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arXiv:1209.1437 · InTech eBooks

Quantum Damped Harmonic Oscillator

Kazuyuki Fujii

Published Apr 3, 2013• 14 citations
arXivHTMLPDF

Abstract

In this chapter we treat the quantum damped harmonic oscillator, and study mathematical structure of the model, and construct general solution with any initial condition, and give a quantum counterpart in the case of taking coherent state as an initial condition. This is a simple and good model of Quantum Mechanics with dissipation which is important to understand real world, and readers will get a powerful weapon for Quantum Physics.

DOI
10.5772/52671

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In this chapter we treat the quantum damped harmonic oscillator, and study mathematical structure of the model, and construct general solution with any initial condition, and give a quantum counterpart in the case of taking coherent state as an initial condition. This is a simple and good model of Quantum Mechanics with dissipation which is important to understand real world, and readers will get a powerful weapon for Quantum Physics.
The damped harmonic oscillator is a workhorse for the study of dissipation in quantum mechanics. However, despite its simplicity, this system has given rise to some approximations whose validity and relation to more refined descriptions deserve a thorough investigation. In this work, we apply a method that allows us to diagonalize exactly the dissipative Hamiltonians that are frequently adopted in the literature. Using this method, we derive the conditions of validity of the rotating-wave approximation (RWA) and show how this approximate description relates to more general ones. We also show that the existence of dissipative coherent states is intimately related to the RWA. Finally, through the evaluation of the dynamics of the damped oscillator, we notice an important property of the dissipative model that has not been properly accounted for in previous works, namely the necessity of new constraints to the application of the factorizable initial conditions.
The study of quantum properties of the harmonic oscillator is important in physics as many real-world systems oscillate harmonically. Motion of systems in a confined space is often modeled as being a quantized harmonic motion or, in first instance, is approximated by a harmonic motion. This chapter helps the reader to solve the equation employing an algebraic operator technique, which is based on the Dirac notation. This approach has several definite advantages and exploits the commutation relations among the operators involved and their properties. The chapter transforms the stationary Schrodinger equation into a second-order differential equation and finds the solution to the equation with the aid of a more advanced mathematical technique that involves special functions. The energy of a harmonic oscillator is quantized, with the sequence of values. The difference in energy between adjacent energy levels is equal to the energy of a single photon, ħω.
In this chapter we introduce the basic concept of a quantum harmonic oscillator, and how such oscillators couple to their environment. Our purpose is not to exhaustively cover all aspects of such systems, but rather to provide the foundation material required to treat the quantum mechanics of mechanical resonators and optical cavities – and their interactions – in later chapters. The quantum fluctuation-dissipation theorem is derived relating the power spectral density of environmental forcing of an oscillator to the rate of energy decay from the oscillator into the environment. This is in direct analogy to the familiar fluctuation-dissipation theorem in classical physics. However, in contrast to the classical case, the noncommutation of position and momentum introduces an asymmetry between the positive and negative frequency components of the power spectrum. This provides a natural definition for the effective temperature of the oscillator. The quantum Langevin approach is introduced as a method to model the open dynamics of quantum systems, both with and without recourse to the rotating wave approximation.
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.
In this short paper, a very brief introduction of the quantum harmonic oscillator has been provided. The understanding of the transition from a classically damped oscillator and the quantum mechanical model of the latter is included via a clear mathematics involving simple calculus. Only the one dimensional time dependent Schrodinger’s equation is being studied. This paper is intended to simply provide an analytical description of the subject, there are no advance calculations implying further suggestions in quantum theory.
The in-in formalism and its influence functional generalization are widely used to describe the out-of-equilibrium dynamics of unitary and open quantum systems, respectively. In this paper, we build on these techniques to develop an effective theory of a quantum damped harmonic oscillator and use it to study initial state-dependence, decoherence, and thermalization. We first consider a Gaussian initial state and quadratic influence functional and obtain general equations for the Green's functions of the oscillator. We solve the equations in the specific case of time-local dissipation and use the resulting Green's functions to obtain the purity and unequal-time two-point correlations of the oscillator. We find that the dynamics must include a nonvanishing noise term to yield physical results for the purity and that the oscillator decoheres in time such that the late-time density operator is thermal. We show that the frequency spectrum or unequal-time correlations can, however, distinguish between the damped oscillator and an isolated oscillator in thermal equilibrium and obtain a generalized fluctuation-dissipation relation for the damped oscillator. We briefly consider time-nonlocal dissipation as well, to show that the fluctuation-dissipation is satisfied for a specific choice of dissipation kernels. Lastly, we develop a double in-out path integral approach to go beyond Gaussian initial states and show that our equal-time results for time-local dissipation are in fact nonperturbative in the initial state.