Two-photon processes based on quantum commutators

We developed a method to calculate two-photon processes in quantum mechanics that replaces the infinite summation over the intermediate states by a perturbation expansion. This latter consists of a series of commutators that involve position, momentum, and Hamiltonian quantum operators. We analyzed several single- and many-particle cases for which a closed-form solution to the perturbation expansion exists, as well as more complicated cases for which a solution is found by convergence. Throughout the article, Rayleigh and Raman scattering are taken as examples of two-photon processes. The present method provides a clear distinction between the Thomson scattering, regarded as classical scattering, and quantum contributions. Such a distinction lets us derive general results concerning light scattering. Finally, possible extensions to the developed formalism are discussed.

Figure 1

The light scattering process. An incident photon with energy $E_1$ and linear polarization vector ${\underline{\epsilon }}_1$ scatters off a target T. The scattered photon has energy $E_2$ and linear polarization vector ${\underline{\epsilon }}_2$. $\theta$ is the scattering angle, while the $xz$ plane is the scattering plane.

Figure 2

Graphical representation of the scattering amplitudes ${\mathcal{A}}_{12}$ and ${\mathcal{A}}_{21}$. The state $|s\rangle$ is the target intermediate state during the scattering process.

Figure 3

Probability density ${\left|\mathcal{M}\right|}^2$ for Raman scattering off one particle trapped by a potential box (see Sec. 5a). Atomic units are used. The potential parameters can be retrieved by analyzing the relative positions of the resonance peaks.

Figure 4

Same as Fig. 3, but relative to the case study presented in Sec. 5b.

Figure 5

Probability density for elastic light scattering (Rayleigh scattering) off a particle bound by a harmonic potential.

Figure 6

Probability density of elastic light scattering (Rayleigh scattering) off a particle vibrating along the $x$ direction, the vibration being modeled by a Morse potential. Light is polarized along the $x$ direction. Parameters are set as $(a,V_0,x_e)=(1/3,1,0)$. ${\mathcal{M}}^{\left(n\right)}$ means scattering amplitude evaluated at the $n\text{th}$ order, with ${\mathcal{M}}^{\left(0\right)}={\underline{\epsilon }}_1·{\underline{\epsilon }}_2^{*}$. Convergence of the series expansion is showed.

Figure 7

A set of two coupled-harmonic oscillators. The binding potential (blue line) is harmonic, while the A-B coupling (red line) can be differently defined.