Parity-time symmetry in coherent asymmetric double quantum wells

Abstract

A coherently prepared asymmetric double semiconductor quantum well (QW) is proposed to realize parity-time (PT) symmetry. By appropriately tuning the laser fields and the pertinent QW parameters, PT-symmetric optical potentials are obtained by three different methods. Such a coherent QW system is reconfigurable and controllable, and it can generate new approaches of theoretically and experimentally studying PT-symmetric phenomena.

Introduction

The non-Hermitian parity-time (PT)-symmetric Hamiltonians, which were firstly proposed by Bender and Boettcher in 1998, have attracted great attention¹. Because of the isomorphism between quantum Schrödinger equations and paraxial-wave equations², optical system becomes an ideal bed for experimentally studying PT symmetry. By balancing gain and loss, optical PT symmetry has been realized in different coupled structures, such as waveguides^3,4, lattices^5,6, micro-cavities^7,8, and can be used in the fields of unidirectional propagating^9,10, perfect absorbers^11,12, photon lasers^13,14, phonon lasers¹⁵ and sensors¹⁶. In addition, several interesting phenomena, such as optical solitons¹⁷, Bloch oscillations¹⁸ and topological insulators¹⁹ have also been investigated in optical PT-symmetric systems. However, most of the work, particularly in experiment, was based on solid-state materials. Once the structures are fabricated, the properties such as the threshold of the system are unable to be changed.

The susceptibility of host semiconductor quantum wells (QWs) can be controlled via electrical field^20,21,22, carrier density²³ and laser field²⁴. The optical response of QWs is significantly enhanced when the frequency of light field is near resonance with the intersubband transition which has large dipole matrix element²⁵. As a result, one can obtain a dramatic change in the complex dielectric constant^{20,21,22,23,24,25}. Combined with electromagnetically induced transparency (EIT)^26,27, both the refractive index and the absorptive coefficient of QWs can be effectively manipulated via atomic coherence and quantum interference.

In this work, we propose to use a coherently prepared asymmetric double semiconductor QWs to obtain PT symmetry. Such QW systems have been proved to have the possibility to realize quantum coherence and interference^28,29,30. We demonstrate that PT-symmetric optical potentials, such as coupled optical waveguides, one-dimensional (1D) and two-dimensional (2D) PT-symmetric optical lattices can be realized by different methods. Besides, it is possible to control the PT-symmetric properties by tuning the laser fields and the QW parameters. The realization of PT symmetry in coherent QW systems by laser fields have several advantages. Firstly, compared with solid-state materials with micro-structures or nano-structures, PT-symmetric properties in this system can be established by different methods, and can be effectively controlled by various parameters. Secondly, PT symmetry has been constructed theoretically^31,32,33,34 and experimentally³⁵ in different atomic systems. Compared with these atomic systems, semiconductor QW systems have designable and flexible of energy levels, and they are easy to be integrated and stable for practical application. Thirdly, in such systems large nonlinearity can be realized assisted by EIT^36,37,38, which makes it possible to observe traveling effects of lights in non-Hermitian nonlinear optical systems^39,40,41.

Models and Equations

We consider asymmetric double semiconductor QWs⁴², which consist of a 51-monolayer (145 Å)-thick wide well (WW) and 35-monolayer (100 Å)-thick narrow well (NW). Between them there is a thickness of a 9-monolayer (25-Å)-thick Al_0.2Ga_0.8As 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 Al_0.2Ga_0.8As buffer layers. All these pairs are sandwiched between nominally undoped 3500-Å-thick Al_0.2Ga_0.8As layers.

Figure 1

(a) Schematic of one pair of asymmetric double QWs with buffer layers. (b) Band diagram of the asymmetric double QWs, and z represent the wafer-growth direction.

Such asymmetric double semiconductor QWs can be treated as a four-level N configuration^28,29,30, 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, arising from the tunneling effect between the WW and NW via the thin barrier, respectively. The probe field E_p with frequency ω_p probes the transition $|1\rangle \leftrightarrow |3\rangle$, while the coupling field E_c with frequency ω_c and the pump field E_d with frequency ω_d act on transitions $|2\rangle \leftrightarrow |3\rangle$ and $|1\rangle \leftrightarrow |4\rangle$ respectively. The Rabi frequency of the probe, coupling, and pump fields are Ω_p = μ₁₃E_p/2h, Ω_c = μ₂₃E_c/2h, and Ω_d = μ₁₄E_d/2h, respectively, where µ_ij is the associated dipole transition matrix element, and the detuning of the probe, coupling, and pump fields are Δ_p = ω_p − ω₃₁, Δ_c = ω_c − ω₃₂, and Δ_d = ω_d − ω₄₂, respectively, where ω_ij is the transition frequency between levels |i〉 and |j〉.

Under the condition of low QW carrier intensity, many-body effects are attributed to electron–electron interactions can be neglected⁴³. In the interaction picture and under the rotating wave approximation, the Hamiltonian of the QW system can be written as (h = 1)

$$\begin{array}{rcl}H & = & ({{\rm{\Delta }}}_{c}-{{\rm{\Delta }}}_{p})|2\rangle \langle 2|-{{\rm{\Delta }}}_{p}|3\rangle \langle 3|-{{\rm{\Delta }}}_{d}|4\rangle \langle 4|\\ & & +\,[{{\rm{\Omega }}}_{p}|1\rangle \langle 3|+{{\rm{\Omega }}}_{c}|2\rangle \langle 3|+{{\rm{\Omega }}}_{d}|1\rangle \langle 4|+{\rm{H}}{\rm{.c}}{\rm{.}}].\end{array}$$

(1)

Here, H.c. is the Hamiltonian complex conjugate.

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

$$\dot{\rho }=-\,i[H,\rho ]-\frac{1}{2}\{\gamma ,\rho \},$$

(2)

where γ is the dissipation matrix. Substituting Eq. (1) into Eq. (2), the density matrix for each element can be obtained:

$${\dot{\rho }}_{22}=i{{\rm{\Omega }}}_{c}({\rho }_{32}-{\rho }_{23})+{{\rm{\Gamma }}}_{42}{\rho }_{44}+{{\rm{\Gamma }}}_{32}{\rho }_{33}-{{\rm{\Gamma }}}_{2}{\rho }_{22},$$

(3a)

$${\dot{\rho }}_{33}=i{{\rm{\Omega }}}_{c}({\rho }_{23}-{\rho }_{32})+i{{\rm{\Omega }}}_{p}({\rho }_{13}-{\rho }_{31})-{{\rm{\Gamma }}}_{3}{\rho }_{33}+{{\rm{\Gamma }}}_{43}{\rho }_{44},$$

(3b)

$${\dot{\rho }}_{44}=i{{\rm{\Omega }}}_{d}({\rho }_{14}-{\rho }_{41})-{{\rm{\Gamma }}}_{4}{\rho }_{44},$$

(3c)

$${\dot{\rho }}_{12}=-\,i{{\rm{\Omega }}}_{c}{\rho }_{13}+i{{\rm{\Omega }}}_{p}{\rho }_{32}+i{{\rm{\Omega }}}_{d}{\rho }_{42}-{\tilde{\gamma }}_{12}{\rho }_{12},$$

(3d)

$${\dot{\rho }}_{13}=-\,i{{\rm{\Omega }}}_{c}{\rho }_{12}+i{{\rm{\Omega }}}_{d}{\rho }_{43}+i{{\rm{\Omega }}}_{p}({\rho }_{33}-{\rho }_{11})-{\tilde{\gamma }}_{13}{\rho }_{13},$$

(3e)

$${\dot{\rho }}_{14}=i{{\rm{\Omega }}}_{p}{\rho }_{34}+i{{\rm{\Omega }}}_{d}({\rho }_{44}-{\rho }_{11})-{\tilde{\gamma }}_{14}{\rho }_{14},$$

(3f)

$${\dot{\rho }}_{23}=-\,i{{\rm{\Omega }}}_{p}{\rho }_{21}+i{{\rm{\Omega }}}_{c}({\rho }_{33}-{\rho }_{22})-{\tilde{\gamma }}_{23}{\rho }_{23},$$

(3g)

$${\dot{\rho }}_{24}=-\,i{{\rm{\Omega }}}_{d}{\rho }_{21}+i{{\rm{\Omega }}}_{c}{\rho }_{34}-{\tilde{\gamma }}_{24}{\rho }_{24},$$

(3h)

$${\dot{\rho }}_{34}=i{{\rm{\Omega }}}_{p}{\rho }_{14}-i{{\rm{\Omega }}}_{d}{\rho }_{31}+i{{\rm{\Omega }}}_{c}{\rho }_{24}-{\tilde{\gamma }}_{34}{\rho }_{34},$$

(3i)

Here ρ₁₁ + ρ₂₂ + ρ₃₃ + ρ₄₄ = 1 and ${\rho }_{ij}={\rho }_{ji}^{\ast }$. Γ_ij is the natural decay rate between levels |i〉 and |j〉. 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 are, Γ₃₁ = Γ₃₂ and Γ₄₁ = Γ₄₂. There is also no decay in the valence band or conduction band, so we conclude that Γ₂₁ = Γ₄₃ = 0. ${{\rm{\Gamma }}}_{i}={\sum }_{j}^{i-1}{{\rm{\Gamma }}}_{ij}$ denotes the total decay rate of level |i〉. We define ${\tilde{\gamma }}_{12}$ = ${\gamma }_{12}+i({{\rm{\Delta }}}_{p}-{{\rm{\Delta }}}_{c})$, ${\tilde{\gamma }}_{13}$ = ${\gamma }_{13}+i{{\rm{\Delta }}}_{p}$, ${\tilde{\gamma }}_{14}$ = ${\gamma }_{14}+i{{\rm{\Delta }}}_{d}$, ${\tilde{\gamma }}_{23}$ = ${\gamma }_{23}+i{{\rm{\Delta }}}_{c}$, ${\tilde{\gamma }}_{24}$ = ${\gamma }_{24}+i({{\rm{\Delta }}}_{c}+{{\rm{\Delta }}}_{d}-{{\rm{\Delta }}}_{p})$, and ${\tilde{\gamma }}_{34}$ = ${\gamma }_{34}+i({{\rm{\Delta }}}_{d}-{{\rm{\Delta }}}_{p})$, where γ_ij = (Γ_i + Γ_j)/2.

In such QW systems, ${{\rm{\Gamma }}}_{3}={{\rm{\Gamma }}}_{3l}+{{\rm{\Gamma }}}_{3}^{{\rm{dph}}}$ and ${{\rm{\Gamma }}}_{4}={{\rm{\Gamma }}}_{4l}+{{\rm{\Gamma }}}_{4}^{{\rm{dph}}}$, where Γ_3l and Γ_4l are the population decay rates of subbands |3〉 and |4〉 respectively, resulting from longitudinal-optical phonon emission events at low temperature, and ${{\rm{\Gamma }}}_{3}^{{\rm{dph}}}$ and ${{\rm{\Gamma }}}_{4}^{{\rm{dph}}}$ are the dephasing decay rates of quantum coherence due to electron–electron scattering, phonon scattering processes; and elastic interface roughness. Based on the previous studies, we assume that Γ_3l can be equal to Γ_4l, and ${{\rm{\Gamma }}}_{3}^{{\rm{dph}}}$ can be equal to ${{\rm{\Gamma }}}_{4}^{{\rm{dph}}}$. Therefore, Γ₃₁ = Γ₃₂ = Γ₄₁ = Γ₄₂ = Γ can be presented. In addition, we assume there is no interference or dephasing between levels |3〉 and |4〉, which can be realized based on the appropriate reduction of the temperature⁴⁴.

The susceptibility of the QW medium can be obtained through the expression $\chi$ = $N{\mu }_{13}/{\varepsilon }_{0}{E}_{p}\cdot {\rho }_{31}$ = $N{\mu }_{13}^{2}/2{\varepsilon }_{0}{{\rm{h}}{\rm{\Omega }}}_{p}\cdot {\rho }_{31}$, where N is the electron density of the QWs, and ρ₃₁ can be obtained by Eq. (3). χ = χ_R + iχ_I, where χ_R describes the dispersion properties of the probe field, while χ_I describes the absorption properties of the probe field with χ_I > 0 (χ_I < 0) indicating loss (gain).

The refractive index of the QW medium can be written as $n=\sqrt{{\varepsilon }_{c}+\chi }$. Here ${\varepsilon }_{c}={n}_{c}^{2}$ corresponding to the complex dielectric constant of the host QW medium, and n_c is its refractive index when light is far detuned from the resonance. χ denotes the change in the susceptibility^22,23, which results from the coherent contract via the coupling and pump fields near the resonance. Generally, χ = ε_c, then we have $n$ ≈ $\sqrt{{\varepsilon }_{c}}+\chi /22\sqrt{{\varepsilon }_{c}}$ = ${n}_{c}+\chi /2{n}_{c}$ = ${n}_{c}+{\chi }_{R}/2{n}_{c}+i{\chi }_{I}/2{n}_{c}$. We define the real and imaginary parts of the refractive index as n_R and n_I, with n_c being the background index of the system. Thus, n = n_c + n_R + in_I, where n_R = χ_R/2n₀ and n_I = χ_I/2n₀. To achieve PT symmetry, the condition of n_R(r) = n_R(−r) and n_I(r) = −n_I(−r) must be satisfied. And for simplicity, we use the unit $N{\mu }_{13}^{2}/4{\varepsilon }_{0}{\rm{h}}{n}_{0}$ in the following calculations.

Results

PT-symmetric optical waveguides

In order to realize PT-symmetric optical waveguides, we use a pair of coupling laser beams to form two coupled waveguides. Such pair of beams propagate in QWs along y direction. Then, a pump and a probe laser beams with wider laser dimension propagate in the same direction as those of the coupling beams. The schematic diagram is shown in Fig. 2. The pair of coupling laser beams have an identical Gaussian intensity profile so that the total spatial intensity distribution of them varying in x direction is

$${I}_{c}(x)=A{e}^{\frac{-{(x-a)}^{2}}{2{\sigma }^{2}}}+A{e}^{\frac{-{(x+a)}^{2}}{2{\sigma }^{2}}},$$

(4)

where A and a are the constant and the half separation between the two waveguides respectively. $2\sqrt{2\,\mathrm{ln}\,2}\sigma$ can describe the full width at half maximum (FWHM) of the waveguides. By choosing two different detuning of the coupling fields, gain can be introduced to one waveguide and loss can be introduced to the other, even though other parameters are identical. In such consideration, the refractive index in each waveguide spatially varies only with the intensity of the coupling.

Figure 2

Schematic diagram of QW system for realizing PT-symmetric optical waveguides. x and y represent the transverse and longitudinal directions of laser beams.

To realize gain and absorption in two waveguides simultaneously, different detuning of the coupling fields need to be found. For sake of it, we calculate the real (dispersion) χ_R and imaginary (gain or absorption) χ_I parts of the susceptibility as a function of Δ_c, and show the results in Fig. 3(a,b), respectively. The intensity of the coupling field in each waveguide is corresponding to Gaussian profile, so χ_I needs to get larger with increasing coupling intensity. For this reason, we only consider a negative value of the coupling detuning. It can be seen from the figures that for different value of Ω_c, χ_R reaches to the maximum value, and χ_I is close to zero at Δ_c = −2.325Γ. At the vicinity of the zero point, gain is obtained on the left side, and absorption is abstained on the right side. This property called refractive index enhancement with vanishing absorption, has been demonstrated in the early papers^{45,46,47,48,49}.

Figure 3

(a) Real part χ_R and (b) imaginary part χ_I of the susceptibility as a function of coupling detuning Δ_c for different coupling Rabi frequency, Ω_c = 0.4 meV (black solid line), Ω_c = 0.3 meV (blue dashed line), and Ω_c = 0.2 meV (red dotted line). (c) Real part χ_R and (d) imaginary part χ_I of the susceptibility as a function of coupling Rabi frequency Ω_c for different coupling detuning, Δ_c = −2.278 meV (blue solid line) and Δ_c = −2.360 meV (red dashed line). The other parameters are ${{\rm{\Gamma }}}_{3}^{{\rm{dph}}}={{\rm{\Gamma }}}_{4}^{{\rm{dph}}}=2.58{\rm{meV}}$, Γ_3l = Γ_4l = 2.07 meV, Γ₃₁ = Γ₃₂ = Γ₄₁ = Γ₄₂ = Γ = 2.325 meV, Ω_p = 0.01Γ, Ω_d = 2Γ, Δ_p = Δ_d = 0.

In order to realize gain and loss in two waveguides simultaneously by using two different coupling detuning and maintain other parameters identical, the relation between the susceptibility and the coupling intensity needs to be established. Therefore, we choose Δ_c = −2.360Γ and Δ_c = −2.278Γ in two waveguides, and draw χ_R and χ_I as a function of Ω_c shown in Fig. 3(c,d). It can be seen that with selected coupling detuning, the curves of χ_R are identical, and the curves of χ_I are matched with slight difference, respectively. It should be noted that, in Fig. 3(d) we flipped the curve of the gain one by multiplying a minus sign to compare the two curves of gain and loss directly.

Based on the above analysis, firstly, we demonstrate the possibility of realizing spatial modulation of the refractive index. Making use of Eq. (4), and choosing the FWHM ($2\sqrt{2\,\mathrm{ln}\,2}\sigma$) as 7 μm and the separation between the two waveguides as 20 μm, we plot the real n_R and the imaginary n_I parts of the refractive index as a function of x shown in Fig. 4(a–c). It can be seen from the figures that the real part of the refractive index n_R is an even function of x, while that of imaginary part n_I is an odd function of x. By changing the electron intensity of QW, The absolute values of n_R and n_I can be modified with equal scale simultaneously with the changes of the electron intensity of QW.

Figure 4

Real part n_R and imaginary part n_I of the refractive index as a function of position x for different cases, (a) Δ_c = −2.350 meV (loss waveguide) and Δ_c = −2.300 meV (gain waveguide), representing the condition of below threshold, (b) Δ_c = −2.386 meV (loss waveguide) and Δ_c = −2.266 meV (gain waveguide), representing the condition of above threshold, (c) Δ_c = −2.325 meV (both waveguides), representing the condition of non-PT symmetry, respectively. (d–f) are filed mode of the probe laser beam according to (a–c). (g–i) are propagation properties of the probe laser beam according to (a–c). The other parameters are the same as in Fig. 3.

By controlling the ratio of the real and imaginary parts of the refractive index, we can adjust the condition of the system below or above the PT-symmetric threshold. It can be seen from Fig. 3(a,b) that χ_R changes less than χ_I at the vicinity of zero point (Δ_c = −2.325Γ). Therefore, we can alter the coupling detuning in the waveguides to control the ratio. For instance, with different values of the coupling detuning, the ratio of the real and imaginary parts is 12 and 80, and the system can be operated below (Fig. 4(a)) or above (Fig. 4(b)) the PT-symmetric threshold respectively. In addition, when the coupling detuning used in both waveguides is −2.325Γ, the system even becomes a non-symmetric one with much larger ratio (Fig. 4(c)).

Secondly, we present the field modes in the waveguides in Fig. 4(d–f) corresponding to Fig. 4(a–c), respectively. When the condition of the system is below the threshold (Fig. 4(a)), the PT symmetry condition is satisfied, the eigenvalues should be real, and the field modes should be symmetric. However, because of the imperfect PT symmetry condition, the eigenvalues have a very small imaginary part, and the two field modes are slightly asymmetric, as shown in Fig. 4(d). For the case of above the threshold (Fig. 4(b)), that is, the PT symmetry is broken, the eigenvalues become complex, where the imaginary part of the refractive index represents the gain or loss for each filed mode. Figure 4(e) illustrates that, the light modes become strongly asymmetric. For the non-symmetric one (Fig. 4(c)), the light modes are perfectly symmetric, which is shown in Fig. 4(f).

Thirdly, we show in Fig. 4(g–i) the propagation characteristics of the probe beam according to the condition of Fig. 4(a–c) respectively. When the condition of the system is below threshold, the probe beam oscillates periodically between the two waveguides, as show in Fig. 4(g). When the case is above the threshold, it can be seen from Fig. 4(h) that an exponentially growing mode occurs, which signifies the onset of PT symmetry breaking. For the passive waveguides, the probe beam also oscillates periodically (Fig. 4(i)), which is similar to the case of below threshold. However, the oscillation period in Fig. 4(i) is shorter than that of in Fig. 4(g).

In a practical system, it is very difficult to realize perfect PT-symmetric condition. So, we evaluate the degrees of asymmetry of the system by analyzing the asymmetry function of Δ(x) = n(x) − n*(−x), and find that the degree of asymmetry is very small. Therefore, it is manageable and has little impact on practical experiments.

PT-symmetric optical lattices—Type I

In this part, we first consider an 1D optical lattices of period Λ_x along x direction, and in each lattices the electron density of QWs is spatially modulated. The electron density in the ith trap can exhibit a Gaussian distribution ${N}_{i}(x,y)=N{e}^{-{(x-{x}_{i})}^{2}/{\sigma }_{x}^{2}}$, where N is the electron density of a homogeneous QW sample, x_i is the ith trap center, and σ_x is the half width. The modulation of electron density of QWs can be achieved by using high order surface grating^50,51. We further consider a pump laser field with periodically modulated intensity ${{\rm{\Omega }}}_{d}={{\rm{\Omega }}}_{d0}+\delta {{\rm{\Omega }}}_{d}\,\sin \,[2\pi (x-{x}_{i})/{{\rm{\Lambda }}}_{x}]$, which can be easily achieved in experiment by using an imperfect standing-wave (SW) field with unequal forward and backward components. We show the schematic diagram of QW system in Fig. 5. The probe field and coupling filed propagate along z direction, and the SW pump fields propagate along x direction.

Figure 5

Schematic diagram of QW system for realizing PT-symmetric optical lattices. Electron density of QW is modulated in x direction.

First, we calculate the real χ_R and imaginary χ_I parts of the susceptibility as a function of Δ_p for varying value of Ω_d by solving Eq. (3) without modulation of pump intensity or the electron density, and the corresponding results are shown in Fig. 6. It can be seen from red dotted line in Fig. 6(b) that when Ω_d = 0.8Γ, a typical EIT with positive value (loss) is realized. With increasing value of Ω_d, at both sides of the EIT windows, χ_I becomes positive (gain), which is knew as coherent Raman gain without population inversion [blue dashed line and black solid line in Fig. 6(b)]. Meanwhile, χ_R changes form positive dispersion to negative dispersion in the vicinity of EIT window, as shown in Fig. 6(a). The figures indicate that it is possible to realize PT symmetry by choosing suitable modulations.

Figure 6

(a) Real part χ_R and (b) imaginary part χ_I of the susceptibility as a function of probe detuning Δ_p for different pump Rabi frequency, Ω_d = 0.8Γ (red dotted line), Ω_d = Γ (blue dashed line), and Ω_d = 1.2Γ (black solid line). The parameters are Ω_c = Γ and Δ_c = Δ_d = 0. The other parameters are the same as in Fig. 3.

Then, with modulation of the pump intensity and electron density, we calculate the real (n_R) and imaginary parts (n_I) of the refractive index as a function of x and show the results in Fig. 7. Figure 7(a) shows the condition of Δ_p = 2.592 meV and δΩ_d = 0.1Γ. The value of Δ_p is chosen according to blue dashed line in Fig. 6(b), where χ_I has zero value around Δ_p = 2.592 meV, and thus it exhibits gain or absorption if Δ_p is slightly tuned away from this point. It can be seen that n_R is an even function of the lattice position x, while n_I is an odd function of lattice position x with balance gain and loss. Such results clearly indicate that PT symmetry is built in QW system by modulating the pump field and the electron density.

Figure 7

Real part n_R and imaginary part n_I of the refractive index as a function of position x for unchanged absolute value and different signs of Δ_p and δΩ_d, respectively, (a) Δ_p = 2.592 meV and δΩ_d = 0.1Γ, (b) Δ_p = 2.592 meV and δΩ_d = −0.1Γ, (c) Δ_p = −2.592 meV and δΩ_d = 0.1Γ, (d) Δ_p = −2.592 meV and δΩ_d = −0.1Γ. The parameters are Ω_d0 = Γ. The other parameters are the same as in Fig. 6.

In Figs 7(b) and 6(d), we further show the cases for the same absolute value of Δ_p and δΩ_d as that of in Fig. 7(a), but with different sign of them. The results show that the PT-symmetric properties are maintained in all cases. However, the relation between n_R and n_I are different. It is found that positive (negative) value of Δ_p results in the positive (negative) value of n_R and have no effect on the profile of n_I. On the other hand, the sign of δΩ_d determines the position of gain and loss in each period, for instance, with positive (negative) value of δΩ_d, n_I is gain (loss) in one half period −0.5 ≤ x/Λ_x ≤ 0. The understanding of impacts of the parameters on spatial features of the refractive index is important because of its potential application, such as asymmetric light diffraction^52,53.

We next check what will happen if the absolute value of Δ_p and δΩ_d is changed. To do so, we plot 2D n_R and n_I as functions of x and Δ_p in Fig. 8(a,b), and as functions of x and δΩ_d in Fig. 8(c,d), respectively. The figures clearly show that n_R and n_I are modulated along x direction for varying Δ_p or δΩ_d. In addition, the profile of n_I is much more sensitive to the parameters than that of n_R. When Δ_p or δΩ_d is detuned from the value used in Fig. 7(a), through n_R is maintained an even property, n_I losses odd property, where the loss in one half period is not equal to the gain in the other half. Therefore, PT symmetry is destroyed in QW system because of the nonlinear response of n_I to Δ_p or δΩ_d.

Figure 8

(a) Real part n_R and (b) imaginary part n_I of the refractive index as functions of position x and probe detuning Δ_p. The parameters are δΩ_d = 0.1Γ. (c) Real part n_R and (d) imaginary part n_I of the refractive index as functions of position x and modulation intensity δΩ_d. The parameters are Δ_p = 2.592 meV. The other parameters are the same as in Fig. 7.

Last, our goal is to realize 2D modulation of refractive index as functions of x and y, and this can be achieved by 2D modulation of the electron density and pump field, which are

$${N}_{i}(x,y)=N{e}^{-{(x-{x}_{i})}^{2}/{\sigma }_{x}^{2}-{(y-{y}_{i})}^{2}/{\sigma }_{y}^{2}},$$

(5a)

$${{\rm{\Omega }}}_{d}={{\rm{\Omega }}}_{d0}+0.5\,\delta {{\rm{\Omega }}}_{d}\{\sin \,[2\pi (x-{x}_{i})/{{\rm{\Lambda }}}_{x}]+\,\sin \,[2\pi (y-{y}_{i})/{\Lambda }_{y}]\},$$

(5b)

We plot in Fig. 9(a,b) the top view of n_R and n_I as functions of x and y using the same parameters in Fig. 7(a), respectively. The results clearly show that n_R is an even functions of x and y, while n_I is an odd functions of x and y, therefore, 2D PT-symmetric optical lattices is realized in QW systems.

Figure 9

(a) Real part n_R and (b) imaginary part n_I of the refractive index as functions of position x and position y. The parameters are Δ_p = 2.592 meV and δΩ_d = 0.1Γ. The other parameters are the same as in Fig. 7.

PT-symmetric optical lattices—Type II

In this part, we show that PT symmetry can also be achieved by modulating both pump and coupling fields. The schematic diagram of the system is shown in Fig. 10, where the probe field propagates along z direction, and the SW coupling and SW pump fields propagate along x direction. Therefore, we have modulated coupling and pump fields, ${{\rm{\Omega }}}_{c}$ = ${{\rm{\Omega }}}_{c0}+\delta {{\rm{\Omega }}}_{c}\,\cos \,[2\pi (x-{x}_{i}){/{\rm{\Lambda }}}_{x}]$, ${{\rm{\Omega }}}_{d}$ = ${{\rm{\Omega }}}_{d0}+\delta {{\rm{\Omega }}}_{d}\,\sin \,[2\pi (x-{x}_{i}){/{\rm{\Lambda }}}_{x}]$.

Figure 10

Schematic diagram of QW system for realizing PT-symmetric optical lattices.

First, we plot the real and imaginary parts of the refractive index (n_R and n_I) as a function of x in Fig. 11 for different cases. Figure 11(a) shows the condition of Δ_p = 2.592 meV and the modulation intensities δΩ_c = 0.3Γ and δΩ_d = 0.1Γ. It can be seen from figure that the PT symmetry is appears, where n_R is an even function of x, and n_I is an odd function of x. The relations of n_R and n_I can also be modified by changing the sign of the detuning Δ_p and the modulation intensities δΩ_c and δΩ_d, and the changing of their sign will not destroy the PT symmetry. There are eight kinds of combinations of these three parameters, corresponding to eight kinds of spatial refractive index, and we plot the other seven kinds in Fig. 11(b–h). It can be found that Δ_p, δΩ_c or δΩ_d have different impacts on the features of n_R and n_I. More specifically, Δ_p determines n_R being positive or negative, and δΩ_c determines the monotonicity of n_R, and δΩ_d determines the positions of the gain and loss in one period.

Figure 11

Real part n_R and imaginary part n_I of the refractive index as a function of position x for unchanged absolute value and different signs of Δ_p, δΩ_c and δΩ_d, respectively, (a) Δ_p = 2.525 meV, δΩ_c = 0.3Γ and δΩ_d = 0.1Γ, (b) Δ_p = 2.525 meV, δΩ_c = 0.3Γ and δΩ_d = −0.1Γ, (c) Δ_p = 2.525 meV, δΩ_c = −0.3Γ and δΩ_d = 0.1Γ, (d) Δ_p = 2.525 meV, δΩ_c = −0.3Γ and δΩ_d = −0.1Γ, (e) Δ_p = −2.525 meV, δΩ_c = 0.3Γ and δΩ_d = 0.1Γ, (f) Δ_p = −2.525 meV, δΩ_c = 0.3Γ and δΩ_d = −0.1Γ, (g) Δ_p = −2.525 meV, δΩ_c = −0.3Γ and δΩ_d = 0.1Γ, (h) Δ_p = −2.525 meV, δΩ_c = −0.3Γ and δΩ_d = −0.1Γ. The parameters are Ω_c0 = 3Γ and Ω_d0 = Γ. The other parameters are the same as in Fig. 6.

We also check the impact of absolute value of Δ_p, δΩ_c or δΩ_d on the properties of PT-symmetric system. Therefore, we plot 2D n_R and n_I as functions of x and Δ_p in Fig. 12(a,b), as functions of x and δΩ_d in Fig. 12(c,d), and as functions of x and δΩ_c in Fig. 12(e,f), respectively. The figures clearly show that n_R and n_I are modulated along x direction for varying Δ_p, δΩ_c or δΩ_d. However, when these parameters are detuned form the value used in Fig. 11(a), the system will loss the PT-symmetric properties.

Figure 12

(a) Real part n_R and (b) imaginary part n_I of the refractive index as functions of position x and probe detuning Δ_p. The parameters are δΩ_c = 0.3Γ and δΩ_d = 0.1Γ. (c) Real part n_R and (d) imaginary part n_I of the refractive index as functions of position x and modulation intensity δΩ_d. The parameters are Δ_p = 2.525 meV and δΩ_c = 0.3Γ. (e) Real part n_R and (f) imaginary part n_I of the refractive index as functions of position x and modulation intensity δΩ_c. The parameters are Δ_p = 2.525 meV and δΩ_d = 0.1Γ. The other parameters are the same as in Fig. 11.

Last, we show that it is possible to realize 2D PT-symmetric optical lattices by applying 2D modulation of the pump and coupling laser fields, which are

$${{\rm{\Omega }}}_{c}={{\rm{\Omega }}}_{c0}+0.5\,\delta {{\rm{\Omega }}}_{c}\{\cos \,[2\pi (x-{x}_{i})/{{\rm{\Lambda }}}_{x}]+\,\cos \,[2\pi (y-{y}_{i})/{{\rm{\Lambda }}}_{y}]\},$$

(6a)

$${{\rm{\Omega }}}_{d}={{\rm{\Omega }}}_{d0}+0.5\,\delta {{\rm{\Omega }}}_{d}\{\sin \,[2\pi (x-{x}_{i})/{{\rm{\Lambda }}}_{x}]+\,\sin \,[2\pi (y-{y}_{i}){/{\rm{\Lambda }}}_{y}]\},$$

(6b)

Using the same parameters in Fig. 11(a), we calculate n_R and n_I as functions of x and y, and show the tope view of n_R and n_I as functions of x and yin Fig. 13(a,b), respectively. The results indicate that 2D PT-symmetric optical lattices can be achieved, with n_R being an even functions of x and y, and n_I being an odd functions of x and y, respectively.

Figure 13

(a) Real part n_R and (b) imaginary part n_I of the refractive index as functions of position x and position y. The parameters are Δ_p = 2.525 meV, δΩ_c = 0.3Γ and δΩ_d = 0.1Γ. The other parameters are the same as in Fig. 11.

Conclusion

In conclusion, we have demonstrated that coherent asymmetric double semiconductor QWs can be an ideal candidate for studying PT symmetry. We have showed that PT-symmetric optical waveguides can be realized by using two coupling fields with different detuning, 1D and 2D PT-symmetric optical lattices can be realized by spatial modulation of pump laser fields and the electron density of QWs, or by spatial modulation of pump and coupling fields. In addition, the PT-symmetric properties realized in QW systems, such as the relation between the real and the imaginary parts of the complex refractive indices, can be controlled by changing the laser fields and the parameters of QWs.