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arXiv:2211.08853 · The Journal of Chemical Physics

Quantum mechanics of open systems: Dissipaton theories

Yao Wang, YiJing Yan

Published Oct 12, 2022• 37 citations
arXivHTMLPDF

Abstract

This Perspective presents a comprehensive account of the dissipaton theories developed in our group since 2014, including the physical picture of dissipatons and the phase-space dissipaton algebra. The dissipaton-equation-of-motion-space (DEOM-space) formulations cover the Schrödinger picture, the Heisenberg picture, and further the imaginary-time DEOM. Recently developed are the dissipaton theories for studying equilibrium and nonequilibrium thermodynamic mixing processes. The Jarzynski equality and Crooks relation are accurately reproduced numerically. It is anticipated that dissipaton theories would remain essential toward a maturation of quantum mechanics of open systems.

DOI
10.1063/5.0123999

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This Perspective presents a comprehensive account of the dissipaton theories developed in our group since 2014, including the physical picture of dissipatons and the phase-space dissipaton algebra. The dissipaton-equation-of-motion-space (DEOM-space) formulations cover the Schrödinger picture, the Heisenberg picture, and further the imaginary-time DEOM. Recently developed are the dissipaton theories for studying equilibrium and nonequilibrium thermodynamic mixing processes. The Jarzynski equality and Crooks relation are accurately reproduced numerically. It is anticipated that dissipaton theories would remain essential toward a maturation of quantum mechanics of open systems.
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This work presents a unified dissipaton-equation-of-motion (DEOM) theory and its evaluations on the Helmholtz free energy change due to the isotherm mixing of two isolated subsystems. One is a local impurity, and the other is a nonlocal Gaussian bath. DEOM constitutes a fundamental theory for such open quantum mixtures. To complete the theory, we also construct the imaginary-time DEOM formalism via an analytical continuation of dissipaton algebra, which would be limited to equilibrium thermodynamics. On the other hand, the real-time DEOM deals with both equilibrium structural and nonequilibrium dynamic properties. Its combination with the thermodynamic integral formalism would be a viable and accurate means to both equilibrium and transient thermodynamics. As illustrations, we report the numerical results on a spin-boson system, with elaborations on the underlying anharmonic features, the thermodynamic entropy vs the von Neumann entropy, and an indication of "solvent-cage" formation. Beside the required asymptotic equilibrium properties, the proposed transient thermodynamics also supports the basic spontaneity criterion.
Quantum mechanics takes various forms. For atomic and nuclear physics, the Schrödinger picture is convenient. For quantum field theory and for correspondence with classical mechanics through Poisson brackets, the Heisenberg picture is useful. The interaction picture is useful in calculations. The Wigner phase-space is convenient for studying the uncertainty relations and their symmetries. The density matrix and probability distribution are necessary for extracting measurable dynamic quantities and are essential for ensemble averages. Mathematical instruments, such as the Lie algebra of the rotation group, the creation and annihilation operators of the Fock space and the Lie algebra governing Dirac’s two oscillators are shown to be derivable from Heisenberg’s commutation relations.
The quest of exact and nonperturbative methods on quantum dissipation with nonlinear coupling environments remains in general a great challenge. In this review we present a comprehensive account on two approaches to the entangled system-and-environment dynamics, in the presence of linear-plus-quadratic coupling bath. One is the dissipaton-equation-of-motion (DEOM) theory that has been extended recently to treat the nonlinear coupling environment. Another is the extended Fokker-Planck quantum master equation (FP-QME) approach that will be constructed in this work, based on its DEOM correspondence. We closely compare these two approaches, with the focus on the underlying quasi-particle picture, physical implications, and implementations.
Open quantum systems describe quantum systems that interact unavoidably with their environments. These notes introduce the physical origins of decoherence and dissipation in realistic settings, developing the theoretical framework of reduced dynamics and quantum master equations within the connection to thermodynamics in the quantum regime. Additionally, a set of relevant exercises are proposed to reinforce the acquisition of knowledge. The text equips readers with the tools to model and analyze open quantum dynamics.