Asymmetric light diffraction of two- dimensional electromagnetically induced grating with PT symmetry in asymmetric double quantum wells

An asymmetric double semiconductor quantum well is proposed to realize twodimensional 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


Introduction
Electromagnetically induced transparency (EIT) [1][2][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][17][18][19][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].
In recent years, semiconductor quantum wells (QWs) have been investigated to realize quantum coherence and the interference effect [48][49][50][51][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.

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 Al 0.2 Ga 0.8 As 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 Al 0.2 Ga 0.8 As buffer layers.All these pairs are sandwiched between nominally undoped 3500-Å-thick Al 0.2 Ga 0.8 As layers.Such asymmetric double semiconductor QWs can be treated as a four-level N configuration [54][55][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 E p with frequency p ω probes the transition Under the condition of low QW carrier density, many-body effects due to electronelectron 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 ( Here, H.c. is the Hamiltonian complex conjugate.The equation of motion for the density matrix of the system under the relaxation process is where γ is the dissipation matrix.Substituting Eq. (1) into Eq.( 2), the density matrix for each element can be obtained: ( ) 24 24 ,  Γ 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 3 4 , , , , by solving Eq. ( 3), the first-order steady-state solution of (1)   31 ρ can be obtained: ( ) where , and Because the separation between levels 3 and 4 is not large, the population decay rate of γ γ γ γ γ γ Γ =Γ .Then, the zeroth-order terms in Eq. ( 4) can be obtained: It should be noted that the condition of 13 23 14 24 34 necessary for the following calculation.Without this condition, (1)   31 ρ 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: where ) and imaginary ( 2 ) parts of the refractive index can be obtained, with n 0 = 1 being the background index of the system.In the following calculations, for simplicity, we use the unit 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 , 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 where = 2 I α πχ λ and = 2 R β πχ 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 ( ) , , where L e α − and i L e β 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 where ( ) x y Λ Λ 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.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]: where , exp 2 sin .

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 (n R and n I ) in Figs.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 n R and n I in Fig. 2 are below 5%, thus, it is manageable in practical systems.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 ( 0 sin , sin 1  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 n R and n I [37,46]., the other parameters are the same as in Fig. 2.
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 p Δ and the modulation amplitudes of the SW coupling and pump fields  .The other parameters are the same as in Fig. 2.
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.Furthermore, the direction of the probe field (along + z-direction with wave vector k z orz-direction with wave vector -k z ) will not affect the diffraction direction of a PT-symmetric EIG.When the probe field with wave vector k z or -k z propagates through the spatially modulated grating (x-y plane), it will also exhibit wave vectors ± k x and ± k y .With the PT symmetry shown in Figs.2(c) and 2(d), only the selected components of + k x and + k y will be reflected at the ends of the grating; therefore, the photons will be diffracted only to the selected domain, domain I ( 0 sin , sin 1 x y θ θ ≤ ≤ ).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 -zdirection.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 ( 0 sin , sin 1

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.

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.

1 3 →
, while the coupling field E c with frequency c ω and the pump field E d with frequency d , where µ ij is the associated dipole transition matrix element, and the detuning of the probe, coupling, and pump fields are ij ω is the transition frequency between levels i and j .

32 Γ 3 dph Γ and 4 dph
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,31 , where 3lΓ and 4l Γ are the population decay rate of subbands 3 and 4 , respectively, resulting from longitudinaloptical phonon emission events at low temperature, and

4 Γ
), and dephasing decay rates of levels 3 and 4 can also be equal ( ), which are reasonable and practical[54][55][56].Therefore, the total decay rate of levels 3 and 4 can be the same ( 3

χ
N is the electron density of the QWs and = describes the dispersion properties of the probe field, while I χ describes the absorption properties of the probe field with gain).To achieve PT symmetry, the refractive index n must satisfy the condition of n(r) = n*(r).The refractive index 1 and n = n 0 + n R + in I .Then, the real ( 2  for the refractive index. the initial amplitude of the SW coupling (pump) field, and cx δΩ and cy δΩ ( dx δΩ and dy δΩ ) are the modulation amplitudes of the SW coupling (pump) field.
θ are the diffraction angles along the x-and y-axes with respect to the z-direction, respectively, and M x and M y are the numbers of spatial periods of the grating.The (m,n)-order diffraction angle is determined by sin x , 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 0 cx cy δ δ Ω = Ω = , n R is asymmetric, and n I is an even function of the lattice position x and y. n R becomes symmetric, that is, an even function of lattice position x and y, and there is no significant change in n I .These results clearly indicate that 2D PT symmetry can be built QWs applying 2D SW coupling and pump fields.

Fig. 2 .
Fig. 2. (a, c) Real parts n R and (b, d) imaginary parts n I of the 2D complex refractive index for TW and SW coupling field, respectively.The parameters for (a) and (b) are 0 cx cy δ δ Ω = Ω = and those for (c) and (d) are 0.1MHz cx cy δ δ Ω = Ω = 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 ( 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.

Fig. 4 .
Fig. 4. Top view of the real part n R of the 2D complex refractive index for the unchanged absolute value and different signs of p Δ , dyδΩ , 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 p Δ , cx δΩ and cy δΩ results in different n R and has little effect on n I , while the sign of dx δΩ and dy δΩ results in different n I and has little effect on n R .We plot n R for different sign of p Δ , cx δΩ and cy δΩ in Fig. 4 (2 3 kinds of combinations), and n I for different sign of dx δΩ and dy δΩ in Fig. 5 (2 2 kinds of combinations), respectively.Therefore, there are total 2 3 × 2 2 kinds of spatial refractive indices, which are all PT symmetry and have different spatial properties.

Fig. 5 .
Fig. 5. Top view of the imaginary part n I of the 2D complex refractive index for the unchanged absolute value and different signs of dx δΩ and

Fig. 6 .
Fig. 6.Diffraction patterns with PT symmetry as a function of ( sin , sin x y θ θ ) in different 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 sin = sin =0.25 x y θ θ [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.

Fig. 7 ..
Fig. 7. Diffraction intensities with PT symmetry as a function of ( sin , sin x y θ θ ) for different