The parameter GLYPH<27>$_{GLYPH<11>}$ controls the strength of the perturba- tion. In Fig. 5, we have plotted the mean work h W i and the variance, var ( W ) = $^{D}$W 2 E GLYPH<0> h W i $^{2}$, of the work distribution for the anharmonically perturbed harmonic oscillator (47), together with the exact result for the un- perturbed oscillator (2). We observe that both quantities are enhanced by the anharmonicity. From the analyti- cal expressions of the transition probabilities (C.3), we see that additional transitions, m = n GLYPH<6> 4, now become possible because of the quartic correction. These ad- ditional transitions lead to a larger mean and variance of the work. Based on our discussion in section 2.2, we can therefore conclude that the anharmonic perturbation increases the degree of nonadiabaticity of the frequency change. Numerical comparison further shows that the e GLYPH<11> ect of A$_{t}$ can be neglected up to a strength of roughly one percent, GLYPH<27>$_{GLYPH<11>}$ ≲ 0 : 01, of the harmonic amplitude GLYPH<10>$_{t}$ . For a standard trap configuration with trap frequencies of the order kHz-MHz, the harmonic assumptions is ful- filled up to energies of the order of eV, see Ref. [26], and the e GLYPH<11> ect of anharmonic corrections are negligible for these energies. A method to experimentally measure the quantum work distribution in modulated ion trap systems has been proposed in Ref. [16]. In these systems, the confin- ing potential is harmonic to a very good accuracy [26]. One attractive feature of linear Paul traps is however the possibility to modify the shape of the potential with the help of external gate voltages. We here investigate the influence of a small quartic anharmonicity on the work distribution P ( W ). As before, we write the total Hamil- tonian as where the first anharmonic correction is given by, The total transition probabilities can then be written as where A t m ; n are the anharmonic interaction matrix el- ements. The analytic transition probabilities p GLYPH<28> m ; n are again given in Appendix C. For the sake of clarity, we will continue with a numerical discussion of the results. For the numerical analysis, we choose the parameteri- zation of ! 2 t to be linear in time, The function GLYPH<21>$_{t}$ = qE$_{t}$ is proportional to the random electric field E$_{t}$ ( q is the charge of the ion) and is taken to be Gaussian distributed with Linear Paul traps are almost perfectly isolated from their surroundings. They however su GLYPH<11> er from the pres- ence of random electric fields that are generated in the trap electrodes [28]. These weak fluctuating fields are the source of motional heating of the charged ions con- fined in the harmonic trap. The Hamiltonian of the quantum oscillator in the presence of the field is Figure 5: Variance and mean (inset) work for an oscillator with weak anharmonic corrections (red) (47) compared with those of the unper- turbed 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 0.01 0.02 0.03 0.04 0.05 var H W L 1 1.025 1.05 w 1 w 0 0.05 0.1 0.15 < W > 5.2. Anharmonic corrections 5.3. Random electric field corrections The heating rate of the trap is related to the spectral den- sity of the noise GLYPH<21>$_{t}$ [29] We first calculate the transition probabilities p GLYPH<28> m ; n for a fixed value of GLYPH<21>$_{t}$ and then average over GLYPH<21>$_{t}$ using Eq. (54). Since the anharmonic corrections are given by the geo- metric set-up and hence directly scale with the angular frequency of the harmonic oscillator, we assume that H$_{t}$ = H$_{0}$ + GLYPH<10>$_{t}$ + GLYPH<3>$_{t}$ (52) where the small perturbation GLYPH<3>$_{t}$ is linear in position, GLYPH<3>$_{t}$ = GLYPH<21>$_{t}$ x : (53) h GLYPH<21>$_{t}$ i = 0 and h GLYPH<21>$_{t}$ GLYPH<21>$_{s}$ i = GLYPH<20>$_{t}$$_{;}$$_{s}$ : (54) h ˙ n i ’ 1 4 M $_{ℏ}$!$_{t}$ + 1 Z GLYPH<0>1 ds exp ( i !$_{t}$ s ) h GLYPH<21>$_{t}$ GLYPH<21>$_{t}$$_{+}$$_{s}$ i : (55) ! 2 t = ! 2 0 + $^{GLYPH<16>}$! 2 1 GLYPH<0> ! 2 0$^{GLYPH<17>}$ t GLYPH<28> : (50) GLYPH<11>$_{t}$ = GLYPH<27>$_{GLYPH<11>}$ $^{GLYPH<16>}$! 2 1 GLYPH<0> ! 2 0$^{GLYPH<17>}$ t GLYPH<28> : (51) 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> $^{GLYPH<16>}$GLYPH<10> t m ; n + A t m ; n$^{GLYPH<17>}$$^{GLYPH<12>}$ GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> 2 ; (49) A$_{t}$ = GLYPH<11>$_{t}$ x 4 : (48) H$_{t}$ = H$_{0}$ + GLYPH<10>$_{t}$ + A$_{t}$ ; (47) 8