# Volkov States in Second Quantization

tl;dr: Is it possible to recover the semi-classical Volkov 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 Volkov 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. 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 $[a,a^\dagger]=1$. With this the full quantum Hamiltonian reads $$H=\hbar\omega a^\dagger a + \sum_{k}\frac{\hbar^2 \vec{k}^2}{2m}c_\vec{k}^\dagger c_\vec{k} -\sum_{k}\frac{e}{m}\sqrt{\frac{\hbar}{2\omega V\varepsilon_0}}\left( a+a^\dagger \right) \vec{\epsilon}\cdot(\hbar\vec{k})c_\vec{k}^\dagger c_\vec{k} +\sum_{k}\frac{e^2}{2m}\frac{\hbar}{2\omega V\varepsilon_0}\left( a^2 + (a^\dagger)^2 + 2a^\dagger a + 1 \right)c_\vec{k}^\dagger c_\vec{k},$$ with fermionic creation and annihilation operators $\{c_\vec{k}, c^\dagger_\vec{k'}\} = \delta_{\vec{k}\vec{k}'}$.

This Hamiltonian is obviously still diagonal in the electron momentum and thus separates in $\vec{k}$-space. This can be traced back to the usage of the dipole approximation.

I was thinking if the Volkov states can be recovered from the full quantum Hamiltonian, now obviously including the photonic degree of freedom? Since the Volkov solutions are non-perturbative solutions, a perturbation theory approach would require the summation of an infinite order perturbation series. I am especially interested if it is somehow possible to directly see the electron dressing in terms of a dressed quasiparticle propagator or even the direct diagonalization of the Hamiltonian in terms of an effective Hamiltonian with new quasiparticle creation and annihilation operators? Since the Hamiltonian is at most quadratic in each degree of freedom this should at least in principle be possible, right? Also since in this case one is only interested in the photonic dressing of the free electron, the dynamics of the photon field itself do not matter and thus one could trace out the quantum fluctuations of the photon field in the limit of large photon numbers?