Constraints on relaxation rates forN-level quantum systems

We study the constraints imposed on the population and phase relaxation rates by the physical requirement of completely positive evolution for open $N$-level systems. The Lindblad operators that govern the evolution of the system are expressed in terms of observable relaxation rates, explicit formulas for the decoherence rates due to population relaxation are derived, and it is shown that there are additional, nontrivial constraints on the pure dephasing rates for $N>2$. Explicit, experimentally testable inequality constraints for the decoherence rates are derived for three- and four-level systems, and the implications of the results are discussed for generic ladder, $\Lambda$, and V systems and transitions between degenerate energy levels.