Open AccessArticle

Nonlinear Optical Potential with Parity-Time Symmetry in a Coherent Atomic Gas

Delvi Antonio Polanco Adames

¹,

Jianpeng Dou

²,

Ji Lin

¹,

Gengjun Zhu

¹ and

Huijun Li

^1,*

College of Physics and Electronic Information Engineering, Zhejiang Normal University, Jinhua 321004, China

Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

Author to whom correspondence should be addressed.

Symmetry 2022, 14(6), 1135; https://doi.org/10.3390/sym14061135

Submission received: 14 April 2022 / Revised: 14 May 2022 / Accepted: 26 May 2022 / Published: 31 May 2022

(This article belongs to the Special Issue Advances in Quantum Information)

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Abstract

We propose a scheme to realize a parity-time (

PT

) symmetric nonlinear system in a coherent atomic gas via electromagnetically induced transparency. We show that it is possible to construct an optical potential with

PT

symmetry due to the interplay among the Kerr nonlinearity stemmed from the atom-photon interaction, the linear potential induced by a far-detuned Stark laser field, and the optical gain originated from an incoherent pumping. Since the real part of the

PT

-symmetric potential depends only on the intensity of the probe field, the potential is nonlinear and its

PT

-symmetric properties are determined by the input laser intensity of the probe field. Moreover, we obtain the fundamental soliton solutions of the system and attain their stability region in the system parameter space. The dependence of the exceptional point (EP) location on the soliton maximum amplitude is also illustrated. The research results reported here open a new avenue for understanding the unique properties of

PT

symmetry of a nonlinear system. They are also promising for designing novel optical devices applicable in optical information processing and transmission.

Keywords:

parity-time symmetry; electromagnetically induced transparency; soliton

1. Introduction

The inherent loss of a practical physical system usually brings the non-Hermiticity of Hamiltonian, making the system an open one, and leads to a complex eigen spectrum. Surprisingly, a special class of non-Hermitian Hamiltonians with parity-time (

PT

) symmetry may have an entire real spectrum [1,2] in a definite parameter range. Thanks to the mathematical equivalence between the Schrödinger equation in quantum mechanics and the Maxwell equation under paraxial approximation in optics, the concept of

PT

symmetry was soon introduced into the content of optics [3,4,5,6,7,8,9,10,11,12,13]. In optics,

PT

symmetry indicates that the real part of a complex optical potential

V (r)

is an even function of space while its imaginary part is an odd function, i.e.,

V (r) = V^{*} (- r)

. Particularly, there is a critical threshold, alias as the exceptional point (EP), at which the system experiences a phase transition from the

PT

symmetry to the broken

PT

symmetry. If the

PT

-symmetric system is a linear one, the location of EP is determined by the ratio between the imaginary and real parts of the

PT

-symmetric potential, which is irrelevant to the intensity of the input laser field. In this case, when the system works in the unbroken

PT

-symmetric phase, all eigenvalues are real and the corresponding eigenmodes of the system are stable; on the other hand, when the system works in the broken

PT

-symmetric phase, some of eigenvalues become complex and the corresponding eigenmodes are unstable. If the

PT

-symmetric system is a nonlinear one, the situation can be very different. In this case, the location of EP is closely related to the intensity of the input laser field. This is because the Kerr nonlinearity can be treated as a part of the optical potential, which can eventually modify the ratio between the imaginary and real parts of the potential.

In recent years, many intriguing phenomena and important applications have been shown and suggested for optical

PT

-symmetric systems, including the loss induced transparency [9], coherent perfect absorbers [10], unidirectional light propagation [6], unidirectional invisibility [5,14], nonreciprocal light propagation [7,8], single-mode microring lasers [15,16], enhanced sensing [17,18], and so on. However, most of above phenomena were studied only for the linear systems. Therefore, it is very interesting to see what will happen when the

PT

-symmetric systems are nonlinear. It is also important to note that compared with solid-state materials [3,4,5,6,7,8,9,10,11,12,13], atomic gases are more advantageous in realization of

PT

-symmetric structures due to the configurable refractive index profiles, leading to active control of the gain, absorption, and Kerr nonlinearities [19,20,21].

In this article, we propose a scheme to realize a

PT

-symmetric nonlinear system in a coherent atomic gas via electromagnetically induced transparency (EIT). The system under study consists of a cold three-level atomic gas interacting with two lasers. We show that by carefully tuning the Kerr nonlinearity stemmed from the atom-photon interaction, the linear potential induced by a far-detuned Stark laser field, and the optical gain originated from an incoherent pumping between the ground and excited levels, it is possible to construct an optical potential with

PT

symmetry. In contrast with previous studies, the real part of the

PT

-symmetric potential in the present system depends only on the intensity of the probe field through the Kerr nonlinear effect. Therefore, the

PT

-symmetric potential is a nonlinear one and its

PT

-symmetric properties are determined by the input laser intensity of the probe field. Moreover, we obtain the fundamental soliton solutions of the system and find their stability region in the system parameter space. Finally, we illustrate that the EP location can be changed and actively controlled by tuning the soliton amplitude. The research results reported here is not only useful for understanding the unique properties of

PT

symmetry of a nonlinear optical potential, but also promising for designing novel optical devices applicable in optical information processing and transmission.

2. Model and the Motion Equation for the Scheme

2.1. Model

We consider a cold

^{87}

Rb atomic gas with

Λ

-type energy state, see Figure 1. The energy levels are chosen from D

_{1}

line of

^{87}

Rb, here

| 1, 2 〉 = | 5 S_{1 / 2}, F = 1, 2 〉

, and

| 3 〉 = | 5 P_{1 / 2}, F = 1 〉

. A weak probe and a strong control light fields

E_{p, c} = e_{x} E_{p, c} (x, z, t) exp [i (k_{p, c} z - ω_{p, c} t)] + c . c .

interact near resonantly with energy levels

| 1 〉 \to | 3 〉

and

| 2 〉 \to | 3 〉

, respectively. Here

e_{j}

and

k_{j}

(

E_{j}

) denote respectively the polarization unit vector along the jth direction and the wave number (envelope) of the jth field, and

E_{c}

is constant. The energy levels

| l 〉

(

l = 1 - 3

) and

E_{p, c}

constitute the well-known

Λ

-type EIT system.

In addition, another far-detuned standing-wave Stark field is introduced into the system, reading

E_{Stark} = e_{y} \sqrt{2} E_{s} (x) cos (ω_{f} t),

(1)

where

E_{s} (x)

and

ω_{f}

are the amplitude and angular frequency. Consequently, an x-dependent Stark energy shift of

E_{j}

appears, i.e.,

E_{j} \to E_{j} + Δ E_{j}

, where

Δ E_{j} = - \frac{1}{2} α_{j} {〈E_{Stark}^{2}〉}_{t} = - \frac{1}{2} α_{j} {| E_{s} (x) |}^{2}

, with

α_{j}

the scalar polarizability and

{〈 \dots 〉}_{t}

the average in an cycle. The concrete form of

E_{s} (x)

will be selected according to the

PT

-symmetric requirement (see Section 3).

In order to obtain a gain for the probe light field, which is required by the condition of

PT

symmetry, an incoherent optical pumping is also introduced in the system. It pumps atoms from the energy level

| 1 〉

to the energy level

| 3 〉

with the pumping rate

Γ_{31}

as shown in Equations (A1) and (A3) of Appendix A. This kind of pumping has been realized in intense atomical resonant lines emitted from the hollow-cathode lamps or from the microwave discharge lamps [22].

In Figure 1a,

Ω_{p} = (e_{x} \cdot p_{13}) E_{p} / ℏ

and

Ω_{c} = (e_{x} \cdot p_{23}) E_{c} / ℏ

are respectively the half Rabi frequencies of the probe and control fields, where

p_{i j}

is the electric dipole matrix element from

| i 〉

| j 〉

and

Δ_{3, 2}

are the one- and two-photon detunings, respectively. A possible experimental setup is plotted in Figure 1b.

2.2. Maxwell-Bloch Equations

Under the electric dipole and rotating wave approximations, the Hamiltonian in the interaction picture reads

{\hat{H}}_{int} = - ℏ \sum_{j = 1}^{3} Δ_{j}^{'} | j 〉 〈 j | - ℏ (Ω_{p} | 3 〉 〈 1 | + Ω_{c} | 3 〉 〈 2 | + h . c .),

where

h . c .

denotes Hermitian conjugate, and

Δ_{j}^{'} = Δ_{j} + 2 ℏ {| E_{s} |}^{2} .

(2)

The Bloch equations, describing the motion of atoms, are given by

\frac{\partial ρ}{\partial t} = - \frac{i}{ℏ} [{\hat{H}}_{int}, ρ] - Γ ρ,

(3)

where

ρ

is a

6 \times 6

density matrix with the density matrix element

ρ_{i j}

(

i, j

= 1, 2, 3), and

Γ = (\begin{matrix} - Γ_{31} & 0 & Γ_{13} \\ 0 & 0 & Γ_{23} & 0 & 0 & 0 \\ Γ_{31} & 0 & - Γ_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & γ_{21} & 0 & 0 \\ 0 & 0 & 0 & 0 & γ_{31} & 0 \\ 0 & 0 & 0 & 0 & 0 & γ_{32} \end{matrix}),

(4)

γ_{j l} = (Γ_{j} + Γ_{l}) / 2 + γ_{j l}^{dph}

(

j l \neq 31

γ_{31} = (Γ_{3} + Γ_{31}) / 2 + γ_{31}^{dph}

, and

Γ_{l} = \sum_{E_{j} < E_{l}} Γ_{j l}

. Here

γ_{j l}^{dph}

denotes the dephasing rates caused by atomic collisions, which can be neglect when the temperature of atoms is low,

Γ_{j l}

is the rate at which population decays from

| l 〉

| j 〉

. Especially,

Γ_{31}

is the incoherent pumping rate from

| 1 〉

| 3 〉

. The explicit form of the equations of motion for

ρ_{i j}

is given in Appendix A.

The evolution of the probe field is described by Maxwell equation, which, under the slowly varying amplitude approximation, is given by

i (\partial z + \frac{1}{c} \frac{\partial}{\partial t}) Ω_{p} + \frac{c}{2 ω_{p}} \frac{\partial^{2}}{\partial x^{2}} Ω_{p} + κ_{13} ρ_{31} = 0,

(5)

where

κ_{13} = N ω_{p} {| e_{p} \cdot p_{13} |}^{2} / (2 ϵ_{0} ℏ c)

, with N the atomic concentration. In addition, for deriving Equation (5), we have assumed that the probe beam is enough wide in the y direction so that a diffraction term in the y direction (the term

\propto \partial^{2} Ω_{p} / \partial y^{2}

) can be safely neglected.

3. Realization of $PT$ -Symmetric Potential

3.1. The Probe Field Envelope Equation

In order to solve the probe field from Equation (5), one need to obtain the solution of

ρ_{31}

, firstly, by solving Equations (A1)–(A6). Further, we neglect all derivative with respect to time in the Maxwell-Bloch (MB) Equations (3) and (5) since we are only interested in the stationary states [23,24].

Since the probe field is much weaker than the control field, we employ the method of multiple scales by introducing the expansions

ρ_{i j} = ρ_{i j}^{(0)} + ϵ ρ_{i j}^{(1)} + ϵ^{2} ρ_{i j}^{(2)} + ϵ^{3} ρ_{i j}^{(3)} + \dots

Ω_{p} = ϵ Ω_{p}^{(1)} + ϵ^{2} Ω_{p}^{(2)} + ϵ^{3} Ω_{p}^{(3)} + \dots

E_{s} (x) = ϵ E_{s}^{(1)} (x)

d_{i j} = d_{i j}^{(0)} + ϵ^{2} d_{i j}^{(2)}

. For avoiding the divergence at each order, we also introduce the multiple scale variables

z_{l} = ϵ^{l} z

(

l = 0, 1, 2

) and

x_{1} = ϵ x

[23,24]. Here,

ϵ

is a small quantity, determined for example by

ϵ = Ω_{p, \max} / Ω_{c}

. Substituting the expansions into Equations (3) and (5), a sets of equations for

ρ_{i j}^{(l)}

and

Ω_{p}^{(l)}

(

l = 0, 1, 2, 3, \dots

) are obtained, and can be solved order by order.

At the zeroth order (

O (ϵ^{0})

-order), we obtain the base state solutions of the system

\begin{matrix} ρ_{11}^{(0)} = \frac{2 γ_{32} Γ_{13} {| Ω_{c} |}^{2}}{D_{0}}, \end{matrix}

(6)

\begin{matrix} ρ_{22}^{(0)} = \frac{2 γ_{32} Γ_{31} {| Ω_{c} |}^{2} + Γ_{31} Γ_{23} {| d_{32}^{(0)} |}^{2}}{D_{0}} \end{matrix}

(7)

\begin{matrix} ρ_{33}^{(0)} = \frac{2 γ_{32} Γ_{31} {| Ω_{c} |}^{2}}{D_{0}}, \end{matrix}

(8)

\begin{matrix} ρ_{32}^{(0)} = - \frac{Γ_{23} Γ_{31} Ω_{c} (d_{32}^{(0)}) *}{D_{0}}, \end{matrix}

(9)

with other

ρ_{i j}^{(0)} = 0

, and

D_{0} = Γ_{31} Γ_{23} | d_{32}^{(0)} |^{2} + 2 γ_{32} (Γ_{13} + 2 Γ_{31}) {| Ω_{c} |}^{2}

At the first order (

O (ϵ)

-order), the solutions are given by

Ω_{p}^{(1)} = F (x_{1}, z_{1}, z_{2}) e^{i K z_{0}}

ρ_{21, 31}^{(1)} = α_{21, 31}^{(1)} e^{i K z_{0}}

(the explicit expressions of

α_{21, 31}^{(1)}

are given in Appendix B), with other

ρ_{i j}^{(1)} = 0

. Here, the wavenumber K is given by

K = κ_{13} \frac{Ω_{c} (ρ_{32}^{(0)}) * + d_{21}^{(0)} (ρ_{11}^{(0)} - ρ_{33}^{(0)})}{D_{1}},

(10)

where

D_{1} = {| Ω_{c} |}^{2} - d_{21}^{(0)} d_{31}^{(0)}

. From the above equation, we see that K is a complex number, with the real (imaginary) part of K representing the dispersion (gain or loss) of the probe field.

Figure 2 gives the imaginary part of K as a function of

Δ_{3}

. From this figure, there is a absorption peak when

Δ_{3} = 0

in the absence of the control field and the incoherent pumping. However, a typical transparency window is opened when

Ω_{c} = 6 \times 10^{7}

Hz and

Γ_{31} = 0

. But a small abortion Im

(K)

= 0.001 cm

^{- 1}

still exists when

Δ_{3} = 0

. After introducing the incoherent pumping, a gain appears as shown by the dashed-dotted line in Figure 2 and its inset. Such a gain is necessary for the realization of

PT

-symmetric potential of the probe field. Here, the system parameters are taken as

γ_{1} = Δ_{1} = 0

2 γ_{2} = 1 \times 10^{3} Hz

2 γ_{3} = Γ_{3} = 36 MHz

, and the tunable parameter

κ_{13} = 1.0 \times 10^{10} {cm}^{- 1} s^{- 1}

At the second order (

O (ϵ^{2})

-order), we find the equation

\frac{\partial_{F}}{\partial z_{1}} = 0

and the solutions

Ω_{p}^{(2)} = ρ_{21}^{(2)} = ρ_{31}^{(2)} = 0

, and

ρ_{i j}^{(2)} = α_{i j}^{(2)} {| F |}^{2}

(

i j = 11, 22, 33, 32

;

α_{i j}^{(2)}

are given in Appendix B).

The nonlinear envelope equation of the probe field can also be derived at the third order (

O (ϵ^{3})

-order). After returning original variables, the equation reads

i \frac{\partial Ω_{p}}{\partial z} + \frac{c}{2 ω_{p}} \frac{\partial^{2} Ω_{p}}{\partial x^{2}} + \tilde{V} (x) Ω_{p} = 0

(11)

with the potential

\tilde{V} (x) = W_{1} | Ω_{p} |^{2} + W_{2} {| E_{s} (x) |}^{2} + K,

(12)

where

Ω_{p} = ϵ F exp (i K z)

W_{1} = κ_{13} [Ω_{c} {(α_{32}^{(2)})}^{*} - d_{21}^{(0)} (α_{33}^{(2)} - α_{11}^{(2)})] / D_{1}

, and

W_{2} = κ_{13} [(α_{3} - α_{1}) d_{21}^{(0)} α_{31}^{(1)} - (α_{2} - α_{1}) Ω_{c} α_{21}^{(1)}] / (2 ℏ D_{1})

For convenience, Equation (11) can be converted into the dimensionless form

i \frac{\partial u}{\partial s} + \frac{\partial^{2} u}{\partial ξ^{2}} + V (ξ) u = 0,

(13)

with

V (ξ) = g_{1} {| u |}^{2} + g_{2} {| v |}^{2} + i g_{3},

(14)

where

u = Ω_{p} exp (- i Re (K) z) / Ω_{p 0}

v = E_{s} / E_{s 0}

s = z / L_{diff}

ξ = x / R

g_{1} = W_{1} Ω_{p 0}^{2} L_{diff}

g_{2} = W_{2} E_{s 0}^{2} L_{diff}

, and

g_{3} = L_{diff} Im (K)

. Here,

L_{diff} \equiv 2 ω_{p} R^{2} / c

is the characteristic diffraction length,

Ω_{p 0}

is the typical Rabi frequency of the probe field, and

E_{s 0}

denotes the typical amplitude of the far-detuned laser field. Consequently, the probe-field susceptibility is

χ (x) = 2 c \tilde{V} (x) / ω_{p}

, and the nonlinear refractive index

n (x) = \sqrt{1 + χ (x)} \approx 1 + c \tilde{V} (x) / ω_{p}

. Therefore, the

PT

symmetry condition

{\tilde{V}}^{*} (- ξ) = \tilde{V} (ξ)

can also be written as

n^{*} (- ξ) = n (ξ)

Here, we would like to make some remarks about the optical potential (14): (1) The potential (14) is a nonlinear one, i.e., it depends on the probe-field intensity,

g_{1} {| u |}^{2}

, which is contributed from the Kerr nonlinearity. In general, the coefficient

g_{1}

is a complex number, which, however, can be treated as a real one if the condition

Δ_{3} ≫ Γ_{3}

can be fulfilled. Particularly, the real part of the potential is contributed by the term

g_{1} {| u |}^{2}

and its symmetry can be preserved if the probe-field intensity has an even profile in the x direction; (2) The imaginary part of the potential is contributed by the term

g_{2} {| v |}^{2}

. In general, the coefficient

g_{2}

is a complex number, which, however, can be treated as a purely imaginary one if the condition

Δ_{1} \approx Δ_{2} \approx 0

is fulfilled. Further,

{| v |}^{2}

can be made as a constant part added by an odd function in the x direction (see below), corresponding to a constant loss and a distribution of balanced gain and loss, respectively; (3) The constant loss can be compensated by the additional gain

i g_{3}

, contributed by the incoherent pumping. Thus, the total effect of

g_{2} {| v |}^{2} + i g_{3}

leads to the distribution of balanced gain and loss only, i.e., the imaginary part of the potential is anti-symmetric.

3.2. The Design of $PT$ Symmetric Potential

Equation (13) is a nonlinear Schrödinger equation (NLSE) with nonlinear optical potential (14). To realize a

PT

-symmetric potential, we assume that the far-detuned Stark laser field is composed of a constant part and two Gaussian functions with

π

-phase difference in the following form

v = 5 + 0.1 v_{0} [e^{- \frac{{(ξ - ξ_{0})}^{2}}{w^{2}}} - e^{- \frac{{(ξ + ξ_{0})}^{2}}{w^{2}}}],

(15)

where

\pm ξ_{0}

are center positions of two Gaussian functions and w is the width of each Gaussian pulse. In an experiment, the far-detuned laser field (15) can be prepared in advance by overlapping a constant wave and a pair of Gaussian beams side by side with identical profile and

π

-phase difference. Moreover, the amplitude of each components can be adjusted according to the demand of symmetry. Finally, the intensity of the far-detuned laser field is

\begin{matrix} {| v |}^{2} & = 25 + v_{0} [e^{- \frac{{(ξ - ξ_{0})}^{2}}{w^{2}}} - e^{- \frac{{(ξ + ξ_{0})}^{2}}{w^{2}}}] + 0.01 v_{0}^{2} {[e^{- \frac{{(ξ - ξ_{0})}^{2}}{w^{2}}} - e^{- \frac{{(ξ + ξ_{0})}^{2}}{w^{2}}}]}^{2} \\ \approx 25 + v_{0} [e^{- \frac{{(ξ - ξ_{0})}^{2}}{w^{2}}} - e^{- \frac{{(ξ + ξ_{0})}^{2}}{w^{2}}}] . \end{matrix}

(16)

For obtaining the value of coefficients, the system parameters are chosen as

2 γ_{2} = 1 \times 10^{3} Hz

2 γ_{3} = 36 MHz

ω_{p} = 2.37 \times 10^{15} s^{- 1}

. Other (tunable) parameters are taken as

κ_{13} = 3.7 \times 10^{12} {cm}^{- 1} s^{- 1}

R = 2.5 \times 10^{- 3} cm

Ω_{c} = 1.65 \times 10^{7} s^{- 1}

Δ_{2} = 0

Δ_{3} = - 1.9 \times 10^{9} s^{- 1}

Ω_{p 0} = 3.8 \times 10^{5} \sqrt{σ} s^{- 1}

E_{s 0} = 2.7 \times 10^{4} V / cm

Γ_{31} = 17 MHz

. Then, we get

L_{diff} = 1.0

cm, and

g_{1} = (1.0 - 0.04 i) σ, g_{2} = 0.02 + 1.0 i, g_{3} = - 25,

(17)

here,

g_{1}

g_{2}

, and

g_{3}

are controlled by

Ω_{p 0}

E_{s 0}

, and

κ_{13}

, respectively. Therefore, the NLSE with the nonlinear optical potential (14) can be further written as

i \frac{\partial u}{\partial s} + \frac{\partial^{2} u}{\partial ξ^{2}} + \{{σ | u |}^{2} + i v_{0} [e^{- \frac{{(ξ - ξ_{0})}^{2}}{w^{2}}} - e^{- \frac{{(ξ + ξ_{0})}^{2}}{w^{2}}}]\} u = 0 .

(18)

Note that the nonlinear optical potential (the part in the brackets of the last term on the left-hand side of Equation (18)) has the

PT

symmetry as long as the probe-field intensity

{| u |}^{2}

has an even profile in the x direction, which can be fulfilled by the fundamental bright soliton solution.

4. EP and Soliton Solutions

Now we discuss the properties of

PT

symmetry of Equation (18) with linear and nonlinear potential, respectively. We suppose

u (ξ, s) = u_{s} (ξ) e^{- i μ s}

, with

μ

the propagation constant, then the stable state can be obtained from the following equation

- \frac{\partial^{2} u_{s}}{\partial ξ^{2}} - \{σ | u_{s} |^{2} + i v_{0} [e^{- \frac{{(ξ - ξ_{0})}^{2}}{w^{2}}} - e^{- \frac{{(ξ + ξ_{0})}^{2}}{w^{2}}}]\} u_{s} = μ u_{s} .

(19)

Here, we take

ξ_{0} = 2

and

1 / w^{2} = 0.45

4.1. Property of Linear $PT$ -Symmetric Potential

When we take

σ = 0

, Equation (19) becomes a linear eigenvalue problem. In Figure 3, we show the relationship between the imaginary part of

μ

and

v_{0}

by searching for the ground state. Since

v_{0}

characterizes the imaginary part of the potential, the breaking of

PT

symmetry can occur by changing

v_{0}

. From this figure, it is obvious that the spectrum is entirely real (Im

(μ) = 0

) when

v_{0} < 0.2

, i.e., the EP locates at

v_{0} = 0.2

. Above EP,

v_{0} > 0.2

, a phase transition appears and the spectrum becomes partially complex.

4.2. Property of Nonlinear $PT$ -Symmetric Potential

When we take

σ = 1

, corresponding to the self-focusing Kerr nonlinearity, Equation (19) becomes a nonlinear ordinary differential equation. In this case, one can only resort to numerical methods for solving this problem. We obtain the stable state solutions corresponding light powers

P = \int_{- \infty}^{+ \infty} {| ψ |}^{2} d ξ

of the probe field by using the Newton conjugate gradient method [25]. Particularly, the probe beam forms a fundamental bright soliton, as shown in the Figure 4. After obtaining the bright soliton solution,

u_{s} (ξ)

, its stability can be studied by introducing perturbations in the solution, i.e.,

u (s, ξ) = {u_{s} (ξ) + ϵ [w_{0} (ξ) e^{λ s} + v_{0}^{*} (ξ) e^{λ^{*} s}]} e^{- i μ s},

(20)

where

w_{0}

and

v_{0}

are perturbation eigenmodes, and

λ

is the eigenvalue. Substituting Equation (20) into Equation (18), we get the equation

i (\begin{matrix} L_{1} & L_{2} \\ - L_{2}^{*} & - L_{1} \end{matrix}) (\begin{matrix} w_{0} \\ v_{0} \end{matrix}) = λ (\begin{matrix} w_{0} \\ v_{0} \end{matrix}),

(21)

where

L_{1} = μ + \partial^{2} / \partial ξ^{2} + 2 {| u_{s} |}^{2} + i v_{0} [e^{- \frac{{(ξ - ξ_{0})}^{2}}{w^{2}}} - e^{- \frac{{(ξ + ξ_{0})}^{2}}{w^{2}}}]

and

L_{2} = {| u_{s} |}^{2}

. Then, we can use the Fourier collocation method [25] to solve this equation and the soliton

u_{s} (x)

is stable if

λ \leq 0

whereas unstable if

λ > 0

. The soliton stability can be further confirmed by propagating the soliton solution, directly, by using the split-step Fourier method [25].

Shown in Figure 4 is the fundamental bright soliton and their evolution result. Figure 4a–c illustrate the power P curves as a function of the propagation constant

μ

with different

v_{0}

. From the results, we find that the solitons only exist in the region of

μ < 0

, and for each curve there exits the minimum (cut-off value) of the power. In Figure 4d–f, the soliton profiles with different values of

v_{0}

and

μ

are displayed. Here, the blue solid (red dashed) lines stand for the real (imaginary) part of the soliton. It is obvious that the real part of

u_{s}

is an even function about

ξ = 0

, whereas, the imaginary part of

u_{s}

is an odd function. The width of soliton is also changed with

v_{0}

and

μ

The soliton stability is further confirmed by a direct simulation of Equation (18) by taking the soliton solution as the initial condition, after adding a random perturbations into the initial condition of evolution. To this end, the initial value is taken as

u (s = 0, ξ) = u_{s} (ξ) (1 + ε f_{1})

, where

ε = 0.05

and

f_{1}

is a random function uniformly distributed in the interval

[0, 1]

. The evolution results of solitons in Figure 4d–f are shown in Figure 4g–i, respectively. With increasing the propagation distance s, we see that the soliton in Figure 4d propagates stably as shown in Figure 4g. However, in Figure 4h, the soliton with

v_{0} = 0.6, μ = - 0.6

experiences a collapse with increasing s, which is unstable. Finally, in Figure 4i, the soliton splits with the increase of s, which is unstable, either.

In order to have a full picture of the existence and stability of the fundamental bright solitons, we calculate the existence and stability regions of solitons in Figure 5a. The region below the blue dashed line denotes that the solitons can be generated in the parameter intervals, we find that

μ

does not change much by changing

v_{0}

, namely, the imaginary part of

PT

-symmetric potential has little effect on the generation of solitons. Furthermore, we are interested in the stability of solitons. Due to the self-focusing Kerr nonlinearity (

σ = 1

), the fundamental bright solitons are stable in the region on the left of the red solid line, as shown in Figure 5a. Therefore, every points on the red solid line can be treated as an EP for the nonlinear

PT

-symmetric potential. Compared with the linear case, for each value of

μ

, the value of

v_{0}

is lager than 0.2. Thus, we conclude that the self-focusing Kerr nonlinearity can delay the occurrence of the

PT

symmetry breaking, which also triggers the instability of solitons. Particularly, we see that stable solitons can exist for

v_{0} = 0.8

marked by the blue pentagram.

Shown in Figure 5b is the maximum amplitude

| u_{s} |_{\max}

of solitons as a function of

v_{0}

. The solitons are obtained by taking the parameters along the red solid line in Figure 5a. Therefore, the curve in this figure tells us the relation between the EP location and maximum amplitude of solitons. From this figure, we find that with the increase of the maximum amplitude, the EP location increases firstly (denoted by the blue dashed line), and then decreases (denoted by the red solid line). The reason behind this behavior is that the soliton profiles change with the increase of maximum amplitude, that is, the shape of the real part of the

PT

-symmetric potential is not fixed. Thus, it is possible to control the location of EP by changing the soliton amplitude, which provides new a method for controlling EP.

5. Conclusions

In conclusion, we have proposed a scheme to realize a nonlinear

PT

symmetric optical potential in a coherent atomic gas. The system under study consists of a cold three-level atomic media driven by two lasers, works under the condition of EIT. We showed that it is possible to construct an optical potential with

PT

symmetry due to the interplay between the Kerr nonlinearity, the linear Stark potential, and the optical gain. Because the real part of the

PT

-symmetric potential depends only on the intensity of the probe field, the potential is nonlinear and its

PT

-symmetric properties are determined by the input laser intensity of the probe field. Moreover, we obtained the fundamental soliton solutions of the system and attained their stability region in the system parameter space. The location of EPs on the soliton maximum amplitude is also illustrated. The research results reported here is not only useful for understanding the unique properties of

PT

symmetry of a nonlinear optical potential, but also promising for designing novel optical devices applicable in optical information processing and transmission.

Author Contributions

Conceptualization, H.L. and J.L.; methodology, D.A.P.A.; software, J.D. and G.Z.; validation, D.A.P.A. and G.Z.; data curation, D.A.P.A. and J.D.; writing—original draft preparation, H.L.; writing—review and editing, H.L.; project administration, H.L. and J.L. All authors have read and agreed to the published version of the manuscript.

Funding

This work was funded by the National Natural Science Foundation of China (Grant No. 12074343, No. 11835011), Natural Science Foundation of Zhejiang Province of China (Grant No. LZ22A050002).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A. Explicit Expression of Equation (3)

Equations of motion for

ρ_{i j}

are given by

\begin{matrix} i \frac{\partial}{\partial t} ρ_{11} + i Γ_{31} ρ_{11} - i Γ_{13} ρ_{33} + Ω_{p}^{*} ρ_{31} - Ω_{p} ρ_{31}^{*} = 0, \end{matrix}

(A1)

\begin{matrix} i \frac{\partial}{\partial t} ρ_{22} - i Γ_{23} ρ_{33} + Ω_{c}^{*} ρ_{32} - Ω_{c} ρ_{32}^{*} = 0, \end{matrix}

(A2)

\begin{matrix} i (\frac{\partial}{\partial t} + Γ_{3}) ρ_{33} - i Γ_{31} ρ_{11} - Ω_{p}^{*} ρ_{31} + Ω_{p} ρ_{31}^{*} - Ω_{c}^{*} ρ_{32} + Ω_{c} ρ_{32}^{*} = 0, \end{matrix}

(A3)

\begin{matrix} (i \frac{\partial}{\partial t} + d_{21}) ρ_{21} + Ω_{c}^{*} ρ_{31} - Ω_{p} ρ_{32}^{*} = 0, \end{matrix}

(A4)

\begin{matrix} (i \frac{\partial}{\partial t} + d_{31}) ρ_{31} + Ω_{p} (ρ_{11} - ρ_{33}) + Ω_{c} ρ_{21} = 0, \end{matrix}

(A5)

\begin{matrix} (i \frac{\partial}{\partial t} + d_{32}) ρ_{32} + Ω_{c} (ρ_{22} - ρ_{33}) + Ω_{p} ρ_{21}^{*} = 0, \end{matrix}

(A6)

with

d_{j l} = Δ_{j}^{'} - Δ_{l}^{'} + i γ_{j l}

Appendix B. Perturbation Expansion of the MB Equations

The coefficients of perturbation expansion are given by

\begin{matrix} α_{21}^{(1)} = - \frac{d_{31}^{(0)} (ρ_{32}^{(0)}) * + Ω_{c}^{*} (ρ_{11}^{(0)} - ρ_{33}^{(0)})}{D_{1}}, \\ α_{31}^{(1)} = \frac{K}{κ_{13}} . \end{matrix}

(A7)

\begin{matrix} α_{11}^{(2)} = {2 Γ_{13} Im (Ω_{c} d_{32}^{(0)} α_{21}^{(1)}) - 2 Im (α_{31}^{(1)}) [Γ_{23} | d_{32}^{(0)} |^{2} + 4 γ_{32} | Ω_{c} |^{2}]} / D_{0}, \\ α_{22}^{(2)} = {- 2 (Γ_{13} + Γ_{31}) Im (Ω_{c} d_{32}^{(0)} α_{21}^{(1)}) + 2 Im (α_{31}^{(1)}) [Γ_{23} | d_{32}^{(0)} |^{2} + 2 γ_{32} | Ω_{c} |^{2}]} / D_{0}, \\ α_{32}^{(2)} = [2 (Γ_{13} + 2 Γ_{31}) | Ω_{c} |^{2} Im (α_{21}^{(1)}) - Γ_{31} Γ_{23} {(d_{32}^{(0)} α_{21}^{(1)})}^{*} - 2 Γ_{23} {(d_{32}^{(0)})}^{*} Ω_{c} Im (α_{31}^{(1)})] / D_{0}, \\ α_{33}^{(2)} = {2 Γ_{31} Im (Ω_{c} d_{32}^{(0)} α_{21}^{(1)}) + 4 γ_{32} Im (α_{31}^{(1)}) | Ω_{c} |^{2}} / D_{0} . \end{matrix}

(A8)

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Figure 1. (a) Energy level diagram used to obtain a

PT

-symmetric model. (b) A possible experimental setup. All symbols can be found in the text.

Figure 1. (a) Energy level diagram used to obtain a

PT

-symmetric model. (b) A possible experimental setup. All symbols can be found in the text.

Figure 2. The imaginary part Im

(K)

as a function of

Δ_{3} / Γ_{3}

for

Δ_{2} = Δ_{3}

. Solid (blue), dashed (red), and dashed-dotted (green) lines correspond to

(Ω_{c}, Γ_{31}) = (0, 0)

, (

6 \times 10^{7}

Hz, 0), and (

6 \times 10^{7}

Hz, 0.48

Γ_{3}

), respectively. For a better illustration, the dashed-dotted (green) line has been shown in the inset.

Figure 2. The imaginary part Im

(K)

as a function of

Δ_{3} / Γ_{3}

for

Δ_{2} = Δ_{3}

. Solid (blue), dashed (red), and dashed-dotted (green) lines correspond to

(Ω_{c}, Γ_{31}) = (0, 0)

, (

6 \times 10^{7}

Hz, 0), and (

6 \times 10^{7}

Hz, 0.48

Γ_{3}

), respectively. For a better illustration, the dashed-dotted (green) line has been shown in the inset.

Figure 3. The imaginary part of propagation constant Im

(μ)

as a function of

v_{0}

for

σ = 0

Figure 3. The imaginary part of propagation constant Im

(μ)

as a function of

v_{0}

for

σ = 0

Figure 4. (a–c) Light power curves of fundamental bright solitons as a function of the propagation constant

μ

with different

v_{0}

. Here,

v_{0} = 0.4

0.6

and

0.75

, respectively. (d–f) The profiles of bright solitons with different

v_{0}

and the propagation constant

μ

. Here

(v_{0}, μ) = (0.4, - 1.5)

(0.6, - 0.6)

, and

(0.75, - 3.0)

in panels (d–f), respectively. The blue solid lines denote the real parts of solitons, and the red dashed lines denote the imaginary parts of solitons. (g–i) The evolution results of bright solitons with different

v_{0}

and

μ

. Here

(v_{0}, μ) = (0.4, - 1.5)

(0.6, - 0.6)

, and

(0.75, - 3.0)

in panels (g–i), respectively.

Figure 4. (a–c) Light power curves of fundamental bright solitons as a function of the propagation constant

μ

with different

v_{0}

. Here,

v_{0} = 0.4

0.6

and

0.75

, respectively. (d–f) The profiles of bright solitons with different

v_{0}

and the propagation constant

μ

. Here

(v_{0}, μ) = (0.4, - 1.5)

(0.6, - 0.6)

, and

(0.75, - 3.0)

v_{0}

and

μ

. Here

(v_{0}, μ) = (0.4, - 1.5)

(0.6, - 0.6)

, and

(0.75, - 3.0)

in panels (g–i), respectively.

Figure 5. (a) The existence region and stable region of solitons as the functions of

v_{0}

and the propagation constant

μ

. It is the existence region of the bright solitons under the blue dashed line, and it is the stability region of bright solitons on the left of red solid line. The blue pentagram denotes the coordinate of point

(0.8, - 1.6)

. (b) The maximum amplitude

| u_{s} |_{\max}

of solitons as a function of

v_{0}

. The solitons are obtained by taking the parameters along the red solid line in the panel (a). In the red solid line,

μ

changes from −4.74 to −1.6, and it changes from 0 to −1.16 along the blue dashed line.

Figure 5. (a) The existence region and stable region of solitons as the functions of

v_{0}

and the propagation constant

μ

(0.8, - 1.6)

. (b) The maximum amplitude

| u_{s} |_{\max}

of solitons as a function of

v_{0}

. The solitons are obtained by taking the parameters along the red solid line in the panel (a). In the red solid line,

μ

changes from −4.74 to −1.6, and it changes from 0 to −1.16 along the blue dashed line.

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Polanco Adames, D.A.; Dou, J.; Lin, J.; Zhu, G.; Li, H. Nonlinear Optical Potential with Parity-Time Symmetry in a Coherent Atomic Gas. Symmetry 2022, 14, 1135. https://doi.org/10.3390/sym14061135

AMA Style

Polanco Adames DA, Dou J, Lin J, Zhu G, Li H. Nonlinear Optical Potential with Parity-Time Symmetry in a Coherent Atomic Gas. Symmetry. 2022; 14(6):1135. https://doi.org/10.3390/sym14061135

Chicago/Turabian Style

Polanco Adames, Delvi Antonio, Jianpeng Dou, Ji Lin, Gengjun Zhu, and Huijun Li. 2022. "Nonlinear Optical Potential with Parity-Time Symmetry in a Coherent Atomic Gas" Symmetry 14, no. 6: 1135. https://doi.org/10.3390/sym14061135

APA Style

Polanco Adames, D. A., Dou, J., Lin, J., Zhu, G., & Li, H. (2022). Nonlinear Optical Potential with Parity-Time Symmetry in a Coherent Atomic Gas. Symmetry, 14(6), 1135. https://doi.org/10.3390/sym14061135

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Nonlinear Optical Potential with Parity-Time Symmetry in a Coherent Atomic Gas

Abstract

1. Introduction

2. Model and the Motion Equation for the Scheme

2.1. Model

2.2. Maxwell-Bloch Equations

3. Realization of $PT$ -Symmetric Potential

3.1. The Probe Field Envelope Equation

3.2. The Design of $PT$ Symmetric Potential

4. EP and Soliton Solutions

4.1. Property of Linear $PT$ -Symmetric Potential

4.2. Property of Nonlinear $PT$ -Symmetric Potential

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

Appendix A. Explicit Expression of Equation (3)

Appendix B. Perturbation Expansion of the MB Equations

References

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Nonlinear Optical Potential with Parity-Time Symmetry in a Coherent Atomic Gas

Abstract

1. Introduction

2. Model and the Motion Equation for the Scheme

2.1. Model

2.2. Maxwell-Bloch Equations

3. Realization of PT -Symmetric Potential

3.1. The Probe Field Envelope Equation

3.2. The Design of PT Symmetric Potential

4. EP and Soliton Solutions

4.1. Property of Linear PT -Symmetric Potential

4.2. Property of Nonlinear PT -Symmetric Potential

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

Appendix A. Explicit Expression of Equation (3)

Appendix B. Perturbation Expansion of the MB Equations

References

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Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

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3. Realization of $PT$ -Symmetric Potential

3.2. The Design of $PT$ Symmetric Potential

4.1. Property of Linear $PT$ -Symmetric Potential

4.2. Property of Nonlinear $PT$ -Symmetric Potential