  We study the sensitivity of a collisional single-atom probe for ultracold
gases. Inelastic spin-exchange collisions map information about the gas
temperature T or external magnetic field B onto the quantum spin-population of
single-atom probes, and previous work showed enhanced sensitivity for
short-time nonequilibrium spin dynamics [1]. Here, we numerically investigate
the steady-state sensitivity of such single-atom probes to various observables.
We find that the probe shows distinct sensitivity maxima in the (B, T )
parameter diagram, although the underlying spin-exchange rates scale
monotonically with temperature and magnetic field. In parameter space, the
probe generally has the largest sensitivity when sensing the energy ratio
between thermal energy and Zeeman energy in an externally applied magnetic
field, while the sensitivity to the absolute energy, i.e., the sum of kinetic
and Zeeman energy, is low. We identify the parameters yielding sensitivity
maxima for a given absolute energy, which we can relate to a direct comparison
of the thermal Maxwell-Boltzmann distribution with the Zeeman-energy splitting.
We compare our equilibrium results to nonequilibrium experimental results from
a single-atom quantum probe, showing that the sensitivity maxima in parameter
space qualitatively prevail also in the nonequilibrium dynamics, while a
quantitative difference remains. Our work thereby offers a microscopic
explanation for the properties and performance of this single-atom quantum
probe, connecting thermodynamic properties to microscopic interaction
mechanisms. Our results pave the way for optimization of quantum-probe
applications in (B, T ) parameter space beyond the previously shown boost by
nonequilibrium dynamics.
