Emergent force and work fluctuations in a bilayer superfluid Bose-Fermi mixture in mixed dimensions

We investigate a system of two-atomic species in mixed dimensions, in which one species is spread in a three-dimensional space and the other species is confined in two parallel layers. The presence of atoms in three-dimensions creates an induced potential for the ones confined in layers. Depending on the scattering (coherence) length and the layer separation, the formation ofp-wave pairing within the same layer ors-wave pairing between different layers has been suggested. It was shown that these pairs cannot coexist when time-reversal symmetry is on, and there appears a transition fromp-wave tos-wave as the ratio of the layer separation and the effective scattering length decreases. Here, we study the thermodynamical signatures of such transition and find that with the formation of the inter-layer pairing, an emergent force to be present at the critical point and show that it can be derived from the ground state energy. This result offers a tool for experimentally realizing such transitions, and can find notable potential in the field of quantum-thermodynamics.