  Quantum sensing is commonly described as a constrained optimization problem:
maximize the information gained about an unknown quantity using a limited
number of particles. Important sensors including gravitational-wave
interferometers and some atomic sensors do not appear to fit this description,
because there is no external constraint on particle number. Here we develop the
theory of particle-number-unconstrained quantum sensing, and describe how
optimal particle numbers emerge from the competition of particle-environment
and particle-particle interactions. We apply the theory to optical probing of
an atomic medium modeled as a resonant, saturable absorber, and observe the
emergence of well-defined finite optima without external constraints. The
results contradict some expectations from number-constrained quantum sensing,
and show that probing with squeezed beams can give a large sensitivity
advantage over classical strategies, when each is optimized for particle
number.
