Spontaneous and stimulated radiative emission of modulated free-electron quantum wavepackets—semiclassical analysis

Our model for the current of the electron that excites the radiation through Maxwell equations is a single electron model where the charge density of the electron wavepacket is the expectation value of the probability density of the electron wavefunction solution of a relativistic Schrodinger equation [35]. The space-charge plasma effects and inter-particle Coulomb interactions are thus neglected. We model the electron wavefunction after its emission from the cathode by a Gaussian wavepacket in momentum space

$$\begin{eqnarray}&&{\rm{\Psi }}(z,t)=\displaystyle \frac{1}{\sqrt{2\pi {\hslash }}}\int {dp}{{\rm{\Psi }}}_{p}\,{e}^{i({pz}-{E}_{p}t)/{\hslash }},\end{eqnarray} \tag{ 2 }$$

with the coefficient

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{p}={(2\pi {\sigma }_{{p}_{0}}^{2})}^{-1/4}\exp \left(-\displaystyle \frac{{(p-{p}_{0})}^{2}}{4{\sigma }_{{p}_{0}}^{2}}\right).\end{eqnarray} \tag{ 3 }$$

To evaluate the evolution of the wavefunction (equation (2)) in time and space, we use the Taylor expansion of the relativistic electron dispersion relation

$$\begin{eqnarray}&&{E}_{p}=c\sqrt{{m}_{0}^{2}{c}^{2}+{p}^{2}}\approx {E}_{{p}_{0}}+{v}_{0}(p-{p}_{0})+\displaystyle \frac{{(p-{p}_{0})}^{2}}{2{m}^{* }},\end{eqnarray} \tag{ 4 }$$

where ${E}_{{p}_{0}}={\gamma }_{0}{{mc}}^{2},{p}_{0}={\gamma }_{0}{{mv}}_{0},{m}^{* }={\gamma }_{0}^{3}m$ are the central averaged energy and momentum and the effective longitudinal mass of electron wavepacket and ${\gamma }_{0}=1/\sqrt{1-{\beta }_{0}^{2}},{\beta }_{0}={v}_{0}/c$. Substitution of (4) and (3) in (2) results in [19, 35]:

$$\begin{eqnarray}&&{\rm{\Psi }}(z,t)=\displaystyle \frac{{e}^{i({p}_{0}z-{E}_{{p}_{0}}t)/{\hslash }}}{{(2\pi {\sigma }_{{z}_{0}}^{2})}^{1/4}\sqrt{1+i\xi t}}\exp \left(\displaystyle \frac{{(z-{v}_{0}t)}^{2}}{4{\sigma }_{{z}_{0}}^{2}(1+i\xi t)}\right)\end{eqnarray} \tag{ 5 }$$

with $\xi ={\hslash }/2{m}^{* }{\sigma }_{{z}_{0}}^{2}$ and the initial wavepacket 'waist' ${\sigma }_{{z}_{0}}={\hslash }/2{\sigma }_{{p}_{0}}$.

The expectation value of the free drifting electron current density can be written in terms of the expectation value of the electron probability density

$$\begin{eqnarray}&&{\bf{J}}=-\displaystyle \frac{e{\hslash }}{2{m}^{* }}({{\rm{\Psi }}}^{* }{\rm{\nabla }}{\rm{\Psi }}-{\rm{\Psi }}{\rm{\nabla }}{{\rm{\Psi }}}^{* })\simeq -{{ev}}_{0}| {\rm{\Psi }}({\bf{r}},t){| }^{2}.\end{eqnarray} \tag{ 6 }$$

In our 1D model, the axial current is given by

$$\begin{eqnarray}&&{J}_{0}(z,t)=-{{ev}}_{0}| {\rm{\Psi }}(z,t){| }^{2}=-{{ev}}_{0}{f}_{e,\perp }({{\bf{r}}}_{\perp }){f}_{e,z}(z-{v}_{0}t),\end{eqnarray} \tag{ 7 }$$

where

$$\begin{eqnarray}&&{f}_{e}(z-{v}_{0}t)=\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{z}(t)}\exp \left(\displaystyle \frac{{(z-{v}_{0}t)}^{2}}{2{\sigma }_{z}^{2}(t)}\right),\end{eqnarray} \tag{ 8a }$$

$$\begin{eqnarray}&&{\sigma }_{z}(t)={\sigma }_{{z}_{0}}\sqrt{1+{\xi }^{2}{t}^{2}},\end{eqnarray} \tag{ 8b }$$

and the transverse expansion of the transverse profile function ${f}_{e,\perp }({{\bf{r}}}_{\perp })$ is neglected approximately in our one-dimensional modelling.

Now transform the coordinates frame, so that the origin z = 0 is at the entrance to the interaction region and the electron wavepacket arrives there at time t_0e after drift time t_D, and further assume that the electron wavepacket dimensions hardly change along the interaction length ${\sigma }_{z}={\sigma }_{z}({t}_{D})$, or correspondingly the duration ${\sigma }_{t}={\sigma }_{t}({L}_{D})$ in time, then we can write

$$\begin{eqnarray}&&{J}_{0}(z,t)=-{{ev}}_{0}{f}_{e,\perp }({{\bf{r}}}_{\perp }){f}_{e,z}(t-{t}_{0}-z/{v}_{0}),\end{eqnarray} \tag{ 9a }$$

$$\begin{eqnarray}&&{f}_{e}(t)=\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{t}({L}_{D})}\exp \left(\displaystyle \frac{{t}^{2}}{2{\sigma }_{t}^{2}({L}_{D})}\right),\end{eqnarray} \tag{ 9b }$$

$$\begin{eqnarray}&&{\sigma }_{t}({L}_{D})={\sigma }_{{t}_{0}}\sqrt{1+{(\xi /{v}_{0})}^{2}{L}_{D}^{2}},\end{eqnarray} \tag{ 9c }$$

Further elaboration is required for describing the wavepacket evolution at the modulation region in the case shown in figure 2 and the subsequent drift hereafter (see appendix A). In this case, the electron wavepacket undergoes multi-photon emission/absorption processes in the short near-field modulation region, and its quantum wavefunction gets modulated in momentum space at harmonics of the photon recoil momentum $\delta p={\hslash }{\omega }_{b}/{v}_{0}$, where ω_b is the frequency of the modulating laser. After the modulation point, the momentum coefficient of electron wavepacket is obtained

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{p}={(2\pi {\sigma }_{{p}_{0}}^{2})}^{-1/4}\displaystyle \sum _{n=-\infty }^{\infty }{{ \mathcal J }}_{n}(2| g| )\exp \left(-\displaystyle \frac{{(p-{p}_{0}-n\delta p)}^{2}}{4{\sigma }_{{p}_{0}}^{2}}\right)\end{eqnarray} \tag{ 10 }$$

where $2| g| \approx (e/{\hslash }{\omega }_{b}){\int }_{0}^{{L}_{I}}{dzF}(z)$ is the averaged exchanged photon number gained from the near-field with the slow-varying spatial distribution F(z) of the near field of the tip illuminated by the laser and the effective interaction length L_I of optical modulation regime. The modulation amplitude of the n-th order multiphoton process is characterized by the Bessel function and the Gaussian envelope of momentum width ${\sigma }_{{p}_{0}}$, shifted relative to central momentum to ${p}_{0}+n\delta p$.

As in the case of the modulated wavepacket, we obtain the spatial evolution of the wavepacket in real space away from the tip (z > L_C) by substituting (10) in (2)

$$\begin{eqnarray}&&{\rm{\Psi }}(z,t)=\displaystyle \frac{{e}^{i({p}_{0}z-{E}_{{p}_{0}}t)/{\hslash }}}{{(2\pi {\sigma }_{{z}_{0}}^{2})}^{1/4}\sqrt{1+i\xi t}}\displaystyle \sum _{n=-\infty }^{\infty }{{ \mathcal J }}_{n}(2| g| )\exp \left(\displaystyle \frac{{(z-{v}_{0}t-n\delta {pt}/2{m}^{* })}^{2}}{4{\sigma }_{{z}_{0}}^{2}(1+i\xi t)}\right){e}^{i\tfrac{n{\omega }_{b}}{{v}_{0}}(z-{v}_{0}t-n\delta {pt}/2{m}^{* })}\end{eqnarray} \tag{ 11 }$$

The evolution of the modulated wavepacket along the drift section is best presented by Wigner distribution in phase space with respect to the relative position $\zeta ={z}_{0}-{v}_{0}t$ and the shifted momentum $p^{\prime} =p-{p}_{0}$ that is defined as

$$\begin{eqnarray}&&W(z,p^{\prime} ,t)=\displaystyle \frac{1}{2\pi }{\int }_{q}{{\rm{\Psi }}}_{p^{\prime} -q/2}^{* }{{\rm{\Psi }}}_{p^{\prime} +q/2}\exp (-i({E}_{p^{\prime} +q/2}-{E}_{p^{\prime} -q/2})/{\hslash }){e}^{{iqz}/{\hslash }}{dq}\end{eqnarray} \tag{ 12 }$$

Figure 3 shows the Wigner distribution $W(\zeta ,p^{\prime} ,{t}_{D}^{{\prime} })$ after drift time ${t}_{D}^{{\prime} }={L}_{D}^{{\prime} }/{v}_{0}$, where ${L}_{D}^{{\prime} }={L}_{D}-{L}_{c}$ is the drift length from the modulation point to the grating (see figure 2). We also show the projected density distributions in both momentum and spatial spaces in figure 4. The shown distribution parameters are ${\beta }_{0}=0.7,2| g| =11.4$ in correspondence to Feist's experiment [30].

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The Wigner function in phase-space demonstrates the turning of momentum modulation into the tight density micro-bunching at an estimated optimal drift time

$$\begin{eqnarray}&&{t}_{D,\max }^{{\prime} }\approx \displaystyle \frac{1}{2}\displaystyle \frac{{T}_{b}}{{\rm{\Delta }}{p}_{m}/{p}_{0}}\end{eqnarray} \tag{ 13 }$$

where T_b = 2π/ω_b is the bunching period, and the maximal effective momentum gain ${\rm{\Delta }}{p}_{m}\approx 2| g| \delta p$ depends on the exchanged photon number at the modulation point. Note that the momentum spectrum does not vary with drift propagation, thus we cannot reveal the evolved micro-bunched structure of the modulated wavepacket by measuring its momentum spectrum alone. Finally, the probabilistic expectation of the current density of the modulated wavepacket (equation (7)) after drift length ${L}_{D}^{{\prime} }$, is found to be periodically modulated in time and space as shown in figure 3, which displays tight and narrow micro-bunching at the maximal bunching drift time ${t}_{D,\max }^{{\prime} }$ with width $2{\sigma }_{b}\simeq 75\mathrm{as}$ (see in the insert). Classically, we can write then the current density distribution in the interaction region, defined in the coordinates range $0\lt z\lt L$

$$\begin{eqnarray}&&J(z,t)=-{{ev}}_{0}| {\rm{\Psi }}(z,t){| }^{2}=-{{ev}}_{0}{f}_{e,\perp }({{\bf{r}}}_{\perp }){f}_{e,z}(t-{t}_{0}-z/{v}_{0}){f}_{\mathrm{mod}}(t-z/{v}_{0})\end{eqnarray} \tag{ 14 }$$

where t₀ is the modulation reference time at the entrance of the interaction region z = 0. Assuming again that the wavepacket dimensions hardly change along the interaction region, the envelop functions ${f}_{e}(t)={(\sqrt{2\pi }{\sigma }_{t}({L}_{D}))}^{-1}{e}^{-{t}^{2}/2{\sigma }_{t}^{2}({L}_{D})}$ are the unmodulated wavepacket envelope with ${\sigma }_{t}={\sigma }_{z}/{v}_{0}$. Since ${f}_{\mathrm{mod}}(t)$ is periodic with period T_b, it can be expanded as a Fourier series in terms of the harmonics of the bunching frequency

$$\begin{eqnarray}&&{f}_{\mathrm{mod}}(t)=\displaystyle \sum _{l=-\infty }^{\infty }{B}_{l}{e}^{-{il}{\omega }_{b}t}\end{eqnarray} \tag{ 15 }$$

where l denotes the lth order harmonic. Substituting (11) in (14) we derive in appendix A the coefficient B_l of the Fourier series expansion after a drift time ${t}_{D}^{{\prime} }$ away from the modulation tip

$$\begin{eqnarray}&&{B}_{l}=\displaystyle \sum _{n=-\infty }^{\infty }{{ \mathcal J }}_{n}(2| g| ){{ \mathcal J }}_{n+l}(2| g| )\exp \left(-\displaystyle \frac{{(n\delta {t}_{D}^{{\prime} })}^{2}}{4{\sigma }_{{t}_{0}}^{2}(1+i\xi {t}_{D})}-\displaystyle \frac{{((n+l)\delta {t}_{D}^{{\prime} })}^{2}}{4{\sigma }_{{t}_{0}}^{2}(1-i\xi {t}_{D})}\right){e}^{i(2n+l)l{\omega }_{b}{t}_{D}^{{\prime} }+{il}{\phi }_{0}}\end{eqnarray} \tag{ 16 }$$

where $\delta ={\hslash }{\omega }_{b}/{m}^{* }{v}_{0}^{2}$ and we kept here the dependence on the initial phase ϕ₀, which is important for the subsequent extension of the analysis to the multi-particle beam case, where all particle wavepackets are modulated coherently. Noted that t_D is the drift time from cathode and ${t}_{D}^{{\prime} }$ is that from the tip after modulation.

We now turn to calculate the radiation emission by the current of a beam of electron quantum wavepackets. We base our analysis on a general radiation-mode excitation formulation for bunched beam superradiance [18]. The radiation field excited by a general finite pulse of current ${\bf{J}}({\bf{r}},\omega )$ is expanded in the frequency domain in terms of a set of orthogonal directional transverse modes $\{{\tilde{{ \mathcal E }}}_{q}({\bf{r}}),{\tilde{{ \mathcal H }}}_{q}({\bf{r}})\}$ that are the transversely confined homogeneous solution of the electromagnetic wave equations of free space or a source-less guiding structure

$$\begin{eqnarray}&&\{{\bf{E}}({\bf{r}},\omega ),{\bf{H}}({\bf{r}},\omega )\}=\displaystyle \sum _{q}{C}_{q}(\omega ,z)\{{\tilde{{ \mathcal E }}}_{q},{\tilde{{ \mathcal H }}}_{q}\},\end{eqnarray} \tag{ 17 }$$

where ${C}_{q}(\omega ,z)$, the slowly growing field amplitude of a radiation mode q at spectral frequency ω along the propagation direction (z) is derived from Maxwell equations [18]. The increment of the field amplitude of mode q (see appendix B) is

$$\begin{eqnarray}&&{\rm{\Delta }}{C}_{q}={C}_{q}^{\mathrm{out}}(\omega )-{C}_{q}^{\mathrm{in}}(\omega )=-\displaystyle \frac{1}{4{{ \mathcal P }}_{q}}\int {\bf{J}}({\bf{r}},\omega )\cdot {\tilde{{ \mathcal E }}}_{q}^{* }({\bf{r}}){d}^{3}{\bf{r}},\end{eqnarray} \tag{ 18 }$$

where ${{ \mathcal P }}_{q}=\tfrac{1}{2}{\mathfrak{R}}\iint ({\tilde{{ \mathcal E }}}_{q}\times {\tilde{{ \mathcal H }}}_{q})\cdot {\hat{{\bf{e}}}}_{z}{d}^{2}{{\bf{r}}}_{\perp }$ is the normalization power.

The spectral radiative energy emission per mode q, derived from Parseval theorem, is given (for ω > 0) by

$$\begin{eqnarray}&&\displaystyle \frac{{{dW}}_{q}}{d\omega }=\displaystyle \frac{2}{\pi }{{ \mathcal P }}_{q}| {C}_{q}^{\mathrm{out}}(\omega ){| }^{2}.\end{eqnarray} \tag{ 19 }$$

Substituting (18) in (19), the emitted spectral radiative energy per mode can be written in terms of three parts:

$$\begin{eqnarray}&&\displaystyle \frac{{{dW}}_{q}}{d\omega }={\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{in}}+{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{SP}/\mathrm{SR}}+{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{ST}-\mathrm{SR}}\end{eqnarray} \tag{ 20 }$$

with, repectively,

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{in}}=\displaystyle \frac{2}{\pi }{{ \mathcal P }}_{q}| {C}_{q}^{\mathrm{in}}(\omega ){| }^{2},\end{eqnarray} \tag{ 21a }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{SP}/\mathrm{SR}}=\displaystyle \frac{2}{\pi }{{ \mathcal P }}_{q}| {\rm{\Delta }}{C}_{q}(\omega ){| }^{2}=\displaystyle \frac{1}{8\pi {{ \mathcal P }}_{q}}{\left|\int {\bf{J}}({\bf{r}},\omega )\cdot {\tilde{{ \mathcal E }}}_{q}^{* }({\bf{r}}){d}^{3}{\bf{r}}\right|}^{2},\end{eqnarray} \tag{ 21b }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{ST}/\mathrm{ST}-\mathrm{SR}}=\displaystyle \frac{4{{ \mathcal P }}_{q}}{\pi }{\mathfrak{R}}\{{C}_{q}^{\mathrm{in}* }(\omega ){\rm{\Delta }}{C}_{q}(\omega )\}=-\displaystyle \frac{1}{\pi }{\mathfrak{R}}\left\{{C}_{q}^{\mathrm{in}* }(\omega )\int {\bf{J}}({\bf{r}},\omega )\cdot {\tilde{{ \mathcal E }}}_{q}^{* }({\bf{r}}){d}^{3}{\bf{r}}\right\},\end{eqnarray} \tag{ 21c }$$

where the first term is the spectral energy of the input radiation wave, the second term corresponds to radiation emission independent of input wave—random spontaneous (SP) or coherent (superradiant) (SR), the third term is stimulated emission/stimulated-superradiance (ST/STSR) corresponding to stimulated interaction (emission/absorption) of the input radiation wave ${C}_{q}^{\mathrm{in}}(\omega )$ and the spectral component of the current. A detailed derivation of the Radiation mode expansion formulation is given in appendix B.

First we consider the simple case of the probabilistic current of a single electron-wavepacket (equation (6)) in the Fourier transform frequency domain

$$\begin{eqnarray}&&{\bf{J}}({\bf{r}},\omega )={\int }_{-\infty }^{\infty }{\bf{J}}({\bf{r}},t){e}^{i\omega t}{dt}\end{eqnarray} \tag{ 22 }$$

We use equation (9) for the current of an unmodulated electron wavepacket, and then, after Fourier transformation (using ${ \mathcal F }\{{e}^{-{t}^{2}/2{\sigma }_{t}^{2}}/\sqrt{2\pi }\}={e}^{-{\omega }^{2}{\sigma }_{t}^{2}/2}$), substitute it into (18)

$$\begin{eqnarray}&&{\rm{\Delta }}{C}_{q}(\omega )=\displaystyle \frac{e}{4{{ \mathcal P }}_{q}}{\tilde{M}}_{q}(\omega ){e}^{-{\sigma }_{t}^{2}{\omega }^{2}/2}{e}^{i\omega {t}_{0e}}\end{eqnarray} \tag{ 23 }$$

where t_0e is the enterance injection time of the wavepacket into the interaction regime, and we defined here an overlap integral parameter (analogous to 'matrix element' in spatial space)

$$\begin{eqnarray}&&{\tilde{M}}_{q}(\omega )=\int {f}_{e}({{\bf{r}}}_{\perp }){\tilde{{ \mathcal E }}}_{{qz}}^{* }({{\bf{r}}}_{\perp }){e}^{{iz}\omega /{v}_{0}}{d}^{3}{\bf{r}}\end{eqnarray} \tag{ 24 }$$

Substituting (23) in (21a), we get the expressions for the spontaneous and stimulated emission of a single wavepacket:

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{{\rm{e}},\mathrm{SP}}=\displaystyle \frac{{e}^{2}}{8\pi {{ \mathcal P }}_{q}}{| {\tilde{M}}_{q}(\omega )| }^{2}{e}^{-{\sigma }_{t}^{2}{\omega }^{2}},\end{eqnarray} \tag{ 25a }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{{\rm{e}},\mathrm{ST}}=-\displaystyle \frac{e}{\pi }{\mathfrak{R}}\{{C}_{q}^{\mathrm{in}* }(\omega ){\tilde{M}}_{q}(\omega ){e}^{i\omega {t}_{0e}}\}{e}^{-{\sigma }_{t}^{2}{\omega }^{2}/2},\end{eqnarray} \tag{ 25b }$$

and we see right away that both spontaneous and stimulated emission expressions are wavepacket-size-dependent in the semi-classical regime, and vanish in the quantum wavepacket limit $\omega {\sigma }_{t}\gg 1$ (equation (1)). We further reduce these expressions in the case where the axial component of the radiation mode is a traveling wave of wavenumber q_z

$$\begin{eqnarray}&&{\tilde{{ \mathcal E }}}_{{qz}}(z,{{\bf{r}}}_{\perp })={\tilde{{ \mathcal E }}}_{{qz}}({{\bf{r}}}_{\perp }){e}^{{{iq}}_{z}z}\end{eqnarray} \tag{ 26 }$$

Then the axial integration in (equation (24)) can be carried out

$$\begin{eqnarray}&&{\tilde{M}}_{q}(\omega )={\tilde{M}}_{q,\perp }{{ \mathcal E }}_{{qz}0}{{Le}}^{i\theta L/2}{\bf{sinc}}(\theta L/2)\end{eqnarray} \tag{ 27 }$$

where we defined a normalized coefficient ${\tilde{M}}_{q,\perp }={({{ \mathcal E }}_{{qz}0})}^{-1}\int {f}_{e,\perp }({{\bf{r}}}_{\perp }){\tilde{{ \mathcal E }}}_{{qz}}^{* }({{\bf{r}}}_{\perp }){d}^{2}{{\bf{r}}}_{\perp }$ describing the transverse overlap between the field of the radiation mode and the electron wavepacket and ${{ \mathcal E }}_{{qz}0}=| {\tilde{{ \mathcal E }}}_{{qz}}^{* }({{\bf{r}}}_{\perp ,0})|$ with ${{\bf{r}}}_{\perp ,0}$ the transverse coordinate of the center of the wavepacket profile. The parameter

$$\begin{eqnarray}&&\theta =\displaystyle \frac{\omega }{{v}_{0}}-{q}_{z}(\omega )\end{eqnarray} \tag{ 28 }$$

is the electron/radiation-wave synchronism (detuning) parameter. In these terms, the spontaneous emission and stimulated emission are given by:

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{{\rm{e}},\mathrm{SP}}={W}_{q}{{\bf{sinc}}}^{2}(\theta L/2){e}^{-{\sigma }_{t}^{2}{\omega }^{2}},\end{eqnarray} \tag{ 29a }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{{\rm{e}},\mathrm{ST}}=-\displaystyle \frac{e{{ \mathcal E }}_{{qz}0}L}{\pi }{\mathfrak{R}}\{{C}_{q}^{\mathrm{in}* }(\omega ){\tilde{M}}_{q,\perp }{e}^{i\omega {t}_{0}+i\theta L/2}\}{\bf{sinc}}(\theta L/2){e}^{-{\sigma }_{t}^{2}{\omega }^{2}/2},\end{eqnarray} \tag{ 29b }$$

$$\begin{eqnarray}&&{W}_{q}=\displaystyle \frac{{e}^{2}{{ \mathcal E }}_{{qz}0}^{2}{L}^{2}}{8\pi {{ \mathcal P }}_{q}}{| {\tilde{M}}_{q,\perp }| }^{2}.\end{eqnarray} \tag{ 29c }$$

The spontaneous and stimulated emission depend on frequency ω through the exponential decay factor of band limit ω < 1/σ_t and through the frequency dependence of Θ (ω) (equation (28)) in the sinc function, that originates from the spatial overlap relation (18). This spatial overlap spectral dependence depends on the specific interaction processes (Smith-Purcell, Cherenkov, transition radiations). It is derived specifically for Smith-Purcell radiation in the following section D. Note that if the transverse wavepacket function is narrow relative to the transverse variation of the field, then ${\tilde{M}}_{q,\perp }=1$. Also note that for a long interaction length L, efficient interaction can take place only if equation (26) is a slow-wave radiation field component (e.g. in Cerenkov radiation or Smith-Purcell radiation), where a synchronism condition ${v}_{{ph}}=\omega /{q}_{z}\approx {v}_{0}$ can be established, so that θ ≃ 0. Wide frequency band emission can take place also in a short interaction length θL ≪ 1 without satisfying a synchronizm condition (e.g. for transition radiation). In the limit, equation (29a) reduces to the classical expression for spontaneous emission of a single point particle [18].

Now let us concentrate on the stimulated emission term. Assume that the interacting field component of the incident wave is a single frequency harmonic wave (e.g. a laser beam field)

$$\begin{eqnarray}&&{E}_{\mathrm{in}}={E}_{0z}\cos ({\omega }_{0}t+{\varphi }_{0})\end{eqnarray} \tag{ 30 }$$

In terms of the continuous spectral formulation (17) and spectral normalization (18) for ω > 0, this corresponds to the input coefficient (see appendix C):

$$\begin{eqnarray}&&{C}_{q}^{\mathrm{in}}{{ \mathcal E }}_{{qz}0}=\pi {E}_{0z}\delta (\omega -{\omega }_{0}){e}^{-i{\varphi }_{0}}\end{eqnarray} \tag{ 31 }$$

where ${{ \mathcal E }}_{{qz}0}$ is the axial component of the normalized mode in equation (17). Then from integration of equation (21c) over ω, the incremental stimulated-emission radiation energy from a single electron wavepacket is given by

$$\begin{eqnarray}&&{({\rm{\Delta }}{W}_{q})}_{{\rm{e}},\mathrm{ST}}=-{{eE}}_{0z}L\cos (\theta L/2+{\omega }_{0}t-{\varphi }_{0}){\bf{sinc}}(\theta L/2){e}^{-{\sigma }_{t}^{2}{\omega }_{0}^{2}/2}\end{eqnarray} \tag{ 32 }$$

This radiative energy gain/loss is in complete agreement with the energy loss/gain of a single electron quantum wavepacket as calculated semiclassically (in the multi-photon interaction limit) [35]. It is also consistent with the classical point-particle limit σ_t ≪ 1/ω₀. This shows that conservation of energy exchange between a coherent radiation field and an electron wavepacket, contained and interacted entirely within the spatial volume of a single radiation mode, is maintained within the minimal spectral phase-space volume representing the coherent single radiation mode,

$$\begin{eqnarray*}&&{({\rm{\Delta }}{W}_{q})}_{{\rm{e}},\mathrm{ST}}^{\mathrm{RAD}}=-{({\rm{\Delta }}{W}_{q})}_{{\rm{e}},\mathrm{ST}}^{\mathrm{GAIN}}.\end{eqnarray*}$$

Another important new result is a universal relation between stimulated emission radiative energy exchange at frequency ω₀ and spontaneous emission spectral radiant energy into the same coherent phase space volume (single radiation mode) at the same frequency. At maximum emission (synchronous) interaction condition θ = 0 and maximum deceleration phase of the electron wavepacket relative to the wave ${\omega }_{0}{t}_{0e}-{\varphi }_{0}=0$, the ratio between the square of the stimulated emission spectral energy (29b) and the spontaneous emission spectral energy (29a) is

$$\begin{eqnarray*}&&\displaystyle \frac{{({{dW}}_{q}/d\omega )}_{{\rm{e}},\mathrm{ST},\max }^{2}}{{({{dW}}_{q}/d\omega )}_{{\rm{e}},\mathrm{SP},\max }}=4{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{in}}\end{eqnarray*}$$

where we used equation (21a) for ${\left(\tfrac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{in}}$. For a coherent input radiation field (30)–(31) this relation can be written in terms of the energy increments within a coherence spectral width ${\rm{\Delta }}{W}_{q}$:

$$\begin{eqnarray}&&\displaystyle \frac{{({\rm{\Delta }}{W}_{q})}_{{\rm{e}},\mathrm{ST},\max }^{2}}{{({\rm{\Delta }}{W}_{q})}_{{\rm{e}},\mathrm{SP},\max }}=4{W}_{q\mathrm{in}}\end{eqnarray} \tag{ 33 }$$

Here ${W}_{q\mathrm{in}}={P}_{q,\mathrm{in}}L/{v}_{0}$, where ${P}_{q\mathrm{in}}$ is the power incident on the interaction region. This is a new universal relation between semi-classical stimulated and spontaneous radiation emission from a single electron that is entirely independent of the interaction structure. It is useful for estimating the incremental radiation energy emission/absorption (and the corresponding electron deceleration/acceleration) ${({\rm{\Delta }}{W}_{q})}_{{\rm{e}},\mathrm{ST}}$ based on measurement of the spontaneous emission into the same mode ${({\rm{\Delta }}{W}_{q})}_{{\rm{e}},\mathrm{SP}}$. Note that this semiclassical 'Einstein relation' between classical spontaneous emission and stimulated emission is different from the relation derived in (15) in a QED model. Of course, the semi-classical analysis of an electron wavepacket cannot produce the quantum spontaneous emission. This aspect is addressed in a companion article based on QED formulation [36].

Secondly, we consider the case of a modulated electron wavepacket. In that case, using (14)–(15), (22) one gets

$$\begin{eqnarray}&&{\rm{\Delta }}{C}_{q}(\omega )=\displaystyle \frac{e}{4{{ \mathcal P }}_{q}}{\tilde{M}}_{q}\displaystyle \sum _{l=-\infty }^{\infty }{B}_{l}{e}^{-{\sigma }_{t}^{2}{(\omega -l{\omega }_{b})}^{2}/2}{e}^{i(\omega -l{\omega }_{b}){t}_{0e}}{e}^{{il}{\omega }_{b}{t}_{0}}=\displaystyle \frac{e{\tilde{M}}_{q}B(\omega )}{4{{ \mathcal P }}_{q}}\end{eqnarray} \tag{ 34 }$$

where $B(\omega )={e}^{{iz}\omega /{v}_{0}}{\sum }_{l=-\infty }^{\infty }{B}_{l}{e}^{-{\sigma }_{t}^{2}{(\omega -l{\omega }_{b})}^{2}/2}{e}^{i(\omega -l{\omega }_{b}){t}_{0}}$. Figure 4 shows the harmonics of current bunching amplitude factor $| {B}_{l}(\omega )|$ as function of frequency (figure 3(a)) and drift time (figure 3(b)) for optically-modulated electron wavepacket, where the coefficient B_l was evaluated from equation 16. Using equations (34) in (21a), the expressions for spontaneous emission by a single electron modulated wavepacket is

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{{\rm{e}},\mathrm{SP}-\mathrm{MOD}}=\displaystyle \sum _{l}{\left(\displaystyle \frac{{{dW}}_{q,l}}{d\omega }\right)}_{{\rm{e}},\mathrm{SP}-\mathrm{MOD}}\end{eqnarray} \tag{ 35 }$$

with

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q,l}}{d\omega }\right)}_{{\rm{e}},\mathrm{SP}-\mathrm{MOD}}={W}_{q}| {B}_{l}{| }^{2}{{\bf{sinc}}}^{2}(\theta L/2){e}^{-{\sigma }_{t}^{2}{(\omega -l{\omega }_{b})}^{2}}\end{eqnarray} \tag{ 36 }$$

where in (35) we assumed that the overlap between the spectral lines of the harmonics l are negligible, as shown in figure 4(a). The incremental stimulated energy emission/absorption of a modulated electron wavepacket in the case of a coherent incident radiation field is

$$\begin{eqnarray}&&{({\rm{\Delta }}{W}_{q})}_{{\rm{e}},\mathrm{ST}-\mathrm{MOD}}=\displaystyle \sum _{l}{({\rm{\Delta }}{W}_{q,l})}_{{\rm{e}},\mathrm{ST}-\mathrm{MOD}}\end{eqnarray} \tag{ 37 }$$

with

$$\begin{eqnarray*}&&{({\rm{\Delta }}{W}_{q,l})}_{{\rm{e}},\mathrm{ST}-\mathrm{MOD}}={{eE}}_{0z}L| {\tilde{M}}_{q,\perp }| | {B}_{l}| \cos (\theta L/2+({\omega }_{0}-l{\omega }_{b}){t}_{0e}+l{\omega }_{b}{t}_{0}-{\varphi }_{0}){\bf{sinc}}(\theta L/2){e}^{-{\sigma }_{t}^{2}{({\omega }_{0}-l{\omega }_{b})}^{2}/2}\end{eqnarray*}$$

where ${\theta }_{0}=\theta ({\omega }_{0})={\omega }_{0}/{v}_{0}-{q}_{z}$. A striking difference between these expressions and the corresponding cases for the unmodulated quantum wavepacket (29a), (32) is that the modulated wavepacket can emit/absorb radiation at frequencies beyond the quantum cut-off condition (1), which occur at all harmonic frequencies $l{\omega }_{b}$ of the wavepacket modulation of significant component amplitude B_l.

A vivid presentation of radiation emission extinction and revival effects of a modulated quantum electron wavepacket is presented here for the case of the Smith-Purcell radiation(SPR) experiment shown in figures 1 and 2. The radiative modes of the SPR grating structure are Floquet modes

$$\begin{eqnarray}&&{\tilde{{ \mathcal E }}}_{q}({\bf{r}})=\displaystyle \sum _{m}{\tilde{{ \mathcal E }}}_{{qm}}({{\bf{r}}}_{\perp }){e}^{i({q}_{{z}_{0}}+{{mk}}_{G})z}\end{eqnarray} \tag{ 38 }$$

where ${k}_{G}=2\pi /{\lambda }_{G}$, λ_G is the grating period,

$$\begin{eqnarray}&&{q}_{{z}_{0}}=\sqrt{{\omega }^{2}/{c}^{2}-{k}_{q\perp }^{2}}=\displaystyle \frac{\omega }{c}\cos {{\rm{\Theta }}}_{q}\end{eqnarray} \tag{ 39 }$$

The angle Θ_q the 'zig-zag' angle of mode q in a waveguide structure. We use in equation (26) ${q}_{z}={q}_{z0}+{{mk}}_{G}$ where m is the mth order space harmonic of Floquet modes, and apply all the expressions for spontaneous and stimulated emission to each of the space harmonics with a detuning parameter (neglecting the interference between the space harmonics)

$$\begin{eqnarray}&&{\theta }_{m}=\displaystyle \frac{\omega }{{v}_{0}}-{q}_{{z}_{0}}-{{mk}}_{G}\end{eqnarray} \tag{ 40 }$$

The spontaneous emissions (29a) for an unmodulated wavepacket and (36)–(37) for a modulated wavepacket, can be modified to include interaction with any space-harmonics m, which can be synchronous with the electron $\omega /({q}_{z0}+{{mk}}_{G})\approx v$,

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{{\rm{e}},\mathrm{SP}}=\displaystyle \sum _{m}{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{{\rm{e}},\mathrm{SP}}=\displaystyle \sum _{m}{W}_{q,m}{{\bf{sinc}}}^{2}({\theta }_{m}L/2){e}^{-{\sigma }_{t}^{2}{\omega }^{2}},\end{eqnarray} \tag{ 41a }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{{\rm{e}},\mathrm{SP}-\mathrm{MOD}}=\displaystyle \sum _{l,m}{\left(\displaystyle \frac{{{dW}}_{q,{lm}}}{d\omega }\right)}_{{\rm{e}},\mathrm{SP}-\mathrm{MOD}}=\displaystyle \sum _{l,m}{W}_{q,m}| {B}_{l}{| }^{2}{{\bf{sinc}}}^{2}({\theta }_{m}L/2){e}^{-{\sigma }_{t}^{2}{(\omega -l{\omega }_{b})}^{2}},\end{eqnarray} \tag{ 41b }$$

with ${W}_{q,m}=\tfrac{{e}^{2}{{ \mathcal E }}_{{qz},m}^{2}{L}^{2}}{8\pi {P}_{q}}| {\tilde{M}}_{q\perp ,m}{| }^{2}$. It is instructive to observe that the proportionality coefficient in equation (41b) can be related to the known 'Pierce interaction impedance' parameter ${K}_{\mathrm{Pierce}}={{ \mathcal E }}_{{qz}0}^{2}/(2{q}_{z}{{ \mathcal P }}_{q})$ [37]. Note that in order to evaluate this parameter in practice (e.g. for the case of Smith-Purcell radiation), one must solve first analytically or numerically the classical electromagnetic problem of the amplitude of the interacting axial field component ${{ \mathcal E }}_{{qz},m}$ relative to the entire mode - ${{ \mathcal P }}_{q}$.

Figure 5(a) displays the SP spectrum in terms of wavelength $\lambda =2\pi c/\omega$ and angle Θ in the classical limit σ_t≪1/ω, where the wavepacket appears as a point-particle. Emission lines appear for arbitrary angles in the range 0 < Θ < π at frequencies or wavelengths corresponding to the synchronism condition θ(ω_m) = 0,

$$\begin{eqnarray}&&{\omega }_{m}({\rm{\Theta }})={{mck}}_{G}/({\beta }^{-1}-\cos {\rm{\Theta }}),\ \mathrm{or}\ \ {\lambda }_{m}({\rm{\Theta }})=\displaystyle \frac{{\lambda }_{G}}{m}({\beta }^{-1}-\cos {\rm{\Theta }})\end{eqnarray} \tag{ 42 }$$

in agreement with the classical SP emission relation [6].

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In figure 5(b), we show the expected SPR spontaneous spectrum of an unmodulated electron wavepacket in the set-up shown in figure 1. The plot shows that at low frequencies ω < 1/σ_t, or long wavelengths $\lambda \beta \gt {\sigma }_{z}$, the low-pass filtering effect of the finite size wavepacket extinction parameter ${e}^{-{\omega }^{2}{\sigma }_{t}^{2}}$ of equation (41a) (figure 5(c), blue curve) keeps the long wavelength part of the classical SPR spectrum of figure 5(a) unaffected, and cuts off the shorter wavelength and higher harmonic sections when $1/{\omega }_{m-1}({\rm{\Theta }}=0)\lt {\sigma }_{t}\lt 1/{\omega }_{m-1}({\rm{\Theta }}=\pi )$, namely,

$$\begin{eqnarray}&&(1-\beta )\displaystyle \frac{{\lambda }_{G}}{2\pi }\lt {\sigma }_{z}\lt (1+\beta )\displaystyle \frac{{\lambda }_{G}}{2\pi }\end{eqnarray} \tag{ 43 }$$

This case is displayed in figure 5(b) for the parameters ${\sigma }_{z}/{\lambda }_{G}=0.22,\beta =0.7,{N}_{G}=L/{\lambda }_{G}=9$. Here the first order SPR harmonic is partly cut-off, the second order harmonic is barely observable and higher harmonics are extinct.

However, a more dramatic change in the spectrum takes place when the wavepacket is modulated (the 'modulating laser' in figure 2 is turned 'ON'). In this case the wavepacket bunching factor ${e}^{-{\sigma }_{t}^{2}{(\omega -l{\omega }_{b})}^{2}}$ in equation (41b) permits resonant emission only at harmonics ω_l = lω_b and this harmonics-spectrum is cut-off only at much higher frequencies by the filtering effect of the narrow micro-bunches $l{\omega }_{b}\lt 1/{\sigma }_{t}$, or

$$\begin{eqnarray}&&l\lt \displaystyle \frac{1}{2\pi }\displaystyle \frac{{T}_{b}}{{\sigma }_{t}}\end{eqnarray} \tag{ 44 }$$

Figure 5(c) displays the (classical) SPR spontaneous spectrum (43) in terms of $\lambda /{\lambda }_{G}$, for the same parameters and wavepacket size ${\sigma }_{z}/{\lambda }_{G}=1.5$ as in the un-modulation case (figure 5(b)). It is seen that the bunching factor exhibits resonance at harmonics of the bunching frequencies ω_b in the frequency range that was cut-off before modulation. This shows up in figure 6 as three resonant spots, reviving the 1st, 2nd and 3rd order SP space harmonics at distinct emission angles.

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Inspection of equation (43) reveals that resonant emission spots will appear, within the spectral range beyond the frequency cut-off of unmodulated beam radiation, ω > 1/σ_t at the frequencies and emission angles in which the narrow band filtering function ${e}^{-{\sigma }_{t}^{2}{(\omega -l{\omega }_{b})}^{2}}$ and ${{\bf{sinc}}}^{2}({\theta }_{m}L/2)$ overlap. The centres of these spots will be at positions $\omega ={\omega }_{l},{\rm{\Theta }}={{\rm{\Theta }}}_{{lm}}$ where ${{\rm{\Theta }}}_{{lm}}$ is the solution of the equation

$$\begin{eqnarray}&&l{\omega }_{b}={\omega }_{m}({{\rm{\Theta }}}_{{lm}})\end{eqnarray} \tag{ 45 }$$

and ω_m is given by equation (42). The spectral width of the spot depends on which filtering function is narrower. The spectral width of the synchronization function is

$$\begin{eqnarray}&&{\left.\displaystyle \frac{{\rm{\Delta }}\omega }{{\omega }_{m}}\right|}_{{\rm{\Theta }}}=\displaystyle \frac{1}{{{mN}}_{G}}\end{eqnarray} \tag{ 46 }$$

and of the bunched wavepacket bunching factor

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}\omega }{{\omega }_{l}}=\displaystyle \frac{2}{\pi }\displaystyle \frac{1}{{{lN}}_{b}}\end{eqnarray} \tag{ 47 }$$

where ${N}_{b}=2{\sigma }_{t}/{T}_{b}$ is the effective amount of micro-bunches in the modulated wavepacket. figure 6 displays the emission spectrum for the two opposite cases ${{mN}}_{G}\gt {{lN}}_{b}$ and ${{mN}}_{G}\lt {{lN}}_{b}$ as functions of λ and Θ.

The same configuration of Smith-Purcell experiment can be used for measurement of stimulated emission/absorption and corresponding deceleration/acceleration with the input laser beam turned 'ON'. In this case, the incremental exchanged energy from the electron wavepacket to the radiation wave and vice versa is given by modified versions of equations (32) and (37) (for the unmodulated and modulated cases respectively)

$$\begin{eqnarray}&&\begin{array}{l}{({\rm{\Delta }}{W}_{q})}_{{\rm{e}},\mathrm{ST}}=\displaystyle \sum _{m}{({\rm{\Delta }}{W}_{q,m})}_{{\rm{e}},\mathrm{ST}}\\ \quad =\,{{eE}}_{0z}L\displaystyle \sum _{m}| {\tilde{M}}_{q\perp ,m}| \cos ({\theta }_{m}L/2+{\omega }_{0}{t}_{0}-{\varphi }_{0}){\bf{sinc}}({\theta }_{m}L/2){e}^{-{\sigma }_{t}^{2}{\omega }^{2}/2}\end{array}\end{eqnarray} \tag{ 48 }$$

$$\begin{eqnarray}&&\begin{array}{l}{({\rm{\Delta }}{W}_{q})}_{{\rm{e}},\mathrm{ST}-\mathrm{MOD}}=\displaystyle \sum _{m,l}{({\rm{\Delta }}{W}_{q,{lm}})}_{{\rm{e}},\mathrm{ST}-\mathrm{MOD}}\\ \quad =\,{{eE}}_{0z}L\displaystyle \sum _{m,l}| {\tilde{M}}_{q\perp ,m}| {B}_{l}\cos ({\theta }_{m}L/2-({\omega }_{0}-l{\omega }_{b}){t}_{0e}+l{\omega }_{b}{t}_{0}-{\varphi }_{0}){\bf{sinc}}({\theta }_{m}L/2){e}^{-{\sigma }_{t}^{2}{({\omega }_{0}-l{\omega }_{b})}^{2}/2}\end{array}\end{eqnarray} \tag{ 49 }$$

Evidently a spectral diagram similar to figures 5–6, with the spectral cutoff effect and re-emerging spots (in case of a modulated beam), would be measured in the incremental energies of the radiation wave and the electrons when the incident laser beam is scanned over wavelengths λ₀ and incident angle Θ.

It can be argued that this scheme and the characteristic spectral map can be used for measuring the electron quantum wavepacket size σ_t at the entrance to the grating. We note, however, that the semi-classical calculation of spontaneous emission from a single electron quantum wavepacket may have limited validity, as discussed in the companion paper [36] based on QED formulation, and its measurement may be plagued by quantum spontaneous emission noise, not inclusive in a semi-classical formulation. On the other hand, the validity of using semi-classical formulation for stimulated radiative interaction is well founded. We therefore assert that such a stimulated interaction experiment can be a way for measuring the quantum wavepacket size with a SPR experiment as shown in figures 1–2. The available control over the input radiation field intensity E₀ can help to overcome expected quantum and other noise factors in the experimental measurement.

Beyond the cases of single electron-wavepacket radiations, we consider now the radiation of a multi-particle wavepackets beam (N ≫ 1).Radiation measurements with single electron wavepacket would be challenging experiments. To get significant wavepacket-dependent measurement, repeated experiments must be performed with careful pre-selection filtering, to assure similarity (or identity) of the wavepackets in successive measurement experiments [38]. We now consider the case where we measure at once a pulse of electron wavepackets that may be correlated at entrance to the radiative interaction region (see figure 7). Now, assume that the e-beam is composed of electron-wavepackets whose current density distribution is given by ${\bf{J}}({\bf{r}},\omega )={\sum }_{j=1}^{N}{{\bf{J}}}_{j}({\bf{r}},\omega )$. Consequently, the spontaneous/superradiant emission and stimulated-superradiant emission of the beam are given by, respectively,

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{SP}/\mathrm{SR}}=\displaystyle \frac{2}{\pi }{{ \mathcal P }}_{q}{\left|\displaystyle \sum _{j=1}^{N}{\rm{\Delta }}{C}_{{qj}}(\omega )\right|}^{2},\end{eqnarray} \tag{ 50a }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{ST}/\mathrm{ST}-\mathrm{SR}}=\displaystyle \frac{4}{\pi }{{ \mathcal P }}_{q}{\mathfrak{R}}\left\{{C}_{q}^{\mathrm{in}* }(\omega )\cdot \displaystyle \sum _{j=1}^{N}{\rm{\Delta }}{C}_{{qj}}(\omega )\right\},\end{eqnarray} \tag{ 50b }$$

where we define in equation (18)

$$\begin{eqnarray}&&{\rm{\Delta }}{C}_{{qj}}(\omega )=-\displaystyle \frac{1}{4{{ \mathcal P }}_{q}}\int {{\bf{J}}}_{j}({\bf{r}},\omega )\cdot {\tilde{{ \mathcal E }}}_{q}^{* }({\bf{r}}){d}^{3}{\bf{r}},\end{eqnarray} \tag{ 51 }$$

and the current ${J}_{j}(r,\omega )$ of the j electron-wavepacket (equation (22)) is

$$\begin{eqnarray*}&&{{\bf{J}}}_{j}({\bf{r}},\omega )={\int }_{-\infty }^{\infty }{{\bf{J}}}_{j}({\bf{r}},t){e}^{i\omega t}{dt}\end{eqnarray*}$$

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We go back to equations (22)–(25a) and consider now the cooperative emission of a pulse of N electrons. Assuming all particle wavepackets are identical (except for arrival time t_0j), the averaged spectral energy of the pulse is then given by

$$\begin{eqnarray}&&\left\langle {\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{SP}/\mathrm{SR}}^{\mathrm{pulse}}\right\rangle =\left\langle {\left|\displaystyle \sum _{j=1}^{N}{e}^{i\omega {t}_{0j}}\right|}^{2}\right\rangle {\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{{\rm{e}},\mathrm{SP}}\end{eqnarray} \tag{ 52a }$$

$$\begin{eqnarray}&&\left\langle {\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{ST}-\mathrm{SR}}^{\mathrm{pulse}}\right\rangle =\left\langle \displaystyle \sum _{j=1}^{N}{e}^{i\omega {t}_{0j}}\right\rangle {\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{{\rm{e}},\mathrm{ST}}\end{eqnarray} \tag{ 52b }$$

where the single electron spontaneous and stimulated spectral energies are given in equation (29). It is evident that when the electron beam pulse is longer than the radiation optical period ${\sigma }_{{pulse}}\gt T=2\pi /\omega$, and t_oj are random, all the phasor terms in (52b) and all the mixed terms in (52a) interfere destructively. One gets then no average stimulated-emission (and no average acceleration) of the random electron beam, but there is a resultant 'classical spontaneous emission' (shot-noise radiation) of the beam, originating from the diagonal terms in the product in equation (52a). Considering equation (29a),

$$\begin{eqnarray}&&\left\langle {\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{SP}}^{\mathrm{pulse}}\right\rangle =N{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{{\rm{e}},\mathrm{SP}}={{NW}}_{q}{{\bf{sinc}}}^{2}(\theta L/2){e}^{-{\sigma }_{t}^{2}{\omega }^{2}}\end{eqnarray} \tag{ 53 }$$

Namely, the spontaneous emission from N particle equals to N times the spectral emission from a single electron wavepacket. In consideration of the high frequency cut-off of the single electron spontaneous emission (1), this suggests that spontaneous 'shot-noise' radiation of an electron beam is not 'white noise', but diminishes in the quantum limit $\omega {\sigma }_{t}\gg 1$ (see [39]).

In the opposite limit of a short electron beam pulse relative to the optical period ${\sigma }_{\mathrm{pulse}}\lt T$, all phasor terms in (52) within the pulse, sum-up with the same phase and since necessarily ${\sigma }_{t}\lt {\sigma }_{\mathrm{pulse}}$, the expontential decay factors in (29) are unity, and the electrons radiate as point particles without any dependence on σ_t. This is actually the classical case of superradiance analyzed in [18], where the collective emission of the electron pulse depends only on the particles arrival time distribution in the coefficient of equation (53),

$$\begin{eqnarray*}&&{b}_{\mathrm{pulse}}=\displaystyle \frac{1}{N}\left\langle \displaystyle \sum _{j=1}^{N}{e}^{i\omega {t}_{0j}}\right\rangle \end{eqnarray*}$$

One can replace the summutation over j by integration over the temporal distribution of the particle in the beam pulse ${f}_{\mathrm{pulse}}(t-{t}_{0,\mathrm{pulse}})$, where ${\int }_{-\infty }^{\infty }{f}_{\mathrm{pulse}}({t}_{0j}){{dt}}_{0j}=1$ (see appendix D). For a Gaussian pulse temporal distribution

$$\begin{eqnarray}&&{b}_{\mathrm{pulse}}={\int }_{-\infty }^{\infty }{f}_{\mathrm{pulse}}({t}_{0j}-{t}_{0,\mathrm{pulse}}){e}^{i\omega {t}_{0j}}{{dt}}_{0j}={e}^{-{\sigma }_{\mathrm{pulse}}^{2}{\omega }^{2}/2}{e}^{i\omega {t}_{0,\mathrm{pulse}}}\end{eqnarray} \tag{ 54 }$$

and

$$\begin{eqnarray}&&| {b}_{\mathrm{pulse}}{| }^{2}\simeq {e}^{-{\sigma }_{\mathrm{pulse}}^{2}{\omega }^{2}}\end{eqnarray} \tag{ 55 }$$

Using these expressions and equation (29) with ${\sigma }_{t}\ll {\sigma }_{\mathrm{pulse}}$ in (52)

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{SR}}={N}^{2}{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{{\rm{e}},\mathrm{SP}}={N}^{2}{W}_{q}{{\bf{sinc}}}^{2}(\theta L/2){e}^{-{\sigma }_{\mathrm{pulse}}^{2}{\omega }^{2}},\end{eqnarray} \tag{ 56a }$$

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{ST}-\mathrm{SR}}=N{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{{\rm{e}},\mathrm{ST}}=-\displaystyle \frac{{eN}{{ \mathcal E }}_{{qz}0}L}{\pi }{\mathfrak{R}}\{{C}_{q}^{\mathrm{in}* }(\omega ){\tilde{M}}_{q,\perp }{e}^{i\omega {t}_{0}+i\theta L/2}\}{\bf{sinc}}(\theta L/2){e}^{-{\sigma }_{\mathrm{pulse}}^{2}{\omega }^{2}/2},\end{eqnarray} \tag{ 56b }$$

which is the same as the single electron expression (29) with respective N² and N factors, and σ_t replaced by σ_pulse. Thus, we obtained in this limit the super-radiance and stimulated super-radiance expressions of classical point-particles [18]. The classical point particle limit of stimulated superradiance (56b) is nothing but the acceleration formula in conventional particle accelerators. The classical superradiance formula (56a) is of interest primarily for THz radiation generation devices, since attainable short electron beam pulse durations (t_pulse) are in the order of ${10}^{-12}s$ [40, 41]. It is of less interest in the present context, since it has no quantum wavepacket dependence. More detailed derivations see appendix D.

Consider a case where all electron wavepackets in an electron beam pulse are modulated in phase correlation by the same coherent laser beam. In this case, the modulated currents of the wavepackets in the pulse (14) are also correlated, and their radiation emissions in the interaction region are phase correlated as well (see figure 7).

Assume that the expectation value of the electron wavepackets probabilistic density of an ensemble of N electrons is modulated coherently at frequency ω_b by a laser beam, as shown in figure 2; the single electron wavepacket density function in (14) is for electron e = j,

$$\begin{eqnarray}&&{{\bf{J}}}_{j}({\bf{r}},t)=-{{ef}}_{e\perp }({{\bf{r}}}_{\perp }){f}_{\mathrm{en}}(t-z/{v}_{0}-{t}_{0j}){f}_{\mathrm{mod}}(t-z/{v}_{0}-{t}_{0})\end{eqnarray} \tag{ 57 }$$

where ${f}_{\mathrm{mod}}(t)$ is the periodic modulation function (15), composed of many harmonics of ω_b. Important to note that the modulation phase ${\omega }_{b}{t}_{0}$ is common to all the wavepackets (i.e., determined by the modulating laser phase). Here ${f}_{\mathrm{en}}(t)$ is the wavepacket density envelope given by the Gaussian (8). The incremental spectral amplitude ${\rm{\Delta }}{C}_{q}(\omega )$ due to interaction with the entire e-beam pulse is then found by setting e = j, and summing up ${\rm{\Delta }}{C}_{{qj}}(\omega )$ (34) over all particles

$$\begin{eqnarray}&&{\rm{\Delta }}{C}_{q}(\omega )=\displaystyle \sum _{j=1}^{\infty }{\rm{\Delta }}{C}_{{qj}}(\omega )=-\displaystyle \frac{{{eE}}_{{q}_{z}0}}{4{{ \mathcal P }}_{q}}{\tilde{M}}_{q}(\omega )\displaystyle \sum _{l}{B}_{l}{e}^{-{\sigma }_{t}^{2}{(\omega -l{\omega }_{b})}^{2}/2}{e}^{{il}{\omega }_{b}{t}_{0}}\displaystyle \sum _{j=1}^{N}{e}^{i(\omega -l{\omega }_{b}){t}_{0j}}\end{eqnarray} \tag{ 58 }$$

We now define the beam bunching factor of lth-order harmonic frequency over the entire pulse

$$\begin{eqnarray}&&{b}_{\mathrm{pulse}}^{(l)}=\displaystyle \frac{1}{N}\displaystyle \sum _{j=1}^{N}{e}^{i(\omega -l{\omega }_{b}){t}_{0j}}\end{eqnarray} \tag{ 59 }$$

and for N ≫ 1 replace the summation with integration, over the pulse temporal distribution function. For a Gaussian distribution ${f}_{\mathrm{pulse}}({t}_{0j})={(\sqrt{2\pi }{\sigma }_{\mathrm{pulse}})}^{-1}{e}^{{t}_{0j}^{2}/2{\sigma }_{\mathrm{pulse}}^{2}}$ (see appendix D),

$$\begin{eqnarray}&&\langle | {b}_{\mathrm{pulse}}^{(l)}| \rangle ={e}^{-{\sigma }_{\mathrm{pulse}}^{2}{(\omega -l{\omega }_{b})}^{2}/2},\end{eqnarray} \tag{ 60a }$$

$$\begin{eqnarray}&&\langle | {b}_{\mathrm{pulse}}^{(l)}{| }^{2}\rangle ={e}^{-{\sigma }_{t}^{2}{(\omega -l{\omega }_{b})}^{2}},\end{eqnarray} \tag{ 60b }$$

Substitutioin of equations (27), (34) in (50) results in

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q}}{d\omega }\right)}_{\mathrm{SP}/\mathrm{SR}-\mathrm{MOD}}^{\mathrm{pulse}}=\displaystyle \sum _{l}{\left(\displaystyle \frac{{{dW}}_{q,l}}{d\omega }\right)}_{\mathrm{SP}/\mathrm{SR}-\mathrm{MOD}}^{\mathrm{pulse}},\end{eqnarray} \tag{ 61 }$$

where

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{{dW}}_{q,l}}{d\omega }\right)}_{\mathrm{SP}/\mathrm{SR}-\mathrm{MOD}}^{\mathrm{pulse}}={N}^{2}{W}_{q}| {B}_{l}{| }^{2}{{\bf{sinc}}}^{2}(\theta L/2){e}^{-{\sigma }_{\mathrm{pulse}}^{2}{(\omega -l{\omega }_{b})}^{2}}\end{eqnarray} \tag{ 62 }$$

for all harmonics $l\ne 0$. The term l = 0 that corresponds to spontaneous emission from a random unmodulated beam, but with a factor ${B}_{0}^{2}$, and just as in equation (53), it decays as ${e}^{-{\sigma }_{t}^{2}{\omega }^{2}}$ and cuts-off for ω > 1/σ_t. All other harmonics can radiate beyond this cutoff (figure 5) with a narrow bandwidth filtering-factor ${e}^{-{\sigma }_{\mathrm{pulse}}^{2}{(\omega -l{\omega }_{b})}^{2}}$ similar to the single electron modulated wavepacket case (36), but with the electron-beam pulse duration σ_pulse replacing the quantum wavepacket size parameter σ_t. Note that even though the modulated wavepacket beam radiates superradiantly beyond the quantum cut-off condition ω > 1/σ_t (44), it still has a high harmonic cut-off that depends on the tightness of the bunching.

This interesting new result suggests that all electrons in the pulse emit in phase superradiantly even if the electrons enter the interaction region sparsely  and randomly , see figure 7. Because all wavepackets are modulated (by the same laser) at the same frequency and phase, they cooperatively radiate constructively at all harmonics $l{\omega }_{b}$ with coherence time equal to the electron pulse duration t_pulse and corresponding spectral linewidth ${\rm{\Delta }}\omega =1/{\sigma }_{\mathrm{pulse}}$. In a modulated wavepacket Smith-Purcell experiment as shown in figure 2 we expect to get a spectrum similar to the one shown in figure 6 but the radiated energy would be enhanced by a factor of N, and the spectral linewidth would be narrower by a factor σ_pulse/σ_t.

Another interesting observation is that the dependence on the wavepacket size disappeared altogether in (61). In fact, the emission is the same as the superradiant emission of a bunched point-particle beam [18]. Here is another expression of the wave-particle duality nature in spontaneous emission—no distinction between the spectral superradiant emission of a bunched point-particles beam and a pulse of phase correlated modulated wavepackets, even if the electron beam is tenuous and the wavepackets are sparse and random (we do not rule out the possibility that the photon statistics may be different).

It is also interesting to point out that the configuration of bunching the electron wavepackets by a laser and measuring their superradiant emission at the laser modulation frequency shown in figure 2 is reminiscent of the Schwartz-Hora experiment [42] and its interpretation by Marcuse [43] as coherent cooperative optical transition radiation. According to Marcuse the signal to noise ratio, based on his model and the reported experimental parameters, is below the measurable level in the experiment [43], and unfortunately there is no independent experimental confirmation of this effect. We suggest that the Smith-Purcell radiation scheme in figure 2 will be a more efficient radiation scheme for observing superradiant emission from a laser modulated electron wavepackets beam.

In stimulated interaction (acceleration/deceleration) of a pulse of modulated wavepackets we sum up the incremental energy contributions ${({\rm{\Delta }}{W}_{q,{lj}})}_{\mathrm{ST}-\mathrm{MOD}}$ (37) of all modulated electron wavepackets j

$$\begin{eqnarray}\begin{array}{rcl}{({\rm{\Delta }}{W}_{q,l})}_{\mathrm{ST}-\mathrm{MOD}}^{\mathrm{pulse}} & = & -{{eE}}_{0z}L| {\tilde{M}}_{q\perp }| | {B}_{l}| {\bf{sinc}}(\theta L/2){e}^{-{\sigma }_{\mathrm{pulse}}^{2}{({\omega }_{0}-l{\omega }_{b})}^{2}/2}\\ & & \times \displaystyle \sum _{j=1}^{N}\cos (\theta L/2+({\omega }_{0}-l{\omega }_{b}){t}_{0j}+l{\omega }_{b}{t}_{0}-{\varphi }_{0})\end{array}\end{eqnarray} \tag{ 63 }$$

Replacing summation with integration over the Gaussian statistical distribution of the electrons in the pulse (see appendix D), one obtains:

$$\begin{eqnarray}&&{({\rm{\Delta }}{W}_{q,l})}_{\mathrm{ST}-\mathrm{MOD}}^{\mathrm{pulse}}=-{{eNE}}_{0z}L| {\tilde{M}}_{q\perp }| | {B}_{l}| {\bf{sinc}}(\theta L/2){e}^{-{\sigma }_{\mathrm{pulse}}^{2}{({\omega }_{0}-l{\omega }_{b})}^{2}/2}\cos ({\theta }_{0}L/2+l{\omega }_{0}{t}_{0}-{\varphi }_{0})\end{eqnarray} \tag{ 64 }$$

Resonant stimulated superradiant emission/absorption (deceleration/acceleration) takes place at synchronizm (θ = 0) if the interaction laser (figure 2) is tuned to one of the beam bunching harmonic frequencies $({\omega }_{0}=l{\omega }_{b})$ at proper deceleration/acceleration phase relative to the bunching - ${\omega }_{0}{t}_{0}-{\varphi }_{0}=0,\pi$. The laser frequency (ω₀) resonant detuning range is determined by the duration of the electron beam pulse - ${\rm{\Delta }}\omega \lt 1/{\sigma }_{\mathrm{pulse}}$.

In quantum mechanics, the current density operator is given by

$$\begin{eqnarray}&&\vec{J}=\displaystyle \frac{q{\hslash }}{2{m}^{* }i}({\psi }^{* }{\rm{\nabla }}\psi -\psi {\rm{\nabla }}{\psi }^{* }).\end{eqnarray} \tag{ A13 }$$

Considering the longitudinal current J_z along the z-direction (equation (6) in the text), we solve the terms

$$\begin{eqnarray*}\begin{array}{rcl}{\rm{\nabla }}\psi & = & \displaystyle \frac{\partial }{\partial z}\psi (z,t)=\left(\displaystyle \frac{{{ip}}_{0}}{{\hslash }}\right)\psi (z,t)+\displaystyle \frac{{e}^{i({p}_{0}z-{E}_{0}t)/{\hslash }}}{\sqrt{2\pi {\hslash }}}\displaystyle \int {dp}^{\prime} \left(\displaystyle \frac{{ip}^{\prime} }{{\hslash }}\right){\psi }_{p^{\prime} }{e}^{\tfrac{{ip}^{\prime} }{{\hslash }}(z-{v}_{0}t-\tfrac{p{{\prime} }^{2}t}{2{m}^{* }})},\\ {\rm{\nabla }}{\psi }^{* } & = & \displaystyle \frac{\partial }{\partial z}{\psi }^{* }(z,t)=\left(\displaystyle \frac{-{{ip}}_{0}}{{\hslash }}\right){\psi }^{* }(z,t)+\displaystyle \frac{{e}^{-i({p}_{0}z-{E}_{0}t)/{\hslash }}}{\sqrt{2\pi {\hslash }}}\displaystyle \int {dp}^{\prime} \left(\displaystyle \frac{-{ip}^{\prime} }{{\hslash }}\right){\psi }_{p^{\prime} }{e}^{-\tfrac{{ip}^{\prime} }{{\hslash }}(z-{v}_{0}t-\tfrac{p{{\prime} }^{2}t}{2{m}^{* }})},\end{array}\end{eqnarray*}$$

Here we assume ${p}_{0}\gg \langle p^{\prime} \rangle$, and then we only consider the contribution of the first terms Thus, we obtain

$$\begin{eqnarray}&&J(\zeta ,t)\simeq -{{ev}}_{0}| \psi (\zeta ,t){| }^{2},\end{eqnarray} \tag{ A14 }$$

with $\zeta =z-{v}_{0}t$. Then the density probability is given by

$$\begin{eqnarray*}\begin{array}{rcl}| \psi (\zeta ,t){| }^{2} & = & \displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{z}(t)}\displaystyle \sum _{n,m=-\infty }^{\infty }{{ \mathcal J }}_{n}(2| g| ){{ \mathcal J }}_{m}(2| g| )\exp \left\{-\displaystyle \frac{{\left(\zeta -\tfrac{n\delta {pt}}{2{m}^{* }}\right)}^{2}}{4{\sigma }_{z}^{2}(1+i\xi t)}-\displaystyle \frac{{\left(\zeta -\tfrac{m\delta {pt}}{2{m}^{* }}\right)}^{2}}{4{\sigma }_{z}^{2}(1-i\xi t)}\right\}\\ & & \times \exp \left\{i(n-m)\delta p\left(\zeta -\displaystyle \frac{(n+m)\delta {pt}}{2{m}^{* }}\right)/{\hslash }\right\},\end{array}\end{eqnarray*}$$

where the spatial wavepacket size of the spreading wavepacket envelope is

$$\begin{eqnarray}&&{\sigma }_{z}(t)=\sqrt{{\sigma }_{{z}_{0}}^{2}(1+{\xi }^{2}{t}^{2})}.\end{eqnarray} \tag{ A15 }$$

Note that in the limit of no modulation ($2| g| \to 0$ in (A12)) reduces to the limit of equation (5) in the text: a Gaussian wavepacket, exhibiting expansion and phase chirp as it drifts with time. Its density probability after drift still is:

$$\begin{eqnarray}&&| \psi (z,t){| }^{2}=\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{z}(t)}\exp \left(-\displaystyle \frac{{(z-{v}_{0}t)}^{2}}{2{\sigma }_{z}^{2}(t)}\right),\end{eqnarray} \tag{ A16 }$$

Also note that the harmonic density modulation that is implied by (A15) is a direct consequence of the nonlinearity of the energy dispersion relation (third term in (A7)). In its absence ($\xi t={\hslash }t/2{m}^{* }{\sigma }_{{z}_{0}}^{2}\ll 1$) the energy modulation does not convert into density bunching:

$$\begin{eqnarray}\begin{array}{rcl}\psi (z,t) & = & \displaystyle \frac{{e}^{i({p}_{0}z-{E}_{0}t)/{\hslash }}}{{(2\pi {\sigma }_{{z}_{0}}^{2})}^{\tfrac{1}{4}}}\exp \left(-\displaystyle \frac{{(z-{v}_{0}t)}^{2}}{4{\sigma }_{{z}_{0}}^{2}}\right)\displaystyle \sum _{n=-\infty }^{\infty }{J}_{n}(2| g| )\exp \left\{i\left(\displaystyle \frac{n\delta p(z-{v}_{0}t)}{{\hslash }}\right)\right\}\\ & = & \displaystyle \frac{{e}^{i({p}_{0}z-{E}_{0}t)/{\hslash }}}{{(2\pi {\sigma }_{{z}_{0}}^{2})}^{\tfrac{1}{4}}}\exp \left(-\displaystyle \frac{{(z-{v}_{0}t)}^{2}}{4{\sigma }_{{z}_{0}}^{2}}\right){e}^{i| g| \sin (z-{v}_{0}t)/{\sigma }_{{z}_{0}}}.\end{array}\end{eqnarray} \tag{ A17 }$$

and the density distribution is then independent of drift, similarly to (A16), without density modulation or expansion, even though the wavepacket remains energy (phase) modulated.

Finally, we explain here that the choice made in the main text analysis that the wavepacket is emitted at its longitudinal waist at t = 0 (equation (5) in the text), or equivalently—with symmetrical momentum distribution without chirp (equation (3) in the text), does not limit the generality of our analysis.

If the initial wavepacket before modulation is emitted from the electron source chirped, or acquires chirp due to transport to the modulation point, or by a controlled process by streaking technique [40, 41], then the chirp acquired due to the dispersive transport after modulation will combine with this prior chirping. The two effects may add together to enhance the wavepacket widening, or with negative chirping—lead to compression of the wavepacket.

Instead of an unchirped momentum distribution wavepacket (equation (3) in the text) one would starts in this case with a complex Gaussian wavepacket:

$$\begin{eqnarray}&&{\psi }_{p}={(2\pi {\sigma }_{{p}_{0}}^{2})}^{-1/4}\exp \left[-\left(\displaystyle \frac{1}{4{\sigma }_{{p}_{0}}^{2}}-{iC}\right){(p-{p}_{0})}^{2}\right]\end{eqnarray} \tag{ A18 }$$

where C is defined as a prior chirp factor and $\tfrac{1}{{\tilde{\sigma }}_{p}^{2}(0)}=\tfrac{1}{{\sigma }_{{p}_{0}}^{2}}-i4C$. Then the Fourier transformation to real space (equation (2) in the main text) results in a modified complex Gaussian wavepacket (equation (5) in the text) with complex wavepacket size at time t = 0:

$$\begin{eqnarray}&&{\tilde{\sigma }}_{z}(0)=\displaystyle \frac{{\hslash }}{2{\tilde{\sigma }}_{p}(0)}\end{eqnarray} \tag{ A19 }$$

and then size spreading in time is given by

$$\begin{eqnarray*}\begin{array}{rcl}{\tilde{\sigma }}_{z}^{2}(t) & = & {\tilde{\sigma }}_{z}^{2}(0)+\displaystyle \frac{i{\hslash }t}{2{m}^{* }}=\displaystyle \frac{{\sigma }_{{z}_{0}}^{2}}{1-i{{\hslash }}^{2}{ \mathcal C }{\sigma }_{{z}_{0}}^{2}}+\displaystyle \frac{i{\hslash }t}{2{m}^{* }}\\ & = & \displaystyle \frac{{\sigma }_{{z}_{0}}^{2}}{1+{{\hslash }}^{2}{{ \mathcal C }}^{2}{\sigma }_{t}^{4}}+i\left(\displaystyle \frac{{{\hslash }}^{2}{{ \mathcal C }}^{2}{\sigma }_{{z}_{0}}^{4}}{1+{{\hslash }}^{2}{{ \mathcal C }}^{2}{\sigma }_{{z}_{0}}^{4}}+\displaystyle \frac{{\hslash }t}{2{m}^{* }}\right).\end{array}\end{eqnarray*}$$

If the prior chirp factor C is selected such that

$$\begin{eqnarray*}&&\displaystyle \frac{{{\hslash }}^{2}{{ \mathcal C }}^{2}{\sigma }_{{z}_{0}}^{4}}{1+{{\hslash }}^{2}{{ \mathcal C }}^{2}{\sigma }_{{z}_{0}}^{4}}=\displaystyle \frac{{\hslash }{t}_{c}}{2{m}^{* }},\end{eqnarray*}$$

then one obtain

$$\begin{eqnarray}&&{\tilde{\sigma }}_{z}^{2}(t)=\displaystyle \frac{{\sigma }_{{z}_{0}}^{2}}{1+{{\hslash }}^{2}{{ \mathcal C }}^{2}{\sigma }_{t}^{4}}+\displaystyle \frac{i{\hslash }(t+{t}_{c})}{2{m}^{* }}\end{eqnarray} \tag{ A20 }$$

With this substitution, we could consider the prior chirp effect for density bunching, as we did in the equation (12) in the context.