Nucleation and kinematics of vortices in stirred Bose Einstein condensates Jonas Rnning and Luiza Angheluta1 1The Njord Center Department of Physics University of Oslo Blindern 0316 Oslo Norway

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Nucleation and kinematics of vortices in stirred Bose Einstein condensates

Jonas Rønning and Luiza Angheluta1

1The Njord Center, Department of Physics, University of Oslo, Blindern, 0316 Oslo, Norway

(Dated: October 27, 2022)

We apply the Halperin-Mazenco formalism within the Gross-Pitaevskii theory to characterise

the kinematics and nucleation of quantum vortices in a two-dimensional stirred Bose Einstein con-

densate. We introduce a smooth defect density ﬁeld measuring the superﬂuid vorticity and is a

topologically conserved quantity. We use this defect density ﬁeld and its associated current density

to study the precursory pattern formations that occur inside the repulsive potential of an obstacle

and determine the onset of vortex nucleation and shedding. We demonstrate that phase slips form

inside hard potentials even in the absence of vortex nucleation, whereas for soft potentials they

occur only above a critical stirring velocity leading to vortex nucleation. The Halperin-Mazenco

formalism provides an elegant and accurate method of deriving the point vortex dynamic directly

from the Gross-Pitaevskii equation.

Keywords: vortices, superﬂuid, Bose-Einstein condensat

I. INTRODUCTION

Topological defects are the ﬁngerprints of broken con-

tinuous symmetries, and are widely encountered in or-

dered systems, such as disclinations in liquid crystals [1,

2], dislocations in solid crystals [3–5], orientational de-

fects in biological active matter [6–8] or quantized vor-

tices in quantum ﬂuids [9–11], or cosmic strings [12]. The

formation and dynamics of topological defects during

phase ordering kinetics through temperature quenches

from the disordered phase have been well studied for

decades [13]. More recent work has been focused on de-

veloping theoretical frameworks to study non-equilibrium

pattern formations and dynamical regimes of ordered sys-

tems from the collective behavior of topological defects

beyond the phase-ordering kinetics.

The topological defects in an atomic Bose-Einstein

condensate (BEC) are quantum vortices where the con-

densate is locally melted while loosing its phase coher-

ence, and this induces persistent superﬂuid vortical cur-

rents outside the vortex cores. Hence, vortices are the

sole carrier of circulation in the condensate [14]. There is

active research both experimentally and theoretically on

understanding and tracking the non-thermal nucleation

and dynamics of quantized vortices in non-equilibrium

Bose Einstein condensates. Two main frameworks are

currently applied to study the creation of vortices either

by rotating the condensate [15–17] or by coupling the

condensate with a moving obstacle [18–22]. The nucle-

ation criterion is based on the energetic argument that

the superﬂuid ﬂow reaches a critical velocity above which

the condensate phase gradient undergoes phase slips. In

rotating BEC systems, vortices are created when the sys-

tem spins at a uniform frequency above a critical thresh-

old determined by the quantized circulation of the vortex.

Vortices of same circulation form at the edge of the con-

densate and migrate into the bulk where they eventually

form vortex latices [15, 23–26].

The vortex nucleation in condensates stirred by a mov-

ing obstacle has also been studied [27–31] and observed

experimentally [21, 22]. Here, the nucleation criterion

relies on the height U0of the repulsive potential repre-

senting the coupling of the stirring obstacle to the con-

densate. A hard potential corresponds to an almost im-

penetrable obstacle when U0> µ, where µis the conden-

sate chemical potential, such that the condensate density

rapidly decreases and nearly vanishes inside the poten-

tial. By contrast, a soft potential corresponds to a pene-

trable obstacle for U0< µ such that the condensate den-

sity is gently depleted inside the obstacle. The onset of

vortex nucleation induced by a hard obstacle occurs when

the local condensate velocity reaches the critical velocity

for phonon emission, whereas for the soft obstacle this

is a necessary, but not a suﬃcient requirement [27, 28].

Stirring obstacles are typically modelled as Gaussian po-

tentials with varying height and width [30, 31], which

approach a Dirac-delta function in the limit of a solid

obstacle. In Ref. [31], the vortex nucleation induced by

a repulsive Gaussian potential of diﬀerent strengths is

studied numerically. It is found that near the critical

velocity for vortex nucleation, the energy gap between

the ground state and the exited state goes to zero as

a power-law, while ghost vortices, i.e. phase slips, are

formed inside the potential. By contrast, no such ghost

vortices develop in the case of soft potentials. In addi-

tion to tuning the degree of permeability of the obstacle,

diﬀerent vortex shedding regimes, from vortex dipoles,

pairs and clusters [32–34], can be induced by varying the

size of the obstacle through the width of the potential

which also changes the critical stirring velocity [21, 35].

Once vortices are being shed into the condensate, they

interact with each other forming dynamic clusters that

sustain energy cascades and two-dimensional quantum

turbulence [11, 34, 36–39].

Even through compressibility eﬀects, due to shock

waves and phonons, are particularly important in the nu-

cleation and the annihilation of vortex dipoles, they are

typically overlooked in the quantum turbulence regime

where turbulent energy spectra and clustering behavior

is attributed mostly to the mutual interactions between

arXiv:2210.14620v1 [cond-mat.quant-gas] 26 Oct 2022

vortices [37, 40]. The point vortex modelling approach

have been employed to characterised quantum turbulence

from the dynamics of the point vortices [41–45]. In point

vortex models, vortices are reduced to charged point par-

ticles with an overdamped dynamics where their veloc-

ity is determined by the mutual interaction potential or

external potentials. For ﬁnite domains, boundary condi-

tions are satisﬁed by adding mirror vortices to the inter-

action potential [45, 46].

An accurate, non-perturbative method of deriving the

velocity of topological defects directly from the evolution

of the order parameter of the O(2) broken rotational sym-

metry has been developed by Halperin and Mazenko [47–

49]. Topological defects are located as zeros in the 2D

vector order parameter, where the magnitude of order

vanishes to regularise the region where the phase of the

order parameter becomes undeﬁned. The defect veloc-

ity is determined by the magnitude of the defect density

current at the defect position. In the frozen phase ap-

proximation, where the phase of the order parameter is

stationary apart from its moving singularities, the vor-

tex kinematics determined by the evolution of the or-

der parameter reduces to a point vortex model [45, 48].

Within the Gross-Pitaevskii theory, the order parameter

is the condensate wavefunction and the frozen-phase ap-

proximation is the regime where the dynamics of phonon

modes can be neglected. This is a versatile formalism

which has been applied to various systems from point

dislocations [4] and dislocation lines [5] in crystals, to

point orientational defects in active nematics [50] and ac-

tive polar systems [51] and disclination lines in nematic

liquid crystals [52].

In this paper, we adopt the Halperin-Mazenko formal-

ism to gain further theoretical insights into the kinematic

and nucleation of vortices in a stirred BEC that is de-

scribed by the Gross Pitaevskii equation [11, 38, 39].

In Section II, we present the Halperin-Mazenko formal-

ism for 2D BECs, and show that the defect density ﬁeld

Drepresents a generalized, smooth vorticity, deﬁned as

the curl of the superﬂuid current, and its evolution de-

termines the vortex velocity. This method circumvents

the need of operating directly with the singularities in

the condensate phase, which are harder to manipulate

both theoretically and numerically. In Section III, we

study how the defect ﬁeld Dand its corresponding cur-

rent density JDare behaving during a nucleation event

for two representative stirring potentials. In the case of

a soft potential, the D-ﬁeld shows the gradual buildup

of generalized vorticity around the the edge of the po-

tential and before any phase-slips occur. This provides

us with an accurate measure of the precursory pattern

leading to a nucleation event. By contrast, the pattern

of the D-ﬁeld for a hard potential is given by a uniform

halo formed around the edges of the potential coexisting

with phase slips inside the potential, also referred to as

ghost vortices. At the onset of nucleation, the superﬂuid

vorticity halo localises into two regions while a dipole of

phase slips migrates and occupies them to form a vortex

dipole which sheds from the edge of the potential. In

Section IV, we derive the point vortex dynamics and dis-

cuss the eﬀect of non-uniform condensate density on the

vortex velocity. Concluding remarks and a summary are

presented in Section V.

50 0 50

0.6

0.4

0.2

0.0

0.2

0.4

0.6

50 25 0 25 50

0.2

0.4

0.6

0.8

1.0

| |2

FIG. 1. Snapshot of the D-ﬁeld (a) and condensate density

(b) in the presence of a stirring Gaussian potential and shed

vortex dipoles. The vector-ﬁelds are the defect current JD

(a) and the normalized superﬂuid current J(b), respectively.

Negative vortices move in the opposite direction of the defect

current JD.

II. VORTICES AS MOVING ZEROS

The superﬂuid ﬂow and the topological structure of a

weakly-interacting BEC are described by the evolution of

its macroscopic wavefunction ψ=|ψ|eiθ, where |ψ|is the

condensate density and θis the condensate phase which

is coherent, i.e. constant at equilibrium. Disturbances in

the condensate phase generate a superﬂuid ﬂow with a

current (momentum) density

J=|ψ|2∇θ= Im(ψ∗∇ψ),(1)

such that gradients in the condensate phase deﬁne the

superﬂuid ﬂow velocity, which is irrotational everywhere

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NucleationandkinematicsofvorticesinstirredBoseEinsteincondensatesJonasRnningandLuizaAngheluta11TheNjordCenter,DepartmentofPhysics,UniversityofOslo,Blindern,0316Oslo,Norway(Dated:October27,2022)WeapplytheHalperin-MazencoformalismwithintheGross-Pitaevskiitheorytocharacterisethekinematicsandnucleation...

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Nucleation and kinematics of vortices in stirred Bose Einstein condensates Jonas Rnning and Luiza Angheluta1 1The Njord Center Department of Physics University of Oslo Blindern 0316 Oslo Norway

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