Optical nonlinearity of cold atomic ensemble driven by strong coherent field in a saturation regime

TL;DR

This work develops a rigorous microscopic framework to compute the optical nonlinearity of a cold-atom dielectric driven by a strong coherent field. Using the Maxwell–Heisenberg equations and the Kubo formalism, it derives a vectorial susceptibility that splits into Kerr-type and parametric (four-wave mixing–driven) components, with the Mollow triplet revealing the underlying quasi-energy structure. In dilute ensembles, the parametric response peaks near intermediate saturation ($s\approx 1$) and exhibits Autler–Townes features in the transverse channels, while the longitudinal response can show amplification in the saturation regime. Interpolation to high density highlights Lorentz–Lorenz local-field effects, but strong Mollow-induced fluctuations in dense media suppress a simple macroscopic enhancement of the parametric nonlinearity, implying fundamental limits for entangled-photon generation in quantum communication applications.

Abstract

We present a microscopic analysis and evaluation of the dielectric susceptibility of a dielectric medium consisting of vector-type two-energy-level atoms responding on a weak probe mode when the atoms are driven by a strong coherent field. Each atom, in an environment of others, exists as a quasiparticle further structuring a bulk medium. In a limit of dilute atomic gas, the dynamics of each atom follows the Mollow-type nonlinear excitation regime, and the medium susceptibility collectivizes the individual atomic responses to the probe mode. We outline how the collective dynamics can be interpolated up to a dense medium, and we argue from general positions that in such a medium the optical nonlinearity and, in particular, its parametric part could be significantly magnified by manipulating both the coherent pump and the sample density. That indicates certain limitations for potential capabilities of quantum communication protocols utilizing the entangled photons, created by a parametric process, as a main resource of quantum correlations.

Figures (6)

• Figure 1: Energy structure of a $V$-type tripod atom constructing the medium used in our simulation of the dielectric sample. The Zeeman states are specified by the angular momentum and its projection. The shaded arrow indicates the strong field quasi-resonant to the reference transition. The three dashed arrows belong to a weak probe impinging on the atom from an arbitrary direction and superposed in the atomic basis. The probe is detuned from both the atomic transition and coherent mode.
• Figure 2: The excitation geometry of an isolated atom, thinkable as an elementary scatterer inside a dilute atomic cloud, and driven by two modes of the strong control and weak probe, as shown in Fig. \ref{['fig1']}. The responding field contains both the spontaneous emission and scattered part of the probe mode.
• Figure 3: The Kerr-type nonlinear susceptibility for $z$-polarized probe and for the control field resonant to the atomic transition, and for different saturation parameters $s$.
• Figure 4: Same as in Fig. \ref{['fig3']} but for the $x$-polarized probe.
• Figure 5: The parametric nonlinear susceptibility for the control field resonant to the atomic transition, and for different saturation parameters $s$. The upper and lower plots respectively show its absolute value and argument.