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 $\mathbf{J}(\omega)$, whose diagonal entries $J_{11}(\omega)$ and $J_{22}(\omega)$ describe the local spectra, while the off-diagonal entries encode cross-correlations. For a maximally correlated reservoir, $\mathbf{J}(\omega)$ has rank one and therefore a \textit{locally} dark coupling direction at each frequency. If $J_{22}(\omega)/J_{11}(\omega)$ varies with frequency, however, the bright and dark directions rotate and $\ker\mathbf{J}(\omega)$ is frequency dependent. We call this \textit{spectral twisting} and quantify it through the Fubini--Study speed $\tau(\omega)$ 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 $\omega_\times$ defined by $J_{11}(\omega_\times)=J_{22}(\omega_\times)$. 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 $P_{\mathrm{leak}}(t)\propto[\omega_\times\tau(\omega_\times)]^2$. These results may guide dark-state engineering in structured reservoirs, with implications for correlated-noise spectroscopy and decoherence-free encodings.