Quantum Zeno and anti-Zeno effects in the Friedrichs model

Abstract

We analyze the short-time behavior of the survival probability in the frame of the Friedrichs model for different formfactors. We have shown that this probability is not necessary analytic at the time origin. The time when the quantum Zeno effect could be observed is found to be much smaller than usually estimated. We have also studied the anti-Zeno era and have estimated its duration.

Diffusion and entanglement of a kicked particle in an infinite square well under frequent measurements

“…An opposite phenomenon, the anti-Zeno effect (decay acceleration by frequent measurements) was recently discovered by Kofman and Kurizki [2,3]. In the theoretical and numerical investigations of quantum chaotic systems, similar predictions have been presented [6,7,8,9,10].…”

Zero energy resonance and the logarithmically slow decay of unstable multilevel systems

“…We can find such form factors in the physical systems for the spontaneous emission process of photons from the Hydrogen atom 15,22 and the quantum dot. 16 On the other hand, the discussion mentioned here immediately implies the fact that if |ψ ∈ M 2 exists, this time it should satisfy…”

“…The Friedrichs model 9,10 describes the system of the finite discrete levels coupled with the continuous spectrum, in which the former can be interpreted as the unstable excited levels of atoms and the latter as the environmental electromagnetic fields. 11,15,16 We emphasize that the model is not restricted to the single level case 7,8,9,10,11,12,13,15,16 but, rather, the N -level case, 10,17,18,19,20,21 In addition, we assume that the square modulus of the form factors vanishes at zero energy with an integer power, 15,16,22 however it is treated without restriction to a specific form to some extent. Furthermore, since we only consider the initial state spanned by the discrete states, it is sufficient for us to see the reduced resolventR(z) that is just the restriction of the resolvent to the subspace spanned by the discrete states.…”

“…Remember that under the assumption on the form factors, Γ (ω) defined by Eq. (19) has an asymptotic form like…”

Unstable Systems and Quantum Zeno Phenomena in Quantum Field Theory

“…the initial quadratic (Zeno) region is much shorter than the Zeno time: in general, the Zeno time does not yield a correct estimate of the duration of the Zeno region. 9,22,30 (Beware of many erroneous claims in the literature!) The approximation P (t) ≃ 1 − t 2 /τ 2 Z holds for times t < Λ −1 ≪ τ Z .…”