Spectral Twisting in a Common Bosonic Reservoir: Fragility of Two-Qubit Dark-State Protection
The interaction of two qubits with a common bosonic reservoir is characterized by the spectral-density matrix $\mathbf{J}(ω)$, whose diagonal entries $J_{11}(ω)$ and $J_{22}(ω)$ describe the local spectra, while the off-diagonal entries encode cross-correlations. For a maximally correlated reservoir, $\mathbf{J}(ω)$ has rank one and therefore a \textit{locally} dark coupling direction at each frequency. If $J_{22}(ω)/J_{11}(ω)$ varies with frequency, however, the bright and dark directions rotate and $\ker\mathbf{J}(ω)$ is frequency dependent. We call this \textit{spectral twisting} and quantify it through the Fubini--Study speed $τ(ω)$ of the bright spectral projector. We investigate how spectral twisting affects coupled two-qubit dynamics and dark-state protection. We quantify protection loss by the survival \textit{leakage} $P_{\mathrm{leak}}(t)$, which can become finite for states that are dark only locally in frequency. By comparing rotating-wave dynamics and untwisted asymmetric reservoirs, together with analyzing qubit detuning, we distinguish twisting from coupling asymmetry, thermal absorption, counter-rotating processes, and Hamiltonian symmetry breaking. For mismatched Drude--Lorentz spectra, our nonperturbative hierarchical-equations-of-motion calculations show that twisting induces leakage from the singlet, which is locally dark at the spectral crossing $ω_\times$ defined by $J_{11}(ω_\times)=J_{22}(ω_\times)$. Twisting also shifts the optimally protected state and accelerates the decay of Werner-state concurrence. At fixed observation time, we find the quadratic weak-twisting scaling $P_{\mathrm{leak}}(t)\propto[ω_\timesτ(ω_\times)]^2$. These results may guide dark-state engineering in structured reservoirs, with implications for correlated-noise spectroscopy and decoherence-free encodings.