Figure 3: In the limit of high Q GLYPH<3> values, the cumulative work function GLYPH<30> ( W ) for the exact discrete case (28) (red) is indistinguishable from the continuous approximation (35) (blue, dashed). Here !$_{0}$ = 1, !$_{1}$ = 1 : 5, ℏ = 1, GLYPH<12> = 0 : 5 and Q GLYPH<3> = 60. Figure 2: In the limit of high temperatures, the cumulative work func- tion GLYPH<30> ( W ) for the exact discrete case (28) (red) is indistinguishable from the continuous approximation (35) (blue, dashed). Here !$_{0}$ = 1, !$_{1}$ = 1 : 5, ℏ = 1, GLYPH<12> = 0 : 05 and Q GLYPH<3> = 3. 0 50 100 150 200 250 W 0.2 0.4 0.6 0.8 1.0 $^{f}$H W L - 20 0 20 40 60 80 100 W 0.2 0.4 0.6 0.8 1.0 $^{f}$H W L of Q $^{GLYPH<3>}$, we therefore expect the nonequilibrium work dis- tribution to become continuous when the temperature is increased. Figure 2 depicts the exact cumulative distri- bution GLYPH<30> ( W ) and the approximate result corresponding to Eq. (35) for the same parameters as in Fig. 1 except the temperature which has been increased by a factor ten; the two curves are indistinguishable. Large Q GLYPH<3> values. Another way to reach a large mean energy (21) is to increase the value of Q $^{GLYPH<3>}$, while keep- ing the temperature constant. The work distribution can thus become continuous when the degree of nonadia- baticity of the process is increased. Figure 3 shows the exact and approximate cumulative distribution GLYPH<30> ( W ) for a value of Q GLYPH<3> twenty times larger than in Fig. 1, all other parameters being the same: the two curves are again not distinguishable. It is interesting to notice that the tran- sition to a continuous work distribution occurs faster when augmenting the temperature than the degree of nonadiabaticity Q GLYPH<3> (a factor two for the parameters of Fig. 1), the reason being that the mean energy (21) does not depend linearly on temperature in contrast to Q GLYPH<3> (the temperature dependence becomes linear only for high temperatures). trast to classical work distributions which are continu- ous. The discreteness of P ( W ) can therefore be regarded as the hallmark of the quantum nature of work. In order to quantitatively investigate the discrete-to-continuous transition, we introduce the cumulative probability dis- tribution GLYPH<30> ( W ) defined as, In the above equation, the constant GLYPH<31> has to be chosen according to the range validity of the work distribution, e.g. in the zero-temperature limit GLYPH<31> = GLYPH<1> "= 2. The cu- mulative distribution (38) is a staircase function in the case of a discrete distribution P ( W ) and turns over to a smooth function in the continuous limit. Figure 1 shows the exact quantum function GLYPH<30> ( W ) for a given set of pa- rameters, together with the continuous classical approx- imation corresponding to Eq. (35). The numerical value of the parameter Q GLYPH<3> has been chosen to describe realis- tic experiments with ion traps [26]. In order to obtain a criterion for the discreteness of the work distribution P ( W ), we consider the mean energy of the harmonic os- cillator as given by Eq. (21). The energy spectrum of the harmonic oscillator is usually considered to be quasi- continuous when the mean quantum number is much larger than the level separation [25], or in other words, when the mean energy is much larger than the energy quantum, h H$_{GLYPH<28>}$ i GLYPH<29> $_{ℏ}$!$_{1}$ . An inspection of Eq. (21) re- veals that this limit can be achieved in two ways: Noninteger ratios "$_{1}$ ="$_{0}$ . One may wonder whether it is possible to have continuous work distributions, for any values of Q GLYPH<3> and GLYPH<12> , by for instance considering non- rational quotients "$_{1}$ ="$_{0}$ . In this limit, the gaps between GLYPH<14> -peaks could be filled and P ( W ) would appear contin- uous. This is not the case, however. We note indeed that for any ratio "$_{1}$ ="$_{0}$ , we do not obtain real contin- uous distributions with respect to the real set. The GLYPH<14> - functions in Eq. (28) only contribute to the total sum if High temperatures. In the limit of high temperatures, $_{ℏ}$GLYPH<12> GLYPH<28> 1, we have coth ( GLYPH<12>= 2 $_{ℏ}$!$_{0}$ ) GLYPH<29> 1. For a fixed value GLYPH<30> ( W ) = W Z GLYPH<31> dW 0 P ( W $^{0}$) : (38) 6