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Motivated by recent advances in the creation of few-body atomic Fermi gaseswith attractive interactions, we study theoretically the few-to-many particlecrossover of pair excitations, which for large particle numbers evolve intomodes describing amplitude fluctuations of the superfluid order parameter (the"Higgs" modes). Our analysis is based on a Richardson-type pairing model thatcaptures interactions between time-reversed fermion pairs. The model isintegrable, which allows a systematic quantitative study of the few-to-manyparticle crossover with only minor numerical effort. We first establish aparity effect in the ground state energy, which is quantified by a so-calledMatveev-Larkin parameter that generalizes the pairing gap to mesoscopicensembles and which behaves quantitatively differently in a few-body and amany-body regime. The crossover point for this quantity is widely tunable as afunction of interaction strength. We then compute the excitation spectrum anddemonstrate that the pair excitation energy shows a minimum that deepens withincreasing particle number and shifts to smaller interaction strengths,consistent with the finite-size precursor of a quantum phase transition to asuperfluid state. We extract a critical finite-size scaling exponent thatcharacterizes the decrease of the gap with increasing particle number.

Few-to-many particle crossover of pair excitations in a superfluid