Luttinger liquid under non-Hermitian quantum sensing

Abstract The interplay between dissipation and interactions could give rise to exotic phenomena. In this work, we apply the non-Hermitian (NH) linear response theory to investigate the density response of a spinless Luttinger liquid under local NH perturbations. We show that NH backward scattering results in interference between correlation functions arising from dissipation and Langevin noise terms, leading to fluctuations in the induced density rate of change as the Luttinger parameter K varies. For odd integer values of K, this rate vanishes, indicating immunity to dissipation. We demonstrate that the steady-state induced density is predominantly determined by Langevin noise, underscoring the necessity of including Langevin noise terms in NH Hamiltonians. For attractive interactions, the response to NH perturbations grows exponentially and exhibits a $\pi /2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:math> phase shift, indicating stronger sensitivity to NH perturbations compared to Hermitian ones. Additionally, we show that NH forward scattering leads to a decay of the induced density. We refer to these peculiar behaviors of density fluctuations under NH perturbations as anomalous Friedel oscillations. Our findings offer new prospects for sensing and understanding dissipation and interactions.