Spectral Twisting in a Common Bosonic Reservoir: Fragility of Two-Qubit Dark-State Protection

Abstract

The interaction of two qubits with a common bosonic reservoir is encoded by a matrix-valued spectral density $\mathbf{J}(\omega)$, whose diagonal entries describe the local spectra, while the off-diagonal entries encode cross-correlations. Even when \(\mathbf{J}(\omega)\) has rank one, a frequency-independent dark channel need not exist because the family \(\{\mathbf{J}(\omega)\}_{\omega}\) may have a trivial common kernel. We term the frequency-dependent rotation of the bright and dark directions \textit{spectral twisting} and quantify it through the Fubini--Study speed $\tau(\omega)$ of the bright spectral projector. We analyze how twisting modifies two-qubit dynamics and quantify the loss of dark-state protection through the leakage \(P_{\mathrm{leak}}(t)\). Comparisons with untwisted asymmetric reservoirs and rotating-wave dynamics, together with detuned and finite-temperature calculations, distinguish spectral twisting from coupling asymmetry, counter-rotating processes, and thermal absorption. For resonant qubits tuned to the crossing of the two local spectra $\omega_\times$, the singlet is locally dark at the transition frequency but couples to off-resonant components whose bright directions are rotated. In the weak-twisting regime, the fixed-time leakage scales as $P_{\mathrm{leak}}(t)\propto[\omega_\times\tau(\omega_\times)]^2$. We test this prediction for mismatched Drude-Lorentz spectra using nonperturbative hierarchical equations of motion generalized to cross-correlated bath forces. These results provide a geometric framework for dark-state engineering in structured reservoirs.