Collective flow of fermionic impurities immersed in a Bose-Einstein Condensate

Fermions coupled to a bosonic bath occur across physics, from electrons interacting with ionic lattice vibrations (polaron physics) to 3He in superfluid 4He, quark–meson models, and 2D electronic materials where exciton–electron interactions are tunable. Ultracold atomic mixtures offer a clean, controllable realization with tunable short-range interactions, enabling direct comparison to theory. Prior studies have explored dual superfluids, phase separation, mean-field collapse, and strong-coupling Bose polarons. The finite-temperature dynamics of fermions immersed in a partially condensed Bose gas are challenging because interspecies interactions include both incoherent, momentum-relaxing collisions with thermal bosons and coherent, momentum-preserving mean-field interactions with the condensate. The interplay dictates collective behavior and transport, especially near the crossover from collisionless to collisional hydrodynamics at strong coupling. Key open questions include whether, at strong Bose–Fermi interactions with a large condensate fraction, fermions can flow without dissipation through the condensate, and how thermal excitations modify fermion transport. This work addresses these questions by measuring collective oscillations of dilute, spin-polarized fermionic impurities immersed in a BEC across interaction strength and temperature. The study tests whether impurity dynamics can be described as collisionless, driven by the condensate’s mean-field, and maps the temperature-induced crossover to collision-dominated hydrodynamic flow.

The paper situates the work within several domains: - Solid-state analogs: electron–phonon coupling and polaron physics (Landau, Pekar), and hydrodynamic electron fluids in correlated materials. - Quantum fluids: dilute 3He in superfluid 4He and two-fluid hydrodynamics (Landau model). - Ultracold mixtures: realization and control of Bose–Fermi mixtures via Feshbach resonances; observations of dual superfluidity, phase separation, collapse, Bose polarons, and sound propagation. - Collective-mode studies: hydrodynamic and collisionless regimes in BECs and Fermi gases; modes in Bose–Fermi mixtures and coupled superfluids; theoretical treatments via sum rules, RPA, kinetic theory (Boltzmann–Vlasov), and phase-segregated states. - Nonlinear excitations: Faraday waves in elongated BECs under parametric driving. These works motivate probing the impurity collective dynamics as a sensitive diagnostic of coherent vs collisional interactions in Bose–Fermi mixtures, especially in the dilute-impurity limit relevant for Bose polarons and low-carrier unconventional superconductors.

Experimental platform: - Species: fermionic 40K impurities (spin-polarized) immersed in a bosonic 23Na BEC. Both in their hyperfine ground states: Na (F=1, mF=1), K (F=9/2, mF=−9/2). - Trap: crossed 1064 nm optical dipole trap with near-cylindrical symmetry. Typical trap frequencies (bosons): ωB/2π ≈ [103(3), 94(2), 12.2(0.3)] Hz; (fermions): ωF/2π ≈ [125(2), 114(2), 15(1)] Hz. Fermion transverse frequency along y relates as ωF ≈ 1.16ωB. - Densities and temperatures: peak nB ≈ 7×10^13 cm^-3; impurity concentration nF/nB ≈ 0.003–0.02. Lowest temperatures T ≈ 30 nK with T/Tc ≈ 0.2 (BEC) and T/TF ≈ 0.6 (K). For temperature scans, T varied up to and above Tc at fixed final temperature by reducing boson number. - Interactions: Interspecies s-wave scattering length aBF tuned using magnetic Feshbach resonances. Procedure: ramp to zero crossing at 80.3 G, stabilize for 5 ms, then quench to target aBF within 10 μs to initiate oscillations. Differential gravitational sag compensated with a magnetic field gradient optimized via maximizing 3-body loss overlap. Collective-mode excitation and detection: - Drive: Modulate optical trap depth sinusoidally at frequency ω for N cycles (typically 10) to excite radial modes; probe time kept short to mitigate 3-body loss. For interaction scans at low T, modulate a single beam at ≈20% depth variation (primarily transverse breathing). For temperature scans at fixed aBF (≈400 a0), modulate both beams 180° out of phase at ≈15% to couple to transverse quadrupole. - Readout: In-situ absorption imaging; extract in-situ widths and doubly integrated line densities along the transverse direction. Spectrograms constructed versus ω; spectral resolution ≈ ω/N; modulation depth chosen to balance SNR and Fourier-limited resolution. - Normalization: Frequencies normalized to geometric mean boson radial trap frequency ωB; for long-term drifts, normalization obtained by fitting BEC breathing response near 2ωB per dataset. Data analysis and theory: - BEC modeled by hydrodynamic scaling ansatz (Castin–Dum): coupled equations for scaling parameters bi(t) of the Thomas–Fermi radii, capturing quadrupole (√2ωB) and breathing (2ωB) modes. - Fermion spectra: phenomenological asymmetric lineshape used to extract peak positions and widths; strong driving and trap anharmonicity can produce broadened “fan” features below 2ωF in decoupled gases. - Mean-field (static) model: effective fermion trap frequency shift from BEC mean-field in TF limit: ω^2 = ω0^2(1 − gBF/β0) (notation per text), capturing linear frequency shift at small |aBF|. - Collisionless kinetic modeling: Boltzmann–Vlasov equation ∂tf + ṙ·∇rf + ṗ·∇pf = Icoll with Icoll=0 at low T; derive a scaling ansatz for fermion widths ci(t) to first order in gBF, including the time-dependent mean-field from the BEC scaling dynamics, to capture BEC-imprinted modes. - Full numerical simulations: collisionless Boltzmann–Vlasov dynamics including finite temperature cloud size, trap anharmonicity, and realistic parameters used to reproduce observed mode structure across all aBF, validating neglect of Icoll at low T. - Finite-temperature regime: Above T/Tc ≳ 0.4, solve coupled Bose–Fermi Boltzmann equations with nonzero collision integrals to describe emergence of collisionless thermal-boson mode (2ω) and fermion mode (2ωF) and the crossover to collisional hydrodynamics; above Tc the mixture becomes collisionless for both species due to reduced boson density. - Faraday-wave protocol: Parametric radial modulation for 8 cycles at 2ω to excite longitudinal density striations; measure period and infer BEC speed of sound; observe imprinting onto fermion density via interactions.

- At strong interspecies interactions (|aBF| ≳ 350 a0), fermionic impurities respond exclusively at the BEC’s superfluid hydrodynamic modes, reproducing the BEC quadrupole (√2ωB) and breathing (2ωB) resonances; the native fermion collisionless mode at 2ωF disappears from the spectra. - Across interaction strengths, three distinct fermion modes are observed: 2ωB, √2ωB (both BEC modes), and 2ωF (fermion mode). For large |aBF|, the BEC modes dominate; at weak coupling, the fermion collisionless mode exhibits a mean-field shift linear in aBF (with sign dependence). - Spectral peaks remain Fourier-limited without additional broadening across strong couplings, indicating absence of momentum-relaxing collisions with the condensate despite a mean-free path as short as lmfp ≈ (4π aBF^2 nB)^−1 ≈ 0.6 μm at strongest interactions, much smaller than the radial size L ≈ 10 μm. - A dispersive (Fano-type) feature appears in the fermion response at the BEC quadrupole mode for repulsive interactions above ≈170 a0, consistent with coherent coupling between fermionic and bosonic modes. - Static mean-field models capture only the initial linear shift but fail to predict BEC-imprinted modes. Including the BEC’s time-dependent hydrodynamic response in a collisionless Boltzmann–Vlasov framework reproduces the full mode structure across all aBF (including finite temperature size and trap anharmonicity effects). - Temperature dependence at fixed aBF ≈ 400 a0: At low T/Tc, fermions respond only at BEC hydrodynamic modes. Starting at T/Tc ≈ 0.4, two additional fermion resonances emerge: at 2ωB (collisionless mode of the bosonic thermal component) and at 2ωF (native fermion collisionless mode). Above Tc, only the fermion collisionless mode at 2ωF persists, consistent with coupled Boltzmann modeling with nonzero collision integrals. - Interpretation: The appearance of a fermion response at the bosonic collisionless mode (for 0.3Tc ≲ T ≲ Tc) indicates a crossover of the mixture from collisionless to collision-dominated hydrodynamic behavior driven by interspecies collisions. - Faraday waves: Parametric drive produces longitudinal density striations in the BEC with period λFar ≈ 28(7) μm; identical striations are imprinted onto the fermions at aBF = 500 a0. From the drive frequency and period, the speed of sound is c ≈ 5.5(1.4) μm/ms, consistent with the Bogoliubov value c = √(μ/mB) ≈ 4.9(2) μm/ms. This constitutes, to the authors’ knowledge, the first observation of Faraday-like spatial patterns in a gas of fermions. - Thresholds and phase behavior: For |aBF| ≳ 350 a0 only BEC modes are seen in fermion spectra; phase separation is predicted at ≈900 a0 (not explored here).

The experiments demonstrate that in a dilute impurity limit and well below the BEC transition, fermionic impurities move collisionlessly through the condensate, with dynamics governed by coherent mean-field interactions. This enables the fermions to “copy” the condensate’s superfluid hydrodynamic modes, from low-lying quadrupole and breathing modes to parametrically excited Faraday waves, without dissipative broadening even at strong coupling where naive mean-free-path arguments would suggest collisional hydrodynamics. Incorporating the condensate’s time-dependent hydrodynamic response into collisionless Boltzmann–Vlasov dynamics quantitatively accounts for the observed spectra, whereas static mean-field models do not. Upon increasing temperature, thermal bosonic excitations provide a channel for momentum-relaxing interspecies collisions, causing a gradual re-emergence of fermion and thermal-boson collisionless modes and signaling a crossover to collision-dominated hydrodynamic flow of the mixture. Above Tc, the reduced boson density returns both components to collisionless behavior. These findings parallel hydrodynamic-to-ballistic crossovers in 2D electron fluids, highlighting universal aspects of transport in coupled Bose–Fermi systems. The results validate a simple, coherent picture of impurity transport in a quantum fluid and underscore the utility of collective modes as sensitive probes of interspecies coupling, dissipation, and emergent hydrodynamics. The ability to imprint high-order BEC excitations onto fermions further illustrates strong coherent coupling and opens avenues for engineering impurity dynamics via bosonic backgrounds.

This work uncovers a regime where dilute, spin-polarized fermionic impurities immersed in a BEC undergo collisionless, mean-field-driven collective flow that faithfully tracks the condensate’s superfluid hydrodynamic modes. By tuning interspecies interactions and temperature, the study maps the transition from collisionless, coherent impurity motion to a collision-dominated hydrodynamic regime as thermal bosons proliferate. The observation of Faraday waves imprinted on the fermions demonstrates coherent transfer of high-order collective excitations across species. These results provide a benchmark for kinetic theories of Bose–Fermi mixtures, establish collective-mode spectroscopy as a sensitive probe of coherent versus collisional dynamics, and suggest pathways to explore induced fermion–fermion interactions mediated by bosons. Future directions include accessing lower temperatures to enhance induced interactions and pursue boson-mediated p-wave pairing of fermions, probing phase separation at stronger repulsion, exploring different mass ratios and geometries, and engineering nonequilibrium protocols to control impurity transport.

- Dilute-impurity limit and specific species (23Na–40K) and trap geometry may limit generality to other mixtures and mass ratios. - Fermions are only moderately degenerate (T/TF ≈ 0.6–2), and three-body loss at strong coupling limits excitation duration and spectral resolution. - Strong driving and trap anharmonicities contribute to asymmetric fermion lineshapes and broadened background features; linear-response deviations are present. - Interaction range explored reaches |aBF| ≲ 500 a0; predicted phase separation at ≈900 a0 is not probed. - Modeling approximations include Thomas–Fermi treatment of the BEC, hydrodynamic scaling ansatz, and collisionless Boltzmann–Vlasov dynamics at low T; finite-temperature coupled Boltzmann treatments above Tc assume specific collision integrals and neglect beyond-mean-field effects. - Temperature scans vary T/Tc via atom-number reduction at nearly fixed absolute T, which may alter density-dependent parameters (e.g., sound speed) and overlap, complicating direct comparisons across T/Tc.