content_image/439527.jpg content_image/439526.jpg content_image/439519.jpg content_image/439525.jpg content_image/439522.jpg content_image/439524.jpg content_image/439517.jpg content_image/439516.jpg content_image/439523.jpg content_image/439521.jpg content_image/439518.jpg
Details about the specific parameterization of the an- gular frequency only enter through di GLYPH<11> erent numerical values of the parameter Q GLYPH<3> given by, 

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From a physical point of view, Q GLYPH<3> can be regarded as a measure of the degree of adiabaticity of the process and will be discussed in more detail in the following sub- section. Among the properties of the generating func- tion P ( u ; v ), it is worth mentioning that the law of total probability P n p GLYPH<28> m ; n = 1 is fulfilled and is equivalent to 

For a constant frequency, !$_{t}$ GLYPH<17> !$_{0}$ , we note that the solution of Eq. (7) is given by 

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The latter imply with Eq. (14) that Q GLYPH<3> = 1 and Eq. (13) thus simplifies to 

which is equivalent to p GLYPH<28> m ; n = GLYPH<14>$_{m}$$_{;}$$_{n}$ , indicating the ab- sence of transitions, as expected. The symmetry relation of the generating function (13), P ( GLYPH<0> u ; GLYPH<0> v ) = P ( u ; v ), further shows that p GLYPH<28> m ; n = 0 if m , n are of di GLYPH<11> erent par- ity. This is an expression of a selection rule m = n GLYPH<6> 2 k , where k is an integer. In subsection 5.1, we rederive this selection rule by means of time-dependent pertur- bation theory. We mention in addition that the transi- tion probabilities are symmetric, p GLYPH<28> m ; n = p GLYPH<28> n ; $_{m}$, following P ( u ; v ) = P ( v ; u ). Explicit expressions for the transitions probabilities p GLYPH<28> m ; n are given in terms of hypergeometric functions in Appendix B. 

## 2.2. Measure of adiabaticity 

The parameter Q GLYPH<3> defined in (14) can be given a sim- ple physical meaning [19]. We base our discussion of adiabaticity on the equivalent classical harmonic oscil- lator (7) since the generating function P ( u ; v ) (13) is fully determined through the classical solutions X$_{t}$ and Y$_{t}$ . For an adiabatic transformation, the action of the oscillator, given by the ratio of the energy to the angu- lar frequency, is a time-independent constant. For qua- sistatic processes we have the two adiabatic invariants, 

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From the definition (14) of the parameter Q $^{GLYPH<3>}$, we see that in this case we simply have Q GLYPH<3> = 1. As mentioned earlier this implies P ( u ; v ) = (1 GLYPH<0> uv ) GLYPH<0> 1 and p GLYPH<28> m ; n = GLYPH<14>$_{m}$$_{;}$$_{n}$ . The latter is an expression of the quantum adiabatic the- orem: For infinitely slow transformations no transitions between di GLYPH<11> erent quantum states occur. For fast trans- formations, on the other hand, we can regard Q GLYPH<3> as a measure of the degree of nonadiabaticity of the process. As an illustration, we evaluate mean and variance of the energy of the oscillator at time GLYPH<28> and express them as a function of Q $^{GLYPH<3>}$. For a transition from initial state j n i to final state j m i , the mean-quantum number of the final state h m i$_{n}$ can be obtained by taking the first derivative of the generating function (13) of p GLYPH<28> m ; $_{n}$, 

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and expanding the left hand side of (19) in powers of u : 

Noting that p 0 n = exp( GLYPH<0> GLYPH<12> E 0 $_{n}$) = Z$_{0}$ , the mean energy of the oscillator at time GLYPH<28> then reads 

Since h m i$_{n}$ GLYPH<21> 0, and hence h H$_{GLYPH<28>}$ i GLYPH<21> $_{ℏ}$!$_{1}$ = 2, the parameter Q GLYPH<3> necessarily satisfies Q GLYPH<3> GLYPH<21> 1 for generic processes. In the zero temperature limit, Eq. (21) reduces to 

The above equation corrects a misprint appearing in Eq. (5.21) of Ref. [19] ( !$_{0}$ should be replaced by !$_{1}$ ). The mean-square quantum number $^{D}$m 2 $^{E}$$_{n}$GLYPH<0> h m i 2 n at time GLYPH<28> can be calculated in a similar way by considering h m ( m GLYPH<0> 1) i$_{n}$ . By di GLYPH<11> erentiating Eq. (13) twice, we have 

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and a series expansion in powers of u leads to 

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