Phase-space description of photon emission

TL;DR

The paper develops a phase-space description of single-photon emission using the Wigner function of the emitted field to capture spatial and temporal features hidden in momentum-space QED. It builds a generalized-measurement framework (including POVMs) for the electron-photon final state, showing how detector choices shape the photon state and enabling tomographic access to emitter coherence. In Cherenkov radiation, the approach predicts novel quantum effects such as negative photon-spreading times and formation lengths, finite emission flash durations tied to the electron packet size, and a quantum arrival-time shift from medium-induced dipole responses, with near-field snapshots that resemble the emitter wave function. The framework extends to dispersive media, yielding dispersion-driven modifications of formation-time scales and angular features, and it offers a path to tomography and diagnostic applications across radiation processes in particle physics and attosecond metrology.

Abstract

Interactions between charged particles and light occur in real space and time, yet quantum field theory usually describes them in momentum space. Whereas this approach is well suited for calculating emission probabilities and cross sections, it is insensitive to spatial and temporal phenomena such as, for instance, radiation formation, quantum coherence, and wave packet spreading. These effects are becoming increasingly important for experiments involving electrons, photons, atoms, and ions, particularly with the advent of attosecond spectroscopy and metrology. Here, we propose a general method for describing the emission of photons in phase space via a Wigner function. Several effects for Cherenkov radiation are predicted, absent in classical realm or in quantum theory in momentum space, such as a finite spreading time of the photon, finite duration of the flash and a quantum shift of the photon arrival time. The photon spreading time turns out to be negative near the Cherenkov angle, the flash duration is defined by the electron packet size, and the temporal shift can be both positive and negative. The characteristic time scales of these effects lie in the atto- and femtosecond ranges, thereby illustrating atomic origins of these macroscopic phenomena. The near-field distribution of the photon field resembles the electron packet shape, thus making ``snapshots'' of the emitter wave function. Our approach can easily be generalized to the other types of radiation and extended to scattering, decay, and annihilation processes, bringing tomographic methods of quantum optics to particle physics.

Figures (15)

• Figure 1: The Cherenkov radiation generated by the atoms, excited by an incident electron in the medium with the refractive index $n(\omega)$. The emitted photon with the momentum ${\bf k}$ is registered by the photon detector. The final electron with the momentum ${\bf p}'$ may or may not be detected in this scheme, which in turn impacts the photon state. The characteristic emission size is defined by Eq. \ref{['emit_size']} and equal to $\beta\gamma\lambda/2\pi$Jackson
• Figure 2: Generation of Cherenkov radiation within the plane-wave approach when the Cherenkov condition, $|{\bm u}_p| > |{\bm u}_k|=1/n$, is met (left upper panel) and when it is not (right upper panel). The photon field is not vanishing within the formation zone even for $|\bm{u}_p|<|\bm{u}_k|$. The lower panel: the wave packet approach, where the real electron packet has a momentum distribution with a width $\sigma \ll m_e$ according to Eq.(\ref{['WPm']}). The Mach angle is defined in Eq.(\ref{['Machth']}). The vector ${\bf r}$ coincides with ${\bm R} = {\bf r} - {\bm u}_p t + (\partial_{{\bf p}} + \partial_{{\bf k}})\zeta_{fi} + \partial_{{\bf p}}\varphi$ from Eq.(\ref{['Rr']}) only for the Gaussian electron packet with $\varphi=0$ at $t=0$ and neglecting the phase $\zeta_{fi} = \text{arg}\,M_{fi}$ of the amplitude, and the difference ${\bm R} - {\bf r}$ is due to the distribution of an electric dipole moment induced in the medium.
• Figure 3: The effective correlation radius, $R_{\rm eff}$, from Eq.(\ref{['tpr']}) for $\beta = 0.99\,(\gamma \approx 7)$, $n=1.7$, $\omega = 10^{-5}m_e$, $\sigma = 10^{-4}m_e$, the corresponding Cherenkov angle is $\theta_{\text{Ch.cl.}} = \arccos 1/\beta n \approx 53.5^\circ$, the Mach angle is $\theta_{\text{Mach}} \approx 143.9^\circ$ (see Eq.(\ref{['Machth']})). When $R_{\text{eff}}(t') \to \infty$ at $t' \gg t_d$, the space-time dependence of the Wigner function vanishes, whereas in the other limit, $R \gg R_{\text{eff}}(t')$, the Wigner function is exponentially suppressed. The upper panels, (a) and (b), are for $\theta_k = \theta_{\text{Ch.cl.}}\approx 53.5^\circ$, the lower panels, (c) and (d) - $\theta_k = 30^\circ$. The left panels, (a) and (c), show the dependence of $R_{\rm eff}$ on the polar angle $\theta_R$ of ${\bm R}$ with the fixed difference in azimuthal angles $\phi_R - \phi_k = 20^\circ$. The right panels, (b) and (d), in turn, are for the fixed $\theta_R = 143.9^\circ$ showing the dependence on $\phi_R - \phi_k$. Clearly, when the photon is emitted not at the Cherenkov angle, the correlation radius is still concentrated in a vicinity of the Mach cone, however it grows much faster with time $t'$ at the angles different from $\theta_{\text{Mach}}$. Here, $\lambda_c = \hbar/m_ec \approx 3.9\cdot 10^{-11}$ cm - the electron Compton wavelength, and $t_c = \lambda_c/c \approx 1.3\cdot 10^{-21}$ s - electron Compton time. Note that $10^6\,t_c \sim 1$ fs, $10^6\,\lambda_c \sim 0.38\,\mu$m.
• Figure 4: The effective correlation radius from Eq.(\ref{['tpr']}) divided by the distance $u_p t'$. Parameters: $n = 1.5$, $\theta_k = \theta_{\rm Ch.cl.}$. (a)$\beta = 0.7\,(\gamma = 1.4),\, \theta_{\text{Ch.cl.}} \approx 17.8^\circ$, $\theta_{\text{Mach}} \approx 107.8^\circ$. (b)$\beta = 0.9999\,(\gamma = 70.7),\, \theta_{\text{Ch.cl.}} \approx 48.2^\circ$, $\theta_{\text{Mach}} \approx 138.2^\circ$. Nearby the Mach angle $\theta_{\text{Mach}}$, the space-time dependence of the photon energy density quickly vanishes within the correlation radius $R < R_{\text{eff}}(t')$. For macroscopic targets of the length $u_p t' > 1$ mm, the correlation radius is always much smaller than the length except for the vicinity of the Mach angle where $R_{\text{eff}} \lesssim u_p t'$. An observer placed at an edge of a target at the distance $R \sim u_p t'$ would only see the photon field at $\theta_R \approx \theta_{\text{Mach}}$, otherwise the Wigner function is exponentially suppressed, see Eq.(\ref{['tpr']}).
• Figure 5: The spreading time $t_d$ from Eq. \ref{['gtp']} of the photon field measured in picoseconds (a) and its inverse (b). The sharp maxima are nearby the Cherenkov angle $\theta_{\text{Ch.cl.}} = \arccos(1/u_p n)$, which is illustrated in panel (c). The Cherenkov condition is not met for the black line in the panel (a), which is why the field quickly spreads during several attoseconds. The panel (b) shows two singular points \ref{['thetainf']}, where $t_d^{-1} = 0$. The width between these points is connected to the quantum recoil ($\omega/\varepsilon \ne 0$, see Eq.\ref{['Dth']}), where the spreading time turns negative. Notice also, that the extremum of the spreading time lies precisely at the Cherenkov angle