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In this work, we present a stochastic variational calculation (SVM) of energies and wave functions of few particle systems coupled to quantum fields in cavity QED. The light-matter coupled system is described by the Pauli-Fierz Hamiltonian. The spatial wave function and the photon spaces are optimized by a random selection process. Examples for a two-dimensional trion and confined electrons as well as for the He atom and the Hydrogen molecule are presented showing that the light-matter coupling drastically changes the electronic states.

Deformed correlated Gaussian basis functions are introduced and their matrix elements are calculated. These basis functions can be used to solve problems with nonspherical potentials. One example of such potential is the dipole self-interaction term in the Pauli-Fierz Hamiltonian. Examples are presented showing the accuracy and necessity of deformed Gaussian basis functions to accurately solve light-matter coupled systems in cavity QED.

The energy and wave function of a harmonically confined two-electron system coupled to light is calculated by separating the wave functions of the relative and center of mass (CM) motions. The relative motion wave function has a known quasi-analytical solution. The light only couples to the CM variable and the coupled equation can be solved with diagonalization without approximations. The approach works for any coupling strength. Examples of wave functions of light-matter hybrid states are presented.

Explicitly Correlated Gaussian basis is used to calculate the energies and wave functions of one dimensional few-electron systems in confinement potentials created by external potentials or coupling to light in cavity. The appearance and properties of Wigner crystal-like structures are discussed. It is shown that one dimensional Wigner crystals can be formed by coupling electrons to light due to the dipole self-interaction term in the light-matter Hamiltonian, provided an additional extremely weak confining potential is present.