Dynamical Critical Behavior of an Attractive Bose-Einstein Condensate Phase Transition

When matter undergoes a continuous phase transition on a finite timescale, the Kibble-Zurek mechanism predicts universal scaling behavior with respect to structure formation. The scaling is dependent on the universality class and is irrelevant to the details of the system. Here, we examine this phenomenon by controlling the timescale of the phase transition to a Bose-Einstein condensate using sympathetic cooling of an ultracold Bose thermal cloud with tunable interactions in an elongated trap. The phase transition results in a diverse number of bright solitons and gray solitons in the condensate that undergo attractive and repulsive interactions, respectively. The power law dependence of the average soliton number on the timescale of the phase transition is measured for each interaction and compared. The results support the Kibble-Zurek mechanism, in that the scaling behavior is determined by universality and does not rely on the interaction properties.

Figure 1

Critical behavior of BEC in an elongated trap. (a) The arrows indicate the phases of the wave function. The average size of independent phase domains present near the critical point is determined by the quench rate. In the repulsive case, gray solitons appear between the domains, while in the attractive case, independent BEC domains themselves become bright solitons. (b) In situ samples of finite temperature attractive BEC and repulsive BEC of ${}^7\mathrm{Li}$ after phase transitions. Multiple bright solitons or gray solitons are produced after fast quenches.

Figure 2

Properties of attractively interacting bosons during quenches. (a) Cooling efficiency measured from TOF imaging. The cloud reaches a final temperature of 120 nK only for evaporations longer than 9 s, indicating a maximum cooling rate of about $20\mathrm{nK}/\mathrm{s}$. (b) Temperature profiles examined from TOF imaging. The quenches start at 300 nK and stop at 120 nK with different total quench times. Linear decreases in temperature can be observed. (c) Condensate fractions. A critical temperature of 190 nK is measured. The condensate fractions increase similarly for all quenches after the phase transition. (d) Sample in situ images and their optical densities. The first row corresponds to atoms merely below the critical temperature. The second row presents growing solitons. The third row shows three, two, and one solitons (from top to bottom) formed at the end of linear quenches.

Figure 3

Counting statistics of spontaneously formed solitons under different quench times. (a) Bright solitons with attractive interaction. The line shows the fitted power law with the exponent $2.12\pm 0.44$. Averaged over 27 data. (b) Gray solitons in repulsive BEC. The line shows the fitted power law with the exponent $2.57\pm 0.46$. Averaged over 24 data. The error bars show the standard error of the mean.