Microscopic Derivation of Joint-Bath Decoherence in Coupled Bosonic Reservoirs

We present a first-principles derivation of the joint-bath decoherence
cross-coupling term introduced phenomenologically in a companion study
(Paper I). Starting from a generalized Caldeira–Leggett model with
mutually coupled bosonic baths, we perform a canonical transformation
to mass-weighted coordinates, diagonalizing the bath Hamiltonian into
independent normal modes. This transformation yields an explicit
analytical expression for the cross-spectral density J₁₂(ω). We show
that the Cauchy–Schwarz inequality provides a fundamental bound on the
normalized cross-correlation coefficient, ensuring |ρ(ω)| ≤ 1.

Numerical validation on a minimal two-mode Ohmic–Drude model indicates
that the phenomenological value ρ ≈ 0.855 (extracted from experimental
interferometry data in Paper I) is consistent with a weak-to-moderate
inter-bath coupling strength λ_opt ≈ 0.095, corresponding to a
physically realistic coupling ‖H_B1B2‖/‖H_Bi‖ ≲ 12%.

Finally, we demonstrate that the bilinear form c·u·v used in the
phenomenological model arises as the leading-order Taylor approximation
of the exact microscopic √(uv) dependence around a nominal operating
point (u₀,v₀), consistent with physical boundary conditions. This work
provides a microscopic foundation for the empirical joint-bath model
and opens pathways for future non-Markovian and solid-state extensions.

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