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.

Figure 1

(Color online) The dynamics of the width of the condensate for $P=100$, $ϵ=0.01$, and $\gamma =0.05$. The condensate starts to respond periodically at ${\tau }_c={\omega }_P∕2\gamma \approx 22$; the time needed to stabilize the amplitude is, however, slightly longer. This example corresponds to 24 500 ${}^{23}\mathrm{Na}$ loaded in a magnetic trap with frequency $159\mathrm{Hz}$ and $\tau$ is measured in ms.

Figure 2

(Color online) The dynamics of the width of the condensate for $P=100$, $ϵ=1.0$, and $\gamma =0.05$. Notice the chaotic response of the condensate and the bouncing ball behavior shown in the inset. Same rescaling as in Fig. 1 so that $\tau$ is measured in ms.

Figure 3

(Color online) The dynamics of the width of the condensate for $P=100$, $ϵ=0.065$, and $\gamma =0.01$. The condensate starts to respond periodically approximately at ${\tau }_c={\omega }_P∕2\gamma \approx 111$ but there are additional superharmonic nonlinear resonances at approximately ${\tau }_c=n{\omega }_P∕2\gamma$, where $n=2,3,4,\dots$. Same rescaling as in Fig. 1 so that $\tau$ is measured in ms.

Figure 4

(Color online) The dynamics of the width of the condensate for $P=100$, $ϵ=0.065$, and $\gamma =0.001$. For such a small $\gamma$ value, the condensate responds periodically precisely at ${\tau }_c=n{\omega }_P∕2\gamma \approx 1115n$. Same rescaling as in Fig. 1 so that $\tau$ is measured in ms.