1. Introduction

Recently, materials with parity-time (𝒫𝒯)-symmetric refractive index (RI) has been widely studied. The refractive index n(r⃗) is said to be 𝒫𝒯 -symmetric provided the condition n(r⃗) = n^* (r⃗) is satisfied. It requires that the real part of the RI n_Re (r⃗) should be an even function of r⃗ while the imaginary part of the RI n_Im (r⃗) should be an odd function of r⃗. Various researches have been carried out to investigate the special properties of 𝒫𝒯 -symmetric materials, such as double refraction, non-reciprocal light propagation, and power oscillations [1]. Some useful schemes based on 𝒫𝒯 -symmetric materials have been proposed to realize optical switchers [2], optical couplers [3], and single-mode amplifiers [4].

One important motivation of investigating these materials is based on the similarity between scalar wave equation with paraxial approximation and Schrödinger equation. The experimental setup is shown in Fig. 1(a). The propagation direction of the probe light is perpendicular to the varying direction of the RI. The RI is analogous to potential and the axial wave vector is analogous to energy due to the formal similarity. Thus studying the propagation of light in the 𝒫𝒯 -symmetrical materials may greatly boost the study of the 𝒫𝒯 -symmetric potentials, which is of great importance in many disciplines, such as quantum field theories [5], non-Hermitian Anderson models [6] and open quantum systems [7].

 

Fig. 1 (a) Basic experimental setup of the study of 𝒫𝒯 -symmetric media based on the similarity between scalar wave equation with paraxial approximation and Schrödinger equation. (b) Basic experimental setup of the study of 𝒫𝒯 -symmetric Bragg structure.

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Among the special properties of 𝒫𝒯 -symmetric potentials, the most significant property is spontaneous breaking of 𝒫𝒯 -symmetry. In some cases the spontaneous breaking of 𝒫𝒯 -symmetry is manifest that the eigenvalue spectrum of the total Hamiltonian will turn to be complex from real values while the imaginary part of the potential exceeds 𝒫𝒯 -symmetry breaking point [8, 9]. This phenomenon has been proved in coupled waveguide system based on Fe-doped LiNiO₃ crystals [10], photonic crystals [11], and Al_xGa_1−xAs [12].

Other studies focus on the properties of 𝒫𝒯 -symmetric Bragg structures with periodic 𝒫𝒯 -symmetric RI. The basic experimental setup is shown in Fig. 1(b). The propagation direction of the light is parallel to the variation direction of the RI. It is well known that the common Bragg structures which have neither gain nor absorption will be highly reflective if the probe light is near the Bragg wavelength. However, when the RI is at the 𝒫𝒯 -symmetry breaking point, the intensity and phase of light incident from one side after propagating through the structure is indistinguishable from the case when there is no grating [13]. This phenomenon is called unidirectional invisibility, which is the most significant characteristic of 𝒫𝒯 -symmetric Bragg structures. This phenomenon has been observed in silicon waveguides [14] and photonic crystals [15]. These systems, as well as other systems possessing nonreciprocity [16, 17], are promising candidates to realize optical isolators. Bragg solitons in nonlinear 𝒫𝒯 -symmetric media are also investigated [18].

Atom vapor is considered to be a promising candidate for RI engineering. Many suggestions has been proposed to uniformly enhance the RI in atomic vapor and to eliminate the absorption simultaneously [19–22]. Conventional absorption-free Bragg reflector can also be achieved by illuminating a standing-wave of a laser field in a homogeneous atomic medium with a con-figuration of equivalent Ξ-system having successive inverted and non-inverted transitions with slightly different transition frequencies [23, 24]. Some proposals using atomic vapors are also suggested to create PT -symmetric materials to study the propagation of light based on the similarity between paraxial propagation equation and Schrödinger equation [25–27].

In this paper, we propose a scheme based on the proposal in [24] in helium or alkaline atoms to achieve a 𝒫𝒯 -symmetric Bragg structure. Using two coherent control lights and a static magnetic field, we can achieve 𝒫𝒯 -symmetric RI. By changing the related parameters of the optical fields, the periodic RI can be conveniently modulated. We can make the system work on the 𝒫𝒯 -symmetry breaking point with proper parameters, hence unidirectional invisibility can be observed.

2. Physical model and analysis

2.1. Proposal using helium atoms

The level configuration to realize the 𝒫𝒯 -symmetric Bragg structure is shown in Fig. 2(a). The coupling light (probe light) couples all the transitions |i〉 → |j〉 satisfying Δm = 0 (Δm = 1) in ³S₁ → ³P₂. Two coherent control lights are applied. Control light 1 couples all the transitions |i〉 → |j〉 in ³S₁ → ³P₁ satisfying Δm = 0 except |³S₁, m = 0〉 → |³P₁, m = 0〉 which is forbidden by selection rules. Control light 2 couples |1〉 → |0〉. All the detunings are set to be far greater than the decoherence rates. The frequencies of probe and coupling lights are set to satisfy ω_p − ω_c ≈ ω₂₁, ω₁₃. A static magnetic field is applied to induce Zeeman effect to control ω₂₁ and ω₁₃. A possible experimental scheme is shown in Fig. 2(b). The probe light can be incident either parallel or antiparallel to +x-direction. The definition of all the parameters are shown in Table 1.

 

Fig. 2 (a) Level configuration in helium or alkaline atoms to realize the 𝒫𝒯 -symmetric Bragg structure. (b) A possible experimental setup. Both control lights are set to be standing wave with phase difference of π/4.

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Table 1. Definition of parameters

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This level configuration can be seen as a summation of two Λ-systems, formed by |1〉, |2〉, |4〉 and |1〉, |3〉, |6〉, respectively, and a simple two-level system formed by |2〉 and |5〉. The Λ-structure formed by |1〉, |2〉 and |4〉 can been seen as a Ξ-system formed by |1′〉, |2′〉 and |4′〉 shown in Fig. 3(a) [22, 23]. The resonance of transition |1′〉 → |2′〉 is ω_2′1′ = ω₄₁ − Δ_c,42 + Δ_s1 and the resonance of the transition |1′〉 → |4′〉 is ω_4′1′ = ω₄₁ + Δ_s2, where ${\mathrm{\Delta }}_{s1}=-{\mathrm{\Omega }}_{1,82}^2/{\mathrm{\Delta }}_{1,82}-{\mathrm{\Omega }}_{c,42}^2/{\mathrm{\Delta }}_{c,42}+{\mathrm{\Omega }}_2^2/{\mathrm{\Delta }}_2+{\mathrm{\Omega }}_{c,61}^2/{\mathrm{\Delta }}_{c,61}$ and ${\mathrm{\Delta }}_{s2}={\mathrm{\Omega }}_{c,42}^2/{\mathrm{\Delta }}_{c,42}+{\mathrm{\Omega }}_2^2/{\mathrm{\Delta }}_2+{\mathrm{\Omega }}_{c,61}^2/{\mathrm{\Delta }}_{c,61}$ are stark shift caused by the coupling light and the control lights. Similarly the Λ-system formed by |1〉, |3〉 and |6〉 is equivalent to the three-level system formed by |1′〉, |3′〉 and |6′〉 as shown in Fig. 3(b). The resonances of transition |3′〉 → |1′〉 and |3′〉 → |6′〉 are ω_1′3′ = ω₆₃ − Δ_c,61 + Δ_s3 and ω_6′3′ = ω₆₃ + Δ_s4, where ${\mathrm{\Delta }}_{s3}=-{\mathrm{\Omega }}_2^2/{\mathrm{\Delta }}_2-{\mathrm{\Omega }}_{c,61}^2/{\mathrm{\Delta }}_{c,61}+{\mathrm{\Omega }}_{1,93}^2/{\mathrm{\Delta }}_{1,93}+{\mathrm{\Omega }}_{c,73}^2/{\mathrm{\Delta }}_{c,73}$ and ${\mathrm{\Delta }}_{s4}={\mathrm{\Omega }}_{1,93}^2/{\mathrm{\Delta }}_{1,93}+{\mathrm{\Omega }}_{c,73}^2/{\mathrm{\Delta }}_{c,73}+{\mathrm{\Omega }}_{c,61}^2/{\mathrm{\Delta }}_{c,61}$.

 

Fig. 3 (a) Equivalent Ξ-system of the first Λ-system. (b) Equivalent Ξ-system of the second Λ-system.

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The susceptibility χ of the medium at the frequency of ω_p is

Since the two photon resonance condition is satisfied, and all lights are far-detuned, the contribution of transition |1′〉 → |4′〉, |3′〉 → |6′〉, and |2〉 → |5〉 to the absorption can be neglected and the contribution to the the dispersion can be approximately written as

2.2. Numerical simulation

We give an example of using helium atoms to realize the 𝒫𝒯 -symmetric Bragg structure. The values of parameters are listed in Table 2 [28]. Based on these parameters we can derive Δ_1,82 = −Δ_1,93 = −50γ₁. Thus it can be verified that the condition |μ₄₁|²ξ₁ (ρ₂ − ρ₁) = −|μ₆₃|²ξ₂ (ρ₁ − ρ₂) is satisfied. By setting the two control lights to be standing wave, we can achieve the 𝒫𝒯 -symmetric Bragg structure. We let Ω₂ = γ₁ cos (kx).

Table 2. Values of parameters

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In Fig. 4 we demonstrate that imaginary part of χ can be controlled via adjusting the intensity of control light 1. When Ω_1,82 = Ω_1,93 = 0, the imaginary part of χ is zero, thus we obtain a conventional Bragg reflector. When Ω_1,82 = Ω_1,93 = 0.7653γ₁ cos (kx ± π/4), the imaginary part of χ reaches the 𝒫𝒯 -symmetry breaking point. In this case, maximum value of real and imaginary part of deviation from ideal 𝒫𝒯 -symmetry χ(x) − χ^* (−x) are 4.5% of the corresponding fluctuation amplitude of the corresponding part of χ and can be neglected. The average susceptibility of the probe light is 0, which ensure that the wavelength of the probe light nearly equals the Bragg wavelength. In Fig. 5 the band structure of the conventional Bragg reflector and the 𝒫𝒯 -symmetric Bragg structure working at breaking point is illustrated. In Fig. 5(a) we show the real part of Brillouin wavenumber k_B. Because the fluctuation of χ is very small the real part of k_B is indistinguishable from Fig. 5(a). However, the difference between the imaginary part of k_B is obvious as shown in Fig. 5(b). k_B of conventional Bragg reflector becomes complex around ω_p/ω₁ = 1, indicating a band gap, but in the case of 𝒫𝒯 -symmetric Bragg reflector working at the breaking point, the band gap almost vanishes, which is the characteristic of 𝒫𝒯 -symmetry breaking point.

 

Fig. 4 Numerical simulation results of χ. When we turn off control light 1 and set ω_p = ω_J2 − 99.9975γ₁, the real (red, dotted) and imaginary (red, dot dashed) part of χ are shown. In this case, we obtain a conventional Bragg reflector. When we set Ω_1,82 = Ω_1,93 = 0.7653γ₁ cos (kx + π/4) and ω_p = ω_J2 − 99.9916γ₁, we can get the real (blue, solid) and imaginary(blue, dashed) part of χ. In this case, unidirectional invisibility can be observed when the probe light is parallel to +x-direction. The case when Ω_1,82 = Ω_1,93 = 0.7653γ₁ cos (kx − π/4) and ω_p = ω_J2 − 99.9916γ₁ is not shown here. In this case χ is the complex conjugate of that when Ω₁ = 0.7653γ₁ cos (kx + π/4) and unidirectional invisibility occurs when the probe light is antiparallel to the +x-direction.

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Fig. 5 The band structure of the conventional Bragg reflector and the 𝒫𝒯 -symmetric Bragg structure working at breaking point. k_B is the Brillouin wavenumber (normalized by k). (a) Real part of k_B of both Bragg reflector and 𝒫𝒯 -symmetric Bragg structure working at breaking point. (b) Imaginary part of k_B of conventional Bragg reflector (blue, dashed) and 𝒫𝒯 -symmetric Bragg structure working at the breaking point.

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When the phase difference between the control light 1 and 2 is π/4, the unidirectional invisibility will occur when the probe light is parallel to +x-direction. The amplitude of the reflectivity when the probe light is parallel to +x-direction (r_P) and when the probe light is antiparallel to +x-direction (r_A) are shown in Fig. 6(a). We can see that r_A is greatly enhanced when the wavelength of the probe light is near the Bragg wavelength, and r_P is nearly zero in a broad frequency range. The amplitude and argument of the transmittance (t) are the same whether the probe light is parallel or antiparallel to the +x-direction, and are shown in Fig. 6(a) and (d) respectively. The transmittance |t| ∼ 1 and the argument of t is nearly zero in a broad frequency band. Thus if the probe light is parallel to the +x-direction, the phase and amplitude will be indistinguishable after propagating through the vapor from the case when no vapor exist and no reflection will occur, i.e., unidirectional invisibility can be observed in the direction parallel to the +x-direction, demonstrating we can realize 𝒫𝒯 -symmetric Bragg structure using atomic vapor.

 

Fig. 6 The phenomenon of unidirectional invisibility. 1000 periods are used to simulate the transmittance and reflectivity, k_p is the wavevector of the probe light. (a) Unidirectional invisibility occurs when the probe light is parallel to the +x-direction. In this case, Ω_1,82 = Ω_1,93 = 0.7653γ₁ cos (kx + π/4) and ω_p = ω_J2 − 99.9916γ₁. r_A, r_P and t are shown. (b) Unidirectional invisibility occurs when the probe light is antiparallel to the +x-direction. In this case, Ω_1,82 = Ω_1,93 = 0.7653γ₁ cos (kx − π/4) and ω_p = ω_J2 − 99.9916γ₁. r_A, r_P and t are shown. (c) The vapor is a conventional Bragg reflector when Ω_1,82 = Ω_1,93 = 0 and ω_p = ω_J2 − 99.9975γ₁. r and t are shown. (d) The argument of transmittance t when unidirectional invisibility occurs (blue, solid) and when the vapor acs as a conventional Bragg reflector (red, dashed).

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When the phase difference between the control light 1 and 2 is −π/4, r_P and r_A exchange and t remains the same as shown in Fig. 6 (b). Now the unidirectional invisibility occurs when the probe light is antiparallel to the +x-direction.

When we turn off control light 1, the scheme turns out to be a conventional Bragg reflector with the imaginary part of χ equals zero as shown in Fig. 4 [24]. The reflectivity (r) and transmittance (t) is the same no matter its direction is parallel or antiparallel to +x-direction as shown in Fig. 6(c). The incident light will be mostly reflected as long as its frequency is in the Bragg band, and will acquire additional phase when its frequency is not in the center of Bragg band after propagating through the atomic vapor as shown in Fig. 6(d).

3. Conclusion

In conclusion, We demonstrate that a 𝒫𝒯 -symmetric Bragg structure can be realized in the helium or alkaline earth atomic vapours. Compared to 𝒫𝒯 -symmetric Bragg structures realized in linear integrated optical settings, this scheme is more flexible because the fluctuation amplitude of RI can be controlled dynamically by adjusting the intensities of the two control lights, and the direction of unidirectional invisibility can be conveniently manipulated by changing relative phase of the two control lights, paving the way for investigating more interesting phenomena of 𝒫𝒯 -symmetric Bragg structures.