Tunnelling system coupled to a harmonic oscillator: an analytical treatment

We give an analytical formula in term of continued fraction expansions for the spectral function of a tunnelling electron, coupled to a local lattice oscillation, in a two-site cluster at non-zero temperature. We also study the spectral function of the polaron, a better defined quasi-particle in the anti-adiabatic regime and at sufficiently low temperature. The exact results obtained allow us to look into a wide range of temperature and coupling. Asymptotic results can be obtained directly from the continued fraction expansions in both adiabatic and anti-adiabatic regimes. In the intermediate/strong anti-adiabatic case, in contrast to the usual Lang–Firsov approximation scheme, we found that there is no shrinking of the polaron band as temperature increases. Polaron bandwidth gets broader by temperature effects.

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