Characterize the mode-frequency dependence of the break-even coupling

Characterize how the break-even collective coupling \(\lambda_{\rm be}=g_0N\) depends on the mode frequency \(\nu\), and determine why it does not collapse onto the static Dicke superradiant threshold when \(\nu\) is varied.

Background

At fixed mode frequency $\nu=5$, the authors find that the break-even ion number is governed by the collective coupling and corresponds to $\lambda_{\rm be}\simeq2.17$. They test the conjecture that this break-even coupling should scale with the static Dicke threshold $\lambda_c$.

Scanning $\nu$ from 3 to 10 produces 29% scatter in $\lambda_{\rm be}$, and rescaling by $\lambda_c$ does not improve the collapse. The frequency dependence therefore remains unresolved.