Ground-State Energy of A Two-Level System with Phonon Coupling

The coupling of a two-level system to a quantized boson mode has been the focus of many researchers for a number of years. Applications to exciton motion, molecular polaron formation, chaos in quantum systems as well as a number of other effects in condensed matter physics have also been studied. Here we investigate the interaction of a single bosonic mode with a two-level fermionic system given by the Hamiltonian% \[ H=-\delta_{0}\sigma_{x}+\sum_{k}\hbar\omega_{k}a_{k}^{\dag}+\sum_{k}% g_{k}\left( a_{k}^{\dag}+a_{k}\right) \sigma_{z}. \] This quantum system is used as a testing ground for a newly developed Generalized Moments Expansion, GMX$\left( m,n\right) $, of which the well-known Connected Moments Expansion (CMX) and Alternate Moments Expansion (AMX) are special cases: \textrm{CMX}$=$\textrm{GMX}$\left( 1,1\right) \mathrm{,}$ \textrm{AMX=GMX}$\left( 1,2\right) $. The convergence and viability of this scheme is discussed and comparisons are made with other methods.