Kibble-Zurek dynamics in a trapped ultracold Bose gas

The dynamical evolution of an inhomogeneous ultracold atomic gas quenched at different controllable rates through the Bose-Einstein condensation phase transition is studied numerically in the premise of a recent experiment in an anisotropic harmonic trap. Our findings based on the stochastic (projected) Gross-Pitaevskii equation are shown to be consistent at early times with the predictions of the homogeneous Kibble-Zurek mechanism. This is demonstrated by collapsing the early dynamical evolution of densities, spectral functions and correlation lengths for different quench rates, based on an appropriate characterization of the distance to criticality felt by the quenched system. The subsequent long-time evolution, beyond the identified dynamical critical region, is also investigated by looking at the behavior of the density wavefront evolution and the corresponding phase ordering dynamics.

Figure 1

Schematic of the homogeneous KZ mechanism, marking the interplay between the diverging system relaxation time $\tau$ (black dashed line) and the time $\left|t\right|=|\epsilon /\dot{\epsilon }|$ to the transition (solid blue lines). The intersection points of these two curves mark the crossover times $-\widehat{t}$ and $+\widehat{t}$.

Figure 2

(a) SPGPE results (black circles) and experimental data [122] (purple diamonds) for the equilibrium condensate fraction $N_0/N$ vs $T_{c,0}$. The dashed orange curve shows the ideal Bose gas prediction as a reference. The vertical yellow band marks the location of the numerically identified critical region in SPGPE simulations, $T_c\sim 445.5\pm 7.3$ nK; in this range, the total particle number is $(5.6\pm 0.1)\times {10}^6$, which corresponds to the ideal gas critical temperature $T_{c,0}=(488\pm 5)$ nK. Background color indicates the value of the chemical potential at each temperature during a quench, with $\mu \left(t\right)$ and $T\left(t\right)$ proceeding from the rightmost to the leftmost point. (b) Filled blue squares: longitudinal correlation length $l_{\mathrm{coh}}$, extracted as in Eq. (13), during a SPGPE simulation of a quench with ${\tau }_Q=150\mathrm{ms}$. Time is measured from the equilibrium critical time $t_c$, i.e., the center of the yellow vertical band [the same as in (a)]. Open black circles: same quantity calculated in SPGPE simulation for equilibrium states with input values $\mu (t-t_c)$ and $T(t-t_c)$. During the quench the growth of $l_{\mathrm{coh}}$ is delayed with respect to the instantaneous equilibrium: we find such delay to correspond to $(t-t_c)\sim 1.3\widehat{t}$ (dotted vertical cyan line), where $\widehat{t}$ (solid vertical cyan line) is the timescale predicted by the KZ mechanism. The scaled deviation, $\delta l_{\mathrm{coh}}$, between dynamical and equilibrium correlation lengths, defined by Eq. (15), is shown by the purple squares and exhibits a very rapid increase in the critical region, followed by a slower decay during the re-equilibration process, which reflects the phase ordering process. The end of the ramp is denoted by the vertical black dashed line. (c) Corresponding characteristic single-trajectory evolution of the Penrose-Onsager condensate density profiles (${\tau }_Q=150\mathrm{ms}$): plotted yellow and green regions respectively map out the density isosurfaces for $0.1\%$ and $3\%$ of the peak value of the final post-quench equilibrium condensate density; purple filaments denote region of high velocity field, corresponding to the location of spontaneously generated vortices.

Figure 3

Types of quench classified by comparing the critical wavefront $a_{x,\perp }$ and ${\widehat{a}}_{x,\perp }$ for times up to $\widehat{t}$. When the quench duration is comparable with $\widehat{t}$, it becomes quasi-instantaneous, with momenta beyond $1/\widehat{\xi }$ excited. Increasing ${\tau }_Q$ leads to a broad homogeneous quench regime (blue), which fully encompasses all experimentally relevant quenches probed in this work, whose range is marked by the vertical arrow. Slower quenches can lead to regimes where the quench is inhomogeneous only in the transverse direction (green), or in both directions (red), but both of these would require ramp durations exceeding 10 s for the current trap. Note that, since the critical wavefront starts from the trap center at $t_c$, the quench is always homogeneous at $t-t_c=0$. The black dashed lines are boundaries estimated by Eqs. (47, 48, 49, 50).

Figure 4

(a) Evolution of the peak spectral function $f(t,\mathbf{k}=0)$ in the SPGPE simulations as a function of $(t-t_c)$; (b) same curves but rescaled according to the KZ scaling predicted by the linearized theory. The scaled curves for different ${\tau }_Q$ collapse to a common scaling function $F[(t-t_c)/\widehat{t},\mathbf{0}]$ in the approximate regime $(t-t_c)/\widehat{t}\in (0,2)$. Furthermore, as predicted by Eq. (35), in the same regime there is a blow-up that begins near/shortly after $(t-t_c)/\widehat{t}\sim 1$, with the light blue vertical solid and dotted lines in the inset respectively marking the positions of $\widehat{t}$ and $1.3\widehat{t}$. The hollow squares mark $\widehat{t}$ as defined in Eq. (22), while the hollow circles mark the end of the linear ramp. The black dashed line plots the Gaussian divergent trend $\propto \exp \left\{{[(t-t_c)/\widehat{t}]}^2\right\}$ in Eq. (35).

Figure 5

Evolution of the spectral function, (31), in SPGPE simulations for different quench times. Left columns: raw data. Right column: same results but rescaled according to Eq. (33). Vertical bands mark the uncertainty of $t_c$ obtained from equilibrium analysis. The scaled curves for different ${\tau }_Q$ (at least for the three slower quenches) collapse to a common scaling function $F[(t-t_c)/\widehat{t},\mathbf{0}]$ in the approximate regime $(t-t_c)\lesssim 2\widehat{t}$. Grey lines in right column represent the parabola $u=(t-t_c)/\widehat{t}-{\widehat{\xi }}^2{k}^2=1$, along which the spectral function is predicted to be constant according to the stronger relation of Eq. (34), with $t\to t-t_c$, within their regime of validity. The horizontal pink dashed line marks the value of $1/\widehat{\xi }$, which corresponds to the largest wave number being excited up to $t-t_c=\widehat{t}$ as discussed in Sec. 3b.

Figure 6

(a) Evolution of the central particle number Eq. (51) in the SPGPE simulations as a function $(t-t_c)$; (b) same curves but rescaled according to the KZ scaling predicted by the linearized theory. Vertical lines and hollow points have the same meaning as in Fig. 4.

Figure 7

Evolution of the longitudinal correlation length defined by Eq. (13) and calculated in the SPGPE simulations for different quench times: (a) raw data as a function of $(t-t_c)$; (b) same curves rescaled according to the KZ scaling predicted by the linearized theory. The light blue vertical solid and dotted lines in the inset respectively mark the positions of $\widehat{t}$ and $1.3\widehat{t}$.

Figure 8

Evolution of (a) the central particle number and (b) scaled density wave fronts plotted in terms of $(t-t_c-1.3\widehat{t})/{\tau }_Q$. (a) Curves corresponding to different values of ${\tau }_Q$ collapse onto a single curve, with the exception of the fastest quench ${\tau }_Q=72\mathrm{ms}$ (red). The slope of this curve matches the slope of ${\mu }_f/g$, according to Eq. (42), which is portrayed by the black dashed line. Hollow circles mark the ends of linear ramps, while faint bands indicate the error bar in the determination of $N_{c,\mathrm{cen}}$. (Inset) Evolution of $l_{\mathrm{coh}}$ with the same time axis clearly demonstrating that coherence growth does not follow the same timescale, except for very slow ramps. Vertical dashed lines label the end of the different ramps. (b) Evolution of the position of the density wave fronts (see Appendix pp3) showing both longitudinal (solid curves) and transverse (dashed curves), scaled according to the corresponding final equilibrium spatial extents ($\sim 114$ and $\sim 11\mu \mathrm{m}$ for longitudinal and transverse directions).

Figure 9

Numerical identification of the critical point from SPGPE equilibrium simulations. We compare our findings based on (a) the Binder cumulant, (b) the correlation length, and (c) the order parameter $m$. The critical point is found to lie in the yellow band, during which region the Binder cumulant rapidly decreases from 2 to 1, by lowering $T$, while the correlation length and order parameter start increasing. For comparison, all plots also depict the corresponding SPGPE condensate fraction already shown in Fig. 2. Insets show zoomed in versions of the main plots. Horizontal lines in (a) indicate limiting values of $C_b$, and corresponding homogeneous critical value. Horizontal dashed line in inset to (b) indicates the value of the thermal de Broglie wavelength ${\lambda }_{dB}$ whose value in the critical region is $\sim 0.55\mu \mathrm{m}$.

Figure 10

(a) Quantities ${\widehat{t}}^{\prime }$ and ${\widehat{\xi }}^{\prime }$ given in Eqs. (B1) as a function of ${\tau }_Q$, compared to $\widehat{t}$ and $\widehat{\xi }$ used in the main text. The former correspond to the exact equilibrium critical exponent $\nu =0.67$; the latter to mean-field value $\nu =0.5$. (b) Longitudinal correlation length, as in Fig. 7, but rescaled by using $(t-t_c)/{\widehat{t}}^{\prime }$ and ${\widehat{\xi }}^{\prime }$.

Figure 11

(a) Density distribution of atoms in the classical field for different quench times, plotted as a function of $(t-t_c-1.3\widehat{t})/{\tau }_Q$. (b) Spectral function (29) for the same cases. In both columns, the dashed red line is the numerically traced position of the density wave front for the density within $[16,20]\mu {\text{m}}^{-3}$, while the pink line is spectral function $f$ of the $\mathbf{k}=\mathbf{0}$ mode. (c) and (d) show the profiles of the density distribution and the spectral function, respectively, at the time $(t-t_c-1.3\widehat{t})/{\tau }_Q\approx 0.9$, revealing the extent of excitation still present for the faster quenches.