Coupling a discrete state to a quasi-continuum: A model quantum mechanical system that interpolates between Rabi oscillations and decay-revival dynamics

TL;DR

The paper introduces a generalized model where a single discrete state $|\varphi\rangle$ is coupled to an infinite ladder of equally spaced states with a Lorentzian profile $v_k$, parameterized by a width $a$ and spacing $\delta$, enabling a continuous interpolation among the Rabi, Bixon–Jortner, Wigner–Weisskopf, and Lorentzian-Fano continua. A semi-analytical solution is developed by reducing the eigenproblem to a transcendental equation for the dimensionless energies $\epsilon_\mu$, with explicit expressions for eigenvalues and eigenvectors in terms of the unperturbed spectrum and the couplings; limiting cases recover the known models and their characteristic dynamics. The paper also derives the continuum-limit dynamics, showing how the discrete-state probability $|\langle \varphi|\psi(t)\rangle|^2$ can exhibit exponential decay, oscillations, revivals, or damped oscillations depending on the regime. Overall, this Lorentzian-Bixon–Jortner framework provides a unifying, tractable model that encompasses a rich variety of quantum-optical behaviors and offers a semi-analytical handle to study interpolating dynamics across multiple canonical systems.

Abstract

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.

Figures (10)

• Figure 1: Schematic representation of the unperturbed eigenvalues of the Bixon-Jortner system. The perturbation $V$ introduces a constant coupling between the discrete state and the ladder.
• Figure 2: The Lorentzian generalization of the Bixon-Jortner system.
• Figure 3: The left hand side (LHS) and the right hand side (RHS) of the transcendental eigenvalue equations \ref{['final-eig']} and \ref{['BJ_eigval']}.
• Figure 4: Graphical representation of the transcendental eigenvalue equation \ref{['final-eig']} for different choices of the parameters $a$ and $v$, keeping $\delta=1$. In each plot, the left hand side of the equation is shown in blue and the right hand side is shown in orange. The intersections of these two give the eigenvalues. To compare to the limiting behavior, we also show the RHS of the Bixon-Jortner limit \ref{['BJ-limit']} in dashed gray, and the Rabi limit \ref{['eq:rabi-limit-eigvals']} in dot-dashed green. The top row of three figures have a small value of the dimensionless resonance width $a$, hence correspond to the decoupled Rabi limit described in Section \ref{['decoupled-rabi']}. In this case the eigenvalues are either integers, or the two Rabi values given in \ref{['eq:rabi-limit-eigvals']}. The solutions in this case lie on either vertical lines or the Rabi curves shown. The bottom three rows have large values of $a$, and thus are well-approximated by the Bixon-Jortner case. Thus in this case, the eigenvalues are on the Bixon-Jortner curves. The middle row has intermediate values of $a$, and has behavior that is specific to the Lorentzian version of the generalized Bixon-Jortner system. In each row, the three plots correspond to three separate values of the overall coupling scale $v$. The case of larger $v$ corresponds to larger values of $v/\delta$, thus approximates the continuum limit.
• Figure 5: Dynamics of the initial discrete state for different width parameters $a$, superposed with pure Rabi oscillations, pure exponential decay, and the Bixon-Jortner decay and revivals. In each plot, the solid blue curves show the dynamics of the generalized BJ-system, and the dashed curves show the superposed limiting systems. These plots correspond to the $\beta=0.5$ plot of the BJ system in barnett2002methods, page 29, which in our notation corresponds to $v\approx 0.16$, $\delta=1$. As $a$ goes from $a=0.1$ to $a=20$, the system behavior interpolates between Rabi oscillations and the BJ revival dynamics given in barnett2002methods.