Quantum solitons in spin-orbit-coupled Bose-Bose mixtures

Recent experimental and theoretical results show that weakly interacting atomic Bose-Bose mixtures with attractive interspecies interaction are stabilized by beyond-mean-field effects. Here we consider the peculiar properties of these systems in a strictly one-dimensional configuration, taking also into account the nontrivial role of spin-orbit and Rabi couplings. We show that when the value of inter- and intraspecies interaction strengths are such that mean-field contributions to the energy cancel, a self-bound bright soliton fully governed by quantum fluctuations exists. We derive the phase diagram of the phase transition between a single-peak soliton and a multipeak (striped) soliton, produced by the interplay between spin-orbit, Rabi couplings and beyond-mean-field effects, which also affect the breathing mode frequency of the atomic cloud. Finally, we prove that a phase imprinting of the single-peak soliton leads to a self-confined propagating solitary wave even in the presence of spin-orbit coupling.

Figure 1

Density distribution of the quantum bright soliton ($\delta g=0$) for different values of spin-orbit $\gamma$ and Rabi $\mathrm{\Gamma }$ couplings, obtained for a fixed intraspecies interaction $g{N}^{1/3}=1$. In the top-left panel we show the square modulus ${\left|\varphi \right|}^2$ of our analytical solution Eq. (9), exactly coincident with the numerical solution of the coupled Eqs. (3) for $\delta g=0$, $\gamma =0$, and $\mathrm{\Gamma }=0$. In the other panels we report the normalized components $|{\psi }_1{|}^2{N}^{-1}$ (black line) and $|{\psi }_2{|}^2{N}^{-1}$ (red dashed line) with nonzero $\gamma$ and $\mathrm{\Gamma }$, which turn out to be coincident. Here the axial coordinate $x$ is rescaled in units of the transverse harmonic-oscillator length $l_{\perp }=\sqrt{\hslash /\left(m{\omega }_{\perp }\right)}$, with ${\omega }_{\perp }$ the transverse frequency of the confining potential, while $g$, $\delta g$, and $\gamma$ are in units of ${\hslash }^2/\left(m{l}_{\perp }\right)$ and $\mathrm{\Gamma }$ is in units of $\hslash {\omega }_{\perp }$.

Figure 2

Phase diagram of the phase transition from a single peak quantum bright soliton to a striped (multipeak) soliton, obtained for a fixed intraspecies interaction strength $g{N}^{1/3}=1$. The couplings $g{N}^{1/3}$, $\gamma$, and $\mathrm{\Gamma }$ are rescaled as in Fig. 1.

Figure 3

Top panel: breathing mode frequency ${\omega }_b$ as a function of $g{N}^{1/3}$, for $\gamma =0$ and $\mathrm{\Gamma }=0$. The symbols are obtained solving numerically Eq. (13), the black solid line represents our Gaussian ansatz Eq. (12), while the green dashed line is obtained with a ${\text{sech}}^2\left(x\right)$ ansatz (see text). Bottom panel: ${\omega }_b$ as a function of $\gamma$, for many values of $\mathrm{\Gamma }$, and $g{N}^{1/3}=1$. Notice that ${\omega }_b$ increases at the transition between a single-peak and a multipeak soliton. The solid lines are a guide to the eye. Here we rescale ${\omega }_b$ in units of the transverse frequency ${\omega }_{\perp }$, while $t$ is in units of ${\omega }_{\perp }^{-1}$ and all the remaining couplings are rescaled as in Fig. 1.

Figure 4

Time evolution of the quantum bright soliton after a phase imprinting in the form of $exp\left(ikx\right)$, with wave vector $k=2\pi /60$. The unperturbed initial conditions are the striped soliton with $g{N}^{1/3}=1$, $\gamma =4$, and $\mathrm{\Gamma }=2$ (top panel) and the single-peak soliton for $g{N}^{1/3}=1$, $\gamma =1$, and $\mathrm{\Gamma }=4$ (bottom panel): notice how the Galilean invariance is violated only for the striped soliton. The physical quantities are rescaled as in Figs. 1 and 3.