Coherence-decoherence interplay in quantum systems due to projective stochastic pulses: The case of Rabi oscillations

The interplay of coherence and decoherence is played out in a three-level quantum system, in which the third level is incoherently coupled to the second one which itself is in coherent interaction with the first level. The study is based on a stochastic scenario in which the coherent, unitary evolution of the system is randomly interrupted by a Poisson-driven pulse sequence. In the absence of an external pulse, the system undergoes coherent, unitary evolution restricted to the subspace spanned by the first level (level $1$) and the second level (level $2$). The application of a pulse induces transitions between the second and the third level (level $3$), thereby introducing non-unitary effects that perturb the otherwise isolated two-level dynamics. The pulses are assumed to have infinitesimal duration, with strengths modeled as random variables that are uncorrelated across different pulses. A representative model for the stochastically-averaged transition (super)operator mimicking the dynamics induced by the application of pulses allows for an analytical derivation of the matrix elements of the averaged density operator. When the system is initially in level $1$, we obtain in particular the temporal behavior of the stay-put probability, that is, the probability $P_1(t)$ that the system is still in level $1$ at time $t$. As a function of time, the quantity $P_1(t)$ exhibits a coherence-to-decoherence crossover behavior. At short times $t \ll 1/\lambda$, where $\lambda$ is the average frequency at which pulses are applied to the system, coherent dynamics dominate. Consequently, $P_1(t)$ displays pronounced Rabi-like oscillations. At long times $t \gg 1/\lambda$, decoherence effects prevail, leading to an exponential decay of the form $P_1(t) \sim \exp(-\lambda t)$.