• fermionic superfluidity
• multimode cavity QED
• dissipation engineering

In recent years, increasing interest has been given to ultracold-atomic platforms. Conceived as quantum simulators for problems in condensed matter and fundamental physics, these systems allow to physically implement in highly controlled experiments specific Hamiltonian models. Indeed, the experimental setups allow for tuning of interaction strengths and range, dimensionality and temperature. This control, in conjunction with suited probing tools enabling high-resolution detection, makes this quantum technology ideal to explore the boundaries of quantum science.

A subset of these relevant physics problems can be traced back to lattice-based models. These are implemented in experimental setups by superimposing to the atomic cloud an optical lattice, typically created by the interference of several laser beams. These lattices already offer the possibility of exploring a wide range of physical phenomena that includes different geometries or dimensionality, disorder, or the action of synthetic gauge fields.

Despite this flexibility, the lattice potentials are essentially insensitive to particle number and their state. Therefore, optical lattices result to be inadequate to describe certain emergent phenomena, like those involving phonon-like effects, dynamics of glassy media and supersolids. To overcome this limitation, one option is to make the atoms interact with a dynamical potential, such as in an optical cavity. This is the origin of the field of quantum many-body cavity QED.

The specific platform of atoms in cavity QED is highly versatile, providing a number of desirable features and applications, ranging from precision measurements for metrological use, to simulations of many-body systems and engineering of dynamical phase transitions in out-of-equilibrium conditions. Indeed, the effective interaction strength among the atoms, mediated by the cavity photons, can be engineered by exploiting the many round-trips of the photons in the resonator, adding a geometric factor called cooperativity. In addition, atoms can be coupled to one or many modes of the cavity. In a single-cavity mode, the many atoms experience infinite-range interactions, and can - for example - be observed to self-organize into superradiant states. Instead, by employing many degenerate modes of the resonator, as in a confocal or concentric cavity, it is possible to tune also the range of the effective atom-atom interactions. Finally, atoms in optical cavities are intrinsically open quantum systems subjected to driven and dissipative processes. Now, dissipation is not necessarily an obstacle in these platforms. Recent studies, in fact, highlight the possibility of engineering noise and dissipation to access quantum phases of matter in ways that could be more convenient than conventional Hamiltonian engineering.

One most challenging quantum state of matter to be realized in an optical cavity is a fermionic superfluid, due to heating.
The interesting research question then arises, whether one might exploit the advantages offered by many-body, multi-mode cavity QED, to reach stable out-of-equilibrium fermionic superfluidity with tunable-range interactions and the use of dissipation.
This question is the object of this thesis work. We explore the behavior of a fluid of fermionic atoms in a multimode cavity as that in Benjamin Lev's Lab in Stanford, under dissipative dynamics. Starting from a microscopic Hamiltonian in the form of a tUJ model for fermions, we exploit the symmetries of the problem to engineer a dissipative mechanism that makes the fermionic superfluid state robust against heating inside the cavity. This work is realized within an international collaboration with Andrew Daley, Jonathan Keeling, François Damanet and Benjamin Lev.

We now summarise the physics in which our proposal is based. Since our fermionic tUJ model displays a multiple symmetries structure, we can favor the emergence of superfluid order by coupling to the symmetry sector that would instead compete to destroy it. This is a very general symmetry-based mechanism, first proposed by Tindall et al. for the Fermi-Hubbard model in Markovian regime. In our case, the tUJ model displays an SU(2) and U(1) symmetry in the charge and spin sectors respectively, and by getting rid of spin ordering, we protect from heating the order in the charge to reach the superfluid state. It is important to note that this mechanism leads to a particular kind of fermionic superfluid, that is in the form of a checkerboard-like superposition of Cooper up and down pairs of negligible radius on the same site, i.e. doublons. Predicted by Yang in 1989 as an excited state of the Fermi-Hubbard Hamiltonian on a bipartite lattice, this is called an eta-condensate. It is characterized by off-diagonal long-range order (ODLRO) in the correlations between a doublon and a hole located in different lattice sites. The local character of the pairs in real space implies that their constituting fermions are found at the Brillouin zone edges. As a result, the center-of-mass momentum of the condensed Cooper pairs is finite.