of the angular frequency !$_{t}$ . The general form of the propagator can be determined from  $_{t}$ ( x ) by noting that 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]. 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. It is explicitly given by [19] 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 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, 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, and denote the complex conjugate of a number z by z $^{GLYPH<3>}$. The quadratic generating function of GLYPH<30> $^{t}$$_{n}$(x ) is the Schr¨odinger equation for the oscillator can be re- duced to a system of three coupled di GLYPH<11> erential equations for the time-dependent coe GLYPH<14> cients a$_{t}$ , b$_{t}$ and c$_{t}$ , which can be calculated by a Fourier expansion of the left-hand side of Eq. (11). The generating function p GLYPH<28> m ; n is then defined as The nonlinear equation (4) is of the Riccati type and is therefore solvable. It can be mapped to the equation of motion of a classical time-dependent harmonic oscilla- tor via a$_{t}$ = M ˙ X$_{t}$ = X$_{t}$ , and we obtain Combining Eqs. (10) and (11), we find The ( u ; v )-dependence of the generating function P ( u ; v ) remains the same for all possible transformations !$_{t}$ . With the solutions of (4)-(7) the Gaussian wave function  $_{t}$ ( x ) (3) is fully characterized by the time-dependence P ( u ; v ) = X m ; n u$^{m}$v$^{n}$p GLYPH<28> m ; n : (12) P ( u ; v ) = p 2 $^{p}$Q $^{GLYPH<3>}$(1 GLYPH<0> u $^{2}$)(1 GLYPH<0> v $^{2}$) + (1 + u $^{2}$)(1 + v $^{2}$) GLYPH<0> 4 uv : (13) d 2 dt 2 X$_{t}$ + ! 2 t X$_{t}$ = 0 : (7) dc$_{t}$ dt = i$_{ℏ}$ M a$_{t}$ GLYPH<0> 1 M b 2 $_{t}$: (6) 1 M da$_{t}$ dt = GLYPH<0> 1 M 2 a 2 t GLYPH<0> ! 2 $_{t}$; (4) db$_{t}$ dt = GLYPH<0> 1 M a$_{t}$b$_{t}$ ; (5)  $_{t}$ ( x ) = exp$^{GLYPH<18>}$ i 2 ℏ $^{h}$a$_{t}$ x 2 + 2 b$_{t}$x + c$_{t}$ i GLYPH<19> ; (3) 1 X n = 0 u n GLYPH<30> $^{t}$$_{n}$(x ) GLYPH<30> GLYPH<3> $^{t}$$_{n}$(x$_{0}$ ) = s M !$_{t}$ $_{ℏ}$GLYPH<25> (1 GLYPH<0> u $^{2}$) exp$^{0}$ B B $_{@}$GLYPH<0> M !$_{t}$ ℏ (1 + u $^{2}$)( x 2 + x 2 $_{0}$) GLYPH<0> 4 uxx$_{0}$ 2(1 GLYPH<0> u $^{2}$) 1 C C A ; (11) p GLYPH<28> m ; n = GLYPH<12> GLYPH<12> GLYPH<12> Z dx$_{0}$ Z dx GLYPH<30> GLYPH<3> GLYPH<28> $_{m}$(x ) U$_{GLYPH<28>;}$$_{0}$ ( x j x$_{0}$ ) GLYPH<30> 0 $_{n}$(x$_{0}$ )$^{GLYPH<12>}$ GLYPH<12> GLYPH<12> 2 ; (10) H$_{t}$ = p 2 2 M + M 2 ! 2 t x 2 : (2) U$_{t}$$_{;}$$_{t}$$_{0}$ = r M 2 GLYPH<25> i$_{ℏ}$X$_{t}$ exp$^{ }$ iM 2 $_{ℏ}$X$_{t}$ h ˙ X$_{t}$x 2 GLYPH<0> 2 xx$_{0}$ + Y$_{t}$x 2 0$^{i}$$^{!}$ ; (9)  $_{t}$ ( x ) = Z dx$_{0}$ U$_{t}$$_{;}$$_{t}$$_{0}$ ( x j x$_{0}$ )  $_{t}$$_{0}$ ( x$_{0}$ ) : (8) 2.1. Method of generating functions 2. Time-dependent harmonic oscillator 2