Thermodynamics of Nanoelectromechanics

Abstract

Taking inspiration from nanoelectromechanical experiments, we introduce a microscopic toy model consisting of a resonant-level quantum dot coupled to two electrodes and a single-mode oscillator. We derive a Fokker-Planck equation that describes the evolution of the oscillator in the quasi-adiabatic limit. The oscillator can be treated as a quantum flywheel, converting electrical power into stored mechanical energy. Calculating the non-equilibrium steady state of the oscillator, we find that energy dependence in the dot-reservoir coupling is required to produce self-sustained oscillations, corroborating previous findings. We investigate the self-sustained oscillations through the lens of work extraction. We characterise the threshold for self-sustained oscillations using two approaches to quantifying work deposition in non-equilibrium quantum thermodynamics: ergotropy and non-equilibrium free energy. We find that ergotropy acts as an order parameter for self-sustained oscillations, becoming non-zero only when a population inversion is achieved.

Using this toy model, we investigate the thermodynamics of autonomous quantum clocks. The periodic mechanical motion of the system behaves as a clockwork, similar to the swinging of a pendulum, while induced oscillations in the electrical current can be used to read out the ticks. We derive formulae calculating the current and current noise in the nanoelectromechanical model, which allows us to infer statistical properties of the clock's ticks from the current auto-correlation function. Similar to previous studies, we find that the electromechanical clock exhibits a tradeoff between accuracy, resolution, and dissipation. However, our work goes beyond the distribution of individual ticks, investigating how clock accuracy varies over different integration times by computing the Allan variance. We observe non-monotonous features of the Allan variance, which can be attributed to temporal correlations between ticks; these correlations yield a precision advantage for timekeeping over the timescales that the correlations persist.

Finally, we investigate the quantum Mpemba effect from the perspective of non-equilibrium quantum thermodynamics by studying the relaxation dynamics of quantum systems coupled to a Markovian heat bath, as described by Davies maps. In discrete systems starting from a state with coherences in the energy eigenbasis, we demonstrate that an exponential speedup to equilibrium will always occur if the state is transformed to a diagonal state in the energy eigenbasis, provided that a complex eigenvalue defines the spectral gap of the generator. When the transformed state has a higher non-equilibrium free energy, we argue using thermodynamic reasoning that this is a genuine quantum Mpemba effect. Furthermore, we show how a unitary transformation on an initial state can always be constructed to yield the effect and demonstrate our findings by studying the dynamics of both the non-equilibrium free energy and the irreversible entropy production in single and multi-qubit examples. In continuous systems, we explore the effect in classical and quantum systems performing an eigenmode decomposition on the generator. We investigate an ergotropic Mpemba effect in quantum Gaussian states; displaced states are found to be a better resource for storing ergotropy than squeezed states due to an exponential slowdown in the relaxation of the ergotropy in displaced states.