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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. 

## 5. Perturbation theory 

We develop in the present section a perturbative approach to determine the quantum work distribution 

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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. 

## 5.1. Selection rule 

We start by rewriting Hamiltonian (2) as 

where we have introduced the unperturbed Hamiltonian 

and a ”perturbation” term 

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], 

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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 

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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. 

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