- 5 0 5 10 15 20 25 W 0.2 0.4 0.6 0.8 1.0 $^{f}$H W L Adiabatic transformations. As discussed in subsection 2.2, adiabatic transformations are characterized by Q GLYPH<3> = 1. By approximating Eq. (30) in the limit of zero- temperature, $_{ℏ}$GLYPH<12> GLYPH<29> 1, we get Equation (32) expresses the deterministic nature of adia- batic processes at zero-temperature: the oscillator starts and ends in its ground state. Work is hence simply given by the di GLYPH<11> erence of final and initial ground state ener- gies. In the opposite classical limit, $_{ℏ}$GLYPH<12> GLYPH<28> 1, we find An example for a frequency parameterization resulting in an unbounded, divergent value of Q GLYPH<3> can be found in Ref. [24]. In the regime where Q GLYPH<3> < Q GLYPH<3> $_{ss}$, the work dis- tribution has been derived by van Zon and coworkers by explicitly evaluating the complex integral of the inverse Fourier transform [23]. It reads in our notation Equation (33) is identical to the classical work probabil- ity distribution derived by Jarzynski [22]. Note that in this case only positive work fluctuations occur, whose magnitude is controlled by the finite temperature. Nonadiabatic transformations. We next consider nona- diabatic transformation which correspond to Q GLYPH<3> > 1. In the zero-temperature limit, $_{ℏ}$GLYPH<12> GLYPH<29> 1, the work distribu- tion can be approximated in the limit of small "$_{1}$ by, The zero-temperature nonadiabatic work distribution (34) is valid when W GLYPH<21> GLYPH<1> "= 2. This condition follows from the existence of the minimal ground state energy of the oscillator. An expansion of Eq. (30) in the classi- cal limit, $_{ℏ}$GLYPH<12> GLYPH<28> 1, leads to where I$_{0}$ ( x ) denotes the modified Bessel function of the first kind. It is worth mentioning that, in contrast to the adiabatic (33) and nonadiabatic (37) results, nega- tive work values can occur in Eq. (35) for large values of the parameter Q GLYPH<3> [23]. We now turn to the analysis of the transition from the discrete work probability density (28) to the con- tinuous analytical approximations Eqs. (33), (34), (35) and (37), and provide a criterion for the observation of the discreteness of P ( W ). The definition (28) makes clear that the nonequilibrium work distribution consists of a discrete sum of delta peaks when the energy spec- trum of the system is quantized. This is in stark con- where GLYPH<0> ( x ) denotes the Euler Gamma function and K$_{GLYPH<23>}$ ( x ) is the Macdonald function, that is the modified Bessel function of the third kind. Equation (35) is well- defined provided the term under the square-root is posi- tive. This implies that the parameter Q GLYPH<3> is larger than the value Q GLYPH<3> ss that corresponds to a sudden (instanta- neous) switch of the frequency from !$_{0}$ to !$_{1}$ , 4. Transition between discrete and continuous nonequilibrium work distributions Figure 1: Cumulative work distribution GLYPH<30> ( W ) (38) for the exact quan- tum expression (28) (red) and the approximated classical form (35) (blue, dashed) for the parameters !$_{0}$ = 1, !$_{1}$ = 1 : 5, ℏ = 1, GLYPH<12> = 0 : 5 and Q GLYPH<3> = 3. The discrete structure of the quantum distribution is eas- ily recognized as compared to the continuous classical distribution. P ( W ) = GLYPH<14> W GLYPH<0> GLYPH<1> " 2 ! : (32) P ( W ) = GLYPH<12>!$_{0}$ GLYPH<1> ! exp$^{GLYPH<18>}$ GLYPH<0> GLYPH<12>!$_{0}$ GLYPH<1> ! W$^{GLYPH<19>}$ GLYPH<2> ( W ) : (33) P ( W ) = exp $^{GLYPH<16>}$GLYPH<0> W GLYPH<0> GLYPH<1> "= 2 ( Q GLYPH<3> GLYPH<0> 1) "$_{1}$ GLYPH<17> p GLYPH<25> ( Q GLYPH<3> GLYPH<0> 1) "$_{1}$ ( W GLYPH<0> GLYPH<1> "= 2) : (34) P ( W ) = s GLYPH<12> $^{2}$! 2 0 ! 2 0 GLYPH<0> 2 Q $^{GLYPH<3>}$!$_{0}$ !$_{1}$ + ! 2 1 GLYPH<2> exp$^{0}$ B B @ Q $^{GLYPH<3>}$!$_{0}$ !$_{1}$ GLYPH<0> ! 2 0 2 Q $^{GLYPH<3>}$!$_{0}$ !$_{1}$ GLYPH<0> ! 2 0 GLYPH<0> ! 2 1 GLYPH<12> W$^{1}$ C C A GLYPH<2> I$_{0}$$^{0}$ B B @ GLYPH<12>!$_{0}$!$_{1}$ $^{p}$Q GLYPH<3> 2 GLYPH<0> 1 GLYPH<12> $_{GLYPH<12>}$2 Q $^{GLYPH<3>}$!$_{0}$ !$_{1}$ GLYPH<0> ! 2 0 GLYPH<0> ! 2 1$^{GLYPH<12>}$ GLYPH<12> j W j$^{1}$ C C A GLYPH<2> ( W ) ; (37) P ( W ) = s GLYPH<12> $^{2}$! 2 $_{0}$=GLYPH<25> 2 2 Q $^{GLYPH<3>}$!$_{0}$ !$_{1}$ GLYPH<0> ! 2 0 GLYPH<0> ! 2 1 GLYPH<2> exp$^{0}$ B B @ Q $^{GLYPH<3>}$!$_{0}$ !$_{1}$ GLYPH<0> ! 2 0 2 Q $^{GLYPH<3>}$!$_{0}$ !$_{1}$ GLYPH<0> ! 2 0 GLYPH<0> ! 2 1 GLYPH<12> W$^{1}$ C C A GLYPH<2> K$_{0}$$^{0}$ B B @ GLYPH<12>!$_{0}$!$_{1}$ $^{p}$Q GLYPH<3> 2 GLYPH<0> 1 GLYPH<12> $_{GLYPH<12>}$2 Q $^{GLYPH<3>}$!$_{0}$ !$_{1}$ GLYPH<0> ! 2 0 GLYPH<0> ! 2 1$^{GLYPH<12>}$ GLYPH<12> j W j$^{1}$ C C A ; (35) Q GLYPH<3> > ! 2 0 + ! 2 1 2 !$_{0}$!$_{1}$ = Q GLYPH<3> ss : (36) 5