Systems of solitary-waves in the 1D Gross-Pitaevskii equation, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. To analyse the soliton-like nature of these solitary-waves, a particle analogy for the solitary-waves is formulated. Exact soliton solutions exist in the absence of an external trapping potential, which behave in a particle-like manner, and we find the particle analogy we employ to be a good model also when a harmonic trapping potential is present. In the case of two solitons, the particle model is integrable, and the dynamics are completely regular. The extension to three particles supports chaotic regimes. The agreement between the particle model and the wave dynamics remains good even in chaotic regimes. In the case of a system of two solitary waves of equal norm, the solitons are shown to retain their phase difference for repeated collisions. This implies that soliton-like regimes may be found in 3D geometries where solitary waves can be made to repeatedly collide out of phase, stabilising the condensate against collapse.
We examine the dynamics of two bright solitary waves with a negative nonlinear term. The observed repulsion between two solitary waves -- when these are in an antisymmetric combination -- is attributed to conservation laws. Slight breaking of parity, in combination with weak relaxation of energy, leads the two solitary waves to merge. The effective repulsion between solitary waves requires certain nearly ideal conditions and is thus fragile.
In this paper we elucidate the physics underlying the fact that both bright and dark solitary waves can arise in a one-dimensional spin-polarized gas of fermionic atoms with attractive three-dimensional p-wave interactions in a hard-wall trap. This is possible since the one-dimensional fermion system can be mapped to a system of bosons described by the Lieb-Liniger model with either repulsive or attractive delta-function interactions which can support solitary waves.
We analyse the rotation of bright solitary waves formed of atomic Bose-Einstein condensates with attractive atomic interactions. By employing a variational technique and assuming an irrotational quadrupolar flow field, we map out the variational solutions in the rotating frame. In particular, we show that rotation has a considerable stabilising effect on the system, significantly raising the critical threshold for collapse of the bright solitary waves.
The collisions of three-dimensional bright solitary matter waves formed from atomic Bose-Einstein condensates are shown to exhibit rich behaviour. Collisions range from being elastic to completely destructive due to the onset of collapse during the interaction. Through a detailed quantitative analysis we map out the role of relative phase, impact speed and interaction strength. In particular, we identify the importance of the collapse time in the onset of unstable collisions and show how the relative phase controls a population transfer between the waves. Our analysis enables us to interpret r…
We analyse the static solutions of attractive Bose-Einstein condensates under transverse confinement, both with and without axial confinement. By full numerical solution of the Gross-Pitaevskii equation and variational methods we map out the condensate solutions, their energetic properties, and their critical points for instability. With no axial confinement a bright solitary wave solution will tend to decay by dispersion unless the interaction energy is close to the critical value for collapse. In contrast, with axial confinement the only decay mechanism is collapse. The stability of a bright…
We model the dynamics of formation of multiple, long-lived, bright solitary waves in the collapse of Bose-Einstein condensates with attractive interactions as studied in the experiment of Cornish et al. [Phys. Rev. Lett. 96 (2006) 170401]. Using both mean-field and quantum field simulation techniques, we find that while a number of separated wave packets form as observed in the experiment, they do not have a repulsive πphase difference that has been previously inferred. We observe that the inclusion of quantum fluctuations causes soliton dynamics to be predominantly repulsive in one dimensiona…
We analyze the physics of bright solitons in two-dimensional dipolar Bose-Einstein condensates. These solitons are not possible in short-range interacting gases. In particular, we discuss the necessary conditions for the existence of stable 2D bright solitary waves. The stability of the solutions is studied by means of the analysis of lowest-lying excitations. Additionally, we study the scattering of solitary waves, which contrary to the contact-interacting case, is inelastic, and could lead to fusion of solitary waves. Finally, the experimental possibilities for observability are discussed.
Motivated by recent experimental observations, we study theoretically multiple bright solitary waves of trapped Bose-Einstein condensates. Through variational and numerical analyses, we determine the threshold for collapse of these states. Under $π$-phase differences between adjacent waves, we show that the experimental states lie consistently at the threshold for collapse, where the corresponding in-phase states are highly unstable. Following the observation of two long-lived solitary waves in a trap, we perform detailed three-dimensional simulations which confirm that in-phase waves undergo …
We study bright solitary waves of three dimensional trapped Bose-Einstein condensates and their collisions. For a single solitary wave, in addition to an upper critical number, we also find a {\em lower} cut-off, below which no stable state can be found. Collisions between solitary waves can be elastic, inelastic with either reduced or increased outgoing speed, or completely unstable due to a collapse instability. A $π$-phase difference between the waves promotes elastic collisions, and gives excellent agreement with recent experimental results over long timescales.
We examine the propagation of solitary waves in elongated clouds of trapped bosonic atoms as the confinement, the strength of the interatomic interaction, and the atom density are varied. We identify three different physical regimes and develop a general formalism that allows us to interpolate between them. Finally we pay special attention to the transition to the Tonks-Girardeau limit of strongly-interacting bosons.
We study the dynamic structure factor of two coherent bright solitary waves in attractive Bose- Einstein condensates confined in a harmonic trap. We demonstrate that the wave function of the two solitary waves with a fixed relative phase shows interference in momentum space. The fringes are shifted depending on the values of the phase. This momentum interference can be extracted from the dynamic structure factor of the system using stimulated two-photon Bragg scattering. Thus our method provides a way to measure the relative phase directly.
Collisions between bright solitary waves in the 1D Gross-Pitaevskii equation with a harmonic potential, which models a trapped atomic Bose-Einstein condensate, are investigated theoretically. A particle analogy for the solitary waves is formulated, which is shown to be integrable for a two-particle system. The extension to three particles is shown to support chaotic regimes. Good agreement is found between the particle model and simulations of the full wave dynamics, suggesting that the dynamics can be described in terms of solitons both in regular and chaotic regimes, thus presenting a paradi…
We present a theoretical treatment of Bragg spectroscopy of an accelerating condensate in a solitary-wave state. Our treatment is based on the Gross-Pitaevskii equation with an optical potential representing the Bragg pulse and an additional external time-dependent potential generating the solitary-wave behaviour. By transforming to a frame translating with the condensate, we derive an approximate set of equations that can be readily solved to generate approximate Bragg spectra. Our analytic method is accurate within a well defined parameter regime and provides physical insight into the struct…
In a benchmark dynamical-lattice model in three dimensions, the discrete nonlinear Schr{ö}dinger equation, we find discrete vortex solitons with various values of the topological charge $S$. Stability regions for the vortices with $S=0,1,3$ are investigated. The S=2 vortex is unstable, spontaneously rearranging into a stable one with S=3. In a two-component extension of the model, we find a novel class of stable structures, consisting of vortices in the different components, perpendicularly oriented to each other. Self-localized states of the proposed types can be observed experimentally in Bo…
The dynamics of quantum vortices in a two-dimensional annular condensate are considered by numerically simulating the Gross-Pitaevskii equation. Families of solitary wave sequences are reported, both without and with a persistent flow, for various values of interaction strength. It is shown that in the toroidal geometry the dispersion curve of solutions is much richer than in the cases of a semi-infinite channel or uniform condensate studied previously. In particular, the toroidal condensate is found to have states of single vortices at the same position and circulation that move with differen…
We show that it is possible to observe matter wave interference patterns in the collision of bright solitons (Bose Einstein condensates) without free ballistic expansion for suitable choices of scattering length and time dependent trap.
We study fundamental and vortical solitons in disk-morphed Bose-Einstein condensates (BECs) subject to strong confinement along the axial direction. Starting from the three-dimensional (3D) Gross-Pitaevskii equation (GPE), we proceed to an effective 2D nonpolynomial Schroeodinger equation (NPSE) derived by means of the integration over the axial coordinate. Results produced by the latter equation are in very good agreement with those obtained from the full 3D GPE, including cases when the formal 2D equation with the cubic nonlinearity is unreliable. The 2D NPSE is used to predict density profi…
We provide a familiy of transparent tight-binding models with non-trivial potentials and site-dependent hopping parameters. Their feasibility is discussed in electromagnetic resonators, dielectric slabs and quantum-mechanical traps. In the second part of the paper, the arrays are obtained through a generalization of supersymmetric quantum mechanics in discrete variables. The formalism includes a finite-difference Darboux transformation applied to the scattering matrix of a periodic array. A procedure for constructing a hierarchy of discrete hamiltonians is indicated and a particular biparametr…
We examine the dynamics of a bright solitary wave in the presence of a repulsive or attractive localized ``impurity'' in Bose-Einstein condensates (BECs). We study the generation and stability of a pair of steady states in the vicinity of the impurity as the impurity strength is varied. These two new steady states, one stable and one unstable, disappear through a saddle-node bifurcation as the strength of the impurity is decreased. The dynamics of the soliton is also examined in all the cases (including cases where the soliton is offset from one of the relevant fixed points). The numerical res…
We derive exact solitonic solutions of a class of Gross-Pitaevskii equations with time-dependent harmonic trapping potential and interatomic interaction. We find families of exact single-solitonic, multi-solitonic, and solitary wave solutions. We show that, with the special case of an oscillating trapping potential and interatomic interaction, a soliton can be localized indefinitely at an arbitrary position. The localization is shown to be experimentally possible for sufficiently long time even with only an oscillating trapping potential and a constant interatomic interaction.
We investigate the dynamics of a spin-orbit (SO) coupled BECs in a time dependent harmonic trap and show the dynamical system to be completely integrable by constructing the Lax pair. We then employ gauge transformation approach to witness the rapid oscillations of the condensates for a relatively smaller value of SO coupling in a time independent harmonic trap compared to their counterparts in a transient trap. Keeping track of the evolution of the condensates in a transient trap during its transition from confining to expulsive trap, we notice that they collapse in the expulsive trap. We fur…
The Hong-Ou-Mandel experiment leads indistinguishable photons simultaneously reaching a 50:50 beam splitter to emerge on the same port through two-photon interference. Motivated by this phenomenon, we consider numerical experiments of the same flavor for classical, {\it wave} objects in the setting of repulsive condensates. We examine dark solitons interacting with a repulsive barrier, a case in which we find no significant asymmetries in the emerging waves after the collision, presumably due to their topological nature. We also consider case examples of two-component systems, where the dark s…
Using variational and numerical solutions of the mean-field Gross-Pitaevskii equation we show that a bright soliton can be stabilized in a trapless three-dimensional attractive Bose-Einstein condensate (BEC) by a rapid periodic temporal modulation of scattering length alone by using a Feshbach resonance. This scheme also stabilizes a rotating vortex soliton in two dimensions. Apart from possible experimental application in BEC, the present study suggests that the spatiotemporal solitons of nonlinear optics in three dimensions can also be stabilized in a layered Kerr medium with sign-changing n…
