Dark state role in time-reversal symmetry breaking

We investigate the role of the global driving phase Φ in the dynamics of driven few-level quantum systems, a central setting in coherent control of atomic, molecular, and solid-state platforms. In particular, we focus on systems with closed-loop couplings, where external driving fields induce interference effects that strongly influence population transfer and symmetry properties of time-evolution. While full time-reversal symmetry requires Φ=0,π, leading to a real Hamiltonian, we focus on a less restrictive transformation, the phase inversion (or complex conjugation of the Hamiltonian), under which population dynamics can remain symmetric even though coherences generally do not. We show that the presence of a dark (spectator) state is a sufficient condition for this population phase symmetry (PΦS), as it constrains the dynamics to reduced subspaces characterized by SU(2) or open-loop SU(3) evolution. We analyze this mechanism in three- and four-level systems and derive general conditions for PΦS that extend to generic n-level configurations, with n even. These findings provide practical guidelines for achieving robust control in quantum systems, with potential applications in quantum information processing and quantum computing.