Entanglements of electrons and cavity-photons in the strong coupling regime

TL;DR

This paper explores the theoretical framework for achieving and understanding strong coupling and entanglement between free electrons and cavity photons, proposing potential applications in quantum communication.

Contribution

It introduces a unified model for strong electron-photon interactions and predicts non-Columbic entanglement between free electrons and photons, advancing quantum information science.

Findings

Predicted non-Columbic electron-photon entanglement

Unified model for strong-coupling electron-photon interactions

Potential for electron-based quantum communication

Abstract

This work sets a road-map towards an experimental realization of strong coupling between free-electrons and photons, and analytically explores entanglement phenomena that emerge in this regime. The proposed model unifies the strong-coupling predictions with known electron-photon interactions. Additionally, this work predicts a non-Columbic entanglement between freely propagating electrons. Since strong-coupling can map entanglements between photon pairs onto photon-electron pairs, it may harness electron beams for quantum communication, thus far exclusive to photons.

Equations

$$|\psi ⟩=\sum _{n=0}^{\infty }\sum _{j=-\infty }^{\infty }{c}_{nj}|E_j,n⟩.$$

$$\overrightarrow{\hat{\mathbf{E}}}\left(\mathbf{r},t\right)=\frac{1}{L^{3/2}}\sum _{\mathbf{k}}\sum _s\sqrt{\frac{\hslash \omega }{2{\epsilon }_0}}\left[i{\hat{a}}_{\mathbf{k},s}\left(0\right){\epsilon }_{\mathbf{k},s}{e}^{i\left(\mathbf{k}\cdot \mathbf{r}-\omega t\right)}+h.c.\right].$$

$$\overrightarrow{\hat{\mathbf{E}}}\left(\mathbf{r},t\right)={\overrightarrow{\hat{\mathbf{E}}}}^{(+)}\left(\mathbf{r},t\right)+{\overrightarrow{\hat{\mathbf{E}}}}^{(-)}\left(\mathbf{r},t\right)$$

$${\overrightarrow{\hat{\mathbf{E}}}}^{(+)}=\hat{a},$$

$$\mathcal{H}=\sqrt{E_{rest}^2+{\left(\hat{P}c\right)}^2}+\hslash {\omega }_0{\hat{a}}^{†}\hat{a}+\xi \left(\hat{b}{\hat{a}}^{†}+{\hat{b}}^{†}\hat{a}\right).$$

$${\varphi }_{\left(E,z\right)}$$

$$={\varphi }_{\left(E,0\right)}+\frac{z}{\hslash c}\sqrt{E^2-E_{rest}^2}$$

$$\stackrel{\left(E=E_0+\Delta E\right)}{\approx }{\varphi }_{\left(E,0\right)}+\frac{z}{\hslash c}\left(\sqrt{E_0^2-E_{rest}^2}+\frac{E_0\left(\Delta E\right)}{\sqrt{E_0^2-E_{rest}^2}}-\frac{E_{rest}^2{\left(\Delta E\right)}^2}{2{\left(E_0^2-E_{rest}^2\right)}^{3/2}}\right).$$

$$z_{dispersion}=\hslash c\cdot {\left|\frac{E_{rest}^2{\left(\Delta E\right)}^2}{2{\left(E_0^2-E_{rest}^2\right)}^{3/2}}\right|}^{-1}.$$

$${\theta }_f=\frac{\Delta P_x}{P_0}$$

$$\approx \frac{q|E_z|L}{v_0{P}_0}.$$

$${\theta }_f$$

$$\hat{S}=$$

$$=$$

$$\hat{b}(t){\hat{a}}^{†}(t)=\hat{b}{e}^{i\omega t}{\hat{a}}^{†}{e}^{-i\omega t}=\hat{b}{\hat{a}}^{†}.$$

$$D\left(\hat{b}\alpha \right)=\exp \left(\alpha \hat{b}{\hat{a}}^{†}-{\alpha }^{\ast }{\hat{b}}^{†}\hat{a}\right),$$

$$-i\frac{\xi \tau }{\hslash }$$

$$⟨E_{-k},n|D_{\left(\hat{b}\alpha \right)}|E_0,0⟩$$

$$=e^{\frac{{|\alpha |}^2}{2}}⟨E_{-k},n|\sum _{m,\ell =0}^{\infty }\frac{{\left(-{\alpha }^{\ast }\right)}^m{\left({\hat{b}}^{†}\right)}^m{\hat{a}}^m}{m!}\frac{{\alpha }^{\ell }{\hat{b}}^{\ell }{\left({\hat{a}}^{†}\right)}^{\ell }}{\ell !}|E_0,0⟩$$

$$=e^{\frac{{|\alpha |}^2}{2}}\sum _{m,\ell =0}^{\infty }⟨E_{-k-m},n+m|\frac{\sqrt{\left(n+m\right)!}}{\sqrt{n!}}\frac{{\left(-{\alpha }^{\ast }\right)}^m}{m!}\frac{{\alpha }^{\ell }}{\ell !}\sqrt{\ell !}|E_{-\ell },\ell ⟩.$$

$$=e^{\frac{{|\alpha |}^2}{2}}\sum _{m=0}^{\infty }\frac{\sqrt{\left(n+m\right)!}}{\sqrt{n!}}\frac{{\left(-{\alpha }^{\ast }\right)}^m}{m!}\frac{{\alpha }^m{\alpha }^n}{\left(n+m\right)!}\sqrt{\left(n+m\right)!}$$

$$=e^{\frac{{|\alpha |}^2}{2}}\frac{{\alpha }^n}{\sqrt{n!}}\sum _{m=0}^{\infty }\frac{{\left(-{\left|\alpha \right|}^2\right)}^m}{m!}=e^{-\frac{{|\alpha |}^2}{2}}\frac{{\alpha }^n}{\sqrt{n!}}{e}^{-{|\alpha |}^2}$$

$$=e^{-\frac{{|\alpha |}^2}{2}}\frac{{\alpha }^n}{\sqrt{n!}}=e^{-\frac{{|\alpha |}^2}{2}}\frac{{\alpha }^k}{\sqrt{k!}}.$$

$$P_k={\left|⟨E_{-k},k|D_{\left(\hat{b}\alpha \right)}|E_0,0⟩\right|}^2=e^{-{|\alpha |}^2}\frac{{\alpha }^{2k}}{k!}$$

$$|{\psi }_i⟩=|E_0⟩\otimes |\beta ⟩=|E_0,\beta ⟩.$$

$$|{\psi }_f^{\text{PINEM}}⟩=\sum _{n=0}^{\infty }\sum _{k=-\infty }^{\infty }{c}_{n,k}|E_k,n⟩.$$

$$c_{n,k}=⟨E_k,n|D_{\left(\hat{b}\alpha \right)}|E_0,\beta ⟩\stackrel{BCH}{=}$$

$$=e^{\frac{{|\alpha |}^2}{2}}⟨E_k,n|e^{-{\alpha }^{\ast }{\hat{b}}^{†}\hat{a}}{e}^{\alpha \hat{b}{\hat{a}}^{†}}|E_0,\beta ⟩$$

$$=e^{\frac{{|\alpha |}^2}{2}}⟨E_k,n|\sum _{m,\ell ,j=0}^{\infty }\underset{e^{-{\alpha }^{\ast }{\hat{b}}^{†}\hat{a}}}{\underbrace{\frac{{\left(-{\alpha }^{\ast }\right)}^m{\left({\hat{b}}^{†}\right)}^m{\hat{a}}^m}{m!}}}\underset{e^{\alpha \hat{b}{\hat{a}}^{†}}}{\underbrace{\frac{{\alpha }^{\ell }{\hat{b}}^{\ell }{\left({\hat{a}}^{†}\right)}^{\ell }}{\ell !}}}\underset{|\beta ⟩}{\underbrace{e^{-\frac{{|\beta |}^2}{2}}\frac{{\beta }^j}{\sqrt{j!}}}}|E_0,j⟩$$

$$=e^{\frac{{|\alpha |}^2-{|\beta |}^2}{2}}\sum _{m,\ell ,j=0}^{\infty }⟨E_{k-m},n+m|\sqrt{\frac{\left(n+m\right)!}{n!}}\frac{{\left(-{\alpha }^{\ast }\right)}^m}{m!}\frac{{\alpha }^{\ell }}{\ell !}\frac{{\beta }^j}{\sqrt{j!}}\sqrt{\frac{\left(j+\ell \right)!}{j!}}|E_{-\ell },j+\ell ⟩.$$

$$⟨E_{k-m},n+m|E_{-\ell },j+\ell ⟩$$

$$m=k+\ell ,n+m=n+k+\ell =j+\ell .$$

$$c_{n,k}=e^{\frac{{|\alpha |}^2-{|\beta |}^2}{2}}\sum _{\ell =0}^{\infty }\sqrt{\frac{\left(n+k+\ell \right)!}{n!}}\frac{{\left(-{\alpha }^{\ast }\right)}^{k+\ell }}{\left(k+\ell \right)!}\frac{{\alpha }^{\ell }}{\ell !}\frac{{\beta }^{n+k}}{\left(n+k\right)!}\sqrt{\left(n+k+\ell \right)!}.$$

$$\boxed{c_{n,k}=e^{\frac{{|\alpha |}^2-{|\beta |}^2}{2}}\frac{{\left(-{\alpha }^{\ast }\right)}^k{\beta }^{n+k}}{\sqrt{n!}}\sum _{\ell =0}^{\infty }\underset{(\ast )}{\underbrace{\frac{\left(n+k+\ell \right)!}{\left(n+k\right)!}}}\frac{{\left(-{\left|\alpha \right|}^2\right)}^{\ell }}{\left(k+\ell \right)!\ell !}}.$$

$${\left|\alpha \right|}^2$$

$$⟨n+k⟩$$

Full text

1. S.1 Basics of the coherent interaction between electronic and photonic states

2. S.1.1 Assumptions

3. S.1.2 Electron-dispersion effects

4. S.1.2.1 Relevant distance for electron-energy dispersion effects

5. S.1.2.2 Estimation of the electron recoil (transverse deflection)

6. S.1.3 The displacement operator - S-matrix approach

7. S.1.3.1 Derivation of the S-matrix as a displacement operator

8. S.2 Effects for photons interacting with an electron-beam

9. S.2.1 EELS as field-less PINEM - strong interaction without a driving field

10. S.2.2 PINEM - electron interaction with a strong laser-field

11. S.2.2.1 Retrieving the experimental PINEM spectrum for weak interactions with strong fields

12. S.2.2.2 Comments on PINEM with nearly classical fields

13. S.2.2.3 Separated expressions for gain and for loss channels of electron-photon interactions

14. S.3 Two-electron interaction mediated by cavity photons

15. S.4 Ladder operators for the relativistic electron

16. S.5 Quantitative evaluation of electron-fiber coupling

17. A Assisting derivations

S.1 Basics of the coherent interaction between electronic and photonic states

S.1.1 Assumptions

• •

The electron state is .

[TABLE]

The electronic part of the state may just be written as $\left|E_{j}\right>$, similar to Feist et al. 2015 [1].

• •

The ladder operators are $\hat{b},\hat{b}^{\dagger}$, with the commutation relation $\left[\hat{b},\,\hat{b}^{\dagger}\right]=0$. This commutation relation holds throughout relevant the energy spectrum. For fast electrons,$\left(\hat{b}\right)^{\dagger}=\hat{b}^{\dagger}$. For very slow electrons, the lowering and lifting operators are not the hermitian conjugate of each other. See details in section S.4.

• •

The electrons interact with a harmonic system, with energy spacing $\omega_{0}$, such that $E_{j+k}=E_{j}+k\hbar\omega_{0}$.

• •

For the photons the ladder operators are the standard $\hat{a}$ and $\hat{a}^{\dagger}$, with $\left[\hat{a},\,\hat{a}^{\dagger}\right]=1$.(See for example quantization in Scully and Zubairy [2] or Mandel and Wolf [3])

• •

The operator of the electric field for free space, is

[TABLE]

That can be written for simplicity as

[TABLE]

Here I just abbreviate

[TABLE]

where the time and space dependency are understood. The quantization of fiber modes has a similar form, see section S.5.

• •

The interaction Hamiltonian is $\mathcal{H}_{e}+\mathcal{H}_{np}+\mathcal{H}_{I}$.

The Hamiltonian for a local interaction is

[TABLE]

The exact form of the electron Hamiltonian, $\mathcal{H}_{e}$, and the operators $b,b^{\dagger}$ is described in section S.4. $\xi$ is the local coupling strength, with units of energy. It is meaningful only in the context of the total interaction strength, as e.g. in eq. (12).

S.1.2 Electron-dispersion effects

S.1.2.1 Relevant distance for electron-energy dispersion effects

The distances that are relevant for dispersion effects can be evaluated from a Taylor expantion of the phase for an electron plane-wave

[TABLE]

The dispersion can be neglected for short propagation distances, $z\ll z_{dispersion}$, where

[TABLE]

For example, electrons at 200 keV, and 11.65 eV bandwidth (10 orders of photons with 1064 nm vacuum-wavelength), $z_{dispersion}\approx 5.3\,mm$.

S.1.2.2 Estimation of the electron recoil (transverse deflection)

Since the optical mode has both parallel field ($E_{z}$ )and transverse field ($E_{x},E_{y}$) components, it can, in principle, deflect the electron and compromise the validity one-dimentional description, as in this work. For the optical-mode parameters described in this work, say in Figure 4 of the manuscript, this deflection is small. Given a similar field components, as in our case, one can assume for simplicity $|E_{x}|\approx|E_{z}|$. During the interaction time $L/v_{0}$, the transverse deflection can be estimated by the classical impulse is the added transverse momentum $\Delta P_{x}=q|E_{x}|(L/v_{0})$. $v_{0}$ is the group velocity of the relativistic electron.

Thus, the final deflection $\theta_{f}$ from the induced transverse momentum is

[TABLE]

using the phase-matched coupling $\alpha=\frac{qE_{z}L}{2\hbar\omega_{0}}$, $\hbar\omega_{0}=1.1eV$, $P_{0}=\gamma m_{0}v_{0}$, and the parameters for 200 keV electrons, $\gamma=1.39$, and $v_{0}=0.7c$, one can write

[TABLE]

The deflection roughly scales with the coupling, and is only few micro-radians. Assuming the acceleration is constant and the deflection trajectory is parabolic, this FINAL deflection correspond to a trajectory $x(z)=\frac{\theta_{f}}{L}\frac{z^{2}}{2}$, which result in a final deflection $x(z=L)\approx 0.2\,nm$ for $\alpha=1$ and $L=100\,\mu m$.

S.1.3 The displacement operator - S-matrix approach

The scattering operator (in the interaction picture), $\hat{S}$ need to be accounted for in full, to be valid for strong couplings,

[TABLE]

Here, I removed the time-ordering operator, $\mathcal{T}$, since the there is no time dependence for the operator product $\hat{b}\hat{a}^{\dagger}$ for interaction lengths short enough to suppress dispersion (see eq. (7))

[TABLE]

$\tau$ is an effective interaction duration. Additional phases, such as temporal-delays accumulated by the electron energy-states, can be neglected below the characteristic dispersion distance, as in eq. (7).

S.1.3.1 Derivation of the S-matrix as a displacement operator

Eq. (12) has the form of the displacement operator,

[TABLE]

with a substitution

[TABLE]

For this to be correct,$\frac{\xi\tau}{\hbar}\in\Re eal$.

Aditional comments:

• •

The only difference of $D\left(\hat{b}\alpha\right)$ from $D\left(\alpha\right)$ is that $\hat{b}$ is an operator. It implies conservation of energy, where every Fock-state is entangled with the corresponding state of an electron energy loss $\left|E_{j-n},n\right>$. The splitting of the energy between the electron channel and photon channel resembles a beam-splitter operator, but with $\hat{b}\hat{a}^{\dagger}$ instead of the $\hat{a}_{1}\hat{a}_{2}^{\dagger}$

• •

If $\frac{\xi\tau}{\hbar}\in\Re eal$, than the translation parameter, $\alpha$, is purely imaginary. This means that when the electrons interact with radiation they change momentum, without instantaneous shifts. Time (or propagation) translates momentum difference to modifications of the probability distribution.

S.2 Effects for photons interacting with an electron-beam

S.2.1 EELS as field-less PINEM - strong interaction without a driving field

A strong interaction depends on the coupling parameter, $\alpha$, with the light field acting only as the initial state. Due to conservation of energy, one expects to end with energy-loss, $E_{-k}$, with $k>0$. The probability amplitude to find an electron in energy $E_{-k}$ can be written as

[TABLE]

Here, I used the Backer-Campbell-Hausdorff formula (BCH) to express the displacement operator(eq. (136)), and used $\left(a^{\dagger}\right)^{m}\left|n\right>=\sqrt{\frac{\left(n+m\right)!}{n!}}\left|n+m\right>$. Using the orthogonality of the electron-energy states and the photon states, one get $\left<E_{-k-m},n+m|E_{-\ell},\ell\right>=\delta_{-k-m,-\ell}\delta_{n+m,\ell}$, so $n=k$ (conservation of energy) and $\ell=m+n$. So, the above is

[TABLE]

This is the expected Poisson distribution (see ref. [4])

[TABLE]

Typical EELS is retrieved for $|\alpha|^{2}\ll 1$, $RightarrowP_{0}\approx\left(1-|\alpha|^{2}\right)$, $P_{1}=|\alpha|^{2}$.

S.2.2 PINEM - electron interaction with a strong laser-field

For PINEM, the initial state $\left|\psi_{i}\right>$, before the electron interacts with light is an uncorrelated state,

[TABLE]

For large $\beta$ the optical coherent state is a good approximation for classical fields. The important quantum numbers for the final state are the final quanta of electron-energy gain, $k$, and the remaining number of photons $n$,

[TABLE]

The PINEM interaction can be written by the displacement operator.

[TABLE]

Orthogonality of the states imposes

[TABLE]

Thus, one remain with a summation over $\ell$,

[TABLE]

After some rearrangements, the final expression for the final-state amplitudes is

[TABLE]

This is an exact expression for the quantum state following PINEM, at any coupling strength. To extend this expression for gain ($k>0$) and for loss ($k<0$) one can replace the factorial operations by Riemann’s gamma function $x!\to\Gamma\left(x+1\right)$, which diverges for negative integers. Since possible negative factorials terms of $\left(k+\ell\right)!\ell!$ diverge in the denominator, their corresponding arguments can be ignored. For $\left(n+k\right)<0$, the term marked $\left(*\right)$, is either $\left(*\right)=1$ for $\ell=0$, or $\left(*\right)=0$ for $\ell>0$, and is thus regularized. Section S.2.2.3 retrieves the explicit PINEM coefficients for gain and loss, using the factorials of explicitly positive integers, rather than Riemann’s Gamma-function. One example of PINEM-like spectrum is in the main text (figure 2c), and here, figureS.1 presents similar spectrograms, for various coupling constants.

S.2.2.1 Retrieving the experimental PINEM spectrum for weak interactions with strong fields

Here, I show how the derivation above retrieves the known PINEM spectrum for the electron, and in what conditions the field is decoupled from the electron state modification. Using eq. (29), in the parameter regime accessible to experiments to date, this derivation should yield the a Bessel-function spectrum - $\propto\left|J_{k}\left(2\left|g\right|\right)\right|^{2}$, with a possible additional phase. In the experiments the coupling is weak, the field is strong, and there are only few quanta of energy exchanged between the photons and the electrons

[TABLE]

Although the summation is up to $\ell\to\infty$, the argument in the sum decays rapidly for $\ell>\left|\alpha\beta\right|^{2}$, so one can compare $\ell$ with other parameters of this system. The comparison to $\left|\beta\alpha\right|$ becomes clearer from eq. (34) and the definition of the Bessel-function of the first kind. By employing eq. (142), and the ratio $\ell/\left(n+k\right)\ll 1$, one can write $(*)\approx\left(n+k\right)^{\ell}$. In that case, the summation arguments acquire the following form

[TABLE]

where $g$ is

[TABLE]

This brings the form of the Bessel-function amplitudes to the energy spectrum,

[TABLE]

The above approximation almost reproduces the Bessel-like amplitudes of PINEM, but it leaves some correlations between $n$ and $k$. To remove these correlations and retrieve classical-field effects, one has to neglect correlations in the coherent state. Specifically when assuming $<j>=<n+k>\approx\left|\beta\right|^{2}$ and $\left\langle\sqrt{n+k}\right\rangle\approx\sqrt{\left\langle n+k\right\rangle}$ the following is simplified

[TABLE]

The photon states and electron states are now separable.

[TABLE]

I used here the relation $g=-g^{*}$, or $\arg{\left(g\right)}=\arg{\left(-g^{*}\right)}$ since $\alpha=-\alpha^{*}$.

S.2.2.2 Comments on PINEM with nearly classical fields

Some points from the above derivation of final state for strong fields interacting weakly with electrons are worth stressing:

• •

$g$ has the same meaning as for classical fields, as in Refs. [1, 5].

• •

The electronic states have the amplitude as in the experiments,

[TABLE]

not only the probabilities.

• •

$g$ is proportional to the electric field and the coupling constant, $g\propto\alpha\left|E\right|$, since $\left|E\right|\propto\sqrt{\left<n+k\right>}=\left|\beta\right|$. This is in agreement with the it’s classical definition.

• •

There is a phase locking between the initial coherent state and the electron state. It appears in the argument $e^{ik\left(\arg\left(\beta g\right)\right)}$.

• •

The locking phase just contributes an linear phase with the energy, that is, it provides for the definition of time-zero.

• •

$\left|g\right|=\left|\alpha\beta\right|$, which means that one can increase the width of the electron spectrum (have many PINEM orders, $\Delta E\propto 2\left|g\right|$). The scaling, for a given cavity, will be linear with the interaction length (via $\alpha$), and linear with the PINEM-driving electric field (via $\beta$).

• •

Since for a coherent state $\left|\beta\right>$, the coupling and $g$ are related by $g=\alpha|\beta|$, it means that the coupling, $\alpha$ can be retrieved from classical calculations of PINEM by

[TABLE]

• •

As mentioned in the main text, the equivalence of the gain and loss channel originates from a small coupling, $\left|\alpha\right|\ll 1$. The mean energy loss is $\left|\alpha\right|^{2}$ in any experimental configuration, EELS, PINEM, weak- or strong-coupling.

• •

In practice, the correlations between the photon states $n$ and the electron energy indices $k$ is negligible for a high-$\beta$ coherent state and weak coupling. This is visualy clear from the calculation in the main text.

S.2.2.3 Separated expressions for gain and for loss channels of electron-photon interactions

One reason to keep the factorials in eq.(29) is the numerical errors induced when evaluating factorials $x!$ through the Riemann Gamma function $\Gamma(x+1)$. Thus, for the calculation presented in the figures of the main text, I explicitly separated the expression to the cases of $k\geq 0$ and $k<0$. This is done by choosing the summation index that spans $0\to\infty$. That is, $\ell$ for $k\geq 0$ and $m$ for $k<0$. The other index is eliminated by the substitution

[TABLE]

Such a separation would explicitly assure that the physical constraint are met $m,\ell,j,n\geq 0$. First, for $k\geq 0$ we have eq. (29), with $\ell$ as the summation index. For $k<0$, the index selection in in eq. (48) is $\ell=m-k$ and $n=j-k$,

[TABLE]

The last expressions are retrieved by changing t $-k\to\left|k\right|$, and using $\alpha=-\alpha^{*}$. Thus, the two expressions for gain and loss, eqs. (29) and (51), respectively, can be combined

[TABLE]

Note that for the $k<0$ part, I just wrote the arbitrary summation index as $\ell$ instead of $m$, and used $-k=\left|k\right|$, while for the $k>0$ part, I substituted $k=\left|k\right|$.

S.3 Two-electron interaction mediated by cavity photons

After an interaction of an electron with a cavity, eq. (20) calculates the final state amplitudes,

[TABLE]

It is a coherent photonic state with parameter $\alpha_{1}$, the strength of the first interaction, and an electronic part that conserves a net energy $E_{0}$. I derive the interaction strength as $\alpha_{1}$ and $\alpha_{2}$ for the first and second electron, respectively, to be able to separate their contributions. Typically, one can take equal interaction strengths, $\alpha_{1}=\alpha_{2}$, as done in the main text. The loss index is changed here to ($s$) to differ the loss of the first electron from the gain index of the second electron, $k$ and the photons’ index $n$. In the manuscript the index $n$ instead of $s$ for brevity. I now consider a second electron with energy $\mathscr{E}_{0}=E_{0}$. The different symbol just marks a difference between the first and the second electrons. Thus, the initial electron-electron-photon state, before the second electron interacts with stored photons is

[TABLE]

The final state can be characterized by the individual electron-electron-photon states,

[TABLE]

The coefficients are given by the projection

[TABLE]

Note that here the operator $\hat{b}$ is acting on the second electron, leaving the first unchanged. The states’ orthogonality imposes

[TABLE]

Inserting the indices selection, and using the states’ orthogonality, eq. (61) is

[TABLE]

The relation $n+k=s$ allows the last equation to be written with as a function of the electron energies only, $k,s$. So, one can write the coefficients $c_{s,k}$

[TABLE]

Factorials of negative numbers diverge according to the Riemann’s Gamma function. The term $\sqrt{\left(s-k\right)!}\geq 0$ diverges for $s<k$. In other words, the highest k is the full conversion energy taken from the $1^{st}$ electron to the $2^{nd}$. Thus, this term nullifies the probability for a non-physical energy-gain of the second electron. Note that in the main text I chose to simplify the system by choosing $\alpha_{1}=\alpha_{2}=\alpha$, which is a realistic realization of eq. (69), when the two electrons share the same beam path and interact with the same cavity.

Similar to the discussion in section S.2.2.3, the indices selection $m,\ell$ differs for the gain- and loss-channels, as used in practice to calculate the 2-particle amplitudes numerically, with the substitutions

[TABLE]

First, for the case of energy gain by the second electron, $k>0$, the two-electron probabilty amplitudes are in eq. (69). For the case of energy loss by the second electron, $k<0$, the proper index to keep is $m$, with the relation $\ell=m-k=m+{\left|k\right|}$ from eq. (71).

[TABLE]

Finally, one can combine the expressions for the energy-gain and energy-loss for the second electron,

[TABLE]

Using the equality $-\alpha_{2}^{*}=\alpha_{2}$, setting $\left|k\right|$ appropriately, and using just either $\ell$ as a summation index, a more compact equation can be written

[TABLE]

Except for the spectrograms in the main text, for the coefficients $c_{s,k}$ I added some here, for different coupling strength, $\alpha=\alpha_{1}=\alpha_{2}$

There are several interesting examples to consider

Checking the limit of no initial gain – if the interaction strength of the first electron is nullified, $\left|\alpha_{1}\right|=0$, the interaction of the second electron should result in the spectrum of a single interaction. Only the coefficients with $s=0$ survive due to the factor $\alpha_{1}^{s}$, and the $\sqrt{s-k}$ term suggests that $k\leq 0$. The second electron therefore, populates no gain states. According to eq. (78), the coefficients $c_{s,k}$ will be

[TABLE]

which is a Poissonian probability distribution, just as in eq. (20) . 2. 2.

The coefficients retrieved from the two-electron interaction (eq. (78)), and for the PINEM interaction (eq. (55)) are equivalent, by just selecting indices. Since the first electron in the two-electron case induces a coherent state, the following stage, which is the interaction of a coherent state with an approaching electron is identical to strong-coupling PINEM. However, the important difference is the quantum numbers. For PINEM, the quantum numbers (in which one may search for entanglements) are $n,k$, while for two electrons, their energy states, $s,k$ are important. Thus, for physically relevant purposes, the two cases are sheared $n=s-k$. Such a shear can be identified by comparing figure 2 and figure 3 in the main text. One can also consider the $1^{st}$ electron spectrum as corresponding to the initial optical state, which differs from the final, non-coherent-state-like photon distribution. 3. 3.

Approaching strong-field PINEM for the $2^{nd}$ electron.

Similar to section S.2.2.2, $c_{s,k}$ can resemble the experimentally measured PINEM for weak coupling limit. For the two-electron case, that requires verri different coupling strengths, $\left|\alpha_{1}\gg 1\gg\left|\alpha_{2}\right|\right|$.For a large energy deposition in the cavity by the first electron, one can assume $\left(\ell+\left|k\right|\right),\ell\ll s$, and approximate

[TABLE]

The factorial approximation is justified in eq. (142). Considering the gain/loss dependent part of (78), including the term $\sqrt{\left(s-k\right)!}$, and changing $k$ to $\left|k\right|$ in a consistent manner for $k<0$ and $k\geq 0$, one can write the following equalities to combine the gain and loss parts

[TABLE]

Incorporating that into eq. (78) gives a separable equation. Same logic here applied as

[TABLE]

where, again, $g=\alpha_{2}\sqrt{s}\approx\alpha_{2}\left|\alpha_{1}\right|$, and this retrieves the experimental Bessel-amplitudes of PINEM. One should note that the above separability is naturally occurring for the quantum numbers $s,k$ since the first electron is unaffected by any detail of the intaraction with a second elctron. It already left the interaction region. In the PINEM case, where the final state is expressed with $n,k$ and $s=n+k$, the two states cannot be separated. A similar treatment leading to eq. (42) would result in $c_{n,k}=\left[e^{-\frac{\left|\beta\right|^{2}}{2}}\frac{\beta^{n+k}}{\sqrt{n+k}}\right]\left[\cdots\right]$, which is clearly not separable. For this reason, the quantum fluctuations remain here, while they have to be mitigated in eq. (42).

S.4 Ladder operators for the relativistic electron

• •

Assuming the energy is allowed in levels $\left|n\right>$, the Hamiltonian comply with $\hat{H}\left|n\right>=E_{n}\left|n\right>$.

• •

In this section, $n$ is the energy level-index of the electron, with respect to the zero-loss energy $E_{n=0}=E_{0}$. The number operator is

[TABLE]

For the zero-loss energy, $\hat{H}\left|0\right>=E_{0}\left|0\right>$. $E_{0}$ relates to the electron rest energy $E_{rest}$ and the zero-loss momentum $P_{0}$ by $E_{n=0}=\sqrt{E_{rest}^{2}+\left(P_{0}c\right)^{2}}$.

• •

For nearly plane-wave electrons, the ladder operators commute $[b^{\dagger},b]=0$. In that case, They cannot construct the Hamiltonian. The Hamiltonian is, by definition, sensitive to the level index, and hence cannot commute with a ladder operator, e.g. $\hat{H}\left(b^{\dagger}\left|n\right>\right)=E_{n+1}\left(b^{\dagger}\left|n\right>\right)\neq b^{\dagger}\left(\hat{H}\left|n\right>\right)=E_{n}b^{\dagger}\left(\left|n\right>\right)$.

• •

The momentum of state $\left|n\right>$ is $P_{0}+P_{n}$. Thus, it can be written as $\left|n\right>=\exp\left[{\frac{i}{\hbar}\left(P_{0}+P_{n}\right)}\right]$

• •

The dispersion relation for the electrons around the zero-loss energy is

[TABLE]

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For relativistic electrons, $P_{n}\ll P_{0}$, the Hamiltonian is linear with the momentum, $P_{n}$. The ladder operators can be written explicitly as

[TABLE]

with $\Delta k=\left(P_{n+1}-P_{n}\right)/\hbar=\omega_{0}/{v_{Zero-loss}}$.

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To show the ladder operators are correct, one needs to show is that $\hat{H}\left(b^{\dagger}\left|n\right>\right)=E_{n+1}\left(b^{\dagger}\left|n\right>\right)$

[TABLE]

Similarly, $\hat{H}\left(b\left|n\right>\right)=E_{n-1}\left(b\left|n\right>\right)$.

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Since $\hat{b},\hat{b}^{\dagger}$ are pure phasors, they reconstruct the relations in Ref. [1],

[TABLE]

This relation applies also for non-relativistic electrons if the underlying assumptions hold.

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Alternatively, the ladder operators can be constructed in a diagonal form

[TABLE]

based on the known values of the wave-vectors $k_{n}$. One can see that, especially for slow electrons $\hat{b}$ and $\hat{b}^{\dagger}$ are not exactly hermitian conjugates of each other. Hence, they may not be convenient for the representation of observables quantities. However, even at acceleration voltages of few keV, the dispersion becomes linear enough to allow the assumption that $\hat{b}$ and $\hat{b}^{\dagger}$ are complex conjugates.

S.5 Quantitative evaluation of electron-fiber coupling

This section is mostly technical, in the form of bullet-points that allows, with the sources to follow the quantitative estimation of the coupling constant (say, per $1\mu m$). It is based on Refs. [6, 7] and assisted by Prof. Elias N. Glytsis notes about Cylindrical Dielectric Waveguides, 2017. Numerical results from finite-elements calculation are in good agreement with the analytic calculation below. (see figure S.4)

In short, I calculate the field distribution for an $HE_{11}$ mode in a round , clad-less, step-index fiber, and estimate the field close to it’s surface, in vacuum. For a given fiber length of $1\,\mu m$, I calculate $g$ and employ eq. (46), to calculate $g$ per photon, that is, the coupling constant $\alpha$. The average number of photons in the classical field of the mode is evaluated as $<n>=U/\hbar\omega$, where $U$ is the total field’s energy, per $\mu m$, and $\hbar\omega$ is the photon energy.

I start with the basic form of the mode. An $HE_{11}$ mode is always guided in a fiber. It is typically given by the the electric and magnetic fields parallel to the fiber , $E_{z}$ and $H_{z}$

[TABLE]

$J_{\ell}(x)$, and $K_{\ell}(x)$ are the Bessel function of the first kind and the modified Bessel function of the second kind. An electron traversing parallel to the fiber will excite one linearly polarized mode, thus, radial function is $\sin\left(\phi\right)$. The normalization for the radial function is already included in the calculation of $A_{1}$.

First, I note that $E_{z}$ is the most relevant field component, as it determines $g$ by accelerating or decelerating the co-propagating electrons.

For $HE_{11}$, find the smallest solution of the propatation constant, $\beta$, from the equation for $\ell=1$.

[TABLE]

Here, $J^{\prime}_{\ell}(u)=\left.\frac{d}{dx}J_{\ell}(x)\right|_{x=u}$, and similarly for $K^{\prime}_{\ell}(w)$. From $\beta$, one finds $u$ and $w$. The remaining coefficients, $B_{1},F_{1},G_{1}$ (assuming an arbitrary $A_{1}=1$ for simplicity) are,

[TABLE]

Now one has a full expression for the mode’s fields. The next step is to find the number of photons $\left\langle n\right\rangle$ per $\mu m$. Once one calculates $g$ for classical field, the coupling constant is quantitatively retrieved from the classical-field calculation by $\left|alpha\right|=\frac{\left|g\right|}{\left|\beta\right|}=\frac{\left|g\right|}{\sqrt{\left\langle n\right\rangle}}$. In practice, one can choose to normalize $A_{1}$ per photon so that $\left\langle n\right\rangle=1$ results (just for the simplicity of the calculation, not for that actual physical case) in the direct form $\alpha=g$.

For a propagating mode, the energy is time-stationary and azimutally uniform, so only the radial distribution requires calculation, at a given time. I choose the time of maximal $E_{z}$, along the axis $\phi=0$. Thus, one can ignore field components that nullify along the axis of $\phi=0$, or those with a temporal phase shift $i$, since their quarter-cycle shift nullifies when $E_{z}$ maximal.

The Field components other than $E_{z}$ are

[TABLE]

The in-phase components, spatially and temporally are boxed. The others do not contribute. It is convenient to express the energy in terms of $\vec{E}$ and $\left(\mu_{0}\omega\vec{H}\right)$, since the factor $\mu_{0}\omega$ comes either from the above ratios or from eq. (112). The energy is

[TABLE]

Using the relations $\varepsilon_{0}=\left(\mu_{0}c^{2}\right)^{-1}$, one can write

[TABLE]

To normalize the fiber mode across some volume, we choose a fiber length $L\,[\mu m]$, with periodic boundary conditions, to allow for a unidirectional mode. This simplification can be easily taken into account in the cavity design, using the cavity effective length an any particular geometry. For example, the mode effective volume would be larger by a factor of $\sqrt{2}$ for a cavity encapsulated between two mirrors. I assume that the energy distribution is independent of $\phi$.

Using these fields, one can require that $A_{1}$ normalizes the energy to that of one photon,

[TABLE]

where one assumes the fields above were initially scaled according to eq. (109). Thus, $A_{1}$ is given by

[TABLE]

This integral is evaluated for the inner and outer segments, $\int_{0}^{a}$ and $\int_{a}^{\infty}$ using $\left(J_{1}(),n_{core}\right)$ and $\left(K_{1}(),n_{clad}=1\right)$ , respectively.

At this point one has the classical field of an $HE_{11}$ fiber mode with an average energy of one photon. To evaluate the coupling constant, one only needs to calculate $g$ via its definition in refs. [1, 5]

[TABLE]

for the electron trajectory $\left(z,t(z)\right)$ (see main text). Only $E_{z}(r>a)$ is relevant to accelerate/decelerate an electron in vacuum.

[TABLE]

The maximal relevant field is, available for electron coupling right at the fiber edge is

[TABLE]

For a phase-matched interaction the electron experience a time-independent field along its path $\left(z,t(z)\right)$, so

[TABLE]

Thus, the maximal PINEM interaction per photon, is on the surface of the fiber, under conditions of full-phase-matching is

[TABLE]

This is, quantitatively, the maximum electron-photon coupling

[TABLE]

One should note that the maximal coupling scales with the cavity length as

[TABLE]

, since the volume normalization included in $A_{1}$, scales as $1/\sqrt{L}$.

For a step index fiber of Si3N4-core in vacuum having length of $100\,\mu m$ with periodic boundary conditions, I calculated the coupling properties vs. the fiber diameter (figure S.3). The calculated properties are the optimal coupling, the distance for $e^{-1}$ decay of the field, the acceleration voltage for phase-matched electrons, and on the right axis, the coherence lengths for acceleration voltages of 200 keV and 300 keV. The calculations for the fields of the electromagnetic mode, and the normalization terms were varified using COMSOL Multiphysics${}^{\text{\textregistered}}$, shown on figure S.4.

Appendix A Assisting derivations

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**Explicit derivation of BCH

**According to BCH, for operators that obey $[X,Y]=const$

[TABLE]

where I just stressed the importance of the sign. For the displacement operator,

[TABLE]

which means that

[TABLE]

• •

Approximating $\left(N+\ell\right)!\approx N!\left(N\right)^{\ell}$ using Stirling’s Formula, $z!\approx\sqrt{2\pi z}\left(\frac{z}{e}\right)^{z}$. Taking the natural logarithm result in

[TABLE]

I will need to convert $ln\left(N+\ell\right)$ to a form with $ln\left(N\right)$, so explicitly

[TABLE]

which by Taylor expantion to the first order ,provides

[TABLE]

Expanding the factorial $\left(N+\ell\right)!$ according to the above,

[TABLE]

Neglecting terms of order $\left(\frac{\ell}{N}\right)^{2}$, this resuls in

[TABLE]

Within a correction of $e^{\left(\frac{\ell}{N}\right)^{2}}$, one gets,

[TABLE]

For simplification, in the approximation of $2\ell^{2}\ll N$, this leads to the final result

[TABLE]

Which we use to simplify the factorial terms in the electron-photon coupling and electron-electron coupling.