Long-Range Ising Interactions Mediated by λϕ^{4} Fields: Probing the Renormalization of Sound in Crystals of Trapped Ions

Abstract

The generating functional of a self-interacting scalar quantum field theory (QFT), which contains all the relevant information about real-time dynamics and scattering experiments, can be mapped onto a collection of multipartite-entangled two-level sensors via an interferometric protocol that exploits a specific set of source functions. Although one typically focuses on impulsive δ-like sources, as these give direct access to n-point Feynman propagators, we show in this work that using always-on harmonic sources can simplify substantially the sensing protocol. In a specific regime, the effective real-time dynamics of the quantum sensors can be described by a quantum Ising model with long-range couplings, the range and strength of which contains all the relevant information about the renormalization of the QFT, which can now be extracted in the absence of multipartite entanglement. We present a detailed analysis of how this sensing protocol can be relevant to characterize the long-wavelength QFT that describes quantized sound waves of trapped-ion crystals in the vicinity of a structural phase transition, opening a new route to characterize the associated renormalization of sound.