Quantum work statistics of linear and nonlinear parametric oscillators arXiv:1001.3055v1 [cond-mat.stat-mech] 18 Jan 2010 Sebastian De GLYPH<11> ner $^{GLYPH<3>}$, Obinna Abah, Eric Lutz Department of Physics, University of Augsburg, D-86135 Augsburg, Germany We consider the nonequilibrium work distribution of a quantum oscillator with modulated angular frequency. We examine the discrete-to-continuous transition of the distribution as the temperature and the degree of nonadiabaticity of the frequency transformation are increased. We further develop a perturbative approach to analyze the e GLYPH<11> ect of weak quartic anharmonicities, as well as of a random electric field on a charged oscillator. We find in both cases that the degree of nonadiabaticity is enhanced by the perturbation. Keywords: nanothermodynamics, nonequilibrium statistics, ion traps PACS: 05.30.-d, 05.70.Ln, 05.40.-a The modern trend of miniaturization leads to the de- velopment of smaller and smaller devices, such as nano- engines and molecular motors [1, 2, 3]. On these very short length scales, thermal as well as quantum fluc- tuations become important, and usual thermodynamic quantities, such as work and heat, acquire a stochastic nature. The traditional framework of thermodynamics, which neglects fluctuations, thus fails to provide a com- plete description of nanosystems. Extensions of the sec- ond law to these small systems have been recently intro- duced in the form of the fluctuation theorem [4, 5] and the Jarzynski equality [6]. The Jarzynski work relation, state at the end of the transformation; for a general pro- cess, the system may be arbitrarily far from equilibrium. For the case of an open system, the final free energy is that of the asymptotic state reached by the system after equilibration with the heat bath, while for an isolated system, it corresponds to that of the hypothetical equi- librium state that would be obtained after coupling to the bath. In Eq. (1) the average of the exponentiated to- tal work is taken over an ensemble of many realizations of the same process, the nonequilibrium fluctuations of the work being characterized by the probability distri- bution P ( W ). The classical Jarzynski equality has been verified experimentally using stretched biopolymers [8], a mechanical torsion pendulum [9], and a colloidal par- ticle in an anharmonic trap [10]. At the quantum level, the Jarzynski relation has been shown to hold for iso- lated [11, 12] as well as for open systems [13, 14], but an experimental investigation is still lacking. This is partly due to the fact that quantum work is not an observable in the usual sense, as it is not described by a Hermi- tian operator in Hilbert space, but by a two-time correla- tion function [15]. A measurement scheme to determine the full quantum work statistics P ( W ), and hence verify the quantum Jarzynski equality, using ultracold trapped ions has been proposed in Ref. [16]. We have recently explicitly evaluated the continuous envelop of the work distribution P ( W ) for a time-dependent harmonic oscil- lator, an analytically solvable model [17]. In the gen- eral quantum case, however, the work distribution is dis- crete, reflecting the quantized nature of the energy spec- permits determination of an equilibrium free energy dif- ference GLYPH<1> F from the fluctuations of the nonequilibrium work W done on the system during an arbitrary trans- formation, quasistatic or not. The classical Jarzynski equality (1) is valid for both isolated and open systems [7]. It should be noted that the system is initially as- sumed to be in a thermal state with inverse temperature GLYPH<12> . However, it is not required to be in an equilibrium $^{GLYPH<10>}$exp ( GLYPH<0> GLYPH<12> W ) GLYPH<11> = exp ( GLYPH<0> GLYPH<12> GLYPH<1> F ) (1) $^{GLYPH<3>}$Corresponding author Email address: sebastian.deffner@physik.uni-augsburg.de (Sebastian De GLYPH<11> ner ) URL: www.physik.uni-augsburg.de/~deffnese (Sebastian De GLYPH<11> ner ) Preprint submitted to Elsevier October 26, 2018 1. Introduction Abstract