Interaction-sensitive oscillations of dark solitons in trapped dipolar condensates

Thanks to their immense purity and controllability, dipolar Bose-Einstein condensates are an exemplar for studying fundamental nonlocal nonlinear physics. Here we show that a family of fundamental nonlinear waves—the dark solitons—are supported in trapped quasi-one-dimensional dipolar condensates and within reach of current experiments. Remarkably, the oscillation frequency of the soliton is strongly dependent on the atomic interactions, in stark contrast to the nondipolar case. Established analytical techniques are shown to not capture the simulated dynamics. These sensitive waves may act as mesoscopic probes of the underlying quantum matter field.

Figure 1

Density profile $n\left(z\right)$ of the quasi-1D dipolar BEC (polarization perpendicular to the axis) featuring a central black soliton, as a function of ${\varepsilon }_{\text{dd}}$. The vdW interactions satisfy (a) $\left|\beta \right|=61$ and (b) $\beta =1500$, and the trap frequency ratio ${\omega }_z/{\omega }_{\perp }=0.0025$. Only the regimes satisfying Eq. (3) are shown, with the line $a_{\mathrm{eff}}=0$ indicated (yellow dashed line). The color scale is normalized to the peak density of the soliton-free BEC, $n_0(z=0)$. The roton-unstable regions extend to ${\varepsilon }_{\text{dd}}=\pm \infty$. (c) Example density profiles, for ${\varepsilon }_{\text{dd}}=-1.7$ (blue dot dashed line), ${\varepsilon }_{\text{dd}}=-74$ (red solid line), and ${\varepsilon }_{\text{dd}}=0$ (yellow dashed line). (d) Soliton-free density profile (solid lines), with the TF prediction of Eq. (4) overlaid (dotted), for the same ${\varepsilon }_{\text{dd}}$ values as in (c).

Figure 2

Density dynamics of a dark soliton in the quasi-1D dipolar condensate for (a) ${\varepsilon }_{\text{dd}}=-1.7$, (b) ${\varepsilon }_{\text{dd}}=0$, and (c) ${\varepsilon }_{\text{dd}}=-5.5$. These values correspond to close to $a_{\mathrm{eff}}=0$, the nondipolar case, and close to the roton instability, respectively. Remaining parameters as in Fig. 1. Time is expressed in units of $\tau =1/{\omega }_z$.

Figure 3

Oscillation frequency of the dark soliton (starting as an off-center black soliton) based on the 1D dipolar GPE (red circles and triangles), 3D dipolar GPE (blue crosses and diamonds). The system parameters are as Fig. 1; the 3D system also assumes 164Dy atoms, ${\omega }_{\perp }=2\pi \times 16$ kHz and $|a_s|=50a_0$, where $a_0$ is the Bohr radius. The 3D system is stable only in the range of markers.

Figure 4

Oscillation frequency and phase diagram for a dark soliton in a ${}^{164}\mathrm{Dy}$ BEC with Feshbach tuning of $a_s$, based on a recent experiment setup [44]. Shown are cases where the soliton is imposed in the initial condition (blue triangles, as per Fig. 1) and imprinted in real time (red crosses). Outside of the dark soliton regime, the condensate is either roton unstable (red), an attractive condensate incapable of supporting dark solitons (grey), or the dark solitons are dimensionally unstable. Parameters: $\theta =\pi /2$, $({\omega }_{\perp },{\omega }_z)=2\pi \times (128,2)$ Hz, $a_{\text{dd}}=132a_0$, and $N=10000$.

Figure 5

Left: Density of the atomic cloud with soliton as a function of time and position. Right: Schematic cut through density profile at two instances of times: $t=0$, situation A, when the soliton is displaced from the trap center by $z_0$ and it has a zero velocity, and $t=\pi /\left(2{\omega }_{\mathrm{s}}\right)$, situation B, when the soliton is at a quarter of period, passing the center of the trap with the velocity ${\omega }_{\mathrm{s}}{z}_0$.