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trum. In the present article, we provide a detailed dis- cussion of the transition from discrete to continuous dis- tributions by introducing the cumulative work function. We further analyze the e GLYPH<11> ect of a weak anharmonicity on P ( W ) by using time-dependent perturbation theory. We finally examine the e GLYPH<11> ect of a weak random electric field on the work statistics of a charged harmonic oscil- lator, a case of high relevance for the thermodynamic study of realistic modulated Paul traps [18]. 

## 2. Time-dependent harmonic oscillator 

We begin by reviewing the solution of the quantum parametric oscillator, following the method developed by Husimi [19]. We use this opportunity to extend the results of Ref. [19] and correct some misprints. The Hamiltonian of a quantum mechanical harmonic oscil- lator with time-dependent angular frequency !$_{t}$ is 

The parameterization !$_{t}$ starts at initial value !$_{0}$ at t = t$_{0}$ and ends at final value !$_{1}$ at t = GLYPH<28> . We denote by GLYPH<30> t n the instantaneous eigenfunctions and by E t n = $_{ℏ}$!$_{t}$ ( n + 1 = 2) the instantaneous eigenvalues of the quadratic Hamiltonian (2). The dynamics of the har- monic oscillator is Gaussian for any !$_{t}$ . By introducing the Gaussian wave function ansatz, 

the Schr¨odinger equation for the oscillator can be re- duced to a system of three coupled di GLYPH<11> erential equations for the time-dependent coe GLYPH<14> cients a$_{t}$ , b$_{t}$ and c$_{t}$ , 

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The nonlinear equation (4) is of the Riccati type and is therefore solvable. It can be mapped to the equation of motion of a classical time-dependent harmonic oscilla- tor via a$_{t}$ = M ˙ X$_{t}$ = X$_{t}$ , and we obtain 

With the solutions of (4)-(7) the Gaussian wave function  $_{t}$ ( x ) (3) is fully characterized by the time-dependence 

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of the angular frequency !$_{t}$ . The general form of the propagator can be determined from  $_{t}$ ( x ) by noting that 

It is explicitly given by [19] 

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where X$_{t}$ and Y$_{t}$ are solutions of Eq. (7) satisfying the boundary conditions X$_{0}$ = 0, ˙ X$_{0}$ = 1 and Y$_{0}$ = 1, ˙ Y$_{0}$ = 0, the latter being an expression of the commutation rela- tion between position and momentum. 

## 2.1. Method of generating functions 

The time variation of the angular frequency (2) in- duces transitions between di GLYPH<11> erent energy eigenstates of the oscillator. We are thus interested in the transition probabilities p GLYPH<28> m ; n from an initial state j n i at t$_{0}$ = 0 to a final state j m i at t = GLYPH<28> . In the following, we use the method of generating functions to evaluate p GLYPH<28> m ; n [19]. We start with the definition, 

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and denote the complex conjugate of a number z by z $^{GLYPH<3>}$. The quadratic generating function of GLYPH<30> $^{t}$$_{n}$(x ) is 

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which can be calculated by a Fourier expansion of the left-hand side of Eq. (11). The generating function p GLYPH<28> m ; n is then defined as 

Combining Eqs. (10) and (11), we find 

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The ( u ; v )-dependence of the generating function P ( u ; v ) remains the same for all possible transformations !$_{t}$ . 

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