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Physica A Statistical Mechanics and its Applications

Constants of motion and non-stationary wave functions for the damped, time-dependent harmonic oscillator

B. Rémaud, E. Susana Hernández

Published Sep 1, 1980• 21 citations
Publisher
DOI
10.1016/0378-4371(80)90206-x

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Multi-mode entanglement is investigated in a system composed of N coupled identical harmonic oscillators interacting with a common environment. We treat the problem generally by working with the Hamiltonian without the rotating-wave approximation and by considering the environment as a non-Markovian reservoir to the oscillators. We invoke an N -mode unitary transformation of the position and momentum operators and find that in the transformed basis the system is represented by a set of independent harmonic oscillators with only one of them coupled to the environment. Working in the Wigner representation of the density operator, we find that the covariance matrix has a block diagonal form that can be expressed in terms of multiples of 3 × 3 and 4 × 4 matrices. This simple property allows us to treat the problem to some extent analytically. We illustrate the advantage of working in the transformed basis using a simple example of three harmonic oscillators and find that the entanglement can persist for long times due to the presence of constants of motion for the covariance matrix elements. We find that, in contrast to what one would expect, a strong damping of the oscillators leads to a better stationary entanglement than in the case of a weak damping.
Abstract We present an exact solution of a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position. The free Hamiltonian of the proposed model has the form of the BenDaniel–Duke kinetic energy operator. The position-dependency of the mass and the angular frequency is such that the homogeneous nature of the harmonic oscillator force constant k and hence the regular harmonic oscillator potential is preserved. As a consequence thereof, a quantization of the confinement parameter is observed. It is shown that the discrete energy spectrum of the confined harmonic oscillator with position-dependent mass and angular frequency is finite, has a non-equidistant form and depends on the confinement parameter. The wave functions of the stationary states of the confined oscillator with position-dependent mass and angular frequency are expressed in terms of the associated Legendre or Gegenbauer polynomials. In the limit where the confinement parameter tends to ∞, both the energy spectrum and the wave functions converge to the well-known equidistant energy spectrum and the wave functions of the stationary non-relativistic harmonic oscillator expressed in terms of Hermite polynomials. The position-dependent effective mass and angular frequency also become constant under this limit.