Solving the qubit coupled with reservoir under time-varying external field with Ket-Bra Entangled State Method
Authors: Ren Yi-Chong  , Fan Hong-Yi  ,   University of Science and Technology of China, Hefei 2300266, China;,   Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
Published: December 2020

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corresponding Lindblad operator can be divided into the generators of Su(2) or Su(1,1) Lie group, the super-operator is no longer applicable.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.