We establish the complete positivity of a quantum dynamical map associated with the quantum optical master equation of the partial secular Bloch-Redfield type, which describes the generation of noise-induced Fano coherences in three-level $\mathrm{\Lambda }$ and V systems driven by incoherent blackbody radiation. To this end, we calculate the equation's Kossakowski matrix by evaluating the action of the Liouvillian on the orthonormal $\mathrm{SU}(N$) basis matrices and then computing a sum of matrix-product traces [M. J. W. Hall et al., Phys. Rev. A 89, 042120 (2014)]. We also show that the Kossakowski matrix can be calculated using an alternative approach based on the Moore-Penrose pseudo-inverse of a rectangular tensor composed of the structure constants of $\mathrm{SU}(N$). Using these approaches, we find that the eigenvalues of the Kossakowski matrix are identical for the $\mathrm{\Lambda }$ and V systems, implying hitherto undetected similarities in their dissipative dynamics, further elucidated by examining the diagonal Lindblad operators constructed from Kossakowski matrix spectra. The approaches explored here enable the extraction of the Kossakowski matrix directly from the Liouvillian in the system eigenstate basis and do not require one to provide an operator form of the quantum master equation. They are expected to be particularly advantageous in situations where quantum master equations are defined only by their Liouvillians, facilitating valuable insights into dissipative processes in open multilevel quantum systems of prime importance in precision measurement, photosynthetic light harvesting, and quantum thermodynamics.

Close

PrevNext

View figure in article

• Quantum master equation