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## RESULTS AND DISCUSSION

Figure 1 represents a conceptual system to investigate coherent CL, which can be implemented within an electron microscope. Optical phase information from a PINEM-driving laser field is imprinted on and carried by the electron over a distance *z*, resulting in a coherent CL emission by interaction with an out-coupling sample system. Such electron transfer of optical coherence can be detected by an interferometric setting that targets either the linear or the nonlinear response of free electrons (Fig. 1, B and C, respectively).

We first consider comb-like electron-energy superposition states combined with the radiation vacuum ∣0〉∣ψin〉=∣0〉⊗∑j=−∞∞cj∣Ej〉=∑j=−∞∞cj∣Ej,0〉(1)where *c _{j}* are the complex probability amplitudes for electron states with energy

*E*, and the index

_{j}*j*runs over electron-energy levels. The coefficients

*c*are normalized as ∑

_{j}*∣*

_{j}*c*∣

_{j}^{2}= 1, and their phases vary with electron propagation in vacuum according to the free-particle dispersion. The quantum-optical and coherence properties of the CL from an electron state with a temporally modulated density depend on the interaction of the electron with an emitter. For simplicity, we consider a single nondegenerate optical band into which CL is emitted. General expressions for the quantum properties of the CL are derived in Materials and Methods, including the radiation continuum, as well as states with definite momentum, energy, and polarization. The photon frequency represents a good quantum number within the interaction bandwidth because it is a single-valued function of the longitudinal momentum. Transverse deflections can be neglected under the nonrecoil approximation (

*34*), which is valid for the examples that we discuss here. The quantum-optical properties of the CL emission can be described by the scattering operator, Ŝ, which, under the above conditions, has the form of a displacement operatorŜ=exp [∫0∞dω(gωb̂ωâω†−gω*b̂ω†âω)](2)as derived for classical and quantum fields (

*20*,

*36*). For every frequency, the scattering operator allows for annihilation and creation of a photon, marked by the operators âω and âω†, where energy is conserved by the corresponding electron-energy ladder operators b̂ω† and b̂ω, respectively. Here,

*g*

_{ω}accounts for the electron-photon coupling at the angular frequency ω, where the photon spectral density of the CL is given by ∣

*g*

_{ω}∣

^{2}. Similarly, ∣

*g*

_{ω}∣

^{2}would be the EELS spectral density in the absence of competing loss mechanisms (e.g., bulk plasmons and incoherent emissivity). The phase of

*g*

_{ω}is arbitrary, and

*g*

_{ω}can be chosen as a non-negative real-valued spectral function. However, in examples such as the radiation into normal modes of a fiber, it is convenient to impose a flat spectral phase on the photonic modes and place the spectral degree of freedom as a complex coupling function. A rigorous derivation of Eq. 2 and the conditions for which it applies are given in Materials and Methods (“Evolution operator” section).

The final quantum state ∣ψf〉=Ŝ∣ψin〉, which can be calculated for an arbitrary coupling strength, is used to obtain the properties of the CL emission (see the “Generalization for strong electron-photon coupling” section). One can gain some intuition from the first-order approximation in the weak electron-photon coupling regime, ∣*g*_{ω}∣^{2} ≪ 1, for which the final state is∣ψf〉≈∑jcj[∣Ej,0〉+∫0∞dωgω∣Ej−ħω,1ω〉](3)

The weak interaction has a small probability amplitude to create a photon with angular frequency ω, represented by state ∣1_{ω}〉, accompanied by a corresponding electron-energy loss. Notably, the CL intensity is unaffected by the specific electron superposition state〈n̂ω〉=〈âω†âω〉=∑j=−∞∞∣cj∣2∣gω∣2=∣gω∣2(4)(see derivation for continuous spectrum in the “Mean number of excitations after interaction” section). The expectation value for an operator refers to the final electron-photon state, 〈Ô〉≡〈ψf∣Ô∣ψf〉. Equation 4 complies with the current understanding of CL and, thus, provides for a solid scientific basis for the predictions in this work. In contrast to the wave function–independent photon emission probabilities, the expectation value for the electric field carries information on the electron temporal structure. The physical electric field at a particular frequency ω and time *t* = 0 is E→ω=〈E→̂ω〉 and can be represented as a sum of two complex components, E→̂ω=E→̂ω(+)+E→̂ω(−), with E→̂ω(−)=(E→̂ω(+))†. The field’s amplitude is proportional to the expectation value of a ladder operator〈Êω(+)〉∝〈âω〉(5)

The proportionality makes it convenient to represent the field with the ladder operator, âω≡âω,(t=0), since âω relates directly to the photon statistics (e.g., shot noise in an interferometer). This procedure is justified if the effects of the spatial distribution and the polarization of the field can be traced out, as in CL into single-mode fibers (see the exact expression in the “Mean electric field after interaction” section). Evaluating the photonic ladder operator for ∣ψ* _{f}*〉 in Eq. 3, we find〈âω〉=∑j,j′=−∞∞cj*cj′gω〈Ej,0∣Ej′−ħω,0〉(6)

We assume that the PINEM-driven state is a comb separated by the photon energy, ħω0, where the electron spectral density distribution is much narrower than the separation of the levels. Thus, one can use discrete level indices (*37*) and write 〈Ej∣Ej′−ħω〉=δEj,Ej′−ħω=δj′,j+n, where *n* is the energy exchange in terms of a harmonic of the fundamental ladder separation, n=ω/ω0. Substituting the Kronecker δ into Eq. 6 yields a simple expression for the field〈âω〉={gnω0∑j=−∞∞cj*cj+nω=nω0 ,n∈ℤ0otherwise(7)

Since *c _{j}* are the amplitudes of the energy states

*E*, they are proportional to the Fourier coefficients of the electron wave function, that is,

_{j}*c*≡ 〈

_{j}*E*∣ψ〉 ∝ ∫ ψ(

_{j}*t*)

*e*

^{ijω0t}

*dt*, which can be used to simplify the CL field〈Êω(+)〉∝〈âω〉=gωFT[∣ψ(t)∣2](ω)(8)

Incidentally, the temporal electron-probability amplitude, ψ(*t*), can be represented spatially along the propagation axis ψ˜(z)=ψ(t=z/v) using the electron group velocity *v*. Equation 8 is a central result of this paper, representing a general property of CL from a structured electron state (see detailed calculation in the “Mean electric field after interaction”…

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Read More:Optical coherence transfer mediated by free electrons