Casimir-Lifshitz interaction between bodies integrated in a microelectromechanical/nanoelectromechanical quantum damped oscillator

TL;DR

The paper develops a theory of Casimir-like, fluctuation-induced forces for bodies embedded in a macroscopic quantum damped oscillator, showing how distance-dependent $\Omega(d)$ and $\gamma(\omega,d)$ yield a force that, under broad conditions, is dominated by low-frequency fluctuations and can be captured using an Ohmic/lumped-element description. It derives a general Matsubara-sum expression for the oscillator-induced force split into $f_{\Omega}$ and $f_{\gamma}$, analyzes limiting cases (including Drude damping) and demonstrates finite, measurable circuit-induced contributions in simple RLC geometries. The work connects to microscopic Zwanzig-Caldeira-Leggett theory, clarifies the role of frequency dispersion in preventing divergences in the lumped description, and provides concrete estimates showing when circuit-induced forces could be detected as corrections to the conventional Casimir-Lifshitz force in planar, spherical, and inductive-capacitive setups. Overall, it establishes conditions under which lumped-element descriptions are valid for fluctuation-induced forces and highlights practical geometries and parameters where circuit-induced effects may be observable in condensed-matter nanodevices.

Abstract

A theory is proposed for the component of the Casimir-like force that arises between bodies embedded in a macroscopic quantum damped oscillator. When the oscillator's parameters depend on the distance between the bodies, the oscillator-induced Casimir-like force is generally determined by a broad spectral range extending to high frequencies, limited by the frequency dispersion of the damping function. Here it is shown that there is a large class of systems in which the low-frequency range dominates the forces. This allows for the use of the Ohmic approximation, which is crucial for extending the theory to the lumped element description of fluctuation-induced forces in electrical circuits. Estimates of the circuit-induced Casimir-Lifshitz force suggest that under certain conditions it can be identified experimentally due to its dependence on various circuit elements.

Figures (1)

• Figure 1: Simplest RLC circuits connected in series (a) and in parallel (b).