A Mixture of Bose and Fermi Superfluids

Abstract: Superconductivity and superfluidity of fermionic and bosonic systems are remarkable many-body quantum phenomena. In liquid helium and dilute gases, Bose and Fermi superfluidity has been observed separately, but producing a mixture in which both the fermionic and the bosonic components are superfluid is challenging. Here we report on the observation of such a mixture with dilute gases of two Lithium isotopes, 6Li and 7Li. We probe the collective dynamics of this system by exciting center-of-mass oscillations that exhibit extremely low damping below a certain critical velocity. Using high precision spectroscopy of these modes we observe coherent energy exchange and measure the coupling between the two superfluids. Our observations can be captured theoretically using a sum-rule approach that we interpret in terms of two coupled oscillators.

• The paper demonstrates precise control over interspecies interactions in ultracold 7Li-6Li mixtures using Feshbach resonances and detailed scattering analyses.
• It employs advanced experimental setups with hybrid optical-magnetic traps and RF state transfer to achieve distinct Bose-Einstein and Fermi superfluid phases.
• A coupled oscillator model combined with Lee-Huang-Yang corrections uncovers critical velocity thresholds and quantum fluctuation effects driving the superfluid dynamics.

A Mixture of Bose and Fermi Superfluids: Analysis and Experimental Findings

The supplementary material to the paper titled "A Mixture of Bose and Fermi Superfluids" by Ferrier-Barbut et al. provides an extensive exploration of the interactions and behaviors of superfluid systems comprised of Bose-Einstein Condensates (BEC) and Fermi gases. The study analyzes a mixture of $^7$Li and $^6$Li isotopes at ultracold temperatures, employing a range of experimental and theoretical approaches to characterize the properties of the superfluid mixtures under varying magnetic fields and other experimental conditions.

Feshbach Resonances and Interactions

The authors investigate the s-wave scattering lengths pertinent to $^7$Li-$^7$Li, $^6$Li-$^6$Li, and $^6$Li-$^7$Li interactions in the magnetic field range of 700 G to 1000 G. Notably, $^7$Li exhibits Feshbach resonances at 845.5 G and 894 G, while $^6$Li exhibits a broad s-wave resonance at 832.18 G. These resonances play a crucial role in tuning the interaction strength between species, with the inter-isotope scattering length calculated as $a_{\rm bf}=40.8\,a_0$, independent of the magnetic field. The study highlights the potential for precise control over these interactions, which is vital for exploring superfluid properties in mixed-species condensates.

Experimental Methodology

The experimental setup involves preparing a $^7$Li-$^6$Li mixture under controlled cooling and trap conditions. Detailed preparation, including initial cooling to 40 µK and RF-induced state transfer, ensures the formation of superfluid phases. The researchers use hybrid trapping combining optical and magnetic potentials to achieve ultracold temperatures, with final temperatures yielding $^7$Li BEC at 30 µK and exhibiting a critical temperature for condensation at 700 nK. The Fermi energy for $^6$Li is noted as 880 nK, illustrating the difference in thermal conditions needed for Bose and Fermi systems.

Oscillation Dynamics and Critical Velocity

A key aspect of the study is the investigation of collective excitations and relative motion damping between the superfluid species. The data demonstrates damped and undamped dipole mode oscillations depending on initial conditions and the resulting velocity relative to the Fermi velocity $v_{\rm F}$. Notably, the authors identify a critical velocity $v_{\rm c}=0.42^{+0.05}_{-0.11}\,v_{\rm F}$, suggesting a velocity threshold indicative of superfluid properties akin to sound propagation in a Fermi gas. This observation corroborates prior findings in isotropic traps and elucidates dynamics specific to elongated trapping geometries.

Mean-Field and Lee-Huang-Yang Evaluations

The authors evaluate the mean-field and beyond-mean-field corrections to the superfluid dynamics using models such as Lee-Huang-Yang (LHY) theory. The LHY corrections account for quantum fluctuations beyond the mean-field approximation, providing predictions for frequency shifts in the small $^6$Li-$^7$Li mixtures as a function of the Fermi-gas interaction parameter $1/k_{\rm F}a_{\rm f}$. These theoretical considerations are pivotal for understanding the interplay of bosonic and fermionic superfluidity, offering congruence with experimental measurements.

Theoretical Modeling: Coupled Oscillators

The dynamics of the superfluid mixture are further contextualized through a coupled oscillator model using the sum-rule approach. Analytical expressions derived from this model allow insights into the interactions and frequency shifts observed in the experiments, delineating the distinct roles of bosonic and fermionic contributions to the mixture's overall dynamics.

Implications and Future Directions

This study's findings on superfluid mixtures extend foundational understanding of quantum gas systems and provide rich potential for exploration in quantum simulation and analog quantum computations. Future directions could include exploring tuning mechanisms on interspecies interactions, exploring anisotropic traps, or exploiting novel nonequilibrium phenomena to reveal further exotic states of matter. Such extensions could elucidate new quantum phases and refine theoretical models for superfluid dynamics in complex multi-component systems.