Kinetic theory of dark solitons with tunable friction

We study controllable friction in a system consisting of a dark soliton in a one-dimensional Bose-Einstein condensate coupled to a noninteracting Fermi gas. The fermions act as impurity atoms, not part of the original condensate, that scatter off of the soliton. We study semiclassical dynamics of the dark soliton, a particlelike object with negative mass, and calculate its friction coefficient. Surprisingly, it depends periodically on the ratio of interspecies (impurity-condensate) to intraspecies (condensate-condensate) interaction strengths. By tuning this ratio, one can access a regime where the friction coefficient vanishes. We develop a general theory of stochastic dynamics for negative-mass objects and find that their dynamics are drastically different from their positive-mass counterparts: they do not undergo Brownian motion. From the exact phase-space probability distribution function (i.e., in position and velocity), we find that both the trajectory and lifetime of the soliton are altered by friction, and the soliton can undergo Brownian motion only in the presence of friction and a confining potential. These results agree qualitatively with experimental observations by Aycock et al. [Proc. Natl. Acad. Sci. USA 114, 2503 (2017)] in a similar system with bosonic impurity scatterers.

Figure 1

The reflection coefficient $R(k,\lambda )$ as a function of $\lambda$, where $\lambda (\lambda -1)=2m_{\mathrm{i}}{g}^{\prime }/m_{\mathrm{c}}g$. $R(k,\lambda )$ is strongly peaked at $k\approx 0$ and is periodic as a function of $\lambda$. When $\lambda$ is an integer, $R(k,\lambda )$ is exactly zero.

Figure 2

The soliton friction coefficient $\gamma$ is periodic as a function of $\lambda$, where $\lambda (\lambda -1)=2m_{\mathrm{i}}{g}^{\prime }/m_{\mathrm{c}}g$. Friction vanishes for integer $\lambda$, indicating that the soliton is reflectionless to scatterers. A system with tunable interactions enables tuning $\gamma$ without changing the number of scatterers. $\gamma$ is calculated in units of ${\mu }_{\mathrm{i}}/c\xi$, where ${\mu }_{\mathrm{i}}$ is the chemical potential of impurities, $c$ is the speed of sound in the condensate, and $\xi$ is the condensate healing length. For impurities of ${}^{173}\mathrm{Yb}$ as we have calculated here, $\gamma$ decreases with increasing temperature. Increasingly dark lines indicate higher temperatures.

Figure 3

Variance in soliton position $D_x(t,\omega )$ as a function of time from the exact expression (20) for $v_{\mathrm{i}}/c=0.02$ [orange (light gray) line] and $v_{\mathrm{i}}/c=0.1$ (black line) with $v_{\mathrm{th}}\approx 0.1$ mm/s and ${\mathrm{\Gamma }}^{-1}\approx 1$ s. Top: Results for a harmonic potential with $\omega =100\mathrm{\Gamma }$. $D_x$ grows linearly in time with additional oscillations due to confinement. The amplitude of oscillation increases with increasing $v_{\mathrm{i}}$. Dotted lines show the average value using the linear approximation in Eq. (21). Bottom: Comparison with $D_x(t,\omega )$ in the limit $\omega \ll \mathrm{\Gamma }$ (dashed lines). In the absence of harmonic confinement, $D_x$ initially grows like $t^3$ for $\mathrm{\Gamma }t\ll 1$, then grows exponentially. There is no diffusive regime.

Figure 4

Soliton lifetime as a function of initial velocity $v_{\mathrm{i}}$, with $\omega =100\mathrm{\Gamma }$ and where $c$ is the condensate speed of sound. Solitons that start at higher initial velocities have a shorter lifetime, which one would intuitively expect. Soliton lifetime is only weakly dependent on trapping frequency $\omega$.

Figure 5

Top: The chemical potential ${\mu }_{\mathrm{i}}$ of fermionic impurities from solving Eq. (A1) numerically for $N_{\mathrm{i}}=1000{}^{173}\mathrm{Yb}$ atoms with $L=250\mu \mathrm{m}$. ${\mu }_{\mathrm{i}}$ decreases slightly as $\lambda$ is increased and there are more bound states in the soliton well. Increasingly dark lines indicate higher temperatures. Bottom: Chemical potential for $T=150\mathrm{nK}$. The steps at each integer indicate an additional bound state in the soliton well. The chemical potential including the phase shift (solid line) is increased slightly from the result without it (dotted line).

Figure 6

Left: Survival probability as defined by the exact expression in Eq. (D4) (gray line) is highly oscillatory due to the dynamics in the trap. The lower bound is given by replacing ${\overline{v}}_{\mathrm{s}}\left(t\right)$ with Eq. (D5) (black line). Soliton lifetime is marked by ${\tau }_{\mathrm{s}}$ on the horizontal axis. Calculated for $v_{\mathrm{i}}=0.1c$ and $\omega =50\mathrm{\Gamma }$. Right: Survival probability for different soliton initial velocities with $\omega =50\mathrm{\Gamma }$. Survival probability falls off more quickly for faster initial velocities, with the fastest initial velocity indicated by the red (dark gray) line.