Memory and thermal amplification in spin--cavity squared commutators

Squared commutators in the Holstein–Primakoff limit of a spin–cavity system provide a compact way to separate propagation from covariance growth in a finite-temperature reservoir with memory. In the finite-temperature NMQSD construction, the linear quadrature commutator is fixed by the retarded spin–cavity propagator, whereas a quadratic commutator carries the same retarded factor together with a covariance factor. For a zero-mean Gaussian state, $C_{R_i^2,R_j}(t)=4\mathrm{\mid }{\kappa }_{ij}(t){\mathrm{\mid }}^2{V}_{ii}(t)$; the symmetrized expression gives the spin-side and mixed channels. Since $\bar{n}$ enters the covariance sector but not the homogeneous retarded kernel, raising $\bar{n}$ from 0 to 1 leaves the linear transfer unchanged while increasing the quadratic signal. Varying the bath-memory rate and the counter-rotating coupling within the stable HP region then shows how stored cavity history changes both the transfer weight and its distribution in time. The calculation separates memory-dependent propagation from thermal covariance growth in collective spin–cavity dynamics.