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Iranian Journal of Physics Research

Quantum dynamics of A damped harmonic oscillator with velocity-dependent coupling

Mohsen Daeimohammad, M. Nili Ahmadabadi

Published Dec 1, 2022• 0 citations
Publisher

Abstract

The quantum theory of a damped harmonic oscillator has been continuously studied since 1939. For quantization, an oscillation oscillator usually uses a reservoir that includes a number of finite and discrete oscillators. However, a set of discrete oscillators is unable to study the quantum theory of an Ohmic damping (damping proportional to velocity). In this research, a continuous set of harmonic oscillators has been used to investigate Ohmic damping. Using a continuum reservoir instead of a discrete reservoir will enrich the results of the dynamic system. The results of this research can be used to study nano-mechanical systems and opto-mechanical systems.

DOI
10.47176/ijpr.22.3.11395

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The quantum theory of a damped harmonic oscillator has been continuously studied since 1939. For quantization, an oscillation oscillator usually uses a reservoir that includes a number of finite and discrete oscillators. However, a set of discrete oscillators is unable to study the quantum theory of an Ohmic damping (damping proportional to velocity). In this research, a continuous set of harmonic oscillators has been used to investigate Ohmic damping. Using a continuum reservoir instead of a discrete reservoir will enrich the results of the dynamic system. The results of this research can be used to study nano-mechanical systems 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.
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.
The quantum model for a damped harmonic oscillator linearly coupled to a reservoir of continuous frequency modes has been exactly diagonalized either with or without the rotating wave approximation. The exact dynamics, formally described in terms of the coupling function, is analyzed for the Drude model. In this scenario, we focus on a class of couplings recovering the Drude form as a limiting case. The exact time evolution of the expectation values of the position, momentum and number operators are described through Fox H -functions, as well as the correlation function of the reservoir. In correspondence with the critical values of the frequency, discontinuities appear in the long time scale dynamics, arbitrarily slowing down the relaxations of the above observables. This effect allows one to enhance arbitrarily the lifetime of the excitations, giving relevant applications in the study of the quantum domain of opto-mechanical and nano-mechanical resonators. The critical frequency approach constitutes a further method of control in the scenario of environment-induced decoherence via reservoir engineering.
We have analysed the quantization schemes presented in references [1, 2] for a damped harmonic oscillator and found out that there are some defects in volved in these schemes. In order to remedy these defects, we assume the coordinate system fluctuating with a velocity g(t) in addition to damping force f(t) proportional to the velocity and suggest a method of quantization for this system based on the full quantum theory of lasers. It is proved that the usual Heisenberg relation xp-px=ih can be maintained, but the quantum term in statistical average value of physical variable does not damp as shown in reference [1].
The squeezing dynamics of a damped harmonic oscillator are studied for different types of environment without making the Markovian approximation. The squeezing dynamics of a coherent state depend on the reservoir spectrum in a unique way that can, in the weak coupling approximation, be analyzed analytically. Comparison of squeezing dynamics for Ohmic, sub-Ohmic and super-Ohmic environments is done showing a clear connection between the squeezing--non-squeezing oscillations and reservoir structure. Understanding the effects occurring due to structured reservoirs is important both from a purely theoretical point of view and in connection with evolving experimental techniques and future quantum computing applications.