Parametric excitation of a Bose-Einstein condensate in a one-dimensional optical lattice

We study the response of a Bose-Einstein condensate to a periodic modulation of the depth of an optical lattice. Using Gross-Pitaevskii theory, we show that a modulation at frequency $\Omega$ drives the parametric excitation of Bogoliubov modes with frequency $\Omega ∕2$. The ensuing nonlinear dynamics leads to a rapid broadening of the momentum distribution and a consequent large increase of the condensate size after free expansion. We show that this process does not require the presence of a large condensate depletion. Our results reproduce the main features of the spectrum measured in the superfluid phase by Stöferle et al. [Phys. Rev. Lett. 92, 130403 (2004)].

Figure 1

Upper panel: the width of the density distribution ${⟨z^2⟩}^{1∕2}$ after $25\mathrm{ms}$ of free expansion is plotted as a function of the modulation time $t_{\mathrm{m}}$, as obtained with $\Omega =1.33E_R∕\hslash$ and $A=0.2$. Lower panel: population $N_q$ of the parametrically excited mode with $q=0.56q_B$ and $\omega =\Omega ∕2$. The different lines refer to different initial states. Solid line: ground state; dashed: ground state plus a weak Bragg pulse; dotted: ground state plus a small white noise.

Figure 2

Density (left) and momentum (right) distributions at different times $A$, $B$, $C$, and $D$ given in Fig. 1.

Figure 3

For each modulation of given frequency $\Omega$, we draw a point at $(q,\omega )$, where $q$ is the wave vector of the parametrically excited mode and $\omega =\Omega ∕2$. The solid line is the lowest Bogoliubov band.

Figure 4

Width ${⟨z^2⟩}^{1∕2}$ of the expanded array of condensates as function of $\Omega$. Top panel: experimental data of Ref. 2, for $s=4$; Solid line in bottom panel: our simulations starting from a small white noise. Empty circles and solid squares: same but with a much larger white noise and thermal-like noise, respectively (see text for explanation).