Appendix B. Exact transition probabilities Appendix A. Analytical expression for Q GLYPH<3> 10 According to the selection rule m = n GLYPH<6> 2 k , l runs over even numbers only, if m , n are even, and over odd num- bers only, if m , n are odd. The explicit expression for Further analysis of the parameter Q GLYPH<3> in the context of vacuum squeezing can be found in Ref. [24]. Here the functions C and S denote the corresponding Mathieu functions [32]. The parameters d$_{1}$ and d$_{2}$ are given by, The matrix elements U GLYPH<28> m ; n can then be obtained by a se- ries expansion of (B.6) in powers of u and v [19] we can write By introducing the complex parameters, to evaluate the generating function of the propagator, and Eq. (A.1) takes the form of the Mathieu equation. The solutions X$_{t}$ and Y$_{t}$ can then be written as, We use again the method of generating functions. We use the linear generating function of GLYPH<30> $^{t}$$_{n}$(x ) [27], We here collect the analytical expressions of the tran- sition probabilities p GLYPH<28> m ; n [19]. Despite its apparent sim- plicity, the generation function P ( u ; v ) (13) cannot be expanded in powers of u and v in an exact series. We thus make use of the p GLYPH<28> m ; n as defined by the matrix ele- ments of the propagator U$_{GLYPH<28>;}$$_{0}$ ( x j x$_{0}$ ), p GLYPH<28> m ; n = GLYPH<12> $_{GLYPH<12>}$U GLYPH<28> m ; n$^{GLYPH<12>}$ GLYPH<12> 2 (10). This matrix elements are given by Closed expressions for the adiabaticity parameter Q GLYPH<3> can be found whenever the classical equation (7) can be solved analytically. Equation (A.1) is of the gen- eral form of a Hill equation which can be solved under various conditions [30, 31]. Equation (A.1) reduces to common di GLYPH<11> erential equations in the case of specific pa- rameterizations !$_{t}$ . Thus for the case of linear parame- terizations (50), the solutions are given in terms of the Airy-functions [17]. On the other hand, for a sinusoidal parameterization U ( u ; v ) = 4 p !$_{0}$ !$_{1}$ p i GLYPH<27>= 2 GLYPH<25> exp$^{ }$ GLYPH<16> u 2 GLYPH<0> 4 i p !$_{0}$ !$_{1}$ uv + GLYPH<16> $^{GLYPH<3>}$v 2 GLYPH<27> ! : (B.6) GLYPH<16> = !$_{1}$ !$_{0}$ X$_{GLYPH<28>}$ GLYPH<0> !$_{0}$ i ˙ X$_{GLYPH<28>}$ + !$_{1}$ iY$_{GLYPH<28>}$ + ˙ Y$_{GLYPH<28>}$ j GLYPH<16> j 2 = 2 !$_{0}$ !$_{1}$ ( Q GLYPH<3> GLYPH<0> 1) (B.4) GLYPH<27> = !$_{1}$ !$_{0}$ X$_{GLYPH<28>}$ GLYPH<0> !$_{0}$ i ˙ X$_{GLYPH<28>}$ GLYPH<0> !$_{1}$ iY$_{GLYPH<28>}$ GLYPH<0> ˙ Y$_{GLYPH<28>}$ j GLYPH<27> j 2 = 2 !$_{0}$ !$_{1}$ ( Q GLYPH<3> + 1) (B.5) U ( u ; v ) = X m ; n GLYPH<25> 2 n + m n ! m ! $_{!}$1 = 2 u$^{n}$v$^{m}$U GLYPH<28> m ; n : (B.3) 1 X n = 1 p GLYPH<25> 2 n n ! $_{!}$1 = 2 z$^{n}$GLYPH<30>$^{t}$$_{n}$(x ) = 4 $^{r}$M !$_{t}$ ℏ exp$^{0}$ B B @ GLYPH<0> M !$_{t}$ 2 ℏ x 2 + 2$^{r}$ M !$_{t}$ ℏ z x GLYPH<0> z 2$^{1}$ C C A ; (B.2) ! 2 t = ! 2 1 GLYPH<0> $^{GLYPH<16>}$! 2 1 GLYPH<0> ! 2 0$^{GLYPH<17>}$ cos$^{GLYPH<18>}$ GLYPH<25> 2 t GLYPH<28> GLYPH<19> (A.2) U GLYPH<28> m ; n = Z dx$_{0}$$^{Z}$ dx GLYPH<30> GLYPH<3> GLYPH<28> $_{m}$(x ) U$_{GLYPH<28>;}$$_{0}$ ( x j x$_{0}$ ) GLYPH<30> 0 $_{n}$(x$_{0}$ ) : (B.1) d 2 dt 2 X$_{t}$ + ! 2 t X$_{t}$ = 0 (A.1) X$_{t}$ = 4 GLYPH<28> GLYPH<25> S [ d$_{1}$ ; d$_{2}$ ; 0] C$^{GLYPH<20>}$ d$_{1}$ ; d$_{2}$ ; GLYPH<25> t 4 GLYPH<28> GLYPH<21> GLYPH<0> C [ d$_{1}$ ; d$_{2}$ ; 0] S$^{GLYPH<20>}$ d$_{1}$ ; d$_{2}$ ; GLYPH<25> t 4 GLYPH<28> GLYPH<21> ! GLYPH<2>$^{ }$ C’ [ d$_{1}$ ; d$_{2}$ ; 0] S [ d$_{1}$ ; d$_{2}$ ; 0] GLYPH<0> C [ d$_{1}$ ; d$_{2}$ ; 0] S’ [ d$_{1}$ ; d$_{2}$ ; 0]$^{$_{!}$GLYPH<0>}$$^{1}$ (A.3) Y$_{t}$ =$^{ }$ S’ [ d$_{1}$ ; d$_{2}$ ; 0] C$^{GLYPH<20>}$ d$_{1}$ ; d$_{2}$ ; GLYPH<25> t 4 GLYPH<28> GLYPH<21> GLYPH<0> C’ [ d$_{1}$ ; d$_{2}$ ; 0] S$^{GLYPH<20>}$ d$_{1}$ ; d$_{2}$ ; GLYPH<25> t 4 GLYPH<28> GLYPH<21> ! GLYPH<2>$^{ }$ C [ d$_{1}$ ; d$_{2}$ ; 0] S’ [ d$_{1}$ ; d$_{2}$ ; 0] GLYPH<0> C’ [ d$_{1}$ ; d$_{2}$ ; 0] S [ d$_{1}$ ; d$_{2}$ ; 0]$^{$_{!}$GLYPH<0>}$$^{1}$ : (A.4) d$_{1}$ = 16 GLYPH<25> 2 ! 2 $_{1}$GLYPH<28> 2 (A.5) d$_{2}$ = 8 GLYPH<25> 2 $^{GLYPH<16>}$! 2 1 GLYPH<0> ! 2 0$^{GLYPH<17>}$ GLYPH<28> 2 : (A.6) U GLYPH<28> m ; n = 4 $^{p}$2 !$_{0}$ !$_{1}$ r n ! m ! GLYPH<16> $^{n}$GLYPH<16> GLYPH<3> m 2 n + m GLYPH<0> $^{1}$i GLYPH<27> n + m + 1 GLYPH<2>$^{min(}$$^{m}$$^{;}$$^{n}$$^{)}$ X l = 0 [ GLYPH<0> 2 i p 2 = ( Q GLYPH<3> GLYPH<0> 1)] l l ! [( n GLYPH<0> l ) = 2]! [( m GLYPH<0> l ) = 2]! : (B.7) and