1. Introduction

The dramatic development of x-ray sources such as the high-brilliant synchrotron radiation and x-ray free-electron laser opens an opportunity to understand the interaction between x-ray photons and matters more deeply from the viewpoint of quantum optics [1–3]. A particularly attractive platform is Mössbauer nuclei which has revealed and reproduced a number of novel fundamental phenomena both experimentally [4–10] and theoretically [11–17], such as the collective lamb shift (CLS) [4], electromagnetically induced transparency (EIT) [5], spontaneously generated coherence (SGC) [6], Fano resonance control and interferometric phase detection [7], group delay tunable for x-ray pulses [8], x-ray and nuclei collective strong coupling in nuclear optical lattice [9], narrow band x-ray pulses stopping [11], collective magnetic splitting in single-photon superradiance [12] and related effects [10,13–17]. An ideal laboratory to study these features is an x-ray planar waveguide [4, 5, 18] or so-called cavity-QED setup [19] which is an ultrathin film of Mössbauer nuclei embedded in a multilayer structure and probed by the hard x-ray pulses. Similar cavities have been extensively used to analyze the elemental impurities by x-ray fluorescence in semiconductor industry [21], focus the x-ray [22] and construct x-ray photonic crystals [23]. The multilayer is made from different materials with high electron density like Pt or Pd and low electron density like carbon, and these different electron densities lead to various refraction indexes. Pt or Pd are employed as mirrors and the C layer in the middle leads to guiding the x-ray. In this way, the multilayer permits coupling of x-ray evanescently into the specific guided modes. Then the interaction between the cavity modes and the nuclei can be achieved by embedding Mössbauer nuclear layer inside this multilayer. Because the real parts of refraction indexes for the hard x-ray are usually slightly smaller than unity, the cavity modes are typically excited and probed at tiny grazing angles.

Previous works indicate that the position of the layer of nuclei should be chosen carefully inside the cavity [1, 4, 4, 19]. The guided mode of the cavity shows a standing wave with a range of maxima and minima, i.e., the antinodes and nodes which mean significant differences of photonic density of states. The photonic density of states is related to the strength of cooperative emission and the CLS directly, so an ensemble of nuclei locates at different positions of the standing wave will cause various decay widths and CLSs [1, 4, 5]. For instance, the design scheme for combining antinode-layer and node-layer to achieve EIT effect in x-ray regime was based on this rule [5]. Another clearer evidence for the importance of the positions of ⁵⁷Fe layers was also shown in this EIT works [5], where only in the node-antinode configuration an EIT-like reflection curve was obtained while no EIT-like reflection curve was observed in the configuration of antinode-node [5,19]. Normally and necessarily, the positions of the ⁵⁷Fe layers are assigned according to the field distribution inside the cavity where the guided modes come into being. The field distribution can be calculated together with the reflectivity curve as a function of the grazing angle θ when taking no account of the Mössbauer nuclear layers, and the specific order for the guided mode can be recognized through finding the dip in the reflectivity curve.

It can be seen that it is fundamentally important to analyze the field distribution inside the cavity. In the condition of the guided mode for the bare cavity, the reflection is strongly suppressed to be near zero. Therefore, the reflection is correlated positively to the emission from the ensemble of nuclei after embedding nuclear layer into the cavity [4], and the reflectance can be improved highly around the nuclear transition energy according to the calculations by both classical matrix formalism [1,5] and full quantum optics model [18], which have been tested by various experiments [4–6,8,9]. Moreover, a logical concomitant phenomenon is that the field distribution inside the cavity should be modified [1, 19]. Heeg and Evers have shown the field redistribution for the EIT work of Röhlsberger et al [5] which embeds two nuclear layers inside the cavity, where the field intensity weakens when the ensemble of nuclei is resonant. They supposed the field redistribution can be view as a superposition of the bare cavity field and the nuclear scattering, but no further analysis was given in their work [19]. On the other hand, inspired by this EIT works, Hu et al have proposed a proof-of-concept platform in the metamaterials domain [20] which is a different research area working below the visible frequency. In their work, the x-ray cavity was replaced by a microstrip which is used to guide the microwave, and the nuclear scatterer was replaced by a split-resonant ring (SRR) which is a classical resonator based on surface current LC resonance. A EIT-like curve and similar field redistribution were given, but the field intensity around the scatterer is very high which is different from the result of Heeg and Evers [19].

In the present work, we study the field redistribution in detail. Firstly, after inserting a single ⁵⁷Fe layer into the bare cavity, the field intensity becomes very weak except when the nuclear layer is embedded at the node point. It can be understood as a destructive interference process between the two dressed states, so the photon dose not excite the cavity nor the ensemble of nuclei. For the specific shape of the field redistribution, we propose that the nuclear layer can play a role of mirror at the resonant energy which can destroy the standing wave. To support this notion, we utilize the resonant ⁵⁷Fe layer to be a bottom mirror layer and construct guided modes that only appear at resonant energy. Subsequently, this idea can also give a self-consistent explanation for the EIT work [5].

2. Methods

To describe the optical properties of the cavity, various theoretical methods have been developed including classical and quantum optical formalism. Based on the Parratt formalism which is an exactly recursive method to obtain the optical parameters of x-ray from the layer structures, Röhlsberger has promoted it in a transfer-matrix formalism to include resonant multipole scattering with its polarization dependence [24–26], and it agrees well with the recent experimental results [4,5,9]. A numerical implementation of this formalism is included in a software package CONUSS, and it can act as a benchmark for developing other theories [18, 27]. The Parratt formalism is good enough to describe the cavity, which is valid in this work since the magnetic hyperfine field is not necessary to be considered here. The typical multilayer structure is shown in Fig. 1, and different layers have various refractive indices and thicknesses. When all the layers and interfaces are parallel, the electric field amplitude of a plan x-ray wave propagating in a medium j with energy ω at a position r can be expressed as [28, 29]

 

Fig. 1 A schematic diagram of a cavity structure constructed by multilayer. The refraction and reflection parts of the electric field amplitude are also shown in each layer.

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The component of the wave vector in z direction can be written as

The recursive relation is run in the IMD software package which is a mature program for modeling the optical properties of multilayer film based on the recursive method [28]. It is well known that the transmissivity of the cavity can not be easily measured in the experiment because the substrate for depositing the multilayer is usually quite thick, but it is possible to be measured if a thin SiN substrate is used for depositing the multilayer. The $E_j^r$ and $E_j^t$ can also be used to give the field intensity

It can be found that n_j is a crucial parameter in the procedure of calculation. The oscillator strength of the nuclear resonance is about two orders of magnitude larger than that of the electronic resonance [24], so the refraction index of ⁵⁷Fe must be amended around the resonant position. Neglecting the hyperfine interaction and the polarization dependence, the refraction index n_N contributed by the ⁵⁷Fe nuclei with the forward scattering amplitude f_N can be rewritten as [24, 33]

The black solid lines in Fig. 2 show that the δ and β of the ⁵⁷Fe diverge from zero, and those of the carbon and platinum are also shown for comparison. The δ and β of carbon and platinum can be regarded as a constant in the energy range of our concern since the γ₀ is extremely narrow. It can be found that the refraction index difference between C and the resonant ⁵⁷Fe nuclei is much larger than the difference between C and Pt, which implies that the ⁵⁷Fe layer can play a role of reflective layer to interdict transmission like Pt mirror layer because the reflective effect is mainly from the δ difference and the larger β can cause more absorption. More clear evidence can be seen in the discussion section.

 

Fig. 2 The real part increment −δ and the imaginary part increment β for n of the ⁵⁷Fe around the nuclear resonant energy, as well as the ones of the mirror medium Pt and the guiding medium C for comparison. The δ and β far from the resonance are not zero due to the contributions from δ_e and β_e.

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3. Results and discussion

As depicted in Fig. 3(a), a single ⁵⁷Fe layer is arranged at the center of the cavity. When we keep the energy detuning far away from zero, the nuclei can be neglected. That is, in the case of off-resonance, the third order guided mode of the bare cavity appears at the grazing angle of 0.203°, the typical standing wave is shown in Fig. 3(b). However, if the energy detuning is around zero, the field of the cavity becomes very weak. Correspondingly, the reflectivity increases prominently as shown by the solid black line in Fig. 3(c). The shift LN is the CLS, and the width Γ_N is much larger than the natural linewidth γ₀ owing to the cooperative emission originating from the single-photon superradiance [34]. The reflectivity curve is in good agreement with the previous results [4, 5] to ensure the effectiveness of our approach. When the ⁵⁷Fe layer is gradually moved from the antinode to the node without changing the actual size of the cavity, the phenomenon of destroying the field always remains except the node one as shown in Fig. 3(d).

 

Fig. 3 (a) The geometry of the cavity is labeled on the top of the cavity schematic, and the first layer of Pt is thin enough (2.5 nm) that the x-ray can penetrate into the cavity evanescently. Compared with Fig. 1, we rotate the coordinate system for convenience. (b) Field distribution inside the cavity for off-resonance or on-resonance. The solid brown bar represents the ⁵⁷Fe layer arranged at the middle of the cavity, i.e., the antinode position. (c) The reflectivity curves for the ⁵⁷Fe layer arranging at the antinode or the node position. (d) The field redistributions for on-resonant nuclear layer at different positions. The thicknesses of the iron are kept as 2 nm. With moving the nuclear layer from antinode to node, the grazing angle should be modulated slightly to ensure the guided mode in the third order, the corresponding grazing angles are 0.203°, 0.203°, 0.201°, 0.198°, 0.196°.

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Focusing on field re-distribution in the present work, the discussion of off-resonance condition is omitted below because the field redistribution only occurs in the on-resonance condition as shown in Fig. 3(b). In the case of on-resonance, the transition energy of the ⁵⁷Fe resonance can match the cavity mode energy exactly, and one synchrotron radiation pulse can only provide at most one resonant photon. It should be noted that there will be more than one photon to excite the cavity, but there is at most one photon in resonance with the nuclei. Considering the interaction between the cavity and the nuclei, it is only necessary to consider this one resonant photon, so one excitation subspace is tenable [35] for this situation, i.e., one cavity mode excitation, or one nuclear ensemble excitation. Therefore, in the system, there are three possible states which can be defined as |g0〉, |g1〉 and |e0〉 [18, 33, 35]. |g0〉 is the ground state that the photon is outside the cavity, neither cavity nor nuclei is excited. |g1〉 represents that the cavity mode appears but the nuclei is still on the ground state. |e0〉 means that the nuclei is excited but no photon in the cavity. The x-ray cavity has a low quality factor [4–6, 8] which corresponds a very large decay constant κ. For the ensemble of nuclei, the decay constant γ₀ is small since the natural linewidth is so narrow. The coupling strength g between one atom and the cavity mode is very small, however, the collective coupling strength between the nuclear ensemble and the cavity mode can increase dramatically due to the Dicke state [34, 36]. Consequently, the coupling strength should be modified as $\sqrt{Ng}$, N is the number of nuclei. The strong coupling regime was not reached in the typical experimental works [9], so the coupling strength $\sqrt{Ng}$ is larger than γ₀ but smaller than κ [18]. Therefore, for the two upper excited states |g1〉 and |e0〉 (undressed states), they can be transformed into two dressed states as below

These two dressed states have same decay width, and it is much larger than the energy splitting between these two states [35]. That means, if we want to excite anyone of these two excited states using one photon, they will destructively interfere with each other. Hence, the system remains at the manifold with the ground state |g0〉. It looks like the photon is still outside the cavity, so as shown in Fig. 3(b) and 3(d) the field inside the cavity is very weak correspondingly. We should note that it is very complicated to give a rigorous theoretical analysis for the destructive interference, but it can explain_√the field redistribution successfully. When the layer is embedded at node, the coupling strength $\sqrt{Ng}$ is near zero, which result in that the destructive interference can not occur. As depicted by the solid brown line in Fig. 3(d), the standing wave is not destroyed, and the reflectivity curve is negligible as shown in Fig. 3(c). Under this situation the collective effect is nonsignificant, so the energy width is comparable with the natural linewidth. A similar field redistribution has been reported by Hu et al [20], where the cavity and the resonator worked in the microwave band. The field inside the microwave cavity is also weakened, but the field intensity around SRR is even higher than the standing wave peak, and it is different from the present result and the result of Heeg et al [19]. Actually in the metamaterials domain, classical resonators such like SRR [37], “I” shape resonator, or even a simple cut-wire resonator based on surface current LC resonance [38], dielectric brick resonator based on Mie-resonance [38], and nano-structure based on surface plasmon resonance [39], are called meta-atom. All of the resonators can localize the field strongly around them when they are resonant, and it is not easy to deal with them with full quantum formalism.

For the specific shape of the field redistribution as depicted in Fig. 3(b), it can be found that the coordinate of the nuclear layer is an obvious critical point. The field is weak in the area between the top Pt layer and the ⁵⁷Fe layer, and the field behind the ⁵⁷Fe layer is zero. Herein the resonant nuclei acts like a mirror layer, and it can cut the size of the cavity. Consequently, the field behind the nuclear layer becomes negligible and the dimension of the cavity becomes half of the initial one. The phase requirement of the standing wave is completely destroyed which leads to a very weak field in front of the ⁵⁷Fe layer. As long as the layer is not at the node position, the phase requirement is always destroyed, so this phenomenon always remains as shown in Fig. 3(d). In the case of node-location, the photonic density of states at the node position is zero, so the coupling strength is too weak to prop up the node-layer as a mirror. On the other hand, the phase requirement is also met. Therefore, the standing wave can not be damaged.

In order to ascertain that the nuclear layer can act as a reflective layer like a mirror, we employ the ⁵⁷Fe film as a doubtless bottom mirror layer as shown in Fig. 4. In this scenario, we arrange the ⁵⁷Fe layer at the node position and retain the Pt and C layers in front of the nuclear layer with their size and the grazing angle fixed. Compared with the off-resonant situation in Fig. 3(b), the photonic density of states at the point of ⁵⁷Fe layer increases slightly which leads to the larger increase of the coupling strength, so the ⁵⁷Fe layer can play a role of mirror. In the meanwhile, the phase condition can be fulfilled when a mirror layer is located at any node position because the phase is changed by an integer multiple of π. As shown in Fig. 4(a), when the x-ray energy is at off-resonance, the reflectance is about 0.43 mainly coming from the top Pt layer. And under the on-resonant circumstance, the reflectivity curve drops to zero and the field redistribution keeps one peak of the standing wave. In order to verify our results, the reflectivity curve is also calculated by CONUSS, the benchmarked simulation [27]. The comparison is depicted in Fig. 4(a) and a good agreement is observed. This phenomenon is not related to the peaks number being left over, e.g., a same reflection curve is shown in Fig. 4(b) corresponding to two antinodes of the retained standing wave.

 

Fig. 4 Reflected curves and the field distributions when the ⁵⁷Fe layer is employed as a mirror. (a) Choose the geometry that keeps one antinode compared with the original cavity, and (b) keeps two antinodes. The schematics of these two samples are shown on the top of the spectra, the size of (b) is larger than the size of (a), and the thicknesses of the iron are kept as 2 nm as the same as in Fig. 3. The grazing angles are all 0.196° as the same as the node condition in Fig. 3(d).

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Moreover, this mirror notion can also give a self-consistent explanation for the EIT effect [5] observed by Röhlsberger et al. In their work, there are two ⁵⁷Fe layers with node-antinode configuration as shown by Fig. 5. As depicted in Fig. 3(b), the field intensity at the original node point is not zero anymore when the antinode-layer is at on-resonance. Therefore the node-layer can restart to play a role of mirror which is much different from the nonsignificant single node-layer as shown in Fig. 3(d). Subsequently, the field behind the node-layer becomes negligible but a standing wave is kept in front of this layer. That is, the cavity mode is driven again which corresponds to a sharp dip in the reflectivity curve [5]. The collective effect of the antinode-layer is strong as shown in Fig. 3(c) and the collective effect of the node-layer is weak as shown in Fig. 4, so a three-level system with two significantly different decay widths is constructed and the EIT spectrum can be obtained. Here it can be inferred that the node-layer can not be excited by the cavity mode directly and the interaction between the node-layer and the cavity mode can only be connected through the antinode-layer. The physics for this x-ray EIT effect is same as the work by Hu et al [20], so a similar field redistribution and a EIT-like curves are also observed. In the metamaterials domain, the node-layer can be called as a dark resonator and the antinode-layer can be called as a light resonator. And this pervasive idea has been widely implemented in this area to achieve EIT-like effect, where the dark resonator and the bright resonator were designed as plane sub-wavelength configuration [37–39] rather than the node-antinode configuration.

 

Fig. 5 The field distributions at off or on-resonance for the x-ray EIT structure, the guided mode at the third order is also driven which is same as the one in Fig. 3(b). The thicknesses of the iron are kept as 2 nm. The present result shows a good agreement with the result of Ref. [19].

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4. Conclusion

In conclusion, the field redistribution in an x-ray cavity-QED setup around the nuclear resonant energy is studied. In the one-excitation subspace, the transitions from the ground state to the two upper dressed states will interfere destructively, so the cavity mode can not be driven which lead to a very weak field intensity inside the cavity. And the resonant nuclear layer can play a role of reflective layer like a mirror which can destroy the guided mode. Therefore, for the specific shape of the field redistribution, the field is very weak in the area between the top Pt layer and the ⁵⁷Fe layer, and the field behind the ⁵⁷Fe layer is zero. Following this viewpoint, we utilize the resonant ⁵⁷Fe layer to be the bottom mirror layer in which the guided modes can only be formed at the resonant energy. At last, the EIT structure based on x-ray cavity with nuclear ensemble is reviewed and a self-consistent analysis for the field redistribution is presented. As a concomitant and an indispensable part, the field redistribution will provide a more comprehensive understanding for the x-ray cavity-QED setup.