X-ray cavity quantum optics of inner-shell transitions

Wang Shu-Xing; Li Tian-Jun; Huang Xin-Chao; Zhu Lin-Fan

doi:10.7498/aps.73.20241218

Abstract

Over the past decade, X-ray quantum optics has emerged as a dynamic research field, driven by significant advancements in X-ray sources such as next-generation synchrotron radiation facilities and X-ray free-electron lasers, as well as improvements in X-ray methodologies and sample fabrication techniques. One of the most successful platforms in this field is the X-ray planar thin-film cavity, also known as the X-ray cavity QED setup. To date, most studies in X-ray cavity quantum optics have focused on Mössbauer nuclear resonances. However, this approach is constrained by the limited availability of suitable nuclear isotopes and the lack of universal applicability. Recently, experimental realizations of X-ray cavity quantum control in atomic inner-shell transitions have demonstrated that cavity effects can simultaneously modify transition energies and core-hole lifetimes. These pioneering studies suggest that X-ray cavity quantum optics based on inner-shell transitions will become a promising new platform. Notably, the core-hole state is a fundamental concept in various modern X-ray spectroscopic techniques. Therefore, integrating X-ray quantum optics with X-ray spectroscopy holds the potential to open new frontiers in the field of core-level spectroscopy.In this review, we introduce the experimental systems used in X-ray cavity quantum optics with inner-shell transitions, covering cavity structures, sample fabrications, and experimental methodologies. We explain that X-ray thin-film cavity experiments require high flux, high energy resolution, minimal beam divergence, and precise angular control, necessitating the use of synchrotron radiations. Grazing reflectivity and fluorescence measurements are described in detail, along with a brief introduction to resonant inelastic X-ray scattering techniques. The review also outlines simulation tools, including the classical Parratt algorithm, semi-classical matrix formalism, quantum optical theory based on the Jaynes-Cummings model, and the quantum Green’s function method. We discuss the similarities and unique features of electronic inner-shell transitions and highlight recent advancements, focusing on cavity-induced phenomena such as collective Lamb shift, Fano interference, core-hole lifetime control, etc. Observables such as reflectivity and fluorescence spectra play a central role in these studies. Finally, we review and discuss potential future directions for the field. Designing novel cavities is crucial for addressing current debates regarding cavity effects in inner-shell transitions and uncovering new quantum optical phenomena. Integrating modern X-ray spectroscopies with X-ray cavity quantum optics represents a promising research frontier with significant application potential. Furthermore, X-ray free-electron lasers, with much higher pulse intensity and shorter pulse duration, are expected to propel X-ray cavity quantum optics into the nonlinear and multiphoton regimes, opening new avenues for exploration.

Keywords:

Authors and contacts

1.
Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
2.
I. Physikalisches Institut, Justus-Liebig-Universität Gießen and Helmholtz Forschungsakademie Hessen für FAIR (HFHF), Campus Gießen, GSI Helmholtzzentrum für Schwerionenforschung, Gießen 35392, Germany
3.
FXE Instrument, European XFEL, Schenefeld 22869, Germany

Corresponding author: Huang Xin-Chao, xinchao.huang@xfel.eu ; Zhu Lin-Fan, lfzhu@ustc.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12334010, U1932207).

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图 1 实验体系原理图, 反射谱和荧光谱插图取自文献[61], 共振发射谱插图取自文献[82]

Figure 1. Sketch of the experimental scheme, the reflectivity and fluorescence maps refer to Ref. [61], and the emission spectra map following inelastic scattering is taken from Ref. [82].

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图 2 WSi₂内壳层白线跃迁示意图

Figure 2. Schematic diagram of inner-shell transitions in WSi₂.

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图 3 PETRA III光源P23线站布局示意图^[90]

Figure 3. Layout of the P23 beamline of the PETRA III synchrotron^[90].

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图 4 Parratt迭代方法示意图. 其中$ n_i $为第i层介质对X射线的折射率, $ d_i $为第i层介质的厚度, $ r_{i-1,i} $和$ t_{i-1,i} $为X射线在介质($i-1 $)与i界面处的反射和透射系数

Figure 4. Illustration of Parratt’s method. $ n_i $ and $ d_i $ are the refractive indices and thickness of the$ i\text{-}\mathrm{th} $ layer, $ r_{i-1,i} $ and $ t_{i-1,i} $ are the Fresnel coefficients for reflection and transmission at the interface between $ (i-1) $ and $ i\text{-}\mathrm{th} $ layers.

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图 5 平面腔中场幅度分布示意图

Figure 5. Sketch map of field amplitudes in the cavity.

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图 6 表面平行度较好(a)和较差的(b)的反射光图像, 入射光斑尺寸横向大于纵向

Figure 6. Reflected X-ray patterns from the cavity samples with (a) flat surface and (b) distorted surface, respectively. The horizontal beam size is larger than the vertical one.

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图 7 (a)结构Pt(2.0 nm)/C(18.0 nm)/WSi₂(2.0 nm)/C(18.0 nm)/Pt(16.0 nm)/Si₁₀₀(infinitely thick)的薄膜平面腔的X射线反射率曲线; (b)不同角度下腔内场强随着z方向深度的分布, 其中白色实线描绘了不同膜层的边界

Figure 7. (a) Rocking curve of the cavity with the structure of Pt(2.0 nm)/C(18.0 nm)/WSi₂(2.0 nm)/C(18.0 nm)/Pt(16.0 nm)/Si₁₀₀(infinitely thick); (b) the field intensity distribution inside the cavity. The white solid lines dipict the boundaries of different layers.

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图 8 入射X射线在共振能量和偏离共振能量下的摇摆曲线

Figure 8. Rocking curves under on-resonance and off-resonance X-ray energies.

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图 9 (a) Parratt迭代、(b) 传输矩阵以及(c) 格林函数方法模拟的平面腔一阶模式角附近的反射率二维谱

Figure 9. Simulated two-dimensional reflectivity maps of the cavity around the first mode angle using (a) the Parratt’s recursion, (b) the transfer matrix method, and (c) the Green’s function framework, respectively.

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图 10 3种方法计算的反射谱对比, 入射角度相对于一阶模式角分别为(a) –0.001°角失谐、(b) 0° 以及(c) 0.001°角失谐

Figure 10. Comparisons of reflectivity spectra at (a) –0.001°, (b) 0° and (c) 0.001° offsets deviate from the first mode angle.

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图 11 薄膜平面腔内的⁵⁷Fe核跃迁的集体兰姆移位与超辐射速率增强效应, 腔结构与图8中使用的一致

Figure 11. Collective Lamb shift and superradiance of Mössbauer transition of ⁵⁷Fe due to the cavity effect. The cavity structure used here is same to Fig. 8.

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图 12 实验测量与理论模拟的X射线反射二维谱　(a)实验测量结果; (b)实验数据扣除吸收边; (c)理论模拟结果; (d)理论模拟扣除吸收边

Figure 12. X-ray reflection two-dimensional spectrum of experimental measurements and theoretical simulations: (a) Experimental reflectivity map; (b) experimental data by exclusion of the absorption edge; (c) simulated reflectivity map; (d) simulated map by exclusion of the absorption edge.

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图 13 3种腔结构下实验测量的模式角度下的反射谱及拟合曲线, 其中数据点和红色虚线分别为实验测量结果与拟合结果, 绿色实线为实验数据去除拟合的吸收边得到的法诺线形, 数据引自文献[61]

Figure 13. Measured reflectivity spectra at the first mode angle. The dots are experimental data, and the dashed lines are the fit to data according to the theoretical model. The solid lines present the Fano profiles in the reflectivity spectra by subtracting the fitted edge components from the experimental data. The squares of $ \mathrm{Im}(q) $ for each data set are also presented. Data are quoted from Ref. [61].

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图 14 不同⁵⁷Fe占比的原子核层在(a)欠耦合腔和(b)过耦合腔中对入射角度为10 μrad负失谐、模式角与10 μrad正失谐的反射谱

Figure 14. Reflectivity spectra at the first mode angle and $ \pm 10\; $μrad offsets of (a) undercritical cavities and (b) overcritical cavities with different fractions of ⁵⁷Fe.

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图 15 SDD探测器采集的荧光全谱

Figure 15. Full fluorescence spectrum collected by the SDD detector.

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图 16 (a) 原子层位置处的场强模拟值与X射线能量及入射角度的关系, 使用的腔结构与图7相同; (b) 根据互易定理模拟的荧光二维谱; (c) 实验测量的荧光二维谱

Figure 16. (a) Simulated field intensity at the atom position for the cavity in Fig. 7; (b) simulated fluorescence 2D map according to the reciprocal theory; (c) the measured fluorescence 2D map.

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图 17 (a)一阶、(b)三阶、(c)五阶模式角以及(d)远离腔模式角度下的荧光谱及拟合曲线, 远离腔模式时, 洛伦兹响应的线宽为原子本身的线宽3.6 eV, 而在模式角度下, 辐射速率受到腔效应增强, 线宽显著增大, 数据引自[60]

Figure 17. Selected fluorescence spectra at the (a) 1^st, (b) 3^rd, (c) 5^th mode angles, and (d) offset angle far from the mode angles. The experimental spectra are fitted by the theoretical model, and the widths of the Lorentzian response are presented. Note that the response features as the natural linewidth of the atomic transition at off-resonant angles while the width is strongly altered by the cavity effect at mode angles. Data are quoted from Ref. [60].

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图 18 远场下不同出射角度的Co K_α荧光辐射强度, 在第1阶和第3阶腔模式角度下观测到了明显的荧光强度增强. 黑色实线为基于互易定理的模拟结果, 蓝色点线是使用X射线激发空穴态, 红色点线是使用电子束激发空穴态, 数据引自[58]

Figure 18. Far-field fluorescence intensities at different emission angles. The directional emission is observed at the first and third cavity modes. The blue and red dotted lines are experimental data resulted from X-ray excitation and electron beam excitation, respectively. The solid black line is the simulation based on the reciprocity theorem. Data are digitized from [58].

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