1. Introduction

In recent years, significant attention has been given to the development of meta-material to bring uncommon and outstanding properties in the world. Such as the Hermitian lattice systems where the transport properties have been studied, but less consideration has been given to non-Hermitian systems. After the discovery of parity-time ($\mathcal {PT}$) symmetric Hamiltonian, a new way has been opened for the physics community to study the non-Hermitian systems [1,2]. It has been developed that non-Hermitian Hamiltonian contains real eigen spectrum in a specific region of space under $\mathcal {PT}$-symmetry. Since then much attention has been paid to the concept of $\mathcal {PT}$-symmetry in various theoretical fields such as quantum field theories, Lattice quantum chromodynamics (QCD) theories, non-Hermitian Anderson models, quantum optics and complex Lie algebras. In contrast to the theoretical investigations, the characteristics of $\mathcal {PT}$-symmetry has explored experimentally [3], and various fascinating features have been studied such as power oscillation [4–6], unidirectional invisibility [7,8], coherent perfect absorbers [9,10] and non-reciprocal light propagation [11]. The $\mathcal {PT}$-symmetry has been studied either theoretically and experimentally in optical systems, for example, in photonic structures [12,13], coupled wave guides [14,15], whispering-gallery micro-cavities [16,17], optomechanical systems [18,19], asymmetric mode switching [20], optical switching [21] and transmission lines [22]. However, some investigations have come into sight in which atomic clouds or atomic vapors are used for the investigation of $\mathcal {PT}$-symmetry [23–29]. In $N$-level atomic configuration [28], $\mathcal {PT}$-antisymmetric modulation of optical susceptibility has been realized in one-dimensional (1D) atomic lattices in the absence of gain property. Very recently, a new scheme has been designed that consists of 1D atomic lattices and investigated $\mathcal {PT}$-symmetry and $\mathcal {PT}$-antisymmetry along the lattice direction of the optical susceptibility [29]. In this article [29], a four-level $N$-type atomic medium has been used and realized $\mathcal {PT}$- and $\mathcal {PT}$-antisymmetric modulation and designed a sinusoidal modulation of a field in terms of intensity. The proposed model also based on Gaussian distribution in each cycle of atomic density in 1D optical lattices. The $\mathcal {PT}$- and $\mathcal {PT}$-antisymmetric optical susceptibility in 1D atomic lattices have been investigated via changing the modulation amplitude.

The phenomenon becomes quite different when a medium satisfies the condition $n(x)=-n^*(-x)$. According to the condition, a medium is said to be $\mathcal {PT}$-antisymmetric in which the real (imaginary) part is odd (even) in space. The phenomenon of $\mathcal {PT}$-antisymmetry has been reported in 1D atomic lattices [28,29]. Under $\mathcal {PT}$-antisymmetry, a refraction-less propagation in warm vapors has been reported experimentally [30]. In addition, for 1D $\mathcal {PT}$-symmetric optical system, the complex optical refractive index must satisfy the condition i.e., $n(x)=n^*(-x)$. It means that the real part of refractive index must be even function of $x$ i.e., $n_r(x)=n_r(-x)$ whereas the imaginary part be an odd function of $x$ such as $n_I(x)=-n_I(-x)$. To extend 1D $\mathcal {PT}$-symmetry to 2D space, we define the complex optical refractive index as $n(x,y)=n^*(-x,-y)$. This shows that the real and imaginary parts of a complex optical refractive index must be even and odd functions of transverse coordinates $x$ and $y$, respectively. The revolutionary research cited above and from recent literature on $\mathcal {PT}$-symmetry covers 1D schemes. Further, motivation comes from recent development to study 2D $\mathcal {PT}$-symmetry lattices such as modulation of gain-loss in 2D materials [31,32] and 2D crystals [33,34]. In such investigations, the gain-loss and refractive index are modulated simultaneously. Nevertheless, none of these investigations [31–34] is $\mathcal {PT}$-symmetric system because they do not fulfill the conditions of $\mathcal {PT}$-symmetry.

In this article, we suggest a scheme that consists of 2D optical lattices for the investigation of coherent control of $\mathcal {PT}$-, non-$\mathcal {PT}$-, and $\mathcal {PT}$-antisymmetry in optical susceptibility along $x$- and $y$-directions of the lattices. In 2D atomic lattices, we also consider a Gaussian distribution of the trapped atoms in each cycle. As discussed above that the $\mathcal {PT}$- and $\mathcal {PT}$-antisymmetry have been reported in 1D atomic lattices [29] using a four-level $N$-type atomic configuration. We follow the same approach as discussed above [29] and consider a three-level atomic medium instead of a four-level $N$-type configuration. The motivation comes from the 1D atomic lattices or monolayer which is only suitable for laboratory scale and these samples are typically tens of microns in size. In contrast with 1D lattices, the 2D materials have high crystallinity and superior electrical properties. In our proposed model, we consider 2D atomic lattices that have atomic coherence and long-range periodicity. We control the $\mathcal {PT}$- to non-$\mathcal {PT}$-symmetry and $\mathcal {PT}$-antisymmetry to non-$\mathcal {PT}$-symmetry without changing the structure of the atomic lattices by modifying the external parameter. The applications of our proposed model can be extended to optical devices such as modulators, detectors, photonic transistors, and diodes. Further, we consider an experimentally viable atomic configuration in our model [35] which has been used earlier for the observation of electromagnetically induced transparency via a microwave field. It means that the coherent control of the $\mathcal {PT}$- to non-$\mathcal {PT}$-symmetry and $\mathcal {PT}$-antisymmetry to non-$\mathcal {PT}$-symmetry can be easily realized experimentally.

2. Model

We consider 2D optical lattices having Gaussian distribution of clusters of cold three-level rubidium ($^{87}$Rb) atoms. Each cluster settled at the bottom of a dipole trap in $x$- and $y$-directions. Two optical fields are interacting with three-level atoms and coupled to the atomic transitions $|a\rangle \longleftrightarrow |b\rangle$ and $|a\rangle \longleftrightarrow |c\rangle$, see the energy level configuration in Fig. 1(a). A microwave field is also applied which coupled to the ground states as $|b\rangle \longleftrightarrow |c\rangle$. The interaction Hamiltonian under the rotating wave and dipole approximation is [35]

 

Fig. 1. (a) Energy-level configuration for three-level atomic configuration. (b) In the $x$- and $y$-directions, three-level atomic medium is tapped in 2D optical lattices in a Gaussian distribution.

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To calculate the optical susceptibility, we need the coherence $\rho _{ab}$ that can be calculated using the steady state regime i.e., $\dot \rho _{ab}=\dot \rho _{cb}=0$. The expression for $\rho _{ab}$ can be calculated as

3. Results and discussion

We start our discussion by studying the real and imaginary parts of the optical susceptibility without considering any spatial modulation. We plot the real and imaginary parts of the optical susceptibility versus probe field detuning $\Delta$ for two values of microwave field i.e., $\Omega _{\mu }=0$ and $0.05\gamma$, see Fig. 2. The other parameters [35] are $\gamma =1$MHz, $\gamma _{ab}=1\gamma$, $\gamma _{cb}=0.01\gamma$, $\Omega _p=0.01\gamma$, and $\Omega _2=1\gamma$. In the absence of the microwave field, we plot the real and imaginary parts of optical susceptibility versus probe field detuning. We get a typical electromagnetically induced transparency (EIT) spectrum as shown in Fig. 2 in black curves. In the typical EIT window, we have a normal dispersion and transparency between two absorption peaks. In our system, the microwave field plays a key role and can change the optical properties of the system. In the presence of the microwave field, a steep positive dispersion appears in the real part of optical susceptibility, whereas gain (amplification) appears in the imaginary part of optical susceptibility, see the red curves in Fig. 2. Under the influence of the microwave field and the relative phase, the transparency window converts into an amplification window. It is also shown in Fig. 2 that the normal dispersion is steeper (slower group velocity) in the amplification window as compared to the EIT window. This kind of behavior has been reported theoretically as well as experimentally [35,36] in the same three-level $\Lambda$ atomic configuration under the influence of microwave field and relative phase. It is also emphasized that by applying the microwave field, we get loss and gain simultaneously in a system, see the imaginary part (red curve) of the optical susceptibility in Fig. 2(b). Following the control of optical susceptibility via microwave field, probe field intensity and relative phase, next we expect the realization of $\mathcal {PT}$-, non- $\mathcal {PT}$-, and $\mathcal {PT}$-antisymmetry via microwave field, probe field intensity and relative phase.

 

Fig. 2. (a) Real and (b) imaginary parts of the optical susceptibility vs probe field detuning $\Delta$. The parameters are presented at the text.

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To realize $\mathcal {PT}$- and $\mathcal {PT}$-antisymmetry, we need gain and loss in a system simultaneously. Also for $\mathcal {PT}$- and $\mathcal {PT}$-antisymmetry, the refractive index or susceptibility must be a function of space. In our system, the susceptibility is a function of position ($x$ and $y$) via Eq. (5) and Eq. (6). To find the gain and loss in a system simultaneously, we must focus on the probe field detuning here. Once we find the value of probe field detuning where the gain and loss balance each other then we can study the $\mathcal {PT}$- and $\mathcal {PT}$-antisymmetry easily. To find the value of probe field detuning, we must show a density plot of the imaginary part of optical susceptibility versus probe field detuning $\Delta$ and $x$- or $y$-direction of the optical lattices, see Fig. 3. It is also emphasized that we should consider a small range of probe field detuning where the gain and loss approximately equal. We encircled (blue circle) the range of probe field detuning in Fig. 2(b). The plot clearly shows that there is gain and loss at positions $x/a_1$ or $(y/a_1)\approx 0.13$ and $x/a_1$ or $(y/a_1)\approx -0.13$. We can easily pick the value of probe field detuning from Fig. 3 at which the gain and loss are approximately equal which is $\Delta =0.48\gamma$ in our case. For the realization of $\mathcal {PT}$-symmetry, we must consider $\Delta =0.48\gamma$ throughout in our analysis.

 

Fig. 3. Density plot of imaginary part of optical susceptibility vs probe field detuning $\Delta$ and lattice position $x/a_1$ or ($y/a_1$). The parameters are $\Omega _{11}=1\gamma$, $\delta \Omega _{2x}=\delta \Omega _{2y}=0.3\gamma$, $\sigma =0.2a_1$, $a_1=0.5\lambda$ and $\Omega _\mu =0.05\gamma$, the other parameters remains the same as presented in the text.

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3.1 Coherent control of $\mathcal {PT}$- and non-$\mathcal {PT}$-symmetry

$\mathcal {PT}$-symmetric regime. To study the spatial modulation of optical susceptibility of 2D optical lattices, first, we make the optical susceptibility of the medium position-dependent. In our system, the optical susceptibility becomes position-dependent via Eq. (5) and Eq. (6). It is also established from Fig. 3 that the gain-loss ratio balances each other simultaneously at probe field detuning $\Delta =0.48\gamma$. Following this idea we consider $\Delta =0.48\gamma$, $\Omega _{\mu }=0.05\gamma$ and the relative phase $\phi =\pi /2$ and plot the real and imaginary parts of the optical susceptibility versus spatial distribution $x/a_1$ and $y/a_1$ as depicted in Fig. 4. The plots show that the real part of the optical susceptibility of 2D optical lattices is an even function of $x$ and $y$, while the imaginary part is an odd one. To analyze the imaginary part of the optical susceptibility as shown in Fig. 4(b), it is seen that the gain and loss ratio balance each other. Therefore, the plots in Fig. 4 fulfills the condition of $\mathcal {PT}$-symmetry, where the imaginary (real) part is an odd (even) function of position $x$ and $y$ as well as the gain-loss ratio also balance each other. The characteristics of the optical susceptibility of 2D optical lattices can be manipulated by changing the external parameters like laser fields. In the following, we only change the modulation amplitudes of a strong coupling field and study the behavior of $\mathcal {PT}$-symmetry. It is emphasized that in such an operation, the characteristics of $\mathcal {PT}$-symmetric medium do not change and experimentally it can be realized easily. We analyze all kinds of optical susceptibility of 2D optical lattices under different combinations of $\delta \Omega _{2x}$ and $\delta \Omega _{2y}$. We show the contour plot of real and imaginary parts of optical susceptibility of 2D optical lattices versus $x$-and $y$-direction as depicted in Fig. 5 and 6, respectively. It is found that the real part of optical susceptibility remains the same and even function of $x$ and $y$ for four different combinations of modulation amplitude i.e., ($\delta \Omega _{2x}=\delta \Omega _{2y}=0.3\gamma$), ($\delta \Omega _{2x}=\delta \Omega _{2y}=-0.3\gamma$), ($\delta \Omega _{2x}=-0.3\gamma ,\delta \Omega _{2y}=0.3\gamma$) and ($\delta \Omega _{2x}=0.3\gamma ,\delta \Omega _{2y}=-0.3\gamma$), see Fig. 5. We also find that the sign of $\delta \Omega _{2x}$ and $\delta \Omega _{2y}$ affect the imaginary part of optical susceptibility as depicted in Fig. 6. It is seen that the gain-loss ratio balance each other in all four combinations of $\delta \Omega _{2x}$ and $\delta \Omega _{2y}$ and also odd functions of $x$ and $y$. It means that the four kinds of optical susceptibilities fulfills the conditions of $\mathcal {PT}$-symmetry. It is also shown that Figs. 6(a,d) and 6(b,c) are mirror images of each other. Here we get four types of optical susceptibilities of 2D optical lattices through the imaginary part, however, all are $\mathcal {PT}$-symmetric. Experimentally, it is very easy to realize the $\mathcal {PT}$-symmetry by considering any combination of modulation amplitudes $\delta \Omega _{2x}$ and $\delta \Omega _{2y}$. From our results, one can see that the optical susceptibilities have such anti-symmetric properties with respect to the x- and y-direction. It can be used to the optical phase sensing for detecting the phase difference along the x- and y-directions.

 

Fig. 4. 3D plot of (a) real and (b) imaginary parts of the optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field detuning $\Delta =0.48\gamma$ and $\Omega _{\mu }=0.05\gamma$, the other parameters remains the same as that in Fig. 3.

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Fig. 5. Real part of optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field detuning $\Delta =0.48\gamma$, $\delta \Omega _{2x}=\pm 0.3\gamma , \delta \Omega _{2y}=\pm 0.3\gamma$, and $\Omega _{\mu }=0.05\gamma$, the other parameters remains the same as that in Fig. 4 .

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Fig. 6. Imaginary part of the optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field detuning $\Delta =0.48\gamma$, (a) $\delta \Omega _{2x}=0.3\gamma , \delta \Omega _{2y}=0.3\gamma$, (b) $\delta \Omega _{2x}=-0.3\gamma , \delta \Omega _{2y}=0.3\gamma$, (c) $\delta \Omega _{2x}=0.3\gamma , \delta \Omega _{2y}=-0.3\gamma$, (d) $\delta \Omega _{2x}=-0.3\gamma , \delta \Omega _{2y}=-0.3\gamma$ and $\Omega _{\mu }=0.05\gamma$, the other parameters remains the same as that in Fig. 4.

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Non-$\mathcal {PT}$-symmetric regime.

For the realization of $\mathcal {PT}$-symmetry, the simultaneous presence of gain and loss in a system plays a key role. In the following analysis, we discuss non-$\mathcal {PT}$-symmetry, where the gain and loss ratio do not balance each other or the real(imaginary) part is not an even(odd) function of space. Again, we study the optical susceptibility of 2D optical lattices by varying the microwave and probe field intensities. Initially, we plot the real and imaginary parts of optical susceptibility versus $x/a_1$ and $y/a_1$ by considering $\Omega _{\mu }=0.01\gamma$ instead of $\Omega _{\mu }=0.05\gamma$ while keeping all the other parameters unchanged, see Figs. 7(a, b). The real(imaginary) part of optical susceptibility is even(even) function of space and there is no gain in the system, therefore, the response of a medium does not fulfill the condition of $\mathcal {PT}$-symmetry and is known as non-$\mathcal {PT}$-symmetry. Further, our optical susceptibility dependent on probe field intensity ($\Omega _p$) and the medium can be switched from $\mathcal {PT}$-symmetry to non-$\mathcal {PT}$-symmetry by varying the intensity of the probe field. We show a contour plot of real and imaginary parts of optical susceptibility versus $x/a_1$ and $y/a_1$ by changing the probe field intensity from $\Omega _p=0.01\gamma$ to $\Omega _p=0.05\gamma$ while keeping all the other parameters unchanged of Fig. 4. The plots in Figs. 7(c, d) show that the imaginary part does not fulfill the condition of $\mathcal {PT}$-symmetry. Again the medium behaves like a non-$\mathcal {PT}$-symmetric regime. A similar switch can also be achieved by changing the relative phase of the fields. It is now obvious from the above investigation that the medium can be switched from $\mathcal {PT}$- to non-$\mathcal {PT}$-symmetric regime and vice versa via changing the intensity of microwave field ($\Omega _{\mu }$) and probe field ($\Omega _p$). It means that coherent control of $\mathcal {PT}$- to non-$\mathcal {PT}$-symmetry and vice versa is investigated via changing the intensities of microwave and probe fields.

 

Fig. 7. (a, b) Real and Imaginary parts of the optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field $\Omega _{p}=0.01\gamma$, (c, d) Real and Imaginary parts of the optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with microwave field $\Omega _{\mu }=0.01\gamma$, all the other parameters remain the same as that of Fig. 4.

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3.2 Coherent control of $\mathcal {PT}$-antisymmetry and non-$\mathcal {PT}$-symmetry

$\mathcal {PT}$-antisymmetry regime.

For $\mathcal {PT}$-antisymmetry condition the optical susceptibility $\chi (x,y)=-\chi ^*(-x,-y)$ in 2D medium, where the real part of the optical susceptibility must be an odd function of $x$ and $y$ while the imaginary part is an even function. To study the $\mathcal {PT}$-antisymmetry of 2D optical lattices, we must analyze the real and imaginary parts of optical susceptibility. Again we are looking for the value of probe field detuning at which the real part of the optical susceptibility has positive and negative values. We show the density plot of real part of the optical sucseptibility versus probe field detuning and $x$ or $y$-direction of the optical lattices by considering the relative phase $\phi =\pi$ as shown in Fig. 8. The plot shows that at around $\Delta =0.57\gamma$, the positive and negative values approximately the same at positions $x$ or $y\approx$ 0.13 and $x$ or $y\approx$ -0.13. We show the real and imaginary parts of optical susceptibility versus $x/a_1$ and $y/a_1$ as shown in Fig. 9. The plots show that the real part of optical susceptibility is an odd function of space while the imaginary part is an even function. This is a clear evidence for the realization of $\mathcal {PT}$-antisymmetry in terms of optical susceptibility of 2D optical lattices. It is also noted that the real part of optical susceptibility can be modified by considering different combination of modulation amplitudes i.e., ($\delta \Omega _{2x}=\delta \Omega _{2y}=0.3\gamma$), ($\delta \Omega _{2x}=\delta \Omega _{2y}=-0.3\gamma$), ($\delta \Omega _{2x}=-0.3\gamma ,\delta \Omega _{2y}=0.3\gamma$) and ($\delta \Omega _{2x}=0.3\gamma ,\delta \Omega _{2y}=-0.3\gamma$), see Fig. 10. We investigate four different real parts of optical susceptibility where each part is an odd function of $x$ and $y$. It is clear from Figs. 10(a,d) and 10(b,c) that these are mirror image of each other. We also show the imaginary part of optical susceptibility versus $x/a_1$ and $y/a_1$ for four possible combinations of modulation amplitudes. The four possible modulation amplitudes do not affect the imaginary part of optical susceptibility. Here, we plot the imaginary part of optical susceptibility for ($\delta \Omega _{2x}=\delta \Omega _{2y}=0.3\gamma$), see Fig. 11. For the remaining combination of modulation amplitudes the imaginary part remains the same as that of Fig. 11. The real and imaginary parts in Fig. 10 and 11 show four kinds of optical susceptibilities corresponding to $\mathcal {PT}$-antisymmetry. After the establishment of four kinds of optical susceptibilities for 2D optical lattices, it is now very easy to realize the $\mathcal {PT}$-antisymmetry experimentally by choosing any combination of modulation amplitudes $\delta \Omega _{2x}$ and $\delta \Omega _{2y}$.

 

Fig. 8. Density plot of real part of optical susceptibility vs probe field detuning $\Delta$ and lattice position $x/a_1$ or ($y/a_1$) by considering $\phi =\pi$. The other parameters remains the same as presented in the text.

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Fig. 9. Real and imaginary parts of the optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field detuning $\Delta =0.57\gamma$ and $\phi =\pi$, all the other parameters remain the same as that in Fig. 4.

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Fig. 10. Real part of the optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field detuning $\Delta =0.57\gamma$, (a) $\delta \Omega _{2x}=0.3\gamma , \delta \Omega _{2y}=0.3\gamma$, (b) $\delta \Omega _{2x}=-0.3\gamma , \delta \Omega _{2y}=0.3\gamma$, (c) $\delta \Omega _{2x}=0.3\gamma , \delta \Omega _{2y}=-0.3\gamma$, (d) $\delta \Omega _{2x}=-0.3\gamma , \delta \Omega _{2y}=-0.3\gamma$ and $\Omega _{\mu }=0.05\gamma$, the other parameters remains the same as that in Fig. 4.

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Fig. 11. Imaginary part of optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with probe field detuning $\Delta =0.57\gamma$, $\delta \Omega _{2x}=\pm 0.3\gamma , \delta \Omega _{2y}=\pm 0.3\gamma$, and $\Omega _{\mu }=0.05\gamma$, the other parameters remains the same as that in Fig. 4

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Non-$\mathcal {PT}$-symmetric regime.

The switch of $\mathcal {PT}$-antisymmetry to non-$\mathcal {PT}$-symmetry can be achieved via changing the relative phase $\phi$ of optical and microwave fields while keeping all the rest of parameters unchanged of Fig. 10(a). We plot the real and imaginary parts of optical susceptibility versus $x/a_1$ and $y/a_1$ by considering $\phi =\pi /2$ instead of $\pi$, see Fig. 12. The real and imaginary parts of optical susceptibility show that neither it fulfills the conditions of $\mathcal {PT}$-antisymmetry nor $\mathcal {PT}$-symmetry and the medium behaves like non- $\mathcal {PT}$-symmetric. In this section, we get the coherent control of $\mathcal {PT}$-antisymmetry and non-$\mathcal {PT}$-symmetry and vice versa via changing the relative phase of fields.

 

Fig. 12. Real and imaginary parts of optical susceptibility vs lattice position $x/a_1$ and $y/a_1$ with relative phase $\phi =\pi /2$, the other parameters remain the same as that in Figs. 10 and 11.

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4. Conclusion

In conclusion, we considered 2D optical lattices and thoroughly studied the realization of $\mathcal {PT}$- non-$\mathcal {PT}$- and $\mathcal {PT}$-antisymmetry. The 2D atomic lattices consist of cold rubidium atoms of three-level configuration trapped in a Gaussian distribution. We generated gain in the system by applying a microwave field in the three-level atomic configuration. The coherent control of the $\mathcal {PT}$- to non-$\mathcal {PT}$-symmetry is achieved via a microwave field. Similarly, the coherent control of $\mathcal {PT}$-antisymmetry to non-$\mathcal {PT}$-symmetry is achieved via a relative phase of optical and microwave fields. In some earlier investigations [23,25,27], in which various parameters are necessary for the investigation of $\mathcal {PT}$ symmetry. Our scheme offers a very simple of all-optical controllable technique to realize the coherent control of $\mathcal {PT}$- non-$\mathcal {PT}$- and $\mathcal {PT}$-antisymmetry. The microwave dependent optical susceptibility in three-level atomic configuration can offer a platform to study the field properties in $\mathcal {PT}$- non-$\mathcal {PT}$- and $\mathcal {PT}$-antisymmetry conditions.