Survival probability of a single resonance

Abstract

An exact single-level resonance formula for the survival probability S(t) in the full time interval, that depends only on the resonance energy r and the decay width Γr and fulfils time-reversal invariance, is used to discuss the non-exponential contributions to decay. At short times the formula behaves as S(t)≈1-ct1/2 with c a constant, whereas at long times it behaves as S(t)≈dt-3, d being a constant. With the time expressed in lifetime units, the onset of non-exponential decay is given at short times by τS≈4/[π(R2 + R + 1/4)] and at long times by τL≈5.41 ln (R) + 12.25, where R = r /Γr. The predictions of the formula are compared with numerical examples and some experimental results searching for non-exponential contributions to decay.