In this paper, we study the one-level Friedrichs model with using the quantum time super-operator that predicts the excited state decay inside the continuum. Its survival probability in long time limit is an algebraically decreasing function and an exponentially decreasing multiplied by the oscillating functions.
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 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 ).
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.
When the initial state of a quantum mechanical system is an excited state, then it is expected that the occupation, or survival, probability of that state will decrease. This is studied numerically within the Bixon-Jortner model, which was introduced to model intramolecular radiationless transitions. Here a finite set of states is used and for a fixed number of states, the parameters of the model are the energy level separation and the strength of the transition matrix element. All three of these are varied to see their effects on the survival probability. After a short interval of time, the survival probability decay is often found to be an exponential. But the survival probability is then found to increase with further time and then decrease in a pattern that continues in time. This repopulation is a general feature when a countable set of states is present.