Spin-boson systems: equivalence between the dilute-blip and the Born approximations

Abstract

We demonstrate the full equivalence between the dilute-blip approximation introduced by Chakravarty and Leggett, and the second-order Born approximation of the general relaxation theory. Besides its simplicity, the latter scheme allows us to derive the explicit dynamics in a direct way, invoking only standard analysis based on Laplace transformation. As an example, the case of the symmetric double-well potential is revisited. In addition, explicit expressions of the coordinate of the particle are derived for some values of the coupling constant a, which serve as illustrations of the three cases : exponentially damped oscillatory relaxation ( a < 1/2 ) , non-exponential and non-oscillatory relaxation ( 1/2 < α < 1) and finally, dynamics above the critical value ( a > 1) .