Orbital angular momentum radiation and polarization of relativistic electrons in magnetic fields

1. Problem Statement

While the Sokolov-Ternov effect—the spontaneous spin polarization of relativistic electrons in storage rings due to synchrotron radiation—is a well-established phenomenon (predicting a polarization limit of $\sim 92.38\%$), the behavior of Orbital Angular Momentum (OAM) in similar radiative processes remains largely unexplored.

• The Gap: Vortex electron beams (carrying intrinsic OAM) are increasingly relevant for high-energy physics and microscopy. However, it is unknown whether synchrotron radiation can induce a net polarization of the OAM degree of freedom, analogous to spin polarization.
• The Challenge: A unified quantum-electrodynamic (QED) treatment is required that simultaneously accounts for both spin and OAM degrees of freedom, particularly in the regime of high electron energy and low photon emission.

2. Methodology

The authors employ a rigorous theoretical framework based on the Dirac equation in a uniform magnetic field.

• Theoretical Foundation:

  • The system is modeled using the Dirac wave function for an electron in a uniform vertical magnetic field ($H$).
  • Quantum states are defined by principal quantum number $n$, radial quantum number $s$, OAM quantum number $\ell$, and spin projection $\zeta$.
  • Transition probabilities for single-photon emission are calculated using time-dependent perturbation theory within the second-quantized formalism.

• Approximation Techniques:

  • High-Energy/Low-Photon-Energy Limit: The analysis focuses on the regime where electron energy $E \gg m_0c^2$ and emitted photon energy $\hbar\omega \ll E$. This corresponds to the emission of "soft" photons where quantum number changes are small ($|\Delta n|, |\Delta \ell| \sim 1$).
  • WKB Approximation: For large quantum numbers ($n, n' \gg 1$), the authors apply the Wentzel-Kramers-Brillouin (WKB) approximation to evaluate the radial matrix elements (involving Laguerre functions). This allows for the derivation of analytical expressions for transition rates that would otherwise require complex numerical integration.
  • Asymptotic Analysis: The behavior of modified Bessel functions ($K_{1/3}, K_{2/3}$) in the small-argument limit is utilized to determine the asymmetry in transition rates.

3. Key Contributions

• Unified Formalism: The paper provides the first comprehensive analytical derivation of OAM transition rates and radiation intensities that explicitly couple spin and OAM dynamics in synchrotron radiation.
• Discovery of OAM Asymmetry: The authors demonstrate that, similar to the spin-flip asymmetry in the Sokolov-Ternov effect, there is a fundamental asymmetry in OAM transitions. Specifically, transitions that decrease the OAM quantum number ($\Delta \ell = -1$) are significantly more probable than those that increase it ($\Delta \ell = +1$).
• Analytical Rate Equations: The study derives explicit formulas for the transition rates $w_+$ (OAM increase) and $w_-$ (OAM decrease) and the resulting relaxation dynamics.

4. Key Results

A. Asymmetry in Transition Rates

In the low-photon-energy limit, the ratio of the transition rates for increasing vs. decreasing OAM is calculated as:
$$\frac{w_+}{w_-} \approx 0.3393$$
This indicates that transitions reducing the OAM are approximately three times more probable than those increasing it. This asymmetry drives the electron beam toward a state of net OAM polarization.

B. Polarization Dynamics and Stationary State

• Spin Polarization: The system reaches a stationary state where the spin polarization $P_{spin}$ approaches 92.38% (with 96.19% of electrons having spins antiparallel to the field), consistent with the classic Sokolov-Ternov limit.
• OAM Polarization: Unlike spin, which collapses into a single state, OAM polarization spreads over the lowest accessible states.
  • For large initial OAM ranges ($\ell_0$), the stationary-state polarization $P_{OAM}$ approaches unity (100%).
  • Approximately 96.09% of the electron population accumulates in the three states nearest the minimal OAM projection.
• Relaxation Time: A critical finding is the timescale difference. The characteristic relaxation time for OAM polarization ($\tau_{OAM}$) is orders of magnitude shorter than the spin polarization time ($\tau_{spin}$).
  • $\tau_{spin}$ is typically on the order of hours for GeV electrons in 1 T fields.
  • $\tau_{OAM}$ is on the order of seconds to minutes.

C. Distribution Characteristics

The stationary OAM distribution is exponential, favoring lower $\ell$ values:
$$n^{(st)}_m \propto \mu^{\ell_0 - m}$$
where $\mu = w_-/w_+ \approx 2.947$. This contrasts with spin, which is a two-level system, whereas OAM involves a multi-level ladder where the beam naturally "drifts" toward the minimum $\ell$.

5. Significance and Implications

• Control of Vortex Beams: The results suggest a mechanism for self-polarizing vortex electron beams in storage rings. Since $\tau_{OAM} \ll \tau_{spin}$, synchrotron radiation can rapidly align the OAM of an electron beam without external manipulation.
• New Physics for Accelerators: This extends the Sokolov-Ternov effect to the orbital degree of freedom, offering new tools for generating and controlling structured charged-particle beams for high-energy accelerator applications.
• Astrophysical Relevance: The findings may have implications for understanding radiation from relativistic electrons in astrophysical magnetic fields (e.g., pulsars), where vortex states might naturally arise and polarize.
• Theoretical Bridge: The work bridges the gap between established spin polarization theory and the emerging field of electron vortex beams, providing a foundational radiation theory for future experimental verification.

In conclusion, the paper establishes that synchrotron radiation is not only a source of spin polarization but also a powerful mechanism for OAM polarization, characterized by a rapid relaxation time and a distinct asymptotic distribution favoring minimal orbital angular momentum.