The power-spectrum tensor in steady-state systems and its role in quantum friction

TL;DR

The paper develops a general framework for the power-spectrum tensor $\underline{S}(\omega)$ of quantum systems in nonequilibrium steady states, extending beyond traditional equilibrium fluctuation-dissipation relations. It derives a structural decomposition of $\underline{S}(\omega)$ into a real symmetric part and a rotation-like term, relates it to the susceptibility $\underline{\alpha}(\omega)$, and introduces a nonequilibrium correction $\underline{J}(\omega)$ that quantifies deviations from the standard FDT. The authors then apply the framework to quantum friction, showing how the power spectrum and the electromagnetic environment determine a velocity-dependent friction force via the Green tensor $\underline{G}_{\Im}$, with symmetry arguments enforcing $F_{\rm fr}(v)$ to be odd in $v$. An exactly solvable 3D isotropic oscillator model illustrates these concepts, providing explicit expressions for $\underline{S}(\omega,v)$ and validating the general results, including crossing relations and the nontrivial role of $\underline{J}(\omega)$. Overall, the work offers a robust, microscope-free approach to analyze nonequilibrium fluctuation phenomena and thermodynamic consistency in fluctuation-induced forces.

Abstract

We derive and classify properties of the power-spectrum tensor for systems in general steady-states, including stationary states not necessarily corresponding to equilibrium configurations. We establish a rigorous connection between the power-spectrum tensor and other quantities that characterize these systems, providing a systematic comparison with their equilibrium counterparts. As a physical application, we investigate the problem of quantum friction, describing the contactless quantum-electrodynamic drag acting on a particle moving in close proximity to material bodies at zero temperature. Specifically, we demonstrate how including additional information about the system's physical properties facilitates the derivation of more precise constraints on the power spectrum and its functional dependencies.

Figures (2)

• Figure 1: a: Sketch of the generic configuration analyzed in this work. A large, potentially driven quantum system composed of a subsystem interacting with an environment. The environment itself can be resulting from the interaction of different subsystems. b: A possible physical implementation of this generic scheme: An externally driven atom coupled to a complex quantum environment, emerging from the self-consistent dynamics of the electromagnetic field in the presence of material bodies. The energy entering the system through the drive can be dissipated away in form of electromagnetic radiation and heat within the material, allowing the systems to reach a nonequilibrium steady state.
• Figure 2: Schematic representation of a particle moving in vacuum close to an arrangement of different, macroscopic objects. The particle moves on a straight line with direction $\mathbf{n}$, which coincides with the direction of translational invariance of the arrangement of macroscopic bodies. In this setup, the quantum fluctuations of the electromagnetic field behave as a viscous medium and the particle's motion is affected by a corresponding drag force. An external force acting on the particle's center of mass (not shown) keeps the motion at constant velocity $v$. The power generated by the external drive is dissipated in the electromagnetic plus dissipative-matter environment Intravaia15Reiche20c.