Photon emission by vortex particles accelerated in a linac

Abstract

We study the photon emission by charged spinless particles with phase vortices and an orbital angular momentum (OAM) projection in longitudinal electric and magnetic fields within the scalar QED. A realistic wave packet of an electron or ion accelerated by a radio-frequency wave locally feels a constant and spatially homogeneous field, which allows us to develop an effective model for losing the angular momentum of the vortex particle due to photon emission. For the fields typical for accelerator facilities, we find that an effective lifetime of the vortex state greatly exceeds the acceleration time. This proves that the acceleration of vortex electrons, ions, muons, and so forth to relativistic energies is possible in conventional linacs, as well as in the wake-field accelerators with higher field gradients, the OAM losses due to the photon emission are mostly negligible, and that the vortex quantum state is highly robust against these losses.

Figures (7)

• Figure 1: The electron is bounded by a magnetic field $\bm{H}$ and accelerated by an electric field $\bm{E}$, both of which are directed along the $z$-axis. During the acceleration the electron undergoes the transition between the Landau states with quantum numbers from $n,l,p_z$ to $n',l',p_z'$. The transition comes with the emission of the photon, thus, the root mean squared transverse radius of the Landau state changes from $\sqrt{\langle\rho^2\rangle} = \rho_{\rm H}\sqrt{2n + l + 1}$ to $\sqrt{\langle\rho'^2\rangle} = \rho_{\rm H}\sqrt{2n' + l' + 1}$ and becomes smaller. The longitudinal size of the electron packet, $\sigma$, remains unchanged during the emission.
• Figure 2: Double-differential probability, $d^{2}W/(dp'_z\,dk_\perp)$, of plane-wave photon emission by an accelerated vortex particle as a function of the photon transverse momentum $k_\perp$. The S-matrix is strongly localized around $k_{\perp\text{max}}\sim 2/\rho_\text{H}$. Parameters: $H=10~\mathrm{T}$, $E=10~\mathrm{MV/m}$, $\sigma=1~\mathrm{nm}$ (coherence length), $p_z=0.5~\mathrm{keV}$ (initial longitudinal electron momentum), $k_z= 3~\mathrm{eV}$ (longitudinal photon momentum), $n=n'=3$, $l = l' = 3$, $t_{\mathrm{out}}=25~\mathrm{ns}$$(L= c t_\text{out} \approx 7~\mathrm{m})$.
• Figure 3: Differential probability of photon emission, $dW/dp'_z$, versus the final longitudinal momentum $p'_z$ for several final OAM projections $l'$ at fixed initial $l=3$. The distribution is peaked near $p'_z=p_z$, i.e., emission is dominated by soft photons $k_z=p_z-p'_z$. Parameters: $H=10~\mathrm{T}$, $E=10~\mathrm{MV/m}$, $\sigma=1~\mathrm{nm}$ (coherence length), $p_z=0.5~\mathrm{keV}$ (initial longitudinal electron momentum), $n=n'=3$, $t_{\mathrm{out}}=25~\mathrm{ns}$$(L=c t_\text{out} \approx7~\mathrm{m})$.
• Figure 4: Lifetime $\tau$ of a vortex state against OAM-changing transitions as a function of the initial OAM $l$ for different magnetic fields $H$. Decreasing $H$ suppresses transverse emission and increases $\tau$. (a) Parameters: $E=10~\mathrm{MV/m}$, $\sigma=1~\mathrm{nm}$ (coherence length), $p_z=0.5~\mathrm{keV}$ (initial longitudinal momentum), $n=n'=3$, $t_{\mathrm{out}}=25~\mathrm{ns}$$(L=c t_\text{out} \approx7~\mathrm{m})$. (b) Same dependence for a higher energy and stronger field: $E=100~\mathrm{MV/m}$, $p_z=100~\mathrm{keV}$, $n=n'=10$, $t_{\mathrm{out}}=55~\mathrm{ns}$$(L=c t_\text{out} \approx15~\mathrm{m})$; the qualitative trend with $H$ remains the same.
• Figure 5: Squared S-matrix element $|S_{fi}|^{2}$ for the transition $n=n'=3$, $l=3\to l'=2$ as a function of accelerating electric field $E$ for different packet lengths $\sigma$. The graph shows the crossover from the narrow-packet regime (weak $E$-dependence) to the wide-packet regime (strong $E$-dependence). Parameters: $H=10~\mathrm{T}$, $p_z=0.5~\mathrm{keV}$ (initial longitudinal electron momentum), $k_z=3~\mathrm{eV}$ (longitudinal photon momentum), $t_{\mathrm{out}}=25~\mathrm{ns}$$(L=7~\mathrm{m})$.