Zeno dynamics in wave-packet diffraction spreading

We analyze a simple and feasible practical scheme displaying Zeno, anti-Zeno, and inverse-Zeno effects in the observation of wave-packet spreading caused by free evolution. The scheme is valid both in spatial diffraction of classical optical waves and in time diffraction of a quantum wave packet. In the optical realization, diffraction spreading is observed by placing slits between a light source and a light-power detector. We show that the occurrence of Zeno or anti-Zeno effects depends just on the frequency of observations between the source and detector. These effects are seen to be related to the diffraction mode theory in Fabry-Perot resonators.

Figure 1

(a) Unobserved diffraction. (b) Observed diffraction.

Figure 2

Normalized power $P_N/P_0$ in the detector slit (a) as a function of the number of slits for a fixed source-detector distance ${\zeta }_{\mathrm{D}}$ and (b) as a function of the distance ${\zeta }_{\mathrm{D}}$ for a few, increasing number of slits within ${\zeta }_{\mathrm{D}}$.

Figure 3

Normalized power $P\left(\zeta \right)/P_0$, where $P\left(\zeta \right)$ is computed according to Eq. (3), as a function of the propagation distance $\zeta$ for different number of intermediate slits within ${\zeta }_{\mathrm{D}}=2.0$. Steps are at ${\zeta }_n=n({\zeta }_{\mathrm{D}}/N)$, $n=0,1\cdots ,N$ for each $N$. The lowest step of each staircase is the power that arrives at the detector in each case.

Figure 4

(a) Open circles: Normalized power $P_N/P_0$ in the detector slit as a function of the number of intermediate slits for some given values of ${\zeta }_{\mathrm{D}}$. Dashed curves: prediction of Eq. (13), valid for $N\ll {\zeta }_{\mathrm{D}}$. (b) Solid curves: Normalized power $P\left(\zeta \right)/P_0$ as a function of the propagation distance $\zeta$ for different number of slits within ${\zeta }_{\mathrm{D}}=15$. The lowest step of each staircase is the power that arrives at the detector in each case. Dashed curves: the same but using the approximate values ${(2N/\pi {\zeta }_{\mathrm{D}})}^n$, $n=0,1\cdots ,N$, at ${\zeta }_n=n({\zeta }_{\mathrm{D}}/N)$.

Figure 5

Scheme for the inverse-Zeno effect. (a) Unobserved diffraction. (b) Observed diffraction.

Figure 6

For the inverse-Zeno scheme of Fig. 5, (a) normalized power $P_N/P_0$ on the detector slit as a function of the number of intermediate, laterally displaced slits for a fixed source-detector distance ${\zeta }_{\mathrm{D}}$; (b) normalized power $P\left(\zeta \right)/P_0$ as a function of the propagation distance $\zeta$ for different number of intermediate slits within ${\zeta }_{\mathrm{D}}$. Steps are at ${\zeta }_n=n({\zeta }_{\mathrm{D}}/N)$, where the laterally displaced slits by $\Delta {\xi }_n=n(2/N)$, $n=0,1\cdots ,N$ for each $N$ are placed. The lowest step of each staircase is the power that arrives at the detector in each case.

Figure 7

Normalized power $P\left(\zeta \right)/P_0$ as a function of the propagation distance $\zeta$ for different large numbers of intermediate slits within ${\zeta }_{\mathrm{D}}=3.0$ (a) as predicted by the model of diffraction modes and (b) evaluated as in Fig. 3, starting with uniform illumination.

Figure 8

Solid curve: amplitude $\left|\psi \right(\xi \left)\right|$ of the fundamental diffraction mode for $N_F\simeq N/\left(2\pi {\zeta }_{\mathrm{D}}\right)\simeq 25.5$, corresponding to $N=480$ slits in ${\zeta }_{\mathrm{D}}=3$. Dashed curves: amplitudes $|{\psi }_n\left(\xi \right)|$ on the intermediate slits $n=0,40,80$, showing the gradual formation of the fundamental diffraction mode explaining the Zeno effect. For better comparison, all peak amplitudes are normalized to unity.