content_image/415124.jpg content_image/415122.jpg content_image/415125.jpg content_image/415126.jpg content_image/415128.jpg content_image/415127.jpg content_image/415123.jpg content_image/415121.jpg content_image/415120.jpg content_image/415118.jpg content_image/415119.jpg content_image/415117.jpg content_image/415129.jpg
## Appendix A. Analytical expression for Q GLYPH<3> 

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 

Eq. (A.1) takes the form of the Mathieu equation. The solutions X$_{t}$ and Y$_{t}$ can then be written as, 

and 

Here the functions C and S denote the corresponding Mathieu functions [32]. The parameters d$_{1}$ and d$_{2}$ are given by, 

and 

Further analysis of the parameter Q GLYPH<3> in the context of vacuum squeezing can be found in Ref. [24]. 

<img src='content_image/415124.jpg'>

<img src='content_image/415122.jpg'>

<img src='content_image/415125.jpg'>

<img src='content_image/415126.jpg'>

<img src='content_image/415128.jpg'>

<img src='content_image/415127.jpg'>

## Appendix B. Exact transition probabilities 

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 

<img src='content_image/415123.jpg'>

We use again the method of generating functions. We use the linear generating function of GLYPH<30> $^{t}$$_{n}$(x ) [27], 

<img src='content_image/415121.jpg'>

to evaluate the generating function of the propagator, 

<img src='content_image/415120.jpg'>

By introducing the complex parameters, 

<img src='content_image/415118.jpg'>

<img src='content_image/415119.jpg'>

we can write 

<img src='content_image/415117.jpg'>

The matrix elements U m ; n can then be obtained by a se- ries expansion of (B.6) in powers of u and v [19] 

<img src='content_image/415129.jpg'>

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 

