Drag-induced dynamical formation of dark solitons in Bose mixture on a ring

Andreev-Bashkin drag plays a very important role in multiple areas like superfluid mixtures, superconductors and dense nuclear matter. Here, we point out that the drag phenomenon can be also important in physics of solitons, ubiquitous objects arising in a wide array of fields ranging from tsunami waves and fiber-optic communication to biological systems. So far, fruitful studies were conducted in ultracold atomic systems where nontrivial soliton dynamics occurred due to inter-component density-density interaction. In this work we show that current-current coupling between components (Andreev-Bashkin drag) can lead to a substantially different kind of effects, unsupported by density-density interactions, such as a drag-induced dark soliton generation. This also points out that soliton dynamics can be used as a tool to experimentally study the dissipationless drag effect.

A conventional superfluid is described by a complex field ψ = √ n e iϕ . The phase gradient can be identified with the superfluid velocity v = m ∇ϕ, where m is the particle mass [10][11][12]33]. In 1976 Andreev and Bashkin demonstrated that in a two-component interacting superfluid mixture the relation between superfluid velocities and superflows becomes very nontrivial due to existence of a dissipationless drag transport effect [34]. Indeed, the corresponding free-energy density takes the form f = α ρ α v 2 α /2 + ρ d v a · v b , where ρ α (v α ) represents a superfluid density (superfluid velocity) of component α ∈ {a, b}, and ρ d is the Andreev-Bashkin (AB) drag coefficient [34]. Consequently, the superflows, i.e., j α = ∂ vα f = ρ α v α + ρ d v β =α , reveal that the component possessing no superfluid velocity, e.g., v a = 0, will still exhibit a nonzero superflow, j a = 0, as long as v b = 0.
The AB effect strongly affects vortex lattices in superfluids [35,36] and can change the nature of topological solitons in superconducotrs [37]. It is also crucial for the understanding of properties of dense nuclear matter [38,39] and observed pulsar dynamics [40][41][42]. At the microscopic level the drag effect originates from intercomponent particle-particle interaction [34,[43][44][45][46][47]. Especially interesting is the case of strongly correlated superfluids which parameters are precisely controllable in optical lattices [48,49]. There the AB drag originates from the interplay between inter-component particleparticle interaction and lattice effects and can be, in relative terms, arbitrarily strong and ρ d can be also negative [43-47, 50? -59]. Interestingly, AB drag signatures have been found in quantum droplets collisions [60].
The drag effect can have various forms. Recently, it was demonstrated that in certain asymmetrical lattices there exists also a perpendicular entrainment referred to as vector drag [61].
Our goal is to investigate the effects of current-current interaction. Hence, in this work we specifically set the well-studied inter-component density-density interaction to zero. However, the effect of the latter is discussed in Supplemental Material (SM) [82]. The corresponding system of dimensionless time-dependent GP-like equations reads (α ∈ {a, b}, γ = α) where the length scales are measured in units of the ring circumference L, time in units of mL 2 , energy in units of bright and dark soliton solutions that for PBC can be expressed analytically in terms of Jacobi functions [83][84][85][86][87]. A stationary bright soliton in ring geometry (PBC) forms spontaneously in the ground state when g α < g c = −π 2 . On the other hand, dark solitons are collective excitations characterized by density notches accompanied by phase slips in phase distribution ϕ and appear for any g α > 0 [87,88]. For finite rings, i.e., L < ∞, a single dark soliton always propagates with some finite velocity because the phase cyclicity condition, ϕ(L)−ϕ(0) = 2πW where the winding number W ∈ Z, requires a nonzero phase gradient to be satisfied in the presence of a solitonic phase slip. In the limiting case of a totally dark, i.e., black, soliton the corresponding density vanishes in the dip where the phase reveals a single-point discontinuity by π. Therefore, to satisfy PBC the phase ϕ has to accumulate at least as ±iπx/L. Solitons with a shallower density notch accompanied by a smooth ϕ(x) are often called gray solitons. Note that two gray solitons revealing identical densities may possess phase distributions characterized by different W and in consequence different average momenta p = −i dx ψ * ∂ x ψ. In Fig. 1 we show typical density and phase distributions of the lowest energy bright soliton and two types of dark solitons: black and gray.
Let us assume that in our system one of the components, say the b-component, exhibits J b = 0 while J a = 0. If the spatial translation symmetry is broken and J αγ = 0, then a dynamic drag-related current generation and a momentum transfer between the components can be expected. To study this problem, we consider the case in which component a is initially prepared in the uniform ground state ψ a0 for repulsive interaction g a > 0. At the same time b component is prepared in the ground state, ψ b0 , but for attractive interactions characterized by g b < g c that is associated with a stationary bright soliton. In such a case p a = p b = 0, J a = J b = 0 and the drag interactions have no impact on these states. To have J b , J αγ = 0 we additionally set the bright soliton in motion such that initially p b , J b = 0.
Basing on the relationship between yrast states and dark solitons in a single component repulsive Bose gas with PBC, one can ask if the drag-related momentum transfer from component b to a can induce a dark soliton formation in the latter component. We argue that preparing component b in a well localized bright soliton state may reduce excitations of kinds other than the collective solitonic ones. That is, the bright soliton would slow down its propagation when transferring the momentum from b to a, while preserving approximately unchanged shape due to strong intra-component attraction. In such a case, there is a chance that most of the energy gained by component a would correspond to the collective motion characterized by the transferred momentum. Thus, excluding the drag interaction energy, the resulting excited state in a component would have energy close to the one possessed by the yrast state with p a . If so, then one may expect an emergence of dark soliton signatures (density notch and phase slip) in the a-component.
Given that the abovementioned scenario takes place, the induced dark soliton is expected to be different depending on the amount of momentum injected into component a-the latter is likely to change over time. One may ask whether or not it is possible for a specific dark soliton to form in component a that would coexist with the bright soliton in the other component for time-scales longer than the period of a single revolution of the anticipated dark soliton along the ring. We suppose that this can happen when both the target dark soliton and the bright soliton propagate with comparable velocities.
The well localized (narrow in comparison to L) bright soliton can be approximately described by the famous sech-shaped soliton wave function [11] which reveals its particle-like behavior. Note that p = dx|ψ| 2 ∂ x ϕ and ϕ(x) = ϕ(0) + mvx/ + S(x), where S(x) encodes other phase features like phase slips. For well localized bright solitons ∂ x S ≈ 0 in the vicinity of the soliton clump and thus such states propagate with the velocity v ≈ p /m. Generally, dx|ψ| 2 ∂ x S is nonnegligible for dark solitons making the relationship between v and p more complicated. The special case is a black soliton (bs), for which ∂ x S = 0 only at the soliton dip where |ψ bs | 2 = 0. Thus, for the black soliton v bs = p bs /m, where p bs / = π/L + 2πn/L with n ∈ Z.
Let us operate with the dimensionless units and restrict our considerations to states ψ α possessing 0 ≤ p α ≤ 2π measured in /L units. We are going to analyze the possibility of a drag-induced formation of the most distinct of dark solitons, namely, the black soliton. We suppose that a long living coexistence of black and bright solitons may be possible when both objects propagate with comparable velocities. Therefore, at t = 0, we set the initial ground state bright soliton (b component) in motion with p b = 2 p bs = 2π. This is done by multiplying ψ b0 (x) by e i2πx , i.e., ψ b (x, t = 0) = ψ b0 (x)e i2πx . Since p b + p a = 2π is a conserved quantity in our system, we expect that if the momentum is transferred from component b to a, the abovementioned coexistence my appear when p b − p a ≈ 0. In such a case p b ≈ p a ≈ π and the corresponding solitons should propagate with comparable velocities.
We prepared the initial bright soliton state ψ b0 (x) by means of an imaginary time evolution of (1) with α = b, g d = 0 and four different g b = −20, −25, −30, −35, separately. These values of g b are all substantially below the critical value g c = −π 2 which guarantees that the resulting bright soliton density is well localized. This state is then set in motion with p b | t=0 = 2π by incorporating a phase factor as previously described. Component a is prepared in a similar way but with g a ∈ {20, 25, . . . , 90} resulting in the lowest energy state ψ a0 = ψ a (x, t = 0) = 1 (up to a global phase). After the states preparation we switch on the AB drag by setting g d = 0.1 while keeping g a and g b fixed. We then numerically evolve Eqs. (1) in real time up to t = 10, a time more than 30 times longer than the characteristic period of the black soliton revolution around the ring T = 1/π ≈ 0.32.
Our results indicate that the bright soliton in b component survives the evolution for all the considered parameters. For each g b = −20, −25, −30, −35 we find a region in the g a parameter where clear dark soliton signatures (density notch and phase slip) emerge in ψ a (x, t). See SM [82] for snapshots of typical system dynamics. Fig. 2 shows the temporal behavior of the overlap | ψ a |ψ bs | 2 and the momentum difference ( p b − p a )/π for different g a and g b = −20, −30. The overlaps | ψ a |ψ bs | 2 are calculated with the analytical black soliton solution ψ bs characterized by the corresponding g a and located at a position of the phase slip recognized in ψ a (x, t). By choosing a specific color code in the overlap plots we discriminate the regions where | ψ a |ψ bs | 2 > 0.9 (red intensity) from those where | ψ a |ψ bs | 2 < 0.9 (gray intensity). Note that overlaps above 0.9 appear when the momenta p b and p a are similar and are maintained for time scales significantly longer than T . We observe that the critical g a above which a dark soliton appears depends on the value of g b . That is, for stronger attraction, i.e., narrower bright soliton in the b component, the regime of the (nearly) black soliton formation shifts to larger g a 's corresponding to the narrower dark solitons. In SM [83] we also analyze how drag-induced states ψ a would evolve if drag is quenched to zero (drag-free dynamics) at a time when | ψ a |ψ bs | 2 ≈ 1. It turns out that such generated states reveal a genuine dark soliton drag-free evolution.
To better understand the system dynamics, in Fig. 3 we closer study cases with g b = −30 and g a = 65, 70, 75. As before, we analyze the time dependence of the overlap | ψ a |ψ bs | 2 and momentum p a /π. Additionally, we monitor the minimum Euclidean distance ∆ along the ring between the bright soliton and the draginduced dark soliton, the minimum reached by an anticipated density notch min(|ψ a (t)| 2 ), as well as the ratio of the bright soliton height to its initial value max(|ψ b (t)| 2 )/max(|ψ b (0)| 2 ). In all the cases an initial momentum transfer leads to the formation of a (nearly) black soliton. Indeed, the overlap | ψ a |ψ bs | 2 increases together with p a , and the density notch is simultaneously being carved as indicated by the decreasing value of min(|ψ a (t)| 2 ). At the same time the distance ∆ reveals an increasing separation between solitons in the two components reaching maximum, ∆ ≈ 0.5, at a time in the middle of the plateau of | ψ a |ψ bs | 2 ≈ 1. The seemingly linear trend in ∆ for ∆ 0.1 reveals a constant relative motion between the spatially separated solitons |v b −v a | ≈ 1 three times slower than the single-component black soliton velocity v bs = π. This behavior of ∆ repeats multiple times during the evolution.
Due to different velocities and assumed ring system geometry, the solitons collide multiple times during the course of evolution. It turns out that the induced (nearly) black soliton state often is substantially disturbed or even completely destroyed when both solitons meet, i.e., when ∆ → 0, which results in an abrupt drop of the overlap value | ψ a |ψ bs | 2 . The dark soliton re-localizes again when ∆ increases. Such a mechanism is the origin of quasi-periodic patterns visible in Figs. 2 and 3. However, as indicated by the behavior of max(|ψ b (t)| 2 )/max(|ψ b (0)| 2 ), the bright soliton remains almost unaffected when passing through the dark one. On the other hand, as shown in Fig. 3(b) for t > 7 and , and (c) shows from top to bottom the dynamics of: the overlap | ψa|ψ bs | 2 , the relative distance ∆ along the ring between the bright soliton (b component) and the phase slip position in ψa(x, t), the average momentum pa /π, as well as the values min(|ψa(t)| 2 ) and max(|ψ b (t)| 2 )/max(|ψ b (0)| 2 ). The results in (a), (b), and (c) correspond to g b = −30 and ga = 65, 70, 75, respectively. The drag-induced dark (nearly black) soliton often is significantly disturbed, or even completely destroyed, when passing through the bright soliton, i.e., when ∆ → 0. In such a case the phase slip in ψa(x, t) is rather tiny or even unrelated to any soliton structure. This is the origin of the narrow spikes observed in the ∆ plots when ∆ → 0 and min(|ψa(t)| 2 ) ≈ 1. Nevertheless, as shown in (b) for t > 7 and in (c) for t > 3, the (nearly) black soliton can survive encounter with the bright soliton. Fig. 3(c) for t > 3, the drag-induced dark soliton can also survive an encounter with the bright soliton. Additionally, in Fig. 3(c) for t ∈ (6.3, 7) and t ∈ (8,9), one can observe signatures of the existence of long living darkbright soliton composites characterized by ∆ ≈ 0. For more intuition, see snapshots of the system evolution in SM [82].
In summary, we have studied the dynamics of a bosonic binary mixture confined in a 1D ring geometry with intra-component contact interactions and intercomponent Andreev-Bashkin drag. Based on the relationship between dark solitons and yrast states characterized by the lowest energy for a given momentum, we formulated and verified the hypothesis concerning a draginduced dark soliton formation process. By numerically computing the system dynamics we tested the scenario where a propagating bright soliton interacts with the other component, prepared in the repulsively interacting uniform ground state. We demonstrated that there exist parameter regimes for which the drag interaction leads to formation of a long living genuine, nearly black, soliton state in the initially uniform component. While we focused on the most distinct black soliton case, the general idea provided here should also allow for generation of gray solitons. Our goal here was to study the effects of current-current interaction on soliton dynamics. An interesting question that warrants further studies is how these effects combine with inter-component densitydensity interactions. This question is beyond the scope of this paper, but in [82] we show that the drag phenomenon is crucial for the dynamical formation of long-living dark solitons, while density-density inter-component coupling does not support this effect in the considered setup. Additionally, we present that the effect at least survives inclusion of not too strong density-density interactions. The discussed phenomenon could guide experiments for a detection of the AB drag effect in binary superfluids. This can open avenue of studying the drag effect directly in a laboratory shedding light on the drag effect in other systems ranging from multicomponent superconductors to superfluids in neutron stars.
In conclusion, previously, soliton physics in binary systems were restricted to the role of density-density interaction. In this paper we report that new kind of soliton dynamics arises in binary system due to current-current coupling. The results indicate that the mixed gradient coupling plays an important role in soliton physics in multi-component systems which warrants further investigation. We expect that competition between the drag effect and density-density inter-component interactions leads to even richer dynamics of multicomponent systems.

I. MONITORING THE SYSTEM DYNAMICS
In Fig. FS1 we present snapshots of representative system dynamics in the setup described in the main text. We show two examples where the dark soliton state is induced in component a due to the drag interactions with component b. That is, on the one hand in (a) the evolution of the system characterized by g a = 50, g b = −20 and g d = 0.1 is monitored in the time frame from t = 0.2 to t = 3.95. On the other hand in (b) we track the dynamics for parameters g a = 70, g b = −30, g d = 0.1 in time between t = 6.2 and 9.95. Note that the draginduced dark soliton in component a for most of the time is very similar to the corresponding black soliton state indicated in the plots for comparison. Nevertheless, it can be significantly disturbed or even completely disappear when passing through the bright soliton in the other component. tions. Here, we also verify the single-component drag-free dynamics of these drag-induced states in right panels of Fig. FS2. That is, by employing Eq. (1) with α = a and g d = 0 we performed the numerical evolution assuming that the initial states are represented by ψ a 's illustrated in FS2(a) and FS2(c). The resulting time dependencies of corresponding densities shown in FS2(b) and FS2(d) reveal genuine dark soliton dynamics-in both cases the well preserved soliton notch propagates with a constant velocity. We would like to stress that also the phase flip signature of the dark soliton survives and always coincides with the corresponding density notch during the evolution.
Let us first consider the case of exclusively densitydensity intercomponent interactions g ab = 0 by setting FIG . FS3. Dynamics with exclusively density-density intercomponent interactions (g d = 0, g ab = 0). From top to bottom the time dependence of: the overlap | ψa|ψ bs | 2 , the relative distance ∆ along the ring between the bright soliton in b and a phase slip position in a component, the average momentum pa /π, as well as the values min|ψa(t)| 2 and max|ψ b (t)| 2 /max|ψ b (0)| 2 . The results in (a), (b), (c), and (d) correspond to g b = −30, ga = 75, g d = 0, and g ab = 3, 5, 7.5, 10, respectively. Note that even if the overlap | ψa|ψ bs | 2 reaches values close to 1, it is never maintained for a noticeable time.