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Europhysics Letters (EPL)

Short-time behaviour of the quantum survival probability

Juan Gonzalo Muga, G. W Wei, Robert F. Snider

Published Aug 1, 1996• 27 citations
Publisher

Abstract

\n \nThe short-time behaviour of the quantum survival probability is\nexamined. The origin of previous discrepancies on the short-time\nbehaviour is explained by means of a discrete decomposition of the\nsurvival amplitude into a sum of decaying exponential terms and\nw-functions. Several possible non-exponential dependences are\ndescribed and exemplified.\n\n

DOI
10.1209/epl/i1996-00561-8

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\n \nThe short-time behaviour of the quantum survival probability is\nexamined. The origin of previous discrepancies on the short-time\nbehaviour is explained by means of a discrete decomposition of the\nsurvival amplitude into a sum of decaying exponential terms and\nw-functions. Several possible non-exponential dependences are\ndescribed and exemplified.\n\n
A detailed analysis of the quantum decay processes shows the survival probability ( t ) can not take that exponential form at any time interval including times smaller than the lifetime τ . We show that for times t ∼ τ and for the times later than τ the form of ( t ) looks as a composition of an oscillating and exponential functions. The amplitude of these oscillations is very small for t ≪ τ and grows with increasing time and depends on the model considered. We also study the survival probability of moving relativistic unstable particles with definite momentum : It turns out that late time deviations of the survival probability of these particles from the exponential-like form of the decay law should occur much earlier than it follows from the classical standard approach resolving itself into replacing time t by t / γ (where γ is the relativistic Lorentz factor) in the formula for the survival probability ( t ).
The temporal behavior of quantum mechanical systems is reviewed. We mainly focus our attention on the time development of the so-called “survival” probability of those systems that are initially prepared in eigenstates of the unperturbed Hamiltonian, by assuming that the latter has a continuous spectrum. The exponential decay of the survival probability, familiar, for example, in radioactive decay phenomena, is representative of a purely probabilistic character of the system under consideration and is naturally expected to lead to a master equation. This behavior, however, can be found only at intermediate times, for deviations from it exist both at short and long times and can have significant consequences. After a short introduction to the long history of the research on the temporal behavior of such quantum mechanical systems, the short-time behavior and its controversial consequences when it is combined with von Neumann’s projection postulate in quantum measurement theory are critically overviewed from a dynamical point of view. We also discuss the so-called quantum Zeno effect from this standpoint. The behavior of the survival amplitude is then scrutinized by investigating the analytic properties of its Fourier and Laplace transforms. The analytic property that there is no singularity except a branch cut running along the real energy axis in the first Riemannian sheet is an important reflection of the time-reversal invariance of the dynamics governing the whole process. It is shown that the exponential behavior is due to the presence of a simple pole in the second Riemannian sheet, while the contribution of the branch point yields a power behavior for the amplitude. The exponential decay form is cancelled at short times and dominated at very long times by the branch-point contributions, which give a Gaussian behavior for the former and a power behavior for the latter. In order to realize the exponential law in quantum theory, it is essential to take into account a certain kind of macroscopic nature of the total system, since the exponential behavior is regarded as a manifestation of a complete loss of coherence of the quantum subsystem under consideration. In this respect, a few attempts at extracting the exponential decay form on the basis of quantum theory, aiming at the master equation, are briefly reviewed, including van Hove’s pioneering work and his well-known “λ2T” limit. In the attempt to further clarify the mechanism of the appearance of a purely probabilistic behavior without resort to any approximation, a solvable dynamical model is presented and extensively studied. The model describes an ultrarelativistic particle interacting with N two-level systems (called “spins”) and is shown to exhibit an exponential behavior at all times in the weak-coupling, macroscopic limit. Furthermore, it is shown that the model can even reproduce the short-time Gaussian behavior followed by the exponential law when an appropriate initial state is chosen. The analysis is exact and no approximation is involved. An interpretation for the change of the temporal behavior in quantum systems is drawn from the results obtained. Some implications for the quantum measurement problem are also discussed, in particular in connection with dissipation.
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)S(t)S(t) has a dependence on the initial state and may behave as either $S(t)=1\ensuremath{-}O({t}^{3/2})$ or $1\ensuremath{-}O({t}^{2})$. 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.
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 Γ r 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- ct 1/2 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 τ S ≈4/[π( R 2 + R + 1/4)] and at long times by τ L ≈5.41 ln ( R ) + 12.25, where R = r /Γ r . The predictions of the formula are compared with numerical examples and some experimental results searching for non-exponential contributions to decay.
The Fock-Krylov formalism for the calculation of survival probabilities of unstable states is revisited paying particular attention to the mathematical constraints on the density of states, the Fourier transform of which gives the survival amplitude. We show that it is not possible to construct a density of states corresponding to a purely exponential survival amplitude. The survival probability P(t)P(t)P(t) and the autocorrelation function of the density of states are shown to form a pair of cosine Fourier transforms. This result is a particular case of the Wiener-Khinchin theorem and forces P(t)P(t)P(t) to be an even function of time which in turn forces the density of states to contain a form factor which vanishes at large energies. Subtle features of the transition regions from the nonexponential to the exponential at small times and the exponential to the power-law decay at large times are discussed by expressing P(t)P(t)P(t) as a function of the number of oscillations nnn performed by it. The transition at short times is shown to occur when the survival probability has completed one oscillation. The number of oscillations depends on the properties of the resonant state and a complete description of the evolution of the unstable state is provided by determining the limits on the number of oscillations in each region.