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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 

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 

<img src='content_image/440437.jpg'>

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 !$_{1}$ , 

<img src='content_image/440435.jpg'>

<img src='content_image/440433.jpg'>

<img src='content_image/440434.jpg'>

<img src='content_image/440438.jpg'>

<img src='content_image/440421.jpg'>

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 

<img src='content_image/440436.jpg'>

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]. 

## 4. Transition between discrete and continuous nonequilibrium work distributions 

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- 

