Relativistic free electrons based quantum physics

Li Jing; Liu Yun-Quan

doi:10.7498/aps.71.20221289

Abstract

The light-matter interaction is one of the fundamental research fields in physics. The electron is the first discovered elementary particle that makes up matter. Therefore, the interaction between electron and light field has long been the research interest of physicists. Electrons are divided into two kinds, i.e. bounded electrons and free electrons. The quantum transition of bounded electron system is constrained by the selection rules with the discrete energy levels, while the free electron systems are not. In the last decade, the experiments of photon-induced near-field electron microscopy (PINEM) have been demonstrated. The experimental setup of PINEM is based on ultrafast electron transmission microscopy (UTEM). The thoeritcal framworks have also been developed to describe the interaction between quantum free electrons and optical fields. Within macroscopic quantum electrodynamics, the concept of photon is extended to photonic quasi-particles. Solutions of maxwell's equations in medium that satisfy certain boundary conditions are called photonic quasiparticles, such as surface plasmon polaritons, phonon polaritons, or even magnetic field. The different dispersion relations of photonic quasi-particles produce abundant phenomena in the interaction between light and matter. The underlying information about the PINEM interaction can be inferred from the electron energy loss spectrum (EELS). It has been used for implementing the near-field imaging in its infancy. By now it is capable of not only realizing time-resolved dynamic imaging, reconstructing the dispersion relation of photonics crystal and its Bloch mode, but also measuring the mode lifetime directly. The PINEM has also been used to study free electron wavepacket reshaping, free electron comb, free electron attosecond pulse train, etc. Recently, this field has entered into the era of quantum optics, and people use PINEM to study novel phenomena in quantum optics, such as entanglement between free electrons and cavity photons, entanglement between free electrons and free electrons, free electron qubits, and preparation of novel light quantum states. In this paper, the theoretical and experimental development of free-electron quantum physics are reviewed. We have disscussed the application scenarios of quantum free electron system. The current difficulties and future development are envisaged.

Keywords:

photon-induced near field electron microscopy /
free electron /
quantum optics /
time-resolved imaging

Authors and contacts

State Key Laboratory for Mesoscopic Physics and Collaborative Innovation Center of Quantum Matter, School of Physics, Peking University, Beijing 100871, China

Corresponding author: Liu Yun-Quan, yunquan.liu@pku.edu.cn

Funds: Project supported by the Key R&D Program of China (Grant No. 2022YFA1604301) and the National Natural Science Foundation of China (Grant Nos. 92050201, 92250306).

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Cited By

图 1 电子、光子和倏逝场相互作用的物理描述^[33]及实验装置^[56]　(a)飞秒激光脉冲到达前($t < 0$), 电子就已经过纳米管时的情况, 目前二者还没有发生时空重叠; (b) QEW、飞秒激光脉冲和倏逝场在碳纳米管上具有最大重叠时($t = 0$); (c) 在相互作用期间和随后瞬间($t > 0$) , 电子获得/失去的能量等于单光子能量的整数倍. 插图是在飞秒激光脉冲中成像电子和光子之间的自由-自由跃迁, 在连续体中可能的最终能量. KE表示动能; (d) UTEM实验装置的实物图

Figure 1. Physical depiction of the interaction among the electron, photon, and the evanescent field^[33], and the picture of experimental set-up^[56]. (a) A frame when the electron packet arrives at the nanotube before the femtosecond laser pulse ($ t < 0 $), no spatial-temporal overlap has yet occurred. (b) The precise moment when the electron packet, femtosecond laser pulse, and evanescent field are at maximum overlap at the carbon nanotube. (c) Illustration of the process during and immediately after the interaction ($t > 0$) when the electron gains/loses energy equal to integer multiples of femtosecond laser photons. Inset, the possible final energies in the continuum due to the free-free transitions between the imaging electron and the photons in the femtosecond laser pulse. KE, kinetic energy. (d) Photograph of the UTEM.

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图 2 光学近场相干非弹性电子散射原理与示意图^[67]　(a) 实验示意图; (b) 非弹性电子散射谱中能量截断的光栅扫描图像; (c) 入射动能谱, 峰位于$ {E}_{0} =120\text{ keV} $, 其半高全宽(full-width at half-maximum, FWHM)为$0.7{\text{ eV}}$; (d) 梯状能级图, 箭头表示连续的多态布居转移(类型I)和量子干涉路径(类型II); (e) 近场相互作用后的能谱示例

Figure 2. Schematic and principles of coherent inelastic electron scattering by optical near-fields^[67]: (a) Experimental scheme; (b) raster-scanned image of the energy cutoff in the inelastic electron scattering spectra; (c) incident kinetic energy spectrum (full-width at half-maximum, $0.7{\text{ eV}}$) centered at ${E_0} = 120{\text{ keV}}$; (d) energy level diagram of ladder states, arrows indicate sequential multistate population transfer (type I) and interfering quantum paths (type II); (e) example of kinetic energy spectrum after the near-field interaction.

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图 3 电子和光子的纠缠模式^[35]　(a) 腔处于真空态时; (b) 腔处于相干态$ \left| {\left. \alpha \right\rangle = } \right|\left. 3 \right\rangle $时出现了丰富的纠缠现象; (c) 腔处于$| \alpha \rangle = $$ | {10} \rangle$且${g_{{\text{Qu}}}} = 0.25$时的弱耦合情况

Figure 3. Electron-photon entanglement patterns^[35]: (a) When the cavity is in a vacuum state; (b) rich entanglement features for an initial coherent state$|\alpha \rangle = |3\rangle$ in the cavity; (c) weak coupling and highly populated cavity (${g_{{\text{Qu}}}} = 0.25$, $\left| {\left. \alpha \right\rangle = } \right|\left. {10} \right\rangle $).

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图 4 (a) 根据量子电子波函数初始参数(${\sigma _{{z_0}}}, {L_{\text{D}}}$)对光与电子相互作用的分类. (b)—(e) PINEM、加速和APINEM在相互作用前(折线椭圆)后(红色和黄色表示正值, 蓝色表示负值) 的相空间表示及其能量(动量)分布. QEW的初始分布$ {W}^{\left(00\right)} $的面积是$h/2$, 单元格面积为普朗克常量$h$. PINEM情况下, 初始分布在时间方向扩展, 而其能量展宽较窄, 其中(b)为无啁啾, (c)为预啁啾; (d) 类粒子加速, 初始时间短且相位明确的QEW具有净动量转移; (e) APINEM情况. 初始量子电子束进行了强啁啾, 产生了量子干涉条纹^[70]

Figure 4. (a) Universal classification of light-matter interaction regimes in terms of the initial parameters of the quantum electron wave function: its minimal axial waist size ${\sigma _{{z_0}}}$ and the pre-interaction drift length from this point, ${L_{\text{D}}}$. (b)–(e) Illustrations of PINEM, acceleration, and APINEM processes in phase-space representation before (broken-line ellipses) and after (positive, red and yellow; negative, blue) interaction and their energy (momentum) distributions. The initial distributions $ {W}^{\left(00\right)} $ of the QEW of area $ h/2 $ are overlaid over a grid of area $h$ (Planck constant) tiles: PINEM case, the initial distribution is temporally (or longitudinally) expanded, and its energy spread is narrow: (b) unchirped; (c) prechirped; (d) particle-like acceleration with net momentum shift for an initially temporally short QEW with well-defined phase; (e) APINEM case. Expanded and strongly prechirped initial QEW, with quantum interference fringes emerging^[70].

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图 5 (a) 周期调制的电子脉冲结构的演化成阿秒脉冲, 其中电子密度作为近场相互作用后传播距离的函数(数值模拟采用$ \left|g\right| = 5.7 $); (b) 电子量子态在时间焦点位置处 (图(a)中传播距离为$1.8{\text{ mm}}$处)光调制一个周期的相空间表示; (c) Wigner函数的动量投影; (d) Wigner函数空间投影的中心部分, 脉冲持续时间(FWHM)仅为$82{\text{ as}}$^[67]

Figure 5. (a) A periodically modulated electron pulse structure evolved into attosecond pulse (electron density) as a function of the propagation distance after the near-field interaction (numerical simulation for $ \left|g\right| = 5.7 $); (b) phase space (Wigner) representation of one period of the light-modulated electron quantum state at the temporal focus position (propagation distance of $1.8{\text{ mm}}$ in panel (a)); (c) momentum projection of Wigner function exhibiting spectral modulations as observed in the experiments; (d) central part of spatial projection. A peak with a duration of only $82{\text{ as}}$ (FWHM) is produced^[67].

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图 6 (a) 拉姆齐型自由电子干涉仪的工作原理: 同一个电子脉冲(绿色)依次作用于空间分离的两个场${g_1}$和${g_2}$; (b) 具有两个相互作用区域的纳米结构的扫描电子显微图(俯视图和侧视图), 金片间距$5{\text{ μm}}$; (c)控制激发纳米结构的实验场景示意图; (d) 局域耦合强度$\left| {{g_{{\text{tot}}}}} \right|$的空间分布图像, 此时激励条件几乎满足角落区域对电子能谱进行完全再压缩^[76]

Figure 6. (a) Working principle of the Ramsey-type free electron interferometer: an electron pulse (green) is acted on at two spatially separated nodes ${g_1}$ and ${g_2}$; (b) scanning electron micrographs of the nanostructure featuring two interaction zones (top and side view), distance between gold paddles is $5{\text{ μm}}$; (c) sketch of the experimental scenario displaying polarization-controlled excitation of the nanostructure; (d) raster-scanned image of the local coupling strength $\left| {{g_{{\text{tot}}}}} \right|$ for excitation conditions near complete recompression in the corner region^[76].

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图 7 时域阿秒脉冲整形的模拟　(a) 模拟所得能谱图, 每个光周期中有$80{\text{ as}}$的时间聚焦, 约占光周期$3{\text{%}} $; (b) 相应的Wigner函数; (c) Wigner函数的时间投影显示出对密度函数的调制, 脉冲持续时间(FWHM)为$531{\text{ as}}$(减去阴影部分顶部的基线后, 脉冲持续时间有效值为$296{\text{ as}}$); (d) 对应的电子能谱(动量投影) ^[78]

Figure 7. Simulation of attosecond temporal reshaping a simulated spectrogram assuming: (a) Energy spectrum obtained by simulation, including a small timing jitter of 80 as (3% of the optical period); (b) corresponding Wigner function; (c) temporal projection of the Wigner function exhibits density modulations with a FWHM pulse duration of $531{\text{ as}}$ (after baseline subtraction, rms pulse duration: $296 \text{ as}$); (d) corresponding electron energy spectrum (momentum projection) ^[78].

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图 8 阿秒电子脉冲序列的实验演示　(a) 使用两个石墨薄片来制备阿秒电子脉冲序列的实验装置示意图, 插图为定制的TEM样品架; (b)多个光周期的实验光谱图(上)以及其中两个周期的特写(下); (c) 重构的Wigner函数; (d) Wigner函数的时间投影展示了对密度的调制, FWHM为$655{\text{ as}}$(减去阴影区域上部基线后, 脉冲有效持续时间$277{\text{ as}}$); (e) 对应的电子能谱(动量投影)^[78]

Figure 8. Experimental demonstration of attosecond electron pulse trains: (a) Sketch of the experimental set-up employing two graphite flakes for the preparation (upper plane) and characterization (lower plane) of attosecond electron pulse trains, and the inset is photograph showing the custom-built TEM sample holder; (b) experimental spectrogram recorded over multiple optical cycles (top) and close-up of two cycles (bottom); (c) reconstructed Wigner function; (d) temporal projection of the Wigner function exhibits density modulations with a FWHM of $655{\text{ as}}$ (after subtraction of a baseline indicated by the grey-shaded area; r.m.s. pulse duration of $277{\text{ as}}$); (e) corresponding electron energy spectrum (momentum projection) ^[78].

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图 9 自由电子与光激发的SPP相互作用的示意图　(a) 在银层中雕刻的一维纳米腔通过光照明产生SPP的示意图; (b) 旋转样品方向, 使得实验测量中电子-光相互作用为零时的能量滤波图像; (c) 实验测量中具有不可忽略的电子-光相互作用的能量过滤图像. 只有在后一种构型中, 传播光和SPP场才会在电子束的作用下产生位置相关的干涉, 从而产生一个空间振荡的场振幅, 可以在真实空间中成像^[10]

Figure 9. Visualization of propagating surface-plasmon polaritons: (a) Schematic representation of the generation of surface plasmon polaritons by optical illumination at the edge of a nanocavity carved in the Ag layer; (b) experimentally measured energy-filtered image for a sample orientation such to have a vanishing electron-light interaction; (c) experimentally measured energy-filtered image for a sample orientation such to have a non-negligible electron-light interaction. Only in the latter configuration a position-dependent interference of the propagating light and SPP fields as mediated by the electron beam occurs giving rise to a spatially oscillating field amplitude that can be imaged in real-space^[10].

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图 10 (a) 实验过程示意图; (b) 非局域全息法的示意图; (c) 电子-SPP相互作用后非弹性散射电子的空间分布, 比例尺为$2{\text{ μm}}$; (d) 利用文中详细介绍的半解析理论计算的实空间电子强度分布, 比例尺为$2{\text{ μm}}$; (e) FDTD模拟得到的界面总电场z分量的相位图(比例尺为$1{\text{ μm}}$)^[83]

Figure 10. (a) Schematic representation of the experimental geometry; (b) schematic representation of the non-local holographic method; (c) experimentally measured spatial distribution of the inelastically scattered electrons following the electron-plasmon interaction, scale bar, 2 μm; (d) calculated real-space electron intensity distribution using the semi-analytical theory detailed in the text (scale bar, 2 μm); (e) simulated phase map of the $z$ component of the total electric field at the interface obtained from FDTD simulations (scale bar, 1 μm)^[83].

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图 11 (a) UTEM中自由电子与光子腔的量子相互作用的五个自由度. (b)—(d) 光子晶体能带结构的重建与Bloch模的直接成像　(b)通过扫描入射光角度和波长而测得的能带结构; (c) 光子晶体和入射泵浦激光脉冲的示意图; (d) 在(b)中标记的角度和波长处测得的光子晶体的Bloch模式. 比例尺:$300{\text{ nm}}$^[54]

Figure 11. (a) The UTEM setup offers five degrees of freedom to measure the interactions. (b)–(d) Reconstruction of band structure and direct imaging of the Bloch modes of the photonic crystal: (b) Band structure measured by scanning over incident laser angles and wavelengths; (c) layout of the photonic crystal and incident pump laser pulse; (d) Bloch modes of the photonic crystal measured at the angles and wavelengths marked in panel (b). Scale bar, $300{\text{ nm}}$^[54].

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图 12 由UTEM观测二维极化激元波包^[55]　(a) 实验装置以及实验过程示意图; (b)样品的色散关系; (c) 自由电子探测hBN内部(TM偏振)传播的PhP波包, 插图为激光开启(左)和关闭(右)时的EELS; (d) 测量不同时间延迟${\tau _{\text{d}}}$下的电子的能量滤波, 显示了PhP波包的传播动力学

Figure 12. Direct observation of two dimensional (2D) polariton wave packets using UTEM^[55]: (a) Experimental setup and the process; (b) dispersion relation of the sample; (c) free electron probing the (TM polarized) propagating PhP wave packet inside the hBN, and the insets show EELS spectra with the laser on (left) and off (right); (d) measurement of the energy-filtered electrons for different time delays ${\tau _{\text{d}}}$ showing the propagation dynamics of the PhP wave packe.

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图 13 强耦合的纠缠特性, 由符合测量概率$ |c_{n, k}^{{\text{e - e}}}{|^2} $表示^[35]　(a) 耦合系数${g_{{\text{Qu}}}} = 1$; (b)耦合系数${g_{{\text{Qu}}}} = 3$

Figure 13. Electron-electron interaction for two distant electrons in a beam, mediated by long-lived photons^[35]. The color map ${\left| {c_{n, k}^{{\text{e - e}}}} \right|^2}$ is the coincident probability: (a) ${g_{{\text{Qu}}}} = 1$; (b) ${g_{{\text{Qu}}}} = 3$.

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图 14 用自干涉法测量电子密度矩阵^[37]　(a) 通过分束器后沿两条不同长度($z$和$z'$)的电子路径探索电子自相关的实验示意图; (b)—(i) 对不同的PINEM光场而言, 电子密度矩阵的实部(左)和虚部(右)作为两个电子各自的位移时间$\tau , \tau '$的函数

Figure 14. Measuring the electron density matrix through self-interference^[37]: (a) Sketch of an experimental arrangement to explore electron auto-correlation by means of a beam splitter and different lengths $\left( z \right.$ and $\left. {z'} \right)$ along the two electron paths before recombination at the detection region; (b)–(i) real (left panels) and imaginary (right panels) parts of the electron density matrix as a function of shifted times $\tau $ and $\tau '$ for different statistics of the PINEM light.

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图 15 集成在电子显微镜中的硅光子器件提供了有效的电子与连续波光的相互作用, 使量子光子统计的检测成为可能^[40]　(a) 透射电镜中电子波函数的连续波调制; (b) 硅-光子纳米结构(扫描电子显微镜图像), 包括一个布拉格镜和一个周期通道; (c) 与相干光态和热光态相互作用后的电子能谱; (d) 由测量光谱重构的相应光子统计量

Figure 15. A silicon-photonics device integrated in an electron microscope provides efficient electron interactions with CW light, enabling the detection of the quantum photon statistics^[40]: (a) CW modulation of electron wave functions in transmission electron microscopy; (b) silicon-photonic nanostructure (scanning electron microscope image), consisting of a Bragg mirror and a periodic channel; (c) electron energy spectrum after the interaction with two types of light states: coherent and thermal; (d) corresponding photon statistics reconstructed from the measured spectra.

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图 16 自由电子-光相互作用在电子能谱上刻印了光子的量子统计^[40]　(a) 电子行走者与光子进行连续的相互作用; (b) 电子行走理论与Q-PINEM理论完全匹配; (c) 相干态和(d) 热态的电子能谱随电场振幅的变化

Figure 16. Free-electron-light interactions imprint the quantum photon statistics on the electron energy spectra, demonstrating the transition from quantum walk to classical random walk of a free electron^[40]: (a) Electron walker performs consecutive interactions with the photons; (b) electron walker theory exactly matches with the Q-PINEM theory; lectron energy spectra for (c) coherent and (d) thermal states evolving with the electric field amplitude.

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图 17 利用Q-PINEM塑造具有新奇量子统计的光子态^[38]　(a) Q-PINEM相互作用的物理实现示意图; (b) 单次Q-PINEM相互作用的方案

Figure 17. Shaping photonic states of novel quantum statistics using Q-PINEM interactions^[38]: (a) Schematic for a physical realization of a Q-PINEM interaction; (b) interaction scheme of a single Q-PINEM interaction.

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图 18 最大对比度下的量子复苏和亚光周期电子显微示意图^[45]　(a) 阿秒电子脉冲的示意以及自由电子量子比特的概念; (b) 能谱和边带相位的演化; (c) 电子的时域波包脉冲; (d) 模拟的电子波包$|\varPsi {|^2}$; (e) 电子脉冲持续时间$\tau $(实线)和时间对比度(虚线)作为$L$的函数

Figure 18. Quantum revivals and sub-light-cycle electron microscopy at maximum contrast^[45]: (a) Concept for exploiting quantum revivals for generating attosecond electron pulses and qubits; (b) evolution of the energy spectrum and sideband phases; (c) wave packets and pulses in the time domain; (d) simulated quantum carpet $|\varPsi {|^2}$ of an electron wave packet; (e) electron pulse duration ${{\Delta }}\tau $ (solid) and temporal contrast (dashed) as a function of $L$.

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图 19 (a) 实验方案; (b) 相互作用后, 每个电子能态被纠缠到不同的光子态; (c) 对于不同的初始参数, 一次相互作用后对能量为${E_0} - \hbar \omega $ (即$k = 1$)的电子进行后选择的概率图; (d) 对能量为${E_0} - \hbar \omega $的电子(即$ k =1 $)进行后选择后两个光量子态之间纠缠熵的图^[47]

Figure 19. (a) A scheme of the proposed experiment; (b) each electron energy is entangled to a different photonic state after the interaction; (c) a map of the probability to post-select the electron with energy ${E_0} - \hbar \omega $ (i.e., $k = 1$) after one interaction for different initial parameters$;$ (d) a map of the entropy of entanglement between the two states of light after post-selecting on electrons with energy ${E_0} - \hbar \omega $ (i.e., $k = 1$)^[47].

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Vol. 71, Issue 23 - 2022-12-05

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