Exploring the non-equilibrium fluctuation relation for quantum mechanical tunneling of electrons across a modulating barrier

Abstract

We study the validity of the fluctuation relation for a non-equilibrium quantum mechanical system, viz., electrons quantum mechanically tunneling across a periodically modulating barrier. Experimentally this system is realized by measuring the fluctuation in the tunneling current between an STM tip and a vibrating gold film deposited on a piezo crystal. The long time series of tunneling signal shows large positive and negative fluctuations. The analysis shows that over finite observation time intervals, the probability distribution of the average rate of work done has a positive mean. About this mean, the distribution is broad and it is spread over not only positive but also negative values. These positive and negative values correspond to work done either on the electron by the external drive or work done by the electron against the drive, as it tunnels across the modulating barrier, respectively. For different driving frequencies, we show that the probability distribution satisfies the non-equilibrium fluctuation relation (NEFR). Thus, we prove that the NEFR is valid for a driven quantum mechanical system. For this tunneling process, we determine the large deviation function (LDF), which is related to NEFR. We see changes in the shape of the LDF as a function of the drive frequency, although NEFR is valid at all these frequencies. Measuring dissipation associated with microscopic irreversible trajectories in non-equilibrium quantum mechanical systems is a challenging task. Here we use NEFR also to obtain a measure of the dissipation associated with the electron tunneling across the modulated barrier.