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Magnetic solitons in binary mixtures of Bose–Einstein condensates

  • Lev P. PitaevskiiEmail author
Classical and quantum plasmas
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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.

In this article I will present a theory of new types of solitons, 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:
$$\begin{aligned} \delta g\equiv g-g_{12}\ll g. \end{aligned}$$
(1)
This means that the mixture is near the boundary of phase separation. Here \( g=\sqrt{g_{11}g_{22}}\). I assume that there is no phase separation, that is that the system is on the stable part of the phase diagram. This implies the condition \(\delta g>0\). For example for the two hyperfine states \( |F=1;\,m_{F}=\pm 1\rangle \) of \(^{23}\)Na isotope, one has due to rotational symmetry \(g_{11}=g_{22}\) and \(\delta g/g\approx 0.07\) (Knoop et al. 2011). The condition (1) ensures that the density dynamics and spin dynamics are decoupled. In particular at this condition the total density \( n=n_{1}+n_{2}\) is only weakly perturbed in the region of the MS. In the following, I will assume that \(n=const\) throughout.

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.

Let us assume that there is no external potential. For a one-dimensional two-component BEC with constant total density the spinor wave function can be conveniently parametrized as follows:
$$\begin{aligned} \left( \begin{array}{l} \psi _{1} \\ \psi _{2} \end{array} \right) =\sqrt{n}\left( \begin{array}{l} \cos (\theta /2)e^{i\varphi _{1}} \\ \sin (\theta /2)e^{i\varphi _{2}} \end{array} \right) , \end{aligned}$$
(2)
where \(\varphi _{1,2}\) are the phases of the two spin components, and \(n=n_1+n_2\) is the one-dimensional total density. According to the general assumption \( n=const\). The spin polarization is in these variables \((n_{1}-n_{2})/n=\cos \theta \). It is useful to go to the relative phase \(\varphi _{A}=\varphi _{1}-\varphi _{2} \) and the total phase \(\varphi _{B}=\varphi _{1}+\varphi _{2}\) of the two order parameters. Without loss of generality one can assume \(\varphi _{1,2}=0 \) at \(z=-\infty \). In terms of these new variables, the Lagrangian is
$$\begin{aligned} {\mathcal {L}}= & {} -\frac{n\hbar }{2}\left( \cos \theta \partial _{t}\varphi _{A}+\partial _{t}\varphi _{B}\right) -\frac{n\hbar ^{2}}{8m}(2\cos \theta \partial _{z}\varphi _{A}\partial _{z}\varphi _{B} \nonumber \\&+\left( \partial _{z}\varphi _{A}\right) ^{2}+\left( \partial _{z}\varphi _{B}\right) ^{2}+\left( \partial _{z}\theta \right) ^{2})+\frac{n^{2}\delta g }{4}\sin ^{2}\theta . \end{aligned}$$
(3)
In the absence of a time dependence this Lagrangian coincides with the one obtained by Son and Stephanov in Ref. Son and Stephanov (2002), but without the coherent RC term. The term \(\partial _{t}\varphi _{B}\) is a derivative and does not contribute to equations of motion. It will be omitted below. Note also that due to the assumption \(n=const\) the Lagrangian depends on the coupling constants only through the combination \(\delta g\).
We are looking for traveling soliton solutions and assume that \(\theta =\theta (z-Vt)\) and \(\varphi _{A,B}=\varphi _{A,B}(z-Vt)\). Then the Lagrangian takes the following form:
$$\begin{aligned} \tilde{{\mathcal {L}}}= & {} \frac{{\mathcal {L}}}{nmc_{s}^{2}}=U(\cos \theta \partial _{\zeta }\varphi _{A})-\frac{1}{2}\left[ 2\cos \theta \partial _{\zeta }\varphi _{A}\partial _{\zeta }\varphi _{B} \right. \nonumber \\&\left. +(\partial _{\zeta }\varphi _{A})^{2}+(\partial _{\zeta }\varphi _{B})^{2}+(\partial _{\zeta }\theta )^{2}\right] +\frac{1}{2}\sin ^{2}\theta . \end{aligned}$$
(4)
Here \(\zeta =(z-Vt)/\xi _{s}\) is the coordinate in units of the spin healing length \(\xi _{s}=\hbar /\sqrt{2mn\delta g}\) and \(U=V/c_{s}\) is the velocity in units of spin sound speed \(c_{s}=\sqrt{n\delta g/2m}\).
First let us perform the variation of the Lagrangian over the total phase \( \varphi _{B} \). This gives the equation \(\partial _{\zeta }\left( \partial _{\zeta }\varphi _{B}+\cos \theta \partial _{\zeta }\varphi _{A}\right) =0\). We are looking for a localized solitonic solution, which demands the boundary condition
$$\begin{aligned} \partial _{\zeta }\varphi _{A,B}=0\ at\ \zeta =\pm \infty . \end{aligned}$$
(5)
Thus we obtain the following equation:
$$\begin{aligned} \partial _{\zeta }\varphi _{B}+\cos \theta \partial _{\zeta }\varphi _{A}=0. \end{aligned}$$
(6)
The variations with respect to \(\varphi _{A}\) and \(\theta \) give now the last two equations. These equations contain the total phase only as \(\partial _{\zeta }\varphi _{B}\). Excluding this quantity with help of Eq. (6) and using the last boundary condition \(n_{1,2}=n/2\), i.e, \(\theta =\pi /2\), at \(\zeta =\pm \infty \) , after some transformations we finally obtain two differential equations:
$$\begin{aligned} \partial _{\zeta }\varphi _{A}=U\frac{\cos \theta }{\sin ^{2}\theta },\qquad \partial _{\zeta }^{2}\theta =U^{2}\frac{\cos \theta }{\sin ^{3}\theta } -\sin \theta \cos \theta . \end{aligned}$$
(7)
An integration of the second Eq. (7) gives the density distributions of the components:
$$\begin{aligned} n_{1,2}=\frac{n}{2}(1\pm \cos \theta )=\frac{n}{2}\left[ 1\pm \frac{\sqrt{ 1-U^{2}}}{\cosh (\zeta \sqrt{1-U^{2}})}\right] . \end{aligned}$$
(8)
The relative phase \(\varphi _{A}\) and the total phase \(\varphi _{B}\) can be calculated as follows:
$$\begin{aligned} \cot \varphi _{A}= & {} -\sinh (\zeta \sqrt{1-U^{2}})/U, \end{aligned}$$
(9)
$$\begin{aligned} \tan (\varphi _{B}+C)= & {} -\sqrt{1-U^{2}}\tanh (\zeta \sqrt{1-U^{2}})/U. \end{aligned}$$
(10)
The 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 \(\zeta =+\infty \).
Fig. 1

Structure of a Magnetic Soliton. a The red solid and blue dashed curves are the densities of the two components as functions of the coordinate. The densities satisfy equation \((n_1+n_2)/n=1\), due to the constance of total density. b The green solid and yellow dashed curves are the relative and total phases. One can see that the jump of the relative phase \(\varphi _{A}\) is equal to \(\pi \) (Adapted from Qu et al. (2016)) (color figure online)

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:
$$\begin{aligned} E=n\hbar c_{s}\sqrt{1-V^{2}/c_{s}^{2}}. \end{aligned}$$
(11)
This function is maximal for a static MS and, like the magnetization, tends to zero at \(V\rightarrow c_{s}\). For small values of the velocity, the soliton behaves like a quasi-particle with a negative effective mass \(m_{ \text {eff}}\equiv 2\partial E/\partial V^{2}=-n\hbar /c_{s}\).
Recall that plane solitons with a negative effective mass are unstable against the deformation of their shape, the so-called snake instability. The growth rate of a long wave length harmonic perturbation, propagating along the soliton is (see Kamchatnov and Pitaevskii 2008):
$$\begin{aligned} \text {Im}\omega =\sqrt{E/\left| m_{\text {eff}}\right| }k, \end{aligned}$$
(12)
where 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.

The simulation begins with imprinting a MS with velocity \(V=0.1c_{s}\) at the center of a two-dimensional elongated trap. The pictures of the distribution of the density after \(t=29\) ms time of evolution is plotted in Fig. 2. The simulation was performed for different numbers of atoms, i. e. for different values of the chemical potential \(\mu = 9.3 \omega _{y},15 \omega _{y}, 25 \omega _{y}\), where \(\omega _{y}\) is the transverse trapping frequency. We present in Fig. 2 the results of the calculations of density distribution \(n_{1}\). One can see in the figure that if the the number of atoms increases, the transverse size of the BEC cloud also increases. For \(\mu =9.3\omega _{y}\), the MS is stable and oscillates in the trap as predicted by 1D calculations. For \(\mu =15\omega _{y}\), the magnetic soliton still moves and oscillates in the trap. However, vortex pairs soon appear in the second component and the system alternatively oscillates between a vortex pair and a MS. For \( \mu =25\omega _{y}\) soliton quickly starts to bend (panel (c). After a longer evolution time the soliton decays into vortices (This stage of the process is not presented in the figure.)
Fig. 2

The density distributions of an oscillating MS in a two-dimensional trap. The density \(n_1 \) is presented for different values of chemical potentials \(\mu \) and aspect ratios \(\lambda =R_z/R_y\): a\( \mu =9.3 \omega _y\), \(\lambda =10\); b\(\mu =15 \omega _y\) , \(\lambda =10\); c\(\mu =25\omega _y \), \(\lambda =5\). The densities are plotted after 29 mc evolution time. (Adapted from Qu et al. (2016))

3 Magnetic solitons in Rabi-coupled Bose–Einstein condensates

In this section we consider a two-component BEC in the presence of the Rabi coupling (RC). This coupling is induced by the electromagnetic field with the frequency corresponding to the energy splitting between components. The coupling is described by a term in the GPE Hamiltonian density, allowing the exchange of atoms between the components:
$$\begin{aligned} {{\mathcal {H}}}_{R}=-\frac{1}{2}\hbar \Omega \left( \psi _{1}^{*}\psi _{2}+\psi _{2}^{*}\psi _{1}\right) . \end{aligned}$$
(13)
The parameter \(\Omega \) is the so-called Rabi frequency. It plays the role of the Rabi coupling constant. We will assume without loss of generality that \( \Omega >0\). We also assume that the RC is weak in the sense that
$$\begin{aligned} \Omega \ll \mu /\hbar \sim gn/\hbar , \end{aligned}$$
(14)
where \(\mu \) is the chemical potential of the mixture. The Rabi term (13) explicitly depends on the phase difference \(\varphi _{A}\) . This means that the coupling is coherent.

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 \(n=0,\pm 1,\ldots \).

We will assume, as before, that the inequality (1) is satisfied. Thus the total density \(n=n_{1}+n_{2}\) can be considered as a constant. Then we can use again the parametrization (2) and, repeating the derivation of the previous section, we obtain instead (7) the system of equations:
$$\begin{aligned}&U\frac{\partial \theta }{\partial \zeta }+2\cos \theta \frac{\partial \theta }{\partial \zeta }\frac{\partial \varphi _{A}}{\partial \zeta }+\sin \theta \frac{\partial ^{2}\varphi _{A}}{\partial \zeta ^{2}} \nonumber \\&\quad -\frac{\omega _{\text {R}} }{3}\sin \varphi _{A}=0, \end{aligned}$$
(15)
$$\begin{aligned}&\quad -U\sin \theta \frac{\partial \varphi _{A}}{\partial \zeta }+\frac{\partial ^{2}\theta }{\partial \zeta ^{2}}-\sin \theta \cos \theta \left( \frac{ \partial \varphi _{A}}{\partial \zeta }\right) ^{2}+\sin \theta \cos \theta \nonumber \\&\quad +\frac{\omega _{\text {R}}}{3}\cos \theta \cos \varphi _{A}=0, \end{aligned}$$
(16)
where \(\omega _{\text {R}}=(3\Omega /n\delta g)\). It is quite useful for yje calculations below that, as it follows from these equations, the quantity
$$\begin{aligned} \tilde{{\mathcal {G}}}\equiv & {} -\frac{1}{2}\left[ \left( \frac{\partial \theta }{\partial \zeta }\right) ^{2}+\sin ^{2}\theta \left( \frac{\partial \varphi _{A}}{\partial \zeta }\right) ^{2}\right] \nonumber \\&+\frac{1}{2}\cos ^{2}\theta +\frac{1}{3}\omega _{\text {R}}(1-\sin \theta \cos \varphi _{A})=0. \end{aligned}$$
(17)
is \(\zeta \)-independent. One can check directly that \(d\tilde{{\mathcal {G}}} /d\zeta =0\) and \(\tilde{{\mathcal {G}}}=0\) at \(\zeta \rightarrow \pm \infty \).

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.

Taking \(U=0\) and \(\theta =\pi /2\), we reduce Eq. (15) to
$$\begin{aligned} \frac{\partial ^2\varphi _A}{\partial \zeta ^2}-\frac{\omega _\text {R}}{3} \sin \varphi _A=0. \end{aligned}$$
(18)
This is the well-known sine-Gordon equation. Its solution is
$$\begin{aligned} \varphi _A=4\arctan e^{\zeta \sqrt{\omega _\text {R}/3}}=4\arctan e^{\kappa z}, \end{aligned}$$
(19)
where \(\kappa ^{-1}=\sqrt{\hbar /2m\Omega }=\xi _\text {s}\sqrt{3/\omega _\text {R}}\) is the width of the SS domain wall.

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.

One can show that, if the Rabi coupling is weak, \(\omega _{\text {R}} \ll 1\), the SS solution can be generalized to a moving soliton simply by changing \( \omega _{ \text {R}}\rightarrow \omega _{\text {R}}\left( 1-U^{2}\right) \) or \(\Omega \rightarrow \Omega /\left( 1-U^{2}\right) \). This means that the moving domain wall becomes thinner and thinner as 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:
$$\begin{aligned} m_{\text {eff}}(U)\equiv \frac{1}{V}\frac{\partial E}{\partial V}=\frac{ 4n\hbar }{ c_{\text {s}}}\sqrt{\frac{\omega _{\text {R}}}{3}}\frac{1}{ (1-U^2)^{3/2}}. \end{aligned}$$
(20)
It is worth noticing that the effective mass is proportional to \(\sqrt{ \omega _{R}}\) which is a small parameter at the condition of applicability of the accepted approximation. The effective mass increases when the velocity 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 boundary conditions for the \(2\pi \) MS are
$$\begin{aligned} \theta (\zeta =\pm \infty )=\frac{\pi }{2},\quad \varphi _{A}(\zeta =-\infty )=0,\quad \varphi _{A}(\zeta =+\infty )=2\pi , \end{aligned}$$
(21)
I restrict my discussion to solutions with the following symmetry properties:
$$\begin{aligned} \varphi _{A}(-\zeta )=2\pi -\varphi _{A}(\zeta ),\quad \theta (-\zeta )=\theta (\zeta ), \end{aligned}$$
(22)
It is not difficult to check that Eqs. (15) and (16) are invariant under this transformation.
This implies in particular that \(\varphi _{A}(0)=\pi ,\)\(\partial _{\zeta }\theta |_{\zeta =0}=0\). It follows from Eq. (17), that there is a connection between the values of \(\varphi _{A}\) and \(\theta _{0} \equiv \theta (\zeta =0)\):
$$\begin{aligned} \left( \frac{\partial \varphi _{A}}{\partial \zeta }\right) _{\zeta =0}^{2}= \frac{\cos ^{2}\theta _{0}+\frac{2}{3}\omega _{\text {R}}(1+\sin \theta _{0}) }{\sin ^{2}\theta _{0}}. \end{aligned}$$
(23)
To find the solutions of Eqs. (15) and (16), we choose \(\theta _{0}\) as a tuning parameter. We carefully tune \(\theta _{0}\) until the solutions of the equations satisfy the boundary conditions in Eq. (21) for the MS. Notice that to obtain a correct solution, the parameter \(\theta _{0}\) must be tuned with very high precision.
Figure 3 shows the densities and the relative and total phases of a \(2\pi \) soliton with \(U=V/c_{\text {s}}=0.28\) velocity.
Fig. 3

Structure of the \(2\pi \) soliton with \(V/c_\text {s}=0.28\), \( m_0=0.48\), and RC \(\omega _\text {R}=0.3\). a Solid red and dashed blue curves show the densities of the components. b Solid green and dashed orange curves present the relative \(\varphi _A\) and total \( \varphi _B\) phases (Adapted from Qu et al. (2017)) (color figure online)

Fig. 4

Structure of a \(0\pi \) soliton with \(V/c_\text {s}=-0.9\), \( m_0=0.89\), and RC \(\omega _\text {R}=0.3\). a Solid red and dashed blue curves present the densities of the components. b Solid green and dashed orange curves show the relative \(\varphi _A\) and total phase \( \varphi _B\). This \(0\pi \) soliton has a negative effective mass and the jump of \(\varphi _A\) is 0 (Adapted from Qu et al. (2017)) (color figure online)

Let us now discuss the properties of \(0\pi \) solitons. Our GPE calculations show that a \(2\pi \) MS transforms into a \(0\pi \) one when the density of one component vanishes at the center of the soliton. At this configuration the phase of this component is not well defined and can be changed by \(2\pi \) without any energy cost. The boundary conditions for \(0\pi \) solitons are
$$\begin{aligned} \theta (\zeta =\pm \infty )=\frac{\pi }{2},\quad \varphi _{A}(\zeta =\pm \infty )=0, \end{aligned}$$
(24)
and \(\varphi \) and \(\theta \) functions posses symmetry
$$\begin{aligned} \varphi _{A}(-\zeta )=-\varphi _{A}(\zeta ),\quad \theta (-\zeta )=\theta (\zeta ), \end{aligned}$$
(25)
This gives \(\varphi _{A}(0)=0,\partial _{\zeta }\theta |_{\zeta =0}=0\). Using Eq. (17), we obtain the slope of the relative phase at the soliton center analogously to Eq. (23), but with the change \( (1+\sin \theta _{0})\) to \((1-\sin \theta _{0})\). The procedure of solution of the equations is the same as the one used for \(2\pi \) solitons.
Figure 4 presents the structure of a \(0\pi \) soliton with \( U=V/c_{\text {s}}=-0.9\). The mixture is magnetized in the core of the soliton. There are two points restricting two oppositely magnetized regions on the wings where \(n_{1}=n_{2}\).
Fig. 5

a Velocity dependence of the magnetization \(m_0\) at the center of the MS. b Velocity dependence of the energy of MS for different RC strengths \(\omega _\text {R}=0.3\)(red solid curve), 1 (dashed black curve) and 2 (dash-dotted blue curve). Curves without circles indicate the \(2\pi \) solitons; curves with circles indicate \(0\pi \) solitons. The green square indicates points where the effective mass of the \( 2\pi \) soliton diverges and the green cross indicates the point of the transformation between \(2 \pi \) and \(0\pi \) solitons for \( \omega _\text {R}=0.36\) (Adapted from Qu et al. (2017)) (color figure online)

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 EV 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.

The \(0\pi \) Rabi MS exists only at large enough V. Its effective mass of \( 0\pi \) soliton is always negative, irrespective of the strength of the Rabi coupling.
Fig. 6

Oscillation of a Rabi MS in a harmonic trap. At \(t=0\), a static SS domain wall is imprinted at the right side (point A) of the trap center. The \(2\pi \) soliton starts moving toward the periphery, and soon after the reflection, it transforms into a \(0\pi \) soliton (point D ). \(2\pi \) solitons are indicated by red circles; \(0 \pi \) solitons, by blue squares. Green crosses indicate the points of transformation between \(2\pi \) and \(0\pi \) solitons (Adapted from Qu et al. (2017)) (color figure online)

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.
Fig. 7

Oscillation of the Rabi MS in an elongated harmonic trap after imprinting a SS domain wall. For three time instants we plot the density \( n_2 \) in the upper panel and the relative phase \(\varphi _A\) in the lower panel: a\(\omega _{\mathrm {ho}} t = 0\), b\(\omega _{ \mathrm {ho}} t = 3.2\), c\(\omega _{\mathrm {ho}} t = 5.6\). Here \(\omega _{\mathrm {ho}}=(\omega _{z}\omega _{y})^{1/2}\). Values of parameters: aspect ratio \(\lambda =10\), RC \(\Omega = 0.5\, \omega _\text {ho}\), coupling constant \(\delta g = 0.4\, g\) (Adapted from Qu et al. (2017))

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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Copyright information

© Accademia Nazionale dei Lincei 2019

Authors and Affiliations

  1. 1.INO-CNR BEC Center and Dipartimento di FisicaUniversità di TrentoPovoItaly
  2. 2.Kapitza Institute for Physical Problems RASMoscowRussia

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