## Abstract

We propose an efficient scheme in helium or alkaline earth atomic vapor to achieve a parity-time symmetric Bragg structure using coherent lights. Unidirectional invisibility can be realized in this scheme, i.e., the atomic vapor shows total transparency for probe light incident from one particular direction, but exhibits enhanced Bragg reflection for probe from the opposite side. By changing the relative phase between the coherent lights, this direction can easily be manipulated, providing a convenient way for investigating special properties of *𝒫𝒯* -symmetric Bragg structures.

© 2014 Optical Society of America

## 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].

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 LiNiO_{3} crystals [10], photonic crystals [11], and Al_{x}Ga_{1−x}As [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 Δ*m* = 0 (Δ*m* = 1) in ^{3}S_{1} → ^{3}P_{2}. Two coherent control lights are applied. Control light 1 couples all the transitions |*i*〉 → |*j*〉 in ^{3}S_{1} → ^{3}P_{1} satisfying Δ*m* = 0 except |^{3}S_{1}, *m* = 0〉 → |^{3}P_{1}, *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}*ω*

_{21},

*ω*

_{13}. A static magnetic field is applied to induce Zeeman effect to control

*ω*

_{21}and

*ω*

_{13}. 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.

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′} = *ω*_{41} − Δ_{c,42} + Δ_{s1} and the resonance of the transition |1′〉 → |4′〉 is *ω*_{4′1′} = *ω*_{41} + 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′} = *ω*_{63} − Δ_{c,61} + Δ_{s3} and *ω*_{6′3′} = *ω*_{63} + 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
${\mathrm{\Delta}}_{s4}={\mathrm{\Omega}}_{1,93}^{2}/{\mathrm{\Delta}}_{1,93}+{\mathrm{\Omega}}_{c,73}^{2}/{\mathrm{\Delta}}_{c,73}+{\mathrm{\Omega}}_{c,61}^{2}/{\mathrm{\Delta}}_{c,61}$.

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

*ξ*

_{1}and

*ξ*

_{2}are small enough to make sure

*γ*

_{2′1′}≈

*γ*

_{1′3′}≈

*γ*

_{2}, and let |

*μ*

_{41}|

^{2}

*ξ*

_{1}(

*ρ*

_{1}−

*ρ*

_{2}) = −|

*μ*

_{63}|

^{2}

*ξ*

_{2}(

*ρ*

_{3}−

*ρ*

_{1}), the susceptibility can be written as

*χ*=

_{N}*N*(

*ρ*

_{1}−

*ρ*

_{2}) |

*μ*

_{41}|

^{2}/(

*h̄ε*

_{0}),

*δ*= (

*ω*

_{2′1′}+

*ω*

_{1′3′})/2 −

*ω*, and

_{p}*δω*= (

*ω*

_{2′1′}−

*ω*

_{1′3′})/2. To satisfy the

*𝒫𝒯*- symmetry condition, i.e.,

*χ*(

*x*) =

*χ*

^{*}(−

*x*),

*δ*and

*δω*should be odd and even function of

*x*. If we let

*ω*

_{1}=

*ω*

_{J1}, then Δ

_{1,28}= −Δ

_{1,39}due to Zeeman splitting caused by the magnetic field, and set the two control lights to be standing wave, i.e., Ω

_{1,82}= Ω

_{1,93}= Ω

_{1}cos (

*kx*±

*π*/4), Ω

_{2}= Ω

_{2}cos (

*kx*), and let

*ω*= [

_{p}*ω*

_{2′1′}(

*x*= 0) +

*ω*

_{1′3′}(

*x*= 0)]/2 we can make sure that

*δ*and

*δω*will be antisymmetric and symmetric with respect to

*x*respectively, thus achieving the

*𝒫𝒯*-symmetric Bragg structures.

#### 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} = −50*γ*_{1}. Thus it can be verified that the condition |*μ*_{41}|^{2}*ξ*_{1} (*ρ*_{2} − *ρ*_{1}) = −|*μ*_{63}|^{2}*ξ*_{2} (*ρ*_{1} − *ρ*_{2}) is satisfied. By setting the two control lights to be standing wave, we can achieve the *𝒫𝒯* -symmetric Bragg structure. We let Ω_{2} = *γ*_{1} cos (*kx*).

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} = 0.7653*γ*_{1} 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*is indistinguishable from Fig. 5(a). However, the difference between the imaginary part of

_{B}*k*is obvious as shown in Fig. 5(b).

_{B}*k*of conventional Bragg reflector becomes complex around

_{B}*ω*

_{p}/ω_{1}= 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.

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*) are shown in Fig. 6(a). We can see that

_{A}*r*is greatly enhanced when the wavelength of the probe light is near the Bragg wavelength, and

_{A}*r*is nearly zero in a broad frequency range. The amplitude and argument of the transmittance (

_{P}*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.

When the phase difference between the control light 1 and 2 is −*π*/4, *r _{P}* and

*r*exchange and

_{A}*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.

## Acknowledgments

This work is supported by the National Science Fund for Distinguished Young Scholars of China (Grant No. 61225003), National Natural Science Foundation of China (Grant No. 61101081), and the National Hi-Tech Research and Development (863) Program.

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