Perturbative expansion for the master equation and its applications

Abstract

We construct generally applicable small-loss rate expansions for the density operator of an open system. Successive terms of those expansions yield characteristic loss rates for dissipation processes. Three applications are presented in order to give further insight into the context of those expansions. The first application, of a two-level atom coupling to a bosonic environment, shows the procedure and the advantage of the expansion, whereas the second application that consists of a single mode field in a cavity with linewidth κ due to partial transmission through one mirror illustrates a practical use of those expansions in quantum measurements, and the third one, for an atom coupled to modes of a lossy cavity shows the another use of the perturbative expansion. [11,12]. The problem can be described generally as interest in the effective dynamics of one subsystem of several interacting subsystems. A formal framework to describe the effective dynamics of such a subsystem is set up in ref. [13], and a short-time perturbative expansion for coherence loss has also been constructed [14]. To some extent(for example, if we are interested in a behavior for finite time), however, time is not as good as the loss rate as a perturbative parameter. Motivated by this and recent experimental developments [15][16][17][18][19][20] as well as the analysis of models related to them [21][22][23][24], we construct generally smallloss rate expansions for dissipation. The results suggest that these are useful in many areas such as high-Q Cavity QED [15-20], quantum computation[11,12,21-26], quantum measurement [27,28], quantum optics [3][4][5][6][7] etc.We consider an open quantum system, the total Hamiltonian describing such a system is expressed aswhere H 0 and H env indicate the free Hamiltonian of the system and of the environment, respectively. H I is the interaction Hamiltonian between the system and the environment. It is well known that the form of the master equation depends on the precise kind of the system-environment interaction. In order to derive * Corresponding address 1

Hall conductance and topological invariant for open systems

“…In an optical lattice setup, such a dissipative dynamics can be engineered by manipulating couplings of the lattice to different atomic species, which play the role of the dissipative bath [17][18][19][20][21][22]. In a recent work [23], we have developed a perturbation theory for open systems based on the Lindblad master equation. In this approach, the decay rate was treated as a perturbation.…”

Dissipation and decoherence in collision processes: Beyond dynamical symmetry and the non-autonomous quantum master equation

“…(16), the eigensolutions of the rate operator are obtained (2), respectively. After some calculation, we obtained …”

“…In order to solve this problem, the usual method is to convert the master equation to a set of differential equations for some quantum statistical moments or expansion coefficients in terms of some bases truncated at a reasonable order [4,7]. Later some other useful approximate and exact methods have been used, such as the short-time expansion [15], the small loss rate expansion [16], and so on [17][18][19]. But these problems were only restricted to the autonomous case.…”

Application of Perturbation Theory to a Master Equation

“…Unfortunately, even this master equation is difficult to treat and to solve in an exact form [13][14][15]; hence, it is often required to apply a perturbative treatment. Many perturbative methods have been developed to solve particular problems modeled by the Lindblad master equation, such as a two-level nonlinear quantum system, a single-mode field in a lossy cavity, two-level atom coupling to a Bose-mode environment, and a single atom coupling to a mode of a lossy cavity [16][17][18]. In fact, it has been shown that even though decoherence takes place, the reconstruction of quasiprobability distribution functions may be achieved in atomfield [19,20] or laser-trapped ion interactions [21].…”