Changing the lift to rotational-energy scaling

PDF p. 4: Final paragraph continuing onto p. 5; PDF p. 5: Opening continuation

Consider the equal-rate benchmark with the selected absorption–emission (Æ) code in the $J=5$ block and the Aydin–Barg code in the $J=4$ block. Replace the original diagonal lift $\mathrm{\Delta }Jm$ by $\mathrm{\Delta }J(J+1)m$, retaining the benchmark's other assumptions and checking-time convention. Adopt the general-lift postulate with $\mathrm{\Delta }(J,m)=\mathrm{\Delta }J(J+1)m$ and the same three downward jump matrices and rate parameter; this fixed-jump continuation after changing the lift is an additional model assumption. Here $J$ labels a spin block, $m=-J,-J+1,\dots ,J$ labels its magnetic basis states, $\mathrm{\Delta }\ge 0$ is the lift parameter, and $\gamma >0$ is the common modeled instantaneous downward rate for states with $J\ge 1$.

At a fixed $J$ and magnetic label $m$, the new lift is $J+1$ times the original lift. For a downward transition $\mathrm{\mid }J,m⟩\to \mathrm{\mid }J-1,m+\delta ⟩$, its contribution to source-minus-destination energy is $\mathrm{\Delta }[2Jm-J(J-1)\delta ]$. Thus, at the same $\mathrm{\Delta }$ and the same magnetic label, the multiplier is $6$ in the Æ host block and $5$ in the Aydin–Barg host block. These energy-shift comparisons alone do not establish an ordering of checking times.

The article reports that this change penalizes the higher-$J$ Æ code, moves the checking-time crossover to approximately $\mathrm{\Delta }\mathrm{/}\gamma =1$, and leaves Aydin–Barg ahead by up to $20\mathrm{\%}$ thereafter. These are attributed predictions of the model, not independently verified numerical inequalities or measured outcomes.

The article passage supplies no separate plotted data or error bars for this alternative-lift comparison. The comparison uses the same nominal $\mathrm{\Delta }$ and equal-rate benchmark; its ordering need not describe a different physical species or a different rate model.