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The Journal of Chemical Physics

Theories of quantum dissipation and nonlinear coupling bath descriptors

Rui‐Xue Xu, Yang Liu, Hou-Dao Zhang, YiJing Yan

Published Mar 15, 2018• 35 citations
Publisher

Abstract

The quest of an exact and nonperturbative treatment of quantum dissipation in nonlinear coupling environments remains in general an intractable task. In this work, we address the key issues toward the solutions to the lowest nonlinear environment, a harmonic bath coupled both linearly and quadratically with an arbitrary system. To determine the bath coupling descriptors, we propose a physical mapping scheme, together with the prescription reference invariance requirement. We then adopt a recently developed dissipaton equation of motion theory [R. X. Xu et al., Chin. J. Chem. Phys. 30, 395 (2017)], with the underlying statistical quasi-particle ("dissipaton") algebra being extended to the quadratic bath coupling. We report the numerical results on a two-level system dynamics and absorption and emission line shapes.

DOI
10.1063/1.4991779

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The quest of an exact and nonperturbative treatment of quantum dissipation in nonlinear coupling environments remains in general an intractable task. In this work, we address the key issues toward the solutions to the lowest nonlinear environment, a harmonic bath coupled both linearly and quadratically with an arbitrary system. To determine the bath coupling descriptors, we propose a physical mapping scheme, together with the prescription reference invariance requirement. We then adopt a recently developed dissipaton equation of motion theory [R. X. Xu et al., Chin. J. Chem. Phys. 30, 395 (2017)], with the underlying statistical quasi-particle ("dissipaton") algebra being extended to the quadratic bath coupling. We report the numerical results on a two-level system dynamics and absorption and emission line shapes.
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.
Accurate and efficient simulation on quantum dissipation with nonlinear environment couplings remains a challenging task nowadays. In this work, we propose to incorporate the stochastic fields, which resolve just the nonlinear environment coupling terms, into the dissipaton-equation-of-motion (DEOM) construction. The stochastic fields are introduced via the Hubbard-Stratonovich transformation. After the transformation, the resulted stochastic-fields-dressed (SFD) total Hamiltonian contains only linear environment coupling terms. On the basis of that, SFD-DEOM can then be constructed. The resultant SFD-DEOM, together with the ensemble average over the stochastic fields, constitutes an exact and nonperturbative approach to quantum dissipation under nonlinear environment couplings. It is also of relatively high efficiency and stability due to the fact that only nonlinear environment coupling terms are dealt with stochastic fields, while linear couplings are still treated as the usual DEOM. Numerical performance and demonstrations are presented with a two-state model system.
We develop an operator formulation of statistical quasiparticle theory for open quantum systems with nonlinear system-bath coupling, extending beyond the linear coupling setting of the standard Gaussian influence functional formulation. For Gaussian baths, the statistical quasiparticles, termed dissipatons, are explicitly constructed as collective operators in the thermofield bath space. Our construction provides a microscopic operator basis for the dissipaton algebra and generates hierarchical equations of motion directly from the Liouville equation. Nonlinear interactions are incorporated through a recursive generalized Wick's theorem, enabling automatic generation of hierarchical equations for polynomial couplings. The resulting framework connects explicit bath operator representations to nonperturbative, non-Markovian dynamics and unifies the treatment of linear and nonlinear system-bath interactions.
In this work we construct a novel dissipaton-equation-of-motion (DEOM) theory in quadratic bath coupling environment, based on an extended algebraic statistical quasi-particle approach. To validate the new ingredient of the underlying dissipaton algebra, we derive an extended Zusman equation via a totally different approach. We prove that the new theory, if it starts with the identical setup, constitutes the dynamical resolutions to the extended Zusman equation. Thus, we verify the generalized (non-Gaussian) Wick's theorem with dissipatons-pair added. This new algebraic ingredient enables the dissipaton approach being naturally extended to nonlinear coupling environments. Moreover, it is noticed that, unlike the linear bath coupling case, the influence of a non-Gaussian environment cannot be completely characterized with the linear response theory. The new theory has to take this fact into account. The developed DEOM theory manifests the dynamical interplay between dissipatons and nonlinear bath coupling descriptors that will be specified. Numerical demonstrations will be given with the optical line shapes in quadratic coupling environment.
The study of dissipative systems holds significant interest in physics.This thesis aims to explore these phenomena in both classical and quantum regimes.In the first part, we utilize the system-reservoir approach to investigate the dynamics of two Brownian particles immersed in the same bath.Two methods are employed to address this problem: one with bilinear coupling and the other with nonlinear coupling between the particles and the bath.The extension of the system to include two particles with bilinear coupling yields unphysical results, such as free-particle motion for the relative coordinate and a lack of interaction between closely spaced particles.To address this issue, authors have been introduced an exponential coupling.In this work, we propose a method to reconcile both linear and nonlinear couplings.We demonstrate how to derive the same nonlinear dissipation rates starting from the bilinear Lagrangian, achieved through a modified spectral function that explicitly depends on the distance between the particles.Additionally, we implement a modified spectral function to mitigate anomalous diffusion observed in the standard nonlinear model, along with a phenomenological model describing hydrodynamic interaction between a pair of Brownian particles in a viscous fluid.In the quantum regime, we adopt the same system-reservoir approach to investigate the dissipative adaptation hypothesis proposed by Jeremy England.This hypothesis proposes a general thermodynamic mechanism that explains the self-organization of systems through the dissipation of work absorbed from an external drive.In the second part of this thesis, we explore the quantum dynamics of systems subjected to an external drive, evaluating the thermodynamic quantities of a self-organization process.To do so, we utilize a time-dependent spin-boson Hamiltonian that characterizes a particle subject to a metastable double potential with time-dependent parameters controlling the asymmetry of the wells.Our objective is to demonstrate that the asymmetric potential can localize the particle in the unstable side of the well and verify that this transition results in the highest energy absorption.In conclusion, we propose further investigations into the driven spin-boson model to establish a comprehensive theory of the system's evolution and its thermodynamic implications.
The recently developed Lindblad approach for quantum dynamics of an open system with linear dissipation has been extended to a model of nonlinear coupling. The coupling term ${H}_{i}={\ensuremath{\sum}}_{k}{C}_{k}{f(x)q}_{k}$ is an arbitrary function of the system coordinate, but remains to be linear in bath coordinates. A Lindblad master equation and its associated stochastic differential equation have been obtained. To study dissipative surface dynamics, a general form of f(x)f(x)f(x) has been proposed. It is shown that such a nonlinear-coupling model is crucial to qualitatively describe gas-surface dynamics, when a gas molecule impinging or leaving a surface goes beyond the linear-coupling regime.