Entanglement of a qubit with a single oscillator mode

We solve a model of a qubit strongly coupled to a massive environmental oscillator mode where the qubit back action is treated exactly. Using a Ginzburg-Landau formalism, we derive an effective action for this well-known localization transition. An entangled state emerges as an instanton in the collective qubit-environment degree of freedom and the resulting model is shown to be formally equivalent to a fluctuating gap model of a disordered Peierls chain. Below the transition, spectral weight is transferred to an exponentially small energy scale, leaving the qubit coherent but damped. Unlike the spin-boson model, coherent and effectively localized behaviors may coexist.