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arXiv:1809.04391 · Physical Review A

Time ordering in the classically driven nonlinear Jaynes-Cummings model

Tobias Lipfert, F. Krumm, Mikhail I. Kolobov, W. Vogel

Published Dec 12, 2018• 4 citations

Abstract

Recently [F. Krumm and W. Vogel, Phys. Rev. A 97, 043806 (2018)], the detuned and nonlinear Jaynes-Cummings model describing the quantized motion of a trapped ion was introduced and its corresponding dynamics was solved via considering the driving laser in a quantized manner. For the dynamics with a classical driving laser an approximation in terms of neglected time ordering was considered to obtain analytical expressions and compared to numerically obtained exact dynamics, which revealed the insufficiency of this approximation. In the present work we reconsider this model and show that it can likewise be solved analytically, i.e., including time ordering, with a classical driving laser field. Using the analytical solution, we investigate the quantum time-ordering effects of the system with respect to nonclassicality and nonsimulatability of the motional states of the ion. Furthermore, we use the Magnus expansion to analyze the impact of certain orders of the time ordering. We derive an exact radius of convergence for these time-ordering corrections beyond the established and only sufficient criterion. Finally, the differences between the solution derived here and the previously found one using a quantized pump are discussed.

DOI: 10.1103/physreva.98.063817

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Recently [F. Krumm and W. Vogel, Phys. Rev. A 97, 043806 (2018)], the detuned and nonlinear Jaynes-Cummings model describing the quantized motion of a trapped ion was introduced and its corresponding dynamics was solved via considering the driving laser in a quantized manner. For the dynamics with a classical driving laser an approximation in terms of neglected time ordering was considered to obtain analytical expressions and compared to numerically obtained exact dynamics, which revealed the insufficiency of this approximation. In the present work we reconsider this model and show that it can likewise be solved analytically, i.e., including time ordering, with a classical driving laser field. Using the analytical solution, we investigate the quantum time-ordering effects of the system with respect to nonclassicality and nonsimulatability of the motional states of the ion. Furthermore, we use the Magnus expansion to analyze the impact of certain orders of the time ordering. We derive an exact radius of convergence for these time-ordering corrections beyond the established and only sufficient criterion. Finally, the differences between the solution derived here and the previously found one using a quantized pump are discussed.

Abstract In a previous paper (Krumm and Vogel 2018 Phys. Rev. A 97 043806) we presented a method to solve the nonlinear Jaynes–Cummings dynamics, describing the quantized motion of a trapped ion exactly, including detuning. Here we investigate this model with respect to nonclassical effects, such as squeezing and sub-Poisson statistics. We show that for the versatile model under study there exist quantum phenomena beyond squeezing and sub-Poisson statistics, such as anomalous quantum correlations of two non-commuting observables. In particular, it is shown that for the excitation of the zeroth sideband neither squeezing nor sub-Poisson statistics occur, but anomalous correlations can be verified. Furthermore, it is shown how these anomalous correlation functions can be derived from measured data.

The Lindblad approach to open quantum systems is introduced to study the dynamics of a single trapped ion prepared in nonclassical motional states and subjected to continuous measurement of its internal population. This results in an inhibition of the dynamics similar to the one occurring in the quantum Zeno effect. In particular, modifications to the Jaynes-Cummings collapses and revivals arising from an initial coherent state of motion in various regimes of interaction with the driving laser are dealt with in detail.

The Jaynes-Cummings model (JCM) is studied in the absence of the rotating-wave approximation (RWA) by a coherent-state expansion technique. In comparison with the previous paper in which the coherent-state expansion was performed only to the third order, we carry out in this paper a complete expansion to demonstrate exactly the dynamics of the JCM without the RWA. Our study gives a systematic method to solve the non-RWA problem, which would be useful in various physical systems, e.g., in a system with an ultracold trapped ion experiencing the running waves of lasers.