS equation can be used—the range of applicability of the semigroup symmetry. Next, we study the behavior of a solution for a weak interaction with an environment and make a distinction between interacting and non-interacting cases. Finally, we prove that there cannot exist oscillating solutions to the FGKLS equation, which would resemble the behavior of a closed quantum system.
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