Quantum Zeno effect by indirect measurement: The effect of the detector

We study the quantum Zeno effect in the case of indirect measurement, where the detector does not interact directly with the unstable system. Expanding on the model of Koshino and Shimizu [Phys. Rev. Lett. 92, 030401 (2004)] we consider a realistic Hamiltonian for the detector with a finite bandwidth. We also take explicitly into account the position, the dimensions, and the uncertainty in the measurement of the detector. Our results show that the quantum Zeno effect is not expected to occur, except for the unphysical case where the detector and the unstable system overlap.

Figure 1

A schematic plot of the TLA and the detector (shaded region) as we consider it. The electromagnetic field emitted by the TLA enters the detector with a penetration depth of $1∕\gamma$. The atom is displaced by $x_D$ and the detector spans from 0 to $x_0$.

Figure 2

The decay rate of the TLA excited state population, coupled to the detector, over the free decay rate, following the KS approach. Parameters are $2\pi \Delta ∕\gamma =100$ and $\eta ∕\gamma =1,10$ for the solid and dashed curve, respectively. Together we show, for the sake of comparison, the decay rate in the absence of the detector (dot-dashed curve). Since we take into account only a finite bandwidth, the decay rate is 0 at $t=0$. Discretisation range: 0 to $2\Omega$, with 100 modes for the electromagnetic field and $100\left(k\right)\times 100\left(\omega \right)$ modes for the detector in the same range.

Figure 3

The decay rate of the TLA excited state population over the free decay rate as its relative position with the detector is varied (inset). We considered four cases, from the TLA being at the center of the detector (a) to the TLA outside of the detector(d). The decay rate changes smoothly from the results obtained in Ref. 1 to the free decay rate. The detector is assumed to have an effective Gaussian profile, with a full width at half maximum of 33 $(\hslash =c=\Omega =1)$, as shown in the inset. Parameters: $2\pi \Delta ∕\gamma =100$,$\eta ∕\gamma =10$, discretization as in Fig. 2, $\mathcal{C}(k,k^{\prime })=0.103e^{-{[(k-k^{\prime })∕5.5]}^2}$.

Figure 4

The intensity profile of the emitted photon for the cases (a) and (d) of Fig. 3. In the top part of each figure we show the corresponding detector profile and position. The distance is measured from the TLA and in units of detector full width at half maximum. Lighter shading stands for higher intensities.