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arXiv:2509.17163 · Physical Review A

Quantum late-time decay and channel dependence

Francesco Giacosa, A. Kolbus, Krzysztof Kyzioł, Magdalena Płódowska, Milena Piotrowska, Karol Szary, Arthur Vereijken

Published Jun 12, 2026• 0 citations
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Abstract

Quantum mechanics predicts deviations from exponential decay at short and long times, yet experimental evidence is limited. We report a power-law tail after $\sim$10 lifetimes in two fluorescent compounds (erythrosine~B and eosine Y), confirmed by two detectors probing distinct bands but yielding different power coefficients. The data match a divergent but normalizable spectral density, and theory predicts oscillations as a future test. A novel and general result is that in multichannel QM (and QFT) decay, the lifetime is universal, but the late-time deviations are channel- (or band-) dependent, a feature consistent with our data.

DOI
10.1103/6r7p-mjxb

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Quantum mechanics predicts deviations from exponential decay at short and long times, yet experimental evidence is limited. We report a power-law tail after ∼\sim∼10 lifetimes in two fluorescent compounds (erythrosine~B and eosine Y), confirmed by two detectors probing distinct bands but yielding different power coefficients. The data match a divergent but normalizable spectral density, and theory predicts oscillations as a future test. A novel and general result is that in multichannel QM (and QFT) decay, the lifetime is universal, but the late-time deviations are channel- (or band-) dependent, a feature consistent with our data.
In quantum mechanics, the properties of an unstable system are closely related to its corresponding energy distribution. One of the predictions of the theory is the emergence of a late-time power law in the survival probability P(t)P(t)P(t) of the unstable state. However, in complex systems, similar effects may emerge due to other phenomena. This is the case for nonextensive statistical models, for some continuous lifetime distribution models, and also for certain energy-migration models applied to fluorescence. In this work, these alternative interpretations are confronted with data from time-resolved fluorescence spectroscopy experiments with erythrosine B and eosine Y as fluorophores. It turns out that some of the features expected within the quantum-mechanical description, such as the sharp transition from the exponential to the power-law regime, cannot be reproduced by the specific statistical and energy-migration parametrizations fitted in the analysis.
In this work, we experimentally study the fluorescence decay of Acridine Orange at late times, in order to test whether a late-time power-law behaviour emerges, a feature expected to be very small but consistent with quantum mechanical and quantum field theoretical predictions. Using two distinct photon detectors, we find that the data are well described by a sum of two exponential functions with lifetimes τ1=1.7331±0.001τ_1 = 1.7331 \pm 0.001τ1​=1.7331±0.001 ns and τ2=5.948±0.012τ_2 = 5.948 \pm 0.012τ2​=5.948±0.012 ns, in agreement with values reported in the literature. While no deviation from the exponential decay law is observed, this study serves as a reliable test for the experimental setup and enables a precise determination of the sample lifetimes.
It is well known, both theoretically and experimentally, that the survival probability for an unstable quantum state, formed at t = 0 , is not a simple exponential function, even if the latter is a good approximation for intermediate times. Typically, unstable quantum states/particles can decay in more than a single decay channel. In this work, the general expression for the probability that an unstable state decays into a certain i -th channel between the initial time t = 0 and an arbitrary t > 0 is provided, both for nonrelativistic quantum states and for relativistic particles . These partial decay probabilities are also not exponential and their ratio turns out to be not a simple constant, as it would be in the exponential limit. Quite remarkably, these deviations may last relatively long, thus making them potentially interesting in applications. Thus, multichannel decays represent a promising and yet unexplored framework to search for deviations from the exponential decay law in quantum mechanical systems, such as quantum tunneling, and in the context of particle decays .
We describe some salient features as well as some recent developments concerning short-time deviations from the exponential decay law in the context of Quantum Mechanics by using the Lee Hamiltonian approach and Quantum Field Theory by using relativistic Lagrangians. In particular, the case in which two decay channels are present is analyzed: the ratio of decay probability densities, which is a constant equal to the ratio of decay widths in the exponential limit, shows in general sizable fluctuations which persist also at long times.
Results of theoretical studies of the quantum unstable systems caused that there are rather widespread belief that a universal feature of the quantum decay process is the presence of three time regimes of the decay process: the early time (initial) leading to the Quantum Zeno (or Anti Zeno) Effects, “exponential” (or “canonical”) described by the decay law of the exponential form, and late time characterized by the decay law having inverse-power law form. Based on the fundamental principles of the quantum theory we give the proof that there is no time interval in which the survival probability (decay law) could be a decreasing function of time of the purely exponential form but even at the “exponential” regime the decay curve is oscillatory modulated with a smaller or a large amplitude of oscillations depending on parameters of the model considered.
Metastable decay exhibits a familiar exponential regime bracketed by early-time deviations and late-time power-law tails. We adopt the real-time, flux-based definition of the decay rate in the spirit of Andreassen et al.\ direct method and present a complete analysis of one-dimensional quantum-mechanical resonance models. We show that the kernel admits a universal pole--plus--branch decomposition and use it to define two computable time scales: a dawn time, when a single resonant contribution starts dominating and exponential decay sets in, and a twilight time, when the branch-cut tail overtakes exponential decay. The latter can be expressed in closed form via the Lambert WWW function, making its parametric dependence manifest without fitting. For square, modified square, and Pöschl--Teller barriers we obtain simple thick-barrier formulas, clarify the relation ΓT=TtransΓT = T_{\text{trans}}ΓT=Ttrans​ between the decay rate ΓΓΓ, oscillation period TTT, and transmission probability TtransT_{\text{trans}}Ttrans​, and indicate how our spectral picture can be naturally extended to quantum field theoretic vacuum decay.