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Abstract
We propose a theoretical scheme in a cold rubidium-87 (87Rb) atomic ensemble with a non-Hermitian optical structure, in which a lopsided optical diffraction grating can be realized just with the combination of single spatially periodic modulation and loop-phase. Parity-time (PT) symmetric and parity-time antisymmetric (APT) modulation can be switched by adjusting different relative phases of the applied beams. Both PT symmetry and PT antisymmetry in our system are robust to the amplitudes of coupling fields, which allows optical response to be modulated precisely without symmetry breaking. Our scheme shows some nontrivial optical properties, such as lopsided diffraction, single-order diffraction, asymmetric Dammam-like diffraction, etc. Our work will benefit the development of versatile non-Hermitian/asymmetric optical devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A system with non-Hermitian Hamiltonian being commutative with the parity-time operator ($[PT, H]=0$), proposed by Bander et al., has a real eigenenergy spectrum and some novel properties under certain conditions [1]. Due to the similarity between Schrodinger’s equation and the optical Helmholtz equation, the optical system with out-of-phase spatial modulation is a good platform to simulate the system with PT symmetry, which is named the non-Hermitian optical system [2]. Corresponding to the potential condition $V(x)=V^{*}(-x)$ satisfied in the PT symmetry system, the non-Hermitian optical system needs to satisfy complex refractive index condition $n(x)=n^{*}(-x)$ [3–6]. Similarly, optical PT antisymmetry is realized in optical media with instead condition $n(x)=-n^{*}(-x)$ [7–11]. In recent years, non-Hermitian optical structures based on discrete systems such as optical waveguide [12,13], hybrid optical micro-cavity [14,15], electrical circuit resonators [16] as well as continuous optical media such as cold atomic ensemble with spatially periodic modulation [5–10], have been implemented successively both in experimental and theoretical works. Moreover, a series of new applications or properties have been reported as optical Bloch oscillation [17,18], photon or phonon laser [19–21], unidirectional/bidirectional invisibility/optical cloaks [22–28], and so on.
The spectroscopic device, such as diffraction grating, has been a significant branch of optical devices since Newton’s era. As an important tool for spectral analysis and optical imaging, it has been playing an important role in many fields such as physics, chemistry, astronomy, biology, etc., [29–32]. Electromagnetically induced grating (EIG) [33–35] based on electromagnetically induced transparency (EIT) [36], makes it possible to tune the diffraction patterns dynamically. Hybrid modulation EIG schemes with traditional amplitude/phase modulations and untraditional ones (nonlinear or nonlocal modulation) have improved diffraction efficiency while greatly expanding the versatility of the grating [37–42]. In recent years, combined with the non-Hermitian optical modulation, many schemes of one-/two-dimensional asymmetric optical diffraction gratings have been proposed serially [43–50], and even similar applications of asymmetric scattering have appeared in the field of acoustics [51].
However, due to rigid realization conditions of PT/APT symmetry, there is still a great hindrance to the realization of precise and flexible dynamic operation, especially for some special optical diffractions. In most previous schemes, dual spatial periodic modulation (via amplitude, detuning of coupling field or atomic density) has been used to achieve two goals, including the realization of PT/APT symmetry and construction of a grating structure. It results in the lack of accurate modulation capabilities with the protection of PT/APT symmetry. Therefore, a method for the preparation of non-Hermitian EIG with simple structure (easy to analyze), dynamic control ability, and protection of optical non-Hermitian symmetry is extremely necessary and desired.
In this paper, we propose a theoretical scheme with a four-level loop-$\mathcal {N}$ configuration in a cold $^{87}$Rb atomic ensemble, in which an EIG with PT-symmetric/-antisymmetric structure can be realized via a single spatially periodic detuning modulation. PT-symmetric and PT-antisymmetric structures can be easily switched by adjusting different relative phases of the applied beams. In addition, the non-Hermitian optical symmetry in our system is robust to variation of coupling amplitude, giving our scheme the ability of dynamic modulation with symmetry protection. Considering the contribution of higher-order scattering, which violates Friedel’s law [52–54], we also discuss the realization condition of asymmetric diffraction. The perfect lopsided diffraction attained here is explained as the cooperative result of higher-order scattering and spatial Kramers-Kronig relation [55–58]. Moreover, we can also achieve single-order and Dammann-like asymmetric diffraction with our scheme.
This work is organized through the following Sec. 2, where we describe the background model, and Sec. 3, where we discuss the robustness of optical non-Hermitian symmetry, far-field Fraunhofer lopsided diffraction arising from the combination of higher-order scattering, and spatial Kramers-Kronig relation also with discussions on special asymmetric diffractions based on our scheme. We summarize, at last, our conclusions in Sec. 4.
2. Model and equations
We start by considering an ensemble of cold $^{87}$Rb atoms driven into an equivalent four-level loop-$\mathcal {N}$ configuration, by four coherent fields with frequencies (amplitudes) $\omega _{p}$ ($\mathcal {E}_{p}$), $\omega _{c}$ ($\mathcal {E}_{c}$), $\omega _{d}$ ($\mathcal {E}_{d}$) and $\omega _{m}$ ($\mathcal {E}_{m}$) as shown in Fig. 1(a). The weak probe field $\omega _{p}$ interacts with transition $|g\rangle \leftrightarrow |e\rangle$, while the strong control fields $\omega _{c}$ and $\omega _{d}$ act upon transitions $|m\rangle \leftrightarrow |e\rangle$ and $|e\rangle \leftrightarrow |b\rangle$, respectively. The dipole-forbidden transition between states $|g\rangle$ and $|m\rangle$ can be interacted by an equivalent microwave field (Raman-type process) with $\omega _{m}$. The half Rabi frequencies are defined as $\Omega _{p} =\mathcal {E}_{p} \cdot \wp _{ge}/2\hbar$, $\Omega _{c} = \mathcal {E}_{c}\cdot \wp _{me}/2\hbar$, and $\Omega _{d} = \mathcal {E}_{d}\cdot \wp _{mb}/2\hbar$ with $\omega _{ij}$ being transition frequencies and $\wp _{ij}$ being dipole moments. In the rotating-wave and electric-dipole approximations, the Hamiltonian for our loop-$\mathcal {N}$-type cold atoms can be written down as $\hat {H}=\hat {H}_{a}+\hat {V}_{af}$ containing an unperturbed atomic Hamiltonian $\hat {H}_{a}$ and an atom-field interaction Hamiltonian $\hat {V}_{af}$:
Fig. 1. An equivalent four-level loop-$\mathcal {N}$ configuration (a) for cold $^{87}$Rb atomic Quasi one-dimensional thin grating driven by a probe field ($\Omega _p$), two coupling fields ($\Omega _c$ and $\Omega _d$) and a microwave field ($\Omega _m$) in $z$ direction. A far-detuned dressing field $(\omega =\omega _{ab}\pm \Delta$ and $\Delta \gg \Omega$) act on transition $|b\rangle \leftrightarrow |a\rangle$ to give state $|b\rangle$ a spatial modulation of detuning $\delta _d(x)=\delta \cdot \sin [C\cdot \pi x]$ with $\delta =\frac {\Omega ^2}{\Delta }$, following the method in Ref. [8] (b$_1$). The dressing field includes SW components ($\frac {\Omega }{\sqrt {2}}$ and $\omega _{ab}-\omega =\Delta$) along $x$ direction (with small angle $\pm \theta _{ab}$) and TW component ($\Omega$ and $\omega _{ab}-\omega =-\Delta$) along $z$ direction (b$_2$). The corresponding initial phases $\phi _{\mu }$ of fields $\Omega _{\mu }$ ($\mu \in \{p,c,m\}$) constitute the loop-phase $\Phi =\phi _m+\phi _c-\phi _p$. The system satisfies PT symmetry (c$_1$) and PT antisymmetry (c$_2$), with $\Phi =n\pi$ and $\Phi =\pi /2,3\pi /2$, respectively, so the non-Hermitian symmetry can be switched by controlling the loop-phase $\Phi$.
Setting the time derivatives as zero, the steady-state solution can be attained by solving Eq. (2) analytically:
Here $\textbf {Re}[\chi _{p}]$ and $\textbf {Im}[\chi _{p}]$ are used to represent the real and imaginary parts of $\chi _p$, describing the dispersion and absorption/gain ($\textbf {Im}[\chi _{p}]>0$/$\textbf {Im}[\chi _{p}]<0$) properties, respectively. Aiming to implement a $nontrivial$ EIG with asymmetric diffraction patterns, $\delta _{d}$ needs to be periodically modulated as:
with AC Stark shift $\delta =\frac {\Omega ^2}{\Delta }$ and period of standing wave field $a=\frac {\lambda _{ab}}{\sin \theta _{ab}}$ [See Fig. 1(b$_1$-b$_2$)]. For a medium of thickness $\mathcal {L}$ along the $z$ direction and modulated in the $x$ direction, the transmission function of a probe beam takes the form where $T_{a}(x)=e^{-k_p\int _0^{\mathcal {L}}\textbf {Im}[\chi _{p}(x,z)]dz}$ ($T_{p}(x)=e^{ik_p\int _0^{\mathcal {L}}\textbf {Re}[\chi _{p}(x,z)]dz}$) denotes the amplitude (phase) component with $k_p=2\pi /\lambda _{p}$ being the probe wave vector and $\lambda _{p}$ being the probe wavelength. The Fourier transformation of $T_{\mathcal {L}}(x)$ then yields the Fraunhofer or far-field intensity diffraction equation3. Results and discussion
In this section, to implement analytical/numerical calculations based on the above equations, the four atomic levels in Fig. 1(a) are assumed as $|g\rangle \equiv |5S_{1/2}, F=2\rangle$ and $|m\rangle \equiv |5S_{1/2}, F=1\rangle$, as well as $|e\rangle \equiv |5P_{1/2}, F=1\rangle$ and $|b\rangle \equiv |5P_{3/2}, F=0\rangle$ for cold $^{87}$Rb atoms. Then the decay rates are $\Gamma _{eg}=\Gamma _{em}=\Gamma _{bm}\simeq 3.0\times 2{\pi }$ MHz, $\gamma _{eg}=\gamma _{em}\simeq 3.0\times 2{\pi }$ MHz, $\gamma _{mb}=\gamma _{gb}=\Gamma _{bm}/2$, and $\gamma _{mg}\simeq 1.0$ kHz. In addition, the atomic medium length is $\mathcal {L}=6.0$ $\mu$m and the density is $N=4.0\times 10^{10}$cm$^{-3}$.
3.1 PT-symmetry/-antisymmetry based on single periodic modulation
In the beginning, we apply two control fields ($\Omega _{c}/2\pi =5.0$ MHz, $\Omega _{d}/2\pi =3.5$ MHz) and an equivalent microwave field ($\Omega _m=0.5\times 2\pi$ MHz), which are all travelling waves (TW). Figure 2 shows the absorption ($\textbf {Im}[\chi _p(\omega )]$) and dispersion ($\textbf {Re}[\chi _p(\omega )]$) spectra of the system, with detunings $\Delta _c=\delta _d=0$ and different loop phases ($\Phi =0,\pi /2,\pi$ and $3\pi /2$). Evidently, $\textbf {Im}[\chi _p(\omega )]$ ($\textbf {Re}[\chi _p(\omega )]$) is an odd (even) function of frequency $\omega$ ($\delta _p$) with loop phase $\Phi =n\pi$ as shown in Fig. 2(a) and (c). While the parity of $\textbf {Im}[\chi _p(\omega )]$ and $\textbf {Re}[\chi _p(\omega )]$ are opposite if the loop phase is chosen as $\Phi =(2n-1)\pi /2$ in Fig. 2(b) and (d), with $n\in \mathbb {Z}$.
Fig. 2. Dispersion $\textbf {Re}[\chi _{p}(\omega )]$ (blue-solid curves) and absorption $\textbf {Im}[\chi _{p}(\omega )]$ (orange-dotted curves) as functions of probe detuning $\delta _p$ with $\Delta _{c}=\delta _d=0$ for (a) $\Phi =0$, (b) $\Phi ={\pi }$, (c) $\Phi ={\pi }/2$, (d) $\Phi =3{\pi }/2$, respectively. Other parameters are chosen as $\Omega _{p}/2\pi =0.01$ MHz, $\Omega _{c}/2\pi =5.0$ MHz, $\Omega _{d}/2\pi =3.5$ MHz and $\Omega _{m}/2\pi =0.5$ MHz.
Figure 3 shows absorption and dispersion spectra via $x$ coordinate when we apply spatial modulation to detuning as in Eq. (5)($\delta =8.0\times 2\pi$ MHz), with different loop phases $\Phi$. Under the three-photon resonance condition ($\delta _p+\Delta _c+\delta _d=0$), it can be attained that $\delta _p(x)=-\delta _d(x)=\delta \cdot \sin [\frac {4\pi (x-x_0)}{a}]$ with $\Delta _c=0$. Then it is easy to get $\chi _p(\omega )=\chi _p[\delta _p(x)]\to \chi _p(x)$, indicating the parity characteristic transfer from the frequency domain to spatial domain [See Appendix A]. It is obvious that $\textbf {Im}[\chi _{p}(x)]$ and $\textbf {Re}[\chi _{p}(x)]$ are odd (even) and even (odd) functions of $x$ in Fig. 3(a$_1$)-(d$_1$) with $\Phi =0,\pi$ ($\Phi =\pi /2,3\pi /2$), respectively, so the system satisfies optical PT-symmetry (PT-antisymmetry).
Fig. 3. Absorption $\textbf {Im}[\chi _{p}(x)])$ (a$_1$-d$_1$) and dispersion $\textbf {Re}[\chi _{p}(x)]$ (a$_2$-d$_2$) versus $x$ and the amplitude of coupling field $\Omega _c$ for different loop phases $\Phi =0, \pi /2, \pi, 3\pi /2$, respectively, and other parameters are same as in Fig. 2 except $\delta _{d}=\delta \sin [{4\pi }(x-x_{0})/a]$ and $\delta /2\pi =8.0$ MHz.
Switching between PT-symmetric modulation and PT-antisymmetric modulation by adjusting the loop phase is convenient in our scheme [See Fig. 3, Eq. (16), and Table 1], which can be not realized in previous work. With the loop phase being chosen as $\Phi =2m\pi$ or $\Phi =(2m+1)\pi$ ($m\in \mathbb {Z}$), the system is under PT-symmetric modulation. The real (imaginary) parts $\textbf {Re}[\chi _{p}(x)]$ ($\textbf {Im}[\chi _{p}(x)]$) of the susceptibilities in these two cases with different phases just have the opposite signs. It is worth noting that except loss case (normal APT) with $\Phi =\frac {(4m+1)\pi }{2}$, PT antisymmetric cases also include a gain case (abnormal APT) with $\Phi =\frac {(4m-1)\pi }{2}$, which is uncommon in the continuous medium schemes. The gain APT structures will provide an alternative scheme with high flexibility for some potential applications which require both considerable gain and out-of-phase modulation.
Table 1. Non-Hermitian spatial modulation
The parity of the system polarization under the loop-phase control makes it possible to achieve both optical PT-symmetry and PT-antisymmetry based on single spatial periodic modulation solely, which is quite different from the previous schemes [5–10,43–50]. Especially for PT-symmetry in these continuous periodic systems, the dual spatially periodic modulation, generally provided by multiple SW fields or atomic lattice systems (spatial density modulation), is necessary. Notably, single spatially periodic modulation dramatically reduces the system’s complexity and makes it possible to analytically analyze the relationship between non-Hermitian optical symmetry (PT-symmetry/-antisymmetry) and asymmetric diffraction in a system with continuously varying complex refractive index.
Robustness of non-Hermitian optical symmetry is necessary to be considered for a system that requires precise control, as the symmetry needs to be satisfied strictly to implement some special optical functions (such as perfect lopsided diffraction) [See Sec. 3.2]. In previous works, the non-Hermitian symmetries are very fragile to even a slight perturbation of parameters such as coupling amplitudes. Here we consider the robustness of our system with varying coupling intensity $\Omega _c$ in Fig. 3. Obviously, the system has a large dynamic tuning range here ($\Omega _c$ $\in$ $[2.2,10.0]\times 2\pi$ MHz), with the protection of non-Hermitian optical symmetry. Similarly, non-Hermitian symmetry robustness to other varying coupling amplitudes ($\Omega _d$ and $\Omega _m$) are shown in Fig. 4. The above conclusion can also be analytically supported by Eq. (15). Compared to the optical depth (OD) modulation (via initial preparation of atomic density $N$ and medium length $\mathcal {L}$) in the previous work [44,49], the amplitude modulation in our scheme is more feasible with a large dynamic modulation range under non-Hermitian optical symmetry protecting, meaning the possibility to achieve some special optical functions.
Fig. 4. Dispersion $\textbf {Re}[\chi _{p}(x)]$ (blue curves) and absorption $\textbf {Im}[\chi _{p}(x)]$ (orange curves) spectra versus $x$ for (a$_1$) $\Phi =0$, (b$_1$) $\Phi ={\pi }$, (c$_1$) $\Phi =3{\pi }/2$, respectively, with the same parameters as in Fig. 2 except $\Omega _d/2\pi =$ 3.0 MHz (solid curves), 3.5 MHz (dashed curves), 4.0 MHz (dashed and dotted curves), and 5.0 MHz (dotted curves). Moreover, panels (a$_2$)-(c$_2$) are the corresponding spectra with the same parameters as in Fig. 2 except $\Omega _m/2\pi =$ 0.3 MHz (solid curves), 0.5 MHz (dashed curves), 0.7 MHz (dashed and dotted curves), and 0.9 MHz (dotted curves).
3.2 Lopsided, single-order, and Dammann-like asymmetric diffractions
After constructing the tunable optical PT/APT symmetry of the system with the loop phase, we try to analyze the diffraction characteristics of our system in this subsection. The optical diffraction characteristics of this non-Hermitian EIG can be studied by injecting a probe field along the $z$-axis which is perpendicular to the direction of spatial periodic modulation ($x$-axis). Figures 5(a$_1$-d$_1$) give dispersion and absorption (or gain) spectra under the out-of-phase (PT-symmetric/-antisymmetric) modulations for different loop-phases, with same parameters as in Fig. 2 except $\delta /2\pi =8.0$ MHz. Figures 5(a$_2$-d$_2$) show that the probe beam is only diffracted into the negative angles in the range $\theta \in (-\pi /6,0)$ with loop phase $\Phi =0,\pi /2,\pi,3\pi /2$, respectively. Figure 5(b$_2$) displays normal APT case, that is, the system is pure dissipative. Instead, Fig. 5(d$_2$) exhibits an unconventional optical PT-antisymmetric case with optical gain, which is also quite different from the PT-symmetric case [See Fig. 5(a$_2$) and Fig. 5(c$_2$)] with the balance of gain and loss.
Fig. 5. Dispersion $\textbf {Re}[\chi _{p}(x)]$ (blue-solid curves) and absorption $\textbf {Im}[\chi _{p}(x)]$ (orange-dotted curves) spectra of gratings with lopsided diffraction patterns versus $x$ with $\delta _{d}=\delta \sin [{4\pi }(x-x_{0})/a]$ for (a$_1$) $\Phi =0$, (b$_1$) $\Phi ={\pi }$, (c$_1$) $\Phi ={\pi }/2$, (d$_1$) $\Phi =3{\pi }/2$, respectively. Moreover, (a$_2$)-(d$_2$) are the corresponding grating diffraction intensity angle spectrum with the same parameters as in Fig. 2 except $\delta /2\pi =8.0$ MHz, $R=6$ and $M=10$.
It is known that diffraction peaks occur at discrete angles $\theta _n$ determined by $k_n=k_p\sin \theta _n=2n\pi /a$ or $n=R\sin \theta _n$, where we choose $R = a/\lambda _p=6$ and $M = \varpi _B/a=10$. Combined with Eq. (6), we could focus on the $n$th-order diffraction by examining $\mathcal {E}_n=\mathcal {E}_p(\theta _n)$ for $n\neq 0$. For simplicity, with $\alpha (x)=k_p\textbf {Im}[\chi _p(x)]\mathcal {L}$, $\beta (x)=k_p\textbf {Re}[\chi _p(x)]\mathcal {L}$, and $\gamma _n(x)=2n\pi x$, we can make a power series expansion of Eq. (8) and obtain
Considering $\gamma _n(-x)=-\gamma _n(x)$, it is not difficult to find $\eta _n\equiv 0$ ($I_{n}=I_{-n}$) with $f_n^{\prime }\cdot g_n^{\prime }=f_n^{\prime \prime }\cdot g_n^{\prime \prime }$, indicating the symmetric diffraction occurs in the system, in the case of spatial even symmetry with $\textbf {Im}[\chi _p(x)]=\textbf {Im}[\chi _p(-x)]$ and $\textbf {Re}[\chi _p(x)]=\textbf {Re}[\chi _p(-x)]$ (in-phase modulation). On the contrary, if the system is deviated from in-phase modulation, with $f_n^{\prime }\cdot g_n^{\prime }\neq f_n^{\prime \prime }\cdot g_n^{\prime \prime }$ ($\eta _n\neq 0$), we can obtain the asymmetric diffraction angle spectra. Moreover, if optical systems satisfy PT-symmetry or PT-antisymmetry (out-of-phase) there will be a large possibility to achieve $\eta _n\to 1$, named the perfect asymmetric diffraction or lopsided diffraction [See Appendix B].
Figure 6(a) shows the asymmetric diffraction angular spectra $I_p(\theta )$ with varying coupling amplitude $\Omega _c$ of our system for the PT-symmetric case ($\Phi =0$). Panels (b$_1$) and (b$_2$) show the $\pm n$th order diffraction intensity $I_p(\pm \theta _{n})$ ($n=\{1,2,3\}$) varying with the coupling amplitude ($\Omega _c/2\pi \in (0,5)$ MHz). Moreover, we show the intensity contrast ratio (asymmetry diffraction coefficient) $\eta _n$ versus $\Omega _c$ in Fig. 6(c). Comparing the three panels of Fig. 6, it is easy to find that, in the range of $\Omega _c/2\pi \in (2.0,3.5)$ MHz, a high degree of asymmetric diffraction is obtained, with one-sided non-zero diffraction intensity. Particularly, we can get $\eta _n>0.93$ and $I_p(\theta _1)>0.3$, which is extremely close to the optimal situation for PT-symmetric case, at the red line position ($\Omega _c/2\pi =2.5$ MHz).
Fig. 6. Diffraction intensity $I_p(\theta )$ versus the diffraction angle and the amplitude of coupling field $\Omega _c$ (a), negative order diffraction intensity $I_p(-\theta _n)$ (b$_1$), positive order diffraction intensity $I_p(\theta _n)$ (b$_2$) and asymmetric diffraction coefficient $\eta _i$ versus $\Omega _c$ (c), with the same parameters as in Fig. 5 except $\Phi =0$. The position of the red line corresponds to $\Omega _c/2\pi =2.5$ MHz.
Next, we further discuss the asymmetric diffraction of the system with different loop phases. The curves of asymmetry diffraction coefficients versus $\Omega _c$ for different diffraction orders $\eta _n$ ($n=\{1,2,3\}$) are shown in Fig. 7 with different loop phases $\Phi$. Comparing the PT symmetric modulation cases in Fig. 7(a) and Fig. 7(c), we can obtain the asymmetry diffraction coefficients $\eta _{1,2}\to 1$ ($0.80<\eta _{3}<0.93$) in the coupling amplitude range $2.0<\Omega _c/2\pi <3.5$ MHz, and values of $\eta _{1,2}$ reach the maximum at $\Omega _c/2\pi =2.5$ MHz, so these cases can be named as near lopsided diffractions in Table 1.
Fig. 7. Asymmetric diffraction coefficient $\eta _n$ versus $\Omega _c$, $n\in \{1,2,3\}$, with the same parameters as in Fig. 5 except loop-phase $\Phi =$0, $\pi /2$, $\pi$ and $3\pi /2$ in panel (a), (b), (c) and (d), respectively.
Figure 7(b) shows asymmetry diffraction coefficients $\eta _n$ in normal PT-antisymmetric case with $\Phi =\pi /2$. Obviously, the perfect lopsided diffraction effect has been achieved, that is, $\eta _{n}\equiv 1$ ($n=\{1,2,3\}$) if the coupling amplitude is chosen as $\Omega _c/2\pi >4.5$ MHz. Employing the same parameters as in Fig. 5(b), for the pure-loss optical system, we can easily obtain
Compared to previous schemes [44,48], lopsided diffraction properties of the grating here can be modulated by coupling fields, being a dynamic modulation, instead of OD depending on the initial condition. With the tunable ability of lopsided diffraction in our scheme, we could discuss some special diffractions further. Single-order diffraction is displayed in Fig. 8(a$_1$) and (b$_1$) for the optical APT case ($\Phi =\pi /2$), with showing $\chi _p$ and diffraction angle spectrum of the one-dimensional case (1D), respectively. Here the parameters are chosen as $\Omega _c=1.056\gamma _0,\Omega _c=3.0\gamma _0$, and $\Omega _c=0.7\gamma _0$ ($\gamma _0=1.0\times 2\pi$ MHz). The probe beam is only diffracted into the $-1$st order with a considerable diffraction efficiency (intensity) $I_{-1}\simeq$ 0.55. In addition, the above conclusion can be extended to the two-dimensional case (2D). We plot the single-order asymmetric diffraction for 2D case in Fig. 8(b$_1$) with spatial periodic modulation in both directions ($\delta _d=\delta _{x}\sin [4\pi (x-x_0)/a]+\delta _{y}\sin [4\pi (y-y_0)/a]$). Here, only a simple case ($\delta _{x}/2\pi =4.0$ MHz and $\delta _{y}/2\pi =0.1$ MHz) is given to illustrate the functionality of our scheme. In fact, in combination with the two-dimensional Hermitian/non-Hermitian hybrid modulation [49], there will be other diffraction modes of single-order asymmetric diffraction.
Fig. 8. Single-order (a$_1$-b$_1$) and Dammann-like asymmetric (a$_2$-b$_2$) diffraction. Dispersion $\textbf {Re}[\chi _{p}(x)]$ (blue-solid curves)/absorption $\textbf {Im}[\chi _{p}(x)]$ (orange-dotted curves) spectra versus $x$ in (a$_1$-a$_2$) and diffraction intensity $I_p(\theta )$ versus $\sin (\theta )$ in (b$_1$-b$_2$) for 1D case, with $\Omega _c/2\pi =1.056$ MHz$,\Omega _c/2\pi =3.0$ MHz$,\Omega _c/2\pi =0.7$ MHz and $\delta /2\pi =6.0$ MHz for (a$_1$-b$_1$); with the same parameters as in Fig. 5 except $\Omega _c/2\pi =1.73$ MHz and $\delta /2\pi =4.0$ MHz for (a$_2$-b$_2$). Diffraction intensity $I_p(\theta _x,\theta _y)$ for 2D case in (b$_{1}^{\text {in}}$) with $\delta _d=\delta _{x}\sin [4\pi (x-x_0)/a]+\delta _{y}\cos [4\pi (y-y_0)/a]$, $\delta _{x}/2\pi =4.0$ MHz and $\delta _{y}/2\pi =0.1$ MHz.
Identically, through continuous parameter modulation for several coupling fields ($\Omega _{c,d,m}$, $\delta _{d}$, $\delta _{p,d}$), the special asymmetric diffraction with multiple equal intensity diffraction orders can be easily accomplished. As the simplest example, the case with two equal-intensity diffraction orders is shown in Fig. 8(a$_2$) and (b$_2$). Here we first check the susceptibilities and diffraction angle spectra in the 1D case to ensure the satisfaction of non-Hermitian optical symmetry and lopsided diffraction conditions. In the same way, following the 2D non-Hermitian hybrid modulation methods [49], an array of diffraction beams with equal intensity will be implemented, named asymmetric Dammann-like diffraction grating, the analogue of the Dammann gratings [61–65] in the asymmetric case. These special asymmetric diffractions and the implementation methods will greatly facilitate the development and application of scattering-type asymmetric optical devices.
4. Conclusion
In summary, the ensemble of cold $^{87}$Rb atoms driven into a four-level loop-$\mathcal {N}$ configuration can provide an interesting venue to realize non-Hermitian EIG with various symmetry features. Firstly, the scheme is proposed to prepare continuous out-of-phase type non-Hermitian optical media (PT-symmetric/-antisymmetric) based on loop-phase ( $\Phi =\phi _m+\phi _c-\phi _p$) and single spatially periodic modulation. PT symmetry and PT antisymmetry can be easily switched by the loop phases. Then robustness of the non-Hermitian optical symmetry of this scheme to the coupling amplitudes is discussed. It shows that our system has a flexible multi-parameter modulation ability with a large dynamic adjusting range while protecting the non-Hermitian optical symmetry, which can not be realized in the previous schemes. In addition, we examine the far-field Fraunhofer diffraction of the atomic grating, including PT symmetry cases as well as loss PT-antisymmetric case (normal APT) and gain PT-antisymmetric (abnormal APT) case. Moreover, we analyze the reason for asymmetric diffraction and discuss the realization condition of the perfect lopsided diffraction, interpreted as the joint contribution of higher-order scattering and the spatial Kramers-Kronig relationship of $\textbf {Re}[\chi _p]$ and $\textbf {Im}[\chi _p]$. Finally, special diffractions, single-order/Dammann-like asymmetric diffraction, are shown to demonstrate the superiority of our scheme.
The introduction of loop-phase modulation reduces the complexity of achieving non-Hermitian optical symmetry in a continuous medium and diversifies the methods of spatial modulation, which will promote and benefit the development of signal-selecting non-Hermitian/asymmetric optical devices.
Appendix A: spatial parity of $\textbf {Re}[\chi _p]$ and $\textbf {Im}[\chi _p]$
The system satisfies $\chi _p(x)=\frac {\mu \mathcal {E}_p}{\hbar \varepsilon _0}\cdot \frac {\varrho _{eg}}{\Omega _p}$, where $\varrho _{eg}$ includes first-order term $\varrho _{eg}^{(1)}=\frac {i\Omega _{p}}{R_{eg}+\frac {\Omega _{c}^{2}}{R_{mg}+\frac {\Omega _{d}^{2}}{R_{bg}}}}$ and the zero-order $\varrho _{eg}^{(0)}=-\alpha _{p} \cdot \frac {e^{i\phi }\Omega _m\Omega _c}{R_{eg}+\frac {\Omega _{c}^{2}}{R_{mg}+\frac {\Omega _{d}^{2}}{R_{bg}}}}$, under the condition $\Omega _p,\Omega _m\ll \Omega _c,\Omega _d$. With dephasing $\gamma _{eg}=\gamma _{em}=\gamma _{dm}=\gamma$, $\gamma _{mg}= \gamma _{m}$ ($\gamma _m\ll \gamma$) and the three-photon resonance condition $\delta _p-\Delta _c+\delta _d(x)=0$, it is obvious that $\delta _p(x)=-\delta _d(x)$ and $\varrho _{eg}(x)$ are the function of $x$ if we choose $\Delta _c=0$. Rewriting the steady-state solutions of Eq. (3) in terms of real and imaginary parts, we can get $\varrho _{eg}^{(0)}(x)=e^{i\Phi }\{\textbf {A}_{0}[\delta _p(x)]+i\textbf {B}_{0}[\delta _p(x)]\}$ and $\varrho _{eg}^{(1)}(x)=\textbf {A}_{1}[\delta _p(x)]+i\textbf {B}_{1}[\delta _p(x)]$. We can obtain that
Moreover, based on Eq. (3) we can easily attain the relationship between $\textbf {Re}[\chi _p]$, $\textbf {Im}[\chi _p]$ and space coordinates $x$:
Here we define $DD=\gamma ^2(\Omega _c^2+\Omega _d^2-\delta _p^2(x))^2+(\gamma ^2+\Omega _d^2)^2\delta _p^2(x)$ and $C_{1\mp }=(\Omega _c^2+\Omega _d^2)(\Omega _p\Omega _d^2\gamma \mp \Omega _m\Omega _c\gamma ^2)$. In the calculation, the constant terms containing $\Omega _p$ can be ignored as $\Omega _p\ll \Omega _m$. The denominator $DD$ is an even polynomial of $\delta _p$ including a nonzero constant term, implying the parity of the above equations is determined by those of the numerators alone.
The real and imaginary parts of $\varrho _{eg}^{(0)}$ introduced by the loop structure finally determine the parity of polarization of the system. Hence the non-Hermitian optical (PT or APT) symmetry can be easily modulated by loop-phase $\Phi$. Significantly, from Eq. (16), we can find the varying coupling amplitudes $\Omega _{\xi _c}$ ($\xi _c=\{c,d,m\}$) have no impact on the parity of polarization of our system under the reasonable condition. Because parity conditions above will be violated if the constant terms or population of the state $|m\rangle$ cannot be neglected, it should be noted that these modulations only hold within specific ranges.
Appendix B: asymmetric diffraction
In this part, we will try to calculate $\eta _n$ according to the spatial symmetry of the system. Firstly, for traditional EIGs, $\textbf {Re}[\chi _p]$ and $\textbf {Im}[\chi _p]$ are even functions of $x$ (in-phase spatial modulation), $\textbf {Im}[\chi _p(x)]=\textbf {Im}[\chi _p(-x)]$ and $\textbf {Re}[\chi _p(x)]=\textbf {Re}[\chi _p(-x)]$. We can obtain $\alpha (x)=\alpha (-x)$ and $\beta (x)=\beta (-x)$, where $\alpha (x)=k_p\textbf {Im}[\chi _p(x)]\mathcal {L}$ and $\beta (x)=k_p\textbf {Re}[\chi _p(x)]\mathcal {L}$. Combined with the above relationship and Eq. (10), the scattering coefficients read:
Apparently, we can get the result $\eta _n=0$ for each $n$th diffraction order, meaning a symmetric diffraction behaviour of the traditional EIGs. Similarly, with out-of-phase spatial modulations, we can get
Thus, we know that when the system satisfies non-Hermitian symmetry, $\eta _n^{\mathcal {PT}}$ ($\eta _n^{\mathcal {APT}}$) is unequal to 0, which causes asymmetric diffraction behaviour of the atomic grating. Moreover, the contribution of high-order scattering can be influenced by adjusting parameters such as coupling amplitudes. It gives a method to achieve adjusting the degree of asymmetry accurately.
Funding
National Natural Science Foundation of China (12104107); Natural Science Foundation of Jilin Province (20220101009JC); Fundamental Research Funds for the Central Universities (2412022ZD046); ``Yucai Project” of Guangxi Normal University.
Acknowledgments
The authors would like to thank the anonymous referee for valuable comments that have much improved the manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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