The interplay of quantum statistics and interactions in Bose-Fermi mixtures creates a complex phase diagram, differing significantly from pure bosonic or fermionic systems. However, studying this diagram is challenging due to rapid interspecies loss when bosons condense. This work aims to investigate the phase transition between a polaronic condensate and a molecular Fermi gas in a density-matched Bose-Fermi mixture. Understanding this transition is crucial, as it offers insights into fundamental many-body physics and provides a route to creating heteronuclear molecules with desirable properties for applications in quantum chemistry and dipolar quantum many-body systems. Previous research has primarily focused on the impurity regime, where one species is dilute. This study delves into the unexplored regime of comparable densities, where strong correlations are expected to dominate, leading to a richer phase diagram potentially featuring supersolidity, charge-density-wave phases, molecular Fermi liquids, and unconventional superconductivity. The matched density regime is particularly promising for efficient heteronuclear molecule association at high phase-space density, which is difficult to achieve due to enhanced loss in the bosonic condensate.

The study of interacting Bose-Fermi mixtures is rooted in the understanding of conventional superconductivity, where electron-phonon coupling, modeled by Fröhlich or Holstein models, leads to effective attraction between electrons. Ultracold atoms and van der Waals materials offer opportunities to realize Bose-Fermi mixtures exhibiting beyond-Fröhlich physics, where bosons and fermions can bind into fermionic molecules. The competition between this binding and mediated interactions enriches the phase diagram. Previous experimental work has focused on the limits of large population imbalance, exploring Bose polarons (fermionic impurities in a bosonic bath) and the Fermi polaron-to-molecule transition (impurities in a Fermi sea). However, the transition in the double-degenerate, density-matched regime remains uncharted, a crucial region for high-density heteronuclear molecule creation and the study of strongly correlated Bose-Fermi mixtures.

The researchers used a density-matched mixture of ²³Na (bosons) and ⁴⁰K (fermions) atoms. A species-dependent dipole trap at 785 nm was employed to achieve density matching, mitigating atomic loss. The system was prepared with comparable densities of ²³Na and ⁴⁰K, achieving double degeneracy. The boson-fermion interaction strength was tuned by varying the magnetic field near a Feshbach resonance. A single magnetic field ramp was used, with its speed optimized to ensure the system stays close to local equilibrium. The condensate fraction was measured as the interaction strength increased, probing the transition from the polaronic phase to the molecular phase. The normalized order parameter, φ = *N*_BEC/(*N*_m + *N*_BEC), was defined to characterize the depletion of the condensate, where *N*_BEC is the number of condensed Na atoms and *N*_m is the number of associated molecules. The number of Feshbach molecules was also measured to independently estimate the transition point. A self-consistent functional renormalization group (fRG) technique was used to predict the theoretical condensate fraction and the polaron-to-molecule transition point. The quantum degeneracy of the resulting molecules was confirmed by analyzing their velocity distribution using a Fermi-Dirac distribution fit. A density decompression technique was used to reduce interspecies loss by adjusting the trapping frequencies for Na and K, leading to a weaker confinement for Na and increasing overlap between the species. The reverse process, starting from a molecular phase and tuning the interactions to the weakly interacting regime, was also studied to examine the reversibility of the phase transition and the degree of coherence preserved.

The experimental results show a clear depletion of the condensate fraction as the attractive boson-fermion interaction strength increases, consistent with a phase transition from a polaronic condensate to a molecular Fermi gas. The order parameter (condensate fraction) vanishes smoothly beyond a critical interaction strength. The transition point, determined from the vanishing of the order parameter and the saturation of Feshbach molecule numbers, is consistent with theoretical predictions from the fRG calculation. High association efficiency (~80%) of Na atoms into Feshbach molecules was achieved by driving the system through the phase transition. Subsequently, these Feshbach molecules were converted to ground-state ²³Na⁴⁰K molecules with a large molecular-frame dipole moment (2.7 Debye) in the quantum-degenerate regime. The experimental data agrees well with theoretical predictions from the NSCT approach, especially regarding the universality of condensate depletion. However, the fRG approach, incorporating an infinite number of particle-hole excitations, provides a more accurate description near the phase transition. The velocity distribution of the molecules confirmed their quantum degeneracy. The study also explored the reverse transition, showing partial restoration of the BEC from the molecular phase, indicating preservation of coherence.

The observed phase transition from a polaronic condensate to a molecular Fermi gas in a density-matched Bose-Fermi mixture represents a new quantum phase transition distinct from the paradigmatic BEC-BCS crossover. The smooth vanishing of the order parameter, unlike the abrupt jump predicted in the impurity limit, highlights the role of finite boson density and temperature. The high molecule association efficiency achieved by driving the QPT is a significant advancement, potentially enabling the creation of large samples of ultracold molecules with significant dipole moments. This opens new avenues for exploring dipolar quantum many-body physics and applications in quantum simulation and quantum chemistry. The agreement between experimental findings and theoretical predictions from both NSCT and fRG, albeit with higher accuracy from fRG close to the transition point, validates the theoretical models and underscores the system’s suitability for testing theoretical frameworks for strongly correlated many-body systems.

This research demonstrates a quantum phase transition from a polaronic condensate to a molecular Fermi gas in a density-matched Bose-Fermi mixture. This transition, distinct from the BEC-BCS crossover, provides a highly efficient mechanism for producing large samples of ultracold, quantum-degenerate heteronuclear molecules with significant dipole moments. The findings significantly advance the field of ultracold molecules and open new avenues for exploring strongly correlated many-body phenomena and applications in quantum technologies. Future work could focus on exploring the detailed dynamics of the transition and investigating other phases predicted in the phase diagram, such as supersolid phases.

The experiment was conducted in a trapped environment, which introduces inhomogeneities in density and interaction strength across the sample. While the authors employed techniques to mitigate this issue, the effects of the trap potential may influence the precise nature of the phase transition. Additionally, the precise order of the phase transition (first or second order) remains challenging to determine definitively due to the interplay of finite temperature and trapping effects. The characterization of the reverse transition using the order parameter was limited due to heating during the dissociation ramp.