Time-dependent adiabatic elimination with center-of-mass motion in matter-wave optics

In matter-wave optics the precise control over atomic center-of-mass and internal degrees of freedom is a key requirement for high-precision sensing devices. The theoretical description of laser-driven matter-wave manipulation often involves multi-photon transitions and the dynamics of multiple internal atomic states. In many applications, however, only a restricted subset of states are populated, while the remaining states are far detuned from transitions.

We generalize a projector-based and time-dependent formalism for adiabatic elimination to systematically separate the dynamics of a relevant subspace from the full space, to include atomic center-of-mass motion and to systems of arbitrary dimension. Our method is applicable, if the spectral gap between the chosen subspace and the eliminated states exceeds their mutual coupling, and when the coupling only varies slowly in time.

Applying the formalism to scenarios typical for matter-wave optics, we show that Doppler shifts associated with eliminated ancilla states give rise to additional contributions in the effective Hamiltonian governing the relevant subsystem.