UA OPTI544 量子光學(xué)7 2-level system approximation的Density Matrix模型

• • Density Matrix的Rabi Equation
  • Non-Hamiltonian Evolution
  • • Elastic Collision
    • Inelastic Collision
    • Spontaneous Decay

這一講我們用Density Operator討論2-level system，在關(guān)于Density Operator的補充內(nèi)容中，我們推導(dǎo)了Density Operator滿足的薛定諤方程：
$$i?\frac{d}{dt}\rho =[H,\rho ]$$

接下來我們代入light-matter interaction的2-level system Hamiltonian推導(dǎo)Density Operator的演化方程。

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Density Matrix及其含義
在正式推導(dǎo)前，我們先搞明白在light-matter interaction的2-level system approximation中，Density Matrix及其含義。首先，量子態(tài)為$|1?,|2?$，讓其作為基向量，可以得到Density Operator的Maxtrix表示為
$$\rho =\left[\begin{array}{cc}{\rho }_{11}& {\rho }_{12}\\ {\rho }_{21}& {\rho }_{22}\end{array}\right]$$

其中${\rho }_{11}+{\rho }_{22}=1$，二者分別代表粒子在$|{\psi }_1?$與$|{\psi }_2?$中被發(fā)現(xiàn)的概率，所以${\rho }_{11},{\rho }_{22}\in \mathbb{R}$，稱其為population；因為$\rho$是厄爾米特算符，所以${\rho }_{12}={\rho }_{21}^{?}$，因此只需要兩個實變量就可以確定這兩個次對角元，稱${\rho }_{12},{\rho }_{21}$為coherence?？偠灾?#xff0c;要確定Density Matrix的演化方程，本質(zhì)上只需要三個實變量的微分方程，這個特點與Bloch Vector非常類似，借助這個思想我們可以建立2-level system approximation的Vector Model以及Maxwell-Bloch方程，但這些內(nèi)容后面再談，這一講推導(dǎo)這幾個矩陣元的方程。

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Density Matrix的Rabi Equation

$\dot{\rho }=?\frac{i}{?}[H,\rho ]=?\frac{i}{?}(H\rho ?\rho H)$,
$$\begin{array}{rl}H\rho ?\rho H& =?\left[\begin{array}{cc}{H}_{11}& H_{12}\\ H_{21}& H_{22}\end{array}\right]\left[\begin{array}{cc}{\rho }_{11}& {\rho }_{12}\\ {\rho }_{21}& {\rho }_{22}\end{array}\right]??\left[\begin{array}{cc}{\rho }_{11}& {\rho }_{12}\\ {\rho }_{21}& {\rho }_{22}\end{array}\right]\left[\begin{array}{cc}{H}_{11}& H_{12}\\ H_{21}& H_{22}\end{array}\right]\\ & =?\left[\begin{array}{cc}{H}_{12}{\rho }_{21}?{\rho }_{12}{H}_{21}& H_{11}{\rho }_{12}+H_{12}{\rho }_{22}?{\rho }_{11}{H}_{12}?{\rho }_{12}{H}_{22}\\ H_{21}{\rho }_{11}+H_{22}{\rho }_{21}?{\rho }_{21}{H}_{11}?{\rho }_{22}{H}_{21}& H_{21}{\rho }_{12}?{\rho }_{21}{H}_{12}\end{array}\right]\end{array}$$

Note that
$$\begin{gathered} H_{12}=?\frac{1}{2}({\chi }_{12}{e}^{?iwt}+{\chi }_{21}^{?}{e}^{iwt}) \\ {H}_{21}=?\frac{1}{2}({\chi }_{21}{e}^{?iwt}+{\chi }_{12}^{?}{e}^{iwt}) \end{gathered}$$

Plug-in and set ${\rho }_{12}={\tilde{\rho }}_{12}{e}^{iwt}$
$$\begin{array}{rl}{\dot{\rho }}_{11}& =?i(H_{12}{\rho }_{21}?{\rho }_{12}{H}_{21})\\ & =?i\left(?\frac{1}{2}({\chi }_{12}{e}^{?i2wt}+{\chi }_{21}^{?}){\rho }_{21}+\frac{1}{2}({\chi }_{21}{e}^{?i2wt}+{\chi }_{12}^{?}){\rho }_{12}\right)\end{array}$$

drop the term $\propto e^{\pm i2wt}$, and set ${\chi }_{21}=\chi ,{\chi }_{21}^{?}={\chi }^{?}$,
$${\dot{\rho }}_{11}=?\frac{i}{2}(\chi {\rho }_{12}?{\chi }^{?}{\rho }_{21})$$

Hence, ${\dot{\rho }}_{22}=?{\dot{\rho }}_{11}=\frac{i}{2}(\chi {\rho }_{12}?{\chi }^{?}{\rho }_{21})$. Since
$$\begin{gathered} H_{11}=0,H_{22}=w_{21} \\ \begin{array}{rl}& H_{11}{\rho }_{12}+H_{12}{\rho }_{22}?{\rho }_{11}{H}_{12}?{\rho }_{12}{H}_{22}=(w_{21}?w){\rho }_{12}?\frac{1}{2}{\chi }^{?}{\rho }_{11}+\frac{1}{2}{\chi }^{?}{\rho }_{22}\end{array} \\ \begin{array}{rl}& H_{21}{\rho }_{11}+H_{22}{\rho }_{21}?{\rho }_{21}{H}_{11}?{\rho }_{22}{H}_{21}=?(w_{21}?w){\rho }_{21}+\frac{1}{2}\chi {\rho }_{11}?\frac{1}{2}\chi {\rho }_{22}\end{array} \end{gathered}$$

Thus,
$$\begin{gathered} {\dot{\rho }}_{12}=i\Delta {\rho }_{12}+\frac{i{\chi }^{?}}{2}({\rho }_{22}?{\rho }_{11}) \\ {\dot{\rho }}_{21}=?i\Delta {\rho }_{21}?\frac{i\chi }{2}({\rho }_{22}?{\rho }_{11}) \end{gathered}$$

綜上，
$$?\begin{array}{l}{\dot{\rho }}_{11}=?\frac{i}{2}(\chi {\rho }_{12}?{\chi }^{?}{\rho }_{21})\\ {\dot{\rho }}_{22}=?{\dot{\rho }}_{11}=\frac{i}{2}(\chi {\rho }_{12}?{\chi }^{?}{\rho }_{21})\\ {\dot{\rho }}_{12}=i\Delta {\rho }_{12}+\frac{i{\chi }^{?}}{2}({\rho }_{22}?{\rho }_{11})\\ {\dot{\rho }}_{21}=?i\Delta {\rho }_{21}?\frac{i\chi }{2}({\rho }_{22}?{\rho }_{11})\end{array}$$

這被稱為Density Matrix的Rabi方程。

Non-Hamiltonian Evolution

在實際應(yīng)用中，上述方程還是過于理想了，因為在粒子的互動中，有可能發(fā)生一些非哈密頓的行為，比如彈性碰撞、非彈性碰撞、自發(fā)衰變等。

Elastic Collision

彈性碰撞中能量是守恒的，所以Bohr頻率在碰撞前后不會改變，但在碰撞過程中會有一定的變化，這時拋開上述方程，只考慮Density Matrix Element ${\rho }_{12}$在短時間內(nèi)由彈性碰撞導(dǎo)致的演化，我們可以寫出下面的微分方程：
$${\dot{\rho }}_{12}=?i(w_{21}+\delta w){\rho }_{12}?{\rho }_{12}(t)={\rho }_{12}(0)e^{?i{w}_{21}t}\underset{碰撞過程導(dǎo)致的相位改變}{e^{?i{\int }_0^t\delta w(t^{\prime })d{t}^{\prime }}}$$

引入下列假設(shè)簡化這個結(jié)果：

• 對所有粒子而言，$\delta w(t)$是獨立同分布的零均值高斯過程（零均值保證彈性碰撞中能量的期望是守恒的）；
• 碰撞具有無記憶性，即$?\delta w(t)\delta w(t^{\prime })?=\frac{2}{\tau }\delta (t?t^{\prime })$（$2/\tau$表示$\delta w(t)$的方差）

根據(jù)這兩個假設(shè)可得
$$?e^{?i{\int }_0^t\delta w(t^{\prime })d{t}^{\prime }}?=e^{?t/\tau }$$

于是
$${\rho }_{12}(t)={\rho }_{12}(0)e^{?i{w}_{21}t}{e}^{?t/\tau }$$

這說明$\tau$的物理含義是彈性碰撞導(dǎo)致的coherence的衰變時間，由此修正它的ODE為
$${\dot{\rho }}_{12}=({\dot{\rho }}_{12}{)}_{S.E.}?\frac{1}{\tau }{\rho }_{12}$$

• $({\dot{\rho }}_{12}{)}_{S.E.}$表示薛定諤方程中coherence的變化率；
• $?\frac{1}{\tau }{\rho }_{12}$表示Elastic Collision Decay

Inelastic Collision

非彈性碰撞相比彈性碰撞會出現(xiàn)atom loss，$Tr(\rho )$不再守恒，用${\Gamma }_1,{\Gamma }_2$表示rate of loss，則方程修正如圖，它對population的影響如下：

其中$e^{?\frac{{\Gamma }_1+{\Gamma }_2}{2}t}$被稱為Inelastic Collision Decay。

由此修正它的ODE為
$${\dot{\rho }}_{12}=({\dot{\rho }}_{12}{)}_{S.E.}?\frac{1}{\tau }{\rho }_{12}?\frac{{\Gamma }_1+{\Gamma }_2}{2}{\rho }_{12}$$

• $?\frac{{\Gamma }_1+{\Gamma }_2}{2}{\rho }_{12}$表示Inelastic Collision Decay

Spontaneous Decay

自發(fā)衰變是由粒子與量子化電磁場的交互導(dǎo)致的，但場的量子化以及量子電動力學(xué)要在Density Matrix討論完后才會學(xué)，所以先跳過這部分直接給結(jié)論。在自發(fā)衰變存在時，population與coherence都會受到影響：
$$\begin{gathered} {\dot{\rho }}_{11}=A_{21}{\rho }_{22} \\ {\dot{\rho }}_{22}=?A_{21}{\rho }_{11} \\ {\dot{\rho }}_{21}=?\frac{A_{21}}{2}{\rho }_{12}={\dot{\rho }}_{21}^{?} \end{gathered}$$

這些效應(yīng)被稱為Spontaneous Decay，其中$A_{21}$代表粒子從高能態(tài)$|2?$向低能態(tài)$|1?$自發(fā)衰變的強度。

現(xiàn)在把Elastic Collision Decay、Inelastic Collision Decay與Spontaneous Decay加入到薛定諤方程中可得2-level system approximation的Density Matrix一般演化方程：
$$?\begin{array}{l}{\dot{\rho }}_{11}=?{\Gamma }_1{\rho }_{11}+A_{21}{\rho }_{22}?\frac{i}{2}(\chi {\rho }_{12}?{\chi }^{?}{\rho }_{21})\\ {\dot{\rho }}_{22}=?{\Gamma }_2{\rho }_{22}?A_{21}{\rho }_{22}+\frac{i}{2}(\chi {\rho }_{12}?{\chi }^{?}{\rho }_{21})\\ {\dot{\rho }}_{12}=(i\Delta ?\beta ){\rho }_{12}+\frac{i{\chi }^{?}}{2}({\rho }_{22}?{\rho }_{11})={\dot{\rho }}_{21}^{?}\\ \beta =\frac{1}{\tau }+\frac{{\Gamma }_1+{\Gamma }_2}{2}+\frac{A_{21}}{2}\end{array}$$

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