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It is clear that integrating over final state energies simply produces a sum over vibrational transitions such that the n ’th term in Eq. (A4) represents T$_{0}$$_{n}$ . Calculating T$_{1}$ ( ε$_{i}$, ε$_{f}$ ) is a bit more involved since the integrand in (A1) now includes the first Laguerre polynomial L$_{1}$ ( x ) = 1 − x . We start by writing 

where 

Taylor expanding the second exponential in the τ integral and performing the integration gives 

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<img src='content_image/420299.jpg'>

<img src='content_image/420298.jpg'>

The first two terms are each others complex conjugated and the last two terms factorizes (s and t integrals) into complex conjugated and the integrals can then be performed. The final result is rather complicated but consist of an infinite number of terms, each of which involves a delta function δ ( ε$_{i}$ − ε$_{f}$ − m ℏ ω$_{0}$ ), where m runs from -1 to infinity. We can thus obtain T$_{10}$ and T$_{11}$ by collecting terms involving m = 1 and m = 0 respectively. The results are 

and 

<img src='content_image/420305.jpg'>

∗ Electronic address: tolsen@fysik.dtu.dk 

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1 H. Park, J. Park, A. K. L. Lim, E. H. Anderson, A. P. 

