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arXiv:1606.06153

Classical analogue of the continuous transition between the Weisskopf-Wigner exponential decay and the Rabi oscillation

Gilles Dolfo, J. Vigué

Published Jun 20, 2016• 1 citation
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Abstract

When a discrete state is coupled to a continuum, the dynamics can be described either by the Weisskopf-Wigner exponential decay or by the Rabi oscillation, depending on the relative magnitudes of the continuum width and of the Rabi frequency. A continuous transition between these two regimes exists, as demonstrated in 1977 by C. Cohen-Tannoudji and P. Avan. Here, we describe a fully analogous transition in classical mechanics, by studying the dynamics of two coupled mechanical oscillators in the presence of damping. By varying the relative magnitudes of the damping and coupling terms, we observe a continuous transition between a regime analogous to the Rabi oscillation and a regime analogous to the Weisskopf-Wigner exponential decay.

DOI
10.48550/arxiv.1606.06153

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When a discrete state is coupled to a continuum, the dynamics can be described either by the Weisskopf-Wigner exponential decay or by the Rabi oscillation, depending on the relative magnitudes of the continuum width and of the Rabi frequency. A continuous transition between these two regimes exists, as demonstrated in 1977 by C. Cohen-Tannoudji and P. Avan. Here, we describe a fully analogous transition in classical mechanics, by studying the dynamics of two coupled mechanical oscillators in the presence of damping. By varying the relative magnitudes of the damping and coupling terms, we observe a continuous transition between a regime analogous to the Rabi oscillation and a regime analogous to the Weisskopf-Wigner exponential decay.
We formulate a quantum-mechanical system consisting of a single discrete state coupled to an infinite ladder of equally spaced states, with the coupling described by a Lorentzian profile. Various limiting cases of this system correspond to well-known models in quantum optics: the narrow-resonance limit yields the Rabi model, the wide-resonance limit yields the Bixon-Jortner model, the wide-resonance true continuum limit yields the Wigner-Weisskopf model, and the fixed-resonance true continuum limit yields a system typically studied using methods developed by Fano. We obtain a semianalytical solution to the eigenvalue problem by reducing it to a transcendental equation and demonstrate the limiting behaviors described above. We then investigate numerically the dynamics of the initial discrete state and show that, in various limiting regimes, the system exhibits a wide range of behaviors predicted by the asymptotic theory, including exponential decay, revivals, Rabi oscillations, and damped oscillations. The ability of the system to interpolate between such a rich variety of behaviors and established model systems, together with the accessibility of a semianalytical solution, makes it a useful model in quantum optics and related fields.
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.
We formulate a quantum-mechanical system consisting of a single discrete state coupled to an infinite ladder of equally spaced states, with the coupling described by a Lorentzian profile. Various limiting cases of this system correspond to well-known models in quantum optics: the narrow-resonance limit yields the Rabi model, the wide-resonance limit yields the Bixon--Jortner model, the wide-resonance true continuum limit yields the Wigner--Weisskopf model, and the fixed-resonance true continuum limit yields a system typically studied using methods developed by Fano. We obtain a semianalytical solution to the eigenvalue problem by reducing it to a transcendental equation and demonstrate the limiting behaviors described above. We then investigate numerically the dynamics of the initial discrete state and show that, in various limiting regimes, the system exhibits a wide range of behaviors predicted by the asymptotic theory, including exponential decay, revivals, Rabi oscillations, and damped oscillations. The ability of the system to interpolate between such a rich variety of behaviors and established model systems, together with the accessibility of a semianalytical solution, makes it a useful model in quantum optics and related fields.
This pedagogical paper synthesizes a coherent conceptual pathway from the mode expansion of free quantum fields to the continuous dynamics of resonant energy exchange in two-level systems. We clarify several recurrent points of confusion: (i) the fixed characteristic frequency of a quantum harmonic oscillator across its number states; (ii) the relation between spatial nodes of energy eigenfunctions and the principal quantum number in atomic orbitals; (iii) the interpretation of complex-conjugate mode functions as counter-propagating waves; (iv) the analogy—and the limits of that analogy—between discrete mode spectra of classical strings and quantum energy quantization; and (v) the continuous, unitary character of radiative transitions mediated by Rabi oscillations. Particular emphasis is placed on the mathematical isomorphism between the Schrödinger dynamics of probability amplitudes and the Newtonian dynamics of a classical harmonic oscillator governed by Hooke’s law. The discussion is intended as a self-contained resource for students and non-specialists seeking a unified intuitive picture that remains faithful to the underlying quantum theory.
The quantum theory of a damped harmonic oscillator has been continuously studied since 1939. For quantization, an oscillation oscillator usually uses a reservoir that includes a number of finite and discrete oscillators. However, a set of discrete oscillators is unable to study the quantum theory of an Ohmic damping (damping proportional to velocity). In this research, a continuous set of harmonic oscillators has been used to investigate Ohmic damping. Using a continuum reservoir instead of a discrete reservoir will enrich the results of the dynamic system. The results of this research can be used to study nano-mechanical systems and opto-mechanical systems.