Quantum Zeno effect for a free-moving particle

Although the quantum Zeno effect takes its name from Zeno's arrow paradox, the effect of frequently observing the position of a freely moving particle on its motion has not been analyzed in detail in the frame of standard quantum mechanics. We study the evolution of a moving free particle while monitoring whether it lingers in a given region of space, and explain the dependence of the lingering probability on the frequency of the measurements and the initial momentum of the particle. Stopping the particle entails the emergence of Schrödinger cat states during the observed evolution, closely connected to the high-order diffraction modes in Fabry-Pérot optical resonators.

Figure 1

(a) Probability $P_N$ that a moving particle remains in its original location at the time that the particle would classically take to leave it, as a function of the number $N$ of intermediate measurements and the action $s$ of the particle. (b) Probability on increasing the number of measurements and fixed action. (c) Probability on increasing the action and fixed number of measurements.

Figure 2

For a particle with $s=8$ initially in $[-1,1]$, (a) probability density before (dashed curve) and after free evolution for a time $\tau =1$ with $N$ intermediate measurements; (b), (c) time evolution of the expected values of the position $\langle \xi \rangle$ and momentum $\langle \kappa \rangle$ during free evolution with $N$ intermediate measurements; (d) probability that the particle remains in $[-1,1]$ at the $n\mathrm{th}$ intermediate measurement as a function of the time $\tau =n/N$ of the measurement.

Figure 3

(a), (b) For a particle with $s=8$ initially in $[-1,1]$, the position probability densities and momentum probability densities before (dashed curves) and after free evolution for a time $\tau =4$ with $N=4096$ intermediate measurements (solid curves). (c), (d) For particles with $s=8$ and $s=14$ initially in $[-1,1]$, the time evolution of the expected values of the position $\langle \xi \rangle$ and momentum $\langle \kappa \rangle$ during free evolution with $N=4096$ intermediate measurements. (e), (f) The same as in (a) and (b) except that $s=14$.

Figure 4

(a) For $s=8$, and for several numbers of total measurements $N$ within $\tau =4$, the probability that the particle remains in $[-1,1]$ at the $n\mathrm{th}$ intermediate measurement as a function of the time $\tau =4n/N$ of the measurement. The solid curves represent the numerically evaluated values and the dashed curves the theoretical predictions. (b) Probability $P_N$ that a moving particle remains in $[-1,1]$ after $N$ intermediate measurements within $\tau =1$ as a function of the action $s$ of the particle. The circles represent the numerically evaluated probabilities and the dashed curves the predictions of Eq. (17).

Figure 5

Wigner functions for (a) the initial and (b) evolved states in Figs. 3 and 3.