Analytical study of quadratic and nonquadratic short-time behavior of quantum decay

Abstract

The short-time behavior of quantum decay of an unstable state initially located within an interaction region of finite range is investigated using a resonant expansion of the survival amplitude. It is shown that in general the short-time behavior of the survival probability S(t) has a dependence on the initial state and may behave as either S(t) = 1 − O(t 3/2 ) or 1 − O(t 2 ). These cases are illustrated by solvable models. The experiment reported by Wilkinson et al. [Nature (London) 387, 575 (1997)] does not distinguish between the above short-time behaviors.

Exact quantum decay of an interacting many-particle system: the Calogero–Sutherland model

“…The scaling dynamics holds exactly for the CS gas and simplifies the ensuing analysis by contrast to other many-body systems such as, e.g., the 1D Bose gas, where only recently moderate progress accounting for its dynamics has been reported [52][53][54][55]. Comparing the first leading terms in a long-time asymptotic expansion, it is found that the power-law (14) sets in when the time of evolution satisfies…”

“…Experimentally, it was first demonstrated in [11] and its existence sets the ground for the quantum Zeno effect [12]. It is a consequence of unitary time-evolution, provided that the first and second moments of the Hamiltonian exist, see [13,14] for exceptional cases. That deviations from exponential decay are as well to be expected at long times was pointed out by Khalfin in 1957, for systems whose energy spectrum is bounded from below [8].…”

Time-domain resonances and the ultimate fate of a decaying quantum state

“…It is of interest to see, in view of (8), that the coefficients C n fulfill the relation [17] Re ∞ n=1 C nCn = 1,…”

“…Khalfin demonstrated that if the energy spectra E of the system is bounded from below, i.e., E ∈ (0,∞), the exponential decay law cannot hold at long times [5]. At short times there is also a departure from the exponential behavior that, however, has a dependence on the initial state of the problem [6][7][8]. The experimental verification of the above departures from the exponential decay law remained elusive for decades but has been finally corroborated by experiment [9,10].…”

Decay properties of unstable Tonks-Girardeau gases from a split trap

“…In practice, it suffices to take into account in Eq. ( 11) only a few terms from the poles p r and inverse power contributions to achieve satisfactory approximations, except for short times [49]. For details regarding the resonance expansion approach, see [10,16,23].…”