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Acta Physica Sinica

QUANTUM THEORY OF A DAMPED HARMONIC OSCILLATOR SIMULATING LASER OPEN SYSTEM

TAN WEI-HAN, Xuewen Wang, XIE CHENG-GANG, ZHANG GUAN-MEI, (1)上海科学技术大学,1981年应届毕业生; (2)中国科学院上海光学精密机械研究所

Published Jan 1, 1982• 0 citations
Publisher

Abstract

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].

DOI
10.7498/aps.31.1569

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We analyse the method of direct quantization suggested in reference [1] for adamped harmonic osillator, in which the quantum condition xp-px=ihe(-(C/M)t) is introduced. It is pointed out that this method can not be generalized to treat the case in which C is a function of time. In order to treat this case in a general approach, Heiaenberg relation xp-px=in must be kept and the force acting upon the osillatormust be supposed to contain a component fR that does not commute with x and satisfies xfR-fRx=ih(C/M).An electronic osillator is analysed as an example to show that ourapproach is consistent with quantum mechanical Langevin theory.

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In the preceding paper (arXiv: 0710.2724 [quant-ph]) we have constructed the general solution for the master equation of quantum damped harmonic oscillator, which is given by the complicated infinite series in the operator algebra level. In this paper we give the explicit and compact forms to solutions (density operators) for some initial values. In particular, the compact one for the initial value based on a coherent state is given, which has not been given as far as we know. Moreover, some related problems are presented.

arXiv:0806.2169•International Journal of Geometric Methods in Modern Physics•Mar 1, 2009•8 citations
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].
Assuming a damping force proportional to the velocity, it is shown that the usual quantum condition between the physical variables, namely the displacement and the momentum, has to be modified by the damping. However, it is possible to change to new variables which are canonical. Then we can apply the usual quantum mechanics and use the usual physical interpretation through the statistical average values. The case of a one-dimensional damped harmonic oscillator under the action of an arbitrary external force has been worked out as an example. No difficulty arises and the results are reasonable.
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 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.
We analyse the method of direct quantization suggested in reference [1] for adamped harmonic osillator, in which the quantum condition xp-px=ihe(-(C/M)t) is introduced. It is pointed out that this method can not be generalized to treat the case in which C is a function of time. In order to treat this case in a general approach, Heiaenberg relation xp-px=in must be kept and the force acting upon the osillatormust be supposed to contain a component fR that does not commute with x and satisfies xfR-fRx=ih(C/M).An electronic osillator is analysed as an example to show that ourapproach is consistent with quantum mechanical Langevin theory.
In the preceding paper (arXiv: 0710.2724 [quant-ph]) we have constructed the general solution for the master equation of quantum damped harmonic oscillator, which is given by the complicated infinite series in the operator algebra level. In this paper we give the explicit and compact forms to solutions (density operators) for some initial values. In particular, the compact one for the initial value based on a coherent state is given, which has not been given as far as we know. Moreover, some related problems are presented.