Fractional Langevin Equation in Quantum Systems with Memory Effect

Abstract

In this paper, we introduce the fractional generalized Langevin equation (FGLE) in quantum systems with memory effect. For a particular form of memory kernel that characterizes the quantum system, we obtain the analytical solution of the FGLE in terms of the two-parameter Mittag-Leffler function. Based on this solution, we study the time evolution of this system including the qubit excited-state energy, polarization and von Neumann entropy. Memory effect of this system is observed directly through the trapping states of these dynamics.

Pseudospectral operational matrix for numerical solution of single and multiterm time fractional diffusion equation

“…In recent years, due to the accuracy of fractional differential equations in describing a variety of engineering and physics fields, such as kinetics [23,24,26,27,34,35,39], solid mechanics [32], quantum systems [38], magnetic plasma [25], and economics [3], many researchers are interested in fractional calculus. In [39] the concepts of fractional kinetic, such as particle dynamics in different potentials, particle advection in fluids, plasma physics, fusion devices, and quantum optics, were discussed.…”

Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart

“…Derivatives of non-integer order have been particularly successful in describing a variety of complex processes with memory effects. These include applications in statistical finance [1], economic modelling [2], image processing [3], quantum systems [4] and kinetics [5][6][7][8][9][10][11]. This paper will focus on the numerical solution of a fractional kinetics description of anomalous diffusion.…”