Coherent-incoherent transition in a Cooper-pair-box coupled to a quantum   oscillator: an equilibrium approach
  Temperature effect on quantum tunneling in a Cooper-pair-box coupled to a quantum oscillator is studied by both numerical and analytical calculations. It is found that, in strong coupling regions, coherent tunneling of a Cooper-pair-box can be destroyed by its coupling to a quantum oscillator and the tunneling becomes thermally activated as the temperature rises, leading to failure of the Cooper-pair-box. The transition temperature between the coherent tunneling and thermo-activated hopping is determined and physics analysis based on small polaron theory is also provided. 
Ying-Hua Huang, Hang Wong, Zhi-De Chen