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Here we present an strategy for the derivation of a time-dependent Dyson mapwhich ensures simultaneously the unitarity of the time evolution and theobservability of a quasi-Hermitian Hamiltonian. The time-dependent Dyson map isderived through a constructed Schr\"{o}dinger-like equation governed by thenon-Hermitian Hamiltonian itself; despite its time-dependence our schemeensures the time-independence of the metric operator, a necessary condition forthe observability of the quasi-Hermitian Hamiltonian. As an illustrativeexample we consider a driven Harmonic oscillator described by a time-dependentnon-Hermitian Hamiltonian. After computing the Dyson map and demonstrating thetime-independence of the associated metric operator, we successfully derive aneigenvalue equation for this time-dependent Hamiltonian which enable us toanalyze the $\mathcal{PT}$-symmetry breaking process.

Unitarity of the time-evolution and observability of non-Hermitian Hamiltonians for time-dependent Dyson maps