Abstract: Using quantum field theory, we calculate Čerenkov radiation from an electron with a vortex-shape wavefunction. When it travels through a photonic waveguide, it emits a photon with a specific frequency, angle, and angular momentum.

When a charged particle travels faster than the phase velocity of light in a medium, it produces Čerenkov radiation (ČR). Although it is now 80 years since Čerenkov’s observation [Nobel 1958], surprisingly little attention was given to the importance of the quantum nature of the charged particles producing the effect. The few works that did study ČR in a quantum formalism considered the electron either as a point particle or as a plane wave, and were focused on classical aspects of the radiation (e.g., mean intensity as a function of angle and frequency), reconfirming the classical results (e.g., [1]). However, many fundamental questions concern the quantum aspects of the emitted radiation, such as angular momentum and entanglement thereof, and the effects of specific electron wavepackets on the ČR. This issue is of great interest today, because it recently became possible to shape the quantum wavefunction of a single electron, imprinting it with orbital angular momentum (OAM) [2]. The promise of such electron beams ranges from manipulating matter on the nano-scale, improving electron microscopy, measuring magnetic responses in biology, and much more [2].

Here, we study the Čerenkov radiation emitted from charged particles described by a quantum wavepacket. For this goal, we develop a matrix elements formalism for scattering of an incoming charged fermion into a fermion and a photon, with each particle characterized by a certain angular spread and orbital angular momentum (OAM). We find an upper frequency cutoff, above which the ČR is zero. Immediately below this cutoff, our formalism predicts a resonance in the emission rate, manifested in a diverging resonant transition rate occurring for particles with velocity close to the critical velocity for ČR. This frequency cutoff and the immediate resonance represent a clear deviation from the conventional ČR theory that displays no such cutoff whatsoever. Moreover, we find complete closed-form expressions for the photon emission rate, showing that the resonance emission is sensitive to the incoming particle spin and the polarization of the emitted photon. This implies that certain spin flip transitions can dominate the process, even when the emitted photon is in the optical range. Moreover, we show that when the electron beam travels through a photonic waveguide, the radiation can be focused to specific discrete angles and frequencies. This controlled ČR, when combined with a helical waveguide, can serve as a source of monochromatic high-frequency vortex beams.

Consider the EM interaction relating a Dirac spinor wavefunction with the EM fields surrounding it. This is naturally done in cylindrical symmetry, where the electron wavefucntion can be spanned by Bessel-like solutions [3]. Previous derivations of ČR treat the electron as a point particle, and find the radiation as a shock front formation. Here instead, we develop a new formalism, starting with a second quantized Dirac Hamiltonian, and calculate the transitions amplitudes M_i→f=∫d⁴x‹e_f,γ|j^μA_μ|e_i,0› between cylindrical spinors described by their spin, linear momentum, and a discrete angular momentum defined around the propagation axis. Surprisingly, we reach a closed-form expression for M_i→f that gives much insight into the transition rules, including all conservation laws, the ČR angle, and the effect of the OAM. This expression shows that the emission angle differs from the well-known ČR angle θ_Čerenkov=acos(c/(nv)), instead following two distinct peaks. Importantly, the effects still remain even for arbitrarily low frequencies, assuming the initial electron parameters are chosen correctly. Finally, by directing the electron though a cylindrical waveguide, where the photonic modes are discretized, we get precise control over the radiation angle and frequency. The dispersion relations of the photons (blue curves) show exactly what transition can occur. These 1D blue lines introduce a very stringent condition that can lead to a tightly controlled emission of a single frequency and angle.

     There are countless scenarios where ČR occurs and even used to observe other effects (nuclear reactors, interactions of high-energy particles in the atmosphere, particle physics experiments, etc.). Hence, these new effects can have promising applications: such as measuring the electron wavefunction through the emitted ČR, creating new sources of coherent radiation, and new methods for particle acceleration by enhanced Inverse-Čerenkov Effect.

[1] J. M. Jauch & K. M. Watson, Phys. Rev. 74, 1485 (1948); S. M. Neamtan, Phys. Rev. 92, 1362 (1953); D. A. Tidman, Nucl. Phys. 2, 289 (1957). 

[2] K. Y. Bliokh, et al., PRL 99, 190404 (2007); M. Uchida and A. Tonomura, Nature 464, 737 (2010); J. Verbeeck, et. al, Nature 467, 301 (2010).

[3] K. Y. Bliokh, M. R. Dennis, and F. Nori, PRL 107, 174802 (2011).