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 nRe (r⃗) should be an even function of r⃗ while the imaginary part of the RI nIm (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.

Download Full Size | PPT Slide | PDF

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 AlxGa1−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 [1922]. 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 [2527].

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 3S13P2. Two coherent control lights are applied. Control light 1 couples all the transitions |i〉 → |j〉 in 3S13P1 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 ωpωcω21, 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.

Download Full Size | PPT Slide | PDF

Tables Icon

Table 1. Definition of parameters

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 ω2′1′ = ω41 − Δc,42 + Δs1 and the resonance of the transition |1′〉 → |4′〉 is ω4′1′ = ω41 + Δs2, where Δs1=Ω1,822/Δ1,82Ωc,422/Δc,42+Ω22/Δ2+Ωc,612/Δc,61 and Δs2=Ωc,422/Δc,42+Ω22/Δ2+Ωc,612/Δc,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 ω1′3′ = ω63 − Δc,61 + Δs3 and ω6′3′ = ω63 + Δs4, where Δs3=Ω22/Δ2Ωc,612/Δc,61+Ω1,932/Δ1,93+Ωc,732/Δc,73 and Δs4=Ω1,932/Δ1,93+Ωc,732/Δc,73+Ωc,612/Δc,61.

 

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

Download Full Size | PPT Slide | PDF

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

χ=N|μ41|2ε0h¯[ξ1(ρ1ρ2)ω21ωp+iγ21+(ρ1ρ4)ω41ωp+iγ41]+N|μ63|2ε0h¯[ξ2(ρ3ρ1)ω13ωp+iγ13+(ρ3ρ6)ω63ωp+iγ63]+N|μ52|2ε0h¯[(ρ5ρ2)ω52+Ω1,822/Δ1,82+Ωc,422/Δ42ωp+iγ1],
where
ξ1=Ωc,422/Δc,422,ξ2=Ωc,612/Δc,612,
and
γ21=(1ξ1)γ2+ξ1γ1,γ41=(1ξ1)γ1+ξ1γ2γ31=(1ξ2)γ2+ξ2γ1,γ63=(1ξ2)γ1+ξ2γ2.

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

χc=N|μ41|2(ρ4ρ1)ε0h¯Δp,41+N|μ63|2(ρ6ρ3)ε0h¯Δp,63+N|μ52|2(ρ5ρ2)ε0h¯Δp,52,
which varies very little when two-photon resonance condition is satisfied. If ξ1 and ξ2 are small enough to make sure γ2′1′γ1′3′γ2, and let |μ41|2ξ1 (ρ1ρ2) = −|μ63|2ξ2 (ρ3ρ1), the susceptibility can be written as
χNε0h¯[|μ41|2ξ1(ρ1ρ2)ω21ωp+iγ21+|μ63|2ξ2(ρ3ρ1)ω13ωp+iγ13]+χc=χN(1δ+δωiγ21δδωiγ2)+χc,
where χN = N (ρ1ρ2) |μ41|2/(h̄ε0), δ = (ω2′1′ + ω1′3′)/2 − ωp, and δω = (ω2′1′ω1′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 ω1 = ωJ1, then Δ1,28 = −Δ1,39 due to Zeeman splitting caused by the magnetic field, and set the two control lights to be standing wave, i.e., Ω1,82 = Ω1,93 = Ω1 cos (kx ± π/4), Ω2 = Ω2 cos (kx), and let ωp = [ω2′1′ (x = 0) +ω1′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 Δ1,82 = −Δ1,93 = −50γ1. Thus it can be verified that the condition |μ41|2ξ1 (ρ2ρ1) = −|μ63|2ξ2 (ρ1ρ2) 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).

Tables Icon

Table 2. Values of parameters

In Fig. 4 we demonstrate that imaginary part of χ can be controlled via adjusting the intensity of control light 1. When Ω1,82 = Ω1,93 = 0, the imaginary part of χ is zero, thus we obtain a conventional Bragg reflector. When Ω1,82 = Ω1,93 = 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 kB. Because the fluctuation of χ is very small the real part of kB is indistinguishable from Fig. 5(a). However, the difference between the imaginary part of kB is obvious as shown in Fig. 5(b). kB of conventional Bragg reflector becomes complex around ωp1 = 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 ωp = ωJ2 − 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 Ω1,82 = Ω1,93 = 0.7653γ1 cos (kx + π/4) and ωp = ωJ2 − 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 Ω1,82 = Ω1,93 = 0.7653γ1 cos (kxπ/4) and ωp = ωJ2 − 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.

Download Full Size | PPT Slide | PDF

 

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

Download Full Size | PPT Slide | PDF

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 (rP) and when the probe light is antiparallel to +x-direction (rA) are shown in Fig. 6(a). We can see that rA is greatly enhanced when the wavelength of the probe light is near the Bragg wavelength, and rP 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, kp is the wavevector of the probe light. (a) Unidirectional invisibility occurs when the probe light is parallel to the +x-direction. In this case, Ω1,82 = Ω1,93 = 0.7653γ1 cos (kx + π/4) and ωp = ωJ2 − 99.9916γ1. rA, rP and t are shown. (b) Unidirectional invisibility occurs when the probe light is antiparallel to the +x-direction. In this case, Ω1,82 = Ω1,93 = 0.7653γ1 cos (kxπ/4) and ωp = ωJ2 − 99.9916γ1. rA, rP and t are shown. (c) The vapor is a conventional Bragg reflector when Ω1,82 = Ω1,93 = 0 and ωp = ωJ2 − 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).

Download Full Size | PPT Slide | PDF

When the phase difference between the control light 1 and 2 is −π/4, rP and rA 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.

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.

References and links

1. K. G. Makris, R. El-Ganainy, and D.N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008). [CrossRef]   [PubMed]  

2. F. Nazari, M. Nazari, and M. K. Morawej-Farshi, “A 2 × 2 spatial optical switch based on PT-symmetry,” Opt. Lett. 36, 4368–4370 (2011). [CrossRef]   [PubMed]  

3. A. A. Sukhorukov, Z. Y. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010). [CrossRef]  

4. M. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmetric laser amplifiers,” Opt. Lett. 37, 764–766 (2012). [CrossRef]   [PubMed]  

5. C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004). [CrossRef]  

6. Y. Goldsheid and B. A. Khoruzhenko, “Distribution of Eigenvalues in Non-Hermitian Anderson Models,” Phys. Rev. Lett. 80, 2897–2900 (1998). [CrossRef]  

7. I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A, 42, 153001 (2009). [CrossRef]  

8. S. Klaiman, N. Moiseyev, and Gunther, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101, 080402 (2008). [CrossRef]   [PubMed]  

9. A. Mostafazadeh, “Pseudo-Hermiticity versus PT-Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 3944 (2002). [CrossRef]  

10. C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010). [CrossRef]  

11. V. Yannopapas, “Spontaneous PT -symmetry breaking in complex frequency band structures,” Phys. Rev. A 89, 013808 (2014). [CrossRef]  

12. A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009). [CrossRef]  

13. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011). [CrossRef]   [PubMed]  

14. L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013). [CrossRef]  

15. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012). [CrossRef]   [PubMed]  

16. V. Yannopapas, “One-way photonic band gaps and optical isolation with three-dimensional photonic crystals of low symmetry,” Phys. Rev. A 88, 043837 (2013). [CrossRef]  

17. C. Wang, C. Zhou, and Z. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express , 19, 26948–26955 (2011). [CrossRef]  

18. M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012). [CrossRef]  

19. M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991). [CrossRef]   [PubMed]  

20. J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett. 70, 1421–1424 (1993). [CrossRef]   [PubMed]  

21. D. D. Yavuz, “Refractive Index Enhancement in a Far-Off Resonant Atomic System,” Phys. Rev. Lett. 95, 223601 (2005). [CrossRef]   [PubMed]  

22. C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011). [CrossRef]  

23. C. O’Brian and O. Kocharovskaya, “Optically Controllable Photonic Structures with Zero Absorption,” Phys. Rev. Lett. 107, 137401 (2011). [CrossRef]  

24. Z. Chen, B. Luo, and H. Guo, “Absorption-free Bragg reflector using Zeeman sublevels in atomic vapor,” Opt. Express , 22, 15564–15570 (2014). [CrossRef]   [PubMed]  

25. C. Hang and G. Huang, “𝒫𝒯 -Symmetry with a System of Three-Level Atoms,” and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013). [CrossRef]  

26. H. Li, J. Dou, and G. Huang, “PT symmetry via electromagnetically induced transparency, Opt. Express ,21, 32053–32062 (2013). [CrossRef]  

27. J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013). [CrossRef]  

28. W. L. Wiese and J. R. Fuhr, “Accurate Atomic Transition Probabilities for Hydrogen, Helium, and Lithium”, Journal of physical and chemical reference data , 38, 565–719 (2009). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. K. G. Makris, R. El-Ganainy, and D.N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [Crossref] [PubMed]
  2. F. Nazari, M. Nazari, and M. K. Morawej-Farshi, “A 2 × 2 spatial optical switch based on PT-symmetry,” Opt. Lett. 36, 4368–4370 (2011).
    [Crossref] [PubMed]
  3. A. A. Sukhorukov, Z. Y. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
    [Crossref]
  4. M. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmetric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).
    [Crossref] [PubMed]
  5. C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
    [Crossref]
  6. Y. Goldsheid and B. A. Khoruzhenko, “Distribution of Eigenvalues in Non-Hermitian Anderson Models,” Phys. Rev. Lett. 80, 2897–2900 (1998).
    [Crossref]
  7. I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A, 42, 153001 (2009).
    [Crossref]
  8. S. Klaiman, N. Moiseyev, and Gunther, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
    [Crossref] [PubMed]
  9. A. Mostafazadeh, “Pseudo-Hermiticity versus PT-Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 3944 (2002).
    [Crossref]
  10. C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [Crossref]
  11. V. Yannopapas, “Spontaneous PT -symmetry breaking in complex frequency band structures,” Phys. Rev. A 89, 013808 (2014).
    [Crossref]
  12. A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
    [Crossref]
  13. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
    [Crossref] [PubMed]
  14. L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
    [Crossref]
  15. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
    [Crossref] [PubMed]
  16. V. Yannopapas, “One-way photonic band gaps and optical isolation with three-dimensional photonic crystals of low symmetry,” Phys. Rev. A 88, 043837 (2013).
    [Crossref]
  17. C. Wang, C. Zhou, and Z. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express,  19, 26948–26955 (2011).
    [Crossref]
  18. M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
    [Crossref]
  19. M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
    [Crossref] [PubMed]
  20. J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett. 70, 1421–1424 (1993).
    [Crossref] [PubMed]
  21. D. D. Yavuz, “Refractive Index Enhancement in a Far-Off Resonant Atomic System,” Phys. Rev. Lett. 95, 223601 (2005).
    [Crossref] [PubMed]
  22. C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
    [Crossref]
  23. C. O’Brian and O. Kocharovskaya, “Optically Controllable Photonic Structures with Zero Absorption,” Phys. Rev. Lett. 107, 137401 (2011).
    [Crossref]
  24. Z. Chen, B. Luo, and H. Guo, “Absorption-free Bragg reflector using Zeeman sublevels in atomic vapor,” Opt. Express,  22, 15564–15570 (2014).
    [Crossref] [PubMed]
  25. C. Hang and G. Huang, “𝒫𝒯 -Symmetry with a System of Three-Level Atoms,” and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
    [Crossref]
  26. H. Li, J. Dou, and G. Huang, “PT symmetry via electromagnetically induced transparency, Opt. Express, 21, 32053–32062 (2013).
    [Crossref]
  27. J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
    [Crossref]
  28. W. L. Wiese and J. R. Fuhr, “Accurate Atomic Transition Probabilities for Hydrogen, Helium, and Lithium”, Journal of physical and chemical reference data,  38, 565–719 (2009).
    [Crossref]

2014 (2)

V. Yannopapas, “Spontaneous PT -symmetry breaking in complex frequency band structures,” Phys. Rev. A 89, 013808 (2014).
[Crossref]

Z. Chen, B. Luo, and H. Guo, “Absorption-free Bragg reflector using Zeeman sublevels in atomic vapor,” Opt. Express,  22, 15564–15570 (2014).
[Crossref] [PubMed]

2013 (5)

C. Hang and G. Huang, “𝒫𝒯 -Symmetry with a System of Three-Level Atoms,” and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

H. Li, J. Dou, and G. Huang, “PT symmetry via electromagnetically induced transparency, Opt. Express, 21, 32053–32062 (2013).
[Crossref]

J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
[Crossref]

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

V. Yannopapas, “One-way photonic band gaps and optical isolation with three-dimensional photonic crystals of low symmetry,” Phys. Rev. A 88, 043837 (2013).
[Crossref]

2012 (3)

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

M. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmetric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).
[Crossref] [PubMed]

2011 (5)

F. Nazari, M. Nazari, and M. K. Morawej-Farshi, “A 2 × 2 spatial optical switch based on PT-symmetry,” Opt. Lett. 36, 4368–4370 (2011).
[Crossref] [PubMed]

C. Wang, C. Zhou, and Z. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express,  19, 26948–26955 (2011).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
[Crossref]

C. O’Brian and O. Kocharovskaya, “Optically Controllable Photonic Structures with Zero Absorption,” Phys. Rev. Lett. 107, 137401 (2011).
[Crossref]

2010 (2)

A. A. Sukhorukov, Z. Y. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[Crossref]

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

2009 (3)

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A, 42, 153001 (2009).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

W. L. Wiese and J. R. Fuhr, “Accurate Atomic Transition Probabilities for Hydrogen, Helium, and Lithium”, Journal of physical and chemical reference data,  38, 565–719 (2009).
[Crossref]

2008 (2)

S. Klaiman, N. Moiseyev, and Gunther, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, and D.N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

2005 (1)

D. D. Yavuz, “Refractive Index Enhancement in a Far-Off Resonant Atomic System,” Phys. Rev. Lett. 95, 223601 (2005).
[Crossref] [PubMed]

2004 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

2002 (1)

A. Mostafazadeh, “Pseudo-Hermiticity versus PT-Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 3944 (2002).
[Crossref]

1998 (1)

Y. Goldsheid and B. A. Khoruzhenko, “Distribution of Eigenvalues in Non-Hermitian Anderson Models,” Phys. Rev. Lett. 80, 2897–2900 (1998).
[Crossref]

1993 (1)

J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett. 70, 1421–1424 (1993).
[Crossref] [PubMed]

1991 (1)

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
[Crossref] [PubMed]

Aceves, A. B.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Almeida, V. R.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Anisimov, P. M.

C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
[Crossref]

Bender, C. M.

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

Bersch, C.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Bowden, C. M.

J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett. 70, 1421–1424 (1993).
[Crossref] [PubMed]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Chen, Y. F.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Chen, Z.

Christodoulides, D. N.

J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
[Crossref]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

M. Miri, P. LiKamWa, and D. N. Christodoulides, “Large area single-mode parity-time-symmetric laser amplifiers,” Opt. Lett. 37, 764–766 (2012).
[Crossref] [PubMed]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Christodoulides, D.N.

K. G. Makris, R. El-Ganainy, and D.N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Dou, J.

Dowling, J. P.

J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett. 70, 1421–1424 (1993).
[Crossref] [PubMed]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

El-Ganainy, R.

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, and D.N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Fegadolli, W. S.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Feng, L.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Fuhr, J. R.

W. L. Wiese and J. R. Fuhr, “Accurate Atomic Transition Probabilities for Hydrogen, Helium, and Lithium”, Journal of physical and chemical reference data,  38, 565–719 (2009).
[Crossref]

Goldsheid, Y.

Y. Goldsheid and B. A. Khoruzhenko, “Distribution of Eigenvalues in Non-Hermitian Anderson Models,” Phys. Rev. Lett. 80, 2897–2900 (1998).
[Crossref]

Gunther,

S. Klaiman, N. Moiseyev, and Gunther, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref] [PubMed]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Guo, H.

Hang, C.

C. Hang and G. Huang, “𝒫𝒯 -Symmetry with a System of Three-Level Atoms,” and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

Huang, G.

C. Hang and G. Huang, “𝒫𝒯 -Symmetry with a System of Three-Level Atoms,” and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

H. Li, J. Dou, and G. Huang, “PT symmetry via electromagnetically induced transparency, Opt. Express, 21, 32053–32062 (2013).
[Crossref]

Jones, H. F.

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

Khoruzhenko, B. A.

Y. Goldsheid and B. A. Khoruzhenko, “Distribution of Eigenvalues in Non-Hermitian Anderson Models,” Phys. Rev. Lett. 80, 2897–2900 (1998).
[Crossref]

Kip, D.

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Kivshar, Y. S.

A. A. Sukhorukov, Z. Y. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[Crossref]

Klaiman, S.

S. Klaiman, N. Moiseyev, and Gunther, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref] [PubMed]

Kocharovskaya, O.

C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
[Crossref]

C. O’Brian and O. Kocharovskaya, “Optically Controllable Photonic Structures with Zero Absorption,” Phys. Rev. Lett. 107, 137401 (2011).
[Crossref]

Kottos, T.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Kovanis, V.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

Li, H.

Li, Z.

LiKamWa, P.

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Lu, M. H.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Luo, B.

Makris, K. G.

K. G. Makris, R. El-Ganainy, and D.N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Makris, K.G.

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Miri, M.

Miri, M. A.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

Miri, M-A.

J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
[Crossref]

Moiseyev, N.

S. Klaiman, N. Moiseyev, and Gunther, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref] [PubMed]

Morandomdotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Morawej-Farshi, M. K.

Mostafazadeh, A.

A. Mostafazadeh, “Pseudo-Hermiticity versus PT-Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 3944 (2002).
[Crossref]

Nazari, F.

Nazari, M.

O’Brian, C.

C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
[Crossref]

C. O’Brian and O. Kocharovskaya, “Optically Controllable Photonic Structures with Zero Absorption,” Phys. Rev. Lett. 107, 137401 (2011).
[Crossref]

Oliveira, J. E. B.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Onishchukov, G.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Peschel, U.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Regensburger, A.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Rostovtsev, Y.

C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
[Crossref]

Rotter, I.

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A, 42, 153001 (2009).
[Crossref]

Rüter, C. E.

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Scherer, A.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Scully, M. O.

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
[Crossref] [PubMed]

Segev, M.

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Sheng, J.

J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
[Crossref]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Sukhorukov, A. A.

A. A. Sukhorukov, Z. Y. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[Crossref]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Wang, C.

Wiese, W. L.

W. L. Wiese and J. R. Fuhr, “Accurate Atomic Transition Probabilities for Hydrogen, Helium, and Lithium”, Journal of physical and chemical reference data,  38, 565–719 (2009).
[Crossref]

Xiao, M.

J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
[Crossref]

Xu, Y. L.

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Xu, Z. Y.

A. A. Sukhorukov, Z. Y. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[Crossref]

Yannopapas, V.

V. Yannopapas, “Spontaneous PT -symmetry breaking in complex frequency band structures,” Phys. Rev. A 89, 013808 (2014).
[Crossref]

V. Yannopapas, “One-way photonic band gaps and optical isolation with three-dimensional photonic crystals of low symmetry,” Phys. Rev. A 88, 043837 (2013).
[Crossref]

Yavuz, D. D.

D. D. Yavuz, “Refractive Index Enhancement in a Far-Off Resonant Atomic System,” Phys. Rev. Lett. 95, 223601 (2005).
[Crossref] [PubMed]

Zhou, C.

J. Math. Phys. (1)

A. Mostafazadeh, “Pseudo-Hermiticity versus PT-Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 3944 (2002).
[Crossref]

J. Phys. A, (1)

I. Rotter, “A non-Hermitian Hamilton operator and the physics of open quantum systems,” J. Phys. A, 42, 153001 (2009).
[Crossref]

Journal of physical and chemical reference data (1)

W. L. Wiese and J. R. Fuhr, “Accurate Atomic Transition Probabilities for Hydrogen, Helium, and Lithium”, Journal of physical and chemical reference data,  38, 565–719 (2009).
[Crossref]

Nat. Phys. (1)

C. E. Rüter, K.G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Nature (1)

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices”, Nature 488, 167–171 (2012).
[Crossref] [PubMed]

Nature Mat. (1)

L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. B. Oliveira, V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Mat. 12(2), 108–113 (2013).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. A (6)

A. A. Sukhorukov, Z. Y. Xu, and Y. S. Kivshar, “Nonlinear suppression of time reversals in PT-symmetric optical couplers,” Phys. Rev. A 82, 043818 (2010).
[Crossref]

V. Yannopapas, “Spontaneous PT -symmetry breaking in complex frequency band structures,” Phys. Rev. A 89, 013808 (2014).
[Crossref]

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A 86, 033801 (2012).
[Crossref]

J. Sheng, M-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88, 041803 (R) (2013).
[Crossref]

V. Yannopapas, “One-way photonic band gaps and optical isolation with three-dimensional photonic crystals of low symmetry,” Phys. Rev. A 88, 043837 (2013).
[Crossref]

C. O’Brian, P. M. Anisimov, Y. Rostovtsev, and O. Kocharovskaya, “Coherent control of refractive index in far-detuned Λ systems,” Phys. Rev. A 84, 063835 (2011).
[Crossref]

Phys. Rev. D (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Extension of PT-symmetric quantum mechanics to quantum field theory with cubic interaction,” Phys. Rev. D 70, 025001 (2004).
[Crossref]

Phys. Rev. Lett, (1)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandomdotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of -Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett, 103, 093902 (2009).
[Crossref]

Phys. Rev. Lett. (9)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional Invisibility Induced by PT-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref] [PubMed]

Y. Goldsheid and B. A. Khoruzhenko, “Distribution of Eigenvalues in Non-Hermitian Anderson Models,” Phys. Rev. Lett. 80, 2897–2900 (1998).
[Crossref]

M. O. Scully, “Enhancement of the index of refraction via quantum coherence,” Phys. Rev. Lett. 67, 1855–1858 (1991).
[Crossref] [PubMed]

J. P. Dowling and C. M. Bowden, “Near dipole-dipole effects in lasing without inversion: An enhancement of gain and absorptionless index of refraction,” Phys. Rev. Lett. 70, 1421–1424 (1993).
[Crossref] [PubMed]

D. D. Yavuz, “Refractive Index Enhancement in a Far-Off Resonant Atomic System,” Phys. Rev. Lett. 95, 223601 (2005).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, and D.N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

S. Klaiman, N. Moiseyev, and Gunther, “Visualization of Branch Points in PT-Symmetric Waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
[Crossref] [PubMed]

C. O’Brian and O. Kocharovskaya, “Optically Controllable Photonic Structures with Zero Absorption,” Phys. Rev. Lett. 107, 137401 (2011).
[Crossref]

C. Hang and G. Huang, “𝒫𝒯 -Symmetry with a System of Three-Level Atoms,” and V. V. Konotop, Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 (a) Basic experimental setup of the study of ���� -symmetric media based on the similarity between scalar wave equation with paraxial approximation and Schrödinger equation. (b) Basic experimental setup of the study of ���� -symmetric Bragg structure.
Fig. 2
Fig. 2 (a) Level configuration in helium or alkaline atoms to realize the ���� -symmetric Bragg structure. (b) A possible experimental setup. Both control lights are set to be standing wave with phase difference of π/4.
Fig. 3
Fig. 3 (a) Equivalent Ξ-system of the first Λ-system. (b) Equivalent Ξ-system of the second Λ-system.
Fig. 4
Fig. 4 Numerical simulation results of χ. When we turn off control light 1 and set ωp = ωJ2 − 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 Ω1,82 = Ω1,93 = 0.7653γ1 cos (kx + π/4) and ωp = ωJ2 − 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 Ω1,82 = Ω1,93 = 0.7653γ1 cos (kxπ/4) and ωp = ωJ2 − 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.
Fig. 5
Fig. 5 The band structure of the conventional Bragg reflector and the ���� -symmetric Bragg structure working at breaking point. kB is the Brillouin wavenumber (normalized by k). (a) Real part of kB of both Bragg reflector and ���� -symmetric Bragg structure working at breaking point. (b) Imaginary part of kB of conventional Bragg reflector (blue, dashed) and ���� -symmetric Bragg structure working at the breaking point.
Fig. 6
Fig. 6 The phenomenon of unidirectional invisibility. 1000 periods are used to simulate the transmittance and reflectivity, kp is the wavevector of the probe light. (a) Unidirectional invisibility occurs when the probe light is parallel to the +x-direction. In this case, Ω1,82 = Ω1,93 = 0.7653γ1 cos (kx + π/4) and ωp = ωJ2 − 99.9916γ1. rA, rP and t are shown. (b) Unidirectional invisibility occurs when the probe light is antiparallel to the +x-direction. In this case, Ω1,82 = Ω1,93 = 0.7653γ1 cos (kxπ/4) and ωp = ωJ2 − 99.9916γ1. rA, rP and t are shown. (c) The vapor is a conventional Bragg reflector when Ω1,82 = Ω1,93 = 0 and ωp = ωJ2 − 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).

Tables (2)

Tables Icon

Table 1 Definition of parameters

Tables Icon

Table 2 Values of parameters

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

χ = N | μ 41 | 2 ε 0 h ¯ [ ξ 1 ( ρ 1 ρ 2 ) ω 2 1 ω p + i γ 2 1 + ( ρ 1 ρ 4 ) ω 4 1 ω p + i γ 4 1 ] + N | μ 63 | 2 ε 0 h ¯ [ ξ 2 ( ρ 3 ρ 1 ) ω 1 3 ω p + i γ 1 3 + ( ρ 3 ρ 6 ) ω 6 3 ω p + i γ 6 3 ] + N | μ 52 | 2 ε 0 h ¯ [ ( ρ 5 ρ 2 ) ω 52 + Ω 1 , 82 2 / Δ 1 , 82 + Ω c , 42 2 / Δ 42 ω p + i γ 1 ] ,
ξ 1 = Ω c , 42 2 / Δ c , 42 2 , ξ 2 = Ω c , 61 2 / Δ c , 61 2 ,
γ 2 1 = ( 1 ξ 1 ) γ 2 + ξ 1 γ 1 , γ 4 1 = ( 1 ξ 1 ) γ 1 + ξ 1 γ 2 γ 3 1 = ( 1 ξ 2 ) γ 2 + ξ 2 γ 1 , γ 6 3 = ( 1 ξ 2 ) γ 1 + ξ 2 γ 2 .
χ c = N | μ 41 | 2 ( ρ 4 ρ 1 ) ε 0 h ¯ Δ p , 41 + N | μ 63 | 2 ( ρ 6 ρ 3 ) ε 0 h ¯ Δ p , 63 + N | μ 52 | 2 ( ρ 5 ρ 2 ) ε 0 h ¯ Δ p , 52 ,
χ N ε 0 h ¯ [ | μ 41 | 2 ξ 1 ( ρ 1 ρ 2 ) ω 2 1 ω p + i γ 2 1 + | μ 63 | 2 ξ 2 ( ρ 3 ρ 1 ) ω 1 3 ω p + i γ 1 3 ] + χ c = χ N ( 1 δ + δ ω i γ 2 1 δ δ ω i γ 2 ) + χ c ,

Metrics