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.
We consider a quantum system that is initially localized at ${\mathbit{x}}_{\text{in}}$ and that is repeatedly projectively probed with a fixed period $\ensuremath{\tau}$ at position ${\mathbit{x}}_{\mathrm{d}}$. We ask for the probability
Fn that the system is detected at ${\mathbit{x}}_{\mathrm{d}}$ for the very first time, where
n is the number of detection attempts. We relate the asymptotic decay and oscillations of
Fn with the system's energy spectrum, which is assumed to be absolutely continuous. In particular,
Fn is determined by the Hamiltonian's measurement spectral density of states (MSDOS)
f(E) that is closely related to the density of energy states (DOS). We find that
Fn decays like a power law whose exponent is determined by the power-law exponent
dS of
f(E) around its singularities
E∗. Our findings are analogous to the classical first passage theory of random walks. In contrast to the classical case, the decay of
Fn is accompanied by oscillations with frequencies that are determined by the singularities
E∗. This gives rise to critical detection periods ${\ensuremath{\tau}}_{c}$ at which the oscillations disappear. In the ordinary case
dS can be identified with the spectral dimension associated with the DOS. Furthermore, the singularities
E∗ are the van Hove singularities of the DOS in this case. We find that the asymptotic statistics of
Fn depend crucially on the initial and detection state and can be wildly different for out-of-the-ordinary states, which is in sharp contrast to the classical theory. The properties of the first-detection probabilities can alternatively be derived from the transition amplitudes. All our results are confirmed by numerical simulations of the tight-binding model, and of a free particle in continuous space both with a normal and with an anomalous dispersion relation. We provide explicit asymptotic formulas for the first-detection probability in these models.
The survival amplitude
G(t) of a nonstationary state decaying into a purely continuous spectrum is treated in terms of an integral transform of an energy distribution with $\ensuremath{\infty}>E\ensuremath{\geqslant}0$. We examine three such distributions. Two are real functions, the Lorentzian
gL(E) and a modified Lorentzian
G(E)=gL(E)E1∕2, and one is the complex version of
gL(E),gcL(E). Real distributions are associated with Hermitian treatments while complex ones result from non-Hermitian treatments. The difference between the two has repercussions on the
G(t) for nonexponential decay (NED) and on the understanding of irreversible decay at the quantum level. For all three distributions, we derive analytically amplitudes (propagators) for NED and then show that these satisfy differential equations, from which additional insight into the decay process for very long and very short times can be obtained. By making analogy with the classical Langevin equation, the terms of the differential equation that are derived when the simpler
gL(E) and
gcL(E) are employed, are interpreted using concepts such as friction and fluctuation. On the other hand, when
gL(E) is multiplied by an energy-dependent factor, as in
G(E), the results are, as expected, more complicated and the interpretability of the differential equation satisfied by the NED propagator loses clarity.
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) 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) 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) as a function of the number of oscillations
n 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.
Abstract Methods based on the use of Green’s functions or Jost functions and the Fock–Krylov method are apparently very different approaches to understand the time evolution of unstable states. We show that the two former methods are equivalent up to some constants and as an outcome find an analytic expression for the energy density of states in the Fock–Krylov amplitude in terms of the coefficients introduced in the Green’s functions and the Jost functions methods. This model-independent density is further used to obtain an analytical expression for the survival amplitude and study its behaviour at large times. Using these expressions, we investigate the origin of the oscillatory behaviour of the decay law in the region of the transition from the exponential to the non-exponential at large times. With the objective of understanding the failure of nuclear and particle physics experiments in observing the non-exponential decay law predicted by quantum mechanics for large times, we derive analytical formulae for the critical transition time, , from the exponential to the inverse power law behaviour at large times. Evaluating for some particle resonances and narrow nuclear states which have been tested experimentally to verify the exponential decay law, we conclude that the large time power law in particle and nuclear decay is hard to find experimentally.