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arXiv:quant-ph/0502076

A minimal coupling method for investigating one dimensional dissipative quantum systems

Fardin Kheirandish, Majid Amooshahi

Published Feb 13, 2005• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0502076

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•Aug 17, 2016•18 citations
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.
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
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.
The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a damped oscillator is determined by an arbitrary effective susceptibility that obeys the Kramers–Kronig relations. This approach offers an alternative description of nano-mechanical oscillators and opto-mechanical systems.
Abstract. The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a damped oscillator is determined by an arbitrary effective susceptibility that obeys Kramers-Kronig relations. This approach offers an alternative description of nano-mechanical oscillators and opto-mechanical systems. PACS numbers: 03.65.-w, 03.65.Yz, 03.70.+k 1.