1. Introduction

In the past few decades, huge effort has been directed toward the investigation of parity-time (PT) symmetry in various physical systems, with the main aim to develop non-Hermitian quantum mechanics [1,2]. Owing to the fact that the Maxwell equation in electrodynamics under paraxial approximation is mathematically equivalent to the Schrödinger equation in quantum mechanics, light propagations in electromagnetic media provide excellent platforms for testing PT-symmetric quantum theory both theoretically [3] and experimentally [4]. In addition, such investigations have already led to many attractive practical applications, including the realization of nonreciprocal and unidirectional invisible light propagations [5], coherent perfect absorbers [6], giant light amplification [7], novel lasers [8], precision measurement [9], quantum computation [10], and optical solitons in PT -symmetric systems [11–15].

It is desirable to have PT-symmetric systems that can work at weak-light level. Recently, it has been shown that several laser fields interacting with atomic gases are promising candidates for realizing optical PT symmetry and achieving new functionalities for non-Hermitian optics that are not available in conventional PT symmetric systems [16–18]. Especially, local weak-light solitons in such PT symmetric systems have been shown possible [19–22]. It will be interesting if one can obtain realistic optical systems that not only possess PT symmetry, but also support stable weak-light solitons, especially two-dimensional stable vortex solitons (VSs) and bipolar solitons (BSs).

In this work, we propose a scheme to realize PT symmetry, 2D optical solitons, and their active manipulation in a cold atomic gas through a spatial modulation of the control laser field and the inclusion of the Kerr nonlinearity of the signal laser field [20,23]. We demonstrate that the space-dependent imaginary part, width and position of the PT potential play key roles for the occurrence of the PT phase transition and the change of the PT phase diagram, which can be actively manipulated. It is shown that the system supports stable 2D optical solitons (VSs and BSs), which can be managed via tuning the PT potential. Furthermore, by taking such PT potential as a defect, the scattering of the optical solitons by the defect displays obvious symmetric behavior, and the stablity of the 2D solitons in propagation is displayed. Finally, collisions controlled mainly by the space-dependent imaginary parts of the PT potentials are shown.

Before proceeding, we note that this work is different from the Refs. [19–22]. First, in our paper, the 2D PT symmetry linear optical potential is realized in a coherent atomic gas. Second, in our work, stable 2D DSs and VSs are found. Third, it is shown that the stability of the 2D solitons depends on the space-dependent imaginary part of the PT potential during the propagation and collisions. The paper is organized as follows. In Sec. 1, the physical model and the realization of the linear PT potential are presented. In Sec. 3, the PT phase transition and the stablity of the 2D solitons in propagation are shown. In Sec. 4, the dynamic characteristics of the symmetric scatter and collisions of solitons are shown. Finally, Sec. 5 summaries the main results obtained in this paper.

2. Model

We start with a cold, dilute N-type four-level atomic gas, which interacts with a weak signal laser field ${E_s}$, a strong control laser field ${E_\textrm{c}}$, and a strong pump-laser field ${E_p}$, coupling to transitions $|3 \rangle \leftrightarrow |1 \rangle$, $|3 \rangle \leftrightarrow |2 \rangle$, and $|4 \rangle \leftrightarrow |1 \rangle$, respectively [shown in Fig. 1(a)]. In order to minimize the doppler effect, all laser fields are assumed to propagate through the atomic ensemble along the z direction. Here an optical pumping is added to provide an active Raman gain to the signal field [24].

 

Fig. 1. (a) Level diagram and the excitation scheme of the N-type four-level atomic gas. $|j \rangle$ are atomic states and ${\Delta _j}$ are detunings ($j = 1\textrm{ - }4$, ${\Delta _1} = 0$); ${\Gamma _3}$ and ${\Gamma _4}$ are, respectively, decay rates of $|3 \rangle$ and $|4 \rangle$; ${\Omega _s}$, ${\Omega _c}$, and ${\Omega _p}$ are, respectively, half Rabi frequencies of the weak signal field (red), strong control field (blue), and strong pump field (purple). (b) (c) Spatial profiles of the real and imaginary parts of the linear potential with ${v_0} = 2$,${v_1} = 0.1$, and ${V_0} = 0$. (d) Real part ${\textrm{Re}} [\chi _s^{(1)}]$ (blue solid line) and imaginary part ${\mathop{\rm Im}\nolimits} [\chi _s^{(1)}]$ (red dashed line) of the susceptibility $\chi _s^{(1)}$ as functions of ${\Delta _2}$. (e) Real part $\textrm{Re}[{\chi_\textrm{s}^{(1)}}]$ as a function of ${\Omega _{\textrm{c0}}}/(2\pi)$ for ${\Delta _2} ={-} 30.11$MHz (blue solid line) and ${\Delta _2} ={-} 30.16$MHz (red dashed line), respectively. (f) The same as (e) but for Imaginary part $\textrm{Im}[{\chi_\textrm{s}^{(1)}}]$ via ${\Omega _{\textrm{c0}}}/(2\pi)$.

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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 ${\hat{H}_{{\mathop{\rm int}} }} ={-} \hbar \sum\limits_{j = 1}^4 {{\Delta _j}|j \rangle \left\langle j \right|- \hbar \left[ {{\Omega _p}|4 \rangle \left\langle 1 \right|+ {\Omega _s}|3 \rangle \left\langle 1 \right|+ {\Omega _c}|3 \rangle \left\langle 2 \right|+ \textrm{H}.\textrm{c}\textrm{.}} \right]}$. Here ${\Delta _2} = {\omega _s} - {\omega _c} - ({{E_2} - {E_1}} )/\hbar$ is two-photon detuning,${\Delta _3} = {\omega _c} - ({{E_3} - {E_1}} )/\hbar$ and ${\Delta _4} = {\omega _p} - ({{E_4} - {E_1}} )/\hbar$ are one-photon detunings, ${\Omega _s} = ({{\textrm{e}_s}{p_{\textrm{13}}}} ){\varepsilon _s}/\hbar$, ${\Omega _c} = ({{\textrm{e}_\textrm{c}} \cdot {p_{23}}} ){\varepsilon _c}/\hbar$,${\Omega _p} = ({{\textrm{e}_p}{p_{\textrm{14}}}} ){\varepsilon _p}/\hbar$ are, respectively, half Rabi frequencies of the signal, control, and pump fields, ${p_{jl}}$ is the electric dipole matrix elements associated with the transition $|j \rangle \leftrightarrow |l \rangle$. The dynamics of atoms is governed by the optical Bloch equation

Being the weak signal field, we can take ${\Omega _s}$ as a small parameter to solve the Bloch Eq. (1) by using a perturbation expansion [25]. With this solution we may obtain the expression of the first-order optical susceptibility of the signal field with the form $\chi _S^{(1 )} = {N_a}{|{{\textrm{e}_\textrm{S}} \cdot {\textrm{p}_{\textrm{13}}}} |^2}a_{31}^{(1 )}/({{\varepsilon_0}\hbar } )$ [23]. As the control field is modulated along $x$ and y directions, we get space-dependent linear optical potential. By combining Eq. (1) and Eq. (2), the dimensionless form of signal field is obtained

3. Linear Gaussian PT-symmetric potential

To realize the linear optical potential with PT symmetry, we illustrate the relation between the linear optical susceptibility of the signal field and the control-field frequency detuning ${\Delta _2}$ for a fixed ${\Omega _c} = {\Omega _{c0}}$. Taking ${\Delta _3} = {\Delta _4} = 0$, ${\Gamma _{12}} = {\Gamma _{34}} \approx 0$, ${\Gamma _{13}} = {\Gamma _{23}} = {\Gamma _{14}} = {\Gamma _{24}} = 5\pi \textrm{MHz}$, ${\Omega _s} = 0.1 \times \pi \textrm{MHz}$, ${\Omega _p} = 4 \times \pi \textrm{MHz}$, and ${\Omega _{c0}} = 2\pi \textrm{MHz}$, we solve Eq. (1) numerically. The blue solid and red dashed lines in Fig. 1(d) are the real part $\textrm{Re}[\chi _s^{(1 )}]$ and imaginary part $\textrm{Im}[\chi _s^{(1 )}]$ of the linear susceptibility, respectively. One can see that $\textrm{Im}[\chi _s^{(1 )}]$ is zero for ${\Delta _2} ={-} 30.14\textrm{MHz}$, i.e., point “P” in Fig. 1(d). It is shown that $\chi _s^{(1 )}$ exhibits absorption on the left side and gain on the right side of point “P”.

Figures 1(e) and 1(f) show $\textrm{Re}[\chi _s^{(1 )}]$ and $\textrm{Im}[\chi _s^{(1 )}]$ vs ${\Omega _{\textrm{c}0}}/2\pi$, resprctively. We notice that $\textrm{Re[}\chi _\textrm{s}^{(1 )}\textrm{]}|{_{{\Delta _2} ={-} 30.11\textrm{MHz}}} \approx \textrm{Re[}\chi _\textrm{s}^{(1 )}\textrm{]}|{_{_{{\Delta _2} ={-} 30.16\textrm{MHz}}}} $, i.e., the real parts of the linear susceptibility is symmetric in Fig. 1(e). On the contrary, it is shown that ${\mathop{\rm Im}\nolimits} [\chi _s^{(1 )}]|{_{{\Delta _2} ={-} 30.11\textrm{MHz}}} \approx{-} {\mathop{\rm Im}\nolimits} [\chi _s^{(1 )}]|{_{{\Delta _2} ={-} 30.16\textrm{MHz}}} $, the imaginary part of the linear susceptibility is antisymmetric. Hence, to meet the condition of PT symmetry, i.e., ${V^ \ast }({ - \xi , - \eta } )= V({\xi ,\eta } )$, we suppose that the control field consists of two identical Gaussian beams with the form

Figures 1(b) and 1(c) show the real and imaginary part of the spatial distribution of the linear potential, respectively. One can see that the linear potential V displays PT symmetry. Unlike Ref. [21], we extend the spatial distribution to two dimensions by considering the diffraction of the signal light on x and y directions.

4. PT phase transitions and their active control solitons

Now, we consider the property of the PT phase transitions and their active control solitons of optically pumped EIT system with PT potential [see Eq. (5)]. Because the system can be actively controlled, the PT phase transition and the propagation of solitons may be manipulated by tuning system parameters.

To numerically obtain nonlinear modes, we suppose the field amplitude is of the form $\psi = \psi {e ^{ibs}}$, with the corresponding soliton norm $U = \int\!\!\!\int {{{|\psi |}^2}d\xi d\eta }$, where $b$ is the propagation coefficient. Submitting it into Eq. (3), we obtain the nonlinear eigenvalue problem ${\partial ^2}\psi /\partial {\xi ^\textrm{2}} + {\partial ^2}\psi /\partial {\eta ^\textrm{2}} - V\psi - {|\psi |^2}\psi = b\psi$. Using the squared-operator [26,27], the eigenvalue problem can be solved numerically. In order to check the stability of solitons, we perform a linear stability analysis [27]. Perturbed solutions are written in the form $\psi = [{\psi _0} + (u + v){e^{i\lambda s}} + ({u^\ast } - {v^\ast }){e^{ - i{\lambda ^\ast }s}}]{e^{ibs}}$, here ${\psi _0}$ is the stationary soliton solution of Eq. (3), $u,v < < {\psi _0}$ are complex functions describing the perturbation profile, and $\lambda$ is the perturbation growth rate. Substituting it into Eq. (3) 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.

Figure 2(a) shows the phase diagram of the PT phase transition in the plane of ${v_1}$ and $d$. The solid border of the blue and red domains in the figure represents the PT phase transition, where the eigenvalue displays a transition from real to complex when the the imaginary amplitude and position of the linear potential are varied. One can see that the blue domain is the phase with broken PT symmetry and the red domain being the phase with PT symmetry. One can also see that the PT phase transition depends not only on ${v_1}$ but also on d. Figure 2(b) shows the phase boundary lines of the PT phase transition in the plane of ${v_1}$ and a. It is shown that the domain of the PT -symmetry phase increases greatly as a decreases.

 

Fig. 2. (a) (b) PT phase diagrams and their active manipulation, PT symmetry (red), broken PT symmetry (blue). (a) PT phase diagram of the system as functions of ${v_1}$ and d with $a = 4.2$ and ${v_0} = 2$, here A (10,0.1) and B (10,0.6). (b) PT phase diagram of the system as functions of ${v_1}$ and a with $d = \textrm{10}$ and ${v_0} = 2$, here C (4.2,0.1), and D (4.2, 0.6). (c)-(f) $U(b)$ dependence. (c) ${v_0} = 2$, ${v_1} = 0.1$, $d = 9.5$,$a = 4,4.2,4.4$. (d) ${v_0} = 2$,$a = 4.2$, $d = 9.5$,${v_1} = 0.1,0.2,0.3$. (e) $a = 4.2$, ${v_1} = 0.1$,$d = 9.5$,${v_0} = 2.5,3,3.5$. (f) ${v_0} = 2$, $a = 4.2$, ${v_1} = 0.1$,$d = 9.1,9.4,9.7$.

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Figures 2(c)–2(f) shows U(b) dependence with different PT-symmetric potential parameters. It is shown that the dynamic characteristics of solitons depend on four parameters (i.e.${v_0}$, ${v_1}$, a, and $d$). From the Figs. 2(c)-(f), one can see that when the potential parameters change, the powers of the solitons oscillate with the propagation coefficient b. From Figs. 2(c) and (d), one can see that b causes a vibration change of soliton power with different values of a or ${v_1}$. Different with Figs. 2(c) and (d), for the same stable interval of b, it is shown that the increase of ${v_0}$ or $d$ causes a decrease of power vibration [see Figs. 2(e) and (f) ]. 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 [28], thus the solitons here are stable [see points “A” and “C” in Fig. 2(c)]. However, the actual stability of numerical solutions is established by the linear stability analysis [see Fig. 3(d),(h)].

 

Fig. 3. Active control of DSs (a-d) and VSs (e-h). (a),(e) Intensity and (b),(f) phase distribution of DSs and VSs in $X - Y$ plane at the propagation distances $z = 6$, respectively. (c), (g) Propagation behaviors of DSs and VSs in the regions $0 < z < 10$(${v_1} = 0.1$) and $10 < z < 20$(${v_1} = 0.6$), respectively. (d),(h) Largest instability growth rate versus the propagation coefficient b with for DSs and VSs [corresponding to the “A"and “C” point in Fig. 2], respectively. Other parameters, ${v_1} = 0.1$,${v_0} = 2$, $d = 10$, and $a = 4.2$.

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Figures 3(a)-(h) display the propagation behaviors of 2D DSs and VSs in the linear PT-symmetric potential. By solving Eq. (3) numerically, we find that the system supports stable DSs and VSs with the parameters ${v_0} = 2$, ${v_1} = 0.1$, $d = 10$,$a = 4.2$ and $b = 1.6$, when the system works in the domain of the PT symmetry[point"A"and “C"in Fig. 2]. It is shown that under a random perturbation, the spatial symmetry DSs and VSs in shape [Figs. 3(a),(d)] and the gradient changes of them in phase are displayed in Figs. 3(b),(f), respectively. For a better understanding of the results, the linear-stability spectra are carried out in Figs. 3(d),(h), respectively. One can see that the real part of eigenvalue $\lambda$ is equal to zero, thus, DSs and VSs are stable with particular parameters in our system.

The active control of the optical soliton can be implemented by adjusting the imaginary part of the linear potentials (characterized by the parameter ${v_1}$). To show this, we focus on the cases (d,${v_1}$)=(10, 0.1) and (10, 0.6) i.e., points"A"and “B” in Fig. 2(a) [(a,${v_1}$)=(4.2, 0.1) and (4.2, 0.6) i.e., points"C"and “D” in Fig. 2(b)]. In these cases, the system works in the PT -symmetry phase for ${v_1} = 0.1$ but in the broken PT -symmetry phase for ${v_1} = 0.6$. Shown in Figs. 3(c) and 3(g) are the results on the propagation of DSs and VSs by tuning the value of ${v_1}$, respectively. It is shown that the propagation behaviors of DSs and VSs in different regions of z. We see that the solitons are stable in the region $0 < z < 10$ (where ${v_1} = 0.1$), while they are not stable in the region $10 < z < 20$ (where ${v_1} = 0.6$).

5. Symmetric scatter and collision of solitons

It is an interesting research topic of the scattering property of solitons in optically pumped EIT system if the PT -symmetric optical potential is taken to be a defect. In our system, parameters of the defect are chosen to be ${v_1} = 0.1$,${v_0} = 2$, $d = 10$, and $a = 4.2$. We assume that the position of the soliton is initially away from the defect, so that there is no interaction between them at $z = 0$. In general, we may have full reflection, transmission, trapping, or some combination of them. These scattering behaviors can be described by the coefficients of reflection (R), transmission (T), and trapping (G), defined, respectively, by $R = \frac{1}{U}\int_{ - \infty }^{ - h} {\int_{ - \infty }^{ - h} {{{|{\psi ({\xi ,\eta } )} |}^2}} d\xi d\eta }$,$T = \frac{1}{U}\int_h^\infty {\int_h^\infty {{{|{\psi ({\xi ,\eta } )} |}^2}} d\xi d\eta }$, and $G = \frac{1}{U}\int_{ - h}^h {\int_{ - h}^h {{{|{\psi ({\xi ,\eta } )} |}^2}} d\xi d\eta }$.

Figure 4(a) [Fig. 4(b)] displays the result of the soliton scattering when the soliton is incident from the left side of the defect with incident velocity $\upsilon = 0.65$($\upsilon = 1.5$). The results are obtained through numerically solving Eq. (3) by using the split-step Fourier method [29] and taking $\psi ({\xi ,\eta } )= {\psi _0}({\xi ,\eta } ){e^{i\upsilon \xi }}$, here ${\psi _0}({\xi ,\eta } )$ is stationary soliton solution of Eq. (3). In the figure, the region between the two vertical dark dashed lines denotes the one where the defect locates; the width of the defect is $x = 10$ in the x direction and $z = \infty$ in the z direction. It is shown that, for smaller (larger) incident velocity, the soliton is completely reflected (transmitted). Figure 4(c) illustrates the result of the reflection coefficient R (blue solid line) and transmission coefficient T (red dash-dotted line) as functions of incident velocity $\upsilon$. The blue dots “a” and “b” in the figure indicate the values of R and T, which correspond to the cases shown in panels (a) and (b), respectively. Different with Ref. [20], we find that when $\upsilon \le 1.1$, soliton is completely reflected, when $\textrm{1}\textrm{.1} < \upsilon < 1.3$, soliton has not only scattering but also transmission, when $\upsilon > 1.3$ the scattering of the soliton changes sharply from a full reflection to a full transmission. There is nearly no trapping of the soliton, i.e.,$G = 0$, during the process of the soliton scattering. For comparison, in Figs. 4(d)-(f), we show the result of the soliton scattering when the soliton is incident from the right side of the defect with the same incident velocities to Fig. 3(a)-(c), similar results can be found. From these results, we conclude that, for the linear PT defect potential, the soliton scattering is reciprocal, i.e. left-right symmetric.

 

Fig. 4. Symmetric soliton scattering by the linear PT -symmetric defect potentials, i.e., ${v_1} = 0.1$. In panels (a), (b), (d), and (e), the region between the two vertical black dashed lines is the one where the defect locates. (a) [(b)] The soliton is incident from the left side of the defect with incident velocity $\upsilon = 0.65$($\upsilon = 1.5$). Here, the point of incidence located in $x ={-} 5$. (c) Reflection coefficient R (blue solid line), [a(0.65,1)] and transmission coefficient T (red dash-dotted line) [b(1.5,1)] as functions of incident velocity $\upsilon$ for the soliton incident from the left side, The dark dots “a” and “b” represent the values of R and T for the cases shown in panels (a) and (b), respectively. (d)-(f) The soliton is incident from the right side with same incident velocity to (a)[(b)]. Other parameters are as same as Fig. 2.

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The collision between two analytic 1D solitons is truly elastic [30] and such solitons pass through each other without deformation at any incident velocities. The collision between two 2D spacial soliton is expected to be inelastic in general with loss of kinetic energy resulting in the deformation of the solitons. Such a collision can at best be quasi-elastic. To test the solitonic nature of the present 2D solitons, we study the frontal head-on collision of two solitons. To set the solitons in motion along the x axis in opposite directions, the respective wave functions are multiplied by exp(±i2x). To illustrate the dynamics upon real z simulation, we place two solitons centered at initial positions (0,${\pm} 10$) in $X - Z$ plane. Figures 5(a)-(d) show the collision process of the two fundamental solitons (FSs) at different propagation distances $s = 0,10,{{40} / {3,20}}$. The dimensionless velocity of a soliton is about 2 and the deviation from elastic collision is found to be small. The simulations reveal four generic outcomes of the collisions. We see that the moving fundamental solitons feature a quasielastic collision. The similar phenomena also occur in VSs[Figs. 5(e)-(h)] and VSs -FSs[Figs. 5(i)-(l)]. It is shown that the collisions lead to partial destruction of two VSs and VSs -FSs.

 

Fig. 5. Collisions of stable 2D optical solitons in $X - Z$ plane. (a)-(d) The process of the two fundamental solitons collision at $y = 0$ and different propagation distances $s = 0,10,{{40} / {3,20}}$ respectively. (e)-(h)The same as panels (a)-(d) but for vortex soliton and fundament soliton. (i)-(l) Two vortex solitons. Other parameters ${v_0} = 2$,$a = 4.2$,${v_1} = 0.1$, and $d = 10$.

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6. Conclusion

In conclusion, we present a physical scheme for realizing PT linear optical potential and optical spacial solitons in an optically pumped EIT system. We demonstrate that the 2D PT-symmetric potential can be produced with the spatial modulation of the control laser field and some relevant atomic parameters. It is found that the imaginary part, the width and the position of the linear PT potential play important roles for the occurrence of the PT phase transition. Furthermore, DSs and VSs are found to be stable in the system, which can be controlled via tuning the PT potential. The dynamic characteristics of the symmetric scatter and collision of solitons are also displayed. The results reported here may have potential applications in optical information processing and transmission, such as: optical switches, optical logic gates, optical diodes, optical beam splitters, optical storage and optical retrieval.