UA OPTI544 量子光学11 Maxwell-Bloch方程
UA OPTI544 量子光學11 Maxwell-Bloch方程
- Maxwell-Bloch方程的推導
- Maxwell-Bloch方程的Steady State
- 光強的表達式
Maxwell-Bloch方程的推導
在光學的semi-classical理論中,我們用Maxwell方程描述場,用量子力學描述介質粒子,為了讓理論自洽,我們需要建立場對粒子的作用(通過interaction potential與Hamiltonian建立薛定諤方程)以及粒子對場的反饋。在前面的推導中,我們一直用的2-level system approximation討論粒子,這一講我們延續這個思路,推導2-level system對場的影響,即Maxwell-Bloch方程。
第一步 場與SVEA
假設場的表達式為
E?(z,t)=?^E(z,t)e?i(wt?kz)\vec E (z,t)=\hat \epsilon \mathcal{E}(z,t)e^{-i(wt-kz)}E(z,t)=?^E(z,t)e?i(wt?kz)
- ?^\hat \epsilon?^是單位向量,代表場的偏振方向;
- E(z,t)\mathcal{E}(z,t)E(z,t)代表wavepacket envelope;
- e?i(wt?kz)e^{-i(wt-kz)}e?i(wt?kz)代表oscillation;
- 場的傳播方向為+z+z+z方向;
引入Slowly-Varying Envelope Approximation(SVEA):E\mathcal EE在時空中為平滑函數,且滿足
{∣?E?z∣<<k∣E∣,∣?E?t∣<<w∣E∣∣?2E?z2∣<<k∣?E?z∣,∣?2E?t2∣<<w∣?E?t∣\begin{cases} |\frac{\partial \mathcal E}{\partial z}|<<k|\mathcal E|, |\frac{\partial \mathcal E}{\partial t}|<<w|\mathcal E| \\ |\frac{\partial^2 \mathcal E}{\partial z^2}|<<k |\frac{\partial \mathcal E}{\partial z}|, |\frac{\partial^2 \mathcal E}{\partial t^2}|<<w |\frac{\partial \mathcal E}{\partial t}|\end{cases}{∣?z?E?∣<<k∣E∣,∣?t?E?∣<<w∣E∣∣?z2?2E?∣<<k∣?z?E?∣,∣?t2?2E?∣<<w∣?t?E?∣?
除ultrafast laser外的所有情形基本上都可以用這個近似。
第二步 Macroscopic Polarization Density
電極化向量為
N?p?^?=N(p?12?a2a1??+p?21?a1a2??)=p?12ρ21e?i(wt?kz)+p?21ρ12ei(wt?kz)N \langle \hat{\vec p} \rangle = N(\vec p_{12} \langle a_2a_1^* \rangle+\vec p_{21}\langle a_1a_2^* \rangle)=\vec p_{12}\rho_{21}e^{-i(wt-kz)}+\vec p_{21}\rho_{12} e^{i(wt-kz)}N?p?^??=N(p?12??a2?a1???+p?21??a1?a2???)=p?12?ρ21?e?i(wt?kz)+p?21?ρ12?ei(wt?kz)
根據這個式子,Physical dipole為
2Re[p?12ρ21e?i(wt?kz)]2Re[\vec p_{12}\rho_{21}e^{-i(wt-kz)}]2Re[p?12?ρ21?e?i(wt?kz)]
記P?=2p?12ρ21e?i(wt?kz)\vec P=2\vec p_{12}\rho_{21}e^{-i(wt-kz)}P=2p?12?ρ21?e?i(wt?kz),代入p?12\vec p_{12}p?12?的表達式,記μ?=p?12??^?\mu^*=\vec p_{12} \cdot \hat \epsilon^*μ?=p?12???^?,
P?=2Nμ?ρ21e?i(wt?kz)\vec P = 2N\mu^*\rho_{21}e^{-i(wt-kz)}P=2Nμ?ρ21?e?i(wt?kz)
第三步 波動方程
根據Maxwell方程,
(?2?z2?1c2?2?t2)E?(z,t)=1?0c2?2?t2P?(z,t)\left( \frac{\partial^2 }{\partial z^2}-\frac{1}{c^2}\frac{\partial^2}{\partial t^2} \right) \vec E(z,t)=\frac{1}{\epsilon_0c^2} \frac{\partial^2}{\partial t^2} \vec P(z,t)(?z2?2??c21??t2?2?)E(z,t)=?0?c21??t2?2?P(z,t)
代入場與電極化的表達式可得
(??z+1c??t)E(z,t)=ik?0Nμ?ρ21\left( \frac{\partial }{\partial z} + \frac{1}{c}\frac{\partial}{\partial t} \right)\mathcal E(z,t)=\frac{ik}{\epsilon_0}N\mu^*\rho_{21}(?z??+c1??t??)E(z,t)=?0?ik?Nμ?ρ21?
綜上,結合OBE與這個微分方程,我們可以得到wavepacket與Bloch Variables的微分方程組,稱這個方程組為Maxwell-Bloch方程:
{(??z+1c??t)E(z,t)=ik2?0Nμ?(u?iv)u˙=?βu+Δv+Im[χ]wv˙=Δu?βv+Re[χ]ww˙=?Im[χ]u?Re[χ]v?1+wΓ1\begin{cases}\left( \frac{\partial }{\partial z} + \frac{1}{c}\frac{\partial}{\partial t} \right)\mathcal E(z,t)=\frac{ik}{2\epsilon_0}N\mu^*(u-iv ) \\ \dot u = -\beta u+\Delta v+Im[\chi]w \\ \dot v = \Delta u-\beta v +Re[\chi]w \\ \dot w = -Im[\chi] u - Re[\chi] v -\frac{1+w}{\Gamma_1}\end{cases}??????????(?z??+c1??t??)E(z,t)=2?0?ik?Nμ?(u?iv)u˙=?βu+Δv+Im[χ]wv˙=Δu?βv+Re[χ]ww˙=?Im[χ]u?Re[χ]v?Γ1?1+w??
Maxwell-Bloch方程的Steady State
所以Envelope的解為
E(z)=E0eaw2ze?iδw2z\mathcal E(z)=\mathcal E_0e^{\frac{aw}{2}z}e^{-i \frac{\delta w}{2}z}E(z)=E0?e2aw?ze?i2δw?z
與經典電動力學中field e?nIkzeinRkze^{-n_Ikz}e^{in_Rkz}e?nI?kzeinR?kz對比可得折射率的表達式為
{nI=?aw2k=?Nw2kσ(Δ)nR=1?δw2k=1?ΔβNw2kσ(Δ)\begin{cases} n_I=-\frac{aw}{2k}=-\frac{Nw}{2k}\sigma(\Delta) \\ n_R= 1-\frac{\delta w}{2k} = 1-\frac{\Delta }{\beta} \frac{Nw}{2k}\sigma(\Delta)\end{cases}{nI?=?2kaw?=?2kNw?σ(Δ)nR?=1?2kδw?=1?βΔ?2kNw?σ(Δ)?
光強的表達式
總結
以上是生活随笔為你收集整理的UA OPTI544 量子光学11 Maxwell-Bloch方程的全部內容,希望文章能夠幫你解決所遇到的問題。
- 上一篇: UA MATH524 复变函数14 La
- 下一篇: UA MATH524 复变函数17 留数