Numerical Analysis for Cavity Quantum Electrodynamics Using Matrix Product States and Numerical Mode Decomposition

The multimode Rabi Hamiltonian describes a cavity quantum electrodynamical (QED) system, involving a two-level atom interacting with multiple quantized electromagnetic modes of the surrounding structure. Since this Hamiltonian is beyond the rotating-wave approximation, it is valid even in the ultrastrong coupling regime. However, there has been a problem of gauge ambiguities with the Rabi Hamiltonian due to the fact that it can be derived from two formally different but physically equivalent fundamental Hamiltonians. This problem has recently been resolved for single electromagnetic mode models. In this work, we mathematically and numerically verify this for multimode models. With this established, we combine the numerical methods, matrix product states (MPS) and numerical mode decomposition (NMD), for analyzing cavity QED systems. The MPS method is used to efficiently represent and time evolve a quantum state, and to this end, the Rabi Hamiltonian is numerically transformed in a stable manner into an equivalent Hamiltonian that has a chain coupling structure, which allows efficient application of MPS. The technique of NMD is used to extract the numerical electromagnetic modes of a general inhomogeneous medium, and this allows one to construct the quantized field operator for an arbitrary electromagnetic environment. As a proof of concept, this combined approach is demonstrated by analyzing 1D cavity QED systems in various settings.