Hill's determinant approach to single-mode spin-boson model

An efficient and stable numerical scheme based on Hill's determinant is proposed for computing the whole spectrum of the single-mode spin-boson model. Its shown that for the low-lying levels a simple version of Hill's determinant method works well, while for high levels a normalization of the determinant is necessary and this normalization results in an entire function whose zeros correspond to the energy levels. It is very convenient to use this entire function to compute all the energy levels numerically. Some high levels of the model are computed and compared with Irish's general rotating-wave approximation. A scaling law in the large photon number limit is numerically tested and several statistical properties of the spectrum are investigated.