Probing Phase Fluctuations in a 2D Degenerate Bose Gas by Free Expansion

We measure the power spectrum of the density distribution of a freely expanding two-dimensional (2D) degenerate Bose gas, where irregular density modulations gradually develop due to initial phase fluctuations in the sample. The spectrum has an oscillatory shape, where the peak positions are found to be independent of temperature and show scaling behavior in the course of expansion. The relative intensity of phase fluctuations is estimated from the normalized spectral peak strength and observed to decrease at lower temperatures, confirming the thermal nature of the phase fluctuations. We investigate the relaxation dynamics of nonequilibrium states using the power spectrum. Free vortices are observed with ring-shaped density ripples in a perturbed sample after a long relaxation time.

Figure 1

Emergence of density fluctuations in a freely expanding 2D Bose gas. Density distributions after (a) $t_e=14\mathrm{ms}$ and (d) 23 ms of time of flight. Density fluctuations gradually develop during expansion, increasing their length scale and visibility. (b), (e) The horizontal density profiles in the center of the samples. The red dashed lines indicate the averaged profiles over 10 individual realizations of the same experiment. The coherent fraction $\eta \approx 30\%$ (see text for details). The power spectrum of density distribution is measured with the magnitude square of its Fourier transform. (c), (f) The averaged power spectra corresponding to (a), (d).

Figure 2

The power spectrum of density fluctuations. (a) 1D power spectra $P(q)$ are obtained by azimuthally averaging the averaged 2D spectra for various expansion times $t_e$. Each spectrum is displayed with an offset for clarity. The dashed lines are the corresponding spectra for the averaged density distributions over 10 individual realizations. (b) $P(q)$ at $t_e=17\mathrm{ms}$ for various temperatures. The spectral peak positions, $q_n$’s, are determined from a fit of multiple Gaussian curves to each spectrum ($n$ is the peak order number). The vertical dashed lines indicate the average peak positions. (c) The empirical function $\hslash q_n^2{t}_e/2\pi m=\alpha n^{\gamma }$ are fit to the peak positions in (a) for each $t_e$. (d) The expansion time dependence of the two free parameters $\alpha$ and $\gamma$.

Figure 3

Temperature dependence of phase fluctuations at thermal equilibrium. The relative intensity of phase fluctuations is estimated from the normalized strength of the first spectral peak $\tilde{P}(q_1)$, and the relative temperature is parameterized with the coherent fraction. The upper insets show the density profiles for different temperatures. The red dashed lines are Gaussian fits to the thermal wings, from which temperatures are estimated (lower inset). Each data point consists of 10 independent measurements, and error bars indicate standard deviation.

Figure 4

Relaxation of a nonequilibrium 2D Bose gas. (a) Time evolutions of various nonequilibrium states in the plane of $\tilde{P}(q_1)$ and $\eta$ ($t_e=17\mathrm{ms}$). The dashed line is the interpolation for the thermal equilibrium data with $t_e=17\mathrm{ms}$ in Fig. 3. (b) The decay curves of the excess phase fluctuations $\tilde{P}(q_1)-{\tilde{P}}_{\mathrm{eq}}(q_1)$, where ${\tilde{P}}_{\mathrm{eq}}(q_1)$ is the equilibrium value corresponding to $\eta$ at $t_h=18\mathrm{s}$ of hold time. Density distributions of the lowest temperature sample at (c) $t_h=0\mathrm{s}$ and (d) 18 s with $t_e=17\mathrm{ms}$, and at $t_h=18\mathrm{s}$ with (e) $t_e=8\mathrm{ms}$ and (f) 23 ms. The samples in (e) and (f) were strongly perturbed to obtain more ring-shaped density ripples.