Solutions of Density Operator Master Equations

Z. Ficek, Mohamed Ridza Wahiddin

In Chapter 9, we derived the master equation for a reduced density operator of a two-level atom coupled to a multi-mode vacuum field. We have seen that the master equation is an operator type equation that, in general, is not easy to solve. We have illustrated the solution of the master equation by representing the density operator in the basis of the two states of the two-level atom. The purpose of the representation of the master equation was to obtain c-number differential equations that are equivalent to the operator equations, but are more readily soluble.

Ravinder Rupchand Puri

F. Gao, Heng Fan, Feng Shuang +1 more

Based on the Kraus-form solution to the master equation describing diffusion we develop an integral form solution by using the method of integration within ordered product of operators, i.e., the evolution law of density operator in diffusion channel can be considered as an integration transformation from an input to its output density operator. It brings much convenience for obtaining the time evolution law in the diffusion process via this formalism.

M. Bönitz

Qian Ye, Chen Qian-Fan, Fan Hong-Yi

Open quantum system, namely system-reservoir model, is described by a master equation of density operator. For example, the Caldeira-Leggett eqaution describes dissipative phenomenon of solid physics. Although some efforts have been made to derive the exact expression of this master equation, so far as we know, it has not been reported in the literature. The purpose of this paper is to provide a new approach to solving the Caldeira-Leggett equation, via this approach the explicit integral-form expression of ρ(t) can be obtained. The main point of this approach is to convert equation of density operator into an equation of density state vector, and then project density state vector into thermo entangled state representation and convert it into wave function by using the technique of integration within an ordered product of operators. Thus the master equation for Caldeira-Leggett model is converted into an differential equation of wave function. Wave function is also a function. The wave function can be obtained via the approach to solving the differential equation in mathematics. It can be converted into a density state vector and density operator. Using the technique of integration within an ordered product of operators again, the integra-form solution of the Caldeira-Leggett equation is obtained.