Recently developed techniques allow for simultaneous measurements of the positions of all ultra-cold atoms in a trap with high resolution. Each such single-shot experiment detects one element of the quantum ensemble formed by the cloud of atoms. Repeated single-shot measurements can be used to determine all correlations between particle positions as opposed to standard measurements that determine particle density or two-particle correlations only. In this paper we discuss the possible outcomes of such single-shot measurements in the case of cloud of ultra-cold noninteracting Fermi atoms. We show that the Pauli exclusion principle alone leads to correlations between particle positions that originate from unexpected spatial structures formed by the atoms.
The best known manifestation of the Fermi-Dirac statistics is the Pauli exclusion principle: no two identical fermions can occupy the same one-particle state. This principle enforces high-order correlations in systems of many identical fermions and is responsible for a particular geometric arrangement of trapped particles even when all mutual interactions are absent. These geometric structures, called Pauli crystals, are predicted for a system of N identical atoms trapped in a harmonic potential. They emerge as the most frequent configurations in a collection of single-shot pictures of the system. Here we study how fragile Pauli crystals are when realistic experimental limitations are taken into account. The influence of the number of single-shots pictures available to analysis, thermal fluctuations and finite efficiency of detection are considered. The role of these sources of noise on the possibility of experimental observation of Pauli crystals is shown and conditions necessary for the detection of the geometrical arrangements of particles are identified.
Exchange statistics endows identical particles with an effective "statistical potential", whose familiar exact form is two-body and purely repulsive for fermions. Here we construct an exact collective many-body form: the thermodynamics of
N trapped ideal fermions maps onto classical distinguishable particles governed by a single potential -- exactly for harmonic confinement at all temperatures, and to leading semiclassical order for arbitrary potentials. The associated force separates canonically into pairwise contributions, which for
N≥3 can turn attractive, governed by a simple geometric criterion. Classical minimization reproduces observed few-body Pauli-crystal symmetries and agrees with the
N=55 ground-state probability maximum at sub-percent shell accuracy. Heating drives discrete structural transitions accompanied by a crossover of the strongest force from attractive to repulsive. Both the potential and its forces are directly computable from existing single-shot imaging data, turning quantum exchange into measurable classical mechanics.