The field redistribution inside an X-ray cavity-QED setup with an embedded 57Fe layer is calculated and studied in detail. The destructive interference between two transitions from the ground state to the two upper dressed states causes that the cavity mode can not be driven. So the field intensity is very weak when the nuclear ensemble is resonant. Moreover, It is found that the resonant nuclear layer can play a role of reflective layer like a mirror and cut the size of the cavity, which will destroy the guided mode. To support this idea, we employ the 57Fe film as the bottom mirror layer of the cavity where a guided mode can only be formed at the resonant energy. Following this perspective, the electromagnetically induced transparency structure based on X-ray cavity-QED setup with nuclear ensemble is reviewed and a phenomenologically self-consistent analysis for the field redistribution is presented.
© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
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) , electromagnetically induced transparency (EIT) , spontaneously generated coherence (SGC) , Fano resonance control and interferometric phase detection , group delay tunable for x-ray pulses , x-ray and nuclei collective strong coupling in nuclear optical lattice , narrow band x-ray pulses stopping , collective magnetic splitting in single-photon superradiance  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  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 , focus the x-ray  and construct x-ray photonic crystals . 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 . Another clearer evidence for the importance of the positions of 57Fe layers was also shown in this EIT works , 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 , 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 , 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  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 . On the other hand, inspired by this EIT works, Hu et al have proposed a proof-of-concept platform in the metamaterials domain  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 .
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 .
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]
The component of the wave vector in z direction can be written as30] 31, 32]
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 . 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 and can also be used to give the field intensity
It can be found that nj 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 , so the refraction index of 57Fe must be amended around the resonant position. Neglecting the hyperfine interaction and the polarization dependence, the refraction index nN contributed by the 57Fe nuclei with the forward scattering amplitude fN can be rewritten as [24, 33]4, 24, 33] 33]. fLM ≈ 0.8 in the ambient temperature is the Lamb-Mössbauer factor, and α = 8.56 is the internal conversion factor. The spins of the nuclear excited and ground states are and , respectively. Δ = ω − ω0 is the energy detuning from the transition energy ω0 = 14.4125 keV, and γ0 = 4.7 neV is the linewidth of the nuclear resonance. At last, add the contribution from the electronic scattering to the refraction index n 
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.
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 ΓN is much larger than the natural linewidth γ0 owing to the cooperative emission originating from the single-photon superradiance . 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).
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  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 , N is the number of nuclei. The strong coupling regime was not reached in the typical experimental works , so the coupling strength is larger than γ0 but smaller than κ . 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 . 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 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 , 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 . Actually in the metamaterials domain, classical resonators such like SRR , “I” shape resonator, or even a simple cut-wire resonator based on surface current LC resonance , dielectric brick resonator based on Mie-resonance , and nano-structure based on surface plasmon resonance , 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 . 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.
Moreover, this mirror notion can also give a self-consistent explanation for the EIT effect  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 . 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 , 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.
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.
The National Natural Science Foundation of China (Grants No. U1332204, U1732133 and 11375130); National Key Research and Development Program of China (Grants No. 2017YFA0402300 and 2017YFA0303500).
Authors thank Ralf Röhlsberger for fruitful discussion.
References and links
1. B. W. Adams, C. Buth, S. M. Cavaletto, J. Evers, Z. Harman, C. H. Keitel, A. Pálffy, Antonio Picón, R. Röhlsberger, Y. Rostovtsev, and K. Tamasaku, “X-ray quantum optics,” J. Mod. Optic. 60, 2–21 (2013). [CrossRef]
2. C. Weninger, M. Purvis, D. Ryan, R. A. London, J. D. Bozek, C. Bostedt, A. Graf, G. Brown, J. J. Rocca, and N. Rohringer, “Stimulated electronic x-ray raman scattering,” Phys. Rev. Lett. 111, 233902 (2013). [CrossRef]
3. S. Shwartz, R. N. Coffee, J. M. Feldkamp, Y. Feng, J. B. Hastings, G. Y. Yin, and S. E. Harris, “X-ray parametric down-conversion in the Langevin regime,” Phys. Rev. Lett. 109, 013602 (2012). [CrossRef] [PubMed]
6. K. P. Heeg, H. C. Wille, K. Schlage, T. Guryeva, D. Schumacher, I. Uschmann, K. S. Schulze, B. Marx, T. Kämpfer, G. G. Paulus, R. Röhlsberger, and J. Evers, “Vacuum-assisted generation and control of atomic coherences at x-ray energies,” Phys. Rev. Lett. 117, 073601 (2013). [CrossRef]
7. K. P. Heeg, C. Ott, D. Schumacher, H. C. Wille, R. Röhlsberger, T. Pfeifer, and J. Evers, “Interferometric phase detection at x-ray energies via Fano resonance control,” Phys. Rev. Lett. 114, 207401 (2015). [CrossRef] [PubMed]
8. K. P. Heeg, J. Haber, D. Schumacher, L. Bocklage, H. C. Wille, K. S. Schulze, R. Loetzsch, I. Uschmann, G. G. Paulus, R. Rüffer, R. Röhlsberger, and J. Evers, “Tunable subluminal propagation of narrow-band x-ray pulses,” Phys. Rev. Lett. 114, 203601 (2015). [CrossRef] [PubMed]
9. J. Haber, K. S. Schulze, K. Schlage, R. Loetzsch, L. Bocklage, T. Gurieva, H. Bernhardt, H. C. Wille, R. Rüffer, I. Uschmann, G. G. Paulus, and R. Röhlsberger, “Collective strong coupling of x-rays and nuclei in a nuclear optical lattice,” Nat. Photon. 10, 445–449 (2016). [CrossRef]
12. X. Kong and A. Pälffy, “Collective radiation spectrum for ensembles with Zeeman splitting in single-photon superradiance,” Phys. Rev. A 96, 033819 (2017). [CrossRef]
14. W. T. Liao and A. Pälffy, “Proposed entanglement of x-ray nuclear polaritons as a potential method for probing matter at the subatomic scale,” Phys. Rev. Lett. 112, 057401 (2014). [CrossRef] [PubMed]
18. K. P. Heeg and J. Evers, “X-ray quantum optics with Mössbauer nuclei embedded in thin-film cavities,” Phys. Rev. A 88, 043828 (2013). [CrossRef]
19. K. P. Heeg and J. Evers, “Collective effects between multiple nuclear ensembles in an x-ray cavity-QED setup,” Phys. Rev. A 91, 063803 (2015). [CrossRef]
20. Y. Hu, W. Liu, Y. Sun, X. Shi, J. Jiang, Y. Yang, S. Zhu, J. Evers, and H. Chen, “Electromagnetically-induced-transparency-like phenomenon with resonant meta-atoms in a cavity,” Phys. Rev. A 92, 053824 (2015). [CrossRef]
21. S. Rai, M. K. Tiwari, G. S. Lodha, M. H. Modi, M. K. Chattopadhyay, S. Majumdar, S. Gardelis, Z. Viskadourakis, J. Giapintzakis, R. V. Nandedkar, S. B. Roy, and P. Chaddah, “Surface and interface study of pulsed-laser-deposited off-stoichiometric NiMnSb thin films on a Si(100) substrate,” Phys. Rev. B 73, 035417 (2006). [CrossRef]
23. K. Okamoto, T. Noma, A. Komoto, W. Kubo, M. Takahashi, A. Iida, and H. Miyata, “X-ray waveguide mode in resonance with a periodic structure,” Phys. Rev. Lett. 109, 233907 (2012). [CrossRef]
24. R. Röhlsberger, Nuclear condensed matter physics with synchrotron radiation, (Springer Tracts in Modern Physics, 2004).
25. R. Röhlsberger, “Theory of x-ray grazing incidence reflection in the presence of nuclear resonance excitation,” Hyperfine Interact. 123, 301–325 (1999). [CrossRef]
26. R. Röhlsberger, J. Evers, and S. Shwartz, “Quantum and nonlinear optics with hard x-rays,” Synchrotron Light Sources and Free-Electron Lasers: Accelerator Physics, Instrumentation and Science Applications, 1197–1229 (Springer, 2016).
27. W. Sturhahn, “CONUSS and PHOENIX: evaluation of nuclear resonant scattering data,” Hyperfine Interact. 125, 149–172 (2000). [CrossRef]
28. D. L. Windt, “IMD–software for modeling the optical properties of multilayer films,” Comp. Phys. 12, 360–370 (1998). [CrossRef]
29. M. Tiwari, G. Lodha, and K. Sawhney, “Applications of the CATGIXRF computer program to the grazing incidence x-ray fluorescence and x-ray reflectivity characterization of thin films and surfaces,” X-Ray Spectrom. 39, 127134 (2010). [CrossRef]
30. A. N. Jens and M. Des, Elements of modern x-ray physics, (John Wiley, 2011).
31. M. Born and E. Wolf, Principles of optics, (Pergamon, 1980).
32. D. G. Stearns, “The scattering of x-rays from nonideal multilayer structures,” J. Appl. Phys. 65, 491–506 (1989). [CrossRef]
33. K. P. Heeg, “X-ray quantum optics with Mössbauer nuclei in thin-film cavities,” Ph.D. thesis, Ruprecht-Karls-Universität Heidelberg, (2014).
34. M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, “Directed spontaneous emission from an extended ensemble of N atoms: timing is everything,” Phys. Rev. Lett. 96, 010501 (2006). [CrossRef] [PubMed]
35. J. Haber, “Hard x-ray quantum optics in thin-film nanostructures,” Ph.D. thesis, Der Universität Hambuer, (2017).
36. P. Herskind, A. Dantan, J. Marler, M. Albert, and M. Drewsen, “Realization of collective strong coupling with ion Coulomb crystals in an optical cavity,” Nat. Phys. 5, 494–498 (2009). [CrossRef]
37. P. Tassin, L. Zhang, T. Koschny, E. N. Economou, and C. M. Soukoulis, “Low-loss metamaterials based on classical electromagnetically induced transparency,” Phys. Rev. Lett. 102, 053901 (2009). [CrossRef] [PubMed]
39. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8, 758–762 (2009). [CrossRef] [PubMed]