Scaling behavior of density fluctuations in an expanding quasi-two-dimensional degenerate Bose gas

We measure the power spectrum of density fluctuations emerging in a freely expanding quasi-two-dimensional (2D) degenerate Bose gas, and we investigate the scaling behavior of the spectrum for the expansion time. The power spectrum shows an oscillatory shape for long expansion times, where the spectral peak positions are observed to be shifted to lower spatial frequencies than the theoretical prediction for a noninteracting expansion case. We find the spectral peak positions in good agreement with the recent numerical simulation presented by Mazets [Phys. Rev. A 86, 055603 (2012)], where the atom-atom interactions are taken into account. We present a mean-field description of the interaction effect in the expansion dynamics and quantitatively account for the observed spectral peak shifts. The spectral shift is intrinsic to the free expansion of a quasi-2D Bose gas due to finite axial confinement. Finally, we investigate the defocusing effect in the power spectrum measurement.

Figure 1

(a) Optical density image of a condensate with quantized vortices after 17-ms time of flight. (b) Average image of the vortex core regions, which are indicated by red boxes in (a). (c) The radial profile of the density-depleted vortex core is obtained by radially averaging the image in (b). $r_v$ is the radial position of the first peak in the profile. (d) $r_v$ measured for various axial positions of the camera in the imaging setup. The optimal camera position $z_c^{*}$ is determined for the minimum $r_v$ from a parabola fit (red line) to the data. (e) $z_c^{*}$ as a function of the expansion time $t_e$. The solid line is a fit line of $z_c^{*}=4.02\left(7\right)(g{t}_e^2/2)$, where $g$ is the gravitational acceleration.

Figure 2

Images of quasi-2D condensates after expanding for (a) $t_e=14$ ms and (d) 21 ms. Density fluctuations emerge in the course of expansion. Parts (b) and (e) are the density fluctuation distributions, $\delta n\left(\vec{r}\right)=n\left(\vec{r}\right)-\overline{n}\left(\vec{r}\right)$, in the center region of the sample for (a) and (d), respectively. Here, $\overline{n}\left(\vec{r}\right)$ is the average density distribution obtained from 30 realizations of the same experiment. Power spectra $P_e(\vec{q};t_e)$ of density fluctuations at (c) $t_e=14$ ms and (f) 21 ms.

Figure 3

Power spectrum of a trapped thermal gas at high temperature. The solid line is a model fit to the spectrum, giving the relative modulation transfer function $M^2\left(q\right)$ of our imaging system (see text for details).

Figure 4

(a) Temporal evolution of the power spectrum of density fluctuations. Each spectrum is displayed with an offset for clarity. (b) The same spectra are replotted as functions of a dimensionless parameter $\hslash q_n^2{t}_e/2\pi m$. The dotted vertical lines indicate the theoretical prediction for the spectral peak positions from Ref. [21].

Figure 5

Accumulated phase $\varphi (q_n,t_e)$ calculated for the spatial frequencies $q_1$ (solid circle), $q_2$ (open square), and $q_3$ (open triangle) of the first, second, and third spectral peaks, respectively, in the power spectrum data.

Figure 6

Scaled power spectrum $[P\left(q\right)-P_m]/[P\left(q_1\right)-P_m]$ as a function of the parameter $\hslash q_n^2{t}_e/2\pi m$. The symbols are the same as in Fig. 4 and the gray dotted lines are the spectra for $t_e<18$ ms. (b) $P_m$ is the minimum value of $P\left(q\right)$ for $q_1<q<q_2$. (c) $S=[P\left(q_1\right)-P_m]/[P\left(q_2\right)-P_m]$.

Figure 7

Defocusing effect in the power spectrum measurement. Power spectra were measured for various camera positions $z_c$ near the optimal position $z_c$. The defocusing distance $d=(z_c-z_c^{*})/M^2$. The dashed lines indicate the spectral peak positions calculated from Eq. (5).