parametric oscillator during a variation of its angular frequency. We give the expressions of its continuous envelop in the limit of high and low temperatures, for adiabatic and nonadiabatic transformations [17]. We further compare our nonadiabatic results with those re- cently derived by van Zon and coworkers in Ref. [23]. In quantum mechanics, the probability density func- tion of work is obtained by considering the di GLYPH<11> erence between final and initial energy eigenstates, E GLYPH<28> m and E 0 $_{n}$, averaged over all possible final and initial states. The probability distribution of the total work done during time GLYPH<28> can thus be written as [15], The mean-square quantum number is then obtained by combining Eqs. (20) and (24): From Eqs. (20), (21) and (25), we can finally write the variance of the energy as The zero-temperature limit, where p 0 n = exp( GLYPH<0> GLYPH<12> E 0 $_{n}$) = Z$_{0}$ is the initial (thermal) occu- pation probability. Equation (28) show that work is a random quantity because of the presence of both ther- mal and quantum uncertainties, encoded respectively in p 0 n and p GLYPH<28> m ; $_{n}$. The characteristic function of the work, defined as the Fourier transform of the probability dis- tribution (28), can be written in closed form in terms of the energies, "$_{0}$ = $_{ℏ}$!$_{0}$ , "$_{1}$ = $_{ℏ}$!$_{1}$ ( GLYPH<1> " = "$_{1}$ GLYPH<0> "$_{0}$ ). Using expression (13) of the generating function, we have is again the correct version of Eq. (5.22) of Ref. [19]. Equation (25) indicates that the parameter Q GLYPH<3> directly controls the magnitude of the variance of the occupa- tion number, $^{D}$m 2 $^{E}$$_{n}$GLYPH<0> h m i 2 $_{n}$. In the adiabatic limit, where Q GLYPH<3> = 1, we readily get h m i$_{n}$ = n and $^{D}$m 2 $^{E}$$_{n}$GLYPH<0> h m i 2 n = 0. We therefore recover that for adiabatic processes the system remains in its initial state, j m i = j n i . On the other hand, for fast nonadiabatic processes, the mean h m i$_{n}$ and the dispersion $^{D}$m 2 E n GLYPH<0> h m i 2 n increase with in- creasing values of Q $^{GLYPH<3>}$, indicating that the quantum oscil- lator ends in a final state j m i which is farther and farther away from the initial state j n i . The latter corresponds to larger and larger final values of the mean energy and energy variance, Eqs. (21) and (26). It is worth mentioning that the above discussion of the adiabaticity parameter Q GLYPH<3> for the parametric oscil- lator is close in spirit to the Einstein criteria for adia- batic processes [21]. Einstein noted that for an adiabatic process, the classical action of the oscillator, h H$_{t}$ i =!$_{t}$ , should remain constant and the number of quanta should therefore remain unchanged. In the present situation, we have h H$_{GLYPH<28>}$ i =!$_{GLYPH<28>}$ / Q $^{GLYPH<3>}$, and the action only remains constant when Q GLYPH<3> = 1. The latter is precisely the con- dition that we derived for an adiabatic transformation. For nonadiabatic processes, the parameter Q GLYPH<3> > 1 thus gives a measure for the increase of the classical action of the oscillator. Further discussions of adiabatic measures can be found in Ref. [20]. In this section, we introduce the probability distri- bution P ( W ) on the nonequilibrium work done on the The above equations for G ( GLYPH<22> ) are exact and fully char- acterizes the work distribution of the time-dependent quantum harmonic oscillator (2) for arbitrary parame- terizations of the angular frequency !$_{t}$ . As mentioned previously, di GLYPH<11> erent realizations of !$_{t}$ merely lead to di GLYPH<11> erent values of the parameter Q $^{GLYPH<3>}$. The direct an- alytic evaluation of the nonequilibrium work distribu- tion P ( W ) by Fourier inverting Eq. (30) is not feasible in the general case due to the nonanalytic denomina- tor of G ( GLYPH<22> ). We here provide analytical approximations in various limits of interest. The detailed derivation of the following probability distributions can be found in Ref. [17]. where the denominator is given by 3. Work probability density function 4 $^{D}$m 2 $^{E}$$_{n}$GLYPH<0> h m i 2 n = 1 2 $^{GLYPH<16>}$Q GLYPH<3> 2 GLYPH<0> 1 GLYPH<17> $^{GLYPH<16>}$n 2 + n + 1 GLYPH<17> : (25) $^{D}$H 2 GLYPH<28> E GLYPH<0> h H$_{GLYPH<28>}$ i 2 = ℏ $^{2}$!2 1 8 csch 2 GLYPH<18> GLYPH<12> 2 $_{ℏ}$!$_{0}$$^{GLYPH<19>}$ GLYPH<2> $^{GLYPH<16>}$1 GLYPH<0> 4 Q GLYPH<3> 2 GLYPH<0> 3 cosh ( GLYPH<12>$_{ℏ}$!$_{0}$ ) + 4 Q GLYPH<3> sinh ( GLYPH<12>$_{ℏ}$!$_{0}$ ) GLYPH<17> : (26) $^{D}$H 2 GLYPH<28> E GLYPH<0> h H$_{GLYPH<28>}$ i 2 = ℏ $^{2}$!2 1 2 $^{GLYPH<16>}$Q GLYPH<3> 2 GLYPH<0> 1 GLYPH<17> ; (27) P ( W ) = X m ; n GLYPH<14> $^{GLYPH<16>}$W GLYPH<0> ( E GLYPH<28> $_{m}$GLYPH<0> E 0 $_{n}$) GLYPH<17> p GLYPH<28> m ; n p 0 $_{n}$; (28) G ( GLYPH<22> ) = Z dW exp ( i GLYPH<22> W ) P ( W ) ; (29) G ( GLYPH<22> ) = s 2 n ( Q $^{GLYPH<3>}$) $^{GLYPH<0>}$1 GLYPH<0> exp ( GLYPH<0> GLYPH<12>"$_{0}$ ) GLYPH<1> exp$^{ }$ i GLYPH<22> GLYPH<1> " 2 ! (30) n ( Q $^{GLYPH<3>}$) = Q GLYPH<3> $^{GLYPH<0>}$1 GLYPH<0> exp (2 i GLYPH<22>"$_{1}$ ) GLYPH<1> $^{GLYPH<0>}$1 GLYPH<0> exp ( GLYPH<0> 2( i GLYPH<22> + GLYPH<12> ) "$_{0}$ ) GLYPH<1> + $^{GLYPH<0>}$1 + exp (2 i GLYPH<22>"$_{1}$ ) GLYPH<1> $^{GLYPH<0>}$1 + exp ( GLYPH<0> 2( i GLYPH<22> + GLYPH<12> ) "$_{0}$ ) GLYPH<1> GLYPH<0> 4 exp ( i GLYPH<22>"$_{1}$ ) exp ( GLYPH<0> ( i GLYPH<22> + GLYPH<12> ) "$_{0}$ ) : (31)