Polariton-Type Dispersion Laws for Four-Level Atoms with Nonequidistant Energy Spectrum That Interact with Three Laser Pulses

Abstract

Characteristic features of polariton dispersion laws are studied for four-level atoms interacting with three pulses of coherent laser radiation with frequencies that are in resonance with optically allowed one-photon transitions 1 $\leftrightarrows$ 2, 2 $\leftrightarrows$ 3, and 3 $\leftrightarrows$ 4 with regard to two-photon transitions 1 $\leftrightarrows$ 3 and 2 $\leftrightarrows$ 4, as well as a direct three-photon transition 1 $\leftrightarrows$ 4. The approximation of a given photon density of three pulses is used. It is shown that the dispersion law consists of four branches the position and shape of which are specified by the Rabi frequencies of the above transitions and the photon densities of the three pulses. The direct consideration of all six optical transitions leads to the dependence of the dispersion law of atomic polaritons on quantum parameters—the phase differences between the Rabi frequencies of the transitions under consideration. Parameter values are found for which the branches of the dispersion law may intersect.

APPENDIX

Expression (16) describes a fourth-degree curve of the form

$$a{{x}^{4}} + b{{x}^{3}} + c{{x}^{2}} + dx + e = 0$$

with real coefficients and a ≠ 0. The nature of the roots of expression (16) is determined by the sign of its discriminant

$$\begin{gathered} {\text{Discr}} = 256{{a}^{3}}{{e}^{3}} - 192{{a}^{2}}bd{{e}^{2}} - 128{{a}^{2}}{{c}^{2}}{{e}^{2}} \\ \, + 144{{a}^{2}}c{{d}^{2}}e - 27{{a}^{2}}{{d}^{4}} + 144a{{b}^{2}}c{{e}^{2}} - 6a{{b}^{2}}{{d}^{2}}e \\ \, - 80ab{{c}^{2}}de + 18abc{{d}^{2}} + 16a{{c}^{4}}e - 4a{{c}^{3}}{{d}^{2}} - 27{{b}^{4}}{{a}^{2}} \\ \, + 18{{b}^{3}}cde - 4{{b}^{3}}{{d}^{3}} - 4{{b}^{2}}{{c}^{3}}e + {{b}^{2}}{{c}^{2}}{{d}^{2}}. \\ \end{gathered} $$

Let us introduce auxiliary polynomials

$$P = 8ac - 3{{b}^{2}},$$

$$R = {{b}^{3}} + 8d{{a}^{2}} - 4abc,$$

$${{D}_{0}} = {{c}^{2}} - 3bd + 12ae,$$

$$D = 64{{a}^{3}}e - 16{{a}^{2}}{{c}^{2}} + 16a{{b}^{2}}c - 16{{a}^{2}}bd - 3{{b}^{4}}.$$

The following variants of roots are possible.

1. If Discr > 0, then all four roots of the equation are real, and, if P < 0 and D < 0, then all four roots are real and distinct.

2. If Discr = 0, then (and only then) the polynomial has a multiple root:

(a) if P < 0, D < 0, and D₀ ≠ 0, then there exist a real double root and two real simple roots;

(b) if D₀ = 0 and D ≠ 0, then there exist a triple root and a simple root, all real;

(c) if P < 0, then there exist two real double roots;

(d) if D₀ = 0, then all four roots are equal to –b/4a.