We investigate the dynamics of a dark-bright soliton in harmonically trapped two-component Bose-Einstein condensates and reveal an interesting spontaneous spatial symmetry breaking driven by nonlinear interactions. When the interaction parameter crosses a threshold value, we find that the dark-bright soliton's motion demonstrates a transition from symmetric periodic oscillation about the origin to asymmetric oscillations offset from the origin. In particular, at the transition point, the effective soliton mass, determined by the ratio of inertial mass to physical mass, diverges. The underlying mechanism is uncovered by constructing trial wave functions and employing the Lagrangian variational method to obtain an effective potential in the quasiparticle picture, which changes from a single well to a double well. The anomalous ``superheavy soliton'' phenomenon is a direct consequence of the dark-bright soliton's physical mass vanishing at the transition point. We obtain the phase diagram of this spatial symmetry-breaking transition. Possible implications of our finding for quantum metrology are discussed.
We investigate the dynamics of a dark-bright soliton in harmonically trapped two-component Bose-Einstein condensates and reveal an interesting spontaneous spatial symmetry breaking driven by nonlinear interactions. When the interaction parameter crosses a threshold value, we find that the dark-bright soliton's motion demonstrates a transition from symmetric periodic oscillation about the origin to asymmetric oscillations offset from the origin. In particular, at the transition point, the effective soliton mass, determined by the ratio of inertial mass to physical mass, diverges. The underlying mechanism is uncovered by constructing trial wave functions and employing the Lagrangian variational method to obtain an effective potential in the quasiparticle picture, which changes from a single well to a double well. The anomalous ``superheavy soliton'' phenomenon is a direct consequence of the dark-bright soliton's physical mass vanishing at the transition point. We obtain the phase diagram of this spatial symmetry-breaking transition. Possible implications of our finding for quantum metrology are discussed.
We consider a binary repulsive Bose-Einstein condensate in a harmonic trap in one spatial dimension and investigate particular solutions consisting of two dark-bright solitons. There are two different stationary solutions characterized by the phase difference in the bright component, in-phase and out-of-phase states. We show that above a critical particle number in the bright component, a symmetry-breaking bifurcation of the pitchfork type occurs that leads to a new asymmetric solution whereas the parental branch, i.e., the out-of-phase state, becomes unstable. These three different states support different small amplitude oscillations, characterized by an almost stationary density of the dark component and a tunneling of the bright component between the two dark solitons. Within a suitable effective double-well picture, these can be understood as the characteristic features of a bosonic Josephson junction (BJJ), and we show within a two-mode approach that all characteristic features of the BJJ phase space are recovered. For larger deviations from the stationary states, the simplifying double-well description breaks down due to the feedback of the bright component onto the dark one, causing the solitons to move. In this regime we observe intricate anharmonic and aperiodic dynamics, exhibiting remnants of the BJJ phase space.
Abstract We investigate the dynamics of a dark-bright soliton in a harmonic potential using a mean-field approach via coupled nonlinear Schrödinger equations appropriate to multi-component Bose–Einstein condensates. We use a modified perturbed dynamical variational Lagrangian approximation, where the perturbation is due to the trap, taken as a Thomas–Fermi profile. The wave function ansatz is taken as the correct hyperbolic tangent and secant solutions in the scalar case for the dark and bright components of the soliton, respectively. We also solve the problem numerically with pseudo-spectral Runge–Kutta methods. We find, analytically and numerically, 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. This result is robust against noise in the initial condition and should, therefore, be experimentally observable.
We study the one-dimensional dynamics of dark-dark solitons in the miscible regime of two density-coupled Bose-Einstein condensates having repulsive interparticle interactions within each condensate (g>0). By using an adiabatic perturbation theory in the parameter g12/g, we show that, contrary to the case of two solitons in scalar condensates, the interactions between solitons are attractive when the interparticle interactions between condensates are repulsive g12>0. As a result, the relative motion of dark solitons with equal chemical potential $\ensuremath{\mu}$ is well approximated by harmonic oscillations of angular frequency ${w}_{r}=(\ensuremath{\mu}/\ensuremath{\hbar})\sqrt{(8/15){g}_{12}/g}$. We also show that, in finite systems, the resonance of this anomalous excitation mode with the spin-density mode of lowest energy gives rise to alternating dynamical instability and stability fringes as a function of the perturbative parameter. In the presence of harmonic trapping (with angular frequency $\mathrm{\ensuremath{\Omega}}$) the solitons are driven by the superposition of two harmonic motions at a frequency given by ${w}^{2}={(\mathrm{\ensuremath{\Omega}}/\sqrt{2})}^{2}+{w}_{r}^{2}$. When g12<0, these two oscillators compete to give rise to an overall effective potential that can be either single well or double well through a pitchfork bifurcation. All our theoretical results are compared with numerical solutions of the Gross-Pitaevskii equation for the dynamics and the Bogoliubov equations for the linear stability. A good agreement is found between them.
We study dark–bright (DB) ring solitons in two-component Bose–Einstein condensates. In the limit of large densities of the dark component, we describe the soliton dynamics by means of an equation of motion for the ring radius. The presence of the bright, 'filling' species is demonstrated to have a stabilizing effect on the ring dark soliton. Near the linear limit, we discuss the symmetry-breaking bifurcations of DB soliton stripes and vortex-bright soliton clusters from the DB ring and relate the stabilizing effect of filling to changes in the bifurcation diagram. Finally, we show that the stabilization by means of a second component is not limited to the radially symmetric structures, but can also be observed in a cross-like DB soliton configuration.
The emergence of vortex-bright soliton dipoles in two-component Bose-Einstein condensates through bifurcations from suitable eigenstates of the underlying linear system is examined. These dipoles can have their bright solitary structures be in phase (symmetric) or out of phase (anti-symmetric). The dynamical robustness of each of these two possibilities is considered and the out-of-phase case is found to exhibit an intriguing symmetry-breaking instability that can in turn lead to tunneling of the bright wave function between the two vortex ``wells.'' We interpret this phenomenon by virtue of a vortex-induced double-well system, whose spontaneous symmetry breaking leads to asymmetric vortex-bright dipoles, in addition to the symmetric and antisymmetric ones. The theoretical prediction of these states is corroborated by detailed numerical computations.
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.