Abstract

An asymmetric double semiconductor quantum well is proposed to realize two-dimensional parity-time (PT) symmetry and an electromagnetically induced grating. In such a nontrivial grating with PT symmetry, the incident probe photons can be diffracted to selected angles depending on the spatial relationship of the real and imaginary parts of the refractive index. Such results are due to the interference mechanism between the amplitude and phase of the grating and can be manipulated by the probe detuning, modulation amplitudes of the standing wave fields, and interaction length of the medium. Such a system may lead to new approaches of observing PT-symmetry-related phenomena and has potential applications in photoelectric devices requiring asymmetric light transport using semiconductor quantum wells.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Electromagnetically induced transparency (EIT) [1–3] is one of the most typical phenomena based on laser-induced atomic coherence. EIT has important applications in several fields such as slow light [4,5], light storage [6,7], optical switching [8,9], enhanced Kerr nonlinearity [10,11], and four-wave mixing [12,13]. When a traveling wave (TW) field is replaced by a standing wave (SW) field in systems with EIT, the absorption and dispersion of the probe field will become spatially periodic, and photonic bandgaps [14,15] and electromagnetically induced gratings (EIGs) [16–20] can thus be formed. In EIGs, the amplitude and phase modulations of the transmission function can be flexibly modulated, and the intensity distribution of different orders of the grating can be manipulated. Therefore, EIGs have a number of potential applications in the field of optical switching and routing [21], optical solitons [22], and photonic Floquet topological insulators [23].

Similarly, the non-Hermitian parity-time (PT)-symmetric Hamiltonian, which was first proposed in 1998 [24], has attracted great attention, and it has been extended to optical systems [25] and observed experimentally [26,27]. In systems with PT symmetry, the profile of the real part of the refractive index is symmetric, while that of the imaginary part is antisymmetric, that is, n(x) = n*(–x). Because of this unique property, several interesting phenomena have been unveiled in systems with optical PT symmetry, including optical solitons [28], non-Hermitian Bloch oscillations [29], unidirectionally propagating photons [30–34], unidirectional diffraction [35–37], perfect absorbers [38,39], photons [40,41] lasers, and phonon lasers [42,43]. In addition, PT symmetry has recently been theoretically [44–46] and experimentally [47] demonstrated in atomic systems.

In recent years, semiconductor quantum wells (QWs) have been investigated to realize quantum coherence and the interference effect [48–52]. Compared to atomic systems, semiconductor QW systems have designable and flexible of energy levels, and they are ease of integration and stable for practical application. Motivated by this research, in this study, we investigate the optical properties of asymmetric double semiconductor QWs driven by one TW probe field and two-dimensional (2D) SW coupling and pump fields. We show that in such a QW system, 2D PT symmetry and a nontrivial 2D EIG can be realized. The probe photons traveling through the QWs can be diffracted into the selected domain depending on the spatial relationship of the real and imaginary parts of the refractive index. We find that such asymmetric diffraction patterns in QWs can be controlled by the sign of the probe field and the modulation amplitudes of the 2D SW coupling and pump fields, and the intensity distribution in the selected domain can be manipulated by varying the interaction length of the QWs.

2. Models and equations

We consider asymmetric double semiconductor QWs [53], which consist of a 51-monolayer (145 Å)-thick wide well (WW) and 35-monolayer (100 Å)-thick narrow well (NW), between which there is a 9-monolayer (25-Å)-thick Al0.2Ga0.8As barrier, as shown in Fig. 1(a). There are ten pairs of QWs (each pair consists of one WW, one NW, and one thick barrier), which are isolated from each other by 200-Å-wide Al0.2Ga0.8As buffer layers. All these pairs are sandwiched between nominally undoped 3500-Å-thick Al0.2Ga0.8As layers.

 figure: Fig. 1

Fig. 1 (a) Schematic of one pair of asymmetric double QWs with buffer layers. (b) Band diagram of the asymmetric double QWs. (c) Geometry of laser beams applied to the asymmetric double QWs along the z-direction, and the corresponding far-field diffraction patterns of the probe field.

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Such asymmetric double semiconductor QWs can be treated as a four-level N configuration [54–56], as shown in Fig. 1(b). Here, levels |1 and |2 are localized hole states in a valence band, while levels |3 and |4 are delocalized bonding and antibonding states in a conduction band, respectively, arising from the tunneling effect between the WW and NW via the thin barrier. The probe field Ep with frequency ωp probes the transition |1|3, while the coupling field Ec with frequency ωc and the pump field Ed with frequency ωd act on transitions |2|3 and |1|4, respectively. The Rabi frequency of the probe, coupling, and pump fields are Ωp=μ13Ep/2, Ωc=μ23Ec/2, and Ωd=μ14Ed/2, respectively, where µij is the associated dipole transition matrix element, and the detuning of the probe, coupling, and pump fields areΔp=ωpω31, Δc=ωcω32, and Δd=ωdω42, respectively, where ωij is the transition frequency between levels |i and |j.

Under the condition of low QW carrier density, many-body effects due to electron–electron interactions can be neglected [57]. In the interaction picture and under the rotating wave approximation, the Hamiltonian of the QW system can be written as (=1)

H=(ΔcΔp)|22|Δp|33|Δd|44|+[Ωp|13|+Ωc|23|+Ωd|14|+H.c.].
Here, H.c. is the Hamiltonian complex conjugate.

The equation of motion for the density matrix of the system under the relaxation process is

ρ=i[H,ρ]12{γ˜,ρ},
where γ˜ is the dissipation matrix. Substituting Eq. (1) into Eq. (2), the density matrix for each element can be obtained:
ρ˙22=iΩc(ρ32ρ23)+Γ42ρ44+Γ32ρ33Γ2ρ22,
ρ˙33=iΩc(ρ23ρ32)+iΩp(ρ13ρ31)Γ3ρ33+Γ43ρ44,
ρ˙44=iΩd(ρ14ρ41)Γ4ρ44,
ρ˙12=iΩcρ13+iΩpρ32+iΩdρ42γ˜12ρ12,
ρ˙13=iΩcρ12+iΩdρ43+iΩp(ρ33ρ11)γ˜13ρ13,
ρ˙14=iΩpρ34+iΩd(ρ44ρ11)γ˜14ρ14,
ρ˙23=iΩpρ21+iΩc(ρ33ρ22)γ˜23ρ23,
ρ˙24=iΩdρ21+iΩcρ34γ˜24ρ24,
ρ˙34=iΩpρ14iΩdρ31+iΩcρ24γ˜34ρ34,
with ρ11+ρ22+ρ33+ρ44=1 and ρij=ρji*. Γijis the natural decay rate between levels |i and |j, and we assume that the decay rates from levels |3 and |4 in the conduction band to levels |1 and |2 in the valence band are identical. There is also no decay in the valence band or conduction band, that is, Γ31=Γ32, Γ41=Γ42, and Γ21=Γ43=0, and Γi=ji1Γij denotes the total decay rate of level |i. We define γ˜12=γ12+i(ΔpΔc), γ˜13=γ13+iΔp, γ˜14=γ14+iΔd, γ˜23=γ23+iΔc, γ˜24=γ24+i(Δc+ΔdΔp), and γ˜34=γ34+i(ΔdΔp), where γij=(Γi+Γj)/2.

In such QW systems, Γ3=Γ3l+Γ3dph and Γ4=Γ4l+Γ4dph, where Γ3l and Γ4l are the population decay rate of subbands |3 and |4, respectively, resulting from longitudinal-optical phonon emission events at low temperature, and Γ3dph and Γ4dph are the dephasing decay rates of quantum coherence due to electron-electron scattering, phonon scattering processes, and elastic interface roughness. For simplicity, we assume there is no interference or dephasing between levels |3 and |4, which can be realized based on the appropriate reduction of the temperature [52].

In the condition of ΩpΩp,Ωd,Γ3,Γ4, by solving Eq. (3), the first-order steady-state solution of ρ31(1)can be obtained:

ρ31(1)=iA(ρ33(0)ρ11(0))+BΩcρ23(0)+CΩdρ14(0)Aγ˜13+BΩc2+CΩd2Ωp,
where A=γ˜12γ˜24γ˜34+γ˜34Ωd2+γ˜12Ωc2, B=γ˜24γ˜34Ωd2+Ωc2, and C=γ˜12γ˜24+Ωd2Ωc2. Because the separation between levels |3 and |4 is not large, the population decay rate of levels |3 and |4 can be equal (Γ3l=Γ4l), and dephasing decay rates of levels |3 and |4 can also be equal (Γ3ldph=Γ4ldph), which are reasonable and practical [54–56]. Therefore, the total decay rate of levels |3 and |4 can be the same (Γ3=Γ4), and thus γ13=γ23=γ14=γ24=γ34/2=Γ3/2=Γ4/2=γ. Then, the zeroth-order terms in Eq. (4) can be obtained:
ρ33(0)=Ωc2Ωd2Ωd2(2Ωc2+γ2+Δc2)+Ωc2(2Ωd2+γ2+Δd2),
ρ11(0)=Ωd2+γ2+Δd2Ωd2ρ33(0),
ρ23(0)=iγ+ΔcΩcρ33(0),
It should be noted that the condition of γ13=γ23=γ14=γ24=γ34/2=Γ3/2=Γ4/2=γ is not necessary for the following calculation. Without this condition, ρ31(1)can also be calculated by Eq. (3), however, it is hardly to give the analytic solution.

Next, the susceptibility of the QW medium can be obtained through the following equation:

χ=Nμ13ε0Epρ31(1)=Nμ1322ε0ρ31(1)Ωp,
where N is the electron density of the QWs andχ=χR+iχI. χR describes the dispersion properties of the probe field, while χI describes the absorption properties of the probe field with χI>0(χI<0) indicating loss (gain). To achieve PT symmetry, the refractive index n must satisfy the condition of n(r) = n*(r). The refractive index n=1+χ=1+χ/2 and n = n0 + nR + inI. Then, the real (nR=χR/2) and imaginary (nI=χI/2) parts of the refractive index can be obtained, with n0 = 1 being the background index of the system. In the following calculations, for simplicity, we use the unit Nμ132/4ε0 for the refractive index.

Then, the interaction length of the QWs probed by the probe field along the z-direction is assumed to be L, of which the unit is ζ=ε0λγ/πNμ132, and λ is the wavelength of the probe field. In the steady-state regime and under the slowly varying envelope approximation, Maxwell’s equation of the probe field propagation is

Epz=(α+iβ)Ep,
where α=2πχI/λ and β=2πχR/λ. Then, the transmission function of the probe field for the interaction length L of the QWs modulated in both the x and y-directions can be given by
T(x,y)=eαLeiβL,
where eαL and eiβL represent the amplitude and phase component, respectively.

To achieve normal 2D EIG with symmetric diffraction patterns, only the pump field needs to be periodically modulated by the method of 2D SW fields in the x- and y- directions [58]. However, our goal here is to achieve a 2D EIG with asymmetric diffraction patterns, so the coupling field must also be periodically modulated, as shown in Fig. 1(c). Therefore, the coupling and pump fields can be written as

Ωc=Ωc0+δΩcxcos[2π(xxi)/Λx]+δΩcycos[2π(yyi)/Λy],Ωd=Ωd0+δΩdxsin[2π(xxi)/Λx]+δΩdysin[2π(yyi)/Λy],
where Λx(Λy) is the space period of the SW coupling (pump) field and can be adjusted in a large range by tuning the angle between each component of the coupling (pump) field. Ωc0(Ωd0) is the initial amplitude of the SW coupling (pump) field, and δΩcx and δΩcy (δΩdx and δΩdy) are the modulation amplitudes of the SW coupling (pump) field.

Considering that the probe field is a plane wave, the Fourier transformation of T(x, y) yields the far-field intensity or Fraunhofer diffraction equation [16]:

I(θ)=|Ep(θ)|2sin2(MxπΛxsinθx/λ)Mx2sin2(πΛxsinθx/λ)sin2(MyπΛysinθy/λ)My2sin2(πΛysinθy/λ),
where
E(θ)=Λx/2Λx/2exp(2πixsinθx/λ)dx×Λy/2Λy/2T(x,y)exp(2πiysinθy/λ)dy.
θx and θy are the diffraction angles along the x- and y-axes with respect to the z-direction, respectively, and Mx and My are the numbers of spatial periods of the grating. The (m,n)-order diffraction angle is determined by sinθx=mλ/Λx and sinθy=mλ/Λy.

3. Results and discussions

First, we show that it is possible to realize 2D PT symmetry in QWs by using 2D SW coupling and pump fields modulated in the x- and y-directions. To make a comparison, we plot the real and imaginary parts of the refractive index (nR and nI) in Figs. 2(a) and 2(b) for the TW coupling field (δΩcx=δΩcy=0), respectively, and in Figs. 2(c) and 2(d) for the SW coupling field (δΩcx0,δΩcy0), respectively. Further, in two cases, the pump field is maintained as a 2D SW in both the x- and y-directions. It can be seen that when δΩcx=δΩcy=0, nR is asymmetric, and nI is an even function of the lattice position x and y. In contrast, when δΩcx0,δΩcy0, nR becomes symmetric, that is, an even function of lattice position x and y, and there is no significant change in nI. These results clearly indicate that 2D PT symmetry can be built in QWs by applying 2D SW coupling and pump fields. Unlike the PT symmetry in atomic lattices [36], the modulation of the carrier density of QWs is not necessary. In a practical system, the ideal conditions for constructing PT symmetry are difficult to reach, especially in atoms or semiconductor materials. Here, we analyze the difference between n(r) and n*(–r) and find that the degrees of asymmetry for nR and nI in Fig. 2 are below 5%, thus, it is manageable in practical systems.

 figure: Fig. 2

Fig. 2 (a, c) Real parts nR and (b, d) imaginary parts nI of the 2D complex refractive index for TW and SW coupling field, respectively. The parameters for (a) and (b) are δΩcx=δΩcy=0 and those for (c) and (d) are δΩcx=δΩcy=0.1MHz. The other parameters are Ωp=0.01MHz, Ωc0=1MHz, Ωd0=3MHz, Δp=3.685MHz, Δc=Δd=0, Γ3dhp=Γ3dhp=2.58meV, Γ3l=Γ4l=2.07meV, Mx=My=5, Λx/λ=Λy/λ=4, and L=500ζ.

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We next consider the probe field impinging upon the QWs perpendicularly to the 2D SW (z-direction). Without PT symmetry, similar to the EIGs in previous studies, the probe field is diffracted into four domains, domain I (0sinθx,sinθy1), domain II (1sinθx0,0sinθy1), domain III (1sinθx,sinθy0), and domain IV (0sinθx1,1sinθy0), as shown in Fig. 3(a). On the contrary, with PT symmetry, it can be seen from Fig. 3(b) that the probe field is mainly diffracted into domain I (0sinθx,sinθy1). Therefore, by spatially manipulating the coupling field, the asymmetric light diffraction with PT symmetry can be constructed from symmetric light diffraction without PT symmetry in QW media.

 figure: Fig. 3

Fig. 3 (a) Diffraction intensities as a function of (sinθx,sinθy) for (a) δΩcx=δΩcy=0 and (b) δΩcx=δΩcy=0.1MHz. The other parameters are the same as in Fig. 2.

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Such asymmetric diffraction can be explained by the interference mechanism between the amplitude and phase functions of the grating. Under the condition of 2D PT symmetry shown in Figs. 2(c) and 2(d), the amplitude and phase functions of the grating can induce constructive interference in domain I and destructive interference in domains II, III, and IV. The constructive interference leads to the increase in the diffraction peak located in domain I, while the destructive interference leads to the decrease in the diffraction peak located in the other three domains. Therefore, asymmetric diffraction is obtained. It can be seen that the complete disappearance of the diffraction peaks in domains II, III, and IV can be realized under the condition of perfect destructive interference. It should be noted that perfect destructive interference and asymmetric diffraction can only be achieved at the exceptional point, which is determined by the ratio of nR and nI [37,46].

Next, we will discuss the relationship between the diffraction direction of a PT-symmetric EIG and the spatial refractive index. The properties of the refractive index can be manipulated by varying the parameters of the laser fields. Here, we only consider changing the sign of the probe detuning Δp and the modulation amplitudes of the SW coupling and pump fields δΩcx, δΩcy, δΩdx, and δΩdy, because such an operation will maintain the PT-symmetric properties and can be easily realized in the experiment. We calculate all kinds of spatial refractive indices under different combinations of these five parameters, finding that the sign of Δp, δΩcx and δΩcy results in different nR and has little effect on nI, while the sign of δΩdx and δΩdy results in different nI and has little effect on nR. We plot nR for different sign of Δp, δΩcx and δΩcy in Fig. 4 (23 kinds of combinations), and nI for different sign of δΩdx and δΩdy in Fig. 5 (22 kinds of combinations), respectively. Therefore, there are total 23 × 22 kinds of spatial refractive indices, which are all PT symmetry and have different spatial properties.

 figure: Fig. 4

Fig. 4 Top view of the real part nR of the 2D complex refractive index for the unchanged absolute value and different signs of Δp, δΩcx and δΩcy, respectively; the parameters are (a) Δp=3.685MHz, δΩcx=0.1MHz and δΩcy=0.1MHz, (b) Δp=3.685MHz, δΩcx=0.1MHz and δΩcy=0.1MHz, (c) Δp=3.685MHz, δΩcx=0.1MHz and δΩcy=0.1MHz, (d) Δp=3.685MHz, δΩcx=0.1MHz and δΩcy=0.1MHz, (e) Δp=3.685MHz, δΩcx=0.1MHz and δΩcy=0.1MHz, (f) Δp=3.685MHz, δΩcx=0.1MHz and δΩcy=0.1MHz, (g) Δp=3.685MHz, δΩcx=0.1MHz and δΩcy=0.1MHz, (h) Δp=3.685MHz, δΩcx=0.1MHz and δΩcy=0.1MHz, the other parameters are the same as in Fig. 2.

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 figure: Fig. 5

Fig. 5 Top view of the imaginary part nI of the 2D complex refractive index for the unchanged absolute value and different signs of δΩdx and δΩdy, respectively; the parameters are (a) δΩdx=0.1MHz and δΩdy=0.1MHz, (b) δΩdx=0.1MHz and δΩdy=0.1MHz, (c) δΩdx=0.1MHz and δΩdy=0.1MHz, (d) δΩdx=0.1MHz and δΩdy=0.1MHz. The other parameters are the same as in Fig. 2.

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These different kinds of spatial refractive indices will result in asymmetric light diffraction in the four different domains (Fig. 6). For instance, under eight kinds of spatial refractive indices (Table 1), the amplitude and phase functions of the grating can induce constructive interference in domain I and destructive interference in domains II, III, and IV, thus, the probe field will mainly be diffracted into domain I [Fig. 6(b)]. Other spatial refractive indices will resulting in the asymmetric diffraction in other domains.

 figure: Fig. 6

Fig. 6 Diffraction patterns with PT symmetry as a function of (sinθx,sinθy) in different domains: (a) domain I (0sinθx,sinθy1), (b) domain II (1sinθx0,0sinθy1), (c) domain III (1sinθx,sinθy0), and (d) domain IV (0sinθx1,1sinθy0).

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Tables Icon

Table 1. Different signs of Δp, δΩcx, δΩcy, δΩdx, and δΩdy.

Furthermore, the direction of the probe field (along + z-direction with wave vector kz or –z-direction with wave vector –kz) will not affect the diffraction direction of a PT-symmetric EIG. When the probe field with wave vector kz or –kz propagates through the spatially modulated grating (x–y plane), it will also exhibit wave vectors ± kx and ± ky. With the PT symmetry shown in Figs. 2(c) and 2(d), only the selected components of + kx and + ky will be reflected at the ends of the grating; therefore, the photons will be diffracted only to the selected domain, domain I (0sinθx,sinθy1). The spatial modulation of the complex refractive index in the x–y plane remains the same under the condition of unchanged spatial distributions of the coupling and pump fields. Thus, the diffracted photons are biased toward domain I regardless of whether the probe field propagates along the + z-direction or –z-direction.

Finally, we consider the intensity distribution of the diffraction depending on the interaction length ζ. As can be seen in Fig. 7(a), when ζ is small, most of the energy is distributed in the (0,0)-order diffraction peak. As ζ increases, because of the enhanced phase modulation depth, the energy is gradually transferred to the higher-order diffraction peak in domain I (0sinθx,sinθy1), as shown in Figs. 7(b) and 7(c). With a further increase in ζ, more energy is transferred to the high-order diffraction peaks, and the highest diffraction peak is the (1,1)-order diffraction peak, which is located in the direction determined by sinθx=sinθy=0.25 [Fig. 7(d)]. Thus, one can control the intensity distribution of the diffraction by adjusting the interaction length ζ when the probe field passes through a QW system.

 figure: Fig. 7

Fig. 7 Diffraction intensities with PT symmetry as a function of (sinθx,sinθy) for different values of the interaction length: (a) L=100ζ, (b) L=300ζ, (c) L=500ζ, and (d) L=700ζ. The other parameters are the same as in Fig. 2.

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4. Conclusions

In summary, we have demonstrated that 2D PT symmetry and a 2D EIG can be realized in an asymmetric double semiconductor QW driven by one TW probe field and 2D SW coupling and pump fields. We have demonstrated that the incident probe field traveling through such a PT symmetry grating can be diffracted to the selected domain, which results from the interference mechanism between the amplitude and phase of the grating. We have found that such asymmetric diffraction patterns are determined by the spatial relationship of the real and imaginary parts of the refractive index and can be controlled by the sign of the probe field and the modulation amplitudes of the 2D SW coupling and pump fields. We have also found that the intensity distribution in the selected angles can be manipulated by varying the interaction length of the medium. Such a QW system can provide a versatile platform for theoretically and experimentally exploring PT-symmetric phenomena and can be used to develop new photoelectric devices requiring asymmetric light transport using semiconductor materials.

Funding

National Natural Science Foundation of China (NSFC) (61774156, 61761136009); External Cooperation Program of the Chinese Academy of Sciences (181722KYSB20160005); Jilin Provincial Natural Science Foundation (20160520095JH, 20180519024JH); The Program of China Scholarships Council (201604910385); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2018249).

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26. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009). [CrossRef]   [PubMed]  

27. 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(3), 192–195 (2010). [CrossRef]  

28. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008). [CrossRef]   [PubMed]  

29. S. Longhi, “Bloch oscillations in complex crystals with PT Symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009). [CrossRef]   [PubMed]  

30. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010). [CrossRef]  

31. 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(21), 213901 (2011). [CrossRef]   [PubMed]  

32. 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,” Nat. Mater. 12(2), 108–113 (2013). [CrossRef]   [PubMed]  

33. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014). [CrossRef]  

34. B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014). [CrossRef]  

35. M. Kulishov, J. Laniel, N. Bélanger, J. Azaña, and D. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005). [CrossRef]   [PubMed]  

36. Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42(21), 4283–4286 (2017). [CrossRef]   [PubMed]  

37. T. Shui, W.-X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A (Coll. Park) 97(3), 033819 (2018). [CrossRef]  

38. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010). [CrossRef]   [PubMed]  

39. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010). [CrossRef]  

40. L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014). [CrossRef]   [PubMed]  

41. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014). [CrossRef]   [PubMed]  

42. H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014). [CrossRef]   [PubMed]  

43. B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A (Coll. Park) 94(3), 031802 (2016). [CrossRef]  

44. C. Hang, G. Huang, and V. V. Konotop, “PT Symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110(8), 083604 (2013). [CrossRef]   [PubMed]  

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

46. J. Sheng, M.-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88(4), 041803 (2013). [CrossRef]  

47. Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016). [CrossRef]   [PubMed]  

48. G. B. Serapiglia, E. Paspalakis, C. Sirtori, K. L. Vodopyanov, and C. C. Phillips, “Laser-induced quantum coherence in a semiconductor quantum well,” Phys. Rev. Lett. 84(5), 1019–1022 (2000). [CrossRef]   [PubMed]  

49. M. Phillips and H. Wang, “Electromagnetically induced transparency due to intervalence band coherence in a GaAs quantum well,” Opt. Lett. 28(10), 831–833 (2003). [CrossRef]   [PubMed]  

50. J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “AC Stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94(15), 157403 (2005). [CrossRef]   [PubMed]  

51. M. D. Frogley, J. F. Dynes, M. Beck, J. Faist, and C. C. Phillips, “Gain without inversion in semiconductor nanostructures,” Nat. Mater. 5(3), 175–178 (2006). [CrossRef]  

52. J.-H. Wu, J.-Y. Gao, J.-H. Xu, L. Silvestri, M. Artoni, G. C. La Rocca, and F. Bassani, “Ultrafast all optical switching via tunable Fano interference,” Phys. Rev. Lett. 95(5), 057401 (2005). [CrossRef]   [PubMed]  

53. H. G. Roskos, M. C. Nuss, J. Shah, K. Leo, D. A. B. Miller, A. M. Fox, S. Schmitt-Rink, and K. Köhler, “Coherent submillimeter-wave emission from charge oscillations in a double-well potential,” Phys. Rev. Lett. 68(14), 2216–2219 (1992). [CrossRef]   [PubMed]  

54. W.-X. Yang and R.-K. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express 16(22), 17161–17170 (2008). [CrossRef]   [PubMed]  

55. C. Zhu and G. Huang, “Slow-light solitons in coupled asymmetric quantum wells via interband transitions,” Phys. Rev. B Condens. Matter Mater. Phys. 80(23), 235408 (2009). [CrossRef]  

56. X. Q. Luo, D. L. Wang, Z. Q. Zhang, J. W. Ding, and W. M. Liu, “Nonlinear optical behavior of a four-level quantum well with coupled relaxation of optical and longitudinal phonons,” Phys. Rev. A 84(3), 033803 (2011). [CrossRef]  

57. D. E. Nikonov, A. Imamoğlu, L. V. Butov, and H. Schmidt, “Collective intersubband excitations in quantum wells: coulomb interaction versus subband dispersion,” Phys. Rev. Lett. 79(23), 4633–4636 (1997). [CrossRef]  

58. Y. Y. Chen, Z. Z. Liu, and R. G. Wan, “Two-dimensional electromagnetically induced grating in coherent atomic medium,” Europhys. Lett. 116(6), 64006 (2016). [CrossRef]  

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    [Crossref] [PubMed]
  27. 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(3), 192–195 (2010).
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  28. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
    [Crossref] [PubMed]
  29. S. Longhi, “Bloch oscillations in complex crystals with PT Symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
    [Crossref] [PubMed]
  30. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
    [Crossref]
  31. 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(21), 213901 (2011).
    [Crossref] [PubMed]
  32. 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,” Nat. Mater. 12(2), 108–113 (2013).
    [Crossref] [PubMed]
  33. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
    [Crossref]
  34. B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
    [Crossref]
  35. M. Kulishov, J. Laniel, N. Bélanger, J. Azaña, and D. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005).
    [Crossref] [PubMed]
  36. Y.-M. Liu, F. Gao, C.-H. Fan, and J.-H. Wu, “Asymmetric light diffraction of an atomic grating with PT symmetry,” Opt. Lett. 42(21), 4283–4286 (2017).
    [Crossref] [PubMed]
  37. T. Shui, W.-X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A (Coll. Park) 97(3), 033819 (2018).
    [Crossref]
  38. Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
    [Crossref] [PubMed]
  39. S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
    [Crossref]
  40. L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
    [Crossref] [PubMed]
  41. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
    [Crossref] [PubMed]
  42. H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
    [Crossref] [PubMed]
  43. B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A (Coll. Park) 94(3), 031802 (2016).
    [Crossref]
  44. C. Hang, G. Huang, and V. V. Konotop, “PT Symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110(8), 083604 (2013).
    [Crossref] [PubMed]
  45. H. J. Li, J. P. Dou, and G. Huang, “PT symmetry via electromagnetically induced transparency,” Opt. Express 21(26), 32053–32062 (2013).
    [Crossref] [PubMed]
  46. J. Sheng, M.-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88(4), 041803 (2013).
    [Crossref]
  47. Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
    [Crossref] [PubMed]
  48. G. B. Serapiglia, E. Paspalakis, C. Sirtori, K. L. Vodopyanov, and C. C. Phillips, “Laser-induced quantum coherence in a semiconductor quantum well,” Phys. Rev. Lett. 84(5), 1019–1022 (2000).
    [Crossref] [PubMed]
  49. M. Phillips and H. Wang, “Electromagnetically induced transparency due to intervalence band coherence in a GaAs quantum well,” Opt. Lett. 28(10), 831–833 (2003).
    [Crossref] [PubMed]
  50. J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “AC Stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94(15), 157403 (2005).
    [Crossref] [PubMed]
  51. M. D. Frogley, J. F. Dynes, M. Beck, J. Faist, and C. C. Phillips, “Gain without inversion in semiconductor nanostructures,” Nat. Mater. 5(3), 175–178 (2006).
    [Crossref]
  52. J.-H. Wu, J.-Y. Gao, J.-H. Xu, L. Silvestri, M. Artoni, G. C. La Rocca, and F. Bassani, “Ultrafast all optical switching via tunable Fano interference,” Phys. Rev. Lett. 95(5), 057401 (2005).
    [Crossref] [PubMed]
  53. H. G. Roskos, M. C. Nuss, J. Shah, K. Leo, D. A. B. Miller, A. M. Fox, S. Schmitt-Rink, and K. Köhler, “Coherent submillimeter-wave emission from charge oscillations in a double-well potential,” Phys. Rev. Lett. 68(14), 2216–2219 (1992).
    [Crossref] [PubMed]
  54. W.-X. Yang and R.-K. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express 16(22), 17161–17170 (2008).
    [Crossref] [PubMed]
  55. C. Zhu and G. Huang, “Slow-light solitons in coupled asymmetric quantum wells via interband transitions,” Phys. Rev. B Condens. Matter Mater. Phys. 80(23), 235408 (2009).
    [Crossref]
  56. X. Q. Luo, D. L. Wang, Z. Q. Zhang, J. W. Ding, and W. M. Liu, “Nonlinear optical behavior of a four-level quantum well with coupled relaxation of optical and longitudinal phonons,” Phys. Rev. A 84(3), 033803 (2011).
    [Crossref]
  57. D. E. Nikonov, A. Imamoğlu, L. V. Butov, and H. Schmidt, “Collective intersubband excitations in quantum wells: coulomb interaction versus subband dispersion,” Phys. Rev. Lett. 79(23), 4633–4636 (1997).
    [Crossref]
  58. Y. Y. Chen, Z. Z. Liu, and R. G. Wan, “Two-dimensional electromagnetically induced grating in coherent atomic medium,” Europhys. Lett. 116(6), 64006 (2016).
    [Crossref]

2018 (1)

T. Shui, W.-X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A (Coll. Park) 97(3), 033819 (2018).
[Crossref]

2017 (1)

2016 (3)

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A (Coll. Park) 94(3), 031802 (2016).
[Crossref]

Y. Y. Chen, Z. Z. Liu, and R. G. Wan, “Two-dimensional electromagnetically induced grating in coherent atomic medium,” Europhys. Lett. 116(6), 64006 (2016).
[Crossref]

2015 (1)

Y. Zhang, Z. Wu, M. R. Belić, H. Zheng, Z. Wang, M. Xiao, and Y. Zhang, “Photonic Floquet topological insulators in atomic ensembles,” Laser Photonics Rev. 9(3), 331–338 (2015).
[Crossref]

2014 (5)

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref] [PubMed]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “PT-symmetric phonon laser,” Phys. Rev. Lett. 113(5), 053604 (2014).
[Crossref] [PubMed]

2013 (4)

C. Hang, G. Huang, and V. V. Konotop, “PT Symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110(8), 083604 (2013).
[Crossref] [PubMed]

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

J. Sheng, M.-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88(4), 041803 (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,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

2011 (4)

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(21), 213901 (2011).
[Crossref] [PubMed]

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[Crossref] [PubMed]

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A 83(3), 033824 (2011).
[Crossref]

X. Q. Luo, D. L. Wang, Z. Q. Zhang, J. W. Ding, and W. M. Liu, “Nonlinear optical behavior of a four-level quantum well with coupled relaxation of optical and longitudinal phonons,” Phys. Rev. A 84(3), 033803 (2011).
[Crossref]

2010 (5)

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref] [PubMed]

S. Longhi, “PT-symmetric laser absorber,” Phys. Rev. A 82(3), 031801 (2010).
[Crossref]

L. E. E. de Araujo, “Electromagnetically induced phase grating,” Opt. Lett. 35(7), 977–979 (2010).
[Crossref] [PubMed]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (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(3), 192–195 (2010).
[Crossref]

2009 (3)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

S. Longhi, “Bloch oscillations in complex crystals with PT Symmetry,” Phys. Rev. Lett. 103(12), 123601 (2009).
[Crossref] [PubMed]

C. Zhu and G. Huang, “Slow-light solitons in coupled asymmetric quantum wells via interband transitions,” Phys. Rev. B Condens. Matter Mater. Phys. 80(23), 235408 (2009).
[Crossref]

2008 (2)

W.-X. Yang and R.-K. Lee, “Controllable entanglement and polarization phase gate in coupled double quantum-well structures,” Opt. Express 16(22), 17161–17170 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

2007 (2)

2006 (2)

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006).
[Crossref] [PubMed]

M. D. Frogley, J. F. Dynes, M. Beck, J. Faist, and C. C. Phillips, “Gain without inversion in semiconductor nanostructures,” Nat. Mater. 5(3), 175–178 (2006).
[Crossref]

2005 (5)

J.-H. Wu, J.-Y. Gao, J.-H. Xu, L. Silvestri, M. Artoni, G. C. La Rocca, and F. Bassani, “Ultrafast all optical switching via tunable Fano interference,” Phys. Rev. Lett. 95(5), 057401 (2005).
[Crossref] [PubMed]

M. Kulishov, J. Laniel, N. Bélanger, J. Azaña, and D. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005).
[Crossref] [PubMed]

J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “AC Stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94(15), 157403 (2005).
[Crossref] [PubMed]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30(7), 699–701 (2005).
[Crossref] [PubMed]

2003 (2)

2002 (2)

A. André and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89(14), 143602 (2002).
[Crossref] [PubMed]

G. C. Cardoso and J. W. R. Tabosa, “Electromagnetically induced gratings in a degenerate open two-level system,” Phys. Rev. A 65(3), 033803 (2002).
[Crossref]

2001 (5)

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87(7), 073601 (2001).
[Crossref] [PubMed]

M. Yan, E. G. Rickey, and Y. Zhu, “Observation of absorptive photon switching by quantum interference,” Phys. Rev. A 64(4), 041801 (2001).
[Crossref]

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88(2), 023602 (2001).
[Crossref] [PubMed]

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86(5), 783–786 (2001).
[Crossref] [PubMed]

2000 (1)

G. B. Serapiglia, E. Paspalakis, C. Sirtori, K. L. Vodopyanov, and C. C. Phillips, “Laser-induced quantum coherence in a semiconductor quantum well,” Phys. Rev. Lett. 84(5), 1019–1022 (2000).
[Crossref] [PubMed]

1999 (2)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[Crossref]

M. Mitsunaga and N. Imoto, “Observation of an electromagnetically induced grating in cold sodium atoms,” Phys. Rev. A 59(6), 4773–4776 (1999).
[Crossref]

1998 (4)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

J. P. Marangos, “Electromagnetically induced transparency,” J. Mod. Opt. 45(3), 471–503 (1998).
[Crossref]

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81(17), 3611–3614 (1998).
[Crossref]

H. Y. Ling, Y.-Q. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998).
[Crossref]

1997 (2)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

D. E. Nikonov, A. Imamoğlu, L. V. Butov, and H. Schmidt, “Collective intersubband excitations in quantum wells: coulomb interaction versus subband dispersion,” Phys. Rev. Lett. 79(23), 4633–4636 (1997).
[Crossref]

1996 (1)

1992 (1)

H. G. Roskos, M. C. Nuss, J. Shah, K. Leo, D. A. B. Miller, A. M. Fox, S. Schmitt-Rink, and K. Köhler, “Coherent submillimeter-wave emission from charge oscillations in a double-well potential,” Phys. Rev. Lett. 68(14), 2216–2219 (1992).
[Crossref] [PubMed]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

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,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

André, A.

A. André and M. D. Lukin, “Manipulating light pulses via dynamically controlled photonic band gap,” Phys. Rev. Lett. 89(14), 143602 (2002).
[Crossref] [PubMed]

Artoni, M.

M. Artoni and G. C. La Rocca, “Optically tunable photonic stop bands in homogeneous absorbing media,” Phys. Rev. Lett. 96(7), 073905 (2006).
[Crossref] [PubMed]

J.-H. Wu, J.-Y. Gao, J.-H. Xu, L. Silvestri, M. Artoni, G. C. La Rocca, and F. Bassani, “Ultrafast all optical switching via tunable Fano interference,” Phys. Rev. Lett. 95(5), 057401 (2005).
[Crossref] [PubMed]

Azaña, J.

Bassani, F.

J.-H. Wu, J.-Y. Gao, J.-H. Xu, L. Silvestri, M. Artoni, G. C. La Rocca, and F. Bassani, “Ultrafast all optical switching via tunable Fano interference,” Phys. Rev. Lett. 95(5), 057401 (2005).
[Crossref] [PubMed]

Beck, M.

M. D. Frogley, J. F. Dynes, M. Beck, J. Faist, and C. C. Phillips, “Gain without inversion in semiconductor nanostructures,” Nat. Mater. 5(3), 175–178 (2006).
[Crossref]

J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “AC Stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94(15), 157403 (2005).
[Crossref] [PubMed]

Behroozi, C. H.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[Crossref]

Bélanger, N.

Belic, M. R.

Y. Zhang, Z. Wu, M. R. Belić, H. Zheng, Z. Wang, M. Xiao, and Y. Zhang, “Photonic Floquet topological insulators in atomic ensembles,” Laser Photonics Rev. 9(3), 331–338 (2015).
[Crossref]

Bender, C. M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Brown, A. W.

Butov, L. V.

D. E. Nikonov, A. Imamoğlu, L. V. Butov, and H. Schmidt, “Collective intersubband excitations in quantum wells: coulomb interaction versus subband dispersion,” Phys. Rev. Lett. 79(23), 4633–4636 (1997).
[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(21), 213901 (2011).
[Crossref] [PubMed]

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref] [PubMed]

Cardoso, G. C.

G. C. Cardoso and J. W. R. Tabosa, “Electromagnetically induced gratings in a degenerate open two-level system,” Phys. Rev. A 65(3), 033803 (2002).
[Crossref]

Chang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Chen, H.

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011).
[Crossref] [PubMed]

Chen, Y. Y.

Y. Y. Chen, Z. Z. Liu, and R. G. Wan, “Two-dimensional electromagnetically induced grating in coherent atomic medium,” Europhys. Lett. 116(6), 64006 (2016).
[Crossref]

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,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Chong, Y. D.

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref] [PubMed]

Christodoulides, D. N.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

J. Sheng, M.-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88(4), 041803 (2013).
[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(21), 213901 (2011).
[Crossref] [PubMed]

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (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(3), 192–195 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[Crossref] [PubMed]

de Araujo, L. E. E.

Ding, J. W.

X. Q. Luo, D. L. Wang, Z. Q. Zhang, J. W. Ding, and W. M. Liu, “Nonlinear optical behavior of a four-level quantum well with coupled relaxation of optical and longitudinal phonons,” Phys. Rev. A 84(3), 033803 (2011).
[Crossref]

Dou, J. P.

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Dutton, Z.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[Crossref]

Dynes, J. F.

M. D. Frogley, J. F. Dynes, M. Beck, J. Faist, and C. C. Phillips, “Gain without inversion in semiconductor nanostructures,” Nat. Mater. 5(3), 175–178 (2006).
[Crossref]

J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “AC Stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94(15), 157403 (2005).
[Crossref] [PubMed]

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(21), 213901 (2011).
[Crossref] [PubMed]

El-Ganainy, R.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (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(3), 192–195 (2010).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[Crossref] [PubMed]

Faist, J.

M. D. Frogley, J. F. Dynes, M. Beck, J. Faist, and C. C. Phillips, “Gain without inversion in semiconductor nanostructures,” Nat. Mater. 5(3), 175–178 (2006).
[Crossref]

J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “AC Stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94(15), 157403 (2005).
[Crossref] [PubMed]

Fan, C.-H.

Fan, S.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

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,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Feng, L.

L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
[Crossref] [PubMed]

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,” Nat. Mater. 12(2), 108–113 (2013).
[Crossref] [PubMed]

Fleischhauer, A.

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86(5), 783–786 (2001).
[Crossref] [PubMed]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

Fox, A. M.

H. G. Roskos, M. C. Nuss, J. Shah, K. Leo, D. A. B. Miller, A. M. Fox, S. Schmitt-Rink, and K. Köhler, “Coherent submillimeter-wave emission from charge oscillations in a double-well potential,” Phys. Rev. Lett. 68(14), 2216–2219 (1992).
[Crossref] [PubMed]

Frogley, M. D.

M. D. Frogley, J. F. Dynes, M. Beck, J. Faist, and C. C. Phillips, “Gain without inversion in semiconductor nanostructures,” Nat. Mater. 5(3), 175–178 (2006).
[Crossref]

J. F. Dynes, M. D. Frogley, M. Beck, J. Faist, and C. C. Phillips, “AC Stark splitting and quantum interference with intersubband transitions in quantum wells,” Phys. Rev. Lett. 94(15), 157403 (2005).
[Crossref] [PubMed]

Gao, F.

Gao, J.-Y.

R.-G. Wan, J. Kou, L. Jiang, Y. Jiang, and J.-Y. Gao, “Electromagnetically induced grating via enhanced nonlinear modulation by spontaneously generated coherence,” Phys. Rev. A 83(3), 033824 (2011).
[Crossref]

J.-H. Wu, J.-Y. Gao, J.-H. Xu, L. Silvestri, M. Artoni, G. C. La Rocca, and F. Bassani, “Ultrafast all optical switching via tunable Fano interference,” Phys. Rev. Lett. 95(5), 057401 (2005).
[Crossref] [PubMed]

Ge, L.

Y. D. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010).
[Crossref] [PubMed]

Gianfreda, M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10(5), 394–398 (2014).
[Crossref]

Goorskey, D.

H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. 87(7), 073601 (2001).
[Crossref] [PubMed]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Ham, B. S.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88(2), 023602 (2001).
[Crossref] [PubMed]

Hang, C.

C. Hang, G. Huang, and V. V. Konotop, “PT Symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110(8), 083604 (2013).
[Crossref] [PubMed]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[Crossref]

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81(17), 3611–3614 (1998).
[Crossref]

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

Hau, L. V.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001).
[Crossref] [PubMed]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[Crossref]

He, B.

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A (Coll. Park) 94(3), 031802 (2016).
[Crossref]

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

Heinrich, M.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Hemmer, P. R.

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88(2), 023602 (2001).
[Crossref] [PubMed]

Hodaei, H.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Hua, S.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Huang, G.

C. Hang, G. Huang, and V. V. Konotop, “PT Symmetry with a system of three-level atoms,” Phys. Rev. Lett. 110(8), 083604 (2013).
[Crossref] [PubMed]

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

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D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86(5), 783–786 (2001).
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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(21), 213901 (2011).
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A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
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D. E. Nikonov, A. Imamoğlu, L. V. Butov, and H. Schmidt, “Collective intersubband excitations in quantum wells: coulomb interaction versus subband dispersion,” Phys. Rev. Lett. 79(23), 4633–4636 (1997).
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H. G. Roskos, M. C. Nuss, J. Shah, K. Leo, D. A. B. Miller, A. M. Fox, S. Schmitt-Rink, and K. Köhler, “Coherent submillimeter-wave emission from charge oscillations in a double-well potential,” Phys. Rev. Lett. 68(14), 2216–2219 (1992).
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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(3), 192–195 (2010).
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Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of Parity-Time Symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
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J. Sheng, M.-A. Miri, D. N. Christodoulides, and M. Xiao, “PT-symmetric optical potentials in a coherent atomic medium,” Phys. Rev. A 88(4), 041803 (2013).
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T. Shui, W.-X. Yang, S. Liu, L. Li, and Z. Zhu, “Asymmetric diffraction by atomic gratings with optical PT symmetry in the Raman-Nath regime,” Phys. Rev. A (Coll. Park) 97(3), 033819 (2018).
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J.-H. Wu, J.-Y. Gao, J.-H. Xu, L. Silvestri, M. Artoni, G. C. La Rocca, and F. Bassani, “Ultrafast all optical switching via tunable Fano interference,” Phys. Rev. Lett. 95(5), 057401 (2005).
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G. B. Serapiglia, E. Paspalakis, C. Sirtori, K. L. Vodopyanov, and C. C. Phillips, “Laser-induced quantum coherence in a semiconductor quantum well,” Phys. Rev. Lett. 84(5), 1019–1022 (2000).
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L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
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L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, “Single-mode laser by parity-time symmetry breaking,” Science 346(6212), 972–975 (2014).
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Phys. Rev. A (Coll. Park) (2)

B. He, L. Yang, and M. Xiao, “Dynamical phonon laser in coupled active-passive microresonators,” Phys. Rev. A (Coll. Park) 94(3), 031802 (2016).
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Figures (7)

Fig. 1
Fig. 1 (a) Schematic of one pair of asymmetric double QWs with buffer layers. (b) Band diagram of the asymmetric double QWs. (c) Geometry of laser beams applied to the asymmetric double QWs along the z-direction, and the corresponding far-field diffraction patterns of the probe field.
Fig. 2
Fig. 2 (a, c) Real parts nR and (b, d) imaginary parts nI of the 2D complex refractive index for TW and SW coupling field, respectively. The parameters for (a) and (b) are δ Ω c x = δ Ω c y = 0 and those for (c) and (d) are δ Ω c x = δ Ω c y = 0.1 MHz . The other parameters are Ω p = 0.01 MHz , Ω c 0 = 1 MHz , Ω d 0 = 3 MHz , Δ p = 3.685 MHz , Δ c = Δ d = 0 , Γ 3 d h p = Γ 3 d h p = 2.58 meV , Γ 3 l = Γ 4 l = 2.07 meV , M x = M y = 5 , Λ x / λ = Λ y / λ = 4 , and L = 500 ζ .
Fig. 3
Fig. 3 (a) Diffraction intensities as a function of ( sin θ x , sin θ y ) for (a) δ Ω c x = δ Ω c y = 0 and (b) δ Ω c x = δ Ω c y = 0.1 MHz . The other parameters are the same as in Fig. 2.
Fig. 4
Fig. 4 Top view of the real part nR of the 2D complex refractive index for the unchanged absolute value and different signs of Δ p , δ Ω c x and δ Ω c y , respectively; the parameters are (a) Δ p = 3.685 MHz , δ Ω c x = 0.1 MHz and δ Ω c y = 0.1 MHz , (b) Δ p = 3.685 MHz , δ Ω c x = 0.1 MHz and δ Ω c y = 0.1 MHz , (c) Δ p = 3.685 MHz , δ Ω c x = 0.1 MHz and δ Ω c y = 0.1 MHz , (d) Δ p = 3.685 MHz , δ Ω c x = 0.1 MHz and δ Ω c y = 0.1 MHz , (e) Δ p = 3.685 MHz , δ Ω c x = 0.1 MHz and δ Ω c y = 0.1 MHz , (f) Δ p = 3.685 MHz , δ Ω c x = 0.1 MHz and δ Ω c y = 0.1 MHz , (g) Δ p = 3.685 MHz , δ Ω c x = 0.1 MHz and δ Ω c y = 0.1 MHz , (h) Δ p = 3.685 MHz , δ Ω c x = 0.1 MHz and δ Ω c y = 0.1 MHz , the other parameters are the same as in Fig. 2.
Fig. 5
Fig. 5 Top view of the imaginary part nI of the 2D complex refractive index for the unchanged absolute value and different signs of δ Ω d x and δ Ω d y , respectively; the parameters are (a) δ Ω d x = 0.1 MHz and δ Ω d y = 0.1 MHz , (b) δ Ω d x = 0.1 MHz and δ Ω d y = 0.1 MHz , (c) δ Ω d x = 0.1 MHz and δ Ω d y = 0.1 MHz , (d) δ Ω d x = 0.1 MHz and δ Ω d y = 0.1 MHz . The other parameters are the same as in Fig. 2.
Fig. 6
Fig. 6 Diffraction patterns with PT symmetry as a function of ( sin θ x , sin θ y ) in different domains: (a) domain I ( 0 sin θ x , sin θ y 1 ), (b) domain II ( 1 sin θ x 0 , 0 sin θ y 1 ), (c) domain III ( 1 sin θ x , sin θ y 0 ), and (d) domain IV ( 0 sin θ x 1 , 1 sin θ y 0 ).
Fig. 7
Fig. 7 Diffraction intensities with PT symmetry as a function of ( sin θ x , sin θ y ) for different values of the interaction length: (a) L = 100 ζ , (b) L = 300 ζ , (c) L = 500 ζ , and (d) L = 700 ζ . The other parameters are the same as in Fig. 2.

Tables (1)

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Table 1 Different signs of Δ p , δ Ω c x , δ Ω c y , δ Ω d x , and δ Ω d y .

Equations (21)

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H = ( Δ c Δ p ) | 2 2 | Δ p | 3 3 | Δ d | 4 4 | + [ Ω p | 1 3 | + Ω c | 2 3 | + Ω d | 1 4 | + H .c . ] .
ρ = i [ H , ρ ] 1 2 { γ ˜ , ρ } ,
ρ ˙ 22 = i Ω c ( ρ 32 ρ 23 ) + Γ 42 ρ 44 + Γ 32 ρ 33 Γ 2 ρ 22 ,
ρ ˙ 33 = i Ω c ( ρ 23 ρ 32 ) + i Ω p ( ρ 13 ρ 31 ) Γ 3 ρ 33 + Γ 43 ρ 44 ,
ρ ˙ 44 = i Ω d ( ρ 14 ρ 41 ) Γ 4 ρ 44 ,
ρ ˙ 12 = i Ω c ρ 13 + i Ω p ρ 32 + i Ω d ρ 42 γ ˜ 12 ρ 12 ,
ρ ˙ 13 = i Ω c ρ 12 + i Ω d ρ 43 + i Ω p ( ρ 33 ρ 11 ) γ ˜ 13 ρ 13 ,
ρ ˙ 14 = i Ω p ρ 34 + i Ω d ( ρ 44 ρ 11 ) γ ˜ 14 ρ 14 ,
ρ ˙ 23 = i Ω p ρ 21 + i Ω c ( ρ 33 ρ 22 ) γ ˜ 23 ρ 23 ,
ρ ˙ 24 = i Ω d ρ 21 + i Ω c ρ 34 γ ˜ 24 ρ 24 ,
ρ ˙ 34 = i Ω p ρ 14 i Ω d ρ 31 + i Ω c ρ 24 γ ˜ 34 ρ 34 ,
ρ 31 ( 1 ) = i A ( ρ 33 ( 0 ) ρ 11 ( 0 ) ) + B Ω c ρ 23 ( 0 ) + C Ω d ρ 14 ( 0 ) A γ ˜ 13 + B Ω c 2 + C Ω d 2 Ω p ,
ρ 33 ( 0 ) = Ω c 2 Ω d 2 Ω d 2 ( 2 Ω c 2 + γ 2 + Δ c 2 ) + Ω c 2 ( 2 Ω d 2 + γ 2 + Δ d 2 ) ,
ρ 11 ( 0 ) = Ω d 2 + γ 2 + Δ d 2 Ω d 2 ρ 33 ( 0 ) ,
ρ 23 ( 0 ) = i γ + Δ c Ω c ρ 33 ( 0 ) ,
χ = N μ 13 ε 0 E p ρ 31 ( 1 ) = N μ 13 2 2 ε 0 ρ 31 ( 1 ) Ω p ,
E p z = ( α + i β ) E p ,
T ( x , y ) = e α L e i β L ,
Ω c = Ω c 0 + δ Ω c x cos [ 2 π ( x x i ) / Λ x ] + δ Ω c y cos [ 2 π ( y y i ) / Λ y ] , Ω d = Ω d 0 + δ Ω d x sin [ 2 π ( x x i ) / Λ x ] + δ Ω d y sin [ 2 π ( y y i ) / Λ y ] ,
I ( θ ) = | E p ( θ ) | 2 sin 2 ( M x π Λ x sin θ x / λ ) M x 2 sin 2 ( π Λ x sin θ x / λ ) sin 2 ( M y π Λ y sin θ y / λ ) M y 2 sin 2 ( π Λ y sin θ y / λ ) ,
E ( θ ) = Λ x / 2 Λ x / 2 exp ( 2 π i x sin θ x / λ ) d x × Λ y / 2 Λ y / 2 T ( x , y ) exp ( 2 π i y sin θ y / λ ) d y .

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