Magnetic solitons in Rabi-coupled Bose-Einstein condensates

We study magnetic solitons, solitary waves of spin polarization (i.e., magnetization), in binary Bose-Einstein condensates in the presence of Rabi coupling. We show that the system exhibits two types of magnetic solitons, called $2\pi$ and $0\pi$ solitons, characterized by a different behavior of the relative phase between the two spin components. $2\pi$ solitons exhibit a $2\pi$ jump of the relative phase, independent of their velocity, the static domain wall explored by Son and Stephanov being an example of such $2\pi$ solitons with vanishing velocity and magnetization. $0\pi$ solitons instead do not exhibit any asymptotic jump in the relative phase. Systematic results are provided for both types of solitons in uniform matter. Numerical calculations in the presence of a one-dimensional harmonic trap reveal that a $2\pi$ soliton evolves in time into a $0\pi$ soliton, and vice versa, oscillating around the center of the trap. Results for the effective mass, the Landau critical velocity, and the role of the transverse confinement are also discussed.

Figure 1

Phase structure of the static Son-Stephanov domain wall. The relative phase ${\phi }_A={\phi }_1-{\phi }_2$ of the two spin states exhibits a $2\pi$ jump when one moves from $-\infty$ to $+\infty$. The width of the domain wall is fixed by the characteristic length of the relative phase ${\xi }_{\text{phase}}=\sqrt{\hslash /2m\mathrm{\Omega }}$, where $\mathrm{\Omega }$ is the Rabi coupling.

Figure 2

Illustration of the dynamics of magnetic solitons in a harmonic trap. At $t=0$, a static Son-Stephanov domain wall, characterized by a $2\pi$ jump in the relative phase, is imprinted at the right (point A) of the trap center. The $2\pi$ soliton starts moving towards the periphery, and soon after the reflection, it evolves into a $0\pi$ soliton (point D; see text). $2\pi$ solitons are indicated by red circles; $0\pi$ solitons, by blue squares. Green $\mathsf{X}$'s are points indicating the transformation between $2\pi$ and $0\pi$ solitons.

Figure 3

Profiles of a $2\pi$ soliton with velocity $V/c_{\text{s}}=0.28, m_0=0.48$, and Rabi coupling ${\omega }_{\mathrm{R}}=0.3$. (a) Solid red and dashed blue lines represent the density distributions of the two components, satisfying $(n_1+n_2)/n=1$. (b) Solid green and dashed orange lines show the relative phase ${\phi }_A$ and total phase ${\phi }_B$ as a function of the coordinate. The jump of the relative phase is $2\pi$, independent of the velocity. This solution is close to the critical velocity where the effective mass diverges (see discussion in Sec. 5).

Figure 4

Profiles of a $2\pi$ soliton with velocity $V/c_{\text{s}}=-0.25, m_0=0.97$, and Rabi coupling ${\omega }_{\text{R}}=0.3$. (a) Solid red and dashed blue lines represent the density distributions of the two components, satisfying $(n_1+n_2)/n=1$. Note that two nodes appear at the wings of the soliton. (b) Solid green and dashed orange lines show the relative phase ${\phi }_A$ and total phase ${\phi }_B$ as a function of the coordinate. This soliton has a negative effective mass and corresponds to the solution very close to the green $\mathsf{X}$ with $m_0=1$ in Fig. 7.

Figure 5

Profiles of a $0\pi$ soliton with velocity $V/c_{\text{s}}=-0.9, m_0=0.89$, and Rabi coupling ${\omega }_{\text{R}}=0.3$. (a) Solid red and dashed blue lines show the density distributions of the two components, satisfying $(n_1+n_2)/n=1$. (b) Solid green and dashed orange lines show the relative phase ${\phi }_A$ and total phase ${\phi }_B$ as a function of the coordinate. This $0\pi$ soliton has a negative effective mass and the asymptotic jump of ${\phi }_A$ is 0.

Figure 6

(a) Plot of $\theta \left(\zeta \right)$ at the transformation point between $2\pi$ and $0\pi$ solitons for Rabi coupling strength ${\omega }_{\mathrm{R}}=0.3$. (a) Dashed lines show the analytical prediction of the behavior of $\theta \left(\zeta \right)$ at small $\zeta \to 0$ [see Eq. (43)]. Relative phases (b) before and (c) after the transformation, which exhibit a $2\pi$ and $0\pi$ phase jump, respectively. Velocities of the solitons are $U=-0.464$ (b) and $U=-0.467$ (c). As $\zeta \to -0$, the relative phase ${\phi }_A\to \pi /2$ (b, c). As $\zeta \to +0, {\phi }_A\to 3\pi /2$ (b) and ${\phi }_A\to -\pi /2$ (c).

Figure 7

(a) Phase diagram of magnetic solitons in the $m_0-U$ plane, where $m_0$ is the magnetization at the center of the soliton and $U=V/c_{\text{s}}$ is the velocity. (b) Velocity dependence of the energy of magnetic solitons for different Rabi coupling strengths: ${\omega }_{\mathrm{R}}=0.3$ (solid red line), ${\omega }_{\mathrm{R}}=1$ (dashed black line), and ${\omega }_{\mathrm{R}}=2$ (dash-dotted blue line). Lines without circles indicate that the solutions are the $2\pi$ solitons; lines with circles, that the solutions are $0\pi$ solitons. The point of origin in (a) corresponds to the solution of the known static Son-Stephanov domain wall with a $+2\pi$ relative phase jump and its energy increases as the Rabi coupling increases [see (b)]. The green square indicates the solution where the effective mass of the $2\pi$ soliton diverges and the green $\mathsf{X}$ indicates the position of the transformation between $2\pi$ and $0\pi$ solitons for ${\omega }_{\mathrm{R}}=0.3$. Note that there exists another series of solutions obtained by changing (a) according to the transformation $V\to -V$; then the solutions are connected to the known static Son-Stephanov domain wall with a $-2\pi$ relative phase jump.

Figure 8

Comparison of the numerically calculated energy and the theoretical prediction for a slowly moving domain wall in the presence of a weak Rabi coupling, ${\omega }_{\mathrm{R}}=0.05$. The dashed black line shows the analytic result [Eq. (35)] and red lines show the numerical results.

Figure 9

Landau critical velocity for the disappearance of $0\pi$ magnetic solitons as a function of the Rabi coupling. The solid line shows the analytic prediction and blue squares represent the numerical results for the points where the energy of the $0\pi$ solitons tends to 0.

Figure 10

Profiles of the $0\pi$ magnetic soliton at velocity $V/c_{\text{s}}=-1.3, m_0=0.405$, and Rabi coupling ${\omega }_{\text{R}}=0.3$. (a) The solid red and dashed blue lines show the density distributions of the two components, satisfying $(n_1+n_2)/n=1$. (b) The solid green and dashed orange lines show the relative phase ${\phi }_A$ and total phase ${\phi }_B$ as a function of the coordinate. For $0\pi$ solitons the asymptotic jump of ${\phi }_A$ is 0. Compared to Fig. 5, there are more oscillations because the velocity of the soliton is close to the Landau critical velocity.

Figure 11

Oscillation of magnetic solitons in a 1D harmonic trap. We imprint a Son-Stephanov domain wall at the initial position $z_0=20\mu \mathrm{m}$; the local Rabi coupling is given by ${\omega }_{\text{R}}\left(z_0\right)=\mathrm{\Omega }/{\mathrm{\Omega }}_{\text{c}}\left(z_0\right)=0.22$. Evolution of the densities and relative phase of the two components after a holding time of (a) ${\omega }_{\text{ho}}t=0$, (b) ${\omega }_{\text{ho}}t=6.3$, and (c) ${\omega }_{\text{ho}}t=12.6$.

Figure 12

(a) In-trap trajectories of magnetic solitons and (b) evolution of the magnetization $m_0$ at the soliton center after imprinting a Son-Stephanov domain wall at $z_0$ for different values of the local Rabi coupling: ${\omega }_{\text{R}}\left(z_0\right)=\mathrm{\Omega }/{\mathrm{\Omega }}_{\text{c}}\left(z_0\right)=0.22$ (solid red line), ${\omega }_{\text{R}}\left(z_0\right)=\mathrm{\Omega }/{\mathrm{\Omega }}_{\text{c}}\left(z_0\right)=0.66$ (dashed black line), and ${\omega }_{\text{R}}\left(z_0\right)=\mathrm{\Omega }/{\mathrm{\Omega }}_{\text{c}}\left(z_0\right)=1.1$ (dash-dotted blue line). Lines without circles indicate $2\pi$ solitons; lines with circles, $0\pi$ solitons. Spin-interaction parameters have been chosen the same as in ${}^{23}\mathrm{Na}$, where $\delta g=0.07g$ and the Thomas-Fermi radius is $R_{\text{TF}}=70\mu \mathrm{m}$. All curves presented here correspond to the time interval of an oscillation period.

Figure 13

(a) In-trap trajectories of magnetic solitons and (b) evolution of the magnetization $m_0$ at the soliton center after imprinting a Son-Stephanov domain wall at different initial positions: $z_0=20\mu \mathrm{m}$ (solid red line) and $z_0=35\mu \mathrm{m}$ (dotted blue line). The corresponding local dimensionless Rabi couplings are ${\omega }_{\text{R}}\left(z_0\right)=0.22$ (red) and ${\omega }_{\text{R}}\left(z_0\right)=0.45$ (blue), respectively. Lines without circles indicate $2\pi$ solitons; lines with circles, $0\pi$ solitons. Spin-interaction parameters have been chosen the same as ${}^{23}\mathrm{Na}$ with $\delta g=0.07g$ and the Thomas-Fermi radius is $R_{\text{TF}}=70\mu \mathrm{m}$.

Figure 14

Evolution of the magnetic solitons in an elongated harmonic trap (aspect ratio $=10$) after imprinting a Son-Stephanov domain wall. For each time instant we show the density of the second component $n_2$ (the brighter the color, the higher the density) in the upper panel and the relative phase ${\phi }_A$ (the color or gray scale changes continuously from 0 to $2\pi$) in the lower panel: (a) ${\omega }_{\mathrm{ho}}t=0.8$, (b) ${\omega }_{\mathrm{ho}}t=3.8$, (c) ${\omega }_{\mathrm{ho}}t=4.2$. Rabi coupling $\mathrm{\Omega }=0.5{\omega }_{\text{ho}}$ and interaction $\delta g=0.4g$. In the top panel we also show a length scale corresponding to $5a_{\mathrm{ho}}$.

Figure 15

Densities of the two spin components and their relative phase ${\phi }_A$ calculated on a cut along the longer axis of the configurations presented in Fig. 14. Soliton (left) before (${\omega }_{\text{ho}}t=3.8$) and (right) after (${\omega }_{}\text{ho}t=4.2$) reflection. We can see that the configuration before the reflection corresponds to a $2\pi$ soliton and the configuration after the reflection corresponds to a $0\pi$ soliton.

Figure 16

Evolution of an imprinted Son-Stephanov domain wall in a spherical harmonic trap. For each time instant we show the density of the second component $n_2$ (the brighter the color, the higher the density) in the upper panel and the relative phase ${\phi }_A$ (the color or gray scale changes continuously from 0 to $2\pi$) in the lower panel: (a) ${\omega }_{\mathrm{ho}}t=0$, (b) ${\omega }_{\mathrm{ho}}t=3.2$, (c) ${\omega }_{\mathrm{ho}}t=5.6$. Rabi coupling $\mathrm{\Omega }=0.5{\omega }_{\text{ho}}$ and interaction $\delta g=0.4g$. In the top left panel we also show a length scale corresponding to $5a_{\mathrm{ho}}$.