The decaying behavior of both the survival S(t) and total P(t) probabilities for unstable multilevel systems at long times is investigated by using the N-level Friedrichs model. The long-time asymptotic-forms of both S(t) and P(t) are obtained for an arbitrary initial-state extending over the unstable levels. It is then clarified how the asymptotic forms depend on the initial population in unstable levels. In particular, a special initial state that maximizes the asymptotic form of both S(t) and P(t) is found. On the other hand, the initial states eliminating the first term of their asymptotic expnasions also exist, which implies that a faster decay rather than expected can be realized. This faster decay for S(t) is numerically confirmed by considering the spontaneous emission process for the hydrogen atom interacting with the electromagnetic field. It is demonstrated that the t-4-decay and a faster decay are realized depending on the initial states, where the latter is estimated as t-8.
The long-time behavior of the survival probability for unstable multilevel systems that follows the power-decay law is studied based on the
N-level Friedrichs model, and is shown to depend on the initial population in unstable states. A special initial state maximizing the asymptote of the survival probability at long times is found and examined by considering the spontaneous-emission process for the hydrogen atom interacting with the electromagnetic field.
We study the short-time and medium-time behavior of the survival probability of decaying states in the framework of the
N-level Friedrichs model. The degenerated and nearly degenerated systems are discussed in detail. We show that in these systems decay can be considerably slowed down or even stopped by appropriate choice of initial conditions. We analyze the behaviour of the multilevel system within the so-called Zeno era. We examine and compare two different definitions of the Zeno time. We demonstrate that the Zeno era can be considerably enlarged by proper choice of the system parameters.
We consider a model of a unstable state defined by the truncated Breit-Wigner energy density distribution function. An analytical form of the survival amplitude
a(t) of the state considered is found. Our attention is focused on the late time properties of
a(t) and on effects generated by the non--exponential behavior of this amplitude in the late time region: In 1957 Khalfin proved that this amplitude tends to zero as
t goes to the infinity more slowly than any exponential function of
t. This effect can be described using a time-dependent decay rate
γ(t) and then the Khalfin result means that this
γ(t) is not a constant but at late times it tends to zero as
t goes to the infinity. It appears that the energy
E(t) of the unstable state behaves similarly: It tends to the minimal energy
Emin of the system as
t→∞. Within the model considered we find two first leading time dependent elements of late time asymptotic expansions of
E(t) and
γ(t). We discuss also possible implications of such a late time asymptotic properties of
E(t) and
γ(t) and cases where these properties may manifest themselves.