Experimental and numerical studies of the velocity field of dark solitons in Bose–Einstein condensates are presented. The formation process after phase imprinting as well as the propagation of the emerging soliton are investigated using spatially resolved Bragg spectroscopy of soliton states in Bose–Einstein condensates of 87 Rb. A comparison of experimental data to results from numerical simulations of the Gross–Pitaevskii equation clearly identifies the flux underlying a dark soliton propagating in a Bose–Einstein condensate. The results allow further optimization of the phase imprinting method for creating collective excitations of Bose–Einstein condensates.
Experimental and numerical studies of the velocity field of dark solitons in Bose–Einstein condensates are presented. The formation process after phase imprinting as well as the propagation of the emerging soliton are investigated using spatially resolved Bragg spectroscopy of soliton states in Bose–Einstein condensates of 87 Rb. A comparison of experimental data to results from numerical simulations of the Gross–Pitaevskii equation clearly identifies the flux underlying a dark soliton propagating in a Bose–Einstein condensate. The results allow further optimization of the phase imprinting method for creating collective excitations of Bose–Einstein condensates.
Summary form only given. Bose-Einstein condensates (BEC) of weakly interacting gases allow for the new regime of nonlinear atom optics. Of special interest are macroscopically excited states like dark solitons or vortices. Solitons, as particle like excitations of matter wave fields, provide a link from condensate physics to fluid mechanics, nonlinear optics and fundamental particle physics. Dark solitons in matter waves are characterized by a local density minimum and, a steep phase gradient of the wavefunction at the position of the minimum. We report on experimental and theoretical investigation of dark solitons in BECs of /sup 87/Rb which are produced by the method of phase imprinting. A highly anisotropic confining potential allows us to be close to the (quasi) 1D situation where dark solitons are expected to be dynamically stable. By monitoring the evolution of the density profile we observe the evolution of density minima travelling at a smaller velocity than the speed of sound. By comparison to analytical and numerical solutions of the 3D Gross-Pitaevskii equation for our experimental conditions we identify these density minima to be moving dark solitons. We have studied in detail the creation and the dynamics of dark solitons as a function of evolution time and imprinted phase.
We systematically construct stationary soliton states in a one-component, two-dimensional, repulsive, Gross-Pitaevskii equation with a ring-shaped targetlike trap similar to the potential used to confine a Bose-Einstein condensate in a recent experiment [R. Mathew, A. Kumar, S. Eckel, F. Jendrzejewski, G. K. Campbell, M. Edwards, and E. Tiesinga, Phys. Rev. A 92, 033602 (2015)]. In addition to the ground-state configuration, we identify a wide variety of excited states involving phase jumps (and associated dark solitons) inside the ring. These configurations are obtained from a systematic bifurcation analysis starting from the linear, small atom density, limit. We study the stability and, when unstable, the dynamics of the most basic configurations. Often these lead to vortical dynamics inside the ring persisting over long time scales in our numerical experiments. To illustrate the relevance of the identified states, we showcase how such dark-soliton configurations (even the unstable ones) can be created in laboratory condensates by using phase-imprinting techniques.
In this thesis, we have studied the behavior of two-component dark-bright solitons in multicomponent Bose-Einstein condensates (BECs) analytically and numerically in different situations. We utilized various analytical methods including the variational method and perturbation theory. By imprinting a linear phase on the bright component only, we were able to impart a velocity relative to the dark component and thereby we obtain an internal oscillation between the two components. We find that there are two modes of the oscillation of the dark-bright soliton. The first one is the famous Goldstone mode. This mode represents a moving dark-bright soliton without internal oscillation and is related to the continuous translational symmetry of the underlying equations of motion in the uniform potential. The second mode is the oscillation of the two components relative to each other. We compared the results obtained from the variational method with numerical simulations and found that the oscillation frequency range is 90 to 405 Hz and therefore observable in multicomponent Bose-Einstein condensate experiments. Also, we studied the binding energy and found a critical value for the breakup of the dark-bright soliton. Building on these results, we have studied another situation where we have the dark-bright soliton oscillate in a harmonic potential. We found for weak trapping the internal modes are nearly independent of center of mass motion of the dark-bright soliton. In contrast, in tighter traps the internal modes couple strongly to the center of mass motion, showing that for dark-bright solitons in a harmonic potential the center of mass and relative degrees of freedom are not independent. We found this result is robust against noise in the initial condition and should, therefore, be experimentally observable. In addition, we have studied the interaction between a moving dark-bright soliton in a uniform background with internal oscillation and a fixed impurity, modeled by a delta function potential. The interaction excites different modes in the system. Our analytical model capture two of these modes: the relative oscillation between the two components, as well as the in-sync oscillation of the widths. The numerical simulations allow further internal modes like out-of-sync oscillations of the soliton widths and even shape deformations of various kinds. We identify regions in parameter space for the transmission, reflection and inelastic scattering of the dark-bright soliton by the potential barrier. We have studied the velocity of dark-bright solitons described with an ansatz that uses one center of mass variable to represent the position of the two components. We found for a dark-bright soliton the maximum velocity is limited by the relative number of atoms in the bright component as compared to the size of the hole or density notch created by the dark component. Above this critical velocity the dark-bright soliton develops internal oscillations, and eventually unbinds and breaks apart.
We present experimental data showing the head-on collision of dark solitons generated in an elongated Bose-Einstein condensate. No discernable interaction can be recorded, in full agreement with the fundamental theoretical concepts of solitons as mutually transparent quasiparticles. Our soliton generation technique allows for the creation of solitons with different depths; hence, they can be distinguished and their trajectories be followed. Simulations of the 1D-Gross-Pitaevskii equation have been performed to compare the experiment with a mean-field description.
The successive dynamical evolution of a Bose-Einstein condensate confined in a cylindrical well is numerically studied in the framework of the time-dependent Gross-Pitaevskii equation. Interference in the nonlinear matter wave leads to concentric density rings. The phase distribution exhibits a discontinuous sequence of plateaulike belts. Abrupt jumps in the phase between adjacent belts imply large radial superfluid velocity at the borderline. This, however, does not mean large particle current because the corresponding superfluid density is nearly zero. The density zeros along with the large gradient are identified as ring dark solitons, which have a brief lifetime before evolving into other soliton states.
The ring dark solitons in spin-1 23Na and 87Rb Bose-Einstein condensates are studied numerically in the framework of the time-dependent Gross-Pitaevskii equations. By simulating the phase engineering technique in real experiments, we explore the roles of the parameters characterizing the far-off resonant laser pulse which can be used to generate the ring dark solitons. The variations of these parameters have a dramatic effect on the lifetime and the decay profiles of the ring dark solitons. If only one ring dark soliton is generated in one component of the condensate, then ring dark solitons in other components are inclined to be induced, resulting in a coexistence state composed of interdependent ring dark solitons coming from different components of the condensate. Ring dark solitons in this coexistence state exhibit dynamical oscillations for hundreds of milliseconds. By studying the lifetime and decaying profiles of ring dark solitons, we explore the similarities and differences of 23Na and 87Rb condensates. In addition, taking into account the fact that the center of the ring may not coincide with that of the trap, we study the dynamics and decaying profiles of the off-centered ring dark solitons in the presence of the symmetry-breaking effect.