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 $Φ$ and connected to an infinite lead with coupling parameter $ε$. Assuming that the initial state $ψ_0$ of the particle is confined inside the loop and evolves freely, we analyse the time evolution of the nonescape probability $P(ψ_0,L,Φ,ε,t)$, which is the probability that the particle will still be inside the loop at some later time $t$.

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