Non-Markovian dynamics in pulsed and continuous wave atom lasers

Nowadays it is a standard technique to produce a BoseEinstein condensate in the laboratory [?,?]. In order to build a coherent source of atoms, an atom laser, a major achievement was the coherent extraction of atoms from an atomic trap. At first a pulsed atom laser was built [?,?], recently also continuous wave atom lasers have been realized [?,?]. A short survey over the experimental situation is given in [?,?]. The theoretical treatment of an atom laser is usually based on the Born-Markov approximation [?,?,?,?,?], which has been used in quantum optics with great success. However, as it has been shown recently by Moy and coworkers [?], this approximation fails in a realistic parameter regime. In this article we will outline a different approach, which is based on the time-convolutionless (TCL) projection operator technique [?,?,?] to study non-Markovian effects resulting from the output coupling. This technique is based on a perturbative expansion in powers of the output coupling strength. The master equation resulting from the perturbative expansion has a form similar to the Born-Markov master equation and is also local in time. This makes the equation of motion easy to solve. As we will see the second order perturbation theory corresponds to the Born-Markov approximation. By taking higher orders of the expansion non-Markovian effects can be studied in a systematic way. It is demonstrated that the perturbative expansion holds in an intermediate coupling regime, which corresponds to realistic parameters, while the Born-Markov approximation fails for these parameters. This paper consists of two parts and is structured as follows. In section ?? we investigate the validity of the time-convolutionless projection operator technique (TCL) for a pulsed atom laser. The model investigated was discussed by Moy et al. [?]. They showed that the Born-Markov approximation fails for realistic parameters. We demonstrate that the time-convolutionless projection operator technique agrees with the exact solution for these parameters. Afterwards we apply the TCL algorithm to a simple model of a continuous wave atom laser in section ??. The numerical results obtained from a simulation using a perturbation expansion including 4th order clearly reveal strong oscillations in the occupation number of the Bose-Einstein condensate (BEC). As will be shown these oscillations can be interpreted as a quantum interference effect which clearly reveals departures from the golden rule and demonstrating the nonMarkovian dynamics of the atom laser. A short summary concludes the article.