Two-level system immersed in a photonic band-gap material: A non-Markovian stochastic Schrödinger-equation approach

It is our aim to study the dynamics of a two-level atom immersed in the modified radiation field of a photonic band-gap material using non-Markovian stochastic Schrödinger equations. Up to now, such methodology has only been applied to toy models and not to physically realistic systems as the one presented here. In order to check its validity, we shall study several of the physical phenomena already described in the literature within non-Markovian master equations, such as the long-time-limit residual population in the excited level of the atom and the population inversion which occurs in the atomic system when applying an external laser field. In addition to the stochastic equation, we propose a non-Markovian master equation derived from the stochastic formalism, which in contrast to the current models of master equation preserves positivity. We propose a correlation function for the radiation field (environment) that captures many of the physically relevant aspects of the problem and describes the short-time behavior in a more accurate way than previously proposed ones. This characteristic permits a correct description of the fluctuations of the electromagnetic field, which in the stochastic formalism are represented by the noise, and a better description of the non-Markovian effects in the atomic dynamics. The methodology presented in this paper to apply stochastic Schrödinger equations can be followed to study more complex systems, like many-level atoms embedded in more complicated photonic band-gap structures.