Double-superradiant cathodoluminescence

Alexey Gorlach, Ori Reinhardt, Andrea Pizzi, Ron Ruimy, Gefen Baranes, Nicholas Rivera, and Ido Kaminer

Phys. Rev. A 109, 023722 – Published 28 February 2024

Abstract

There exist two families of superradiance phenomena: relying on free electrons or on correlated emission by ensembles of atoms. Here we investigate emission from ensembles of atoms driven by coherently shaped free electrons. This interaction creates superradiance emerging from both the atoms’ and the electrons’ coherence—with emission intensity that scales quadratically in both the number of atoms and number of electrons. This phenomenon enables electrons to become probes of quantum correlations in matter, with high temporal and spatial resolution.

Received 26 August 2022
Accepted 26 January 2024

DOI:https://doi.org/10.1103/PhysRevA.109.023722

Physics Subject Headings (PhySH)

Electron beams & optics Light-matter interaction Superradiance & subradiance

Clusters Electrons

Electron microscopy

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Alexey Gorlach ^1,*, Ori Reinhardt ^1,*, Andrea Pizzi ^2,3,*, Ron Ruimy ¹, Gefen Baranes^1,3,4, Nicholas Rivera^3,4, and Ido Kaminer ¹

¹Department of Electrical Engineering and Solid State Institute, Technion–Israel Institute of Technology, 32000 Haifa, Israel
²Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom
³Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
⁴Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

^*These authors contributed equally to this work.

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Vol. 109, Iss. 2 — February 2024

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Figure 1
Double-superradiant cathodoluminescence. (a,b) Emission intensity of the nonsuperradiant atomic system after the interaction with unshaped ( $σ しぐま = 0.0$ ) and comb electrons ( $σ しぐま = 3.0$ ), respectively. The emission intensity from a single electron ( $N_{e} = 1$ ) does not depend on the electron shape. The atomic system is nonsuperradiant: the atoms are excited independently and are not in phase, leading to incoherent thermal emission. (c,d) Emission intensity of the superradiant atomic system after the interaction with unshaped ( $σ しぐま = 0.0$ ) and comb electrons ( $σ しぐま = 3.0$ ), respectively. The atomic emission from unshaped electron excitation has no phase, but the atoms are correlated. This leads to an incoherent Poissonian Wigner function of the emitted light. Comb electrons induce a coherent excitation. For a small number of comb electrons, the emission is a coherent state as shown in (d). However, a large number of comb electrons take the atomic system close to the fully excited state, and the emitted light gets close to a Fock state. The plots assume $g = 0.1$ and $N_{a} = 15$ [the coupling constant $g$ is defined below after Eq. (3)]. Note: the Wigner functions shown here are schematic and for illustrative purposes only.
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Figure 2
Analogy between electron-driven and light-driven superradiance. The excitations of the superradiant atomic system by a Fock and a coherent light state are analogous to the excitations by an unshaped and a comb electron, respectively. This analogy is correct because unshaped free electrons and Fock states of light entangle with the atomic system, while comb electrons and coherent light do not entangle with the atomic system. In cases of entanglement, the atomic system loses its coherence after the interaction (diagonal density matrix, top row). In cases of no entanglement, the information about the phases between different atomic states remains unchanged after the interaction (off-diagonal elements in the density matrix, bottom row).
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Figure 3
The effect of electron shaping on inducing correlations between atoms. (a) The electron energy spectrum before the interaction for different energy bandwidths $σ しぐま$ of the shaped electron. A wider bandwidth corresponds to stronger bunching. (b) The atomic density matrix after the electron interaction for different bandwidths $σ しぐま$ . (c) The average atomic excitation energy following an interaction with $N_{e}$ electrons as a function of their energy bandwidth $σ しぐま$ . The narrow energy bandwidth electron excitation scales as $N_{e}$ , while the wide energy bandwidth electron excitation scales as $N_{e}^{2}$ . In the limit of a perfect energy comb ( $σ しぐま \to \infty$ ), the excitation energy is the largest possible and can be computed exactly (dashed line). Parameters are as in Fig. 1.
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Figure 4
Energy of excitation of the atomic system for a shaped electron. (a) The electron comb has energy width $σ しぐま$ and is composed of many peaks of width $Δ でるた$ equispaced in energy by $ℏ ω おめが$ . (b) Average energy $E$ of the atomic system after the interaction with $N_{e} = 15$ comb electrons ( $σ しぐま = 3$ ), as a function of the detuning between the atomic transition ${ω おめが}_{0}$ and the energy period of the comb $ω おめが$ . The curve has a peak with width $2 Δ でるた$ . Therefore, the atoms’ excitation will effectively occur even in the presence of detuning, as long as $| {ω おめが}_{0} - ω おめが | / ω おめが ≪ Δ でるた$ (in fact, atoms are excited even if the detuning is large, although less effectively). Thus, the two-level model that we are using for the emitters will be correct when the emitter's energy levels have more than $Δ でるた$ separation. The average energy is normalized to the maximal energy of the excitation $E_{\max} = N_{a} ℏ {ω おめが}_{0}$ . (c) Average energy of the excitation as a function of the number of electrons $N_{e}$ and width of the comb $σ しぐま$ . We observe that comb electrons excite the system to much higher energies than the unshaped electrons do. (d) Average energy as a function of $σ しぐま$ for fixed $N_{e} = 15$ [this plot corresponds to the purple line in (c)]. (e,f) Average energy as a function of $N_{e}$ for two different widths $σ しぐま = 0$ and $σ しぐま = 3$ , corresponding to the red lines in (c). These plots show that for unshaped electrons the excitation is proportional to $N_{e}$ , while for comb electron the excitation energy is proportional to $N_{e}^{2}$ .
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Figure 5
Experimental schemes for measuring double-superradiant cathodoluminescence. Schematic setups of an ultrafast (a) transmission or (b) scanning electron microscope (TEM or SEM). A pulse of electrons can be photoinduced and shaped using a laser and a coupling structure, e.g., dielectric laser accelerator structures [61, 62] or a set of mirrors [52]. Each shaped electron pulse excites an ensemble of atoms. The atoms then relax through superradiant emission, which can be collected and characterized in various ways, such as a streak camera for combined spectral and temporal information. The electrons’ energy spectra can be simultaneously measured for additional information, as shown recently in [63, 64].
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Figure 6
The comparison of the numerically calculated average excitation energy (red line) as a function of electrons’ number $N_{e}$ and analytically obtained average energy (red line) according to Eq. (A13) for different $σ しぐま$ of the electron comb. The calculations show that the analytical formula completely coincides with numerics. The energy is normalized to $N_{\max} = N_{a} ℏ {ω おめが}_{0}$ .
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