Time as a dynamical variable in quantum decay

Jan Dereziński, Christian Gérard

Vladan Panković

In this work we consider a new example of the recently introduced quantum Hamlet effect. We consider an especial, abstract, unstable quantum system whose evolution during a small time interval is interrupted by frequent measurements. Here three different final situations exist. First one corresponds to quantum Zeno effect, second one - to quantum anti-Zeno effect and third one - to so-called quantum Hamlet effect. By quantum Zeno effect final non-decay probability is function of number of the decay measurements variable and dynamical degree parameter equivalent to two. When measurements number tends toward infinity non-decay probability has the one limit, or, it tends analytically toward one and system stands non-decayed. By quantum anti-Zeno effect final non-decay probability is function of number of the decay measurements variable and dynamical degree parameter equivalent to one. When measurements number tends toward infinity non-decay probability has the zero limit, or, it tends analytically toward zero and system becomes decayed. By quantum Hamlet effect, final non-decay probability is function of two variable, number of the decay measurements and dynamical degree. When measurements number tends toward infinity and dynamical degree toward one, final non-decay probability depends not only of final value of given variables, but, also, on the ways on which given variables tends toward their final values. It means that final no-decay probability has not (analytical) limit, or that there is no {\it analytical} prediction on the final no-decay probability. To be decayed or no-decayed that is analytically unsolvable question for given quantum system.

Rafael de la Madrid

We present a theoretical analysis of quantum decay in which the survival probability is replaced by a decay rate that is equal to the absolute value squared of the wave function in the time representation. The wave function in the time representation is simply the Fourier transform of the wave function in the energy representation, and it is also the probability amplitude generated by the Positive Operator Valued Measure of a time operator. The present analysis endows time with a dynamical character in quantum decay, and it is applicable only when the unstable system is monitored continuously while it decays. When the analysis is applied to the Gamow state, one recovers the exponential decay law. The analysis allows us to interpret the oscillations in the decay rate of the GSI anomaly, of neutral mesons, and of fluorescence quantum beats as the result of the interference of two resonances in the time representation. In addition, the analysis allows us to show that the time of flight of a resonance coincides with its lifetime.

I. Joichi, S. Matsumoto, M. Yoshimura

A finite number of harmonic oscillators coupled to infinitely many environment oscillators is fundamental to the problem of understanding quantum dissipation of a small system immersed in a large environment. Exact operator solution as a function of time is given to this problem, by using diagonalized dynamical variable of the entire system, the small system plus the environment. The decay law of prepared initial configuration is worked out in greatest detail. A clear separation of the exponential- and the power-law decay period is made possible by our method. Behavior of physical quantities at asymptotically late times can be understood in terms of the overlap probability of the system variable with the diagonal variable of the entire system.

Valia Allori