Effect of Cross-Spectral Correlations on Qubit Dynamics: Coherence Revival and Relaxation Modulation

We investigate the reduced dynamics of a qubit subject to correlated longitudinal and transverse noise arising from its coupling to a shared bosonic bath. The environmental fluctuations are characterized by a positive-semidefinite matrix-valued spectral density, whose complex off-diagonal elements encode correlations between dephasing and relaxation channels in the frequency domain. Within the second-order time-convolutionless framework, we derive closed time-local equations for the Bloch-vector components of the reduced density matrix. The numerical implementation is validated against the exact pure-dephasing solution and the established behavior of the transverse-coupling spin-boson model. When both noise channels are present, the cross-spectral terms couple the otherwise distinct dephasing and relaxation sectors, producing dynamics that cannot be reproduced by adding independent noise contributions. In particular, the correlations generate non-monotonic population relaxation and a transient revival of coherence following its initial decay. The strength, bandwidth, delay, and phase of the cross spectrum provide control parameters for the magnitude and temporal structure of these effects. Our results demonstrate that correlated multi-axis noise can redistribute coherence loss and energy relaxation in time, thereby providing finite temporal windows of enhanced coherence or suppressed relaxation within the weak-coupling regime.