Quasiparticles of widely tuneable inertial mass: The dispersion relation of atomic Josephson vortices and related solitary waves

First of all, we would like to thank both referees for the time and effort they have invested in reviewing our manuscript, as well as the constructive feedback and interesting questions. Please find below a response to all points raised by the referees and a summary of the changes made.

Referee 1:

(1) The referee comments that discussion of the figures should to be improved and should occur near where the figures are placed in the manuscript. We have thus moved old Fig. 2 (now Fig. 5) and Figs. 4-6 (now Figs. 6-8) to more relevant places in the manuscript. We have also made small adjustments to the discussion of the figures in the text and, in particular, have added cross-references in the text where figures are (first) mentioned to the section where the main discussion of the figures is located (this is relevant also to the old Figs. 11 and 13, which are now Figs. 12 and 14).

Further, the referee suggests replacing old Figs. 3 & 7 by one-dimensional plots rather than iso-surfaces. We do quite like the 3D plots, which encode the full information (phase in the colour and density in the transverse extent of the iso-surface), for their intuitive appeal but have decided to show them alongside more conventional line plots in the new figures 2, 3, and 4 (previously 3 and 7).

(2) The referee points out the article Magnetic solitons in Rabi-coupled Bose-Einstein condensates by Qu et al. Indeed this paper is highly relevant and we thank the referee for pointing it out. We only became aware of it after submitting our manuscript for publication. We have now incorporated this reference into the literature review in the introduction.

Finally, the referee is correct in their interpretation of the meaning of the angular momentum ΔP when the two condensates are in a double-well potential. This is a useful quantity because only the Josephson vortices have a non-zero ΔP, while dark and staggered solitons do not. In fact, the property of ΔP≠0 can be used to distinguish vortex and soliton excitations. A note to this end has been added at the end of section 7.

Referee 2:

The referee poses four interesting questions, to which we now reply:

1) The miscible/immiscible threshold in our system occurs at γ=0 (previously mentioned in passing under equation (51)), while the polarised/unpolarised threshold at υ=0. For υ>0 the ground state has equal densities in the two components and, in spinor notation, is proportional to the vector (1,1). Since this is an eigenstate of the σx operator, the state is polarised in the x-direction. This is reflected in the fact that the phases of the two condensates are linked: they must be pairwise equal at each edge of the box. This is no longer the case if υ=0, when the phases of the two strands become completely independent. Moreover, all states on the equator of the Bloch sphere are then degenerate, making for a degenerate mean-field ground state. We constrained our investigation to the regime where the ground state is miscible (γ>0) and polarised (υ>0). A comment to this effect has been added at the start of section 4 where the background solution is introduced.

2) Regarding the magenta dash-dotted curve in Fig. 14 (old Fig. 13), we have indeed used the variational solution for the Josephson vortex around the centre of the dispersion relation to compute mI. Finite velocity solutions are necessary because mI is defined as a derivative with respect to velocity of the dispersion, as per equation (13), but once the derivative is taken, it is evaluated at zero velocity. The missing particle number, on the other hand, can be computed directly for the stationary vortex, with no need to call upon the approximate moving solutions.

We have added a short note specifying that the plots pertain to zero velocity in the captions of Figs. 12, 13, 14 (old Figs. 11, 12, 13) to eliminate confusion.

3) We do not believe that there is a simple explanation in terms of renormalisation group flow due to the presence of another (gapless) excitation sector, which is absent in the sine-Gordon equation. These two sectors asymptotically decouple in the limit υ→0, but since we do not have strong analytical arguments to justify our term selection on this basis, we prefer to keep the current presentation based on a posteriori justification. A comment clarifying the situation has been added in section 11.

4) The referee raises the excellent question of the stability of our excitations in the presence of a thermal cloud. Of course all ultra-cold atom experiments are done at a finite temperature, and it is well-known that over a long period of time, collisions between the dark soliton and the thermal atoms lead to the decay of the soliton. However, this process is comparatively slow, allowing one to observe the soliton for prolonged periods of time. This is due to the fact that the dark soliton possesses a macroscopic feature in the form of a phase step, spanning the entire cloud, so momentum exchange between the condensate and thermal atoms is suppressed. Precisely such collisions, however, are needed in order to allow the phase to progressively unwind, breaking translational invariance, speeding up the soliton until it eventually disappears at the speed of sound. Note that this occurs due to the concave-down dispersion relation of dark solitons.

On the other hand, in the regime where the Josephson vortex branch possesses a local minimum in the dispersion relation, vortex excitations around this minimum should be stabilized by a thermal cloud. Overall, we think it should still be possible to see both dark solitons and Josephson vortices in the regime of bistability, however the lifetime of any excitation with a negative inertial mass will be shortened by the thermal cloud. We have added a brief comment in this spirit at the end of section 13.