A Quantum Fluctuation Theorem

We consider a quantum system strongly driven by forces that are periodic in time. The theorem concerns the probability P (e) of observing a given energy change e after a number of cycles. If the system is thermostated by a (quantum) thermal bath, e is the total amount of energy transferred to the bath, while for an isolated system e is the increase in energy of the system itself. Then, we show that P (e)/P (−e) = e, a parameter-free, model-independent relation. In the past few years there has been a renewed interest in the study of quantum systems out of equilibrium, to a large extent stimulated by the design of new experimental settings and by the construction of new devices. If a system is well out of equilibrium, as for example when it is strongly driven by periodic forces, then linear response theory (understood as linear perturbations around the Gibbs measure) is insufficient. Even in the context of classical mechanics not many generic results are available beyond linear response. An interesting new development consists of a number of relations for strongly out of equilibrium systems, mainly regarding the distribution of work and entropy production. The first of such fluctuation theorems was discovered by Evans et.al [1], who understood that the basic ingredient was time-reversal symmetry. Two important further steps, made by Gallavotti and Cohen widened the scope and interest of the subject. On the one hand, it was realized [2] that the fluctuation theorems are indeed the far from equilibrium generalisations of the well-known equilibrium theorems (fluctuation-dissipation and Onsager reciprocity). Most intriguingly, a byproduct of their proof [3] was that, just as the validity of the fluctuation-dissipation relation is a strong indication of equilibration, the fact that a fluctuation formula holds in a driven stationary system strongly hints that the system can be considered ‘as ergodic as possible’ — with all the implications this entails. These results concern deterministic systems. If instead a finite system is in contact with a stochastic thermal bath, then the ‘ergodicity’ questions become trivial, and all the fluctuation formulae are extremely simple to prove [4, 5, 6].