Propagation of sound in a Bose-Einstein condensate in an optical lattice

We study the propagation of sound waves in a Bose-Einstein condensate trapped in a one-dimensional optical lattice. We find that the velocity of the propagation of sound wave packets decreases with increasing optical lattice depth, as predicted by the Bogoliubov theory. The strong interplay between nonlinearities and the periodicity of the external potential generates phenomena that are not present in the uniform case. Shock waves, for instance, can propagate slower than sound waves, due to the negative curvature of the dispersion relation. Moreover, nonlinear corrections to the Bogoliubov theory appear to be important even with very small density perturbations, inducing a saturation of the amplitude of the sound signal.

Figure 1

GP simulation of wave packets produced by a fast $\left(T_p=1\right)$ perturbation of the type (3) in the presence of an optical lattice ($s=1$, $gn=0.5E_R$). Both the first, the second, and the third bands are excited, leading to the formation of two pairs of wave packets. For a slow perturbation ($T_p⪢1$) only the slow pair composed of phonons of the lowest band is created. Here, we plot the relative density $n(x,t)∕n(t,0)$ as defined in Sec. 2.

Figure 2

Typical result of a GP simulation for small perturbations (linear regime). Two sound packets move symmetrically outward at the Bogoliubov sound velocity. The relative density $n(x,t)∕n(x,0)$ (thin line), the convoluted signal (thick line), and the phase difference ${\varphi }_{\ell +1∕2}$ are shown. The signal amplitudes of the density and relative phase are denoted by $\Delta n$ and $\Delta \varphi$, respectively.

Figure 3

GP simulation for the relative density $n(x,t)∕n(x,0)$ at $t=480\hslash ∕E_R$ with $gn=0.5E_R$ for different lattice depths: (a) $s=0$, (b) $s=10$, and (c) $s=20$.

Figure 4

Sound velocity as a function of lattice depth $s$. The Bogoliubov prediction (solid line) and the results of the simulation (circles) with respective signal amplitudes $\Delta n$ for $gn=0.5E_R$. The signal amplitude $\Delta n$ is defined as indicated in Fig. 2.

Figure 5

DNLS simulation of the density $n_{\ell }\left(t\right)$ and relative phase ${\varphi }_{\ell +1∕2}$ for a deep lattice $\left(\delta ∕U={10}^{-2}\right)$ in the shockwave regime $\left(T_{p}b∕w\sim 5.5\right)$. Upper panels: early stage $\left(t=16\hslash ∕E_R\right)$; lower panels: late stage $\left(t=200\hslash ∕E_R\right)$.

Figure 6

DNLS simulation of the density $n_{\ell }\left(t\right)$ and relative phase ${\varphi }_{\ell +1∕2}$ for a deep lattice $\left(\delta ∕U={10}^{-2}\right)$ in the saturation regime $\left(T_{p}b∕w\sim 7\right)$. Upper panels: early stage $\left(t=16\hslash ∕E_R\right)$; lower panels: late stage $\left(t=200\hslash ∕E_R\right)$.

Figure 7

Results for the density and relative signal amplitudes $\Delta n$ and $\Delta \varphi$ obtained from DNLS simulations for varying perturbation parameters $T_p,b,w$ [see Eq. (3)], tunnel coupling $\delta$, and on-site interaction $U$ [see Eq. (12)] with $\delta ⪡U∕3$.

Figure 8

The signal amplitude $\Delta n$ as a function of the perturbation parameter $b{T}_p∕w$ at $gn=0.5E_R$. Dashed lines: GP simulation for $s=0$ and $gn=0.5E_R$ at $t=100\hslash ∕E_R$ (upper dashed line) and $t=200\hslash ∕E_R$ (lower dashed line). Solid lines: GP simulation for $s=15$ and $gn=0.5E_R$ at $t=100,200,300,400\hslash ∕E_R$ for $s=15$. The corresponding four lines exactly overlap.