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Communications in Theoretical Physics

Padé approximations of quantized-vortex solutions of the Gross–Pitaevskii equation

Weiru Chen, Shan-Quan Lan, Xiyi Liu, Jie-Xiong Mo, Xiao-Bao Xu, Gu-Qiang Li

Published May 6, 2021• 3 citations

Abstract

Abstract Quantized vortices are important topological excitations in Bose–Einstein condensates. The Gross–Pitaevskii equation is a widely accepted theoretical tool. High accuracy quantized-vortex solutions are desirable in many numerical and analytical studies. We successfully derive the Padé approximate solutions for quantized vortices with winding numbers ω = 1, 2, 3, 4, 5, 6 in the context of the Gross–Pitaevskii equation for a uniform condensate. Compared with the numerical solutions, we find that (1) they approximate the entire solutions quite well from the core to infinity; (2) higher-order Padé approximate solutions have higher accuracy; (3) Padé approximate solutions for larger winding numbers have lower accuracy. The healing lengths of the quantized vortices are calculated and found to increase almost linearly with the winding number. Based on experiments performed with 87 Rb cold atoms, the healing lengths of quantized vortices and the number of particles within the healing lengths are calculated, and they may be checked by experiment. Our results show that the Gross–Pitaevskii equation is capable of describing the structure of quantized vortices and physics at length scales smaller than the healing length.

DOI: 10.1088/1572-9494/abfe52

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Abstract Quantized vortices are important topological excitations in Bose–Einstein condensates. The Gross–Pitaevskii equation is a widely accepted theoretical tool. High accuracy quantized-vortex solutions are desirable in many numerical and analytical studies. We successfully derive the Padé approximate solutions for quantized vortices with winding numbers ω = 1, 2, 3, 4, 5, 6 in the context of the Gross–Pitaevskii equation for a uniform condensate. Compared with the numerical solutions, we find that (1) they approximate the entire solutions quite well from the core to infinity; (2) higher-order Padé approximate solutions have higher accuracy; (3) Padé approximate solutions for larger winding numbers have lower accuracy. The healing lengths of the quantized vortices are calculated and found to increase almost linearly with the winding number. Based on experiments performed with 87 Rb cold atoms, the healing lengths of quantized vortices and the number of particles within the healing lengths are calculated, and they may be checked by experiment. Our results show that the Gross–Pitaevskii equation is capable of describing the structure of quantized vortices and physics at length scales smaller than the healing length.

Abstract The dynamics of vortices in Bose–Einstein condensates of dilute cold atoms can be well formulated by Gross–Pitaevskii equation. To better understand the properties of vortices, a systematic method to solve the nonlinear differential equation for the vortex to very high precision is proposed. Through two-point Padé approximants, these solutions are presented in terms of simple rational functions, which can be used in the simulation of vortex dynamics. The precision of the solutions is sensitive to the connecting parameter and the truncation orders. It can be improved significantly with a reasonable extension in the order of rational functions. The errors of the solutions and the limitation of two-point Padé approximants are discussed. This investigation may shed light on the exact solution to the nonlinear vortex equation.

Padé approximants are used to find the approximate vortex solutions of any winding number in the context of Gross-Pitaevskii equation for the uniform condensate and condensates with axisymmetric trapping potentials. The approximate solutions are used to elucidate the annihilation of pair of vortices of opposite circulation by finding the approximation to the streamfunction of the flow generated by such a pair. The critical velocity and minimal separation of vortices are calculated. Keywords: Gross-Pitaevskii model, Padé approximations, vortex solutions. 1

We develop a multiscale technique to describe excitations of a Bose-Einstein condensate, whose characteristic scales are comparable to the healing length, thus going beyond the conventional hydrodynamical approximation. As an application of the theory, we derive an approximate explicit vortex and other solutions. The dynamical stability of the vortex is discussed on the basis of the mathematical framework developed here, the result being that its stability is granted at least up to times on the order of seconds, which is the condensate lifetime. Our analytical results are confirmed by numerical simulations.

The dynamics and interaction of quantized vortices in Bose-Einstein condensates (BECs) are investigated by using the two-dimensional Gross-Pitaevskii equation (GPE) with/without an angular momentum rotation term. If all vortices have the same winding number, they would rotate around the trap center but never collide. In contrast, if the winding numbers are different, their interaction highly depends on the initial distance between vortex centers. The analytical results are presented to describe the dynamics of the vortex centers when β=0. While if β≠0, there is no analytical result but some conclusive numerical findings are provided for the further understanding of vortex interaction in BECs. Finally, the dynamic laws describing the relation of vortex interaction in nonrotating and rotating BECs are presented.

Pade approximants are used to find approximate vortex solutions of any winding number in the context of Gross-Pitaevskii equation for a uniform condensate and condensates with axisymmetric trapping potentials. Rational function and generalised rational function approximations of axisymmetric solitary waves of the Gross-Pitaevskii equation are obtained in two and three dimensions. These approximations are used to establish a new mechanism of vortex nucleation as a result of solitary wave interactions.

We consider a Gross-Pitaevskii(GP) model of a Bose-Einstein Condensate(BEC). We study a single vortex line in the presence of non-local s-wave interactions. We show that the width of such a vortex is a microscopic length scale of the order of s-wave scattering length and is independent of the healing length. This analysis fixes the problem of arbitrariness in the selection of the width of vortices which is so far approximately considered to be the healing length

\xi_{0}

despite not being selected explicitly. We present a variational method to construct analytical solution for vortices of any quantum of circulation from the core to any finite extent. Using our analysis we predict hydrogen BEC to produce the thinnest vortex line.