Resources for bosonic metrology: quantum-enhanced precision from a superselection rule perspective

Abstract: Quantum optics and atomic systems are prominent platforms for exploiting quantum-enhanced precision in parameter estimation. However, not only are quantum optical and atomic systems often treated separately, but even within quantum optics, identifying optimal probes (quantum states) and evolutions (parameter-dependent dynamics) typically relies on case-by-case analyses. Mode, and sometimes only particle entanglement, can yield quantum enhancement of precision in continuous- and discrete-variable regimes, yet a clear connection between these regimes remains elusive. In this work, we present a unified framework for quantum metrology that encompasses all known precision-enhancing regimes using bosonic resources. We introduce a superselection rule compliant representation of the electromagnetic field that explicitly incorporates the phase reference, enforcing total particle number conservation. This approach provides a description of the electromagnetic field which is formally equivalent to the one employed in atomic systems, and we show how it encompasses both the discrete and the continuous limits of quantum optics. Within this framework, we consistently recover established results while offering a coherent physical interpretation of the quantum resources responsible for precision enhancement. Moreover, we develop general strategies to optimize precision using arbitrary multimode entangled probe states. Finally, our formalism readily accommodates noise, measurement strategies and non-unitary evolutions, extending its applicability to realistic experimental scenarios.