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Molecular Physics

Interference effects in sequential decay

Abraham Nitzan, Joshua Jortner, B. J. Berne

Published Aug 1, 1973• 43 citations
Publisher

Abstract

In this paper we utilize the Green's function method to study sequential decay processes which involve interference effects. The model system involves a zero-order discrete state coupled to a set of continua, which are themselves coupled together, while the coupling matrix elements are energy independent. Interference effects in parallel and consecutive decay involving two continua result in the retardation of the decay rate of the initial state and in the reversal of the branching ratio for the population of the two continua. Finally, a general solution was provided for the problem of sequential decay involving multiple continua.

DOI
10.1080/00268977300101631

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The dynamics of a single quantum state embedded in one or several (quasi-)continua is one of the most studied phenomena in quantum mechanics. In this work we investigate its discrete analogue and consider short and long time dynamics based on numerical and analytical solutions of the Schrödinger equation. In addition to derivation of explicit conditions for initial exponential decay, it is shown that a recent model of this class [Phys. Rev. A 95, 053821, (2017)], describing a qubit coupled to a phonon reservoir with energy dependent coupling parameters is identical to a qubit interacting with a finite number of parallel regularly spaced band of states via constant couplings. As a consequence, the characteristic near periodic initial state revivals can be viewed as a transition of probability between different continua via the reviving initial state. Furthermore, the observation of polynomial decay of the reviving peaks is present in any system with constant and sufficiently strong coupling.

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In this paper we utilize the Green's function method to study sequential decay processes which involve interference effects. The model system involves a zero-order discrete state coupled to a set of continua, which are themselves coupled together, while the coupling matrix elements are energy independent. Interference effects in parallel and consecutive decay involving two continua result in the retardation of the decay rate of the initial state and in the reversal of the branching ratio for the population of the two continua. Finally, a general solution was provided for the problem of sequential decay involving multiple continua.
A calculation is presented of population trapping by an infinite sequence of bound states and continua, with results that differ from both the pure exponential decay exhibited by a single bound state, and from the combination of exponential decay and no decay that is obtained when there are two bound states coupled to a single continuum. Instead of either of these, the theory predicts a rule for the decay that becomes, for time t large (but not so large that events are unobservable), an inverse power law in t 1 2/. Possible experimental realizations of this problem are discussed, which also indicate that the Floquet form of the wavefunction does not apply.
We study the dynamics of a simple model for quantum decay, where a single state is coupled to a set of discrete states, the pseudocontinuum, each coupled to a real continuum of states. We find that for constant matrix elements between the single state and the pseudocontinuum the decay occurs via one state in a certain region of the parameters, involving the Dicke and quantum Zeno effects. When the matrix elements are random, several cases are identified. For a pseudocontinuum with small bandwidth there are weakly damped oscillations in the probability to be in the initial single state. For intermediate bandwidth one finds mesoscopic fluctuations in the probability with amplitude inversely proportional to the square root of the volume of the pseudocontinuum space. They last for a long time compared to the nonrandom case.
For pt.I see ibid., vol.15, p.2157 (1982). A system consisting of discrete states and continuum states (which form a so-called internal reservoir) is treated, illustrating the theory of sequential quantum processes in Liouville space developed in the preceding paper. The populations and coherences associated with the discrete states satisfy Markovian master equations when the interaction matrix elements between discrete and continuum states are significant over a broad band of continuum states. The population of a single discrete state decays exponentially with time, whilst the population of two coupled discrete states (one only coupled to the continuum states) may exhibit Rabi oscillations. For the latter case of two coupled discrete levels, the population of particular continuum states approaches a two-peak form for long times (Autler-Townes splitting).
The dynamics of a single quantum state embedded in one continuum or several (quasi)continua is one of the most studied phenomena in quantum mechanics. In this paper we investigate its discrete analog and consider short- and long-time dynamics based on numerical and analytical solutions of the Schr\"odinger equation. In addition to derivation of explicit conditions for initial exponential decay, it is shown that a recent model of this class [L. Guo, A. Grimsmo, A. F. Kockum, M. Pletyukhov, and G. Johansson, Phys. Rev. A 95, 053821 (2017)], describing a qubit coupled to a phonon reservoir with energy dependent coupling parameters, is identical to a qubit interacting with a finite number of parallel regularly spaced bands of states via constant couplings. As a consequence, the characteristic near periodic initial-state revivals can be viewed as a transition of probability between different continua via the reviving initial state. Furthermore, polynomial decay of the reviving peaks is present in any system with constant and sufficiently strong coupling.
The dynamics of a single quantum state embedded in one or several (quasi-)continua is one of the most studied phenomena in quantum mechanics. In this work we investigate its discrete analogue and consider short and long time dynamics based on numerical and analytical solutions of the Schrödinger equation. In addition to derivation of explicit conditions for initial exponential decay, it is shown that a recent model of this class [Phys. Rev. A 95, 053821, (2017)], describing a qubit coupled to a phonon reservoir with energy dependent coupling parameters is identical to a qubit interacting with a finite number of parallel regularly spaced band of states via constant couplings. As a consequence, the characteristic near periodic initial state revivals can be viewed as a transition of probability between different continua via the reviving initial state. Furthermore, the observation of polynomial decay of the reviving peaks is present in any system with constant and sufficiently strong coupling.