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arXiv:2512.13817 · Physical Review A

Fractional power-law decay in the spontaneous emission of a two-level system

Hiroki Nakabayashi, Hayato Kinkawa, Takano Taira, Naomichi Hatano

quant-phSubmitted Dec 15, 2025• Announced Dec 17, 2025• 0 citations• Revised Aug 13, 2026 (v2)
v1 · Dec 15, 2025v2 · Aug 13, 2026
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Abstract

It has been shown for various systems that the decay rate of an unstable quantum system deviates from exponential behavior in short- and long-time regimes associated with memory effects. In particular, it is widely believed that in the short-time regime, the decay is quadratic, inducing the quantum Zeno effect, in which the decay is suppressed by rapidly repeated measurements. In our study, we find that when the environment of the unstable system has an energy spectrum with a lower bound but without an upper one, the decay rate in both regimes is scaled in terms of the spatial dimension and the exponent of the energy dispersion of the environment. Surprisingly, we find that in the short-time regime the decay exhibits fractional scaling, which leads to a quantum Zeno effect with a different scaling of the Zeno time.

Journal reference
Phys. Rev. A 114, L020202 (2026)
DOI
10.1103/wv4z-ddxz
Comments
7+17 pages, 3+1 figures

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It has been shown for various systems that the decay rate of an unstable quantum system deviates from exponential behavior in short- and long-time regimes associated with memory effects. In particular, it is widely believed that in the short-time regime, the decay is quadratic, inducing the quantum Zeno effect, in which the decay is suppressed by rapidly repeated measurements. In our study, we find that when the environment of the unstable system has an energy spectrum with a lower bound but without an upper one, the decay rate in both regimes is scaled in terms of the spatial dimension and the exponent of the energy dispersion of the environment. Surprisingly, we find that in the short-time regime the decay exhibits fractional scaling, which leads to a quantum Zeno effect with a different scaling of the Zeno time.
The short-time behavior of an unstable particle is examined in a realistic field theoretical model. The onset of linear decreasing of the nondecay probability is shown to be extremely rapid, so rapid as to exclude any relevance of regimes quadratic in time (``Zeno'' quantum paradox). The result applies to super-renormalizable, renormalizable, and nonrenormalizable cases, particularly to the proton decay problem. We discuss also the deviations from the exponential decay law in small-Q-value decays.
The characteristic short-time behavior of the survival probability of a decaying state, which underlies the quantum Zeno effect (QZE), is shown to follow quite simply from the Schr\"odinger equation even when the spectrum of the Hamiltonian has no finite lower bound, provided the mean energy of the initial state is finite. The failure of the Breit-Wigner form of energy distribution to predict QZE is traced to the nonexistence of a well-defined mean energy. On the basis of a suitably modified distribution, it is explicitly demonstrated that the existence of a finite lower bound to the energy distribution only affects the asymptotic decay law. For short times, the survival probability depends quadratically on time if the variance of the distribution is also finite, not otherwise. For a sufficiently well-behaved energy distribution, the asymptotic decay must be as ${t}^{{\ensuremath{-}}^{n}}$ if the spectrum has a finite lower bound, or faster than any inverse power of t if there is no lower bound.
The quantum Zeno effect--suppression of decay by frequent measurements--was believed to occur only when the response of the detector is so quick that the initial tiny deviation from the exponential decay law is detectable. However, we show that it can occur even for exactly exponentially decaying systems, for which this condition is never satisfied, by considering a realistic case where the detector has a finite energy band of detection. The conventional theories correspond to the limit of an infinite bandwidth. This implies that the Zeno effect occurs more widely than expected thus far.
We investigate the spontaneous emission of a two-level system coupled to a photonic waveguide, showing that dynamical dephasing in the photon modes profoundly alters the decay law. In the absence of dephasing, the emitter displays conventional exponential decay followed by a long-time power-law tail -- observable only at extremely low survival probabilities. Strikingly, when dephasing is introduced, a robust power-law decay emerges already at short times, driven by photon diffusion in the dynamically disordered environment rather than spectral edge effects. These results reveal a novel, decoherence-induced mechanism for non-exponential spontaneous emission in waveguide QED platforms.
Repeated measurements in quantum mechanics can freeze (the quantum Zeno effect) or enhance (the quantum anti-Zeno effect) the time-evolution of a quantum system. In this paper, we present a general treatment of the quantum Zeno and anti-Zeno effects for arbitrary driven open quantum systems, assuming only that the system-environment coupling is weak. In particular, we obtain a general expression for the effective decay rate of a two-level system subjected to arbitrary driving fields as well as periodic measurements. We demonstrate that the driving fields change the decay rate, and hence the quantum Zeno and anti-Zeno behavior, both qualitatively and quantitatively. We also extend our results to systems consisting of more than one two-level system, as well as a two-level system strongly coupled to an environment of harmonic oscillators, to further illustrate the non-trivial effect of the driving fields on the quantum Zeno and anti-Zeno effects.
The time evolution of quantum states for unstable particles can be conveniently divided into three domains: the very short time where Zeno's paradox is relevant, the intermediate interval where the exponential decay holds more or less, and the very long time where the decay is governed by a power law. In this work, we reexamine several questions relating to the deviations from the simple exponential decay law. On the basis of general considerations, we demonstrate that deviations from exponential decay near t=0t=0t=0 are inevitable. We formulate general resonance models for the decay. From analytic solutions to specific narrow-width models, we estimate the time parameters T1{T}_{1}T1​ and T2{T}_{2}T2​ separating the three domains. The parameter T1{T}_{1}T1​ is found to be much much less than the lifetime ${\ensuremath{\Gamma}}^{\ensuremath{-}1}$, while T2{T}_{2}T2​ is much greater than the lifetime. For instance, for the charged pion decay, ${T}_{1}\ensuremath{\sim}\frac{{10}^{\ensuremath{-}14}}{\ensuremath{\Gamma}}$ and ${T}_{2}\ensuremath{\sim}\frac{190}{\ensuremath{\Gamma}}$. A resolution of Zeno's paradox provided by the present consideration and its limitaions are discussed.