<PERSON> States in Second Quantization
tl;dr: Is it possible to recover the semi-classical <PERSON> states from the full second quantized electron-photon Hamiltonian in the limit of large photon numbers? What could be an possible technical approach?
The interaction of an free electron with an electromagnetic field in the nonrelativistic limit is given by the semi-classical Hamiltonian
$$H=\frac{1}{2m}\left( \hat{\vec{p}} - e\vec{A}(t) \right)^2 \approx \frac{\hat{\vec{p}}^2}{2m} - \frac{e}{m}\vec{A}(t)\cdot \hat{\vec{p}} + \frac{e^2}{2m}\vec{A}^2(t).$$
In the dipole approximation this Hamiltonian seperates in $\vec{k}$-space and is excatly diagonalizable in terms of so-called <PERSON> states:
$$\left|\psi_{\vec{k}}(t)\right> = \exp\left{ -\frac{i}{\hbar} \int_{0}^{t}\mathrm{d}t'\left( \hbar\vec{k} - e\vec{A}(t') \right) \right}\left|\vec{k}\right>$$
For a single-mode plane wave field $\vec{A}=A_0 \vec{\epsilon}\cos(\omega t)$ the integral can be evaluated exactly, giving
$$\left|\psi_{\vec{k}}(t)\right> = \exp\left{ -\frac{i}{\hbar} \left( \left( \frac{\hbar^2\vec{k}^2}{2m} + \frac{e^2A_0^2}{4m}\right)t - \frac{e A_0}{m\omega}\vec{\epsilon}\cdot(\hbar\vec{k})\sin(\omega t) + \frac{e^2 A_0^2}{8 m \omega}\sin(2 \omega t) \right)\right}\left|\vec{k}\right>.$$
Here, the first additional term is the ponderomotive energy (vacuum limit of the quadratic AC Stark shift). The second term is the time-dependant position displacement of the electron due to the electromagnetic field, giving raise to the so called quiver motion. 281
The last term is a time dependant phase-shift with, as fas as I know, no intuitive explanation.
These states are usually interpreted as photon-dressed electron states, containing all virtual interactions between the free electron and the photons and thus giving raise to an effective electron energy or mass. However, no real transitions are involved, since a free electron can't absorb real photons due to energy and momentum conservation.
Usually one recovers at least some of the photon particle picture by expanding the $\sin{\omega t}$ and $\sin{2 \omega t}$ terms by using the Jacobi-Anger identity $$ \exp(iz\sin\theta) = \sum_{n=-\infty}^{\infty} J_n(z)\exp(in\theta) $$ The simplicity of the problem drastically changes in the framework of second quantization. The single mode EM field in second quantized form is $$ \hat{\vec{A}} = \sqrt{\frac{\hbar}{2\omega V\varepsilon_0}}\vec{\epsilon}\left( a+a^\dagger \right), $$ with time dependant bosonic creation and annihilation operators $[a,a^\dagger]=1$. 526