Quantum Čerenkov Radiation: Spectral Cutoffs and the Role of Spin and Orbital Angular Momentum

We show that the well-known Čerenkov effect contains new phenomena arising from the quantum nature of charged particles. The Čerenkov transition amplitudes allow coupling between the charged particle and the emitted photon through their orbital angular momentum and spin, by scattering into preferred angles and polarizations. Importantly, the spectral response reveals a discontinuity immediately below a frequency cutoff that can occur in the optical region. Near this cutoff, the intensity of the conventional Čerenkov radiation (ČR) is very small but still finite, while our quantum calculation predicts exactly zero intensity above the cutoff. Below that cutoff, with proper shaping of electron beams (ebeams), we predict that the traditional ČR angle splits into two distinctive cones of photonic shockwaves. One of the shockwaves can move along a backward cone, otherwise considered impossible for conventional ČR in ordinary matter. Our findings are observable for ebeams with realistic parameters, offering new applications including novel quantum optics sources, and opening a new realm for Čerenkov detectors involving the spin and orbital angular momentum of charged particles.

Popular Summary

When a charged particle travels faster than the phase velocity of light in a medium, a shockwave of light known as Čerenkov radiation is produced; this radiation is responsible for the famous “bluish glow” in nuclear reactors. During the 80 years that have elapsed since its discovery, the Čerenkov effect has appeared in almost all fields of physics: Čerenkov detectors in high-energy physics, quantum cascade lasers, nonlinear optics, metamaterials, and biological imaging. However, despite the immense progress that has been made in these fields, researchers still regard the original Čerenkov effect classically—as radiation emitted by a point charge—as they first did in 1937. Here, we predict new effects, occurring because of the quantum wave nature of a charged particle, that create unexpected deviations from the conventional Čerenkov theory.

Our work considers the particle as a wave packet instead of assuming it is a classical particle or approximating it as a plane wave. We find a frequency cutoff in the spectrum of the emitted radiation because of the recoil of the electron by the emission of a single photon quantum. Furthermore, we study how the spin of the electron and the polarization of the emitted photon mix with the orbital angular momentum of the electron and of the photon, together giving rise to complex structure. Namely, the emitted photons couple to the electron through their orbital angular momenta and polarization, resulting in preferred angles of emission along with specific selection rules. Importantly, our work implies that the recently predicted and observed electron vortex beams could emit Čerenkov radiation through a double cone instead of a single one, thus splitting the famous Čerenkov angle into two.

The first person to explore the Čerenkov effect using a quantum formalism was the Russian Nobel Laureate Vitaly Ginzburg in 1940. However, at that time, it was believed that the quantum picture “coincides with the classical expression with accuracy up to infinitely small terms.” Our work may drastically change this view, particularly given the promise of modern materials and more precise instruments that facilitate observing features such as the spectrum and orbital angular momentum of photons and electrons.

Figure 1

Illustration of the ČR process. The incoming (outgoing) electron is drawn in blue (red), and the emitted photon in green.

Figure 2

Matrix-element amplitudes for the ČR process, as a function of the photon wavelength $\lambda$ and emission spread angle ${\theta }_{ph}$. The color map shows the spatial part of the matrix element [Eq. (3)] that vanishes outside of the permitted zone, bounded by the blue, red, and green dashed curves. Along these curves, the amplitude diverges; thus, we use a saturated color scale, with darker colors corresponding to higher transition amplitudes. The divergences are highlighted by the cross-section plots below the maps, also showing the nodal lines of zero amplitude (marked by black circles) between high amplitude “stripes.” These distinct stripes depend on the OAM [$l_{ph}=4$, 8 in (a) and (b)], showing the coupling between the OAM and preferred emission angles inside the permitted zone (which is independent of angular momentum). The black dashed curve denotes the angle of the conventional ČR [Eq. (1a)], changing with $\lambda$ in (a) because of the silica dispersion. The spectral cutoffs are marked by solid purple lines—beyond these wavelengths, no photons are emitted. At the points tangential to this line, ${\theta }_{\mathrm{Č}\mathrm{R}}=0$; hence ${\theta }_{ph}={\theta }_i$ [equals 10.3° in (a) or 0.1° in (b)]. The 4-order-of-magnitude difference between the two panels results from the 2-order-of-magnitude difference in ${\theta }_i$, and the 2-order-of-magnitude difference in ${\theta }_{ph}$ (as shown by the horizontal axes), both of which contribute to the calculation of the area $S_{\mathrm{\Delta }}$. All figures have $\beta =0.685$ and the refractive index of silica including its material dispersion (a) or constant $n=1.45986$ (b).

Figure 3

Deviations between classical and quantum ČR theories—conventional vs QED photon emission rates, demonstrated with Bessel ebeams (solid curves) and Gaussian ebeams (dashed curves). Dashed black lines: Conventional ČR result, according to Frank-Tamm [Eq. (1b)]. Green and orange lines: Quantum ČR for emission into azimuthal and radial polarizations, according to Eq. (S41) (red curve is for their sum). Blue line: The Čerenkov angle [in (e) and (f)], showing a spectral cutoff where it becomes zero and spectral kinks where it crosses ${\theta }_i$, both marked by gray dotted lines. Unlike the conventional Čerenkov theory, the quantum theory predicts ČR that involves electron spin flip [panels (a) and (b)] having discontinuities at both cutoffs. The dominant part of the ČR emission [no spin flip, panels (c) and (d)] matches the conventional theory in (c) but deviates from it near ${\omega }_{\text{cutoff}}$ in (d). The kinks (abrupt change in slope), which are also emphasized in the insets, are a unique feature of the Bessel ebeam that is smoothened out in Gaussian ebeams—compare solid and dashed curves in (c) and (d). For the current choice of parameters, panels (a) and (b) are practically agnostic to ${\theta }_i$. Panels (a) and (c) present ČR in silica for a Bessel ebeam with ${\theta }_i=10.3°$ or Gaussian with spread FWHM of 10.3°, both showing a “trivial” spectral cutoff at $\lambda =550\mathrm{nm}$ (occurs since $n$ drops below $1/\beta$). Panels (b) and (d) present ČR assuming a constant $n=1.45986$ for a Bessel ebeam with ${\theta }_i=0.1°$ or Gaussian with spread FWHM of 0.1°, both showing a new spectral cutoff at $\lambda =244\mathrm{nm}$ [occurs because of the quantum correction to ČR from electron recoil, as shown by ${\omega }_{\text{cutoff}}$ in Eq. (5)]. A different choice of $n$ causes ${\omega }_{\text{cutoff}}$ to shift, yet its exact value only matters at the proximity of the cutoff: no need for a constant $n$ over a wide spectrum. All ebeams have $\beta =0.685$ ($\sim 190\mathrm{keV}$); while the Bessel ebeams are mono-energetic, the Gaussians have an energy uncertainty of $\mathrm{\Delta }E=0.5\mathrm{eV}$. The predicted effects are not special for the current choice of parameters.