Phonon Scharnhorst effect: Acoustic analogue of the vacuum-fluctuation-boosted speed of light

The Scharnhorst effect in quantum electrodynamics predicts a superluminal velocity shift $\delta c>0$ at low photon frequencies due to vacuum fluctuations between two mirrors. Although theoretically robust (no causality violation), the effect is extraordinarily small ($\delta c\mathrm{/}c\approx {10}^{-32}$ for mirrors $1\mu \mathrm{m}$ apart) and outside current experimental reach. Here we propose a condensed matter analogue, replacing the photons in the electromagnetic vacuum by phonon excitations of a Bose-Einstein condensate. In an idealized geometry, the two-dimensional surface of a cylinder (circumference $L$) at zero temperature, the axial speed of sound is increased due to zero-point fluctuations of the condensate by $\delta c\mathrm{/}c\simeq 5\times {10}^{-3}\tilde{g}$ at $L\simeq 2\xi$, to first order in the dimensionless interaction constant $\tilde{g}\simeq {10}^{-1}$ of the Bose gas of healing length $\xi \simeq 1\mu \mathrm{m}$. This is 28 orders of magnitude larger than the photonic effect, basically because phonons have a self-interaction while photons only interact via massive electrons. We discuss the prospects for observation of the acoustic Scharnhorst effect, the main restriction being the low-temperature requirement.