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arXiv:quant-ph/0610133

Dissipation in Quantum Mechanics, Scalar and Vector Field Theory

Fardin Kheirandish, Majid Amooshahi

Published Oct 17, 2006• 3 citations
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Abstract

A new minimal coupling method is introduced. A general dissipative quantum system is investigated consistently and systematically. Some coupling functions describing the interaction between the system and the environment are introduced. Based on coupling functions, some susceptibility functions are attributed to the environment explecitly. Transition probabilities relating the way energy flows from the system to the environment are calculated and the energy conservation is explecitly examined. This new formalism is generalized to the dissipative scalar and vector field theories along the ideas developed for the quantum dissipative systems

DOI
10.48550/arxiv.quant-ph/0610133

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A new minimal coupling method is introduced. A general dissipative quantum system is investigated consistently and systematically. Some coupling functions describing the interaction between the system and the environment are introduced. Based on coupling functions, some susceptibility functions are attributed to the environment explecitly. Transition probabilities relating the way energy flows from the system to the environment are calculated and the energy conservation is explecitly examined. This new formalism is generalized to the dissipative scalar and vector field theories along the ideas developed for the quantum dissipative systems
Quantum mechanics of a general one dimensional dissipative system investigated by it's coupling to a Klein-Gordon field as the environment using a minimal coupling method. Heisenberg equation for such a dissipative system containing a dissipative term proportional to velocity obtained. As an example, quantum dynamics of a damped harmonic oscillator as the prototype of some important one dimensional dissipative models investigated consistently. Some transition probabilities indicating the way energy flows between the subsystems obtained.
The exact response theory based on the Dissipation Function applies to general dynamical systems and has yielded excellent results in various applications. In this article, we propose a method to apply it to quantum mechanics. In many quantum systems, it has not yet been possible to overcome the perturbative approach, and the most developed theory is the linear one. Extensions of the exact response theory developed in the field of nonequilibrium molecular dynamics could prove useful in quantum mechanics, as perturbations of small systems or far-from-equilibrium states cannot always be taken as small perturbations. Here, we introduce a quantum analogue of the classical Dissipation Function. We then derive a quantum expression for the exact calculation of time-dependent expectation values of observables, in a form analogous to that of the classical theory. We restrict our analysis to finite-dimensional Hilbert spaces, for the sake of simplicity, and we apply our method to specific examples, like qubit systems, for which exact results can be obtained by standard techniques. This way, we prove the consistency of our approach with the existing methods, where they apply. Although not required for open systems, we propose a self-adjoint version of our Dissipation Operator, obtaining a second equivalent expression of response, where the contribution of an anti-self-adjoint operator appears. We conclude by using new formalism to solve the Lindblad equations, obtaining exact results for a specific case of qubit decoherence, and suggesting possible future developments of this work.
Caldeira-Leggett model of reservoir is generalized to a reservoir modeled by a continuum of real Klein-Gordon fields, instead of harmonic oscillators. A quantum Langevin type dissipative equation is obtained for the scalar field. The susceptibility of the medium is defined in terms of the reservoir Green's function and the coupling function satisfying causality condition. The connection between the coupling function and the susceptibility of the medium is found to be a Hankel transform from which the coupling function can be determined in terms of the susceptibility of the medium. Noise currents and their fluctuation-dissipation relation are obtained. In a homogeneous medium or reservoir, explicit form of the quantum scalar field, and its large-time limit, are found.
We consider a resistance element (R) as an environment of an LC circuit, which is treated as the main system modeled by a Hamiltonian containing an oscillator with a continuous range of frequencies. Using a minimal coupling method which is appropriate for dissipative quantum systems, we investigate the quantum dynamics of this system. The Heisenberg equation for such a dissipative system contains a dissipative term proportional to Ohmic damping. Some transition probabilities indicating the way energy flows between the subsystems are obtained.