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Energy exchange in coupled forced and damped quantum harmonic oscillators at zero temperature

Е.А. Михин, П.А. Головинский, А.А. Дробышев

Published Jan 1, 2019• 0 citations

Abstract

Исследуются особенности динамики двух связанных квантовых осцилляторов в поле лазерной волны. Построена теоретическая модель этой системы, в которой учтена диссипация энергии. Особое внимание в работе уделено выявлению условий наиболее эффективного протекания процесса обмена экситонами между осцилляторами. Учёт поля лазерной волны и диссипативных сил был произведён с использованием гамильтониана Бейтмана-Кальдирола-Канаи. The forced oscillations in a system of two coupled quantum oscillators of finite Q-factor of zero temperature are considered. The subject of the study was the process of energy exchange between oscillators in this system. The presence of friction and external forces was taken into account by using the Bateman-Caldirola-Kanai Hamiltonian. Expressions are obtained for the time evolution of the average energy of each oscillator for various external forces and parameters of a quantum system.

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Исследуются особенности динамики двух связанных квантовых осцилляторов в поле лазерной волны. Построена теоретическая модель этой системы, в которой учтена диссипация энергии. Особое внимание в работе уделено выявлению условий наиболее эффективного протекания процесса обмена экситонами между осцилляторами. Учёт поля лазерной волны и диссипативных сил был произведён с использованием гамильтониана Бейтмана-Кальдирола-Канаи. The forced oscillations in a system of two coupled quantum oscillators of finite Q-factor of zero temperature are considered. The subject of the study was the process of energy exchange between oscillators in this system. The presence of friction and external forces was taken into account by using the Bateman-Caldirola-Kanai Hamiltonian. Expressions are obtained for the time evolution of the average energy of each oscillator for various external forces and parameters of a quantum system.

Abstract By selecting a right generalized coordinate X, which contains the general solutions of the classical motion equation of a forced damped harmonic oscillator, we obtain a simple Hamiltonian which does not contain time for the oscillator such that Schrödinger equation and its solutions can be directly written out in X representation. The wave functin in x representation are also given with the help of the eigenfunctions of the operator

\hat X

in x representation. The evolution of

\left\langle {\hat x} \right\rangle

is the same as in the classical mechanics, and the uncertainty in position is independent of an external influence; one part of energy mean is quantized and attenuated, and the other is equal to the classical energy.

The damped harmonic oscillator is used to represent the damping processes in nature. Therefore it is necessary to find the Hamiltonian for the damped oscillator. The Hamiltonian is developed for the harmonic oscillator with frictional force proportional to velocity. The solution of the Schr\"odinger equation gives nonstationary states. The quantum friction coefficient is derived to be zero for massless particles. It is thought that velocity-dependent forces exist near the nucleus and therefore, the scattering amplitude for the deep inelastic process is derived.

Previous years researchers began to simulate open quantum system, taking into account the interaction between system and the environment. One approach to deal with this problem is to use the density matrix within the Liouville-von-Neumann formalism or the Markovian variant the Lindblad equations. Another way is to use a stochastic approach where a random force is added to the system. The benefit of the stochastic approach is to solve the dynamics of the system with less time and memory than the density matrix approaches. In this project we want to develop a stochastic approach that can deal with the stochastic wave functions approach. We did this on a 2-level system and found that it works well when comparing to a density matrix approach. Next, we tested a quantum particle connect to a bath of harmonic oscillators using the stochastic approach. We found that a friction term is necessary and applied it. Like in the classical Langevin equations the friction constant and the random force fluctuations are related by the fluctuation-dissipation constant. We showed that with friction the dynamics decays to an ensemble with energy of

E_{gs}+k_BT

. However, we also found here are problems. The system seems to absorb energy indefinitely if the temperature is higher than the zero point energy or if the system is a Morse oscillator. Thus more research is required to make this method work.