The latter can be considered small for small frequency changes. In first order time-dependent perturbation the- ory, the transition probabilities between initial state j n i and final state j m i are given by [27], and a ”perturbation” term where we have introduced the unperturbed Hamiltonian We start by rewriting Hamiltonian (2) as their argument is zero. The only permitted work values are therefore of the form, Now since m and n are integers, only countably infinite values of W can occur, implying that the work distribu- tion is discrete in any case. The latter follows from the fact that rational numbers are densely distributed within the real set. E GLYPH<11> ect of the selection rule. Finally, we examine the ef- fect of the selection rule, m = n GLYPH<6> 2 k , noted in sec- tion 2 on the work distribution P ( W ). The existence of this selection rule limits the possible values that the step sizes of the cumulative distribution GLYPH<30> ( W ) can take. For simplicity, we consider a harmonic oscillator which is initially prepared in a given energy eigenstate j n$_{0}$ i . A direct consequence of Eq. (39) and of the selection rule is then that the allowed work values are, where k are integers. For two neighboring work values, the step size is constant and reads, In Fig. 4 we illustrate the e GLYPH<11> ect of the selection rule by plotting the corresponding cumulative distribution with the same parameters as in Fig. 1. We develop in the present section a perturbative approach to determine the quantum work distribution with the Kronecker-delta GLYPH<14>$_{m}$$_{;}$$_{n}$ . Equation (46) shows that only transitions that satisfy m = n GLYPH<6> 2 are possible, which is precisely the selection rule noted earlier. It should be emphasized that this selection rule is at vari- ance with usual textbook examples which contain the selection rule m = n GLYPH<6> 1. The latter applies to a quan- tum oscillator driven by a small perturbation linear in the position, whereas we here deal with a perturbation (44) which is quadratic in x . The full expression of the transitions probabilities (45) that follow from Eq. (46) is given in Appendix C. where $_{ℏ}$!$_{m}$$_{;}$$_{n}$ = E 0 m GLYPH<0> E 0 n denotes the di GLYPH<11> erence of the unperturbed energy eigenvalues and GLYPH<10> t m ; n = h m j GLYPH<10>$_{t}$ j n i are the corresponding interaction matrix el- ements. By expressing the position operator, x = p $_{ℏ}$= 2 M !$_{0}$ $^{GLYPH<16>}$a y + a $^{GLYPH<17>}$, in terms of the usual ladder oper- ators, a $^{y}$j n i = p n + 1 j n + 1 i and a j n i = p n j n GLYPH<0> 1 i , the interaction matrix elements can be written explicitly as GLYPH<10> t m ; n = GLYPH<0> ℏ 4 !$_{0}$ $^{GLYPH<16>}$! 2 0 GLYPH<0> ! 2 t$^{GLYPH<17>}$ h p n + 1 p n + 2 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{+}$$_{2}$ + (2 n + 1) GLYPH<14>$_{m}$$_{;}$$_{n}$ + p n GLYPH<0> 1 p n GLYPH<14>$_{m}$$_{;}$$_{n}$$_{GLYPH<0>}$$_{2}$ $^{i}$; (46) W = $_{ℏ}$( !$_{1}$ GLYPH<0> !$_{0}$ ) n$_{0}$ + 1 2$^{!}$ GLYPH<6> $_{ℏ}$!$_{1}$ 2 k ; (40) GLYPH<1> W = 2 $_{ℏ}$!$_{1}$ : (41) 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<12>}$$^{GLYPH<12>}$$^{GLYPH<12>}$$^{GLYPH<12>}$ GLYPH<12> GLYPH<12> 2 ; (45) GLYPH<10>$_{t}$ = GLYPH<0> M 2 $^{GLYPH<16>}$! 2 0 GLYPH<0> ! 2 t$^{GLYPH<17>}$ x 2 : (44) H$_{0}$ = p 2 2 M + M 2 ! 2 $_{0}$x $^{2}$; (43) H$_{t}$ = H$_{0}$ + GLYPH<10>$_{t}$ ; (42) W = "$_{1}$ m + 1 2$^{!}$ GLYPH<0> "$_{0}$ n + 1 2$^{!}$ : (39) 7 5. Perturbation theory 5.1. Selection rule Figure 4: Cumulative work distribution (38) for an initially pure state j n$_{0}$ i , illustrating the sum rule, m = n GLYPH<6> 2 k , which limits the possible sizes of the work steps ( n$_{0}$ = 3, "$_{0}$ = 1, "$_{1}$ = 1 : 5 and Q GLYPH<3> = 3). 0 10 20 30 40 W 0.2 0.4 0.6 0.8 1.0 $^{f}$H W L P ( W ). We treat in detail the case of a small anharmonic correction to the potential, as well as the e GLYPH<11> ect of a small external fluctuating electric field on a charged harmonic oscillator. Both situations are motivated by the exper- imental study of the quantum work statistics in linear Paul traps [16]. We begin by giving a simple perturba- tive derivation of the sum rule of section 2, which does not require the full solution of the Schr¨ odinger equation.