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Snell's Law for a vortex dipole in a Bose-Einstein condensate

by Michael M. Cawte, Xiaoquan Yu, Brian P. Anderson, Ashton S. Bradley

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Submission summary

As Contributors: Ashton Bradley · Michael MacCormick Cawte · Xiaoquan Yu
Arxiv Link: (pdf)
Date submitted: 2018-11-07
Submitted by: Bradley, Ashton
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Atomic, Molecular and Optical Physics - Theory
Approach: Theoretical


A quantum vortex dipole, comprised of a closely bound pair of vortices of equal strength with opposite circulation, is a spatially localized travelling excitation of a planar superfluid that carries linear momentum, suggesting a possible analogy with ray optics. We investigate numerically and analytically the motion of a quantum vortex dipole incident upon a step-change in the background superfluid density of an otherwise uniform two-dimensional Bose-Einstein condensate. Due to the conservation of fluid momentum and energy, the incident and refracted angles of the dipole satisfy a relation analogous to Snell's law, when crossing the interface between regions of different density. The predictions of the analogue Snell's law relation are confirmed for a wide range of incident angles by systematic numerical simulations of the Gross-Piteavskii equation. Near the critical angle for total internal reflection, we identify a regime of anomalous Snell's law behaviour where the finite size of the dipole causes transient capture by the interface. Remarkably, despite the extra complexity of the surface interaction, the incoming and outgoing dipole paths obey Snell's law.

Ontology / Topics

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Bose-Einstein condensates (BECs) Dipolar interactions
Current status:
Has been resubmitted

Submission & Refereeing History

Resubmission 1811.02110v4 on 26 February 2019
Submission 1811.02110v1 on 7 November 2018

Reports on this Submission

Anonymous Report 2 on 2018-12-21 Invited Report

  • Cite as: Anonymous, Report on arXiv:1811.02110v1, delivered 2018-12-21, doi: 10.21468/SciPost.Report.760


The paper is well written and the results are very clear and interesting. The agreement between the theory and numerical results are excellent, indicating that the proposed “Snell’s law” for vortex dipoles is reasonable.


Some corrections are needed (written in the report).


The authors of the present paper investigate the dynamics of a vortex dipole incident upon the interface of superfluid with different densities. They derive the relation between the incident and refraction angles, as Snell’s law in optics, which is numerically confirmed by solving the Gross-Pitaevskii equation. They also find an interesting intermediate phenomenon, where the incident vortex dipole travels along the interface before it is reflected.

The paper is well written and the results are very clear and interesting. The agreement between the theory and numerical results are excellent, indicating that the proposed “Snell’s law” for vortex dipoles is reasonable. I recommend this paper for publication in SciPost Phyiscs.

Minor comments:
1) Captions in Figs. 4 and 7: Do the solid lines represent Eq. (19)? The caption should state it explicitly.

2) Caption of Fig. 5: The meaning of the colored arrows should be written.

3) In Fig. 5(b), the traveling distance of the vortex dipole along the interface increases with the incident angle for the red lines, whereas it decreases for the yellow and white lines. Is there some explanation for this behavior?

4) The present results may be related with PRA 85, 023618 (2012), in which the dynamics of vortex dipoles across the interface between binary BECs are studied.

Requested changes

Written in the report.

  • validity: high
  • significance: good
  • originality: good
  • clarity: high
  • formatting: good
  • grammar: excellent

Anonymous Report 1 on 2018-12-4 Invited Report

  • Cite as: Anonymous, Report on arXiv:1811.02110v1, delivered 2018-12-04, doi: 10.21468/SciPost.Report.700


1- Provides a useful guide for the behavior of vortex dipoles (which are self-propelled localized excitations) as they pass through interfaces of sharply changing density.

2- A thorough numerical investigation with some analytic insight


The text could be streamlined and clarified in a few places. But this is only a very minor issue and didn’t really slow my reading to any significant degree.


The manuscript by Cawte et al. studies the motion of vortex dipoles as they cross a step interface between regions of differing density. The system is in a quasi-2D square well configuration and the different densities are created by a step in the potential along the bottom of the well. They numerically analyze the reflection/refraction versus incident angles and compare it to a Snell’s law-like analytical prediction. They find that this prediction is obeyed well before and long after the interaction with the surface, even when for some incident angles there can be complicated intermediate behavior such as surface capture.

I recommend this manuscript for publication in SciPost after some minor revisions. In my view this work is interesting and could guide future experiments on vortices in flattened geometries. The numerical study is thorough, while at the same time being clear and easy to read.

Requested changes

1- The derivation of vortex dipole momentum [eq (11)] is not easy to follow. Why is the 2D integral reduced to a pair of line integrals along the top and bottom borders of the system? Also, why are the limits of the integral only from -d/2 to d/2. I would have expected nonzero contributions to $P_y$ for $x$ values outside of these limits too. Some further explanation in the text would be beneficial.

2- In Eq. (19b) does $\xi_f = \xi_i$ for the present work? Shouldn’t this always be the case when in equilibrium? A comment on this would be useful since sometimes people refer to local effective chemical potentials that depend on the local potential.

3- Do the authors know what is the reason for long surface capture times? What is changing during the capture time that finally allows a dipole to be released? Has it something to do with the reorganization of superfluid flow fields at short to moderate distances? Do you think there are any conditions that could cause a dipole to be captured indefinitely? Discuss.

4- It looks like a couple of typos on the second line of section 4.5. Shouldn’t it read $\rho_i>\rho_f$ and $\rho_2/\rho_1 =$ … ?

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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