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Healing length independent vortex in Bose-Einstein Condensate

Abhijit Pendse, A. Bhattacharyay

Published Feb 17, 2016• 0 citations
OpenAlex

Abstract

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.

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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 ξ0\xi_{0}ξ0​ 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.
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.
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.
Quantized vortices are a hallmark of superfluidity. However, in a photon Bose--Einstein condensate, the photon-photon interaction is so weak that in a homogeneous system the healing length exceeds the experimentally achievable size of the condensate itself. Here we show that a harmonic confinement regularizes the vortex size to experimentally achievable scales. Moreover, such an external confinement closely resembles the standard experimental setup of a pumped dye-filled microcavity. We model the condensate via a complex Gross--Pitaevskii equation and obtain an approximate dynamical single-vortex solution by applying the recently proposed projection optimization method. The latter generalizes the variational approach of closed systems to open-dissipative systems without assuming a specific form for the condensate phase. The radial photon flow, which is characteristic for driven-dissipative systems, yields a definition of the vortex core size based on the competition of gain and loss. However, this condition reproduces the heuristic closed system definition now based on physical grounds. A subsequent linear stability analysis shows that interaction, as well as pumping and loss can drive the system to an unstable regime. In this way one can fundamentally distinguish between closed and open-dissipative systems.
We add a minimal correction term to the local Gross–Pitaevskii equation to represent non-locality in the interactions. We show that the effective minimal non-locality can make the healing length decrease more rapidly with the increase of s-wave scattering length leaving the expression of the velocity of sound unaltered. We discuss the implication of this result for a Bose–Einstein condensate being used as an analogue gravity system. The presented result is important in the context of condensed matter physics as well because one can considerably change the size of a quantized vortex at finite s-wave scattering length by tuning the healing length.
The aim of this paper is to derive the equations of motion of quantum vortices in a weakly non-ideal Bose–Einstein condensate (BEC) at finite temperatures and in the presence of macroscopic superfluid and normal velocities. The motivation for this work is that the dynamics of quantum vortices in superfluid helium has been studied at the phenomenological level, and a number of issues require clarification, which can be addressed by studying the corresponding problem in BEC. In BEC, quantum vortices are located at points where the macroscopic wave function (MWF) ψ(r,T), which describes the BEC dynamics, vanishes ψ(r,T)=0. The intersection of the surfaces Reψ(r,T)=0 and Imψ(r,T)=0 forms a one-dimensional line, which is just the quantum vortex line. During the evolution of the MWF, the position of the zero surfaces changes, that is, the vortex lines move. Conversely, the motion of these vortex filaments determines the evolution of the MWF. Thus, there is a mutual correspondence, and the quantity ψ(r,T) can be considered as a functional of the entire vortex configuration, i.e., {s(ξ,t)}. This viewpoint allows us to derive the equations of motion for vortices from the Gross–Pitaevskii equation (GPE), which describes the dynamics of MWF. The key finding is the confirmation that the equation derived from the fundamental quantum description (the GPE) for a BEC has the same mathematical structure as the equation derived from macroscopic, phenomenological reasoning for superfluid helium.
We consider a Bose-Einstein condensate (BEC) with non-local inter-particle interactions. The local Gross-Pitaevskii (GP) equation is valid for the gas parameter [Formula: see text], but for [Formula: see text], the BEC is described by a modified GP equation (MGPE). We study the exact solutions of the MGPE describing bright and dark solitons. It turns out that the width of these non-local solitons has qualitatively similar behaviour as the modified healing length due to the non-local interactions of the MGPE. We also study the effect of the non-locality and gas parameter (ν) on the stability of the solitons using the Vakhitov-Kolokolov (VK) stability criterion. We show that these soliton solutions are stable according to the VK criterion. Further, the stability of these soliton solutions gets enhanced due to the non-locality of interactions.