1. Introduction

Photonic band gap (PBG) is a phenomenon where certain wavelengths of light are prohibited from propagating through a medium due to interference effects, resulting in a band of frequencies where light cannot pass. Researchers are increasingly excited by the wide range of applications offered by PBGs, causing their thorough studies across a wide spectrum of frequencies, ranging from lower to higher frequencies. Substantial advancements have been achieved in attaining lower frequency PBGs [1–6], and more recently, efforts have been directed towards higher frequency PBGs, specifically in the extreme ultraviolet [7–10] or X-ray domain [11,12].

Electromagnetically induced transparency (EIT)-based PBGs have significant work in the lower frequency range [1–4]. Where $\Lambda$-type rubidium ($\text {}^{87}\text {Rb}$) and cesium ($\text {Cs}$) three-level atomic systems are frequently used. Due to the degenerate wavelength of driving and probing light fields, only the first-order band gaps are formed at the limit of the lower frequency range. On the other hand, work has been done on high-frequency PBGs [10–12]. Where it is observed, high-frequency PBGs can be constructed when the frequency of the driving field is $n$th-order lower than the probe field. It is noticed that ultraviolet and soft X-ray PBGs can be formed via $\text {}^{85}\text {Rb}$ and $\text {Be}^{2+}$ ions by $37^{\text {th}}$ and $61^{\text {st}}$ order lower frequency control, respectively.

The PBGs produced by the high-frequency light reflector via nth-order lower-frequency light control have been used for the first time to construct a high-frequency reflector (mirror) within the limits of ultraviolet and soft X-ray [10]. In this case, a superradiance lattice (SL) has been utilized. However, challenges still existed in achieving maximum light reflection and maintaining a consistent range of PBGs relative to probe detuning. Further work on high-frequency PBGs using modified superradiance lattice (MSL)has been done, which are able to give maximum reflection of light with a constant range with respect to probe detuning [12]. Besides this, SL-based high-frequency PBGs have also been formed in the presence of quantum phase fluctuations [11]. Where quantum fluctuations reduce the reflectivity of PBGs but bring the scheme closer to reality. Moreover, SL has been experimentally realized in ultracold atoms [13].

In this paper, we use the MSL [14] with a loop configuration of coupling fields, where relative phase can be used to control the PBGs. Generally, it is observed that relative phase creates absorption, amplification, and zero absorption at resonance or near resonance in the absorption spectrum of the probe field [15]. So, this feature can provide more control over PBGs, which will be more futuristic. Furthermore, we would like to point out that both lower- and higher-frequency PBGs can be effectively obtained by using this atomic scheme. The idea of MSL comes from SL. However, the concept of SL is based on a standing wave (SW)-coupled EIT system [16]. Here, we would like to mention that EIT is a widely observed optical phenomenon applied across various domains of quantum optics [17–24]. In the case of MSL, we take modified standing wave (MSW) instead of SW. The MSW has already been used in certain quantum optical phenomena. MSW are used to obtained fully PBG with negligible absorption everywhere [1]. Double PBG can also be obtained by using two modes of MSW [2], where both PBGs are completely tuned by detuning and relative phase between two modes of MSW. Similarly, triple PBGs are attained by single modes of MSW [3]. Further, perfect reflectivity is attained for ultracold $^{87}$Rb atoms with little loss and deformation [11] whose frequency components are lies within the PBGs. MSW have also been used to create MSL, which is used for efficient quantum memory [14].

2. Model and equation

We adopt a similar methodology to that described in [16], wherein atoms are chosen at random from a region considerably larger than the wavelength of the first excited state. This selection allows the atoms to absorb photon through phase correlations among the excited atoms. This particular technique has been employed in the Dicke model, which involves $N$ two-level atoms coupled with a single-mode electromagnetic field [25]. The expression for the first excited state, capable of capturing the momentum of the absorbed photon, is given by [16]

Initially, we will examine the absorption spectra of a single atom. Subsequently, we will calculate the average absorption spectra of $N$ atoms, which are distributed randomly over an area significantly larger than the wavelength of the probing field. Therefore, we can calculate the Hamiltonian for a single atom in the interaction picture using equation $V=e^{-\frac {i}{\hbar }H_0t}H_{I}e^{\frac {i}{\hbar }H_0t}$ where

This leads to the resulting Hamiltonian in the interaction picture, which is expressed as

 

Fig. 1. Three-level atoms are randomly distributed in real space. The probe field having mode $\textbf {k}_{p}=k_p \hat {x}$ creates excitation from $| g \rangle$ to $| e\rangle$. The standing wave having modes $\textbf {k}_{1}=k_{1} \hat {x}$ and $\textbf {k}_{2}=-k_{2} \hat {x}$ couples the transition between $| e \rangle$ and $| s\rangle$, whereas microwave field having mode $\textbf {k}_{m}=k_m \hat {x}$ couples $| s \rangle$ and $| g\rangle$.

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If the perturbation conditions are used for both the probe and the counter-propagating control fields, i.e., the former is significantly weaker than the latter, we can choose $\Omega _{1,2}\gg \Omega _{p}$. Under the initial conditions, i.e., $\rho ^{(0)}_{gg}=1$ and $\rho ^{(0)}_{ee}=\rho ^{(0)}_{ss}=\rho ^{(0)}_{se}=\rho ^{(0)}_{es}=0$, the associated density matrix elements, i.e., $\rho ^{(1)}_{eg}$ and $\rho ^{(1)}_{sg}$ are calculated as

Generally, SL is realized by the SW control field, which is responsible for the hopping of excitation from $i$th to $j$th atom. While using the SW control field, there is no intensity at the nodes, and the population distribution is required in the metastable state to avoid disturbing lattice dynamics [13]. However, using MSW instead of SW overcomes the need for population distribution. The reason being that MSW has quasi-nodes (control field intensity is low but never zero) instead of nodes. On the other hand, MSW is not unusual and has been taken into account in a number of areas, including PBGs [1–4,12,14].

Here, we use MSW in place of SW. To generate MSW, a retro-reflected control field is used, which is impinging from a mirror having reflectivity $R_{m}$ [12,14]. The resulting controlled field Rabi frequency is

The dispersion relation and density of states ($D(\varepsilon )$) associated with MSW can be defined as

The absorption spectrum is spatially periodic, and the spatial periodicity is taken as half of the control field wavelength.

3. Modified superradiance lattice

According to [16], for SL while using SW, the probe absorption is zero at resonance (where probe detuning is zero, i.e., $\Delta _{p}=0$), which corresponds to the nodes of the SW. However, a careful review reveals that absorption becomes non-zero when $\Delta _{p}\neq 0$. Beside this, there is also the need to have a certain range of $\Delta _{p}$ where the probe absorption remains zero, i.e., in PBGs [1–4,12,14]. It can be realized by exchanging SW with MSW, which contains quasi-nodes, such that the probe absorption is minimum for a specific range of $\Delta _{p}$) along with resonance ($\Delta _{p}=0$).

We plot the dispersion relation $\varepsilon _{\pm }(x)/\varepsilon _{max}$ versus $kx$ in Figs. 2(a), (b), and (c) (upper row) for $R_{m} = 1.0$, $0.9$, and $0.8$, respectively. We also plot the absorption spectrum ($A(\Delta _p)$) (black solid line) as well as the density of states $D(\varepsilon )$ of SL/MSL (green dashed line), as shown in the lower row of Fig. 2. It is noticed that when $R_{m}$ decreases from 1.0 (pure SW case) to 0.9 and then finally to 0.8 (MSW case), we may attain zero probe absorption not only at $\Delta _{p}=0$ but also within a specific small range of $\Delta _{p}$, as shown by the Grey vertical bar in Figs. 2(a)-(c). Further, we observe that for $R_{m}=1$, $A(\Delta _p)$ overlaps with $D(\varepsilon )$ except at resonance $\Delta _{p}=0$, which is consistent with [16]. It is also observed that this behavior of the absorption spectrum remains the same for $R_{m}=0.9$ and $R_{m}=0.8$ except at resonance, as shown in Figs. 2 (b) and (c). As a result, Figs. 2(a), (b), and (c) explicitly satisfy the SL conditions [16]. Furthermore, Figs. 2 (b) and (c) confirm the realization of the SL utilizing MSW, which we refer to as the MSL.

 

Fig. 2. The dispersion relation of one-dimensional superradiance lattice (upper row). The density of states (black dashed lines) verses $\epsilon /\epsilon _{max}$ and the absorption spectrum of standing wave coupled electromagnetically induced transparency (green solid lines) verses probe field detuning ($\Delta _{p}$) (lower row). $\gamma _{eg}= 0.06\Omega _{0}$. $\gamma _{sg}\approx 0$. (a) $R_m=1$, (b) $R_m=0.9$, and (c) $R_m=0.8$.

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The additional microwave field that couples the energy levels $| s \rangle$ and $| g\rangle$ in Fig. 1 creates a closed-loop configuration and provides another tunable parameter, namely the relative phase $(\phi )$ corresponding to the coupling fields. Generally, certain values of relative phase, namely $0$, $\pi /2$, $\pi$, and $3\pi /2$, are taken into account in loop configuration. Further, it has been noticed in the absorption spectrum of a $\Lambda$-type, three-level EIT medium [15] that phases $\phi =\pi /2$ and $3\pi /2$ produce absorption and amplification, respectively, whereas phases $\phi =0$ and $\pi$ constitute mirror images of each other with respect to probe field detuning. We plot absorption spectrum ($A(\Delta _p)$) versus probe field detuning ($\Delta _p$) for different values of relative phase $\phi$ in Fig. 3. Here, we notice that phases $\phi =0$ and $\pi$ form mirror images, while phases $\phi =\pi /2$ and $3\pi /2$ show absorption and amplification, respectively; however, these results are consistent with [15]. In Fig. 3, one can notice the effect of relative phase on specific region of probe field detuning where probe absorption approaches zero. The dependence of PBGs on absorption spectrum has already been discussed in Fig. 2. As a result, Fig. 3 shows that the PBGs can be significantly affected by the effect of the relative phase. In addition, Fig. 3 also shows a comparison between the effects of a microwave field with varying relative phase values (red solid lines) and the case where the microwave field is zero (black dashed lines).

 

Fig. 3. Absorption spectrum of modified standing wave coupled electromagnetically induced transparency verses probe field detuning ($\Delta _{p}$). Black curve for $\Omega _{m}=0$ and red curve for $\Omega _{m}=0.5\gamma _{eg}$ with (a) $\phi =0\pi$, (b) $\phi =\pi /2$, (c) $\phi =\pi$, and (d) $\phi =3\pi /2$. $\Omega _{p}=\gamma _{eg}$, $R_m =0.8$, $\gamma _{eg}= 0.06\Omega _{0}$, $\gamma _{sg}\approx 0$, and $\Omega _{0}=1$.

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4. Reflection and transmission of light by modified superradiance lattice

After establishing the concept of MSL with relative phase, we use it in one of the applications of the SL, i.e., PBGs [10–12]. The susceptibility Eq. (12) in terms of $m$th order Fourier components can be written as [10–12]

Above equation shows that one cycle of emission and absorption takes time $1/|\Omega _{0}|$, and the entire process should be finished within the decoherence time $\tau _{eg}>n/|\Omega _{0}|$. Equation (18) can be viewed to represent the conservation of momentum. The momentum of probe photon can be reversed when ensemble emits a $n$ coupling photon in the forward mode and absorbs a $n$ coupling photon in the backward mode. Furthermore, Eq. (18) suggests that higher-order Fourier components contribute to susceptibility when $\Omega _{0}$ is much greater than the detuning and decoherence rates.

When phase matching requirements are met, $\textbf {k}_{r}=\textbf {k}_{f}-n\textbf {k}_{c}+n\textbf {k}_{c}-\textbf {k}_{m}+\textbf {k}_{m}$, then three-mode approximation is allowed. However, when the incident photon has mode $\textbf {k}_{p}$, the nth-order Bragg mode $-\textbf {k}_{p}$ is obtained via 2(n+1)th-order coherence $\chi ^{2n+1}$. As a result, $n$ takes the place of index $m$ in Eq. (17). The dynamics of slowly varying amplitudes of the arriving $E_{1}(x)$ and reflecting $E_{2}(x)$ probe fields are represented by [10–12,26,27]

5. Results and discussions

We plot reflectivity $R$ versus probe field detuning $\Delta _{p}$ for different values of relative phase $\phi$, as shown in Fig. 4, when the $5$th order Fourier component is taken at $R_m = 0.8$. We observe that for $\phi =0$ and $\phi =\pi$, PBGs whose reflectivity approaches 100% are formed on the right and left sides of zero probe field detuning, respectively, as shown in Fig. 4(a) and Fig. 4(c). These results behave like mirror images with respect to probe field detuning and are consistent with Fig. 3(a) and Fig. 3(c). The reason is that PBGs are formed where probe absorption is minimum and shift from the right to the left side of the detuning by following the region of minimum probe absorption due to a change in relative phase. Further, minimum reflectivity (which approaches zero) and amplified reflectivity (greater than 100%) are observed for $\phi =\pi /2$ and $\phi =3\pi /2$, respectively as shown in Fig. 4(b) and Fig. 4(d). These results are again following Fig. 3(b) and Fig. 3(d), i.e., for $\phi =\pi /2$, due to absorption of probe field reflectivity approaching zero, and for $\phi =3\pi /2$, greater than 100% reflectivity is observed due to amplification in the probe field.

 

Fig. 4. The reflectivity $R$ versus probe field detuning $\Delta _{p}$ due to $5^{\text {th}}$-order Fourier components for (a) $\phi =0$, (b) $\phi =\pi /2$, (c) $\phi =\pi$, and (d) $\phi =3\pi /2$. Other parameters are $\Omega _{0}=200\gamma _{eg}$, $\Omega _{p}=1\gamma _{eg}$, $\Omega _{m}=0.1\gamma _{eg}$, $\delta k_{n}=0$, $\gamma _{sg}=0$, $\gamma _{eg}=1$, $\mathcal {N}$ = 0.01, and $R_m = 0.8$. The sample length is $10^6 \lambda _{p}$.

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In Fig. 5, we illustrate the relationship between reflectivity $R$ and transmissivity $T$ with respect to probe field detuning. Our investigation goes into the practical application of relative phase ($\phi$), focusing on three input channels of probe field identified in the Grey region of the Fig. 5. Our findings indicate that when $\phi =0$, the scheme reflects light from channel 1 and transmits light from channel 3, as demonstrated in Fig. 5(a). Conversely, when $\phi =\pi$, channel 1 transmits light while channel 3 reflects it as shown in Fig. 5(c). Notably, if we replace $\phi =0$ with $\phi =\pi$, we observe an exchange in reflection and transmission in channels 1 and 2, respectively. Such control over two light channels through relative phase holds potential for use in communication optics. However, when we select $\phi =3\pi /2$, due to amplification in the probe field, more than 100% reflection is obtained for all channels, as shown in Fig. 5(d). Here, one thing is noted that maximum amplified reflection is achieved at resonance. Besides this, transmission approaches zero for all channels. Furthermore, when $\phi =\pi /2$, reflection as well as transmission both approach zero due to the absorption of the probe field as shown in Fig. 5(b). The results obtained in Fig. 5 are based on different values of the relative phase only, and are different from each other with respect to transmission and reflection. This feature shows potential in optics, i.e., optically efficient switches and beam splitters.

 

Fig. 5. The reflectivity (red solid curve) and transmissivity (blue solid curve) versus probe field detuning $\Delta _{p}$ due to $5^{\text {th}}$-order Fourier components for (a) $\phi =0\pi$, (b) $\phi =\pi /2$, (c) $\phi =\pi$, and (d) $\phi =3\pi /2$. Others parameters are $\Omega _{0}=200\gamma _{eg}$, $\Omega _{p}=1\gamma _{eg}$, $\Omega _{m}=0.1\gamma _{eg}$, $\delta k_{n}=0$, $\gamma _{sg}=0$, $\gamma _{eg}=1$, $\mathcal {N} = 0.01$, and $R_m = 0.8$. The sample length is $10^6 \lambda _{p}$. Gray regions represent three input channels of probe field.

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Next, for different orders of Fourier components ($n$), we plot reflectivity $R$ versus probe detuning $\Delta _{p}$ in Fig. 6. Here, we take $\phi =0$, and therefore PBGs are formed on the left side of probe detuning ($\Delta _{p}=0$), like in Fig. 4(a). We notice that PBGs exist with maximum reflection for even comparatively higher orders. In this case, we get reflectivity approaches to 99%, 98%, 96% and 94% for $n$ is equal to $10$, $20$, $50$ and $100$, respectively. This characteristic gives the possibility to control high-frequency light using $n$th order lower-frequency light with negligible loss. Further, PBGs will be formed for $\phi =\pi$ on the right side of the probe field detuning, similar to Fig. 4(c), but they will behave exactly as in Fig. 6 for higher order. Besides this, for $\phi =\pi /2$ and $\phi =3\pi /2$, negligible and greater than 100% reflectivity will be obtained, respectively, like in Fig. 4(b) and Fig. 4(d) for higher-order Fourier components.

 

Fig. 6. The reflectivity $R$ versus probe field detuning $\Delta _{p}$ due to different order of Fourier components at (a) $\Omega _{0}=200\gamma _{eg}$, (b) $\Omega _{0}=290\gamma _{eg}$, (c) $\Omega _{0}=440\gamma _{eg}$, and (d) $\Omega _{0}=630\gamma _{eg}$. $R_{m}=0.95$ and $\phi =0$. Other parameters are same as Fig. 5.

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In Fig. 7, we plot reflectivity $R$ for two different values of probe field incident angle $\delta k_n$. A notable observation is that, in comparison to the case presented in Fig. 4(a), when $\delta k_{n}=-3\times 10^{-5}\nu _{p}/c$ (refer to the black curve in Fig. 7(a)), the PBG exhibits a broadening and shifts away from resonance. Conversely, when $\delta k_{n}=3\times 10^{-5}\nu _{p}/c$, the PBG becomes narrower and shifts towards resonance, as illustrated by the green curve in Fig. 7(a). This same behavior of the PBG is also evident in Fig. 7(c) when compared to Fig. 4(c), for both $\delta k_n$ values of $-3\times 10^{-5}\nu _{p}/c$ and $3\times 10^{-5}\nu _{p}/c$. However, it is also observed that there is minimum change in the PBG with varying values of $\delta k_n$, as shown in both Fig. 7(b) and Fig. 7(d), when compared to the case where $\delta k_n=0$ in Fig. 4(b) and Fig. 4(d). Consequently, Fig. 7 shows that by manipulating $\delta k_n$ while keeping all other parameters constant, one can use additional control over the PBGs.

 

Fig. 7. The reflectivity $R$ versus probe field detuning $\Delta _{p}$ due to $5^{\text {th}}$-order Fourier components for (a) $\phi =0$, (b) $\phi =\pi /2$, (c) $\phi =\pi$, and (d) $\phi =3\pi /2$. Black solid curve for $\delta k_{n}=-3\times 10^{-5}\nu _{p}/c$ and green solid curve for $\delta k_{n}=3\times 10^{-5}\nu _{p}/c$. Other parameters are $\Omega _{0}=200\gamma _{eg}$, $\Omega _{p}=1\gamma _{eg}$, $\Omega _{m}=0.1\gamma _{eg}$, $\gamma _{sg}=0$, $\gamma _{eg}=1$, $\mathcal {N}$ = 0.01, and $R_m = 0.8$. The sample length is $10^6 \lambda _{p}$.

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According to Fig. 6, the $(2n + 1)$th-order coherence $\chi ^{(2n+1)}$ can produce the higher-order Bragg mode $-\textbf {k}_p$ with maximum reflection or minimal loss. So, our considered scheme can be directly applied experimentally to a three-level atomic configuration of beryllium $\text {Be}^{2+}$ ions [10,28] with energy levels, $|s\rangle =|1s2p^{1}P_{1}\rangle$, $|e\rangle =|1s^{21}S_{0}\rangle$, and $|g\rangle =|1s2s^{1}S_{0}\rangle$. The corresponding transition energies are $\hbar \omega _{eg}$ = 123.7 eV($\lambda _{eg}$ = 10 nm) and $\hbar \omega _{es}$ =2.02 eV ($\lambda _{es}$ = 614 nm). The corresponding decoherence rates are $\gamma _{eg} = \Gamma _{eg}/2=6\times 10^{10}\text {s}^{-1}$ and $\gamma _{sg} =9\times 10^{3}\text {s}^{-1}$, which is negligible. Using this atomic configuration we can obtain the $61^{\text {st}}$-order ($\lambda _{es}/\lambda _{eg}=61.4$) photonic PBG, which lies in the soft X-ray regime. Further, we have obtained a photonic band gap for beryllium ions ($\text {Be}^{2+}$) with reflectivity equal to 96%. Beside this, it is observed that our scheme can also be applied to three-level atomic configuration of rubidium $^{85}\text {Rb}$ atoms [10,29] having energy levels, namely, $|e\rangle =|5^{2}S_{1/2}\rangle$, $|s\rangle =|8^{2}S_{1/2}\rangle$, and $|g\rangle =|8^{2}P_{3/2}\rangle$. The decoherence rates are $\gamma _{eg} = 1.25\times 10^{10}\text {s}^{-1}$ and $\gamma _{sg} =3.25\times 10^{6}\text {s}^{-1}$. The transition wavelengths are $\lambda _{eg}$ = 335 nm and $\lambda _{es}$ = 12.40 $\mu$m. This atomic configuration can be used to obtain the $37^{\text {th}}$-order ($\lambda _{es}/\lambda _{eg}=37.01$) photonic PBG, which lies in the ultraviolet regime. Additionally, we have achieved a photonic band gap for rubidium $^{85}\text {Rb}$ atoms, resulting in a reflectivity of 80%. The reduction in reflectivity is due to the larger value of decoherence rate, $\gamma _{sg} =3.25\times 10^{6}\text {s}^{-1}$.

6. Conclusion

In the study of modified superradiance lattice-based photonic band gaps, we have observed that the relative phase between coupling fields plays a fascinating role in controlling the probe field. Notably, this relative phase has the capability to simultaneously address three input channels of the probe field, offering significant potential applications in the field of quantum optics. Furthermore, this approach exhibits promise in both low-frequency and high-frequency domains, yielding maximum reflection. To achieve photonic band gaps in the soft X-ray and ultraviolet regions, beryllium ions ($\text {Be}^{2+}$) and rubidium atoms ($\text {}^{85}\text {Rb}$), respectively, can be utilized as excellent options. Additionally, rubidium atoms ($\text {}^{87}\text {Rb}$) are effective in the low-frequency or infrared range. This innovative scheme can be used to construct highly efficient optical switches and beam splitters.