1. Quantization of a single-mode field
2. Quantum fluctuations 波动of a songle-mode field
3. Quadrature 积分 operators for a single-mode field
4. Multimode fields
5. Thermal 热 fields
6. Vacuum fluctuations and the zero-point energy
7. The quantum phase

the quantization of the electromagnetic field.

mode~photon.

phton number stats.

the field observable.

quantum description of the phase of the quantized electromagnetic field.

A single-mode field

Model:

a one-dimensional cavity along z-axis with perfectly condunting walls. The electric field must vanish on the boundaries and will take the form of a standing waves. The field is assumped to be polarized along the x-direction.

$$ \boldsymbol E(\boldsymbol r,t)=\boldsymbol e_xE_x(z,t)=\boldsymbol e_xAq(t)\sin(kz) $$

$\sin(kz)$→ standing wave $k=\omega/c$, $q(t)$→ time-dependent fctor, $A$← energy.

boundary condition→$k=m\pi/L,\ m=1,2,...$

Using Maxwell equatoin:

$$ \boldsymbol B(\boldsymbol r,t)=\int-\nabla\times\boldsymbol E \mathbf{d}t=\boldsymbol e_yB_y(z,t)\\=-\boldsymbol e_y Aks(t)\cos(kz),\qquad \dot{s}(t)=q(t) $$