Quantum dynamics of incoherently driven V-type system: Analytic solutions beyond the secular approximation

We present closed-form analytic solutions to non-secular Bloch-Redfield master equations for quantum dynamics of a V-type system driven by weak coupling to a thermal bath. We focus on noise-induced Fano coherences among the excited states induced by incoherent driving of the V-system initially in the ground state. For suddenly turned-on incoherent driving, the time evolution of the coherences is determined by the damping parameter $\zeta=\frac{1}{2}(\gamma_1+\gamma_2)/\Delta_p$, where $\gamma_i$ are the radiative decay rates of the excited levels $i=1,2$, and $\Delta_p=\sqrt{\Delta^2 + (1-p^2)\gamma_1\gamma_2}$ depends on the excited-state level splitting $\Delta>0$ and the angle between the transition dipole moments in the energy basis. The coherences oscillate as a function of time in the underdamped limit ($\zeta\gg1$), approach a long-lived quasi-steady state in the overdamped limit ($\zeta\ll 1$), and display an intermediate behavior at critical damping ($\zeta= 1$). The sudden incoherent turn-on generates a mixture of excited eigenstates $|e_1\rangle$ and $|e_2\rangle$ and their in-phase coherent superposition $|\phi_+\rangle = \frac{1}{\sqrt{2\bar{r}}}(\sqrt{r_1} |e_1\rangle + \sqrt{r_2}|e_2\rangle)$, which is remarkably long-lived in the overdamped limit (where $r_1$ and $r_2$ are the incoherent pumping rates). Formation of this coherent superposition {\it enhances} the decay rate from the excited states to the ground state. In the strongly asymmetric V-system where the coupling strengths between the ground state and the excited states differ significantly, we identify additional asymptotic quasistationary coherences, which arise due to slow equilibration of one of the excited states. Finally, we demonstrate that noise-induced Fano coherences are maximized with respect to populations when $r_1=r_2$ and the transition dipole moments are fully aligned.