Quantum dynamics of a damped deformed oscillator

The interaction of a quantum deformed oscillator with the environment is studied deriving a master equation whose form strongly depends on the type of deformation. PACS number(s): 03.70.+k, 02.50.-r Quantum groups [1] introduced as a mathematical description of deformed Lie algebras have given the possibility to generalize the notion of creation and annihiliation operators of the usual oscillator and to introduce q-oscillators [2, 3]. Soon after, several attempts have been made to give a physical interpretation of the q-oscillators. In Ref. [4] they appear in connection with the relativistic oscillator model, while in Ref. [5] q-oscillators have been used in the generalized Jaynes-Cummings model. Instead, the nature of q-oscillators of the electromagnetic field is clarified by the nonlinearity of the field vibrations [6]. This q-nonlinearity reflects the exponential growth of the frequency of vibrations with the amplitude. This observation suggests that there might exist other types of nonlinearities for which the frequency of oscillation varies with the amplitude via a generic function f ; this leads to the concept of f -deformed oscillators [7]. Then, the generation of coherent states for a class of f -deformed oscillators enter in the real possibilities of trapped systems [8]. By virtue of possible further physical realizations of f -oscillators, it could be interesting to study the environmental effects on such oscillators. The aim of this paper is to describe the behavior of a nonlinear oscillator, like the f -oscillator, plunged in a bath modeled by an assembly of harmonic oscillators. A master equation approach is developed to consider several types of reservoir, restricted however to the case of small damping. The free Hamiltonian of an f -deformed oscillator may be taken, in the case of unitary frequency and natural units, as [7] H0 = 1 2 ( AA + AA )