Source-dependent versus uniform Zeeman shifts

PDF p. 3: Paragraph following Eq. (11); PDF p. 1: Introduction, Kerr-anharmonicity background (local conditional calculation)

Assume the Axiom of Choice, Countable Choice and Dependent Choice, and take the finite angular-momentum emitter model with its phenomenological diagonal lift. Its orthonormal states are $\mathrm{\mid }J,m⟩$, where $J$ labels a retained spin block and $m$ its magnetic state. For a transition from $\mathrm{\mid }J,m⟩$ to an existing state $\mathrm{\mid }J-1,m+\delta ⟩$, with $\delta \in \{-1,0,1\}$ and real lift strength $\mathrm{\Delta }$, define transition frequency as source energy minus destination energy in units with $\mathrm{\hslash }=1$.

The lift with level shifts $\mathrm{\Delta }Jm$ changes this transition frequency by $\mathrm{\Delta }[m-(J-1)\delta ]$. For fixed $J$ and $\delta$, this shift differs for distinct allowed source labels $m$ when $\mathrm{\Delta }\mathrm{\ne }0$. By contrast, a diagonal perturbation assigning the level shift $\mathrm{\Delta }m$ to every $\mathrm{\mid }J,m⟩$ changes the transition frequency by $-\delta \mathrm{\Delta }$, independent of $m$. Thus this latter shift alone supplies no source-label distinction through transition frequency within a fixed polarization channel $\delta$. Neither calculation alone determines the full environmental record.

Independently, assume Countable Choice and the prescribed one-mode Kerr model on the finite occupation span, with real parameters $\omega ,K$, occupation states $\mathrm{\mid }n⟩$ for integers $n\ge 0$, and assigned energies $E_n=\omega n+\frac{K}{2}n(n-1)$. The adjacent downward transition frequencies $g_n=E_n-E_{n-1}=\omega +K(n-1)$ satisfy $g_{n+1}-g_n=K$ for every $n\ge 1$. Hence $K\mathrm{\ne }0$ separates each pair of consecutive downward transition frequencies; when $K=0$, all these frequencies equal $\omega$. This case uses Countable Choice for its Fock-space supplier and does not use the finite-emitter or regulated-bath hypotheses. These energy differences alone specify no environmental record or recovery performance.

Phenomenological angular-momentum degeneracy lift

Calibrated diagonal lifts and transition phase differences

Bosonic loss, Kerr shifts and finite-code comparison scope

The Axiom of Countable Choice (AC_ω)

The article describes the first shift as distinguishing decay pathways. The calculation above establishes the stated source dependence of their frequencies; how much information a particular environment acquires requires additional dynamical and measurement assumptions. The independent Kerr calculation is proved here from the prescribed one-mode model.