Vortex solitons in fractional Schrödinger equation with cold Rydberg atoms and \mathcal{P}\mathcal{T}-symmetric potentials

Abstract

A two-dimensional (2D) fractional Rydberg atomic system with parity−time (\mathcal{P}\mathcal{T}) symmetric potentials is studied numerically in this work. We find a family of stable vortex solitons (VSs) with their topological structure and dynamical stability are analyzed in detailed. Control parameters are the Kerr nolinear coefficient, Lévy index, the Rydberg−Rydberg interaction coefficient, and the depth of the \mathcal{P}\mathcal{T} symmetry potential, which affect the distribution and stability range of the stable 2D VSs. The variation of Lévy index impacts the properties of nonlinear vortex states evidently, including the types, existence domain, power, and stability. Stability domains highlight rich topological phase structures, including fundamental vortices and multi-core configurations. These findings advance the understanding of soliton dynamics in non-Hermitian environments and offer insights for topological photonic applications.

Graphical abstract

Keywords

vortex soliton / fractional Schrödinger equation / Rydberg atomic system / ${\color{khaki}{ {\cal{PT}} }}$-symmetric potential

1 Introduction

The dynamics of generation, propagation, and interactions of vortex solitons (alas vortices or VSs) in nonlinear systems have garnered extensive attention across various branches of physics, including nonlinear optics [1-5], Bose−Einstein condensates (BECs) [6-9], quantum droplets [10-12], optical cavities [13], and electron beams [14]. Optical vortices, characterized by their nonzero angular momentum and nontrivial phase profiles around topological singularities [2], hold significant promise for applications in all-optical data processing, where they can act as conduits for weak optical signals [1]. Additionally, their unique properties enable applications such as optical tweezers for particle manipulation and rotation [15, 16], as well as advancements in optical trapping, microscopy, and quantum information technologies [17].

The pursuit of stable media for the propagation of VSs faces persistent challenges, primarily due to their tendency to collapse in Kerr-type focusing materials [18, 19]. To mitigate this instability, researchers have developed strategies to counterbalance the attractive nonlinearity inherent in such systems. A promising approach involves incorporating higher-order nonlinearities to offset the destabilizing effects of lower-order terms [20]. This method has successfully enabled the generation of stable VSs in media with quadratic [21, 22], quadratic−cubic [23-26], and cubic−quintic [27, 28] nonlinearities, as well as in nonlocal materials [29] and Rydberg electromagnetically induced transparency (Rydberg-EIT) systems [30-34]. Additionally, photonic lattices have been demonstrated to suppress azimuthal modulation instabilities, thus stabilizing VSs [35]. Recent advancements include experimental demonstrations [36-44], theoretical analyses [45-48] as well as deep leaning methods [49, 50] aimed at deepening the understanding of these stabilization mechanisms.

Over the past two decades, \mathcal{P}\mathcal{T}-symmetric [51-54] and space-fractional systems [55-57] have garnered significant interest for extending the boundaries of standard quantum mechanics. Optics has emerged as a versatile platform for experimentally realizing \mathcal{P}\mathcal{T}-symmetric [58] and space-fractional [59, 60] concepts. The dynamics of light propagation in \mathcal{P}\mathcal{T}-symmetric systems and nonlinear fractional Schrödinger equations (NFSEs) have been extensively investigated [61-64]. The significance of \mathcal{P}\mathcal{T} symmetry stems from its ability to yield entirely real eigenvalue spectra, provides the complex potential satisfies V(r)=V(-r) and the gain-loss component remains below a critical symmetry-breaking threshold [51]. In \mathcal{P}\mathcal{T}-symmetric linear optical lattices, stable multipole solitons have been observed in cubic-quintic media [65, 66] and hybrid linear−nonlinear lattices [67, 68]. Kartashov et al. demonstrated that the evolution of VSs in \mathcal{P}\mathcal{T} ratchet-like structures depends on the sign of their topological charge [69]. Notably, in ring-shaped \mathcal{P}\mathcal{T}-symmetric configurations, VSs with distinct topological charges exhibit stable propagation, even when the system exceeds the symmetry-breaking point [70]. To date, two-dimensional (2D) VSs have been explored in \mathcal{P}\mathcal{T}-symmetric azimuthal potentials [69, 71] and twisted circular waveguide arrays [72, 73], while stable three-dimensional (3D) vortex light bullets have been predicted in Rydberg atomic systems [74, 75] and Kerr nonlinear optical media [76].

The propagation of vortex beams in NFSEs has recently garnered increasing interest. In a real 2D optical lattice, stable off-site and on-site VSs have been observed in the NFSE [77, 78]. These VSs persist in the semi-infinite gap and remain stable within moderate power regimes. Notably, their stability region contracts as the Lévy index decreases. Additionally, stable in-phase solitons with four, six, and eight poles have been identified in the NFSE with a real optical lattice [79]. The propagation dynamics of nonlinear VSs under fractional diffraction have also been explored in NFSEs with azimuthal \mathcal{P}\mathcal{T}-symmetric potentials [80, 81].

Despite recent progress, the dynamics of VSs in NFSE, which are supported by the interplay of Kerr nonlinear and Rydberg−Rydberg interaction in the presence of the \mathcal{P}\mathcal{T}-symmetric potential, remain incompletely understood. This study aims to elucidate the unique characteristics of VSs in NFSEs under the combined influence of Kerr nonlinearity, Rydberg interactions, and \mathcal{P}\mathcal{T} symmetry. The paper is structured as follows: Section 2 presents the model and numerical simulation methods, Section 3 reports the main findings, and Section 4 concludes the work.

2 The model

In this study, we investigate the properties of VSs in the NFSE with a \mathcal{P}\mathcal{T}-symmetric lattice potential, incorporating both Kerr nonlinearity and Rydberg long-range interactions. A cold four-level atomic system with an inverted-Y type configuration is adopted in this work. The supposed diagram and energy levels are depicted in Fig.1(a) and (b), respectively. The black plate in Fig.1(a) serves as the reflecting mirror, and spatial light modulator (SLM) is responsible for dynamically modulating parameters of the optical field such as phase, amplitude, and polarization, enabling real-time processing of optical information and complex optical field manipulation. The probe (with half Rabi frequency {\mathrm{\Omega }}_p) and control (with half Rabi frequency {\mathrm{\Omega }}_c) laser fields are coupled to the transitions |1⟩ to |3⟩ and |2⟩ to |3⟩, respectively. {\mathrm{\Gamma }}_{ij} are spontaneous emission decay rates from states |j⟩ to |i⟩. The probe and control laser fields, states |1⟩, |2⟩, and |3⟩ constitute the electromagnetically induced transparency (EIT) which is dressed by a high-lying Rydberg state |4⟩ and the auxiliary field (with half Rabi frequency {\mathrm{\Omega }}_a). {\mathrm{\Delta }}_1, {\mathrm{\Delta }}_2, and {\mathrm{\Delta }}_3 are detunings between these laser fields and atomic states. An incoherent pumping with pumping rate {\mathrm{\Gamma }}_{21} is used to pump atoms from state |1⟩ to |2⟩, providing a gain for the probe field [31].

Under the slowly varying amplitude approximation, the model is described by the following dimensionless NFSE:

(1)\begin{array}{rl}\mathrm{i}\frac{\mathrm{\partial }\psi }{\mathrm{\partial }z}=& \frac{1}{2}{\left(-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}\right)}^{\alpha /2}\psi +V(x,y)\psi +\gamma |\psi {|}^2\psi \\ & +\sigma \int V_{\mathrm{v}\mathrm{d}\mathrm{w}}({\mathit{r}}^{\mathrm{\prime }}-\mathit{r}){\mathrm{d}}^2{\mathit{r}}^{\mathrm{\prime }}|\psi {|}^2\psi ,\end{array}

where \psi (x,y,z) is the complex wave function representing the optical field, \mathit{r} and {\mathit{r}}^{\mathrm{\prime }} are positions of two Rydberg atoms with r=\sqrt{x^2+y^2}, x and y are transverse coordinates, z is the propagation distance, and is the Lévy index characterizing the fractional Laplacian operator {\left(-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}\right)}^{\alpha /2} which accounts for the diffraction effects in the system. The coefficient \gamma is a constant characterizing the strength of the Kerr nonlinearity describing the local nonlinearity in the medium.

The potential V(x,y) is adopted as a \mathcal{P}\mathcal{T}-symmetric lattice potential given by V(x,y)=V_0\left[{\cos }^2(x)+{\cos }^2(y)\right]+\mathrm{i}{V}_1\left[\sin (2x)+\sin (2y)\right], which satisfies the relations that the real and imaginary parts are the even and odd functions of the coordinates. Here, V_0 and V_1 are real parameters representing the strengths of the gain/loss distribution, respectively. The \mathcal{P}\mathcal{T} symmetry of the potential ensures that the system can exhibit entirely real spectra under certain conditions, despite being non-Hermitian [51, 52].

The last term represents Rydberg interaction which is included to account for the long-range van der Waals potential

(2)V_{\mathrm{v}\mathrm{d}\mathrm{w}}({\mathit{r}}^{\mathrm{\prime }}-\mathit{r})={\displaystyle \frac{\hslash C_6}{\left[R_b^6+{\left|{\mathit{r}}^{\mathrm{\prime }}-\mathit{r}\right|}^6\right]}},

where the reduced Planck constant is taken as \hslash =1 in the numerical calculations, is the dispersion parameter, R_b={\left[\left|C_6{\mathrm{\Delta }}_3\right|/{\left|{\mathrm{\Omega }}_c\right|}^2\right]}^{1/6} is the Rydberg blockade radius [31]. Here, \sigma is a constant representing the nonlocal strength of the Rydberg interaction. This term is essential for describing the behavior of Rydberg atoms in optical lattices, where the interactions between Rydberg states can significantly influence the dynamics of the system.

To obtain the steady-state solutions of the system, we assume a solution of the form \psi (x,y,z)=\varphi (x,y){\mathrm{e}}^{\mathrm{i}bz+\mathrm{i}m\theta }, where \varphi (x,y) is the spatial profile of the soliton whose trial solution is adopted as the Gaussian type, b is the propagation constant, m and \theta are topological charge and the corresponding azimuthal angle, respectively. Substituting this form into the NFSE (1), the stationary solution for \varphi (x,y) can be expressed as follow:

(3)\begin{array}{rl}& \frac{1}{2}{\left(-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{y}^2}\right)}^{\alpha /2}\varphi +V(x,y)\varphi +\gamma |\varphi {|}^2\varphi \\ & +\sigma \int \frac{C_6}{\left[R_b^6+{\left|{\mathit{r}}^{\mathrm{\prime }}-\mathit{r}\right|}^6\right]}{\mathrm{d}}^2{\mathit{r}}^{\mathrm{\prime }}|\varphi {|}^2\varphi =-b\varphi .\end{array}

To solve this stationary Eq. (3), we employ the modified squared-operator iteration method (MSOM), which is an iterative numerical technique designed to find solitary wave solutions [82]. In this study, the power of VSs is defined as U=\iint |\psi (x,y){|}^2\mathrm{d}x\mathrm{d}y. The stability of VS is analyzed by performing direct numerical simulation of NFSE (1) with the initial condition set to the stationary soliton solution plus a small random noise. To further evaluate the stability of VSs during propagation, we introduce the fidelity

(4)J={\displaystyle \frac{{\left|{\displaystyle \iint \psi (x,y,z)\psi (x,y,z=0)\mathrm{d}x\mathrm{d}y}\right|}^2}{{\displaystyle \iint |\psi (x,y,z){|}^2\mathrm{d}x\mathrm{d}y{\displaystyle \iint |\psi (x,y,z=0){|}^2\mathrm{d}x\mathrm{d}y}}}},

where J\to 1 represents a high-fidelity along propagation evolution. Stable VS is defined as the one remains fidelity at or above 95% after propagating a distance of z=3000.

3 Results and discussion

In this study, we systematically investigate the dynamical properties of VSs in a NFSE framework, incorporating a \mathcal{P}\mathcal{T}-symmetric lattice potential with combined Kerr nonlinearity and Rydberg long-range interactions. The intensity and phase distributions of stable VSs are analyzed across a range of parameters: and various topological charges, as illustrated in Fig.2. A typical ring-shape VSs with topological charges up to m=3 are uncovered in this system. The radius of VS ring is defined as

(5)R={\displaystyle \frac{{\displaystyle \iint \sqrt{x^2+y^2}|\psi {|}^2\mathrm{d}x\mathrm{d}y}}{{\displaystyle \iint |\psi {|}^2\mathrm{d}x\mathrm{d}y}}}.

It is found that VSs would expand with increasing Lévy index \alpha as well as topological charge m, as shown in Fig.3(a). This behavior arises from the enhanced diffraction-like spreading induced by \alpha values and the centrifugal energy associated with higher m. Notably, the phase distributions of these VSs exhibit a central singularity, characteristic of vortex beams, with an m-dependent helical modulation. For m=1, a single 2\pi phase winding is observed, while higher charges (m=2,3) manifest as 4\pi and 6\pi windings, respectively, confirming the topological charge conservation.

To systematically elucidate the formation mechanisms and modulation characteristics of VSs in NFSE with \mathcal{P}\mathcal{T}-symmetric lattice potentials, we conduct a comprehensive parametric analysis including Lévy index \alpha, propagation constant b, local and nonlocal nonlinearities (\gamma and \sigma), and depth of \mathcal{P}\mathcal{T} potential (V_0 and V_1), as shown in Fig.3(b)−(f). The U-b relation in Fig.3(b) demonstrates a monotonic decrease in the power of VSs U with increasing b. Notably, the stable and unstable states of VSs are represented by solid and dashed lines, respectively. Here unstable solutions undergo rapid collapse during propagation. The modulation of power based on local and nonlocal nonlinearities show different features: the power U increases monotonously with \gamma while it keeps almost unchanged with changing \sigma, as shown in Fig.3(c) and (d), respectively. And the local nonlinearity coefficient \gamma can only take negative values while the nonlocal nonlinearity coefficient \sigma is positive. The findings suggest that VSs within this system exhibit insensitivity towards nonlocal Rydberg−Rydberg long-range interactions which primarily function to counterbalance the attractive and repulsive forces during the formation VSs. Fig.3(e) and (f) display the dependence of U with respect to V_0 and V_1, respectively. The results suggest limited tunability through real and imaginary potential modifications. The increasing of topological charge m systematically reduces the viable parameter space for stable VSs. This topological charge-dependent parameter restriction mechanism suggests fundamental limitations in high-m VSs generation, potentially arising from azimuthal instability.

The stability of these VSs is further evaluated by their persistent ring-shaped intensity profiles and coherent phase structures across the propagation distances z, as shown in Fig.4. The Lévy index is fixed as \alpha =1.3, and the topological charges are taken as m=1,2, and 3 for comparison. The results reveal that these VSs maintain their vortex configuration remarkably well during propagation, exhibiting robust stability. Notably, VSs with lower topological charges (e.g., m=1 and m=2) demonstrate exceptional stability, preserving their structural integrity and phase coherence throughout the propagation distance z>3000. VSs with higher topological charges (m=3) still propagate steadily without much distortion in the intensity rings and subtle phase irregularities. This suggests that while topological charge influences stability, the Lévy-index-mediated propagation framework effectively sustains stability of VSs. The fidelity of VSs during propagation remains above 95% (Fig.5), confirming their high stability.

Upon the removal of the local and nonlocal nonlinearities from Eq. (1), this equation becomes a linear equation with various eigenvalues and associated linear eigenmodes. Nonlinear modes can bifurcate from linear modes in the nonlinear case [83]. Thus, it is possible to gain a deeper understanding of VSs from eigenvalue spectrum of the linear system which is shown in Fig.6(a) and (b). Here, the real and imaginary parts of b are displayed in Fig.6(a) and (b), respectively. It is found that when the depth of the external potential V_1\le 0.036, the imaginary part of the propagation constant b_{\mathrm{i}\mathrm{m}}=0, i.e., b is purely real. Under this condition, the non-Hermitian system is \mathcal{P}\mathcal{T}-symmetric. However, when V_1>0.036, the eigenvalues b with topological charges of \pm 1 and \pm 2 are no longer purely real, indicating that the system transitions to a \mathcal{P}\mathcal{T}-symmetry broken state. The critical points are referred to as \mathcal{P}\mathcal{T}-symmetry breaking points, marked as points \mathrm{A} and {\mathrm{A}}^{\prime } in Fig.6(a) and (b). Further investigation reveals that the value V_1 of the \mathcal{P}\mathcal{T}-symmetry broken point would vary with changing V_0 in the nonlinear condition, as shown in Fig.6(c). The red and light blue regions in the figure represent the \mathcal{P}\mathcal{T}-broken and \mathcal{P}\mathcal{T}-symmetric states, respectively.

Based on the stability definition of VSs (profile deformation <5% over 3000 propagation units) above, the stability domains (b,\gamma ) and (b,V_1) planes are displayed in Fig.7, where the green and white areas represent stable and unstable VSs, respectively. The stability domain in (b-\gamma ) plane shows that the value of \gamma can only take negative values with the range of b firstly decrease and then increase slightly. In contrast, the stability domain in (b-V_1) plane exhibits strong V_1-dependence: maximal b-range occurs near V_1\to 0, which progressively narrows at higher V_1 values. Crucially, cooperative parameter tuning enables stable VSs even at relatively bigger V_1 strengths, though with reduced b-ranges.

To elucidate topological structure of VSs in non-Hermitian fractional systems, we analyze the high-resolution mapping of the stability domain in (b,\alpha ) plane, as shown in Fig.8(a). It is interesting that six distinct topological structures, ranging from fundamental vortices to multi-core configurations, are uncovered in Fig.8(b1)−(b6). Lower Lévy indices (\alpha) enable complex multi-core structures [Fig.8(b1)−(b5)] through enhanced fractional diffraction, while higher \alpha values stabilize simpler VSs [Fig.8(b5)−(b6)] via conventional diffraction. The stability of these six types VSs are verified by propagation evolution, as shown in Fig.9. All configurations exhibit stability, with deformation remaining below 5% during propagation. This stability is attained through the balanced modulation of \mathcal{P}\mathcal{T}-potential and the interplay of local/nonlocal nonlinearities. The different topological structure of VSs can be modulated by tuning Lévy index and propagation constant, and the phase structure is not changed during the modulation. This results demonstrates precise control of photonic orbital angular momentum through fractional dimension engineering, enabling tailored topological states in \mathcal{P}\mathcal{T}-symmetric potentials.

4 Conclusion

In this study, we have comprehensively explored the properties and stability dynamics of VSs within the framework of NFSE incorporating a \mathcal{P}\mathcal{T}-symmetric lattice potential, Kerr nonlinearity, and Rydberg long-range interactions. By leveraging the squared-operator iteration method under the slowly varying amplitude approximation, we derived solitary wave solutions and validated their stability through numerical simulations using the step-split Fourier method. Our findings demonstrate the emergence of stable, ring-shaped VSs with topology charge up to 3. This system exhibits \mathcal{P}\mathcal{T}-symmetric behavior under specific parameter regimes, wherein the propagation constant remains purely real. The variation of Lévy index impacts the properties of nonlinear VSs evidently, including the types, the existence domain, power, and stability. Stability domains under control parameters are generated, revealing rich topological phase transitions across the parameter space. Notably, multiple topological phases are identified, ranging from fundamental vortices to complex multi-core configurations, underscoring the topological diversity inherent in non-Hermitian fractional systems. There are primarily two significant effects associated with the introduction of the fractional Schrödinger equation incorporating the Lévy index. Firstly, the fractional Laplacian leads to a reduction in the radius of VSs. Secondly, it gives rise to diverse VS types. These findings contribute to a deeper understanding of soliton dynamics in non-Hermitian environments and provide a theoretical foundation for exploring topological structures with potential applications in advanced photonic devices. Future investigations could focus on extending this framework to include higher-order effects or alternative lattice symmetries, further expanding the topological landscape of such systems.

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