q.h.o 量子谐振子：

• 波动方程的解（谐振子的类比），满足量子力学条件（薛定谔方程+？）

• 坐标表示，Fock states表示

• 电动力学到量子场论，二次量子化的电磁场部分

  • 两个假设：量子场的系综平均是经典场；哈密顿量是光（电磁场）的能量
  • A - 通过两个假设和算符替代证明量子场（矢势算符、电场算符、磁场算符）满足薛定谔方程、boundary conditions and the fundamental commutation relation+？这些量子力学条件
  • B - 直接将矢势分解波模（平面波模、高斯波模等，依据在傅里叶变换,form a complete set and obey the laws of electromagetism），$A_k$ are complex function of space and time coordinates, but not operator; are classcial;$A_k^*$ are part of the complete set of classcial waves since Maxwell’s equations are real. The coefficients $\hat a$ and its Hermitian (quantum vector potential is Hermitian) are operators carrying the quantum properties of light.

    • 选择符合q.h.o.的波模：

      Bose commutation relations $\begin{bmatrix}\hat{a}k,\hat{a}{k'}^\dagger\end{bmatrix}=\delta_{kk'},\quad\begin{bmatrix}\hat{a}k,\hat{a}{k'}\end{bmatrix}=0.$

    • Monochromatic 单色modes.分解波模允许波包（modes that contain a range of optical frequency and hence a range of colors）单色模有结论$\hat{H}=\sum_k\hbar\omega_k\Big(\hat{a}_k^\dagger\hat{a}_k+\frac{1}{2}\Big).$

    • 分解成无穷个波模，总能量为无穷大。the zero point enerfy $E_{0}=\sum_{k}\frac{\hbar\omega_{k}}{2}=\infty$ results in perfectly finite and experimentally confirmed facts.(is constant, with finite energy density, for reason as yet unkown Appendix A)

• electromagnetic oscillator

  • quadrature states
  • Fock state
  • thermal state
  • coherent state describing coherent laser light

  all states are introduced as eigenstate

  • MUS minimum uncertainty states

Quasiprobability distributions

In classcial optics: a phase space distribution $W(q,p)$ quantifies the probability of finding a particular pair of $q,p$ values in their simultaneous measurement.[W→all statistical quantities]

In quantum mechanics:

• Heisenberg’s uncertainty principle

but classcial-like fashion of making statistical predictions

Just one postulate turns out to be sufficient: W behaves like a joint probability distribution for $q,p$+postulate that the position probability distribution after an arbitrary phase shift $\theta$ should equal the projection $\begin{aligned}\operatorname{pr}(q,\theta)& \equiv\langle q|\hat{U}(\theta)\hat{\rho}\hat{U}^{\dagger}(\theta)|q\rangle   \\&=\int_{\infty}^{+\infty}W(q\cos\theta-p\sin\theta,q\sin\theta+p\cos\theta)\mathrm{d}p.\end{aligned}$

Wigner’s representation of quantum mechanics

• whenever quantum corrections to classcial laws were of interest
• Basic properties
  • overlap formula $\mathrm{tr}\{\hat{F}1\hat{F}2\}=2\pi\int{-\infty}^{+\infty}\int{-\infty}^{+\infty}W_1(q,p)W_2(q,p)\mathrm{d}q\mathrm{d}p$
    • calculating expectation values
    • reveals both the similarities and differences between a classcial probability distribution and the Wigner function$\left|\langle\psi_1|\psi_2\rangle\right|^2=2\pi\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}W_1(q,p)W_2(q,p)\mathrm{d}q\mathrm{d}p$
    • quantify the purity of a quantum state$\mathrm{tr}\{\hat{\rho}^{2}\}=2\pi\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}W(q,p)^{2}\mathrm{d}q\mathrm{d}p.$
    • entropy ~W
    • represent the density matrix elements in a given basis in terms of W $\langle a'|\hat{\rho}|a\rangle=\mathrm{tr}\{\hat{\rho}|a\rangle\langle a'|\}=2\pi\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}W(q,p)W_{a'a}(q,p)\mathrm{d}q\mathrm{d}p.$
  • the value of W may range between only $\pm\pi^{-1}$，$\left|W(q,p)\right|\leq\frac{1}{\pi}.$

Other quasiprobability distributions

• Q function
• P function
• s-parameterized quasiprobability distributions$W(q,p;s)\equiv\frac{1}{(2\pi)^2}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\widetilde{W}(u,v;s)\exp(\mathrm{i}uq+\mathrm{i}vp)\mathrm{d}u\mathrm{d}v.$

Simple optical instruments& some experiments