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The generating functional of a self-interacting scalar quantum field theory(QFT), which contains all the relevant information about real-time dynamics andscattering experiments, can be mapped onto a collection ofmultipartite-entangled two-level sensors via an interferometric protocol thatexploits a specific set of source functions. Although one typically focuses onimpulsive delta-like sources, as these give direct access to $n$-point Feynmanpropagators, we show in this work that using always-on harmonic sources cansimplify substantially the sensing protocol. In a specific regime, theeffective real-time dynamics of the quantum sensors can be described by aquantum Ising model with long-range couplings, the range and strength of whichcontains all the relevant information about the renormalisation of the QFT,which can now be extracted in the absence of multi-partite entanglement. Wepresent a detailed analysis of how this sensing protocol can be relevant tocharacterise the long-wavelength QFT that describes quantised sound waves oftrapped-ion crystals in the vicinity of a structural phase transition, openinga new route to characterise the associated renormalisation of sound.

Long-range Ising interactions mediated by $\lambda\phi^4$ fields: probing the renormalisation of sound in crystals of trapped ions