Generating, dragging, and releasing dark solitons in elongated Bose-Einstein condensates

We theoretically analyze quasi-one-dimensional Bose-Einstein condensates under the influence of a harmonic trap and a narrow potential defect that moves through the atomic cloud. Performing simulations on the mean-field level, we explore a robust mechanism in which a single dark soliton is nucleated and immediately pinned by the moving defect, making it possible to drag it to a desired position and release it there. We argue on a perturbative level that a defect potential which is attractive to the atoms is suitable for holding and moving dark solitons. The soliton generation protocol is investigated over a wide range of model parameters and its success is systematically quantified by a suitable fidelity measure, demonstrating its robustness against parameter variations, but also the need for tight focusing of the defect potential. Holding the soliton at a stationary defect for long times may give rise to dynamical instabilities, whose origin we explore within a Bogoliubov–de Gennes linearization analysis. We show that iterating the generation process with multiple defects offers a perspective for initializing multiple soliton dynamics with freely chosen initial conditions.

Figure 1

Effective potentials $W\left(x_0\right)$ for the soliton position coordinate $x_0$ as obtained from soliton perturbation theory. In each case, the underlying atomic potential consists of a harmonic trap and a Gaussian of variable width $\sigma$ and fixed attractive amplitude $V_0=-1$ centered at $x=0$ (black circles for repulsive potential $V_0=1$), $\mu =1$ throughout.

Figure 2

Generating, dragging, and releasing a dark soliton. The white line indicates the trajectory of the Gaussian impurity (parameters $V_0=-12,\sigma =0.1$). At $t=129$ the impurity is switched off. (a) Density ${\left|\psi (x,t)\right|}^2$. (b) Snapshots of the phase angle profile at different times.

Figure 3

Black soliton generation fidelity $\overline{S}$ as a function of the potential strength $V_0$ and width $\sigma$. The Gaussian enters the cloud at a velocity $v=0.0925$, the initial chemical potential $\mu =1$.

Figure 4

Density evolution ${\left|\psi (x,t)\right|}^2$ for some of the simulations underlying Fig. 3. The parameters of the Gaussians are $(V_0,\sigma )=(-12,0.1)$ (a), $(-7,0.28)$, (b) and $(-8,0.19)$ (c), respectively.

Figure 5

Black soliton creation fidelity as a function of the impurity velocity for two different sets of parameters of the Gaussian potential.

Figure 6

Controlled creation of a dark soliton, followed by a long hold time. The parameters of the Gaussian are $V_0=-12, \sigma =0.08, v=0.0925$ (before reaching $x=0$). (a) Density evolution ${\left|\psi (x,t)\right|}^2$. The piecewise linear trajectory of the impurity is marked by a white line. The white box highlights the first region of dynamical instability. (b) Position of the density minimum (as a measure of the soliton center) as a function of time in the time interval marked by the box in (a).

Figure 7

BdG linearization spectra of the dark soliton state in the presence of a Gaussian potential in its center. All BdG frequencies are given in units of ${\omega }_{\perp }$. Blue crosses, red circles, and green asterisks denote the normal, anomalous, and complex modes, respectively. The chemical potential is fixed at $\mu =1.15$, yielding the same norm as the ground state at $\mu =1$. (a) $V_0=-12$, revealing a complex mode at $\sigma =0.08$, corresponding to the parameter set of Fig. 6. (b) $V_0=-0.25$, including also the linearization frequency predicted from the soliton perturbation theory (black dots).

Figure 8

Generating two solitons with two impurity potentials moving independently on the trajectories indicated by white lines. Colors encode the density ${\left|\psi (x,t)\right|}^2$. For both Gaussian potentials, $V_0=-14, \sigma =0.09$.