Quantum Zeno Effect for Exponentially Decaying Systems

The quantum Zeno effect—suppression of decay by frequent measurements—was believed to occur only when the response of the detector is so quick that the initial tiny deviation from the exponential decay law is detectable. However, we show that it can occur even for exactly exponentially decaying systems, for which this condition is never satisfied, by considering a realistic case where the detector has a finite energy band of detection. The conventional theories correspond to the limit of an infinite bandwidth. This implies that the Zeno effect occurs more widely than expected thus far.

Figure 1

Temporal evolution of $1-s(t)$ (solid line), $ϵ(t)$ (dotted line), and $r(t)$ (dashed line). The parameters are chosen as $2\pi \Delta /\gamma =100$ and $\eta /\gamma =1.5$. It is seen that $r(t)$ follows $1-s(t)$ with a delay time about $\tau \equiv {\eta }^{-1}$.

Figure 2

Plot of $|g_{\mu }{|}^2$. $\eta /2\pi \Delta$ is chosen at 0.01 (solid line), 0.1 (dotted line), and 1 (dashed line).

Figure 3

Plot of $\ln s(t)/(\gamma t)$, i.e., the time dependence of the decay rate. $\{2\pi \Delta /\gamma ,\eta /\gamma \}$ are chosen at $\{100,100\}$ (solid line), $\{100,10\}$ (dotted line), and $\{1000,1000\}$ (broken line). The decay rate changes from the free rate $\gamma$ to the suppressed rate $2\pi |g_{\Omega }{|}^2$ at $t\sim {\Delta }^{-1}$.