Mode locking of a driven Bose-Einstein condensate

We consider the dynamics of a driven Bose-Einstein condensate with positive scattering length. Employing an accustomed variational treatment we show that when the scattering length is time modulated as $a\{1+ϵ\sin \left[\omega \left(t\right)t\right]\}$, where $\omega \left(t\right)$ increases linearly in time, i.e., $\omega \left(t\right)=\gamma t$, the response frequency of the condensate locks to the eigenfrequency for small values of $ϵ$ and $\gamma$. A simple analytical model is presented which explains this phenomenon by mapping it to an auto-resonance, i.e., close to resonance the reduced equations describing the collective behavior of the condensate are equivalent to those of a virtual particle trapped in a finite-depth energy minimum of an effective potential.