Electric dipole approximation for perturbation: What is this symbol and how was it part of this derivation?

Question

I'm reading electrodynamics physics notes that describe the polarization of a medium. The notes describe the active atoms as having two levels $a$ and $b$, separated by energy $\hbar \omega$ and represented by a density matrix $\rho$. The atoms are stationary.

The equation of motion of the density matrix is

$$\dot{\rho} = -i[H, \rho] - \dfrac{1}{2}\left(\Gamma \rho + \rho \Gamma \right) + \lambda,$$

where

$$\rho = \begin{bmatrix} \rho_{aa} & \rho_{ab} \\ \rho_{ba} & \rho_{bb} \end{bmatrix}, \ \ \ \ \ H = \begin{bmatrix} W_{a} & V \\ V & W_{b} \end{bmatrix}, \\ \Gamma = \begin{bmatrix} \gamma_{a} & 0 \\ 0 & \gamma_{b} \end{bmatrix}, \ \ \ \ \ \lambda = \begin{bmatrix} \lambda_{a} & 0 \\ 0 & \lambda_{b} \end{bmatrix}$$

The notes then state that the perturbation Hamiltonian is $\hbar V$, and the unperturbed energies of the levels are $\hbar W_a$ and $\hbar W_b$. Furthermore, the two levels decay with damping constants $\gamma_a$ and $\gamma_b$, and are populated by pumping at rates $\lambda_a$ and $\lambda_b$.

Therefore, using the Fourier expansion of the electric field $E(z, t) = \sum\limits_n A_n(t) u_n(z)$, where $u_n(z) = \sin(k_n z)$ and $k_n = \dfrac{n \pi}{L}$, the notes claim that the electric dipole approximation for the perturbation becomes

I've included the image of the equation because I don't actually understand what that symbol between $A(t)$ and $u(z)$ is (I can't find it as a LaTeX symbol, and I've never seen it before). What is this symbol and how was it part of this derivation?

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Relevant: Quantum Mechanical Electric Dipole Hamiltonian

$$V(t) = - \left( \sum_j \dfrac{q_j}{m_j} \left( \hat{\epsilon} \cdot \hat{p}_j \right) \right) \dfrac{E_0}{\omega} \sin(\omega t) \tag{7.3.13}$$

Answer 1

Energy of a dipole $p$ is minus $p$ times electric field (https://unlcms.unl.edu/cas/physics/tsymbal/teaching/EM-913/section4-Electrostatics.pdf), so your symbol probably denotes the electric dipole moment (which is closely related to polarization, as @ZeroTheHero suggested).

EDIT (Jan 20, 2022) Looks like the derivation of the energy of an electric dipole in the electric field is given in Journal of Modern Optics (2004) vol. 51, no. 8, 1137–114, Section 2, so the symbol is indeed the electric dipole moment.

Answer 2

Forming my comments into an answer.

The odd symbol you're looking at is most certainly the (negative of the) electric dipole moment of an individual atom, or the electric dipole moment density of the uniform gas. I've never seen that symbol used before, so it's quite odd the author didn't clearly state what this symbol meant.

You're wondering how to derive the electric dipole approximation, which is essentially the equation in that picture, from the equations up above (the evolution equation for the density matrix, and the electric field Fourier decomposition). However, the dipole approximation is a separately derived approximation, and by itself has nothing to do with those equations. The author of those lecture notes is presumably using the dipole approximation as a tool to write a simplified evolution equation for $\rho$.