Thermalization, condensate growth, and defect formation in an out-of-equilibrium Bose gas

We experimentally and numerically investigate thermalization processes of a trapped ${}^{87}\mathrm{Rb}$ Bose gas, initially prepared in a nonequilibrium state through partial Bragg diffraction of a Bose-Einstein condensate (BEC). The system evolves in a Gaussian potential, where we observe the destruction of the BEC due to collisions and subsequent growth of a new condensed fraction in an oscillating reference frame. Furthermore, we occasionally observe the presence of defects, which we identify as gray solitons. We simulate the evolution of our system using the truncated Wigner method and compare the outcomes with our experimental results.

Figure 1

(a) A sequence of concatenated time-of-flight absorption images showing the momentum evolution of the system in the $y$ dimension as a function of in-trap hold time $t$. Each image is an average of three experimental realizations. (b) A numerical simulation of the same system evolution using the truncated Wigner method, as described in Sec. 5. (c) The average of 100 experimental time-of-flight images for $t=0$, with an $s$-wave scattering halo containing $\sim 25\%$ of the atoms formed. (d) A time-of-flight image following 16-ms of in-potential evolution, where the system has thermalized to form a new condensed fraction. (e) Integrating along the $x$ dimension of panel (d) clearly shows the bimodal nature of the system. Here, we fit a Thomas-Fermi profile to the condensed fraction and a Bose-enhanced Gaussian to the remaining atoms.

Figure 2

(a) Atomic density integrated within a $0.45\hslash k_{\ell }$ region about the $|1\hslash k_{\ell }\rangle$ center-of-momentum state as a function of in-potential evolution time $t$ for the experimental images shown in Fig. 1 (blue circles). We model this growth accounting for Bose enhancement (red dashed line), from which we are able to extract a characteristic growth rate of $\gamma /\left(2\pi \right)=160\pm 15$ Hz. A similar integration is performed for the numerical simulation previously shown in Fig. 1 (black line). (b) Growth of the momentum variance in the $x$ dimension indicating the presence of cross-dimensional thermalization, fit with a growth curve from which we find that ${\gamma }_x/\left(2\pi \right)=115\pm 20$ Hz. (c) Growth rates of the $|1\hslash k_{\ell }\rangle$ state extracted for various systems which thermalize above the critical temperature $T_c$ (blue dots). The growth rates for systems that thermalize below $T_c$ are shown in red, with the data from panel (a) shown as a cross.

Figure 3

Snapshots of the system evolution following the Bragg pulse using Eq. (2) in (a) real space, integrating over $z$ with horizontal axis $x$, and (b) momentum space, integrating over $k_z$ with horizontal axis $k_x$, with $N=2\times {10}^4$, $({\omega }_x,{\omega }_y,{\omega }_z)/\left(2\pi \right)=(264,220,440)$ Hz, and $w=35$ $\mu \mathrm{m}$. The white circle shown in the first image of panel (b) indicates the region where noise is added to the truncated Wigner simulation. (c) Experimental time-of-flight images corresponding to the theoretical images. The images selected do not exactly correspond to multiples of ${\omega }_y/4$ due to the 200-$\mu \mathrm{s}$ time steps of the experiment. The experiment also involves a 10-ms free expansion during which the system will continue to evolve, hence care must be taken when making comparisons with the momentum images of panel (b).

Figure 4

(a) Experimental time-of-flight absorption image featuring a gray soliton of near to 100% contrast. The image is taken after $t=15$ ms of in-potential evolution with a trap frequency of ${\omega }_y/\left(2\pi \right)=280$ Hz. (b) Integrating a strip of the atomic density reveals the nature of the soliton. We perform a fit to the density profile consisting of a Thomas-Fermi profile for the condensed atoms, a Bose-enhanced Gaussian for the noncondensed atoms [30, 31], and a characteristic gray soliton profile.