Phenomenological angular-momentum degeneracy lift

PDF p. 3: Eq. (11); PDF p. 2: Fig. 1(b); PDF p. 13: Final paragraph of S3

Use the angular-momentum emitter model in units with $\mathrm{\hslash }=1$. For a real lifting strength $\mathrm{\Delta }$, add the phenomenological term $H_{\mathrm{\Delta }}=\mathrm{\Delta }{\sum }_{J,m}Jm\mathrm{\mid }J,m⟩⟨J,m\mathrm{\mid }$ to the emitter Hamiltonian. Here $J$ labels a spin block and $m=-J,-J+1,\dots ,J$ labels its magnetic states. The term is separate from the spontaneous-decay coupling; it is introduced to study the effect of lifting degeneracy on decay-path distinguishability.

For the finite-emitter realization, the sum runs over the finitely many complete spin blocks retained in the emitter model. A state's lift is $\mathrm{\Delta }Jm$. Thus a transition from $\mathrm{\mid }J,m⟩$ to $\mathrm{\mid }J-1,m+\delta ⟩$ has lift contribution $\mathrm{\Delta }[m-(J-1)\delta ]$ to its source-minus-destination energy. In particular, for fixed $J$ and $\delta$, this contribution changes with the source label $m$ when $\mathrm{\Delta }\mathrm{\ne }0$. The article's Fig. 1(b) illustrates $J=4$ to $J=3$ transitions; its marked $6\mathrm{\Delta }$ and $9\mathrm{\Delta }$ are the signed lift shifts of the $J=3$, $m=2$ and $m=3$ levels relative to their unlifted energies.

This postulate specifies energy shifts, not an environmental measurement or a complete decay channel. Different transition energies alone do not establish perfect path discrimination or a measured recovery performance. The article's unrestricted sum is model notation; no domain or self-adjointness claim for an infinite-block operator is made here.

Angular-momentum emitter, photon bath, and dipole approximation

Calibrated diagonal lifts and transition phase differences

Finite spin blocks, spherical tensors and Clebsch–Gordan phases

By [F1] and [F2], the finite sum has an orthonormal basis of eigenvectors and real eigenvalues $\mathrm{\Delta }Jm$; it therefore defines a Hermitian diagonal operator. Adding it to the emitter term while leaving the bath coupling as specified is the physical modeling assumption, rather than a derivation of a particular field or apparatus.

By [F3], the shift for the stated transition is $\mathrm{\Delta }[Jm-(J-1)(m+\delta )]=\mathrm{\Delta }[m-(J-1)\delta ]$. For $J=3$, the shifts at $m=2$ and $m=3$ are respectively $6\mathrm{\Delta }$ and $9\mathrm{\Delta }$, matching the figure's labels. These are signed shifts; their magnitudes are $6\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }$ and $9\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }$. The frequency calculation does not specify the bath's conditional states, so the distinguishability motivation does not assert perfect discrimination.