We modify the theory of the Quantum Zeno Effect to make it consistent with the postulates of quantum mechanics. This modification allows one, throughout a sequence of observations of an excited system, to address the nature of the observable and thereby to distinguish survival from non-decay, which is necessary whenever excited states are degenerate. As a consequence, one can determine which types of measurements can possibly inhibit the exponential decay of the system. We find that continuous monitoring taken as the limit of a sequence of ideal measurements will only inhibit decay in special cases, such as in well-controlled experiments. Uncontrolled monitoring of an unstable system, however, can cause exponentially decreasing non-decay probability at all times. Furthermore, calculating the decay rate for a general sequence of observations leads to a straightforward derivation of Fermi's Golden Rule, that avoids many of the conceptual difficulties normally encountered. When multiple decay channels are available, the derivation reveals how the total decay rate naturally partitions into a sum of the decay rates for the various channels, in agreement with observations. Continuous and unavoidable monitoring of an excited system by an uncontrolled environment may therefore be a mechanism by which to explain the exponential decay law.
The exponential decay of an unstable quantum state, as described by standard Markovian theories such as Fermi's Golden Rule, assumes a simple, structureless environment. However, in complex environments characterized by disorder, long-range interactions, or strong fluctuations, local decay rates fluctuate, leading to non-Markovian dynamics and power-law ``long-time tails.'' In this paper, we apply the recently proposed \textit{hyperstatistics} framework to the problem of quantum decay in such complex environments. By considering a
γ-distribution of local decay rates across mesoscopic domains, we derive a macroscopic survival probability governed by a
q-exponential function. We then use the
q-generalized Gamma function, defined via the Mellin transform of the
q-exponential, to calculate the moments of the decay-time distribution. We show that the mean quantum lifetime is
⟨t⟩=∫0∞P(t)dt=[⟨Γ⟩(2−q)]−1, and is therefore finite for
q<2. The convergence of the second moment, and hence of the lifetime variance, instead requires the stricter condition
q<3/2. These results provide a refined physical interpretation: for
1<q<3/2 both the mean lifetime and its variance are finite; for
3/2≤q<2 the mean lifetime remains finite but lifetime fluctuations become infinitely broad; and for
q≥2 the mean lifetime itself diverges, corresponding to a truly trapped, localized, or Griffiths-like regime.
We study a class of Friedrichs Hamiltonians, that is, operators describing an excited state coupled to a bath through a rank-one perturbation, in the case where the bath Hamiltonian has purely discrete spectrum. We consider sequences of such Hamiltonians for which the spectral measures associated to the coupling functions by the bath Hamiltonians converge weakly to a limiting measure that is absolutely continuous with a Hölder continuous density near the excited energy. Under this assumption, we show that the survival probability of the excited state decays in an approximately exponential manner on suitable time scales and under favourable conditions, with a decay rate given by Fermi's golden rule for the limiting measure. The error is estimated in terms of the coupling strength and the Lévy distance between the discrete spectral measures and the limiting measure. As an application, we treat a two-level atom coupled to a massless bosonic field confined to a large cavity in the rotating-wave approximation.
A simple quantum mechanical model consisting of a discrete level resonantly coupled to a continuum of finite width, where the coupling can be varied from perturbative to strong (Fano-Anderson model), is considered. The particle is initially localized at the discrete level, and the time dependence of the amplitude to find the particle at the discrete level is calculated without resorting to perturbation theory. The deviations from the exponential decay law, predicted by the Fermi's Golden Rule, are discussed. We also study analytic structure of the Green's function (GF) for the model. We analyze the GF poles, branch points and Riemann surface, and show how the Fermi's Golden Rule, valid in perturbative regime for not to large time, appears in this context. The knowledge of analytic structure of the GF in frequency representation opens opportunities for obtaining easy for numerical calculations formulas for the GF in time representation, alternative to those using the spectral density.
Fermi's golden rule applies in the limit where an initial quantum state is weakly coupled to a continuum of other final states overlapping its energy. Here we investigate what happens away from this limit, where the set of final states is discrete, with a nonzero mean level spacing; this question arises in a number of recently investigated many-body systems. For different symmetry classes, we analytically and/or numerically calculate the universal crossovers in the average decay of the initial state as the level spacing is varied, with the golden rule emerging in the limit of a continuum. Among the corrections to the exponential decay of the initial state given by Fermi's golden rule is the appearance of the spectral form factor in the longtime regime for small but nonzero level spacing.
In Jaynes's semiclassical radiation theory, the decay of an excited atomic state depends on its initial population. For the radiative decay of the
2P level in atomic hydrogen, produced by pulsed excitation from the
2S level, the theory predicts a decay rate $\ensuremath{\Gamma}=(1\ensuremath{-}f){\ensuremath{\Gamma}}_{0}+\ensuremath{\beta}P$, where $\ensuremath{\beta}=f{\ensuremath{\Gamma}}_{0}$. Here
f is the initial
2S population, ${\ensuremath{\Gamma}}_{0}$ is the natural
2P decay rate, and
P is the
2P population relative to
f. For $f=0.086\ifmmode\pm\else\textpm\fi{}0.016$, and
P values in the range 0.15 to 0.64, we have measured $\ensuremath{\beta}=\ensuremath{-}(2.8\ifmmode\pm\else\textpm\fi{}2.8)\ifmmode\times\else\texttimes\fi{}{10}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$. The corresponding value from the semiclassical theory is $\ensuremath{\beta}=+(53.8\ifmmode\pm\else\textpm\fi{}10.0)\ifmmode\times\else\texttimes\fi{}{10}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$.