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Journal of Physics Conference Series

FDT for fluctuations of a special perturbated harmonic oscillator and its application in q-deformed Harmonic Oscillator

Huangcheng Yin

Published Jul 1, 2023• 0 citations
Publisher

Abstract

Abstract In this paper, we will use fluctuation dissipation theorem on the harmonic oscillator with time correlation creation annihilation operator as the perturbated term and through calculation, we find that the exact value of the dissipation associated Imχ of this special perturbation harmonic oscillator is exactly equal to π. Next, we give its potential application in the q-deformed harmonic oscillator, which means connect the q-deformed harmonic oscillator with fluctuation dissipation.

DOI
10.1088/1742-6596/2543/1/012010

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Abstract In this paper, we will use fluctuation dissipation theorem on the harmonic oscillator with time correlation creation annihilation operator as the perturbated term and through calculation, we find that the exact value of the dissipation associated Imχ of this special perturbation harmonic oscillator is exactly equal to π. Next, we give its potential application in the q-deformed harmonic oscillator, which means connect the q-deformed harmonic oscillator with fluctuation dissipation.
The dynamics of the harmonic oscillator, with a fluctuating, time-dependent frequency, is studied in the quantum framework. For periodic fluctuations of period \ensuremath{\tau}, it is found that the mean energy diverges exponentially in time, provided \ensuremath{\tau} falls into certain ``bands,'' centered around ``quasiresonant'' values, for which the rate of increase is maximum. In agreement with the correspondence principle, this rate is shown to coincide exactly with the one obtained for the classic oscillator, in the same conditions [Loris Ferrari, Phys. Rev. B 56, 593 (1997)]. The notion of adiabaticity has been revisited in the case of exponentially diverging mean energy. It is found that the Heisenberg form of the Hamiltonian operator can be expressed as a new effective Hamiltonian, which is still harmonic, but has exponentially diverging coefficients. It is argued that the same results hold in general (that is, without any quasiresonance condition) if the frequency fluctuation in time is disordered. A brief discussion is presented about nonharmonic interactions leading to dissipative effects. Applications to real physical systems are outlined, with special reference to the electrons in a uniform and constant magnetic field, to which a small fluctuating component is added.
We discuss the superoscillators in thc supersymmetric quantum systems, and give the solution to the system. By making use of the annihilation and creation operators and their q -deformed operators, we also give the q -deformed quantum harmonic oscillator model in the supersymmetric quantum systems. The q -supercoherent states associated with the q -deformed supersymmetric quantum harmonic oscillators are constructed explicitly, and their properties are investigated. The uncertainty relations for q -supercoherent states are discussed as well.
The question of the "physical meaning" and "origin" of the Bose-Einstein (BE) factor in the fluctuation-dissipation theorem (FDT) is often raised and this term is sometimes interpreted as originating from a real harmonic oscillator composition of the physical system. Such an interpretation, however, is not really founded. Inspired by the famous work of Caldeira and Leggett, we have been able to show that, whenever linear response theory is applicable, which is the main hypothesis under which the FDT is established, any generic bosonic and/or fermionic system at temperature TTT can be mapped onto a fictitious system of harmonic oscillators so that the suscettivity and the mean square of the fluctuating observable of the real system coincide with the corresponding quantities of the fictitious one. We claim that it is in this sense, and only in this sense, that the BE factor can be interpreted in terms of harmonic oscillators, no other physical meaning can be superimposed to it. At the best of our knowledge, this is the first time that such a mapping is explicitly worked out.
This work addresses the study of the oscillator algebra, defined by four parameters ppp, qqq, ααα, and ννν. The time-independent Schrödinger equation for the induced deformed harmonic oscillator is solved; explicit analytic expressions of the energy spectrum are given. Deformed states are built and discussed with respect to the criteria of coherent state construction. Various commutators involving annihilation and creation operators and their combinatorics are computed and analyzed. Finally, the correlation functions of matrix elements of main normal and antinormal forms, pertinent for quantum optics analysis, are computed.
Here a special case of perturbation in quantum harmonic oscillator is studied. Here we assume the perturbed potential to be a Harmonic Oscillator that has been shifted in the position space.We construct the new creation and annihilation operators for the new Hamiltonian to find out its energy eigenstates. What is interesting about the solution of this system is that we find out that the ground state of this system is a coherent state of the ground state of the original harmonic oscillator. We know that the Hamiltonian of electromagnetic field is similar to that of the harmonic oscillator where the electric field and magnetic field are expressed in terms of time evolution of annihilation and creation operators. According to the quantum electromagnetic field theory, photon states are expressed as the coherent states of the ground states of the unperturbed harmonic oscillator.In the previous research works,it has been established how coherent states provide a better picture of Electric field compared to the energy eigenstates. This paper mainly deals with why the coherent states provide a more complete picture of electric and magnetic field theory compared to just energy eigenstates. Here we find how our perturbation act as a coherent state generator and adds a new term in the Hamiltonian. The Perturbation is thought to arise from the only available interaction i.e. observation. The perturbation also explains how observation leads to coherence of photon states. We will find out how the new perturbation affects the time evolution of electromagnetic field. This also explains how the classical electromagnetic field theory takes into account the factor of observation, due to which the classical laws are in perfect agreement with the quantum laws.
The author considers a dynamic system interacting with a special dissipative system composed of many harmonic oscillators. Using the Feynman disentangling theorem and a bath approximation he derives differential equations for correlation functions of dynamic operators (CFD) concerning the maximum time argument t which in general occurs in more than one operator. They must be solved with appropriate initial conditions at time t', the second in height time argument in the CFD. This procedure allows the successive calculation of CFD with an arbitrary time arrangement (e.g. CFD of three operators, where the central one carries the smallest time argument). The exact equations of motion for dynamic operators may also be written as Langevin equations. The author defines the corresponding fluctuation operators and derives compact expressions for their correlation functions (CFF). In general the fluctuation operators show no Gaussian behaviour. CFF of second order are delta -correlated and CFF of third order are only delta -correlated with respect to the two highest time arguments.