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Birkhäuser Basel eBooks

Application of ATS in a Quantum-optical Model

Anatolii Alekseevich Karatsuba, Ekatherina Anatol'evna Karatsuba

Published Jan 1, 2009• 10 citations
Publisher

Abstract

The problem of the interaction of a single two-level atom with a single mode of the quantized electromagnetic field in a coherent state in an ideal resonator in the resonance case is considered. The evolution in time of the atomic inversion, represented by the Jaynes-Cummings sum, is studied. On the basis of the application of the theorem on the approximation of a trigonometric sum by a shorter one (ATS), a new efficient method for approximating the Jaynes-Cummings sum is constructed. New asymptotic formulas for the atomic inversion are found, which approximate it on various time intervals, defined by relation between the atom-field coupling constant and the average photon number in the resonator field before the interaction of the field with the atom. The asymptotics that we obtain give the possibility to predetermine the details of the process of the inversion in the Jaynes-Cummings model depending on the field characteristics.

DOI
10.1007/978-3-7643-9906-1_13

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The problem of the interaction of a single two-level atom with a single mode of the quantized electromagnetic field in a coherent state in an ideal resonator in the resonance case is considered. The evolution in time of the atomic inversion, represented by the Jaynes-Cummings sum, is studied. On the basis of the application of the theorem on the approximation of a trigonometric sum by a shorter one (ATS), a new efficient method for approximating the Jaynes-Cummings sum is constructed. New asymptotic formulas for the atomic inversion are found, which approximate it on various time intervals, defined by relation between the atom-field coupling constant and the average photon number in the resonator field before the interaction of the field with the atom. The asymptotics that we obtain give the possibility to predetermine the details of the process of the inversion in the Jaynes-Cummings model depending on the field characteristics.
A new method is devised to study the atomic inversion in the model of a two-level atom interacting with a single quantized mode of the (initially coherent) electromagnetic field in an ideal resonant cavity. The method is based on number-theoretic results applied to the approximation of special series, specifically, on the functional equation for Jacobi theta functions and the ATS theorem. New asymptotic formulas are derived, with the help of which the behavior of the atomic inversion function on various time intervals can be determined in detail depending on the parameters of the system.
We present a new approach to the study of the function of atomic inversion in the model of interaction of a single two-level atom with a single mode of the quantized electromagnetic field in the coherent state in an ideal resonator. The approach suggested is based on an application of certain number-theoretic results to the approximation of the trigonometric sums of a special form, in particular of the functional equation of the Jacobi theta function. New asymptotic formulae for the atomic inversion are found. The asymptotics that we obtain make it possible to predetermine the details of the behavior of the inversion on various time intervals depending on the parameters of the system.
We extend the standard Jaynes-Cummings model to include transient effects due to a modulation of the atom-field coupling. The exact solution for such a generalized model is found in terms of Bessel functions for the atom-cavity mode interaction modulated by the exponential function. This function is incorporated into the basic Hamiltonian phenomenologically for modelling the processes of adiabatic switching on or switching off the interaction of a two-level atom (at rest) with a quantized mode. The dynamical behaviour of the atomic inversion for such processes is plotted and compared with the atomic dynamics for the standard model. If we take into account that, in micromaser experiments, Rydberg atoms move through a cavity with a spatially inhomogenous field, we deal with the Jaynes-Cummings model with a spatial modulation the form of which is defined by the mode-shape function depending, in turn, on the geometrical form of resonator. In the cases of a rectangular and a spherical cavity the analytical solutions for atomic inversion are found in the explicit form in the weak-coupling limit. We consider in detail the time evolution of the atomic inversion while the atom passes through a rectangular cavity. Contrary to the standard model the initial inversion can be exactly restored during the atomic transit depending on the relation between the detuning, the atomic velocity and the number of half-wavelengths of the chosen mode. The numerical solutions of the basic equations are also obtained and plotted for rectangular and spherical resonators.
Jaynes–Cummings (JC) models are by far the most widely used to study the dynamics of two-level systems, such as atoms or molecules under suitable conditions, coupled to quantized electromagnetic fields. Nowadays, analytical solutions of JC models are more useful than ever for understanding and controlling radiation-matter interactions relevant to quantum information processing, quantum computing, and other quantum technologies. This study presents the analysis and generalization of a method for the solution of the dynamical problem in the time-independent JC model (Carbonaro et al., 1979) that enables an exact description of the atom-field dynamics in the resonant time-dependent JC model and in a nontrivial class of off-resonance time-dependent JC models. Our analysis produces the ready-to-use forms of the evolution operator in the resonant and non-resonant cases, which are then used to derive the evolution of the atomic population inversion for some types of atom-field coupling. The central idea behind the proposed approach offers opportunities for its extension to other time-dependent cases and its generalization to JC-related models.
An exact solution is found for the collective model of two identical two-level atoms that resonantly interact with a two-mode quantum electromagnetic field in an ideal cavity via two-photon nondegenerate transitions. In the case under study, at the initial moment, both field modes are in the coherent state and atoms are in the excited state. The time dependences of the atomic probabilities, the mean number of photons in the modes, and the statistics and squeezing of the photon modes are studied based on the exact solution.