Introduction

The formation of three-dimensional (3D) nonlinear localized states is a problem of fundamental importance in diverse areas of physics. It keeps drawing much interest, revealing various mechanisms which support the formation of such states1,2,3,4,5. The creation of 3D solitons is a significantly more challenging problem than the generation of low-dimensional ones, as the usual focusing cubic nonlinearity causes the critical and supercritical wave collapse in 2D and 3D geometries, respectively6,7. Various methods have been elaborated, chiefly in a theoretical form, to remedy this situation and stabilize multidimensional solitons, in the form of fundamental, multipolar, and vortical states8,9. Stable 3D solitons may be formed in settings with saturable or competing nonlinearities10,11,12, nonlocal interactions13,14, spatially modulated nonlinearity15, PT-symmetric optical lattices16,17, waveguide arrays and lattices imprinted in different materials18,19,20, and binary Bose–Einstein condensates (BECs) subject to the action of spin–orbit coupling (SOC), which lend the system specific symmetry properties21,22. Accordingly, many kinds of solitons in SOC BECs, defined by their intrinsic symmetry, were predicted, such as semi-vortices (SVs) and mixed modes (MMs)23,24,25, vortex-bright solitons26,27,28, and filled-core vortices29.

In particular, SOC BECs, composed as mixtures of atoms in two different hyperfine states, demonstrate coupling between the pseudospin degree of freedom and spatial structure of the condensate5,30,31,32. SOC notably modifies the dispersion of the system33,34, breaks the Galilean invariance35, and thus substantially impacts the properties of solitons in the free space36,37. Quite interesting is also the impact of SOC on BEC in external potentials, where possible symmetries of self-sustained solitons and their dynamics are determined by the symmetry of the potential38,39,40,41,42. A conclusion is that BECs under the action of SOC offer a versatile platform for the investigation of nonlinear phenomena in the presence of synthetic fields43 and gauge potentials44.

Theoretical45 and experimental46 studies have revealed that strong effective nonlinearities can be induced by the long-range Rydberg-Rydberg interaction (RRI) between remote atoms. To this end, RRI is mapped into a nonlocal optical nonlinearity through electromagnetically-induced transparency (EIT) at the single-photon level47. This option provides an important platform for the study of optical soliton dynamics with tunable parameters48,49. In particular, Rydberg gases are proven to be an effective medium for generating stable solitons with low energies under the action of the strong long-range RRI-induced nonlinearity50,51,52,53,54. Rydberg atomic gases are controllable in an active way through tunable parameters55,56, such as atomic levels, detuning, laser intensities, etc. Furthermore, long lifetimes of the Rydberg atomic states (~ tens of microseconds) guarantee that the induced nonlinearities are quite robust57. Thus, Rydberg-EIT settings provide a fertile ground for realizing quantum nonlinear optics48 and developing new photon devices, such as single-photon switches and transistors58,59, quantum memories, and phase gates56,60.

Although studies of various solitons in the context of BEC constitute a mature field, the existence and stability of the solitons in the framework of the mean-field theory, which is based on Gross-Pitaevskii equations (GPEs) under the action of SOC, contact interactions, and long-range spherically symmetric attractive RRIs remain an area of active work. The present paper aims to predict 3D solitons in binary atomic condensates combining SOC and RRI, taking into account the underlying symmetries and utilizing systematic numerical simulations of the corresponding model. In particular, we construct the solitons of the SV or MM types. Stability regions for the solitons are identified via the linear stability analysis and direct simulations of their perturbed evolution.

The model and numerical method

As said above, the subject of the analysis is the 3D Rydberg-dressed binary BEC under the action of SOC. The scheme of the respective three-level atomic system is shown in Fig. 1a. We consider the ultra-cold atomic gas of \(N\) atoms, each possessing a ground state \(\left| g \right\rangle\) and an excited one \(\left| {n^{\prime}P} \right\rangle\). An excited Rydberg state, composed of two Rydberg atoms at a distance \({\mathbf{r}}_{ij} = {\mathbf{r}}_{i} - {\mathbf{r}}_{j}\), is denoted \(\left| e \right\rangle\). Two probe pulses with Rabi frequency \(\Omega_{1/2}\) and detuning \(\Delta_{1/2}\) couple the three-level states. In the mean-field approximation, the dynamics of the 3D spinor wave function \(\Psi = \left( {\Psi_{ + } ,\Psi_{ - } } \right)\) in the binary BEC with RRI and SOC of the Rashba type with strength \(\lambda\)61 is governed by the system of coupled scaled GPEs, which are written in the scaled form:

$${\text{i}}\frac{{\partial \Psi_{ + } }}{\partial t} = - \frac{1}{2}\nabla^{2} \Psi_{ + } + \left( {\left| {\Psi_{ + } } \right|^{2} + \left| {\Psi_{ - } } \right|^{2} } \right)\Psi_{ + } + V_{1} \Psi_{ + } - i\lambda (\partial_{x} \Psi_{ - } - i\partial_{y} \Psi_{ - } + \partial_{z} \Psi_{ + } )$$
(1a)
$${\text{i}}\frac{{\partial \Psi_{ - } }}{\partial t} = - \frac{1}{2}\nabla^{2} \Psi_{ - } + \left( {\left| {\Psi_{ - } } \right|^{2} + \left| {\Psi_{ + } } \right|^{2} } \right)\Psi_{ - } + V_{2} \Psi_{ - } - i\lambda (\partial_{x} \Psi_{ + } + i\partial_{y} \Psi_{ + } - \partial_{z} \Psi_{ - } )$$
(1b)

where \(\nabla^{2} \equiv \partial_{x}^{2} + \partial_{y}^{2} + \partial_{z}^{2}\), the coefficients of the contact self- and cross-repulsion are set to be 1 by scaling (in most cases, these coefficients are nearly equal). The last term is the Rashba SOC which is obtained as \(V_{{{\text{SOC}}}} = \lambda {\mathbf{p}} \cdot {{\varvec{\upsigma}}}\)62, where \({\mathbf{p}} = - i\nabla\) is the momentum operator and \({{\varvec{\upsigma}}} = (\sigma_{x} ,\sigma_{y} ,\sigma_{z} )\) are Pauli matrices. The potential \(V_{1} = - \int {{\text{d}}^{3} {\mathbf{r}}^{\prime}\left[ {U_{11} ({\mathbf{r}}^{\prime})\left| {\Psi_{ + } } \right|^{2} + U_{12} ({\mathbf{r}}^{\prime})\left| {\Psi_{ - } } \right|^{2} } \right]}\) and \(V_{2} = - \int {{\text{d}}^{3} {\mathbf{r}}^{\prime}\left[ {U_{21} ({\mathbf{r}}^{\prime})\left| {\Psi_{ + } } \right|^{2} + U_{22} ({\mathbf{r}}^{\prime})\left| {\Psi_{ - } } \right|^{2} } \right]}\) represent the interaction between remote Rydberg atoms with

$$U_{ij} = {{C_{ij} }/{(R_{c}^{6} + {\mathbf{r}}_{ij}^{6} )}}$$
(1c)

is the van der Waals potential, where \(R\,_{c}\) is the soft-core radius, and

$$C_{ij} = (\Omega /2\Delta )^{4} C_{6}^{(ij)} ,$$
(1d)

where \(C_{6}^{(ij)}\) are dispersion parameters which determine the intra- and inter-species couplings in the two-component BEC system15,16. Here \(\Omega = \frac{{\Omega_{1} \Omega_{2} }}{{2\Delta_{1} }}\) is the effective Rabi frequency of the system, and \(\Delta = \Delta_{1} + \Delta_{2}\), where \(\Omega_{1}\) and \(\Omega_{2}\) are Rabi frequencies of two laser fields, \(\Delta_{1}\) and \(\Delta_{2}\) being the respective detunings. Note that, for a small Rydberg radius (\(R_{\,c} \ll r_{ij}\)), Eq. (1c) takes the form of \(U_{ij} \approx {{C_{ij} } /{r_{ij}^{6} }}\), i.e., the RRI potential may be treated as a spherically symmetric s-wave scattering pseudopotential. In this work, by tuning system parameters, the values of \(C_{ij}\), viz., \(C_{11}\), \(C_{12} = C_{21}\) and \(C_{22}\), range from -1000 to 1000.

Figure 1
figure 1

(a) The schematic of the Rydberg-dressed three-level atomic system, with the laser coupling between the ground state \(\left| {n_{0} S} \right\rangle\) and the Rydberg state \(\left| {nS} \right\rangle\). For \(\Delta_{1} \gg \Omega_{1}\), the system reduces to an effective two-level atom, with states \(\left| g \right\rangle = \left| {n_{0} S} \right\rangle\) and \(\left| e \right\rangle = \left| {nS} \right\rangle\) coupled by the two-photon Rabi frequency \(\Omega\) and detuning \(\Delta\). Density isosurfaces of 3D solitons (b–e) and phase structures in \((x,y)\) plane (f–i) for the SV +, SV−, MM +, and MM− types.

Full size image

Our main focus herein is on discussing the effects of RRI and SOC on the formation of solitons. For \(\Delta_{1} \gg \Omega_{1}\), the system reduces to an effective two-level atom, with states \(\left| g \right\rangle = \left| {n_{0} S} \right\rangle\) and \(\left| e \right\rangle = \left| {nS} \right\rangle\) coupled by a two-photon Rabi frequency \(\Omega\) and detuning \(\Delta\). For experimental considerations, a suitable candidate for the realization of the setup is the gas of \(^{{{87}}} {\text{Rb}}\) atoms with \(\Delta = 2\pi \times 32 \; {\text{MHz}}\) and \(\Omega = 2\pi \times 1 \;{\text{MHz}}\)12,63.

Characteristic physical units related to the scaled ones in Eq. (1) are chosen as follows. The number of atoms in the binary BEC is characterized as \(\approx 10^{4} N\) where \(N\) is the scaled norm of the wave function64,65,66. The scaled length and time units are \(10\,\,{\upmu m}\) and \(100\,{\text{ms}}\), respectively. For example, \((x,y) = (5,5)\) corresponds to the spatial domain of size \((50\,\,{\upmu m},\,\,50\,\,{\upmu m})\), and \(t = 10\) corresponds to \(1000\,{\text{ms}}\). A typical distance between two Rydberg atoms is of the order \(r_{ij} \approx 10\,{\upmu m}\), while the unit for softcore size is estimated as \(\approx 5.6\,{\upmu m}\)66.

Stationary spinor wave functions with chemical potential \(\mu\) are looked for as \(\Psi_{ \pm } = \psi_{ \pm } e^{ - i\mu t}\). In cylindrical coordinates \(\left( {r,\varphi ,z} \right)\), solutions with symmetries corresponding to two different soliton species, SV and MM, are seeded by the following initial guesses for the stationary wave function21:

$$\psi_{SV + } = (A_{1} + iB_{1} z)e^{{ - \alpha_{1} r^{2} - \beta_{1} z^{2} }} ,\;\;\psi_{SV - } = (iA_{2} + B_{2} z)re^{{ - \alpha_{2} r^{2} - \beta_{2} z^{2} + i\varphi }}$$
(2a)

for the SV (semi-vortex), and

$$\psi_{MM + } = \cos \theta \psi_{SV + } - \sin \theta \psi_{SV - }^{*} ,\;\;\psi_{MM - } = \cos \theta \psi_{SV - } + \sin \theta \psi_{SV + }^{*}$$
(2b)

for the MM (mixed mode), where * stands for the complex conjugate.

The energy corresponding to Eqs. (1a) and (1b) is

$$\begin{aligned} E_{tot} & = E_{{{\text{kin}}}} + E_{{\text{int}}} + E_{{{\text{Ryd}}}} + E_{{{\text{SOC}}}} , \hfill \\ E_{{{\text{kin}}}} & = \int {\frac{1}{2}(\left| {\nabla \psi_{ + } } \right|^{2} + \left| {\nabla \psi_{ - } } \right|^{2} ){\text{d}}{\mathbf{r}}} , \hfill \\ E_{{\text{int}}} & = \int {\left( {\frac{1}{2}\left| {\psi_{ + } } \right|^{4} + \frac{1}{2}\left| {\psi_{ - } } \right|^{4} + \left| {\psi_{ + } } \right|^{2} \left| {\psi_{ - } } \right|^{2} } \right){\text{d}}{\mathbf{r}}} , \hfill \\ E_{{{\text{Ryd}}}} & = - \iint\limits_{{}} {{\text{d}}^{3} {\mathbf{r}}{\text{d}}^{3} {\mathbf{r^{\prime}}}}\left[ {\frac{1}{2}U_{11} \left( {{\mathbf{r}} - {\mathbf{r^{\prime}}}} \right)\left| {\psi_{ + } \left( {\mathbf{r}} \right)} \right|^{2} \left| {\psi_{ + } \left( {{\mathbf{r^{\prime}}}} \right)} \right|^{2} + \frac{1}{2}U_{22} \left( {{\mathbf{r}} - {\mathbf{r^{\prime}}}} \right)\left| {\psi_{ - } \left( {\mathbf{r}} \right)} \right|^{2} \left| {\psi_{ - } \left( {{\mathbf{r^{\prime}}}} \right)} \right|^{2} } \right. \hfill \\ & \quad \left. { + U_{12} \left( {{\mathbf{r}} - {\mathbf{r^{\prime}}}} \right)\left| {\psi_{ + } \left( {\mathbf{r}} \right)} \right|^{2} \left| {\psi_{ - } \left( {{\mathbf{r^{\prime}}}} \right)} \right|^{2} } \right], \hfill \\ E_{{{\text{SOC}}}} & = \lambda \int {{\text{d}}^{3} {\mathbf{r}}\left( {\psi_{ + }^{*} \hat{D}_{ - } \psi_{ - } + \psi_{ - }^{*} \hat{D}_{ + } \psi_{ + } } \right)} . \hfill \\ \end{aligned}$$
(3)

where SOC operators, with their respective symmetry structures, are \(\hat{D}_{ \mp } = \partial_{x} \mp i\partial_{y} \mp \partial_{z}\), and terms \(E_{{{\text{kin}}}}\), \(E_{{{\text{int}}}}\), \(E_{{{\text{Ryd}}}}\), and \(E_{{{\text{SOC}}}}\) represent the kinetic energy, inter- and intra-species interaction, RRI, and SOC interaction, respectively. Below, the RRI parameters \(C_{ij}\), radius \(R_{c}\) and SOC strength \(\lambda\) are varied to study the soliton dynamics.

To explore the stability of solitons, perturbed solutions are taken as

$$\begin{aligned} \Psi_{ + } & = (\psi_{ + } + g_{1} e^{bt} + h_{1}^{ * } e^{{b^{ * } t}} )e^{ - i\mu t} \hfill \\ \Psi_{ - } & = (\psi_{ - } + g_{2} e^{bt + ik\varphi } + h_{2}^{ * } e^{{b^{ * } t - ik\varphi }} )e^{ - i\mu t + i\varphi } , \hfill \\ \end{aligned}$$
(4)

where \(g_{1,2}\) and \(h_{1,2}\) are amplitudes of small perturbations, \(b \equiv b_{r} + ib_{i}\) is the corresponding instability growth rate, and \(k\) is the integer azimuthal index, \(\varphi\) is the azimuthal angle. The ansatz for SVs permit both non-vortex and vortex modes, corresponding to the \(\Psi_{ + }\) and \(\Psi_{ - }\). For \(\Psi_{ - }\), the perturbation includes the azimuthal angle \(\varphi\). The soliton solutions may be stable if \({\text{Re}}(b) = 0\) for all eigenvalues. The linearization of Eqs. (1) with respect to the small perturbations produces the system of the respective Bogoliubov – de Gennes equations:

$$\begin{aligned} - ibg_{1}& = \left( {\frac{1}{2}\nabla^{2} + \mu - V_{1} + i\lambda \frac{\partial }{\partial z} - \left| {\psi_{ - } } \right|^{2} - 2\left| {\psi_{ + } } \right|^{2} } \right)g_{1} - \psi_{ + }^{2} h_{1} \hfill \\ & \quad + \left[ {i\lambda \left( {\frac{{e^{ik\varphi } \partial }}{\partial x} + \frac{{\partial e^{ik\varphi } }}{\partial x} - i\frac{{e^{ik\varphi } \partial }}{\partial y} - i\frac{{\partial e^{ik\varphi } }}{\partial y}} \right) - \psi_{ + } \psi_{ - }^{ * } e^{ik\varphi } } \right]g_{2} - \psi_{ + } \psi_{ - } e^{ik\varphi } h_{2} , \hfill \\ \end{aligned}$$
(5a)
$$\begin{aligned} - ibh_{1} & = \psi_{ + }^{{ *^{2} }} g_{1} + \left[ { - \frac{1}{2}\nabla^{2} - \mu + V_{1} + i\lambda \frac{\partial }{\partial z} + \left| {\psi_{ - } } \right|^{2} + 2\left| {\psi_{ + } } \right|^{2} } \right]h_{1} \hfill \\ & \quad + \psi_{ + }^{ * } \psi_{ - }^{ * } e^{ik\varphi } g_{2} + \left[ {i\lambda \left( {\frac{{e^{ik\varphi } \partial }}{\partial x} + \frac{{\partial e^{ik\varphi } }}{\partial x} + i\frac{{e^{ik\varphi } \partial }}{\partial y} + i\frac{{\partial e^{ik\varphi } }}{\partial y}} \right) + \psi_{ + }^{ * } \psi_{ - } e^{ik\varphi } } \right]h_{2} , \hfill \\ \end{aligned}$$
(5b)
$$\begin{aligned} - ibg_{2} & = \left[ {i\lambda \left( {\frac{\partial }{\partial x} + \frac{{e^{ - ik\varphi } \partial e^{ik\varphi } }}{\partial x} + i\frac{\partial }{\partial y} + i\frac{{e^{ - ik\varphi } \partial e^{ik\varphi } }}{\partial y}} \right) - \psi_{ + }^{ * } \psi_{ - } } \right]e^{ - ik\varphi } g_{1} - \psi_{ + } \psi_{ - } e^{ - ik\varphi } h_{1} \hfill \\ & \quad + \left( {\frac{1}{2}\nabla^{2} + \mu - V_{2} - i\lambda \frac{\partial }{\partial z} - \left| {\psi_{ + } } \right|^{2} - 2\left| {\psi_{ - } } \right|^{2} } \right)g_{2} - \psi_{ - }^{2} h_{2} , \hfill \\ \end{aligned}$$
(5c)
$$\begin{aligned} - ibh_{2} & = \psi_{ + }^{ * } \psi_{ - }^{ * } e^{ - ik\varphi } g_{1} + \left[ {i\lambda \left( {\frac{\partial }{\partial x} + \frac{{e^{ - ik\varphi } \partial e^{ik\varphi } }}{\partial x} - i\frac{\partial }{\partial y} - i\frac{{e^{ - ik\varphi } \partial e^{ik\varphi } }}{\partial y}} \right) + \psi_{ + } \psi_{ - }^{ * } } \right]e^{ - ik\varphi } h_{1} \hfill \\ & \quad + \psi_{ - }^{{ *^{2} }} g_{2} + \left( { - \frac{1}{2}\nabla^{2} - \mu + V_{2} - i\lambda \frac{\partial }{\partial z} + \left| {\psi_{ + } } \right|^{2} + 2\left| {\psi_{ - } } \right|^{2} } \right)h_{2} . \hfill \\ \end{aligned}$$
(5d)

Results

In the present system, the total norm \(N = \iiint {(\left| {\psi_{ + } } \right|^{2} + \left| {\psi_{ - } } \right|^{2} )dxdydz}\) is composed of the spin-up and spin-down terms, \(N^{ + } = \iiint {\left| {\psi_{ + } } \right|^{2} dxdydz}\) and \(N^{ - } = \iiint {\left| {\psi_{ - } } \right|^{2} dxdydz}\), respectively. They determine the respective norm shares, \(F_{ \pm } = {{N_{ \pm } }/N}\).

Stationary solutions for the wave function seeded by the initial guesses (2) were obtained using the squared-operator method67 and also by the imaginary-time evolution method41. As a result, two-component solitons are produced, \({\text{SV }} = \left\{ {\text{SV+,SV}}- \right\}{,}\,\,{\text{and}}\,\,{\text{MM = }}\left\{ {\text{MM+,MM}}- \right\}\). Examples of density isosurfaces and phase structure of these components of stable solitons are plotted in Fig. 1b–i, respectively. For the mode with the SV symmetry, the norm is distributed along a ring, and the phase represents the vortex structure. On the other hand, in both components of the MM mode, the phase patterns are also vortical, while the density distributions seem like those representing “distorted vortices”, cf. qualitatively similar density patterns in the 3D SV and MM solitons produced by the GPE system with the contact-only cubic nonlinearity and SOC terms of the Weyl type21.

Families of the SV solitons

Figure 2 summarizes the numerical solutions for the 3D solitons of the SV type, in the form of dependences of their energy, total norm, and the norm share of the SV + component, on the Rydberg interaction strengths, \(C_{ij} (i,j = 1,2)\), and the softcore radius, \(R\,_{c}\). It is seen that the energy, being vanishingly small for \(C_{ij} < 0\), rapidly increases with the increase of \(C_{ij} > 0\), seen in Fig. 2a–c. Solid and dashed lines in Fig. 2 denote stable and unstable zones of the SV solutions in the panels. The results also indicate that the stability zone is smaller for larger \(R\,_{c}\).

Figure 2
figure 2

The modulation of solitons of the SV type by altering the Rydberg coefficients, \(C_{ij}\). The first to the third rows display, respectively, the total energy \(E\)(a–c), the total norm \(N\) of the solitons (d–f), and the norm share of the spin-up component, \(F^{ + }\)(g–i). The blue, red, and black lines pertain to Rydberg soft-core radii \(R_{c} = 1.0,1.5,2.0\). Solid and dashed lines represent, respectively, quasi-stable and unstable solitons. Points A and B in (a) represent the stable and unstable states with \(C_{11} = 300\) and 150, respectively. The fixed parameters are \(\lambda = 1\), \(C_{11} = 200\), \(C_{12} = C_{21} = 0\), \(C_{22} = 0\), \(R_{c} = 1\) and \(N = 5\).

Full size image

Stable SV solitons exist in an interval of the total norm

$$N_{\min } < N < N_{\max } ,$$
(6)

which is shown in Fig. 2d–f for different values of \(C_{ij}\). It is found that the dependences of \(N_{\min } \,{\text{and }}N_{\max }\) on the RRI coefficients \(C_{11}\), \(C_{12}\) and \(C_{22}\) are completely different, demonstrating the absence of symmetry between different RRI constituents. The norm decreases monotonously with \(C_{11}\), has a peak around \(C_{12} = 0\), and features a sharp drop at \(C_{22} > 250\).

As seen in Fig. 2d–f, one finds that the total norm rapidly decreases at \(C_{ij} > 1\), indicating that the solitons in this BEC system vanish for large detunings. The norm share of the spin-up component of SVs is shown in Fig. 2g–i. It increases monotonously with \(C_{11}\) and is the dominant component of the norm at \(C_{11} > 500\). In contrast to the dependence on \(C_{11}\), \(F^{ + }\) decreases monotonously with \(C_{12}\) and \(C_{22}\). Note that the increase of the Rydberg radius leads to a reduction of the stability zone of SVs, which is also observed in Fig. 2a–c.

Chemical potential \(\mu\) as a function of \(C_{ij}\) is plotted in Fig. 3a–c for different Rydberg soft-core radii \(R_{c}\). It is seen that \(\mu\) monotonously increases with \(C_{11}\), and monotonously decreases with \(C_{12}\) and \(C_{22}\). The other difference between the dependences on \(C_{11}\),\(C_{12}\) and \(C_{22}\) is the stability range. When \(R_{c} = 1(5.6\;\upmu {\text{m}})\), the possibly stable solitons are relatively large, located in the ranges of \(C_{11} \in (150,1000)\), \(C_{12} \in ( - 500,1000)\) and \(C_{22} \in ( - 500,1000)\). Note that, when the Rydberg soft-core radius is relatively large (\(R_{c} = 2.0\)), the stability zones for \(C_{12}\) and \(C_{22}\) are much smaller.

Figure 3
figure 3

Chemical potential \(\mu\) for SVs with different values of the system’s parameters, including \(C_{ij}\), \(N\), and \(\lambda\). Solid and dashed lines represent, respectively, quasi-stable and unstable soliton families. (a–c) The first row is the \(\mu - C_{ij}\) relation for different Rydberg soft-core radii \(R_{c} = 1.0,1.5,2.0\). (d–f) The second row is the \(\mu - N\) relation for different \(C_{ij}\) when \(R_{c} = 1.0\). (g) The \(\mu - \lambda\) relation for different values of \(R_{c}\), where points C and D stand for very narrow zones of quasi-stability at \(R_{c} = 2.0\). (h) The \(\mu - N\) relation for different \(R_{c}\). Solid and dashed lines represent, respectively, stable and unstable solitons. (i) The real part of the stability eigenvalue vs. the chemical potential when \(\lambda = 1\), \(C_{11} = 200\), \(C_{12} = C_{21} = 0\), \(C_{22} = 0\), \(R_{c} = 1\) and \(N = 5\). Other parameters are the same as in Fig. 2.

Full size image

The dependence of chemical potential \(\mu\) on SOC strength \(\lambda\) is shown in Fig. 3g. It is seen that \(\mu\) is much larger if the SOC is absent, \(\lambda \approx 0\), but the solitons are unstable (dashed lines) in that case.

Similar to Fig. 2, solid and dashed lines in panels of Fig. 3 denote stable and unstable zones of the SV solutions. The stability zone of the solitons decreases dramatically with the growth of \(R_{c}\). When \(R_{c} = 1\), two stability zones are observed, divided by the area of \(\lambda \approx 0\). With the increase of \(R_{c}\), the stability zone shrinks. For \(R_{c} = 2.0\) it is compressed into a very narrow area labeled by points A and B. This \(R_{c}\) is a key factor that determines the soliton stability in the system. For the Rashba SOC, the actual particle current consists of both the canonical part, related to the superfluid velocity, and the SOC-induced gauge part, cf. Ref.24. Zhang et al. studied SOC BECs loaded into a toroidal trap and found that, for the counter-circling flow, these two parts have the same magnitude but opposite signs, creating a quasi-1D Rashba ring68. In the 3D BEC system with SOC, we find that the stable SV solitons and their chemical potentials show a nearly-symmetric response with respect to the substitution \(\lambda \to - \lambda\). It is noticed that SV solitons cannot be generated at \(\left| \lambda \right| < 0.6\).

The stability of the solitons may be also evaluated using the “anti-Vakhitov-Kolokolov” (anti-VK) criterion, \({{{\text{d}}N}/{{\text{d}}\mu }} > 0\), which is a necessary but not sufficient condition for the stability of solitons supported by repulsive (defocusing) nonlinearities69. To this end, the chemical potential \(\mu\) is shown, as a function of \(N\), in Fig. 3d–f for different \(C_{ij}\), and in Fig. 3h for different \(\lambda\), demonstrating that an anti-VK criterion holds. In the full form, the stability of SVs is determined by the eigenvalues for small perturbations, produced by the numerical solution of Eq. (5). The real part of the eigenvalues is shown in Fig. 3i. One observes that \({\text{Re}} (b)\) is close to zero (\(\sim 10^{ - 5}\)) in broad intervals, where the SVs are quasi-stable states. The stability of solitons of the MMs type is similarly determined by the eigenvalues.

To explore the dynamics of SVs, profiles of SVs and MMs were produced by direct simulations of their perturbed evolution. In Fig. 4, SV + and SV− with two sets of parameters are shown. The magnitudes of the three-layer isosurface displayed in the figure are \({(0}{\text{.95,0}}{.5,0}{\text{.05)|}}\psi {|}_{\max }\), where \({|}\psi {|}_{\max }\) is the amplitude of the wave field. It is observed in Fig. 4 that the times during which the SVs maintain their integrity are different, depending on the initial inputs. In the first and second columns of Fig. 4, the soliton is quasi-stable, while in the third and fourth columns, it is definitely unstable.

Figure 4
figure 4

Isosurfaces of the SV+ and SV− components as produced by the direct simulations. The first and second columns represent the quasi-stable SVs (point A in Fig. 2a) with \({\text{Re}} (b) = 2.0 \times 10^{ - 5}\), while the third and fourth columns pertain to strongly unstable SVs (point B in Fig. 2a) with \({\text{Re}} (b) = 1.39 \times 10^{ - 2}\).

Full size image

The stability of these solitons was evaluated by the real part of the eigenvalue \({\text{Re}} (b)\), as shown in Fig. 3i. The SVs are generated at \(t = 0\) and evolve in time. \({\text{Re}} (b)\) is different depending on the initial inputs. In Fig. 4, SV + and SV− in the first and second columns demonstrate a small \({\text{Re}} (b) = 2.0 \times 10^{ - 5}\). However, it is larger in the third and fourth columns, viz., \({\text{Re}} (b) = 1.39 \times 10^{ - 2}\). Accordingly, the respective soliton collapses faster. Thus, to maintain the soliton’s stability (or keep the instability weak enough), it is important to optimize parameters of the setting.

Solitons of MM type

The dynamics of MMs is nearly the same as that of SVs. Figure 5 shows the perturbed evolution of the solitons of this type, shown by three-layer isosurface configurations. The (in)stability of these solitons is also characterized by \({\text{Re}} (b)\). The solitons displayed in the first (MM +) and second (MM-) columns of Fig. 5 are identified as quasi-stable ones with \({\text{Re}} (b) = 2.53 \times 10^{ - 5}\), while the third (MM +) and fourth(MM-) columns show their strongly unstable counterpart with \({\text{Re}} (b) = 5.85 \times 10^{ - 2}\). Thus, the quasi-stable and strongly unstable states are produced by the analysis.

Figure 5
figure 5

Isosurfaces of the MM + and MM- components, as produced by the direct simulations. The first and second columns represent the quasi-stable MMs for parameters \(\lambda = 1\), \(C_{11} = 300\), \(C_{12} = C_{21} = 0\), \(C_{22} = 0\), \(R_{c} = 1\), the respective instability growth rate being \({\text{Re}} (b) = 2.53 \times 10^{ - 5}\), while the third and fourth columns pertain to \(\lambda = 1\), \(C_{11} = 100\), \(C_{12} = C_{21} = 0\), \(C_{22} = 0\), \(R_{c} = 1\), and \({\text{Re}} (b) = 5.85 \times 10^{ - 2}\).

Full size image

To quantify the evolution of the solitons, the average width in three directions is defined as

$$W_{j} (t) = \left( {\frac{1}{N}\iiint {j^{2} \left[ {\left| {\psi_{ + } (x,y,z;t)} \right|^{2} + \left| {\psi_{ - } (x,y,z;t)} \right|^{2} } \right]dxdydz}} \right)^{1/2} ,j = x,y,z.$$
(7)

Thus, the anisotropy of the solitons is defined as the width ratio \({{W_{x} }/{W_{y} }}\), and its area is \(S = \pi W_{x} \times W_{y}\). These characteristics of the SVs and MMs are shown, as a function of time, in Fig. 6. It is seen that the width ratio keeps values \({{W_{x} } /{W_{y} }} < 1\) and \({{W_{x} }/{W_{y} }} > 1\) for the SV and MM modes, respectively. The area in the \((x,y)\) plane increases monotonously, indicating gradual spread of the wave functions in the course of the propagation. Though the SVs and MMs are not completely stable, their instability growth rates may be small, allowing long survival times. Such unstable nonlinear states can be regarded as practically stable objects, taking into regard time limitations in experiments.

Figure 6
figure 6

The evolution of the asymmetry ratios and areas of the SV and MM solitons, defined as per Eq. (7), in the course of their propagation. (a) and (b) are width ratio \({{W_{x} } \mathord{\left/ {\vphantom {{W_{x} } {W_{y} }}} \right. \kern-0pt} {W_{y} }}\) and area \(\pi W_{x} W_{y}\), respectively. The parameters for stable SVs and MMs are \(\lambda = 1\), \(C_{11} = 200\), \(C_{12} = C_{21} = 0\), \(C_{22} = 0\), \(R_{c} = 1\).

Full size image

Conclusion

The Rydberg-dressed binary BEC with SOC is proposed here to produce quasi-stable 3D solitons. The three-level atomic scheme is constructed by coupling the two-level atomic structure to the excited Rydberg state. Three-dimensional Gross-Pitaevskii equations are introduced to govern the dynamics of the system. Solitons with SV and MM symmetries are obtained by tuning the system’s parameters, such as Rydberg interaction coefficients \(C_{ij}\), soft-core radius \(R_{C}\), and SOC strength \(\lambda\). These solitons are proven to be quasi-stable by means of the linearized analysis and direct simulations. The quasi-stability zones for the solitons of the SV and MM types are mainly determined by \(R_{C}\), and the dynamics can be effectively controlled by \(C_{ij}\) and \(\lambda\).

The work can be extended for the consideration of interactions between solitons. It may also be relevant to analyze the possibility of the existence of light bullets in the SOC-Rydberg medium.