Angular momentum conservation in two-atom superradiance

Technical Summary: Angular Momentum Conservation in Two-Atom Superradiance

Problem Statement
This paper investigates the conservation of angular momentum in a system of two identical atoms, each possessing a $J=0$ ground state and a $J=1$ excited state. The atoms are positioned on the $x$-axis and excited by a $\sigma^+$-polarized laser pulse propagating in the $z$-direction. While the energy expectation value of the radiated field is known to equal the initial atomic energy, the behavior of angular momentum presents a more complex scenario. Specifically, when atoms lie on the $z$-axis, the initial angular momentum ($2\hbar\hat{z}$ for a fully inverted state) is fully transferred to the radiation field. However, when atoms lie on the $x$-axis, the vacuum radiation field induces an exchange between magnetic sublevels (e.g., transferring excitation from atom 1's $m=1$ state to atom 2's $m=-1$ state). This exchange alters the internal angular momentum of the atomic system during decay. The central problem addressed is determining the final angular momentum of the radiated field in this $x$-axis configuration and identifying the mechanism by which total angular momentum is conserved, given that the field's angular momentum does not simply equal the initial atomic angular momentum.

Methodology
The authors employ two complementary theoretical approaches to calculate the final angular momentum in the field ($L_f$) following the decay of the atoms:

1. Source-Field Approach: This method calculates the expectation value of the angular momentum using the Poynting vector derived from electric and magnetic field operators. A critical modification to conventional source-field theory is implemented:

   • The calculation retains intermediate-zone terms in the field operators. While radiation-zone terms alone yield a Poynting vector parallel to the propagation direction (resulting in zero angular momentum), the inclusion of intermediate-zone terms is necessary to capture the correct angular momentum flux.
   • The finite displacement of the second atom (located at $R_{21}\hat{x}$) is explicitly accounted for in the field operator expressions. Corrections arising from this displacement, often negligible in single-atom calculations or energy calculations, are retained as they are essential for angular momentum conservation.
   • The Rotating Wave Approximation (RWA) and Weisskopf-Wigner Approximation (WWA) are applied during the evaluation of operator expectation values, but not in the initial derivation of the source-field expressions.

2. Schrödinger Picture Approach: To verify the results, the authors utilize a method proposed in Ref. [4] to calculate the field angular momentum directly from the quantum state evolution. This approach requires retaining terms that might be expected to vanish at first glance, ensuring consistency with the source-field results.

The system is modeled using "molecular states" (eigenkets of the combined two-atom system) rather than a "bare-atom" basis. This diagonalization simplifies the dynamics, revealing that the vacuum field induces coupling between states such as $|1g\rangle$ and $|g,-1\rangle$, leading to a complex decay dynamics involving coherences between different molecular states.

Key Contributions and Results

• Non-Conservation of Field Angular Momentum: For atoms on the $x$-axis, the final angular momentum in the field ($\langle L_{field}(t=\infty) \rangle$) is not equal to the initial angular momentum of the two-atom system ($2\hbar\hat{z}$).
  • In the case of a fully inverted initial state (Double Excitation, DE), if the atoms are separated by much less than a wavelength, the final field angular momentum is calculated to be $\hbar\hat{z}$.
  • In the case of a single excitation (SE) where atom 1 is in $m=1$ and atom 2 is in the ground state, the final field angular momentum is calculated to be zero for sub-wavelength separations.
• Mechanism of Exchange: The discrepancy arises from the rapid exchange of excitation between magnetic sublevels ($m=1$ and $m=-1$) mediated by the vacuum field. As the first photon is emitted, the system undergoes a transition where atom 2 can be excited into a superposition of $m=\pm 1$ states. Consequently, when the second photon is emitted, no net angular momentum is transferred to the radiation field in the same manner as the initial state.
• Orbital Angular Momentum Compensation: The "missing" angular momentum is shown to appear as orbital angular momentum of the two-atom system. The exchange interaction exerts a torque on the atoms (specifically, transferring excitation from $m=1$ to $m=-1$ exerts a torque on the receiving atom), thereby changing the mechanical angular momentum of the atomic pair. The sum of the field angular momentum and the change in the atoms' orbital angular momentum equals the initial total angular momentum, satisfying conservation laws.
• Theoretical Necessity of Specific Terms: The paper demonstrates that accurate calculation requires including terms in both the source-field and Schrödinger approaches that are typically neglected in single-atom or energy-only calculations. Specifically, the displacement corrections to the field operators and intermediate-zone field contributions are indispensable for capturing the correct angular momentum balance.

Significance
The paper provides the first analysis of angular momentum conservation specifically within the context of two-atom Dicke superradiance, noting that the authors are unaware of any prior papers focusing on the angular momentum of fields radiated by the specific two-atom system considered here. While extensive literature exists on Dicke superradiance and angular momentum in single-atom radiation, this work addresses the fundamental problem of how angular momentum is partitioned between the radiation field and the mechanical motion of the atoms in a multi-atom system. The findings highlight that in superradiant systems with degenerate excited states, the internal state exchange can fundamentally alter the angular momentum transfer to the field, necessitating a re-evaluation of conservation laws in terms of total system (field + atomic orbital) angular momentum rather than just the field or internal atomic spin. The results underscore the importance of geometric configuration (relative orientation of atoms) in determining the radiative properties of quantum systems.