VORTEX EXCITATIONS UNDER COLLISION BETWEEN A SUPERFLOWING RING-SHAPED BEC AND A CENTERED STATIC BEC

Qingli Zhu. Department of information engineering, nanjing normal university taizhou college, taizhou 225300, china. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History Received: 12 September 2019 Final Accepted: 14 October 2019 Published: November 2019

Since the quantized vortices have been successfully created in a trapped Bose-Einstein Condensate(BEC) of ultracold atomic due to their experimental versatility [1], more and more theoretical and experimental researches are attracted by BEC to investigate the characteristics of quantum vortices. The research progress of the ultracold atomic gases in recent years have also demonstrated BEC is an ideal platform for studying the static and dynamical properties of vortex. In experiment, several methods have been mainly employed to create quantum vortices such as stirring BEC with a laser beam [2][3], trap rotation [4][5], and phase imprintings [6][7].
Except for the studies on static vortices [8][9], numerous studies have been performed to evaluate the dynamical properties including the nucleation [10] and decay of an off-centered vortex [11][12]. However, there still have little studies on the dynamical behaviors of BEC collision. In Ref [13][14] , the behavior of collision and interference dynamics of BEC is found to be related with the initial displacement and phase difference. The collision dynamics of vortices in a two-dimensional spinor Bose-Einstein condensate and vortex dipoles propagating in opposite directions are also investigated [15].
In this paper, by solving the time-dependent Gross-Pitaevskii (GP) equations, we study both the dynamical generation and evolution of vortex excitations under collision between a superflowing ring-shaped BEC and a centered static BEC. We take two cases to perform our calculations according to the atoms being identical or not. Quantized vortices are found to be created symmetrically and their evolution with mainly depends on the atomic interactions.
647 Model:-We consider a single BEC described by the macroscopic wave function ) , ( t r   . In the mean-field framework, the dynamics of a system with N identical atoms close to thermo-dynamic equilibrium and subject to weak dissipation can be described by the dissipative Gross-pitaevskii(GP) equation [16]: is the axially symmetric harmonic trap potential. We focus in this paper on a pancake-shaped situation with t z    . In this extreme limit, the axial dimension is sufficiently thin that the motion along z direction can be neglected and atoms can move only within the xy plane. In actual computation, we discrete the x-y plane into a square lattice points. The space scale is much larger than the TF radius of the condensate. a is assumed to be the lattice constant whose value must be much less than the characteristic length oscillator, we will thus obtain the following lattice-version of the GP equations: It is convenient to introduce dimensionless parameters Since vortices are topological defects, they can be only created in pairs or can enter a system from its boundary. Generally, collisions between BECs can result in the creation and motion of vortices or vortex-antivortex pairs. In this section, we study another dynamical process of vortex generation, where a superflowing ring-shaped condensate with N1atoms collides with a static trapped BEC at center with N2 atoms. This is schematically shown in the inset of Fig.1. The superflowing condensate can be realized by a rotating weak link or obstcle [17]. The wave function for it is characterized by its winding number s, and can be given by   r . If the condensate is then released into the harmonic trap, which can be assumed that the infinite well is suddenly switched off, it would collide with a static trapped BEC at center. The centered static BEC is assumed to be in the ground state of the harmonic trap, with a TF radius slightly less than 1 r . To simplify the discussion, we first assume that the two BECs consist of the same identical atoms, and so both two should be described by a single condensate wave function. When the collision happens, the superflowing atoms from the ring-shaped region tend to move inside, and by merging with the inner atoms, also tend to form a new condensate confined by the trap potential. Under this competition, s symmetric snake-like density valleys are formed quickly, each extending from the trap center to the condensate edge. Each density valley contains one singly quantized vortex, which can be seen from the phase profiles in Fig.1 vortex or antivortex passes across the circle. Soon each density valley will be broken into pieces, many of which are composed of vortex-antivortex pairs. Within a relatively short period of time, these antivortices will evolve out of the condensate, leaving s symmetrically positioned singly quantizd vortices near the original interface of the two BECs. Finally, these vortices will move gradually in a spiral way to the condensate edge and then disappear, leaving behind a new static ground state of BEC confined by the trap. These are illustrated in the upper panels of Fig.1, where the ratio between the numbers of atoms of the two BECs N1/N2 is fixed to be 1.
The dynamical evolution processes for s from 1 to 20 have been investigated. It is found that the above dynamical behavior is generically unchanged regardless of the s value. Quantitatively, upon increasing s, the time intervals for the existence of density valleys become shorter and the relaxation time needed to the new ground state also become shorter. Fluctuation effect has also been taken into account, and the above behaviors are preserved against small fluctuations so long as 4 10 1     . Now we consider the situation where the two BECs consist of different identical atoms. The two BECs are thus described by two different condensate wave functions. Let the interactions between atoms in the same BECs be identical and denoted by g and that between atoms in the different BECs denoted by 12 g .
Generically, the ratio g g / 12 determines the phase regimes of the condensate. In the phase-mixing regime 1 / 12  g g , when the collision happens, the atoms of the ring-shaped BEC first move inside towards the trap center, accompanied by vortices formation. Then the time evolution process takes on 4-fold symmetry. In the phase-separation regime 1 / 12  g g , the two BECs always occupy complementary spaces. While the ringshaped BEC is broken into several pieces with most of the vortices escaping out, the inner BEC has sharp corners accordingly. Eventually, the two BECs evolve into a phase-separated state composed of two semicircular parts, which is the ground state of the system. In the critical regime 1 / 12  g g , the dynamical process is more like that of the phase-separation regime, apart from that the final state of the system is stabilized at an excited state, where the original ring-shaped BEC still preserve the shape and is superflowing with a smaller winding number. These are demonstrated in Fig.2. The calculation is done by the square-lattice discretization and the comparison is also made with that by the triangular-lattice discretization. We find that apart from the 4-fold symmetry related behavior shown in Fig.2(a), these phenomena are independent of the lattice discretization.

Conclusion:-
We have performed numerical calculations of the quai-two-dimensional GP equation to investigate the creation and dynamical evolution of vortices in a harmonic trap potential. As two GOPs move and collide head on, we find extra vortex dipoles and a pair of solitons are nucleated symmetrically after the two GOPs separating from each other which enrich the phase profiles of vortex excitations. when a trapped SO coupling BEC is in a plane-wave phase, it is favorable for vortex creation under the influence of SO coupling effect. The velocity of a vortex-antivortex is much smaller than that without SO coupling.