Quantisation of particle motion in dissipative harmonic environment

J. Geicke

In order to avoid the known problems of canonical quantisation of the dissipative equation mx+ alpha x n + delta V/ delta x=0, a modified approach is proposed. It is based on the classically conserved quantity E=mx 2 /2+V(x)+ alpha integral x n dx rather than on a Hamilton function. Linear, quadratic and cubic damping (n=1, 2 and 3) are discussed explicitly. For n=1 and n=3 normalisable (one-particle) states cannot be stable. However, for quadratic friction, which is classically a reversible process, the author shows, using the example of the harmonic oscillator, that stable normalisable eigenstates can exist.

M. C. Valsakumar

Subhashish Banerjee, Rupamanjari Ghosh

The dynamics of an open quantum system exhibiting the quantum Brownian motion is analyzed when the coupling between the system and its environment is nonlinear, and the system and the reservoir are initially correlated. For couplings quadratic in the environment variables, the influence functional for the system is obtained perturbatively up to second order in the coupling constant, and then the propagator is explicitly evaluated when the particle is under the influence of a harmonic potential and an additional anharmonic potential, the so-called washboard potential. As an application of the propagator, the master equation and the Wigner equation are obtained for the quantum Brownian particle moving in a harmonic potential for the generalized correlated initial condition, and then for the specific case of the simplified "thermal" initial condition. The system is shown to obey the corresponding fluctuation-dissipation theorem.

Ronit Mahapatra, Asrarul Haque

M. Hron, Mohsen Razavy

A wave equation formulation of the problem of quantum tunneling in a dissipative medium is developed by considering a many-body system in which the central particle is subject to an arbitrary force law, and at the same time is coupled to a bath of noninteracting harmonic oscillators. For the motion of the central particle it is possible to obtain an effective Lagrangian and Hamiltonian by eliminating the degrees of freedom of the oscillators. However both of these operators are nonlocal, and it is difficult to derive a wave equation for this motion. As an alternative method one can write a many-body Schrödinger equation for the whole system, and then eliminate the wave functions of all of the oscillators. This result is a many-channel Schrödinger equation for the motion of the central particle. By truncating this set of coupled equations, one can solve the problem for different force laws. In particular, in this work, the cases of dissipative tunneling, hopping, and quantum coherence are studied in detail. It is also shown how this approach can be generalized to multidimensional dissipative systems.

Alberto Acevedo, Manuel Berrondo, Jean-François S. Van Huele