content_image/383192.jpg content_image/383193.jpg content_image/383197.jpg content_image/383198.jpg content_image/383191.jpg content_image/383194.jpg content_image/383196.jpg
the matrix elements U GLYPH<28> m ; n then reads for even elements 

<img src='content_image/383192.jpg'>

and for odd elements 

<img src='content_image/383193.jpg'>

We have here introduced the hypergeometric function $_{2}$F$_{1}$ [32] in order to simplify the sums and write the matrix elements U GLYPH<28> m ; n in closed form. GLYPH<0> ( x ) denotes the Euler Gamma function. Combining everything, we get the explicit expressions for the transition probabilities which reads for even transitions 

<img src='content_image/383197.jpg'>

and for odd transitions 

<img src='content_image/383198.jpg'>

## Appendix C. Perturbational transition probabili- ties 

In this appendix, we provide the approximate first- order transition probabilities calculated with the help of time-dependent perturbation theory. 

## Appendix C.1. Isolated harmonic oscillator 

Following Eqs. (45) and (46), the transition probabil- ities for an isolated harmonic oscillator are 

<img src='content_image/383191.jpg'>

The selection rule m = n GLYPH<6> 2 finds its origin in the pres- ence of the Kronecker deltas. 

## Appendix C.2. Anharmonic corrections 

The interaction matrix element A t m ; n (48) are given by 

<img src='content_image/383194.jpg'>

The transition probabilities (49) for a harmonic oscilla- tor with quartic corrections then read 

<img src='content_image/383196.jpg'>

where additional transitions m = n GLYPH<6> 4 become possible. 

