Three-dimensional solitons in Rydberg-dressed cold atomic gases with spin–orbit coupling

We present numerical results for three-dimensional (3D) solitons with symmetries of the semi-vortex (SV) and mixed-mode (MM) types, which can be created in spinor Bose–Einstein condensates of Rydberg atoms under the action of the spin–orbit coupling (SOC). By means of systematic numerical computations, we demonstrate that the interplay of SOC and long-range spherically symmetric Rydberg interactions stabilize the 3D solitons, improving their resistance to collapse. We find how the stability range depends on the strengths of the SOC and Rydberg interactions and the soft-core atomic radius.

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 condensate [5,[30][31][32].SOC notably modifies the dispersion of the system [33,34], breaks the Galilean invariance [35], and thus substantially impacts the properties of solitons in the free space [36,37].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 potential [38][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 fields [43] and gauge potentials [44].
Theoretical [45] and experimental [46] 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 level [47].This option provides an important platform for the study of optical soliton dynamics with tunable parameters [48,49].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 nonlinearity [50][51][52][53][54]. Rydberg atomic gases are controllable in an active way through tunable parameters [55,56], 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 robust [57].Thus, Rydberg-EIT settings provide a fertile ground for realizing quantum nonlinear optics [48] and developing new photon devices, such as single-photon switches and transistors [58,59], quantum memories, and phase gates [56,60].
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. 1(a).We consider the ultra-cold atomic gas of N where 2 2 2 2 x y z        , 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 SOC V   p σ [62], where i    p is the momentum operator and ( , , ) are Pauli matrices.The potential represent the interaction between remote Rydberg atoms with 66 () is the van der Waals potential, where c R is the soft-core radius, and where 6 ij C () are dispersion parameters which determine the intra-and inter-species couplings in the two-component BEC system [15,16].Here     [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 for the SV (semi-vortex), and * cos sin for the MM (mixed mode), where * stands for the complex conjugate.
The energy corresponding to Eqs. (1a) and (1b) is where SOC operators, with their respective symmetry structures, are ˆx To explore the stability of solitons, perturbed solutions are taken as where 12 g ， and 12 h ， are amplitudes of small perturbations, (1) with respect to the small perturbations produces the system of the respective Bogoliubovde Gennes equations:

Results
In the present system, the total norm Stationary solutions for the wave function seeded by the initial guesses (2) were obtained using the squared-operator method [67] and also by the imaginary-time evolution method [41].As a result, two-component solitons are produced,

   
SV= SV+,SV-, and MM= MM+,MM-.Examples of density isosurfaces and phase structure of these components of stable solitons are plotted in Figs.1(b,c,d,e) and 1(f,g,h,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 type [21].Stable SV solitons exist in an interval of the total norm min max Note that the increase of the Rydberg radius leads to a reduction of the stability zone of SVs, which is also observed in Fig. 2(a,b,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 ring [68].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   .It is noticed that SV solitons cannot be generated at 0.6   .

Families of the SV solitons
The stability of the solitons may be also evaluated using the "anti-Vakhitov-Kolokolov" (anti-VK) criterion, d d 0 N   , which is a necessary but not sufficient condition for the stability of solitons supported by repulsive (defocusing) nonlinearities [69].To this end, the chemical potential  is shown, as a function of N , in Figs.3(d,e,f) for different ij C , and in Fig. 3(h) for different  , 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 Eqs.
(5).The real part of the eigenvalues is shown in Fig. 3(i).One observes that Re( ) b is close to zero(

Solitons of MM type
The dynamics of MMs is nearly the same as of SVs. .Thus, the quasi-stable and strongly unstable states are produced by the analysis.To quantify the evolution of the solitons, the average width in three directions is defined as

Conclusion
The Rydberg-dressed binary BEC with SOC (spin-orbit coupling) 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 (semi-vortex) and MM (mixed-mode) symmetries are obtained by tuning the system's parameters, such as Rydberg interaction coefficients

11 C , 12 21 CC  and 22 C
frequencies of two laser fields,1  and 2  being the respective detunings.Note that, for a small Rydberg radius ( may be treated as a spherically symmetric s-wave scattering pseudopotential.In this work, by tuning system parameters, the values of ij C , viz., , range from -1000 to 1000.Our main focus herein is on discussing the effects of RRI and SOC on the formation of solitons.For 11  , the system reduces to an effective two-level atom, with states 0 coupled by a two-photon Rabi frequency  and detuning  .For experimental considerations, a suitable candidate for the realization of the setup is the gas of 87 Rb atoms with

N
is the scaled norm of the wave function[64][65][66].The scaled length and time units are 10 μm and 100 ms , respectively.For example, ( , )(5,5) xy  corresponds to the spatial domain of size (50 μm, 50 μm) , and 10 t  corresponds to 1000ms .A typical distance between two Rydberg atoms is of the order 10μm ij r  , while the unit for softcore size is estimated as 5.6μm  [66].

Fig. 1 (
Fig. 1 (a) The schematic of the Rydberg-dressed three-level atomic system, with the laser coupling

E
represent the kinetic energy, inter-and intra-species interaction, RRI, and SOC interaction, respectively.Below, the RRI parameters ij C , radius c R and SOC strength  are varied to study the soliton dynamics.
rate, and k is the integer azimuthal index,  is the azimuthal angle.The ansatz for SVs permit both non-vortex and vortex modes, perturbation includes the azimuthal angle  .The soliton solutions may be stable if0 ) ( Re  bfor all eigenvalues.The linearization of Eqs.
determine the respective norm shares, F N N   .

Figure 2
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, ( , 1,2) ij C i j  , and the softcore

Fig. 2 .. 11 C
Fig. 2. The modulation of solitons of the SV type by altering the Rydberg coefficients, ij C .The

5 10 
) in broad intervals, where the SVs are quasi-stable states.The stability of solitons of the MMs type is similarly determined by the eigenvalues.

Fig. 3 . 1  5 N 5 Re
Fig. 3.Chemical potential  for SVs with different values of the system's parameters, including

Fig. 4 . 5 Re
Fig. 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. 2(a)) with

Figure 5 5 Re
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 byRe( )  b .The solitons displayed in the first (MM+) and second (MM-) columns of Fig.5are identified as quasi-stable ones with

Fig. 5 . 1 
Fig. 5. Isosurfaces of the MM+ and MM-components, as produced by the direct simulations.The

and 1 xyWW
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 1 xy WW for the SV and MM modes, respectively.The area in the ( , ) xy plane increases monotonously, indicating gradual spread of the wave functions in the course of the propagation.Though the SVs and MMs are 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.

Fig. 6
Fig.6The evolution of the asymmetry ratios and areas of the SV and MM solitons, defined as per

ijC
, soft-core radius C R , and SOC strength  .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 C R , and the dynamics can be effectively controlled by ij C and  .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.
Thus, the anisotropy of the solitons is defined as the width ratio xy WW , and its area is