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Physical Review A

Exponential decay, recurrences, and quantum-mechanical spreading in a quasicontinuum model

Peter W. Milonni, Jay R. Ackerhalt, H. W. Galbraith, Mei-Li Shih

Published Jul 1, 1983• 62 citations
Publisher

Abstract

We consider a single quantum state coupled equally to each of a set of evenly spaced quasicontinuum (QC) states. We obtain a delay differential equation for the initial-state probability amplitude, and this equation is solved analytically. When the QC-level spacing goes to zero, the initial-state probability decays exactly exponentially. For finite QC-level spacings, however, there are recurrences of initial-state probability. We discuss Tolman's "quantum-mechanical spreading" of probability and also a classical analog of our model.

DOI
10.1103/physreva.28.32

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We consider a single quantum state coupled equally to each of a set of evenly spaced quasicontinuum (QC) states. We obtain a delay differential equation for the initial-state probability amplitude, and this equation is solved analytically. When the QC-level spacing goes to zero, the initial-state probability decays exactly exponentially. For finite QC-level spacings, however, there are recurrences of initial-state probability. We discuss Tolman's "quantum-mechanical spreading" of probability and also a classical analog of our model.
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.
Suppose an initial state is coupled to a continuum of energy states. The population of the initial state is expected to decrease with time, but is the decrease monotonic? The occupation probability of the initial state is the survival probability and the question is equivalent to asking if there are intervals of time where the survival probability increases. Such regrowth is also referred to as regeneration or recurrence and it occurs in systems with a countable number of discrete states. Regrowth is investigated with a simple model that allows transitions between the initial state and continuum states, but transitions between continuum states are not permitted. The model uses the solution of Schroedinger's Equation for a full energy continuum. Such a continuum runs from -infinity to +infinity and is found to have only exponential decay in time. However, the survival probability for a truncated continuum turns out to have a wide variety of behaviors. Generally, the survival probability decreases by several orders of magnitudes, often as an exponential, and then has limited regrowth.
We study a model where two discrete levels are coupled to a band of closely spaced levels forming a quasicontinuum (QC). This model describes how the initial-stage coherence in multiphoton excitation of polyatomic molecules interferes with the recurrent time dynamics due to the QC coupling. The time evolution of the system is solved exactly assuming the QC to be infinite, flat, and uniform; and a detailed investigation of different special cases is carried out. Weak QC excitation is found to exhibit features which are connected to few level dynamics rather than to an irreversible decay, and we study how this modifies the absorption cross section and the Rabi oscillations. The conditions for coherent trapping are discussed.
We formulate a quantum mechanical system consisting of a single discrete state coupled to an infinite ladder of equally-spaced states, the coupling between the two being given by a Lorentzian profile. Various limits of this system correspond to well-known models from quantum optics, namely, the narrow resonance limit gives the Rabi system, the wide resonance limit gives the Bixon-Jortner system, the wide resonance, true continuum limit gives the Wigner-Weisskopf system, and the fixed resonance, true continuum limit gives a system that is typically studied by methods developed by Fano. We give a semi-analytical solution of the eigenvalue problem by reducing it to a transcendental equation, and demonstrate the aforementioned limiting behaviors. We then study the dynamics of the initial discrete state numerically, and show that it gives a wide range of behaviors in various limiting cases as predicted by our asymptotic theory including exponential decay, revivals, Rabi oscillations, and damped oscillations. The ability of this system to interpolate between such a rich set of behaviors and existing model systems, and the accessibility of a semi-analytical solution, make it a useful model system in quantum optics and related fields.
The photoexcitation of a quasicontinuum of atomic or molecular levels by a laser pulse with a fast rise-time is considered. We first generalize non-perturbative treatments of quasicontinuum excitation. A second general theoretical method is developed in which resonant states are considered non-perturbatively whereas background non-resonant states are included through a perturbative approximation. As a simple example we show how our method describes the photoexcitation of a quasicontinuum of equally spaced levels, exhibiting Rabi oscillations periodically perturbed by background states. We then examine the case of a Rydberg series of highly excited states coherently excited from a low-lying initial state. The time evolution of the coherent superposition of Rydberg levels is shown to be a perturbed Rabi oscillation, but without the periodic disruption of the equally spaced quasicontinuum. Finally, we include the effect of further excitation to a second continuum.