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Approximate Solutions for the Ion-Laser Interaction in the High Intensity Regime: Matrix Method Perturbative Analysis

Braulio Misael Villegas-Martínez, H. M. Moya-Cessa, Francisco Soto

Published Apr 19, 2021• 0 citations
Publisher

Abstract

Abstract We provide an explicit expression for the second-order perturbative solution of a single trapped-ion interactingwith a lser field in the strong excitation regime. From the perturbative analytical solution, based on a matrix methodand a final normalization of the perturbed solutions, we show that the probability to find the ion in its excited statefits well with former results.

DOI
10.21203/rs.3.rs-416083/v1

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Abstract We provide an explicit expression for the second-order perturbative solution of a single trapped-ion interactingwith a lser field in the strong excitation regime. From the perturbative analytical solution, based on a matrix methodand a final normalization of the perturbed solutions, we show that the probability to find the ion in its excited statefits well with former results.
We present a second-order perturbative analysis of the model describing a two-level trapped ion interacting with a travelling laser field, in the Lamb–Dicke regime. Unlike the customary approach, based on the interaction picture and the rotating wave approximation, we reduce the original Hamiltonian to a time-independent model, unitarily equivalent to a spin-boson model, which features a form well suited to apply the perturbation theory. We determine the first- and second-order corrections of eigenstates and eigenvalues. By varying the interaction parameters we identify four regimes characterized by different eigenvalue distributions (possibly including level doublets) and suggest the presence of an intermediate regime where the spectrum undergoes macroscopic changes. In this regime, where the perturbation approach does not hold, our reduced Hamiltonian is shown to reproduce the Jaynes–Cummings model.
A high-intensity single-mode gas laser is analyzed by several mathematical methods. Exact numerical results are best obtained by solving for the coefficients of a Fourier series solution by using a backward recurrence scheme or a successive convergents method for evaluating a continued fraction. Numerical integration techniques can also be used to convert the periodic boundary conditions to initial conditions and then solve the resulting initial-value problem. Direct analytic integration leads to exact solutions in terms of Bessel functions for the special case of zero detuning, equal decay rates, and neglecting of collision broadening. For general single-mode operation, an improved analytic approximation [the second recursion relation approximation (2RCRA)] is obtained by taking an additional term in the recursion relations for the Fourier coefficients. This approximation predicts the existence of one of the secondary resonances in graphs of the spatially averaged population inversion and quadrature polarization coefficient densities and matches the exact solution from the position of the first resonance and outward independent of the intensity. The 2RCRA also matches the exact detuning curve except at small detunings. The first-order ("rate equation") approximation, by comparison, does not predict any secondary resonances and does not match the exact solution at higher intensities. A physical interpretation for the appearance of the secondary resonances is presented in terms of multiphoton interactions and a prediction for the positions of the resonances for the special case discussed above is obtained. As predicted by Greenstein the exact detuning curves show quantitatively that the Lamb dip decreases and disappears as the intensity increases for high excitations.