Microcavity electron-hole-photon systems are long anticipated to exhibit a crossover from Bose-Einstein condensate (BEC) to Bardeen-Cooper-Schrieffer (BCS) superfluid, when carrier density is tuned to reach the Mott transition density. Yet, theoretical understanding of such a BEC-BCS crossover largely relies on the mean-field framework and the nature of the carriers at the crossover remains unclear to some extent. The BEC-BCS crossover is also known to occur in ultracold atoms confined in optical traps. In this talk, I will introduce some recent results on the *universal* BEC-BCS crossover physics from the theory group at Swinburne University.

In the first part of the talk, motivated by the recent demonstration of a BCS polariton laser [1], I will discuss the properties of *a strongly interacting BCS polariton condensate* at thermal equilibrium [2]. Based on a simplified short-range description of the electron-hole attraction, we examine the role of quantum fluctuations in an exciton-polariton condensate and determine the number of different type carriers at the crossover beyond mean-field. Near Mott density and with ultra-strong light-matter coupling, we find an unexpectedly large phase window for the strongly correlated BCS polariton condensate, where both fermionic Bogoliubov quasi-particles and bosonic excitons are significantly populated and strongly couple to photons. We predict its absorption spectrum and show that the upper polariton energy gets notably renormalized, giving rise to a high-energy side-peak at large carrier density, as observed in recent experiments.

In the second part, I will talk about *spin-1/2* *ultracold Fermi gases confined in two dimensions* with a tight harmonic trapping potential along the *z*-direction. Here, virtual excitations along the tightly-confined direction play the same role of photons in microcavities. In the case of an additional harmonic trap added on the *xy*-plane, the Fermi cloud shows an interesting violation of a classical scale invariance (i.e., the so-called quantum anomaly), as indicated by its breathing mode frequency. We present theoretical understanding of such a quantum anomaly from both few-body [3] and many-body perspective [4]. We also find the formation of cluster states at a positive interaction range [5], similar to biexcitons, triexcitons and bipolaritons observed in microcavities.

**References**

[1] J. Hu, Z. Wang, S. Kim, H. Deng, S. Brodbeck, C. Schneider, S. Höfling, N. H. Kwong, and R. Binder, *Signatures of a Bardeen-Cooper-Schrieffer Polariton Laser*, arXiv:1902.00142v1.

[2] H. Hu and X.-J. Liu, *Quantum fluctuations in a strongly interacting BCS polariton condensate at thermal equilibrium*, arXiv:1910.06494v1.

[3] X.-Y. Yin, H. Hu, and X.-J. Liu, *Few-body perspective of quantum anomaly in two-dimensional interacting Fermi gases*, arXiv:1907.10195v1.

[4] H. Hu, B. C. Mulkerin, U. Toniolo, L. He, and X.-J. Liu, Phys. Rev. Lett. **122**, 070401 (2019).

[5] X.-Y. Yin, H. Hu, and X.-J. Liu, Phys. Rev. Lett. **123**, 073401 (2019).