Atomic systems interacting with quantized modes of the electromagnetic field (cavity QED) have been extensively studied in the literature. The set of field–matter states (polaritons) can be described by the Pauli–Fierz (PF) Hamiltonian, which is derived from the minimal coupling scheme in the Coulomb gauge and the second quantization of the transverse components of the electromagnetic field under confinement. The longitudinal field components are accounted for through interaction potentials between charged particles. Despite the significant simplifications involved in its derivation, the PF equation has a broad range of validity, enabling the description of optical, molecular, and condensed matter systems.
After the seminal work of E. Jaynes and F. Cummings, the literature on the subject is rich with analytical models and simple numerical calculations. However, even with exact knowledge of the material (atomic) states and the field states (cavity modes), a rigorous treatment requires a full -or at least a converged- consideration of all possible field-matter configurations. Furthermore, the longitudinal field component describing interactions between material particles must be consistent with the transverse modes, which are affected by the confinement; this interaction is not necessarily the simple Coulomb potential.
In this work, we present a Configuration–Interaction (CI) method for the eigenstates of the Pauli–Fierz Hamiltonian for atomic systems in spherical cavities. The longitudinal field components (matter interactions) are consistent with the field's boundary conditions inside the conductor. In the case of spherical confinement, this results in a substantial modification of the Coulomb interactions doubt to the coupling with surface polarization, as well as an image potential for the interaction between each charge and the surface. For large cavities, the radial components of the material states tend towards well-defined surface image states, and the material dynamics become 2D, restricted to the surface of the sphere.
