Nonequilibrium dynamics of an ultracold dipolar gas

We study the relaxation and damping dynamics of an ultracold, but not quantum degenerate, gas consisting of dipolar particles. These simulations are performed using a direct simulation Monte Carlo method and employing the highly anisotropic differential cross section of dipoles in the Wigner threshold regime. We find that both cross-dimensional relaxation and damping of breathing modes occur at rates that are strongly dependent on the orientation of the dipole moments relative to the trap axis. The relaxation simulations are in excellent agreement with recent experimental results in erbium. The results direct our interest toward a less explored regime in dipolar gases where interactions are dominated by collision processes rather than mean-field interactions.

Figure 1

(a) The initial state of the gas, where the atoms occupy the equilibrium state of an approximately cylindrically symmetric trap, elongated in the $y$ direction. The dipole alignment direction is given by $\widehat{\varepsilon }$. (b) The experiment begins when the trap frequency in the $y$ direction is suddenly increased, sending the system out of equilibrium. Rethermalization dynamics depends on the angle $\beta$ between the dipole alignment direction and the $y$ axis.

Figure 2

A comparison between the experimentally measured rethermalization process and the results from the DSMC simulation. (a–c) The solid (red) line shows the result of the DSMC simulation, calculated analagously to Eq. (28), but along the $z$ axis. Experimental data points are shown by (blue) squares with error bars. The agreement is reasonable, especially considering that there was no postprocessing of the experimental data or any adjustments to the theory in order to produce these fits (no free parameters).

Figure 3

Pseudotemperatures along the $x$ axis [dashed (red) line], $y$ axis [dot-dashed (blue) line], and $z$ axis [solid (green) line] defined analogously to Eq. (28). In the first 14 ms (the ramp time), the temperature along the $y$ axis increases as a result of the rapid change in the trap frequency along this direction. After 14 ms, the system relaxes down towards the new equilibrium state. The rate at which $T_x,T_y$, and $T_z$ return to an equilibrium value displays a strong dependence on $\beta$, which we explore further in Sec. 5b. An unexpected feature we observe is the nonmonotonic path by which $T_z$ returns to equilibrium near $\beta ={45}^{\circ }$. The effect is shown in greater detail in (l) by zooming in on the relevant part of (f). This is certainly an interesting consequence of the anisotropic dipole differential scattering, but note that the behavior only occurs along one of the coordinate axes (the $z$ axis in this case), and overall there is $\mathit{no}$ violation of Boltzmann's $H$ theorem.

Figure 4

$\alpha$ (number of collisions required for rethermalization) as a function of the angle $\beta$ along (a) the $x$ direction [dashed (red) line] and $y$ direction [solid (blue) line] and (b) the $z$ direction. Data in (b) were taken from the experiment in Ref. [15] (data not taken in the $x$ and $y$ directions).

Figure 5

The rapid change in trapping frequency along the $y$ axis generates a large breathing mode along this direction. This is here in plots of ${\mathcal{T}}_y$ versus time for a variety of values of $\beta$. These breathing modes exist also in the momentum distribution, ${\mathcal{T}}_{p_y}$, and look identical to these plots except that the oscillations are exactly $\pi$ radians out of phase (leading to the monotonic behavior in $T_y$ shown in Fig. 3). The dashed (red) line in each of the figures represents $T_y$. We use Eq. (30) as a fit to the decay of this breathing mode. The breathing mode dynamics along the $x$ and $z$ axes are barely noticeable in our simulations.

Figure 6

The breathing mode along the $y$ axis is damped over a time scale ${\tau }_{\mathrm{osc}}$ found from Eqs. (30) and (31). The dependence on $\beta$ is shown. Note the qualitative similarity of ${\alpha }_{\mathrm{osc}}$ (shown above) to ${\alpha }_y$ in the solid (blue) line in Fig. 4. However, the oscillations take considerably longer to damp than the envelope, as ${\alpha }_{\mathrm{osc}}>{\alpha }_y$.

Figure 7

Comparison between the DSMC simulation for an instantaneous quench and the analytic formulas in Eqs. (34). (a) Covariance between position and momentum space; (b) variance in position space (proportional to ${\mathcal{T}}_y$). These particular data are for the dipole alignment direction of $\beta =0$. The DSMC simulation is shown by the solid (blue) line; the analytic formulas, by the dashed (red) line. The analytic formulas do an excellent job of correctly predicting the amplitude and phase of the oscillations. For this ratio of collision-to-trap frequency, the damping becomes appreciable, of the order of several trap periods.

Figure 8

The (maximum) number of particles in a volume element of phase space equal to $h^3$ as a function of time for two separate dipole-alignment angles: (a) $\beta =0$ and (b) $\beta =45$. The phase-space density decreases as the system equilibrates to a higher final temperature. From this, we estimate that quantum many-body effects are indeed small enough to be neglected (at least as a first approximation).

Figure 9

Pseudotemperatures along the $x$ axis [dashed (red) line], $y$ axis [dot-dashed (blue) line], and $z$ axis [solid (green) line] as a function of time for the bosonic dipole scattering cross section. Experimental data were not taken for this case, but we observe that the rethermalization rates show a strong dependence on $\beta$ (particularly along the $x$ axis).

Figure 10

(a) ${\alpha }_x$, (b) ${\alpha }_y$, and (c) ${\alpha }_z$ (number of collisions required for rethermalization) as a function of $\beta$ in the case of bosons. (d) Maximum phase-space density as a function of time for $\beta ={30}^{\circ }$.

Figure 11

Decay of the breathing mode along the $y$ axis in the case of bosons. Again, there is a strong qualitative similarity between this curve and the curve in Fig. 10, but an important quantitative difference is that ${\alpha }_{\mathrm{osc}}$ is larger by a factor of $\approx 2$.