In complete analogy to Eq. (49), we obtain with the interaction matrix elements GLYPH<3> t m ; n given by, The explicit expression of the transition probabilities can be found in Appendix C.3. After averaging over all possible GLYPH<21>$_{t}$ , the transition probabilities can be divided into two distinct contributions coming from the para- metric variation of the frequency ( GLYPH<10>$_{t}$ in Eq. (52)) and the noise term ( GLYPH<3>$_{t}$ in Eq. (52)), We have considered the statistics of the nonequilib- rium work of a quantum oscillator when its angular fre- quency is varied in time. Due to the quantized nature of the energy spectrum, the work probability distribution is discrete. We have analyzed the discrete-to-continuous transition of the work distribution in various limits by introducing the cumulative function. We have shown that the cumulative work distribution becomes smooth in the limit of high temperatures and of large values of the adiabaticity parameter Q $^{GLYPH<3>}$; in both regimes, the mean energy of the oscillator is much larger than the energy separation, and the spectrum can be considered quasi- continuous. We have moreover developed a perturbative approach to investigate the e GLYPH<11> ects of small quartic an- harmonicities on the work distribution. We have found that the latter increase both the mean and the variance of the final energy of the oscillator, indicating an augmen- tation of the nonadiabaticity of the frequency change. In a similar way, we have studied the influence of a weak electric noise on a charged harmonic oscillator and ob- tained an analogous enhancement of the degree of nona- diabaticity. Our results permit an accurate description of measured quantum work distributions in modulated Paul trap under realistic experimental conditions. Equation (49) shows that the e GLYPH<11> ect of the random elec- tric field is to renormalize the adiabaticity parameter Q GLYPH<3> ! Q GLYPH<3> + Q GLYPH<3> $_{GLYPH<21>$_{t}$}$. Both the mean and the variance of the work distribution are increased as depicted in Fig. 6. The fluctuating field thus enhances the degree of nona- diabaticity. This e GLYPH<11> ect can be understood by noting that the perturbation GLYPH<3>$_{t}$ generates additional transitions be- tween states (the latter obey m = n GLYPH<6> 1). We observe that the variance is more sensitive to the perturbation than the mean, since it depends quadratically on Q GLYPH<3> and not linearly. For the numerical analysis we have chosen a white noise of the form, Here the parameter Q GLYPH<3> GLYPH<21>$_{t}$ is defined as Similarly, we can separate the mean final energy into a deterministic and a stochastic part, Acknowledgements We would like to thank Peter Talkner for discussions and comments. This work is based on OA’s project within the NIM summer research program, and was supported by the Emmy Noether Pro- gram of the DFG (contract LU1382 / 1-1) and the cluster of excellence Nanosystems Initiative Munich (NIM). where the relative noise strength is given by GLYPH<27>$_{GLYPH<21>}$ . As for the case of the anharmonic perturbation, we note that one can neglect the influence of the electric noise up to a relative strength of roughly one percent, GLYPH<27>$_{GLYPH<21>}$ ≲ 0 : 01. 9 GLYPH<20>$_{t}$$_{;}$$_{s}$ = GLYPH<27>$_{GLYPH<21>}$ $^{GLYPH<16>}$! 2 0 GLYPH<0> ! 2 1$^{GLYPH<17>}$ GLYPH<14> ( t GLYPH<0> s ) ; (62) with Q GLYPH<3> GLYPH<21>$_{t}$ = h H$_{GLYPH<28>}$ i$_{GLYPH<21>}$$_{t}$ $_{ℏ}$!$_{1}$ = 2 coth ( GLYPH<12>= 2 $_{ℏ}$!$_{0}$ ) (60) h H i$_{GLYPH<21>}$$_{t}$ = X n ; m $_{ℏ}$!$_{1}$ m + 1 2$^{!}$ p GLYPH<28> m ; n ( h GLYPH<21>$_{t}$ GLYPH<21>$_{s}$ i ) p 0 n : (61) h H$_{GLYPH<28>}$ i = X m ; n E GLYPH<28> m $^{GLYPH<16>}$p GLYPH<28> m ; $_{n}$( !$_{t}$ ) + p GLYPH<28> m ; n ( h GLYPH<21>$_{t}$ GLYPH<21>$_{s}$ i ) GLYPH<17> p 0 n = $_{ℏ}$!$_{1}$ 2 $^{GLYPH<16>}$Q GLYPH<3> + Q GLYPH<3> GLYPH<21>$_{t}$ $^{GLYPH<17>}$coth GLYPH<18> GLYPH<12> 2 $_{ℏ}$!$_{0}$$^{GLYPH<19>}$ : (59) $^{D}$p GLYPH<28> m ; n$^{E}$ GLYPH<21>$_{t}$ = p GLYPH<28> m ; $_{n}$( !$_{t}$ ) + p GLYPH<28> m ; n ( h GLYPH<21>$_{t}$ GLYPH<21>$_{s}$ i ) : (58) p GLYPH<28> m ; n = GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<14>$_{m}$$_{;}$$_{n}$ + 1 i$_{ℏ}$ GLYPH<28> Z 0 dt exp $^{GLYPH<0>}$i !$_{m}$$_{;}$$_{n}$ t $^{GLYPH<1>}$( GLYPH<10> t m ; n + GLYPH<3> t m ; $_{n}$)$^{GLYPH<12>}$ GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> 2 ; (56) GLYPH<3> t m ; n = GLYPH<21>$_{t}$ s ℏ 2 M !$_{0}$ GLYPH<16> p n + 1 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{+}$$_{1}$ + p n GLYPH<14>$_{m}$$_{;}$$_{n}$$_{GLYPH<0>}$$_{1}$ GLYPH<17> : (57) 6. Conclusion Figure 6: Variance and mean (inset) work for a charged oscillator with weak electric noise (red) (47) compared with those of the unperturbed oscillator (blue, dashed) (2) ( !$_{0}$ = 0 : 5, GLYPH<28> = 1, GLYPH<12> = 0 : 5, M = 1, ℏ = 1 and GLYPH<27>$_{GLYPH<11>}$ = 0 : 025). 1.00 1.01 1.02 1.03 1.04 1.05$^{w}$ 1 w 0 0.005 0.010 0.015 var H W L 1 1.025 1.05 w 1 w 0 0.05 0.1 < W >