Observing the evolution of a quantum system that does not evolve

This article deals with the problem of gathering information on the time evolution of a single metastable quantum system whose evolution is impeded by the quantum Zeno effect. It has been found it is in principle possible to obtain some information on the time evolution and, depending on the specific system, even to measure its average decay rate, even if the system does not undergo any evolution at all.

Figure 1

A schematization of higher order processes in the simple case of a two levels system. The system interacts four times with the environment at $t=t_1$, $t_2$, $t_3$, and $t_4$. The projective measurements are performed at $t=0$ and $t=\tau$ and both of them find the system in its excited state. The solid lines show the trajectory of the system. The first one corresponds to a zeroth order process, with no jumps, the second and the third to a second order process, and the fourth to a fourth order process. Clearly, if in the end the system is to be found in its initial state, only even order processes are possible.

Figure 2

The value of $P_e^{NQ}$, with $Q=100$, is plotted using the exact $P_e$ (top panel) and its second order development (bottom panel).