# 物理代写|量子光学代写Quantum Optics代考|OSE6347

## 物理代写|量子光学代写Quantum Optics代考|Drude–Lorentz and Drude Models

The Drude-Lorentz model is one of the most simple description schemes for a dielectric function. It is based on a harmonic oscillator model, which can be modelled in the form of two oppositely charged particles attached to a spring, and the system is driven by an external electric field $E(t)$. Without specifying any details, we assume that a displacement $x(t)$ leads to a dipole moment $p(t)=e x(t)$. We start by writing down Newton’s equations of motion for the driven oscillator
$$m \ddot{x}=-m \omega_0^2 x-m \gamma \dot{x}+e E(t) .$$
Here $m$ and $e$ are the mass and charge of the oscillator, respectively, $\omega_0$ is the resonance frequency, and $\gamma$ the damping constant. We assume a time-harmonic electric field of the form
$$E=E_0 e^{-i \omega t} .$$
After an initial stage, the system oscillates with the same frequency $\omega$. Then, Eq. (7.4) can be rewritten in the form
$$-\omega^2 x=-\omega_0^2 x+i \gamma \omega x+\frac{e E_0}{m},$$
where we have cancelled the common exponential terms. The displacement can thus be written in the form
$$x=\frac{e}{m} \frac{1}{\omega_0^2-\omega^2-i \gamma \omega} E_0$$
This is the usual resonance dependence for a harmonic oscillator. It becomes largest when the frequency $\omega$ hits the resonance frequency $\omega_0$. On resonance, the amplitude is governed by the damping constant $\gamma$ and the amplitude of the driving field.
From the above expression we can compute the dipole moment $p=e x$ and, in turn, the polarization $P=$ nex by multiplying with the density of oscillators $n$ (remember that the polarization is a dipole density). We thus get
$$P=n e x=\frac{n e^2}{m} \frac{1}{\omega_0^2-\omega^2-i \gamma \omega} E_0=\varepsilon_0 \chi_e E_0$$

## 物理代写|量子光学代写Quantum Optics代考|From Microscopic to Macroscopic Electromagnetism

Electrodynamics in matter relies on the concepts of polarization and magnetization, which can be associated with electric and magnetic dipole densities. This approach works because the characteristic length scale of electromagnetic waves is on the order of micrometers, possibly a few tens to hundreds of nanometers for evanescent waves, whereas the relevant length scale for matter is in the nanometer range. For this reason, the fine details of matter play no important role for the dynamics of electromagnetic waves, which only interact with some kind of averaged matter state.

There exists no clear-cut definition of how an averaging over the microscopic charge and current distributions should be done. One could argue that this is because averaging is so robust that one always gets the same result irrespective of the starting expression. On the other hand, the whole issue of averaging is quite unrewarding as one finally has to end up with the macroscopic Maxwell’s equations anyhow. Our discussion of the topic closely follows the book of Jackson [2] and intends to make the reader familiar with the assumptions underlying such averaging, but also to raise awareness that it might fail at small dimensions where an explicit microscopic description might be needed.
We start our analysis with the microscopic Maxwell’s equations,
\begin{aligned} \nabla \cdot \boldsymbol{e} & =\frac{\varrho}{\varepsilon_0}, & \nabla \times \boldsymbol{e} & =-\frac{\partial \boldsymbol{b}}{\partial t} \ \nabla \cdot \boldsymbol{b} & =0, & \nabla \times \boldsymbol{b} & =\mu_0 \boldsymbol{j}+\mu_0 \varepsilon_0 \frac{\partial \boldsymbol{e}}{\partial t}, \end{aligned}
where we use $\boldsymbol{e}, \boldsymbol{b}$ for the true microscopic fields and reserve $\boldsymbol{E}, \boldsymbol{B}$ for the averaged ones. $Q$ and $j$ are the microscopic charge and current distributions that we will describe in a semiclassical framework, although there would be no fundamental differences for a quantum mechanical description.

## 物理代写|量子光学代写Quantum Optics代考|Drude–Lorentz and Drude Models

Drude-Lorentz 模型是介电函数最简单的描述方案之一。它基于谐振子模型，可以用两个带相反电 荷的粒子附着在弹簧上的形式建模，系统由外部电场驱动 $E(t)$. 在不指定任何细节的情况下，我们 假设位移 $x(t)$ 导致偶极矩 $p(t)=e x(t)$. 我们首先写下驱动振荡器的牛顿运动方程
$$m \ddot{x}=-m \omega_0^2 x-m \gamma \dot{x}+e E(t) .$$

$$E=E_0 e^{-i \omega t} .$$

$$-\omega^2 x=-\omega_0^2 x+i \gamma \omega x+\frac{e E_0}{m},$$

$$x=\frac{e}{m} \frac{1}{\omega_0^2-\omega^2-i \gamma \omega} E_0$$

$$P=n e x=\frac{n e^2}{m} \frac{1}{\omega_0^2-\omega^2-i \gamma \omega} E_0=\varepsilon_0 \chi_e E_0$$

## 物理代写|量子光学代写Quantum Optics代考|From Microscopic to Macroscopic Electromagnetism

$$\nabla \cdot \boldsymbol{e}=\frac{\varrho}{\varepsilon_0}, \quad \nabla \times \boldsymbol{e}=-\frac{\partial \boldsymbol{b}}{\partial t} \nabla \cdot \boldsymbol{b} \quad=0, \nabla \times \boldsymbol{b} \quad=\mu_0 \boldsymbol{j}+\mu_0 \varepsilon_0 \frac{\partial \boldsymbol{e}}{\partial t},$$

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