# Magnetic solitons in binary mixtures of Bose–Einstein condensates

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## Abstract

Solitons, that is stable localized perturbations of a medium, are the topological excitations of nonlinear systems. They can be stable and live for a long time and may have promising applications for telecommunication. The basic one is the Tsuzuki dark soliton, which can be described by an analytical solution of the Gross–Pitaevskii equation (GPE). Ultracold Bose–Einstein condensed (BEC) gases are an important example for the investigation of solitons which can be created by phase and density imprinting. New possibilities arise in mixtures of different hyperspin states of ultra-cold gases, where the so-called magnetic solitons (MS), that is localized magnetized regions, can exist. We will see that these MS permit an analytical description. New peculiar phenomena can take place in the presence of a coherent Rabi coupling between the spin states, where two different type of solitons exist—so-called \(2\pi \) and \(0\pi \) solitons. \(2\pi \) solitons, unlike the usual Tsuzuki solitons, have at small velocity a positive effective mass and consequently do not undergo the snake instability. Solitary waves can oscillate in BEC gases along elongated traps. The theoretical description of this motion requires the knowledge of the effective soliton mass and the effective number of particles in the soliton. These quantities are calculated.

## Keywords

Bose–Einstein condensates Solitons Gross–Pitaevskii equation Snake instability Rabi coupling Domain wall Effective mass## 1 Introduction

Solitary waves play a special role among the excited states of condensed matter. They are localized perturbations that, due to the competition of nonlinearity and dispersion, can move in the medium without deformation. As a result they can have important applications in information processing. They can exist, in addition to those found in condensed matter theory (Emori et al. 2013; Su et al. 1979), in a wide variety of different branches of science including classical hydrodynamics, astrophysics (Ryutova et al. 1998), cosmology (Kibble 1976), and optics (Mollenauer et al. 1980). Trapped BEC gases are important objects for the investigation of solitons which can be created by phase and density imprinting, quantum quenches, etc. (Tsuzuki 1971; Burger et al. 1999; Denschlag1 et al. 2000). Dark and bright solitons, that is regions where the density is decreased or increased, have been already observed in repulsive (Burger et al. 1999; Denschlag1 et al. 2000) and in attractive (Khaykovich et al. 2002) BEC.

*magnetic solitons*(MS), in mixtures of hyperfine states of Bose–Einstein condensates (BEC). My presentation is based on research publications of the BEC center in Trento (Qu et al. 2016, 2017). A reader can also find in these papers references to preceding works. The MS is a region of localized spin polarization \(n_{1}-n_{2}\), where \(n_{1,2}\) are the densities of the two components. It propagates over a polarized background. An important assumption of the theory is that in typical mixtures of hyperfine states of alkali BEC the interaction constants satisfy the following inequality:

## 2 Magnetic solitons in BEC in the absence of Rabi coupling

I will consider a two-component BEC at \(T=0\). In this section I will assume that there is no Rabi coupling (RC) between components. This means that the number of atoms of every spin state is conserved. The mixture can be described by a system of two coupled GPEs. One can obtain these equations using a proper Lagrangian density (see, for example, Pitaevskii and Stringari (2016), Section 21). However, I will simplify the problem taking into account that the total density is constant.

*C*should be chosen so as to satisfy the condition \(\varphi _{B}(\zeta =-\infty )=0\). An important property of this type of soliton follows from Eq. (9): the relative phase \(\varphi _{A}\) always exhibits a \(\pi \) jump between \(\zeta =-\infty \) and \(\zeta =+\infty \).

An example of coordinate dependence of the densities of the components in the MS core is shown in Fig. 1a for the velocity \( U=V/c_s=0.6\). As it should be, the spin polarization tends to zero at large distance from the soliton. According to Eq. 8 it should be \(0\le |U|\le 1\), that is the velocity of the MS is smaller than the spin sound velocity \(c_{s}\). The magnetization \((n_1-n_2)/n \) at the center of the soliton is equal \(m_{0}=\sqrt{1-U^{2}}\). When *V* approaches \(c_{s}\), the magnetization tends to zero. However, the total magnetization \( \int _{-\infty }^{+\infty } dz (n_1-n_2)/n\) does not depend on the velocity and is given by a simple equation \(\pi \xi _s\). The width of the MS increases when \(V \rightarrow c_{s}\) as \(1/\sqrt{1-U^2}\).

Figure 1b presents the phases of the MS for \(U=0.6\). One can see that, in accordance with Eq. (9), the jump of the relative phase \(\varphi _A\) is exactly \(\pi \). However, the slope of \(\varphi _A \) at the soliton center (\(\partial _{\zeta }\varphi _{A}|_{\zeta =0}\)) is steeper for a slower MS and becomes a step function for a static MS. Contrary to \(\varphi _{A}\), the asymptotic phase jump of \(\varphi _{B}\) is velocity-dependent and \(\varphi _{B} \rightarrow 0\) for \(U\rightarrow 1\).

*Energy and effective mass*From the point of view of the theory of superfluidity a MS is a quasi-particle. Its energy

*E*plays the part of its Hamiltonian and one can use the law of energy conservation to describe the dynamics of the soliton in an external field, for example in a harmonic trap (Busch and Anglin 2000; Konotop and Pitaevskii 2004). The energy

*E*should be calculated for a uniform BEC as the difference between the grand canonical energy potentials in the presence and in the absence of the MS (see for example Pitaevskii and Stringari (2016), Chap. 5). Direct calculation gives the following:

*k*is the wave vector of the perturbation, which for the MS should satisfy the inequality \( k\xi _{s}\ll 1\).

Knowledge of the soliton energy permits us to justify the main assumption of this theory: the total density is weakly perturbed in the MS core. The problem is simple for a static MS with \(V=0\). Then \(E=\hbar n^{3/2}\sqrt{ \delta g/2m} \) and one can calculate the depletion of the total number of atoms due to the presence of the soliton using the thermodynamic relation \( N_{D}\equiv \int _{-\infty }^{\infty }\left[ n(z)-n\right] \mathrm{d}z=-\partial E/\partial \mu ,\) where \(\mu =ng\) is the chemical potential of the mixture in the absence of the soliton. One obtains \(N_{D}=-3n\xi _{s}(\delta g/2g)\). \(N_{D}\) can be considered as an effective number of atoms in the soliton. It is possible to estimate the density depletion inside the soliton core as \( \left| n(z)-n\right| \sim \left| N_{D}\right| /\xi _{s}\sim n\delta g/g\ll n\) (Calculation of the density perturbation for a moving soliton is a more complicated problem, see Pitaevskii 2016).

*Soliton oscillations in a harmonic trap* A natural way for comparing the theory of MS with experiment is by observing its oscillation in an elongated harmonic trap. Let us assume that the trapping frequency along the axis is \(\omega _z\). Remember that the frequency of oscillation period of a Tsuzuki dark soliton in a single component BEC is \(\omega _{z}/ \sqrt{2}\) (Busch and Anglin 2000; Konotop and Pitaevskii 2004).

In the general case the law of conservation of energy gives for the frequency of small oscillations \((\omega/\omega_{z}= \sqrt{mN_{D}m_{\text {eff }}})\). Using the parameters calculated above for the MS, we find \( \omega /\omega _{z}=2\sqrt{3\delta g/4g}\). Thus the frequency of small-amplitude oscillations of the MS depends on the interaction, in contrast to the usual dark solitons.

Solitons of this type were observed in experiments (Danaila et al. 2016). The oscillation frequency is in qualitative agreement with the present theory. The reason for a quantitative disagreement is probably the violation of the balance of the densities of the two components. In the experiment \(n_{1} \ne n_{2}\).

*Numerical simulation in two dimensions* So far we discussed the properties of the MS as a one-dimensional problem, assuming that all quantities that characterize the soliton depend only on one spatial coordinate *z*. Actually, a soliton is confined in a three-dimensional trap. A one-dimensional approximation is reasonable if the transverse trap frequency is large and the transverse size of the BEC cloud is small in comparison to the spin healing length \(\xi _{s}\). As we already noted, inasmuch as the effective mass of the magnetic soliton is negative, the MS exhibits the snake instability when the transverse size becomes larger than the width of the soliton. In this relation the MS is similar to the Tsuzuki dark soliton. It is important, however, that the spin healing length in BEC mixtures under the condition (1) is larger than the density healing length \(\xi _{d}=\hbar /\sqrt{2mng}\). As a result the MS are more robust against the snake instability than the dark solitons whose width is smaller being fixed by \(\xi _{d}\). For the same reason the MS can be wide enough to be observed directly in in situ measurements. Further, the width of magnetic solitons increases with their velocity. Consequently, a fast moving MS is more robust than a slow moving one. It is impossible to construct an analytic solution describing the complicated phenomena which arise due to the presence of the snake instability and the transverse confinement. To investigate this problem we performed numerical simulations of the MS in a two-dimensional trap using the system of the GPE equations without the \(n=\) const simplification.

## 3 Magnetic solitons in Rabi-coupled Bose–Einstein condensates

We will see that the solitons in the presence of RC are completely different from ones in the absence of RC, which were considered in the previous section. It was shown there that the relative phase of the two components of MS in the absence of RC has a \(\pi \) phase jump. In the presence of the RC, the relative phase \(\varphi _{A}\) to minimize energy (13) should satisfy the condition \(\cos \varphi _{A}=1\) at \(z\rightarrow \pm \infty \). This means that the asymptotic jump of \(\varphi _{A}\) must be equal to \(2n\pi \) with \(n=0,\pm 1,\ldots \).

## 4 Son–Stephanov domain wall

Son and Stephanov in their pioneering article Son and Stephanov (2002) constructed a static \(2\pi \) soliton in a RC binary condensate. One can consider this object as a static domain wall with a \(2\pi \) jump of the relative phase. This static solution of equations (15)–(16) at \(U=0\) is characterized by the absence of magnetization of the soliton core, \(n_{1}=n_{2}\) for all \(\zeta \). This solution is a metastable one if the condition \(\omega _{\text {R}}<1\) is satisfied. For larger values of \( \Omega \) the static domain wall does not correspond to a local minimum of the energy functional and the soliton must be unstable. However, one should keep in mind that the stability of the Rabi MS has not been investigated so far (Note one of the first works on this subject Usui and Takeuchi 2015).

The absence of magnetization of the static domain wall makes its experimental detection difficult. However, if the Son–Stephanov (SS) soliton moves, the motion induces magnetization. This opens up good perspectives for the observation of the MS. In this section we recover the static SS solution.

It is necessary to note here that the applicability of the \(n=const\) approximation can be violated near the crossing point of the dispersion curves of the spin and of the density excitations. It takes place at quite small wave vectors \(k_{\text {R}}\sim \kappa \sqrt{\hbar \Omega /gn}\). Such wave vectors play no role in the theory presented here.

*U*increases, contrary to the MS in the absence of RC. One can calculate the energy of the moving domain wall and finally the effective mass:

*U*increases. However, the equations of this section are wrong if \(1-U^2\) is very small. For a low velocity we find \(m_{\text {eff}}/m=8n\xi _{\text {s}} \sqrt{\omega _{\text {R}}/3}\). The positiveness of the effective mass means the absence of snake instability of a static domain wall. At some value of the velocity the effective mass diverges. (However, in an exact solution it occurs before the velocity approaches the \(U=1\) point, contrary to the approximate Eq. (20). When

*V*increases further, \(m_\text {eff} \) becomes negative and the soliton should exhibit the snake instability.

## 5 Numerical solutions for Rabi MS

In this section, we discuss the numerical solutions for \(2\pi \) and \(0\pi \) types of the “Rabi” MS in uniform matter. The two types of MS must satisfy the same differential equations (15)–(16). However, different boundary conditions are valid for these two types of solitons.

The relative phase \(\varphi _{A}\) of \(2\pi \) MS has the same asymptotic jump as in the SS static soliton. However, for \(V \ne 0\) the spin densities are not equal in the soliton core. Motion polarizes the soliton.

The results of the systematic investigation of the energy and magnetization of the Rabi solitons are presented in Fig. 5. One should pay attention to the solid red curves which describe MS at weak RC. The presence of a peculiar loop on the energy curve for small *V* means that the equations have different types of solution at given *V*. The lower point on the curve at \(V=0\) and \(m_0=0\) corresponds to the SS static domain wall. One can conclude from Fig. 5b that this solution is a local minimum of the *E*(*V*) curve as long as \(\omega _\text {R}< 1\). However, the equations have two extra solutions, which have finite magnetization even at \( V=0\) and correspond to a local maximum of the *E*–*V* curve. These two solutions are different in the directions of their magnetizations. The effective mass of solitons [see Eq. (20)] is related to the slope of the *E*(*V*) line. Such a behaviour results in a hysteresis-like bend on \(m_{0}\) curve. The effective mass of a stable solution is positive, of an unstable is negative. At \(\omega _\text {R}> 1\) the equations have at given *V* only one solution, which is unstable. Then the effective mass of an unstable \(2\pi \) soliton is negative.

*V*. Its effective mass of \( 0\pi \) soliton is always negative, irrespective of the strength of the Rabi coupling.

## 6 Numerical simulation of oscillations of Rabi MS in a trap

*Numerical simulation in 1D*In order to generate moving MS in a one-dimensional trap we imprinted the phase of the static SS domain wall, with its center displaced from the center of the trap (see Fig. 6), and simulated evolution of this configuration using GPE. Initially there is no magnetization. Once the domain wall moves, the core of the wall is polarized, transforming the wall in a moving \(2\pi \) soliton. As time evolves the soliton moves toward the periphery of the trapped gas and increases its velocity because its effective mass is positive. Before reaching the border of the condensate the soliton slows down because at some intermediate point, labelled “B” in Fig. 6, the effective mass diverges and becomes negative. Eventually, the soliton reaches zero velocity at the point “C” in Fig. 6 and is reflected towards the center of the trap. When the RC is sufficiently small, the \(2\pi \) soliton is transformed into a \(0\pi \) one. The \(0\pi \) soliton is then accelerated towards the center of the trap and slows down when it approaches the region of lower density, on the opposite side of the trap. The \(0\pi \) soliton cannot reach zero velocity and at some point is transformed again into a \(2\pi \) soliton which eventually reaches zero velocity, to be reflected again. This highly non trivial behaviour is presented in Fig. 6 , where the position of the soliton is shown as a function of time.

*Role of the transverse confinement* In this section we simulate the MS motion in a two-dimensional trap, using numerical solutions of the GPE. In 2D (and also in 3D), MS with a negative effective mass should be unstable in relation to the snake instability. However, for elongated enough geometry, it is still possible to observe persistent oscillations of the solitons. One can roughly estimate the condition on the parameters of the MS to ensure its stability as \(R_{\perp } < \xi _{\mathrm {\ phase}}\). If one uses in calculations \(\Omega = 0.5\, \omega _\text {ho}, \mu \approx 50\, \hbar \omega _\text {ho}\), and \(\delta g = 0.4\, g\), this gives an aspect ratio \( \lambda \equiv \omega _\perp / \omega _{\mathrm {ho}} > 2 \sqrt{\mu \Omega / (\hbar \omega _{\mathrm {ho}}^2)} = 10\). The GPE simulation shows that a \(2\pi \) MS, which was initially imprinted across the trap with a small displacement from the center, begins to move along the longitudinal axis towards the edge of the BEC cloud. When the \(2\pi \) soliton moves to the turning point, it develops a magnetization and induces two vortices at its ends (see Fig. 7). Then, it moves back towards the center of the trap as it takes place for the 1D solution, but now we observe that the soliton is fragmented into two pieces and no longer extends through the whole transverse dimension [see Fig. 7c]. As discussed in Tylutki et al. (2016) and Calderaro et al. (2017) the end of a finite domain wall is always associated with a vortex in one of the two spin components. This ensures the proper behaviour of the phase around the end point. In the region between the vortices the soliton core is magnetized. This solution corresponds to the \(0\pi \) MS discovered in the one-dimensional calculations. The \(0\pi \) soliton continues to move and it survives for a long time while oscillating and repeatedly transforming to \(2\pi \) solitons.

All results presented in this article have been obtained in collaboration with my colleagues Sandro Stringari, Chunlei Qu and Marek Tulutki. It is a pleasure for me to express my gratitude to them. I thank the Accademia Nazionale dei Lincei for the invitation to the conference “Classical and quantum plasmas: matter under extreme conditions”, where this talk was presented.

## Notes

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