Pulsed and continuous measurements of exponentially decaying systems

We study the influence of a detector on the decay law of a quantum state whose undisturbed survival probability is purely exponential. In particular, we consider a detector with a finite energy band of detection, i.e., it interacts only with decay products having an energy within a certain range of values. In one case, we assume that the detector performs many repeated measurements at short time intervals in all of which a collapse of the wave function occurs (bang-bang or pulsed-type measurements). In the second case, we assume a continuous measurement, which preserves unitarity. We confirm the slowing down of the decay in presence of a measuring apparatus, the quantum Zeno effect, but the outcomes of the detector are in general qualitatively and quantitatively different in the two cases. In turn, this implies that the so-called Schulman relation (the equivalence of pulsed and continuous measurements) does not hold in this case and that it is in principle possible to experimentally access how a certain detector performs a measurement.

Figure 1

The ratio between the decay probability $w_{\lambda }\left(t\right)$ and $t$ is displayed as a function of time for different values of $\lambda$. Only in the limit $\lambda \to \infty$ the decay probability $w_{\lambda }\left(t\right)$ is linear in time while for finite values of $\lambda$ it shows a quadratic behavior at short times. Note, the limits $\lambda \to \infty$ and $t\to 0$ do not commute. In the insert we show $w_{\lambda }\left(t\right)$, notice that for $t\to \infty$ it saturates to a value smaller than one if $\lambda$ is finite.

Figure 2

We display the saturation value of the no-click probability in the continuous case [right-hand side of Eq. (15), valid in the limit of large $\sigma$] as a function of $\lambda$. We also show the bang-bang case [see Eq. (6)] as a function of $\lambda$ and for different values of $\tau$.

Figure 3

$p_{\text{no-click}}$ as a function of time for continuous and bang-bang measurements for two values of the detection efficiency $\sigma$ and their relative values of $\tau$ as prescribed by the Schulman rule; also the exponential decay is shown for comparison. The results of the two types of measurements are qualitatively different, the difference becoming larger for larger values of $\sigma$.