Sufficient conditions for the anti-Zeno effect

Abstract

The ideal anti-Zeno effect means that a perpetual observation leads to an immediate disappearance of the unstable system. We present a straightforward way to derive sufficient conditions under which such a situation occurs expressed in terms of the decaying states and spectral properties of the Hamiltonian. They show, in particular, that the gap between Zeno and anti-Zeno effects is in fact very narrow.The Zeno effect which means that an unstable system will never decay if we monitor its decay perpetually is known for decades. For the first time it was formulated explicitly in this context by Beskow and Nilsson [3] and soon after a mathematical analysis [5,14] revealed sufficient conditions under which it exists; it became truly popular after the authors of [20] coined its present name. Recently the effect attracted a new wave of mathematical [8,9,19,22] and physical [11,10,12,13,16,18] interest; in the mentioned papers one can find a more complete bibliography.Although the opposite situation, in which a frequent measurement can on the contrary speed up the decay, or ideally to lead to an immediate disappearance of the unstable system, was also noticed early [6], it attracted attention only recently -see, e.g., [1,2,17,21] and also [22] and references therein. As in the case of the Zeno effect, the problem can be tackled from two points of view. The more practical one concerns the increase of the measured lifetime in case when the measurement are performed with a certain frequency. On the other hand, theoretically one can ask what happens if the period between two successive measurements tend to zero. The distinction between the two 1

Solvable Models of Resonances and Decays

“…if dim H u = ∞ the involved series can easily be seen to converge using Parseval relation. Using next the normalization ∞ −∞ dω jk (λ) = δ jk we arrive after a simple calculation [Ex05] at the formula (6.1)…”

The decay law can have an irregular character

“…The effect is expected to be present for any α > 0; however, for a numerical demonstration we seek here it is reasonable to choose a large value at which the particle leaks out only slowly 7. Another striking deviation from the exponential decay law is that the decay rate explodes as t → 0, which is due to the fact that the energy distribution of the state ψ decays too slowly at high energies, cf[6].…”

“…The decay of an unstable quantum systems is one of the effects frequently discussed and various aspects of such processes were considered. To name just a few, recall the long-time deviation from the exponential decay law [1], the short-time behavior related to the Zeno and anti-Zeno effects [3,4,5,6], revival effects such the classical one in the kaon-antikaon system, etc. In all the existing literature [7], however, the decay law is treated as a smooth function, either explicitly or implicitly, e.g.…”

“…Now we are ready to compute the decay law for a given initial state. Without loss of generality we may put R = 1, we choose the value α = 500 for numerical evaluation 6 and replace the infinite series by a cut-off one with |n| 1000; this guarantees numerical stability of the result. As the first example we consider initial wavefunction constant within the well, i.e.,…”