Bound mode of an atom laser

Summary form only given. Hope, Savage and Moy have presented a model Hamiltonian for an atom mirror. In conjunction with a Bose-Einstein condensate such a device can model the output coupler for an atom laser. The basic structure of the Hamiltonian is identical to the optical output coupler. A cavity mode at a particular wavevector is coupled to a set of output modes over a range of wavevectors. The difference is that the parameter in the Hamiltonian which would be the frequency in the optical case is the square of the wavevector for atoms. This has the consequence that the cavity mode is only coupled to one of the two superposition (sum and difference) modes of the output. We present a Fano diagonalisation of this output coupler model. The Hamiltonian can be expressed in terms of the diagonal operators together with operators representing a bound mode. This mode appears when the coupling is strong compared to the energy of the cavity mode. Any states of the bound mode represent excitations which will not propagate out of the cavity. We give the criterion for the existence of the mode in the cases where the interaction strengths are real and complex.