1. Introduction

High-dimensional spatiotemporal solitons (STSs) are solitary nonlinear waves localized in two-spatial dimensions and one-time axis. STSs have attracted broad interest in many fields, such as Bose-Einstein condensates (BECs), plasma physics, and nonlinear optics [1–3]. In nonlinear optics, spatiotemporal optical solitons (SOSs) generally suffer severe instability as Kerr nonlinearity leads to supercritical collapse [4,5]. It has become the goal of many scientific researchers to obtain stable SOSs in the past decades [6,7].

The first step is the identification of proper optical media with tunable dispersion, diffraction, and nonlinearities that can support the generation of stable SOSs. It is encouraging that a number of such setups have been proposed, such as materials with special nonlinear interactions [8], saturable nonlinearities [9], quadratic nonlinearities [10,11], competing [12,13] or nonlocal [14–16] nonlinearities, waveguide arrays [17,18], spin-orbit coupling systems [19,20], PT-symmetric optical lattices [21–24], especially electromagnetically induced transparency (i.e., a typical quantum interference effect occurring in resonant multilevel atomic systems) in cold Rydberg atomic gases [25–27].

Recently, stable local weak-light solitons are found in systems with optical parity-time potentials [28–30]. These systems are constructed by the coupling of several laser fields and atomic gases. Among these works, Qin et al. [30] realized PT linear and nonlinear optical potentials in the atomic gas system and found stable optical solitons. It is great interesting that a combined one-dimensional (1D) linear and nonlinear PT-symmetric potential can be constructed through the spatial modulation of the control laser and the inclusion of the Kerr nonlinearity of the signal laser.

The introduction of PT-symmetric optical lattices and EIT in cold gases has significantly promoted the stability of SOSs in nonlinear optical systems [22]. It is an open question how to obtain 3D vector SOSs in a cold atomic system with PT-symmetric linear and nonlinear potentials. It will be interesting if one can obtain realistic optical systems that not only possess combined PT-symmetric linear and nonlinear potential, but also support weak-light and slow velocity 3D vector SOSs. Subsequently, we will present a systematic study to address this question.

In this paper, we propose a realistic scheme for physically realizing combined linear and nonlinear optical potentials with PT symmetry by using a coherent atomic gas. We shall show that the space-dependent imaginary part of the nonlinear PT potential plays a key role for the occurrence of the PT phase transition, which can be actively manipulated. We also investigate the existence and stability of these 3D vector SOSs supported by the linear and nonlinear PT potentials.

Before proceeding, we note that this work is different from that in Qin et al. [30]. First, the systems studied are different. In Qin et al. [30], the system is built with N-type four-level atomic gases which interact with three laser fields, a weak signal laser field, a strong control laser field and a strong pump-laser field. In our paper, we consider a cold, lifetime-broadened atomic gas with a tripod-type level configuration, interacting resonantly with two pulsed probe laser fields and a strong continuous-wave control laser field. This construction effectively allows us to create two EIT structures in the absorption spectrum of the system. Second, the potentials are different. In Qin et al. [30], the PT-symmetric potentials are produced in 1D space. In our paper, we construct of the 2D combined linear and nonlinear PT-symmetric potentials, especially, the system contains cross nonlinear potentials. Third, in Qin et al. [30], the system supports 1D stable optical solitons. In our paper, 3D vector SSs and VSs have been found and proven to be stable with certain parameters. Finally, in our paper, the frontal head-on collisions of two vector solitons are displayed, and the stability of these solitons is also observed after the collisions.

The paper is organized as follows. In Sec. II, we describe the tripod-type level configuration with PT-symmetric optical linear and nonlinear potentials. In Sec. 3, we display the PT phase transitions and the active controlled solitons. In Sec. 4, we furnish a summary of the results obtained in this work.

2. Theoretical model

We consider a cold, lifetime-broadened atomic gas with a tripod-type level configuration, interacting resonantly with two probe laser fields (with half-Rabi frequency ${\kern 1pt} {\Omega _{{p_1}}}$ and ${\kern 1pt} {\Omega _{{p_2}}}$, respectively), and a linear-polarized, strong continuous-wave control laser field (with half-Rabi frequency ${\kern 1pt} {\Omega _c}$) in Fig. 1(a). Two weak probe fields, ${\kern 1pt} {E_{p1}}$ and ${\kern 1pt} {E_p}_2$, couple to the transition between state ${\kern 1pt} |1 \rangle$ (or ${\kern 1pt} |2 \rangle$) and state ${\kern 1pt} |4 \rangle$, respectively. The strong control field ${\kern 1pt} {E_\textrm{c}}$ couples to the transition between ${\kern 1pt} |3 \rangle$ and ${\kern 1pt} |4 \rangle$. The electric field vector of the system is ${\kern 1pt} {\mathbf E}({\mathbf r},t) = \sum\nolimits_{l = {p_{1(2)}},c} {{{\mathbf e}_l}{\varepsilon _l}\exp [i({{\mathbf k}_l} \cdot {\mathbf r} - {\omega _l}t)]} + \textrm{c}\textrm{.c}.$, where ${\kern 1pt} {{\mathbf e}_l}({{\mathbf k}_l})$ is the unit polarization vector of the electric field component with envelope ${\kern 1pt} {\varepsilon _l}$.

 

Fig. 1. (a) Tripod-type atomic level diagram and excitation scheme. ${\Omega _{{p_{1(2)}}}}$ and ${\Omega _c}$ are half Rabi frequencies of the probe and control fields, respectively. ${\Gamma _{jl}}$ is the spontaneous emission decay rate from ${\kern 1pt} |l \rangle$ to ${\kern 1pt} |j \rangle$. (b) Geometry of the system. The probe and control fields propagate along the ${\kern 1pt} z$ direction.

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Under electric-dipole and rotating-wave approximations in the interaction picture, the Hamiltonian of this optically pumped EIT system is given by ${\hat{H}_{{\mathop{\rm int}} }} ={-} \hbar \sum\limits_{j = 1}^4 {{\Delta _j}|j \rangle \left\langle j \right|- \hbar \left[ {{\Omega _{{p_1}}}|4 \rangle \left\langle 1 \right|+ {\Omega _{{p_2}}}|4 \rangle \left\langle 2 \right|) + {\Omega _c}|3 \rangle \left\langle 4 \right|+ \textrm{H}\textrm{.c}\textrm{.}} \right]}$. Here ${\Omega _{{p_{1(2)}}}} = ({{e_p} \cdot {p_{\textrm{1(2)4}}}} ){\varepsilon _p}/\hbar$, ${\Omega _c} = ({{\textrm{e}_\textrm{c}} \cdot {p_{34}}} ){\varepsilon _c}/\hbar$, ${\Delta _2} = ({{E_2} - {E_1}} )/\hbar$, ${\Delta _3} = {\omega _p} - {\omega _c} - ({{E_3} - {E_1}} )/\hbar$ and ${\Delta _4} = {\omega _p} - ({{E_4} - {E_1}} )/\hbar$, ${E_j}$ is the energy eigenvalue of level ${\kern 1pt} |j \rangle$, ${p_{jl}}$ is the electric dipole matrix elements associated with the transition ${\kern 1pt} |j \rangle \leftrightarrow |l \rangle$. The dynamics of atoms is governed by the optical Bloch equation [30]

When the probe fields are propagating in the medium, the spatial variation is relatively slow. The amplitude of the light field varies with time is much less than that of the optical frequency, and the amplitude of the light field varies with the space is much less than that of the wave vector ${k_{{p_{1(2)}}}}$, shown as ${{\partial {\Omega _{{p_{1(2)}}}}} / \partial }t \ll{-} i{\omega _{{p_{1(2)}}}}{\Omega _{{p_{1(2)}}}}$, ${{\partial {\Omega _{{p_{1(2)}}}}} / {\partial x(y,z)}} \ll i{k_{{p_{1(2)}}}}{\Omega _{{p_{1(2)}}}}$. In this way, we can ignore the higher-order derivatives of the equations. Under the paraxial and slowly varying envelope approximation, we obtain the equations for ${\kern 1pt} {\Omega _{{p_1}}}$ and ${\kern 1pt} {\Omega _{{p_2}}}$ [31]

Being the weak probe fields, we can take ${\kern 1pt} {\Omega _{{p_{1(2)}}}}$ as small parameters to solve the Maxwell-Bloch Eqs. (1) and (2) by using the perturbation expansion [32,33]. Here, we assume ${\Omega _{{p_{1(2)}}}} = \varepsilon \Omega _{{p_{1(2)}}}^{(1)} + {\varepsilon ^2}\Omega _{{p_{1(2)}}}^{(2)} + {\varepsilon ^3}\Omega _{{p_{1(2)}}}^{(3)} + \cdots ,$ and ${\rho _{jl}} = \rho _{jl}^{(0)} + \varepsilon \rho _{jl}^{(1)} + {\varepsilon ^2}\rho _{jl}^{(2)} + \cdots ,$ where $\varepsilon$ is a small parameter characterizing the size of the probe pulse, and ${\rho _{jl}}$ are the density-matrix elements. To obtain a divergence-free expansion, all quantities on the right hand side of the expansion are considered as functions of the multiscale variables ${z_l} = {\varepsilon ^l}z(l = 0,1,2)$, $({x_1},{y_1}) = \varepsilon (x,y)$ and ${t_l} = {\varepsilon ^l}t(l = 0,1)$. Substituting the above expansions into Eqs. (1) and (2), and comparing the coefficients of ${\varepsilon ^l}(l = 1,2,\ldots )$, we obtain a set of linear but inhomogeneous equations which can be solved order by order [30].

At zero order, we obtain $\rho _{11}^{(0)} = \frac{{{J_{12}}{G_2} - {J_{22}}{G_1}}}{{{J_{12}}{J_{21}} - {J_{11}}{J_{22}}}}$, $\rho _{22}^{(0)} = \frac{{{J_{21}}{G_1} - {J_{11}}{G_2}}}{{{J_{12}}{J_{21}} - {J_{11}}{J_{22}}}}$, $\rho _{44}^{(0)} = \frac{{{X_1} - i{\Gamma _{31}} - {X_1}\rho _{11}^{(0)} - {X_2}\rho _{22}^{(0)}}}{{{X_4}}}$, $\rho _{43}^{(0)} = \frac{{ - {\Omega _C}(1 - \rho _{11}^{(0)} - \rho _{22}^{(0)} - 2\rho _{44}^{(0)})}}{{(\omega + {d_{43}})}}$, and $\rho _{33}^{(0)} = 1 - \rho _{11}^{(0)} - \rho _{22}^{(0)} - \rho _{44}^{(0)}$.

At the first order, we obtain the solution $\Omega _{{p_1}}^{(1)} = {F_1}\exp [i({K_1}{z_0} - \omega {t_0})]$, $\Omega _{{p_2}}^{(1)} = {F_2}\exp [i({K_2}{z_0} - \omega {t_0})]$, where ${K_{1(2)}}(\omega )$ is the linear dispersion relation, ${F_{1(2)}}$ is an envelope function of the slow variables ${x_1}$, ${y_1}$, ${z_1}$, ${t_1}$, and ${z_2}$, to be determined yet. At the second order, we obtain the equation $i(\frac{{\partial {F_l}}}{{\partial {z_l}}} + \frac{1}{{{V_{gl}}}}\frac{{\partial {F_l}}}{{\partial {t_l}}}) = 0(l = 1,2)$, with ${V_{gl}} = {(\partial {K_l}/\partial \omega )^{ - 1}}$ being the group velocity of the probe fields.

With the above results, we proceed to the third order solutions. The divergence-free condition in this order yields the nonlinear envelope equations for ${F_1}$ and ${F_2}$.

In the paper, we suppose the control field consists of two identical Gaussian beams with the form ${\Omega _c}({x,y} )= {\Omega _{{c_0}}}\left[ {{e^{ - \frac{{{{({x + {x_0}} )}^2} + {{({y + {y_0}} )}^2}}}{{2{\sigma^2}}}}} + {e^{ - \frac{{{{({x - {x_0}} )}^2} + {{({y - {y_0}} )}^2}}}{{2{\sigma^2}}}}}} \right]$, where $\sigma$ is the width of each beam. The linear and nonlinear optical potentials ($V$ and W, respectively) are [30]:

 

Fig. 2. Spatial profiles of linear PT potential V and nonlinear PT potential ${W_{ij}}$. (a) and (b) the real and (c) and (d) the imaginary parts of the linear and nonlinear potentials. Parameters: ${v_0} = 2$, ${v_1} = 0.1$, ${w_0} = 0.8$, ${w_1} = 0.1$, $d = 9.5$ and $a = \textrm{4}\textrm{.2}$.

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Before presenting the vector spatiotemporal solitons, we make an estimate of the numerical values of the coefficients in our system. We choose the D2 line of $^{87}\textrm{Rb}$ atomic gas [35] with energy levels and coefficients selected as ${\tau _0} = 1.1 \times {10^{ - 7}}\textrm{s}$, ${\kappa _{14}} \approx {\kappa _{24}} \approx 1.2 \times {10^{10}}\textrm{c}{\textrm{m}^{ - 1}}{\textrm{s}^{ - \textrm{1}}}$, ${\Delta _4} = 5.0 \times {10^8}\textrm{Hz}$, ${\Delta _3} = 2.0 \times {10^6}\textrm{Hz}$, ${\Delta _2} = 1.5 \times {10^4}\textrm{Hz}$, ${\Gamma _{31}} \approx {\Gamma _{32}} = 100\textrm{Hz}$, ${\Gamma _{13}} \approx {\Gamma _{23}} = {\Gamma _3}/2$, ${\Gamma _{14}} \approx {\Gamma _{24}} \approx {\Gamma _{34}} = {\Gamma _4}/3$, and ${\Omega _0} = 2.6 \times {10^8}\textrm{Hz}$.

3. PT phase transitions and their active controlled solitons

Now, we consider the property of the PT phase transitions and the vector solitons in optically pumped EIT system with linear and nonlinear PT potentials. Because the system can be actively controlled, the PT phase transitions and the propagation of vector solitons may be manipulated by tuning the system parameters.

To numerically obtain the nonlinear modes, we suppose the field amplitude is of the form ${\psi _{1(2)}} = {\psi _{1(2)}}{e ^{ibs}}$, with the corresponding soliton energies ${U_{1(2)}} = \int\!\!\!\int {{{|{{\psi_{1(2)}}} |}^2}d\xi d\eta }$ and total energy $U = {U_1} + {U_2}$, where $b$ is the propagation coefficient. Using the squared-operator [20,36], the eigenvalue problem of Eq. (4) can be solved numerically. In order to check the stability of these solitons, we perform a linear stability analysis [20]. Perturbed solutions are written in the form ${\psi _{1(2)}} = [{\psi _{01(2)}} + ({u_{1(2)}} + {v_{1(2)}}){e^{\lambda s}} + (u_{1(2)}^\ast{-} v_{1(2)}^\ast ){e^{{\lambda ^\ast }s}}]{e^{ibs}}$, here ${\psi _{0(1,2)}}$ is the stationary soliton solution of Eq. (4), ${u_{1(2)}},{v_{1(2)}} < < {\psi _{01(2)}}$ are complex functions describing the perturbation profiles, and $\lambda$ is the perturbation growth rate. Substituting them into Eq. (4) and linearizing this equation, we get the linear eigenvalue. If $\textrm{Re}(\lambda ) = 0$, the soliton solutions can be stable; otherwise, the solitons would become linearly unstable.

PT phase transitions in our optically pumped EIT system can be manipulated by adjusting four parameters ${v_1}$, ${w_1}$, a, and d in the combined linear and nonlinear potentials, this manipulations are displayed in Fig. 3. Figure 3(a) shows the phase diagram of the system as a function of ${v_1}$ and ${w_1}$ with $a = 4.2$ and $d = 9.5$. One can see that the blue domain is the phase with PT symmetry and the blank domain being the phase with PT broken symmetry. The solid border of the blue and blank domains represents the PT phase transition, where the eigenvalue translates from real to complex when the imaginary amplitudes of the linear and nonlinear potentials are changed. Figure 3(b) shows the phase boundary lines of the PT phase transition by adjusting the width a. It is shown that the domain of the PT-symmetry phase increases slightly as a decreases. On the contrary, the domain of the PT-symmetry phase increases with d in Fig. 3(c).

 

Fig. 3. PT phase diagrams and their active manipulation. (a) The PT phase diagram of the system as a function of ${v_1}$ and ${w_1}$, with $a = 4.2$, $d = 9.5$, ${w_0} = 0.8$, and ${v_0} = 2$. The left-bottom (blue) domain is the phase with PT symmetry, while the upper-right (blank) domain is the phase with PT symmetry breaking. (b) PT phase diagram with $a = 3.8$ (the boundary of PT phase transition is represented by a black dotted line) and $a = 4.2$ (the boundary of PT phase transition is represented by a red solid line). (c) The PT phase diagram with $d = 9.5$ (the boundary of the PT phase transition is represented by a red dotted line and a blue area) and $d = 10$ (the boundary transition of the PT phase is represented by a black solid line and the green area).

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Owing to the double EIT structure, two vector solitons can be obtained simultaneously. The simulations also indicate that under certain parameters, 3D vector solitons with energy ${U_1}$ and ${U_2}$ can be detected in this system. Based on the parameters given above, one finds the propagating velocity ${V_{{g_{1(2)}}}} \approx 2.36 \times {10^{ - 6}}c$ and the maximum average power density by using Poynting vector [5], which is estimated to be ${P_{1(2)}} = 1.6\textrm{nw}$ (seeing point A in Fig. 4). Thus, this optical pulses move very slowly and possess very small power. Figures 4(a)–4(c) show the energy of one single soliton ${U_1}$ with respect to parameter b. When changing the potential parameters, the energies of the solitons monotonically increase with b. It is evident that the positive slope of the dependence ${{dU} / {db > 0}}$ can be observed in the focusing media. This satisfies the Vakhitov-Kolokolov (VK) criterion, thus the solitons here are stable [see points “A” in Figs. 4(a)–4(c)]. Furthermore, the actual stability of numerical solutions i4s established by the linear stability analysis [see Figs. 5(d) and 5(h)]. It is also shown that the dynamics characteristics of vector solitons depend on six parameters (i.e., ${v_0}$, ${w_0}$, ${v_1}$, ${w_1}$, a, and $d$). For different values of these parameters, the changes of energies are shown in Figs. 4(d)-(f). In Fig. 4(d), one can see that ${U_1}$ decreases rapidly with the increase of a and then gradually increases to the maximum, nevertheless, ${U_2}$ shows the opposite properties. However, the total energy U of the two vector solitons keeps the same. Furthermore, ${U_2}$ decreases rapidly to near zero and ${U_1}$ increases to maximum when d increases. From Fig. 4(e), one can see that ${U_1}$ decreases rapidly to near zero as the real (${v_0}$) or imaginary (${v_1}$) parts of the linear potential increase. On the contrary, the energy ${U_2}$ of the other vector soliton increases rapidly to maximum. The modulations of energy ${U_{1(2)}}$ by tuning the nonlinear potential parameters ${w_0}$ and ${w_1}$ are different with the linear case [see Figs. 4(e) and 4(f)]. As shown in Fig. 4(f), ${U_{1(2)}}$ are basically unchanged with the real part of nonlinear potential parameter ${w_0}$. However, ${U_1}$ (${U_2}$) increases (decreases) monotonically with the imaginary part ${w_1}$. The possibility of the transition between the two probe fields can be achieved by changing the parameters, i.e., ${v_0}$, a, ${v_1}$, ${w_1}$ and d. This result can be applied in optical diodes and switches.

 

Fig. 4. (a)-(c) ${U_1}$ versus b with parameters: (a) ${v_0} = {w_0} = 0.8$, ${v_1} = 0.1$, ${w_1} = 1$. (b) ${w_0} = 0.8$, ${w_1} = 1$, $d = 9$, $a = 4.2$. (c) ${v_0} = 0.8$, ${v_1} = 0.1$, $d = 9$, $a = 4.2$. Here, point $\textrm{A}(0.304,8)$. (d)-(f) Energy flow of dual probe lights with parameters: ${v_0} = {w_0} = 0.8$, ${v_1} = 0.1$, ${w_1} = 1$, $d = 9$, $a = 4.2$ and $b = 0.304$.

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Fig. 5. Profiles of stable semilunar solitons (a)-(c) and vortex solitons (e)-(g) under a random perturbation at $s = 20$ (dot A in Figs. 3 and 4). (a) and (e) Isosurfaces. (b) and (f) The field modulus in $\xi - \eta$ plane. (c) and (g) The phases in $\xi - \eta$ plane. (d) and (h) The linear-stability spectra with partial enlarged perspective. Other parameters are the same as Fig. 3.

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Figure 5 display the propagation behaviors of 3D vector semilunar solitons (SSs) and vortex solitons (VSs) in the linear and nonlinear PT-symmetric potentials. By solving Eq. (4) numerically, we find that the system supports stable vector SSs and VSs with parameters ${v_0} = {w_0} = 0.8$, ${v_1} = 0.1$, ${w_1} = 1$, $d = 9$ and $a = 4.2$ when the system works in the PT-symmetry domain (point “A” in Fig. 3). Under a random perturbation, the spatial asymmetric SSs and spatial symmetric VSs in shape are displayed in Figs. 5(a), 5(b), 5(e), and 5(f), respectively. The vortex changes of them in phase are shown in Figs. 5(c) and 5(g), respectively. For a better understanding of the results, the linear-stability spectra are carried out in Figs. 5(d) and 5(h). One can see that the real part of eigenvalue $\lambda$ is equal to zero, thus, vector SSs and VSs are stable with particular parameters in our system.

To illustrate the dynamics of vector solitons, we study the frontal head-on collisions of two vector solitons. The solitons, initially located at the positions $( - 10, + 10)$, are supposed to move along the s axis in the opposite directions, and the respective wave functions are multiplied by $\textrm{exp}({\pm} 1.2s)$. Figures 6(a1)-(a5) show the collision process and the energies of the two vector fundamental solitons (FSs) at different propagation distances $\varDelta s = 0,10,15,20$. The dimensionless velocity of the soliton is about 1.2. The energy is not strictly conserved in the collision process, as the maximum amplitude of energy change is measured as ${{\varDelta U} / {({U_1} + {U_2}) = }}0.5{\%}$. The deviation from elastic collision is found to be small, and the energy distributions of the two solitons remain stable with subtle changes in shape. So the FS-FS collision features a quasi-elastic behaviour. The similar phenomena also occur in VS -FS [ Figs. 6(b1)–6(b5)] and VS-VS [Figs. 6(c1)–6(c5)] collisions. It is shown that the collisions lead to partial destruction of VS -FS and VS-VS.

 

Fig. 6. The collision graphs and energies of two vector solitons at different propagation distances $\varDelta s = 0,10,15,20$. (a1)-(a5) Two vector fundamental solitons, (b1)-(b5) a vector vortex soliton and a fundamental soliton, (c1)-(c5) two vector vortex solitons. In column 1 to 4, the upper part is the isosurface distribution and the lower part is the field modulus in $\xi - \eta $plane. (a5), (b5) and (c5) are the energies of two solitons during the collisions. The initial positions of two solitions are $s( - 10,10)$. The other parameters ${v_0} = 2$, $a = 4.2$, ${v_1} = 0.1$, and $d = 10$.

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4. Conclusion

In this work, we have proposed a scheme for realizing 3D vector SSs and VSs by utilizing EIT in combined linear and nonlinear PT-symmetric potentials. We have shown that the system can support stable vector 3D SSs and VSs, which have extremely low generation power, very slow propagation velocity and constant total energy. The frontal head-on collisions of two vector solitons feature quasi-elastic collisions. Further, the dynamics characteristics of these vector solitons depend on the linear and nonlinear PT-symmetric potential parameters. In particular, the imaginary parts of PT potentials play crucial roles for the PT phase transition, which can be actively tuned in our system. The research presented here opens a route for developing non-Hermitian nonlinear optics, especially for manipulating the high-dimensional vector optical solitons with controllable PT-symmetric optical linear and nonlinear potentials, and may find applications in optical information processing and transmission.