Dynamics of Polar-Core Spin Vortices in Inhomogeneous Spin-1 Bose-Einstein Condensates

The authors show the qualitative differences in the dynamics of different topological vortices.

In this manuscript, the authors have investigated the dynamics of polar-core vortices in a spin-1 ferromagnetic Bose-Einstein condensate (BEC) with non-uniform density. The polar-core vortex is the vortex with spin singularity. Under an easy-plane anisotropy due to the quadratic Zeeman effect, the $m=1$ and $-1$ components have opposite phase winding at the same position with the $m=0$ component filling the core. The authors consider the motion of such a polar-core vortex under a non-uniform density distribution in a confining potential. In the case of a scalar BEC, it is known that the inhomogeneous density profile induces an effective force to the vortices, where vortices with opposite circulation rotate in the opposite direction around the trap center. Thus, the vortices in the $m=1$ and $-1$ components move in the opposite direction. In addition, differently from the case of scalar BEC, the vortices feel an additional force due to the spin interaction. The authors have analytically derived the equation of motion of the relative and mean positions of the vortices in $m=\pm1$ components. The result shows that the polar-core vortex moves to the lower-density region, which the authors confirm by numerical calculation. In the case of a simple harmonic trap, the vortex goes out of the condensate. Then, the authors introduce a density dip at the trap center and show that the vortex is pinned and oscillates around the density dip.

This work extends the analysis of vortex dynamics in a scalar BEC to a spinor one. The authors clearly show the difference in the behavior of the phase vortex in a scalar BEC and the polar-core vortex in a spinor BEC. The fact that such a difference in the topology of vortices leads to different dynamics sounds quite interesting, and it is of interest to a wide range of researchers concerning topology. Although several points are unclear (see below), I’d like to recommend the publication of this manuscript in SciPost Physics.

1. The authors consider the motion of phase singularities in the $m=\pm1$ components and show that the relative position of the vortices becomes larger as the mean position goes down the density gradient. This reads as if the polar-core vortex is unstable against the splitting. However, the polar-core vortex should be stable even in the setup the authors considered. The polar-core vortex should be defined as a singularity of magnetization, so the position of the polar-core vortex can be traced at least in the numerical simulation. The authors should clarify this point in the manuscript. I feel showing the profiles of $|{\bf F}|/n$ is informative.
2. The separation of the two vortices in the $m=\pm1$ components is the appearance of the longitudinal magnetization around the polar core. What is the origin of the longitudinal magnetization? How does the magnetization flow under a density gradient and the spin current around the vortex?
3. I could not understand how Eq. (9) is derived.
4. $\kappa$ seems to be restricted to $\pm1$ after Sec. 3. This should be clarified.

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1. Nice study of potential relevance to current/future experiments with spinor atomic condensates.

2. Combined numerical study with supporting analytical model providing further qualitative understanding.

3. Clearly written manuscript with nice figures and extended referencing.

1. Some findings could be better analysed/quantified.

2. I am generally not in favour of short-hand notations ("PCV" used here for polar core vortex).

This is an interesing paper discussing the dynamics of polar-core spin vortices in spinor atomic Bose-Einstein condensates. Main results are that such a vortex structure will accelerate radially towards the edge of a harmonically-trapped condensate, whereas emission and reabsorption of axial spin waves leads to oscillations about the trap centre in a trap with a local density minimum. Numerical simulations based on mean-field theory (Gross-PItaevskii) are consistent with a simpler point-vortex-based model, which further explains the underlying physical process.

While such work is not really ground-breaking, it can nonetheless be seen as opening a new experimentally-relevant direction in vortex dynamics in spinor condensates: as such, I am happy to support its publication, subject to a few further comments/clarifications.

1. The authors use a strongly dissipative Gross-Pitaevskii model for the initial state, followed by Gross-Pitaevskii dynamical propagation: it was not clear to me why they chose to include (1-i) as opposed to simply -i for the initial relaxation. Moreover, given that the code has such direct capability, I wonder whether the authors tried adding a small damping of the form (1-i gamma) [e.g. with a small gamma value in the typical range 10^(-3)-10^(-2) or so] to investigate to what extent their findings are significantly affected, and what the qualitative nature of the resulting changes would be to the dynamical evolution of such a vortex.

2. The authors choose to work in the parameter regime g_n/|g_s| = 10 to focus on the spin degrees of freedom. While fully acceptable, could they comment more on how a different choice might affect their findings (at the very least qualitatively)?

3. Is there any reason why the t=0 dynamics is not included in Fig. 3 (as it is in Fig. 1)? I feel it might be helpful. Also, perhaps a zoom-in of the region near x=0 might be useful to include, to better highlight the +5 xi_s initial condition (unless I am missing something here).

4. Moreover, might it be useful to more clearly highlight what is plotted, in terms of the +1 and -1 densities?

5. What is the specific role of the boundaries, besides acting so as to reflect the emitted waves, and are the findings at all sensitive to the way they are modelled? In particular, would a different choice of underlying trap (e.g. harmonic trap with a Gaussian barrier in the middle) produce qualitatively very similar results, or would the process be affected by distinct propagation of emitted waves in such an underlying trap?

6. Some more clarifications around Eq. (22) would be welcome: are these values somehow anticipated, or purely fitting extracted? If the latter, is this a unique parameter choice, and can those numbers be more physically interpreted?

7. I was slightly confused about the changing value of the decay exponent alpha after Eq. (23). While I find such analysis, and discussion of the box dependence useful, could something more be extracted from this? Are these and earlier results insensitive to the initial condition choice (e.g. polar-core vortex location?)

8. In single-component condensates, vortex motion has been carefully characterised in 2d. Would the authors expect to ultimately find similar controlled monitoring in spinor systems, or are there genuine challenges prohibiting that? If so, perhaps it might be useful to give some plausible experimental numbers for such observation, as an indication. The one aspect I am unsure of is the survival timescale of such defects, so it would be interesting to see (presumably in a future publication) the effect that some small fluctuations in the initial condition might actually have on the ensuing motion.

Publish (meets expectations and criteria for this Journal)

1. Provides a path that can lead to further interesting models of the hydrodynamics of spinor Bose-Einstein condensates.
2. The results can in principle inspire immediate experiments using existing setups.
3. The manuscript is focussed and well written.

1. The referencing is possibly a little narrowly focussed and while it captures the immediately important works may miss some relevant context from other recent experiments.

2. There is a small number of typographical errors in the manuscript, including inline equations overflowing into the margins and one or two instances of strange spacing in inline equations. This is not any scientific criticism, but included here since the journal specifically asks about paper formatting.

In their manuscript, the authors give a detailed description of what they term a polar-core spin vortex (PCV) in a spin-1 Bose-Einstein condensate (BEC) in two experimentally relevant contexts: a harmonically trapped condensate and a condensate in a box trap with a pinning potential. They find that in the former case, the vortex moves down the density gradient without exhibiting the familiar precession of a vortex in a scalar BEC. In the latter case, the vortex is found to oscillate around the pinning site, exibiting stretching of the vortex-core region. The study is interesting and well presented and should be of interest for both theorists and experimenters. One may quibble whether the study really represents "a groundbreaking theoretical/computational discovery", but it certainly presents a pathway for analysing spinor-vortex dynamics that could be built upon, e.g., to understand spinor vortex interactions, leading to hydrodynamic models. For this reason I support publication, provided the authors consider a number of relatively minor questions.

1. The authors work in what they call the easy-plane phase of a condensate with ferromagnetic interactions. This corresponds to the broken-axisymmetry phase of Ref. 25 at zero longitudinal magnetisation such that the condensate spin has no z-component. The authors correctly state that the condensate spin becomes density dependent, such that the condensate reaches the polar phase beyond some radius in the harmonic trap. What is the spin magnitude at the centre of the trap/at the initial position of the vortex.

2. In the easy-plane phase, with the condensate spin confined to the x,y-plane, one can easily see that the order-parameter space should correspond to U(1) x SO(2). In this case, the vortex with no U(1) charge and a 2pi winding of the in-plane spin vector, leading to a first homotopic group Z x Z and the pure spin vortices considered here. However, it is clear from the presented simulations that the condensate spin does not remain fully in the x,y-plane, but rather develops a non-negligible z-component. How does this modify the understanding of the nature of the vortices? Do they pick up a mass circulation?

3. For the same reason, once the vortex core stretches and the regions of non-zero F_z appear, beta in Eq. (19) aquires a spatial dependence. Can the the vortex spinor be brought back on this form through a change of spinor basis, similar to the corresponding ferromagnetic vortex in Lovegrove et al., PRA 86, 013613 (2012)?

4. The authors consider a pinning potential together with the box trap, but not the harmonic trap. While box traps are becoming increasingly interesting for experiments, the harmonic trap remains the common choice. Should one expect the results observed for the pinning in the box trap to hold also if one were to introduce a pinning potential in a harmonic trap?

5. The authors begin their numerical simulation by including "a strong damping dt -> (1+i)dt", i.e., equal real and imaginary parts for the time step. Is there a reason this step is not done as relaxation in imaginary time (with no real part)? The authors then proceed to integrate the GPEs forward in time with no dissipation. How do results change if a small dissipation is included (i.e., complex time dt -> (1 + eta i)dt for eta ~10^-2 or 10^-3, as a rough model of the small dissipation present in experiments?

6. In their introduction, the authors cite Refs. 4-11 for vortex pinning. However, not all of these seem to be examples of topological defects (but rather, e.g., classical vortices) as implied by the context of the rest of the beginning of the sentence. Can the authors clarify?

7. The immediately relevant spinor-BEC experiments using in situ detection in the group of Y.-i. Shin are appropriately cited in the introduction. However, there have been several other experimental realisations of different types of spinor vortices in the last several years, including in condensates with ferromagnetic interactions (such as Rb-87), for which the results of the present manuscript may be equally relevant, and which ought to provide relevant context.

Finally a very minor detail:

8. The authors model the box trap using the expression given below Eq. (21). How well does this tanh^{10} form model existing box traps such as that of Ref. 72? The expression also includes a constant factor coth^{10}(1). There is presumably a reason for writing this constant on this form?

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