Fluctuating and dissipative dynamics of dark solitons in quasicondensates

The fluctuating and dissipative dynamics of matter-wave dark solitons within harmonically trapped, partially condensed Bose gases is studied both numerically and analytically. A study of the stochastic Gross-Pitaevskii equation, which correctly accounts for density and phase fluctuations at finite temperatures, reveals dark-soliton decay times to be lognormally distributed at each temperature, thereby characterizing the previously predicted long-lived soliton trajectories within each ensemble of numerical realizations [S. P. Cockburn et al., Phys. Rev. Lett. 104, 174101 (2010)]. Expectation values for the average soliton lifetimes extracted from these distributions are found to agree well with both numerical and analytic predictions based upon the dissipative Gross-Pitaevskii model (with the same ab initio damping). Probing the regime for which 0.8 $k_{B}T<\mu <1.6$ $k_{B}T$, we find average soliton lifetimes to scale with temperature as $\tau \sim T^{-4}$, in agreement with predictions previously made for the low-temperature regime $k_{B}T\ll \mu$. The model is also shown to capture the experimentally relevant decrease in the visibility of an oscillating soliton due to the presence of background fluctuations.

Figure 1

Form of the equilibrium dissipation $\gamma \left(z\right)$ at $T=250$ nK when the SGPE density (black solid line) and the Thomas-Fermi density (blue dot-dashed line) are used in the mean-field potential $U\left(z\right)$. The vertical dotted lines indicate the $T=0$ Thomas-Fermi radius $R$.

Figure 2

Illustrative dissipative dark-soliton dynamics in the (top) presence or (bottom) absence of stochastically sampled fluctuations for $T\approx 1.9T_{\phi }\approx 0.5T_{\mathrm{qc}}$. Dissipative effects lead to the decay of the soliton in each case, while fluctuations (in the stochastic simulations) additionally lead to reduced soliton visibility and less regular oscillations. Data here are for ${}^{87}$Rb atoms with ${\omega }_z=2\pi \times 10$ Hz, ${\omega }_{\perp }=2\pi \times 320$ Hz, $\mu =30\hslash {\omega }_z$, and $T=104$ nK.

Figure 3

(a) Average density profile from an ensemble of stochastic simulations, with each realization containing an initially static, black soliton. Thermal diffusion of individual solitons leads to the initially very deep notch average density becoming gradually filled. Snapshots are shown at several times between the initial state (black solid line) and the final state (green dot-dashed line). (b) Growth of the average central density tends toward the equilibrium value, as the soliton is lost in the average density profile, even though the solitons are still present within individual runs (see Fig. 4).

Figure 4

(a) Average density profile containing a moving soliton ($|v_{\mathrm{input}}|=0.25c$); (b) same data as (a), but focused on the soliton region. The average density is a sum over many individual density profiles, each containing a soliton. The notch in the average density profile becomes filled quickly because the solitons in each of the individual realizations follow slightly different paths and quickly end up at different positions. (c) Average of the individual soliton depths: the soliton is first located within each individual stochastic realization, and then its depth is measured. The average of these depths remains nearly constant, which shows a deep soliton remains within each individual run, despite this information being lost in the average density profile. Information on the soliton depths is lost in the average density profile because the individual soliton density notches in each run are, in general, at different spatial points. This implies that soliton information should be extracted from each realization prior to any statistical analysis.

Figure 5

Comparison of the average measured soliton velocity (black circles) to the input velocity (dashed brown line) for various temperatures. The error bars indicate the standard deviation of the measured velocities.

Figure 6

Scatterplots showing decay times, scaled to the ensemble average ($y$ axis), vs initial soliton velocity, scaled to the input velocity ($x$ axis), when $T$ is (a) 150 nK, (b) 175 nK, (c) 200 nK (d) 250 nK, and (e) 300 nK. (f) The standard deviation in the decay times (solid blue line) and initial velocities (dashed green line) vs $T$.

Figure 7

(top) Histogram of stochastic soliton decay times and (bottom) two representative sets of trajectories from the highlighted bins. The solid lines in the bottom plots show stochastic trajectories, while the dashed green lines is the purely dissipative result. The left, middle, and right plots have decay times within the corresponding left, middle, and right highlighted histogram bins. The trajectories from the middle highlighted bin have a decay time close to the mean value, and we find the trajectories to be qualitatively similar to that of the dissipative case.

Figure 8

Histogram of soliton decay times for an ensemble of few hundred SGPE runs (solid black circles) and a fit to a lognormal distribution (dotted blue line).

Figure 9

Lognormal probability distributions obtained from fitting soliton decay time histograms obtained at the temperatures indicated. The inset shows the same data with a log-scale $x$ axis; it is the logarithm of the decay time, which is Gaussian distributed.

Figure 10

Analysis of the decay time distributions: (a) Expectation value of the SGPE soliton decay time (black circles; error bars denote one standard deviation) vs temperature. Corresponding DGPE decay times are also shown (solid red diamonds) along with a fit to the function $\propto T^{-4}$ (brown dashed line). Green squares show analytical predictions from the analysis of Sec. 4; the inset shows the cumulative probability distribution: the fraction of solitons that have decayed vs time for each temperature. (b) SGPE expectation value (solid black line) compared to the geometric mean [brown (gray) solid line] and modal (green dot-dashed line) values for each temperature; (c) standard deviation and (d) skewness of $\tau$ vs temperature.

Figure 11

A comparison of $\gamma \left(z\right)$ as given by Eq. (2) (solid black line), the fit given by Eq. (8) (dashed brown line), and the approximation of Eq. (15) (dot-dashed blue line) for $T=150$ nK. The dotted red curve depicts the mean value of $\gamma \left(z\right)$ over $\left|z\right|\ll R/2$, while the vertical dotted green lines indicate the points $z=\pm R$, which define the rims of the TF cloud. The inset shows a magnification of this figure for the spatial interval $[-R/2,R/2]$: there, the agreement between the fit to $\gamma \left(z\right)$ [Eq. (8); dashed brown line] and the approximate form of this [Eq. (15); dot-dashed blue line] is excellent, as the corresponding curves essentially coincide and become indistinguishable.

Figure 12

Spatiotemporal evolution of the density of a BEC confined in a harmonic trap with $\Omega =0.05$ (${\omega }_z=2\pi \times 10$Hz) and $\mu =1$ [in the scaled units of Eq. (9)], with a dark (black) soliton initially placed at $z_0=2$ for the supercritical case with $\widetilde{{\gamma }_0}/{\gamma }_{cr}=1.89$ (top panel; $a=84.44$, $c=21.67$, and $d=93.97$), critical case with $\widetilde{{\gamma }_0}/{\gamma }_{cr}=1$ (middle panel; $a=43.67$, $c=22.19$, and $d=59.16$), and subcritical case with $\widetilde{{\gamma }_0}/{\gamma }_{cr}=0.09$ (bottom panel; $a=4.53$, $c=23.89$, and $d=19.72$). Here ${\gamma }_{cr}=0.106$. The white dashed lines across the soliton trajectories correspond to the analytical prediction of the linearized equation of motion Eq. (19), while the black dashed lines are obtained by numerically solving the nonlinear equation of motion Eq. (18).

Figure 13

Soliton trajectories as found by the numerical integration of the DGPE (solid black line), the SGPE (dash-dotted red line; single run with a decay time from the mean bin), and the nonlinear ordinary differential equation (ODE), Eq. (18) (dashed green line) for $T=150$ nK. Here $a=0.0219$, $c=26.63$, and $d=3.128$.

Figure 14

Numerical results for the visibility as a function of time for two example SGPE single runs (noisy turquoise curve, top plot, and noisy brown curve, middle plot) vs the DGPE (smooth red curve in top and middle plots); the insets show the corresponding soliton trajectories. The bottom plot shows the analytical result (green squares) vs the average visibility from the SGPE (black circles) and the visibility from the DGPE (red diamonds); the latter two data sets correspond to the local minima of oscillations, which can be seen in the top and middle plots. At this temperature ($T=175$ nK), the analytical prediction is very good up to the point that the soliton can be tracked, although it uses the approximate form for $\gamma \left(z\right)$ of Eq. (15) while the SGPE and DGPE use Eq. (2). The stochastic results are examples of solitons with decay times close to the ensemble mean, indicated by the vertical dashed line in all plots.