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arXiv:1607.00201 · Journal of Physics A Mathematical and Theoretical

Dynamics of trapped interacting vortices in Bose–Einstein condensates: a role of breathing degree of freedom

Katsuhiro Nakamura, Doniyor Babajanov, Davron Matrasulov, Michikazu Kobayashi, Paulsamy Muruganandam

Published Jun 30, 2016• 6 citations
arXivHTMLPDF

Abstract

Abstract With use of a variational principle, we investigate a role of breathing width degree of freedom in the effective theory of interacting vortices in a trapped single-component Bose–Einstein condensates in two dimensions, under strong repulsive cubic nonlinearity. For the trial function, we choose a product of two vortex functions, assuming a pair interaction, and employ the amplitude form of each vortex function in the Padé approximation, which accommodates a hallmark of the vortex core. We obtain the Lagrange equation for the interacting vortex-core coordinates coupled with the time-derivative of width and also its Hamilton formalism by having recourse to a non-standard Poisson bracket. By solving the Hamilton equation, we find rapid radial breathing oscillations superposed on the slower rotational motion of vortex cores, consistent with numerical solutions of the Gross–Pitaevskii equation. In higher-energy states of two vortex systems, the breathing width degree of freedom plays the role of a kicking in the kicked rotator, and generates chaos with a structure of sea-urchin needles. The by-products of the present variational approach include: (1) the charge-dependent logarithmic inter-vortex interaction multiplied with a pre-factor, which depends on the scalar product of a pair of core-position vectors; (2) the charge-independent short-range repulsive inter-vortex interaction and spring force.

DOI
10.1088/1751-8113/49/31/315102

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Abstract With use of a variational principle, we investigate a role of breathing width degree of freedom in the effective theory of interacting vortices in a trapped single-component Bose–Einstein condensates in two dimensions, under strong repulsive cubic nonlinearity. For the trial function, we choose a product of two vortex functions, assuming a pair interaction, and employ the amplitude form of each vortex function in the Padé approximation, which accommodates a hallmark of the vortex core. We obtain the Lagrange equation for the interacting vortex-core coordinates coupled with the time-derivative of width and also its Hamilton formalism by having recourse to a non-standard Poisson bracket. By solving the Hamilton equation, we find rapid radial breathing oscillations superposed on the slower rotational motion of vortex cores, consistent with numerical solutions of the Gross–Pitaevskii equation. In higher-energy states of two vortex systems, the breathing width degree of freedom plays the role of a kicking in the kicked rotator, and generates chaos with a structure of sea-urchin needles. The by-products of the present variational approach include: (1) the charge-dependent logarithmic inter-vortex interaction multiplied with a pre-factor, which depends on the scalar product of a pair of core-position vectors; (2) the charge-independent short-range repulsive inter-vortex interaction and spring force.
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.
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.
We study the stability of vortices in a binary system of Bose-Einstein condensat es, with their wave functions modeled by a set of coupled, time-dependent Gross-Pitaevskii equations. Beginning with an effective two-dimensional system, we identify miscible and immiscible regimes characterized by the inter- and intra-atomic interactions and the initial configuration of the system. We then consider a binary system of Bose-Einstein condensates placed in a rotating harmonic trap and study the single united vortex state in this system. We derive an approximate form for the energy of a single united vortex in the binary system and the critical angular velocity for the global stability of a vortex at the center of the trap. We also compute the metastability onset angular velocity for the local stability of a vortex at the center of the trap. Numerical solutions to the Gross-Pitaevskii equations support these expressions. These rotational results inform us of a novel bimodal formation within the miscible regime of the binary condensate system. We thus demonstrate the nontrivial aspects of vortex stability in interacting binary Bose-Einstein condensates as a result of their nonlinear interactions.
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.
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.
In this study, we investigate the evolution of vortex in harmonically trapped two-component coupled Bose–Einstein condensate with quintic-order nonlinearity. We derive the vortex solution of this two-component system based on the coupled quintic-order Gross–Pitaevskii equation model and the variational method. It is found that the evolution of vortex is a metastable state. The radius of vortex soliton shrinks and expands with time, resulting in periodic breathing oscillation, and the angular frequency of the breathing oscillation is twice the value of the harmonic trapping frequency under infinitesimal nonlinear strength. At the same time, it is also found that the higher-order nonlinear term has a quantitative effect rather than a qualitative impact on the oscillation period. With practical experimental setting, we identify the quasi-stable oscillation of the derived vortex evolution mode and illustrated its features graphically. The theoretical results developed in this work can be used to guide the experimental observation of the vortex phenomenon in ultracold coupled atomic systems with quintic-order nonlinearity.