Suppression of Zeno effect for distant detectors

We describe the influence of continuous measurement in a decaying system and the role of the distance from the detector to the initial location of the system. The detector is modeled first by a step absorbing potential. For a close and strong detector, the decay rate of the system is reduced; weaker detectors do not modify the exponential decay rate but suppress the long-time deviations above a coupling threshold. Nevertheless, these perturbing effects of measurement disappear by increasing the distance between the initial state and the detector, as well as by improving the efficiency of the detector.

Figure 1

Schematic representation of the model including a hard wall at $x=0$, a delta function potential at $x=1$ and a detector for $x⩾X_c$. The wave packet is initially confined into the inner region [0, 1].

Figure 2

$\ln \left[S\left(t\right)\right]$ for different absorptive step potentials, see Eq. (1). $V_c=0$ (thick solid line), 0.1 (dashed line), 0.3 (thin solid line), and 0.5 (dots). $X_c=1$ and $\eta =5$.

Figure 3

$\ln \left[S\left(t\right)\right]$ for stronger absorptive potentials (compared to Fig. 2): $V_c=0.75$ (solid line, still with a visible long-time deviation), 1 (dashed line), 10 (dots), and 100 (dotted-dashed line). $X_c=1$ and $\eta =5$, as in Fig. 2.

Figure 4

Transition time $t_{\mathrm{tran}}$ versus $V_c$ for $\eta =5$ and $X_c=1$. The critical value $V_c^{\mathrm{thre}}$ is marked with a vertical dashed line.

Figure 5

$\ln \left[S\left(t\right)\right]$ versus time for $\eta =5$, $V_c=100$, and different values of $X_c$: 1 (triangles), 2 (thick dashed line), 100 (dotted-dashed line), and 200 (thin solid line). The reference curve for $V_c=0$ (thick solid line) is also shown. The dots correspond to a better detector placed at $X_c=100$, see Eq. (12) and related text, with $E_{\min }=0.003$ and $L=100$.

Figure 6

Manolopoulos potential for $X_c=100$, see Eq. (12) and related text, with $E_{\min }=0.003$ and $L=100$.

Figure 7

Monotonic increase of the lifetime versus $V_c$ for $\eta =5$ and two detector distances: $X_c=3$ (a); $X_c=1$ (b).

Figure 8

Behavior of the survival probability versus time in units of the lifetime $\tau$ at short times for $\eta =5$ and $V_c=0$ (dashed line). The continuous line exhibits, for comparison, pure exponential decay.