Escape behaviour of a quantum particle in a loop coupled to a lead

We consider a one-dimensional loop of circumference $L$ crossed by a constant magnetic flux $\Phi$ and connected to an infinite lead with coupling parameter $\epsilon$. Assuming that the initial state $\psi_0$ of the particle is confined inside the loop and evolves freely, we analyse the time evolution of the nonescape probability $P(\psi_0,L,\Phi,\epsilon,t)$, which is the probability that the particle will still be inside the loop at some later time $t$.