Creating solitons with controllable and near-zero velocity in Bose-Einstein condensates

Established techniques for deterministically creating dark solitons in repulsively interacting atomic Bose-Einstein condensates (BECs) can only access a narrow range of soliton velocities. Because velocity affects the stability of individual solitons and the properties of soliton-soliton interactions, this technical limitation has hindered experimental progress. Here we create dark solitons in highly anisotropic cigar-shaped BECs with arbitrary position and velocity by simultaneously engineering the amplitude and phase of the condensate wave function, improving upon previous techniques which explicitly manipulated only the condensate phase. The single dark soliton solution present in true one-dimensional (1D) systems corresponds to the kink soliton in anisotropic three-dimensional systems and is joined by a host of additional dark solitons, including vortex ring and solitonic vortex solutions. We readily create dark solitons with speeds from zero to half the sound speed. The observed soliton oscillation frequency suggests that we imprinted solitonic vortices, which for our cigar-shaped system are the only stable solitons expected for these velocities. Our numerical simulations of 1D BECs show this technique to be equally effective for creating kink solitons when they are stable. We demonstrate the utility of this technique by deterministically colliding dark solitons with domain walls in two-component spinor BECs.

Figure 1

Experimental concept. (a) Far-detuned laser light illuminates the surface of a DMD, which is programmed to reflect the light with the desired pattern. The DMD patterns are demagnified and imaged onto the atoms, which are represented as the red cloud. The trap laser beams are not shown in this figure. (b) Time sequence used to create solitons with the DMD pattern. The laser power during each phase of the experiment (vertical axis) was controlled using an acousto-optic modulator. The insets depict the potential resulting from the DMD patterns, accounting for the $2.8\mu \mathrm{m}$ (see footnote 1) resolution of our imaging system (because of the finite aperture, but neglecting any aberrations). (c) Absorption image of a typical BEC after TOF without solitons. (d) A BEC created with similar experimental conditions including a soliton.

Figure 2

Oscillation of dark solitons created using a $1.8\left(1\right)\pi$ phase imprint showing: (a) the standard protocol and (b) the improved protocol. In both cases, the top panel plots 1D longitudinal slices taken through our TOF-expanded BECs as a function of soliton evolution time for a single realization of the experiment at each time. The bottom panels plot the resulting soliton positions (symbols), with three realizations of the experiment for each time. The curve in (a) is a sinusoidal fit to the data described in the main text. Because the data in (b) reveals a stationary soliton, we used the procedure described in Appendix pp3 to plot a sinusoidal oscillation with amplitude $A=0.19\mu \mathrm{m}$ and ${\omega }_{\mathrm{s}}=2\pi \times 3.0\mathrm{Hz}$. Dashed lines represent the edges of our BECs.

Figure 3

Soliton oscillation amplitude (a) and velocity (b) for different imprinted phases. The imprinted phase (top axis) is related to the imprinting time (bottom axis) by our calibration of the DMD potential. Solid curves show the results of numerical simulations, and filled symbols are experimental data for the standard (orange) and improved (green) protocols.

Figure 4

Soliton crossing a domain wall. (a) Domain wall formed by atoms in different spin states. Absorption images of two-component BECs: (b) without a soliton present, (c) with a soliton in the $|f=1,m_f=-1\rangle$ component, and (d) with a soliton in the $|f=1,m_f=+1\rangle$ component. (e) Cross-sectional images resolving the domain wall and the soliton motion. (f) Extracted soliton and domain position averaged over three repetitions of the experiment.

Figure 5

Solitons oscillating in condensates including a domain wall. Position of solitons oscillating in a single-spin condensate (top) and a condensate with a domain wall present (center). For each time there are ten realization of the same experiment, and fewer points indicate that no soliton was observed in one or more realization. Solid curves are fits to $e^{\gamma }sin(\omega t+\varphi )$, where $\gamma$ accounts for the possibility that soliton decay is associated with change in the oscillation amplitude (if these were kink solitons and if the decay mechanism were to involve reduction in the soliton depth, this would increase the soliton velocity and hence its oscillation amplitude). Dashed lines represent the edges of our BECs. The survival probability obtained from these ten realizations is shown in the bottom panel. Solid curves are the fit to the survival probability function.

Figure 6

Spectral analysis for solitons created using our improved protocol. Top panel shows in a color scale the amplitude for each frequency ${\omega }_{\mathrm{s}}^{\prime }$ vs the imprinted phase. Vertical red line is placed at the imprinted phase that creates stationary solitons. Bottom panel shows the individual plots with a vertical displacement to avoid overlapping the curves.