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Dissipation evidence for the quantum damped harmonic oscillator via pseudo-bosons

Fabio Bagarello

Published Jan 1, 2012• 5 citations
OpenAlex

Abstract

It is known that a self-adjoint, time-independent hamiltonian can be defined for the quantum damped harmonic oscillator.We show here that the two vacua naturally associated to this operator, when expressed in terms of pseudo-bosonic lowering and raising operators, appear to be non square-integrable.This fact is interpreted as the evidence of the dissipation effect of the classical oscillator at a purely quantum level.

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It is known that a self-adjoint, time-independent hamiltonian can be defined for the quantum damped harmonic oscillator.We show here that the two vacua naturally associated to this operator, when expressed in terms of pseudo-bosonic lowering and raising operators, appear to be non square-integrable.This fact is interpreted as the evidence of the dissipation effect of the classical oscillator at a purely quantum level.
Quantization of a damped harmonic oscillator leads to so called Bateman's dual system. The corresponding Bateman's Hamiltonian, being a self-adjoint operator, displays the discrete family of complex eigenvalues. We show that they correspond to the poles of energy eigenvectors and the corresponding resolvent operator when continued to the complex energy plane. Therefore, the corresponding generalized eigenvectors may be interpreted as resonant states which are responsible for the irreversible quantum dynamics of a damped harmonic oscillator.
In a recent paper a slightly modified version of the Bateman system, originally proposed to describe a damped harmonic oscillator, was proposed. This system is really different from the Bateman’s one, in the sense that this latter cannot be recovered for any choice of its parameters. In this paper we consider this system and we show that, at a quantum level, it is not necessarily dissipative. In particular we show that the Hamiltonian of the system, when quantized, produces different behaviors, depending on some relations between its parameters. In fact, it gives rise to either a two dimensional (standard) harmonic oscillator, or to two independent oscillators, one of which is again standard , and a second one which is an inverted oscillator. The two cases are analyzed in terms of bosonic or pseudo-bosonic ladder operators, and the appearance of distributions for the inverted oscillator is commented.
A previously developed novel theory for the formal canonical quantization of classically dissipating systems will be the starting point for a detailed discussion of the quantum statistical aspects of the simple linearly damped harmonic oscillator. The formalism essentially involves complex classical canonical coordinates and momenta, and thus quite naturally leads to the possibility of creation and annihilation operators. Furthermore, the occurrence of quantal noise operators appears to be of principal importance for the conservation of the fundamental commutator in the course of time, as will be expressed in a simple fluctuation-dissipation relation. Making a canonical transformation back to the real, Cartesian Hermitian position and momentum an "effective" Hermitian Hamiltonian will be derived, with which a transformation is made from the Heisenberg frame to the Schr\"odinger frame where the density operator equation will be computed. This will make it obvious that no proper Schr\"odinger equation exists for the dissipative subsystem on its own, thus reflecting an incomplete knowledge. The master equation will then be translated into its Wigner representation. The intimate connection between the diffusion coefficients in the resulting Fokker-Planck equation and the uncertainty relation will be demonstrated in a clear fashion.
The multidimensional damped harmonic oscillator is treated by means of a non-self-adjoint Hamiltonian with complex potential. The propagator referring to the evolution semigroup is evaluated from the Lie–Trotter formula. The one-dimensional case is discussed in detail with the following results: (a) the nondamped limit gives the correct propagator including the Maslov phase factor, (b) for some initial conditions, the classical limit of the solution can differ from the behavior of the classical damped oscillator, the difference being negligible in the case of weak damping, and (c) the point spectrum of the considered pseudo-Hamiltonian is found.
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.