Density matrix of the system of identical particles, interacting via oscillator forces

Boris Volodumurovich Bondarev

Vladimir I. Man’ko, Luigi Rosa, Patrizia Vitale

We study the symmetry properties under permutation of tomograms describing the states of a system of identical particles in the probability representation. Starting from the action of the permutation group on the density matrix, we define its action on the tomographic probability distribution. Explicit calculations are performed for the two-dimensional harmonic oscillator.

G. N. Brodimas, A. D. Jannussis, N. Patargias

Е. А. Титов

Exact expressions for the statistical sum of the grand canonical ensemble and the one-particle density matrix are derived based on the definition of the density matrix for a system of N identical noninteracting Bose particles in an oscillator potential as a sum with respect to the symmetric exchange of the density matrix coordinates of distinguishable particles. A quasi-classical scenario is analyzed in detail.

Jan F. Maćkowiak, O. Missuno, S. Pruski

Mohammad Attrash, Roi Baer

Previous years researchers began to simulate open quantum system, taking into account the interaction between system and the environment. One approach to deal with this problem is to use the density matrix within the Liouville-von-Neumann formalism or the Markovian variant the Lindblad equations. Another way is to use a stochastic approach where a random force is added to the system. The benefit of the stochastic approach is to solve the dynamics of the system with less time and memory than the density matrix approaches. In this project we want to develop a stochastic approach that can deal with the stochastic wave functions approach. We did this on a 2-level system and found that it works well when comparing to a density matrix approach. Next, we tested a quantum particle connect to a bath of harmonic oscillators using the stochastic approach. We found that a friction term is necessary and applied it. Like in the classical Langevin equations the friction constant and the random force fluctuations are related by the fluctuation-dissipation constant. We showed that with friction the dynamics decays to an ensemble with energy of $E_{gs}+k_BT$. However, we also found here are problems. The system seems to absorb energy indefinitely if the temperature is higher than the zero point energy or if the system is a Morse oscillator. Thus more research is required to make this method work.