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Institute of Physics eBooks

Dynamics of scalar Bose–Einstein condensates with short-range interactions

Paulsamy Muruganandam, Ramaswamy Radha

Published Nov 1, 2025• 0 citations
Publisher

Abstract

Chapter 5 is devoted to the investigation of scalar between Bose–Einstein condensates with short-range interactions using quasi-one-dimensional Gross–Pitaevskii-type equations in time-independent and transient harmonic trapping potentials. Matter wave interference pattern obtained by virtue of the correlation between transient trap frequency and temporal binary interaction in the collisional dynamics of bright solitons is brought out. In addition, the impact arising by virtue of the reinforcement of three-body interaction with binary interaction is also studied.

DOI
10.1088/978-0-7503-5447-9ch5

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•Jan 1, 2021•0 citations
Chapter 5 is devoted to the investigation of scalar between Bose–Einstein condensates with short-range interactions using quasi-one-dimensional Gross–Pitaevskii-type equations in time-independent and transient harmonic trapping potentials. Matter wave interference pattern obtained by virtue of the correlation between transient trap frequency and temporal binary interaction in the collisional dynamics of bright solitons is brought out. In addition, the impact arising by virtue of the reinforcement of three-body interaction with binary interaction is also studied.
We review the recent developments which had taken place in the domain of quasi one dimensional BECs from the viewpoint of integrability. To start with, we consider the dynamics of scalar BECs in a time independent harmonic trap and observe that the scattering length (SL) can be suitably manipulated either to compress the bright solitons to attain peak matter wave density without causing their explosion or to broaden the width of the condensates without diluting them. When the harmonic trap frequency becomes time dependent, we notice that one can stabilize the condensates in the confining domain while the density of the condensates continue to increase in the expulsive region. We also observe that the trap frequency and the temporal SL can be maneuvered to generate matter wave interference patterns indicating the coherent nature of the atoms in the condensates. We also notice that a small repulsive three body interaction when reinforced with attractive binary interaction can extend the region of stability of the condensates in the quasi-one dimensional regime. On the other hand, the investigation of two component BECs in a time dependent harmonic trap suggests that it is possible to switch matter wave energy from one mode to the other confirming the fact that vector BECs are long lived compared to scalar BECs. The Feshbach resonance management of vector BECs indicates that the two component BECs in a time dependent harmonic trap are more stable compared to the condensates in a time independent trap. The introduction of weak time dependent Rabi coupling rapidly compresses the bright solitons which however can be again stabilized through Feshbach resonance or by finetuning the Rabi coupling while the spatial coupling of vector BECs introduces a phase difference between the condensates which subsequently can be exploited to generate interference pattern in the bright or dark solitons.
We study the formation and dynamics of bright solitons in a quasi-one-dimensional reduced mean-field Gross-Pitaevskii equation of a dipolar Bose-Einstein condensate with repulsive short-range interactions. The study is carried out using a variational approximation and a numerical solution. Plots of chemical potential and root mean square (rms) size of solitons are obtained for the quasi-one-dimensional model of three different dipolar condensates of 52Cr, 168Er and 164Dy atoms. The results achieved are in good agreement with those produced by the full three-dimensional mean-field model of the condensate. We also study the dynamics of the collision of a train of two solitons in the quasi-one-dimensional model of every condensate above. At small velocities (zero or close to zero) the dynamics is attractive for a phase difference δ = 0, the solitons coalesce and these oscillate forming a bound soliton molecule. For a phase difference δ = π the effect is repulsive. At large velocities the collision is independent of the initial phase difference δ. This is quasi-elastic and the result is two quasi-solitons.
A one-dimensional (1D) Gross-Pitaevskii (GP) equation which governs the dynamics of Bose-Einstein condensates (BECs) with time-dependent scattering length in a harmonic potential is considered. Through a modified phase imprint transformation into the macroscopic order parameter, we suitably generate a derivative cubic-quintic Gross-Pitaevskii (DCQ-GP) equation that describes the dynamics of condensates in the harmonic potential for both twoand three-body interatomic interactions in the presence of nonlinear dispersion. Employing a modified lens-type transformation, we reduce the DCQ-GP equation into a derivative cubic-quintic Ginzburg-Landau (DCQ-GL) equation that allows us to investigate the modulational instability (MI) of the system and study, through the analytical exact solutions of the considered 1D GP equation, the effects of the three-body interactions and nonlinear dispersion on the dynamics of the BECs. We observe that for BECs with repulsive two-body interatomic interactions, the attractive three-body interactions and the nonlinear dispersion have opposite effects on the condensate: (i) The attractive three-body interactions enhance the instability, while the nonlinear dispersion softens the instability of the system; (ii) the attractive three-body interactions contributes to the ejections of atoms from the condensates while the nonlinear dispersion contributes to the accumulation of atoms in the condensates; (iii) the attractive three-body interactions contributes to the broadening of the matter wave solitons, while the nonlinear dispersion contributes to the compression of the matter wave solitons.
We present two families analytical solutions to the extended Gross-Pitaevskii equation (GPE), which describe the dynamics of bright and dark solitons in Bose-Einstein condensations(BECs) with varying coefficients of two and three-body interactions in a parabolic and complex potential. The properties of the dynamical evolutions of the bright and dark matter-wave solitons are investigated. It is interesting that the matter-wave can be controlled by applying the external magnetic field and a background.
Bright solitons in a quasi-1D Bose–Einstein condensate can be used to enhance precision in matter-wave interferometry, due to their inherent robustness and support against dispersion. Such a soliton interferometer typically relies on a potential barrier used to split a single soliton into two smaller coherent solitons which can then be recombined on the same barrier. In this thesis we examine two extensions to this scheme. Firstly, we investigate a binary BEC system consisting of two bright solitons which are coupled through a mutual nonlinear interaction term. We derive a set of conditions under which the two components can be separated on a potential barrier and use numerical simulations to probe the regimes beyond which this mathematical treatment is applicable. We then use the numerical simulations to look at the effect of the nonlinear coupling on the dynamics of the binary solitons interacting with the barrier. We also look at the interference behaviour found by doubling the simulation time in either a ring trap or a harmonic trapping potential (to ensure recombination on the barrier); as well as the case where the solitons start spatially separated on either side of the barrier in order to find conditions under which the solitons will combine on the barrier. We find a good agreement between the analytical predictions and the results of simulations. Beyond the regions of parameter space where the predictions are expected to hold, we find complex transmission and interference behaviour as a result of nonlinear effects. The second part of this thesis consists of an examination of the prospect of using a subwavelength barrier scheme in a soliton interferometry experiment. This involves using two resonant coupling beams in a Λ-system with a spatially varying intensity. Under certain conditions, this can be used to form an effective potential barrier with a width which is not diffraction-limited. We look at suitable parameter regimes for such a barrier to split and recombine solitons in an interferometer and probe the effects of possible complications such as misalignment in the beams and different scattering lengths in the different states. We simulate the soliton interferometer using the full three component GPE as well as the single component analogue with the effective potential in order to characterise the soliton behaviour and the dependence of the interferometer sensitivity on the system parameters. We find a trend towards an idealised soliton interferometer with a decreasing value of the parameter, w, controlling the barrier characteristics. Also, we demonstrate an agreement between the relevant three component GPE and the analogous single component GPE, in the limit of strong coupling fields.