Magnetic solitons in an immiscible two-component Bose-Einstein condensate

We investigate magnetic solitons in an immiscible binary Bose-Einstein condensate (BEC), where the intraspecies interactions are slightly weaker than the interspecies interactions. While their density and phase profiles are analogous to dark-bright solitons, other characteristic properties such as maximum velocities, widths, total density depletions, and in-trap oscillations are different. In the low-velocity regime, a magnetic soliton reduces to a traveling pair of magnetic domain walls. Collisional behaviors of the solitons are also briefly discussed. We further demonstrate that these solitonic states can be realized in a quasi-one-dimensional spin-1 ferromagnetic BEC with a weak spin interaction, e.g., a ${}^{87}\mathrm{Rb}$ BEC.

Figure 1

(a), (b) Profile of a magnetic soliton with $U=0.3$ and $\mathrm{\Phi }=0$. The blue solid and red dashed curves are (a) density or (b) phase profiles of the two components, respectively. (c) Spin density of a magnetic soliton with various velocities. The purple solid line, green dashed line, and orange dotted line represent the spin densities of a soliton with $U={10}^{-5}$, 0.3, and 0.8, respectively. (d) Dependence of the soliton phases on the soliton velocity. The solid and the dashed lines are slopes of ${\varphi }_1,{\varphi }_2$ at the center of the soliton, respectively. The dotted curve is the phase jump $\mathrm{\Delta }{\varphi }_1={\varphi }_1(\zeta =+\infty )-{\varphi }_1(\zeta =-\infty )$. (e) Total population of the component 2 in a magnetic soliton. (f) Full width half maximum (FWHM) of the magnetic soliton (dashed line) and the dark-bright soliton (solid line). (g) Density depletion of the magnetic soliton. The solid lines and points represent analytical and numerical results, respectively.

Figure 2

Oscillations of a magnetic soliton and a dark-bright soliton in a harmonic trap are compared. The blue circles and red squares are numerical results for the magnetic soliton and the dark-bright soliton, respectively. The black curve is the local density approximation (LDA) prediction for the magnetic soliton. $V_0$ is the soliton velocity at the center of the condensate. In (a) we show the oscillation amplitude $L$ normalized to the Thomas-Fermi radius $R_z=\sqrt{2n_{0}g/M{\omega }_z^2}$, where ${\omega }_z$ and $n_0$ are the trapping frequency and density at the center. In (b) we show the oscillation period normalized to the trap period $T_z=2\pi /{\omega }_z$.

Figure 3

Soliton collisions in a uniform system. Plots show the normalized spin density $F_z/n=(n_1-n_2)/n$ as a function of space and time. The timescale is $t_s={\xi }_s/V_s$. In (a) and (b), $\left|U\right|=0.3$ for both solitons. The phase differences are (a) $\mathrm{\Delta }\mathrm{\Phi }=0$, and (b) $\mathrm{\Delta }\mathrm{\Phi }=\pi$. (c) shows the formation of a magnetic soliton dimer after collision, where initially $\left|U\right|=0.1$ for both solitons and the separation is $5{\xi }_s$. (d) is in comparison with (c) where $\left|U\right|=0.1$ but the separation is $10{\xi }_s$. (e) shows the collision between a magnetic soliton with $\left|U\right|=0.3$ and a static domain wall. (f) shows the collision between two magnetic solitons with $\left|U\right|=0.3$ and $\left|U\right|={10}^{-5}$.

Figure 4

Proposal to generate a magnetic soliton in a quasi-1D ${}^{87}\mathrm{Rb}$ BEC. (a) Local population transfer from $m=1$ to $m=-1$. A Raman laser pulse coupling the $5S_{1/2},|F=1,m=\pm 1\rangle$ states through the $5P_{1/2}$ state illuminates the center of the condensate. The pulse duration is controlled to transfer a desired fraction of atoms. (b) Magnetic shadow. An enlarged laser beam is imaged onto half of the condensate. The laser frequency is tuned to the “magic frequency” so that it only induces a vector ac Stark shift. The laser beam is pulsed such that a finite phase jump is generated. (c) Oscillation of the generated magnetic soliton in a harmonic trap. The plot shows the normalized spin density $[n_{+1}(z,t)-n_{-1}(z,t)]/\mathfrak{n}(z,t)$ as a function of space and time, where $n_{\pm 1}(z,t)$ are the densities of the $m=\pm 1$ components. (d) Plot of the normalized density depletion defined as $[\mathfrak{n}(z,t)-{\mathfrak{n}}_g\left(z\right)]/{\mathfrak{n}}_g\left(z\right)$, where ${\mathfrak{n}}_g\left(z\right)$ is the ground-state density distribution.

Figure 5

Initial state for generating a magnetic soliton.

Figure 6

Density and phase profiles at $\tilde{t}=3.77$.