Master Equation For Density Operator Research Articles

Solution of the Master Equation in Laser Process Under the Action of Linear Resonance Force

The master equation of density operator in laser process under action of linear resonance force can be concisely solved by virtue of thermo-entangled state representation and the technique of integration within an ordered product of operators. We obtain the infinitive operator-sum representation of density operator. As the application of this method, the evolution of thermal field and that of vacuum state in this model are discussed. The results show that thermal field maintains its original character, and vacuum state evolves thermal field in laser process.

Solving the qubit coupled with reservoir under time-varying external field with Ket-Bra Entangled State Method

In this paper, we first make a brief review of the general method of solving master equation of density operator, which includes the C-number method method and the super-operator method. The C-number can transform quantum master equation into Fokker-Plank equation or the differential equation of density matrix elements, and this method has a wide applicable range but the Fokker-Plank equation and differential equation are difficult to solve. Besides, the solution is not always applicable for any initial condition. The super-operator method can solve master equation efficiently compared with C-number method, however the solving process of super-operator method mostly depends on the characteristics of Lie algebra. For instance, if the corresponding Lindblad operator can be divided into the generators of Su(2) or Su(1,1) Lie group, the super-operator is no longer applicable. Thus although super-operator is more efficiently than C-number method, it has a narrow applicable range. Furthermore, other researchers have made much effort to develop super-operator method, for instance, S.J. Wang proposed the left and right action operator, the left operator is the same as the general operator, while the right action operator from the right side acts on other general operator, thus the explicit formation of super-operator can be given by this method. Fan proposed the thermal entangled state representation which can convert operator between real mode and fictitious mode. All these developments depend on Lie algebra, thus they all have a narrow applicable range just like super-operator method. We introduce a new Ket-Bra entangled state (KBES) method in this paper, which can transform master equation into Schrodinger-like equation with the corresponding Ket-Bra entangled state. Then one can use the method of Schrodinger equation such as time evolution method, perturbation method, etc. to solve the master equation. Compared with C-number method and super-operator method, the KBES method has several merits. 1) A wide applicable range, KBES method is applicable for any master equation of finite-level system in theory. 2) Compatibility with computer programming, the most crucial procedure is to calculate the exponent of Lindblad operator eFt which needs the diagonalization of F, and all this can be finished by computer. 3) Most mature methods of Schrodinger equation can be used to solve master equation because of the KBES method can transform master equation into Schrodinger-like equation. Then we study the model which two-level qubit is coupled with reservoir under time-varying external field, the corresponding master equation is deduced and solved by KBES method. Furthermore, we analyze the decoherence evolution of density operator and we consider the entanglement evolutions of two uncoupled qubit cases. We find that the external field seriously influences the decoherence process. The off-diagonal elements 10(t) become damply oscillated when the external field exists, and the frequency of oscillate keeps growing along with . Besides, the dynamic evolution of concurrence is also influenced by the external field, which leads to the occurrence of both entanglement sudden death and entanglement sudden birth, while the last ESB phenomenon only happens under the external field. Thus, we thought that one can suppress the decoherence and disentanglement process by exerting suitable time-varying external field on the open system, of course, the suitable external field can also be obtained by our KBES method in theory.

Integral-form solution of the Caldeira-Leggett density operator equation obtained by virtue of thermo entangled state representation

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.

Tomographic representation of evolution equation for density matrix of open system

The master equation for density operator of open system is written in tomographic representation in which the quantum states are described by standard positive probability distribution. Partial cases of systems with damping including known case of oscillator with friction terms are considered as example. Purity parameter evolution for different damping regime is studied.