the matrix elements U GLYPH<28> m ; n then reads for even elements 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 The selection rule m = n GLYPH<6> 2 finds its origin in the pres- ence of the Kronecker deltas. Appendix C.2. Anharmonic corrections Appendix C. Perturbational transition probabili- ties Appendix C.1. Isolated harmonic oscillator Following Eqs. (45) and (46), the transition probabil- ities for an isolated harmonic oscillator are and for odd elements The transition probabilities (49) for a harmonic oscilla- tor with quartic corrections then read where additional transitions m = n GLYPH<6> 4 become possible. In this appendix, we provide the approximate first- order transition probabilities calculated with the help of time-dependent perturbation theory. and for odd transitions p GLYPH<28> m ; n =$^{GLYPH<12>}$ GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<14>$_{m}$$_{;}$$_{n}$ + i 4 !$_{0}$ GLYPH<28> Z 0 dt exp $^{GLYPH<0>}$i !$_{m}$$_{;}$$_{n}$ t GLYPH<1> $^{GLYPH<16>}$! 2 0 GLYPH<0> ! 2 t$^{GLYPH<17>}$ GLYPH<2>$^{GLYPH<20>}$ p n + 1 p n + 2 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{+}$$_{2}$ + (2 n + 1) GLYPH<14>$_{m}$$_{;}$$_{n}$ + p n p n GLYPH<0> 1 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{GLYPH<0>}$$_{2}$ GLYPH<21> GLYPH<12> GLYPH<12> GLYPH<12> 2 : (C.1) U GLYPH<28> 2 GLYPH<22>; 2 GLYPH<23> = r 2 GLYPH<23> !2 GLYPH<22> ! 2 GLYPH<23> + 2 GLYPH<22> GLYPH<0> $^{1}$i GLYPH<2> s GLYPH<16> 2 GLYPH<23> GLYPH<16> GLYPH<3> 2 GLYPH<22> GLYPH<27> 2 GLYPH<23> + 2 GLYPH<22> + 1 4 p 2 !$_{0}$ !$_{1}$ GLYPH<0> ( GLYPH<22> + 1) GLYPH<0> ( GLYPH<23> + 1) GLYPH<2>$_{2}$ F$_{1}$$^{ }$ GLYPH<0> GLYPH<22>; GLYPH<0> GLYPH<23> ; 1 2 ; 2 1 GLYPH<0> Q GLYPH<3> ! (B.8) U GLYPH<28> 2 GLYPH<22> + 1 ; 2 GLYPH<23> + 1 = GLYPH<0> s 8 i (2 GLYPH<23> + 1)!(2 GLYPH<22> + 1)! ( Q GLYPH<3> GLYPH<0> 1) 2 GLYPH<23> + 2 GLYPH<22> + 1 GLYPH<2> s GLYPH<16> 2 GLYPH<23> + $^{1}$GLYPH<16> GLYPH<3> 2 GLYPH<22> + 1 GLYPH<27> 2 GLYPH<23> + 2 GLYPH<22> + 1 4 p 2 !$_{0}$ !$_{1}$ GLYPH<0> ( GLYPH<22> + 1) GLYPH<0> ( GLYPH<23> + 1) GLYPH<2>$_{2}$ F$_{1}$$^{ }$ GLYPH<0> GLYPH<22>; GLYPH<0> GLYPH<23> ; 3 2 ; 2 1 GLYPH<0> Q GLYPH<3> ! : (B.9) A t m ; n = GLYPH<11>$_{t}$ ℏ 2 M !$_{0}$ $_{!}$2 GLYPH<2>$^{GLYPH<20>}$ p n + 1 p n + 2 p n + 3 p n + 4 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{+}$$_{4}$ + (4 n + 6) p n + 1 p n + 2 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{+}$$_{2}$ + $^{GLYPH<16>}$6 n 2 + 6 n + 3 GLYPH<17> GLYPH<14>$_{m}$$_{;}$$_{n}$ + (4 n GLYPH<0> 2) p n p n GLYPH<0> 1 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{GLYPH<0>}$$_{2}$ + p n p n GLYPH<0> 1 p n GLYPH<0> 2 p n GLYPH<0> 3 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{GLYPH<0>}$$_{4}$ GLYPH<21> : (C.2) The interaction matrix element A t m ; n (48) are given by p GLYPH<28> m ; n =$^{GLYPH<12>}$ GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<14>$_{m}$$_{;}$$_{n}$ + 1 i$_{ℏ}$ GLYPH<28> Z 0 dt exp $^{GLYPH<0>}$i !$_{m}$$_{;}$$_{n}$ t GLYPH<1> GLYPH<2>$^{(}$ GLYPH<0> ℏ 4 !$_{0}$ $^{GLYPH<16>}$! 2 0 GLYPH<0> ! 2 t$^{GLYPH<17>}$$^{GLYPH<20>}$ p n + 1 p n + 2 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{+}$$_{2}$ + (2 n + 1) GLYPH<14>$_{m}$$_{;}$$_{n}$ + p n p n GLYPH<0> 1 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{GLYPH<0>}$$_{2}$ GLYPH<21> + GLYPH<11>$_{t}$ ℏ 2 M !$_{0}$ $_{!}$2 GLYPH<2>$^{GLYPH<20>}$ p n + 1 p n + 2 p n + 3 p n + 4 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{+}$$_{4}$ + (4 n + 6) p n + 1 p n + 2 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{+}$$_{2}$ + $^{GLYPH<16>}$6 n 2 + 6 n + 3 GLYPH<17> GLYPH<14>$_{m}$$_{;}$$_{n}$ + (4 n GLYPH<0> 2) p n p n GLYPH<0> 1 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{GLYPH<0>}$$_{2}$ + p n p n GLYPH<0> 1 p n GLYPH<0> 2 p n GLYPH<0> 3 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{GLYPH<0>}$$_{4}$ GLYPH<21> ) GLYPH<12> GLYPH<12> GLYPH<12> 2 (C.3) p GLYPH<28> 2 GLYPH<22>; 2 GLYPH<23> = 2 1 = 2 ( Q GLYPH<3> + 1) 1 = 2 $^{ }$Q GLYPH<3> GLYPH<0> 1 Q GLYPH<3> + 1$^{$_{!}$GLYPH<22>}$$^{+}$$^{GLYPH<23>}$ GLYPH<2> GLYPH<0> (1 = 2 + GLYPH<22> ) GLYPH<0> (1 = 2 + GLYPH<23> ) GLYPH<25> GLYPH<0> (1 + GLYPH<22> ) GLYPH<0> (1 + GLYPH<23> ) GLYPH<2> " $_{2}$F$_{1}$ GLYPH<0> GLYPH<22>; GLYPH<0> GLYPH<23> ; 1 2 ; 2 1 GLYPH<0> Q GLYPH<3> $_{!#}$2 (B.10) p GLYPH<28> 2 GLYPH<22> + 1 ; 2 GLYPH<23> + 1 = 2 7 = 2 ( Q GLYPH<3> + 1) 3 = 2 $^{ }$Q GLYPH<3> GLYPH<0> 1 Q GLYPH<3> + 1$^{$_{!}$GLYPH<22>}$$^{+}$$^{GLYPH<23>}$ GLYPH<2> GLYPH<0> (3 = 2 + GLYPH<22> ) GLYPH<0> (3 = 2 + GLYPH<23> ) GLYPH<25> GLYPH<0> (1 + GLYPH<22> ) GLYPH<0> (1 + GLYPH<23> ) GLYPH<2> " $_{2}$F$_{1}$ GLYPH<0> GLYPH<22>; GLYPH<0> GLYPH<23> ; 3 2 ; 2 1 GLYPH<0> Q GLYPH<3> $_{!#}$2 : (B.11) 11