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.
Long time behaviour of the survival probability for unstable multilevel systems is studied based on the N-level Friedrichs model in the presence of zero energy resonance, that means the divergence of the resolvent at zero energy. With resort to the technique developed by Jensen and Kato [Duke Math. J. 46, 583 (1979)], the zero energy resonance of this model is characterized by the zero energy eigenstate that does not belong to the Hilbert space. It is then shown that for some kinds of the form factors, logarithmically slower decay of the survival probability can be realized.
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.
The behavior of both the survival
S(t) and the nonescape
P(t) probabilities at long times for the one-dimensional free-particle system is shown to be closely connected to that of the initial wave packet at small momentum. We prove that both
S(t) and
P(t) asymptotically exhibit the same power-law decrease at long times, when the initial wave packet in momentum representation behaves as
km with
m=0 or
1 at small momentum. On the other hand, if the integer m becomes greater than
1, S(t) and
P(t) decrease in different power laws at long times.
The long time behaviour of the survival probability of the initial state and its dependence on the initial states are considered, for the one-dimensional free quantum particle. We derive the asymptotic expansion of the time evolution operator at long times, in terms of the integral operators. This enables us to obtain the asymptotic formula for the survival probability of the initial state ψ( x ), which is assumed to decrease sufficiently rapidly at large | x |. We then show that the behaviour of the survival probability at long times is determined by that of the initial state ψ at zero momentum k = 0. Indeed, it is proved that the survival probability can exhibit the various power decays such as t −2 m −1 for an arbitrary non-negative integer m as t → ∞, corresponding to the initial states with the condition ( k ) = O ( k m ) as k → 0.
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.