Minimum critical velocity of a Gaussian obstacle in a Bose-Einstein condensate Haneul Kwak1Jong Heum Jung1and Y. Shin1 2 3 1Department of Physics and Astronomy Seoul National University Seoul 08826 Korea

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Minimum critical velocity of a Gaussian obstacle in a Bose-Einstein condensate

Haneul Kwak,1Jong Heum Jung,1and Y. Shin1, 2, 3, ∗

1Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea

2Center for Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Korea

3Institute of Applied Physics, Seoul National University, Seoul 08826, Korea

When a superﬂuid ﬂows past an obstacle, quantized vortices can be created in the wake above

a certain critical velocity. In the experiment by Kwon et al. [Phys. Rev. A 91 053615 (2015)],

the critical velocity vcwas measured for atomic Bose-Einstein condensates (BECs) using a moving

repulsive Gaussian potential and vcwas minimized when the potential height V0of the obstacle was

close to the condensate chemical potential µ. Here we numerically investigate the evolution of the

critical vortex shedding in a two-dimensional BEC with increasing V0and show that the minimum

vcat the critical strength V0c≈µresults from the local density reduction and vortex pinning eﬀect

of the repulsive obstacle. The spatial distribution of the superﬂow around the moving obstacle just

below vcis examined. The particle density at the tip of the obstacle decreases as V0increases to

Vc0and at the critical strength, a vortex dipole is suddenly formed and dragged by the moving

obstacle, indicating the onset of vortex pinning. The minimum vcexhibits power-law scaling with

the obstacle size σas vc∼σ−γwith γ≈1/2.

I. INTRODUCTION

A superﬂuid can ﬂow without friction but only below

a certain critical velocity. Above the critical velocity,

the superﬂuid becomes dynamically unstable, generating

excitations such as phonons and quantized vortices [1].

Understanding the critical dynamics and critical veloc-

ity of a superﬂuid is of fundamental and practical impor-

tance for the study of the transport properties of a super-

ﬂuid system [2–4]. The key questions are what induces

the instability of the superﬂuid ﬂow and how the energy

dissipation evolves with the increasing ﬂow velocity. At

substantially high velocities, turbulent states would be

developed in the superﬂuid system with a complex tan-

gle of vortex lines, namely, quantum turbulence [5, 6].

In recent experiments with atomic Bose-Einstein con-

densates (BECs), a localized optical potential formed by

focusing a laser beam was adopted as a movable obsta-

cle [7]. Various superﬂuid dynamics were investigated

by controlling the movement of the obstacle in a sam-

ple. From the onset of energy dissipation with increasing

obstacle speed, critical velocities of various atomic super-

ﬂuid gases were demonstrated [7–12], where the measure-

ment results tested theoretical predictions [13–18] and

revealed the details of the dissipation mechanisms [9–

12, 19, 20]. For a fast obstacle above the critical velocity,

the vortex shedding in the wake of the moving obsta-

cle was investigated [21–25]. A remarkable observation

was that vortex clusters consisting of like-sign vortices

are regularly shed from a uniformly moving obstacle in

atomic BECs [22]. This is analogous to the von K´arm´an

vortex street in the classical viscous ﬂuids in the transi-

tion to turbulence [24, 25].

For the optical obstacle, there are two regimes with

respect to the relative magnitude of the obstacle’s peak

∗yishin@snu.ac.kr

potential V0to the chemical potential µof the BEC. The

particle density at the obstacle position is suppressed be-

cause of the repulsion of the obstacle. However, when

V0< µ, the condensate can penetrate the obstacle and

a zero-density region is not induced in the condensate.

In this penetrable case, vortices can be created only in

the form of a dipole consisting of two vortices of opposite

circulations. When V0> µ, which is referred to as impen-

etrable, a density-depleted hole is formed in the system,

and it would signiﬁcantly alter the characteristics of the

vortex shedding dynamics by allowing the generation of

vortex clusters [22]. In the experiment by Kwon et al. [9],

the critical velocity vcfor vortex shedding was measured

as a function of V0, and vcwas minimized sharply at a

certain critical strength V0cthat was close to µ. This

implies that the onset behavior of the vortex shedding,

which we refer to as critical vortex shedding, undergoes a

certain transition as the obstacle strength changes from

penetrable to impenetrable.

In this paper, we numerically study the critical vor-

tex shedding of a Gaussian obstacle in a two-dimensional

(2D) BEC and investigate its evolution with increasing

obstacle strength. We verify that the critical velocity is

minimized at a critical obstacle strength V0cclose to µ

and show that it arises from the start of vortex pinning

as V0increases above V0c. We examine the spatial distri-

bution of the superﬂow around the moving obstacle just

below vc. At the critical strength, the superﬂow distri-

bution suddenly changes to form a vortex dipole that is

pinned at the tip of the obstacle. When V0is further

increased, a density-depleted region develops and the co-

moving, pinned vortex dipole becomes virtual and is ab-

sorbed in the region. The minimum vcat V0=V0cde-

creases with increasing the obstacle size σ. We ﬁnd that

it exhibits a power-law scaling of vc∼σ−γ, with γ≈1/2,

which is in reasonable agreement with the experimental

results of Ref. [9]. Our results demonstrate the existence

of the minimum critical velocity for a Gaussian obstacle

and elucidate the transition of the critical vortex shed-

arXiv:2210.04403v2 [cond-mat.quant-gas] 14 Feb 2023

ding from the penetrable to impenetrable regime.

The remainder of this paper is organized as follows. In

Sec. II, we describe a theoretical model to study the vor-

tex shedding dynamics in a BEC based on the 2D Gross-

Pitaevskii equation. In Sec. III, we present numerical re-

sults, including a comparison of the shedding dynamics

for penetrable and impenetrable obstacles, and the char-

acterization of the critical vortex dipole state generated

by the moving obstacle at the critical strength. Finally,

in Sec. IV, a summary of this work and the outlooks for

future studies are provided.

II. THEORETICAL MODEL

We consider a situation where an obstacle moves in a

homogeneous BEC with a constant velocity v. In the

mean-ﬁeld theory, the BEC dynamics is described by the

Gross-Pitaevskii equation (GPE),

i~∂Ψ

∂t =−~2

2m∇2+V(r−vt) + g|Ψ|2−µΨ,(1)

where Ψ(r, t) is the macroscopic wave function of the

BEC, ~is Planck’s constant divided by 2π,mis the atom

mass, V(r) is the obstacle potential, and gis the nonlin-

ear coupling coeﬃcient. Taking the unitary transforma-

tion Ψ(r, t) = exp[−vt·∇]ψ(r, t), Eq. (1) is transformed

into the reference frame moving with the obstacle as

i~∂ψ

∂t =−~2

2m∇2+i~v∂

∂x +V(r) + g|ψ|2−µψ(2)

with v=vˆ

x. The characteristic length and time scales of

the system are given by the healing length ξ=~/√2mµ

and tµ=~/µ, respectively. Using the change in vari-

ables, ˜

r=r/ξ and ˜

t=t/tµ, the equation can be ex-

pressed in a dimensionless form as

i∂˜

t˜

ψ=−˜

∇2+i√2˜v∂˜x+˜

V(˜

r) + |˜

ψ|2−1˜

ψ(3)

with ˜

ψ=n−1/2

0ψ, ˜v=v/cs,˜

∇=ξ∇, and ˜

V=V/µ.

Here n0=µ/g is the particle density of the BEC without

the obstacle and cs=pµ/m is the speed of sound.

In this work, we study the BEC dynamics for a Gaus-

sian obstacle in two dimensions. This is motivated by

the recent experiments using highly oblate atomic sam-

ples [9–12, 23], where the vortex line dynamics along

the tight conﬁning direction is energetically irrelevant.

Hence, the shedding dynamics can be well described in

2D. In a hydrodynamic approximation, the dimensional

reduction is carried out by integrating the wave function

component along the short axis. It eﬀectively modiﬁes

the speed of sound in Eq. (3) [26, 27]. The potential of the

Gaussian obstacle is given by V(r) = V0exp[−2(r2/σ2)],

where r=px2+y2and σis the 1/e2radius of the obsta-

cle. The obstacle is located at the origin of the reference

frame.

We numerically solve Eq. (3) in the xy plane with pe-

riodic boundary conditions, using the pseudo spectral

method [28]. In the simulation of vortex shedding for

v > vc, we set the initial state to be a stationary solution

for a velocity vislightly below vc. Next, we increase the

obstacle speed up to the target velocity vfor an accelera-

tion time ta= 200tµ[29]. The initial stationary solution

is obtained using the imaginary-time method where tis

replaced by −iτ [30, 31]. To realize a constant stream

at the front boundary of the obstacle, we adopt the nu-

merical method described in Ref. [25], where damping

zones with a thickness of 20ξare set at the boundary to

attenuate the wake of the BEC and recover the constant

uniform ﬂow at the front boundary. In the calculation of

a stationary solution using the imaginary-time propaga-

tion method, the damping zone is inactivated.

III. RESULTS AND DISCUSSION

A. Determination of critical velocity

Figures 1(a) and 1(b) display the density distribu-

tions of the BEC, n(x, y) = |ψ|2, at t/tµ= 1200 for

two diﬀerent velocities, v/cs= 0.25 and 0.28, respec-

tively [29]. The obstacle size and strength are σ/ξ = 20

and V0/µ = 0.8. When the speed is lower than the

threshold value of vc≈0.26cs, no vortices are generated.

The BEC remains stationary [Fig. 1(a)]. By contrast,

when the obstacle velocity increases above the threshold

velocity, vortices are emitted from the obstacle in a pe-

riodic manner [21]. The periodic vortex shedding is also

examined by inspecting the drag force exerted by the

obstacle Fx=−R˜

ψ∗(∂˜x˜

V)˜

ψd2˜

rusing the Ehrenfest re-

lation [25]. We verify that for v > vcthe force oscillates

in time, corresponding to the periodic vortex emission.

For v < vcit is stationary and remains approximately

zero [Fig. 1(c)].

We determine the critical velocity vcfrom the exis-

tence of a stationary ground state solution via the imagi-

nary time propagation method [15]. The imaginary time

method gives a converging stationary solution for v < vc

or an oscillating solution otherwise. In the oscillating

solution, a pair of vortices is created by the obstacle.

They move away from each other along the y-direction

and are annihilated at the system’s boundary due to

the periodic boundary conditions. This process is re-

peated over an imaginary time. In the calculation of

stationary solutions, we employed a spatial domain of

(Lx, Ly) = (400,400)ξwith (Nx, Ny) = (600,600) grids

and took a time step of ∆τ/tµ= 0.04. We decided the

convergence of a solution through its behavior up to the

imaginary time τ/tµ= 4000. The critical velocities de-

termined from our imaginary time method are identical

to the threshold values from the simulation of the real-

time evolution within an error of 0.02cs.

Figure 2displays the numerical results of the critical

velocities over a range of obstacle strength 0.1≤V0/µ ≤

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MinimumcriticalvelocityofaGaussianobstacleinaBose-EinsteincondensateHaneulKwak,1JongHeumJung,1andY.Shin1,2,3,1DepartmentofPhysicsandAstronomy,SeoulNationalUniversity,Seoul08826,Korea2CenterforCorrelatedElectronSystems,InstituteforBasicScience,Seoul08826,Korea3InstituteofAppliedPhysics,SeoulNational...

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Minimum critical velocity of a Gaussian obstacle in a Bose-Einstein condensate Haneul Kwak1Jong Heum Jung1and Y. Shin1 2 3 1Department of Physics and Astronomy Seoul National University Seoul 08826 Korea.pdf

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Minimum critical velocity of a Gaussian obstacle in a Bose-Einstein condensate Haneul Kwak1Jong Heum Jung1and Y. Shin1 2 3 1Department of Physics and Astronomy Seoul National University Seoul 08826 Korea

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