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NONLINEAR DYNAMICS OF WAVE PACKETS AND VORTICES IN BOSE-EINSTEIN CONDENSATES Chaos and Billiard Motion

Kaoru Nakamura

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

We study the dynamics of single and multi-component Bose-Einstein condensates (BECs) in two dimensions with and without a harmonic trap by using various variants of non- linear Schrodinger (or Gross-Pitaevskii) equation. Firstly, we examine the three-component repulsive BEC with cubic nonlinearity in a harmonic trap, and see the conservative chaos based on a picture of vortex molecules. We obtain an effective nonlinear dynamics for three vortex cores, which are equivalent to three charged particles under the uniform magnetic field with the repulsive inter-particle potential quadratic in the inter-vortex distance r ij on short length scale and logarithmic in r ij on large length scale. The vortices here acquire the inertia in marked contrast to the standard theory of point vortices since Onsager. We then explore chaos in the three-body problem in the context of vortices with inertia. Secondly, by choosing the nonlinear Schrodinger equation with saturable nonlinearity, we investigate the single and multi-component WP dynamics within the hard-walled square and stadium billiards with neither a harmonic trap nor driving field. We analyze the stability of WPs by using the variational (collective-coordinate) method. By emitting the radiation the Gaussian WP becomes deformed to a bell-shaped one and then stabilized. As the velocity increases, WPs tend to be stable against many collisions with billiard walls.

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Physical Review E•Jul 8, 2013•25 citations
We study the dynamics of single and multi-component Bose-Einstein condensates (BECs) in two dimensions with and without a harmonic trap by using various variants of non- linear Schrodinger (or Gross-Pitaevskii) equation. Firstly, we examine the three-component repulsive BEC with cubic nonlinearity in a harmonic trap, and see the conservative chaos based on a picture of vortex molecules. We obtain an effective nonlinear dynamics for three vortex cores, which are equivalent to three charged particles under the uniform magnetic field with the repulsive inter-particle potential quadratic in the inter-vortex distance r ij on short length scale and logarithmic in r ij on large length scale. The vortices here acquire the inertia in marked contrast to the standard theory of point vortices since Onsager. We then explore chaos in the three-body problem in the context of vortices with inertia. Secondly, by choosing the nonlinear Schrodinger equation with saturable nonlinearity, we investigate the single and multi-component WP dynamics within the hard-walled square and stadium billiards with neither a harmonic trap nor driving field. We analyze the stability of WPs by using the variational (collective-coordinate) method. By emitting the radiation the Gaussian WP becomes deformed to a bell-shaped one and then stabilized. As the velocity increases, WPs tend to be stable against many collisions with billiard walls.
Bose-Einstein condensate(BEC) provides a nice stage when the nonlinear Schrödinger equation plays a vital role. We study the dynamics of multi-component repulsive BEC in 2 dimensions with harmonic traps by using the nonlinear Schrödinger (or Gross-Pitaevskii) equation. Firstly we consider a driven two-component BEC with each component trapped in different vertical positions. The appropriate tuning of the oscillation frequency of the magnetic field leads to a striking anti-gravity transport of BEC. This phenomenon is a manifestation of macroscopic non-adiabatic tunneling in a system with two internal(electronic) degrees of freedom. The dynamics splits into a fast complex spatio-temporal oscillation of each condensate wavefunctions together with a slow levitation of the total center of mass. Secondly, we examine the three-component repulsive BEC in 2 dimensions in a harmonic trap in the absence of magnetic field, and construct a model of conservative chaos based on a picture of vortex molecules. We obtain an effective nonlinear dynamics for three vortex cores, which represents three charged particles under the uniform magnetic field with the repulsive inter-particle potential quadratic in the inter-vortex distance rij on short scale and logarithmic in rij on large scale. The vortices here acquire the inertia in marked contrast to the standard theory of point vortices since Onsager. We then explore “the chaos in the three-body problem” in the context of vortices with inertia.
With use of the nonlinear Schr\"odinger (or Gross-Pitaevskii) equation with strong repulsive cubic nonlinearity, dynamics of multicomponent Bose-Einstein condensates (BECs) with a harmonic trap in two dimensions is investigated beyond the Thomas-Fermi regime. In the case when each component has a single vortex, we obtain an effective nonlinear dynamics for vortex cores (particles). The particles here acquire the inertia, in marked contrast to the standard theory of point vortices widely known in the usual hydrodynamics. The effective dynamics is equivalent to that of charged particles under a strong spring force and in the presence of Lorentz force with the uniform magnetic field. The interparticle (vortex-vortex) interaction is singularly repulsive and short ranged with its magnitude decreasing with increasing distance of the center of mass from the trapping center. ``Chaos in the three-body problem'' in the three-vortice system can be seen, which is not expected in the corresponding point vortices without inertia in two dimensions.
To show a mechanism leading to the breakdown of a particle picture for the multi-component Bose-Einstein condensates(BECs) with a harmonic trap in high dimensions, we investigate the corresponding 2-ddd nonlinear Schr{ö}dinger equation (Gross-Pitaevskii equation) with use of a modified variational principle. A molecule of two identical Gaussian wavepackets has two degrees of freedom(DFs), the separation of center-of-masses and the wavepacket width. Without the inter-component interaction(ICI) these DFs show independent regular oscillations with the degenerate eigen-frequencies. The inclusion of ICI strongly mixes these DFs, generating a fat mode that breaks a particle picture, which however can be recovered by introducing a time-periodic ICI with zero average. In case of the molecule of three wavepackets for a three-component BEC, the increase of amplitude of ICI yields a transition from regular to chaotic oscillations in the wavepacket breathing.
The nonlinear Schrodinger equation models a wide variety of different physical phenomena ranging from nonlinear optics, water waves, magnetization of ferromagnets to Bose-Einstein condensates (BEC). The structure of the equation supports existence of topologically non-trivial solutions - vortices. Surprisingly, we demonstrate that the Landau-Lifshitz magnetization equation which is formally also a nonlinear Schrodinger equation does not admit such solutions. On the other hand, the contrary is true for the Gross-Pitaevskii equation which describes the mean-field approximation of BEC. We investigate stability of vortex solutions by means of a very reliable, sensitive and robust technique - the Evans function. This method, although limited to two dimensions, allows us to study rotating axisymmetric BEC for large particle numbers which can be unattainable by other means.We found a singly-quantized vortex linearly stable.The linear stability of multi-quantized vortices depends on the diluteness of a condensate, with alternating intervals of stability and instability. This work justifies previous results in the literature obtained by less reliable methods and opens up a few interesting questions.
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 two-dimensional (2D) nonpolynomial Schr\"odinger 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 the density profiles and dynamical stability of repulsive and attractive BECs with zero and finite topological charges in various planar trapping configurations, including the axisymmetric harmonic confinement and one-dimensional periodic potential. In particular, we find a stable dynamical regime that was not reported before, viz., periodic splitting and recombination of trapped vortices with topological charges 2 or 3 in the self-attractive BEC.
We study vortex dynamics in a trapped Bose-Einstein condensate with tunable two- and three-body interactions. The dynamics is governed by two-dimensional cubic-quintic Gross-Pitaevskii equation. A time-dependent variational method has been used to obtain critical rotational frequency and surface mode frequency analytically and are compared with numerical simulation results. An imaginary time propagation method and Crank-Nicolson scheme for discretization have been used for numerical simulation. The numerically calculated average value of the angular momentum per particle shows very clearly its dynamical relation with the time development of the vortex formation. The rotational frequency dependence of the variation of average value of the angular momentum per particle with time shows that vortices form much faster in time for higher rotational frequency. Similarly, the vortex forms much faster in time with an increase of the strength of the repulsive three-body interaction. The simulation of the vortex lattice formation in the condensate shows that the presence of the three-body interactions does not alter the vortex lattice pattern but it helps in the shape deformations of the condensate thereby leading to vortex lattice formation. Likewise, the three-body interactions enable the vortex lattice formation in Bose-Einstein condensates even with attractive two-body interactions and in purely quintic BEC.