Confinement-induced modifications of many-body observables in the strongly interacting quasi-2D BEC-BCS crossover

The 2015 Heidelberg pair-condensation experiment (Ries et al., Phys. Rev. Lett. 114, 230401) mapped the phase diagram of a strongly interacting quasi-two-dimensional (quasi-2D) Fermi gas across the BEC-BCS crossover, interpreting the data through a strictly 2D theory with a 2D scattering length. The experiment explicitly left open how the finite axial confinement and the three-dimensional internal structure of the pairs modify many-body observables — the pair-condensation transition temperature $T_c$, the equation of state, and other thermodynamic quantities — beyond the two-body sector, especially in the regime where the dimer binding energy satisfies $E_B \approx \hbar\omega_z$; how such confinement effects could be separated experimentally from many-body physics; and two measurement-level questions (the origin of a systematic temperature-fit discrepancy and access to the Tan contact). Later work has produced approximate many-body treatments (effective-range effective field theory, Gaussian pair fluctuation theory, functional renormalization) with only qualitative agreement, and a quantum Monte Carlo benchmark exists only on the deep BEC side; the strongly interacting quasi-2D regime remains without a controlled quantitative description, and no experiment has isolated confinement effects there. The problem is open.

Background

Reducing a fermionic superfluid to two dimensions connects cold-atom physics to layered superconductors, where pairing and condensation are still not fully understood. In an ultracold two-component Fermi gas, the BEC-BCS crossover is the continuous path from a weakly paired Bardeen-Cooper-Schrieffer (BCS) state of large, overlapping Cooper pairs to a Bose-Einstein condensate (BEC) of tightly bound dimers, tuned via a magnetic Feshbach resonance that controls the s-wave scattering length. In two dimensions the interaction strength is encoded in the 2D scattering length $a_{2D}$, and the crossover parameter is $\ln(k_Fa_{2D})$ with Fermi wave vector $k_F$; there is a two-body bound state for all attraction strengths. Because thermal and quantum fluctuations destroy true long-range order in a uniform 2D system, the superfluid transition instead proceeds by the Berezinskii-Kosterlitz-Thouless (BKT) mechanism, in which vortex-antivortex pairs bind below a critical temperature.

Experiments realize 2D not exactly but as a 'quasi-2D' gas: a tight harmonic confinement along one axis (frequency $\omega_z$, oscillator length $\ell_z=\sqrt{\hbar/m\omega_z}$) freezes the axial motion ($\mu, k_BT \lesssim \hbar\omega_z$), but the gas retains a third dimension at short distances. Two-body scattering in this geometry is fully characterized by the 3D scattering length $a_{3D}$ together with $\ell_z$: they determine the effective 2D scattering length $a_{2D}$ and a confinement-induced dimer binding energy $E_B$ that interpolates between the 3D molecular binding energy (for $E_B \gg \hbar\omega_z$, dimers much smaller than $\ell_z$, unresolved by the confinement) and the 2D value (for $E_B \ll \hbar\omega_z$, dimers wide and flat). The two-body sector can therefore be mapped exactly onto a 2D contact-interaction model; mean-field treatments of the quasi-2D crossover keeping all harmonic-oscillator transverse modes existed at the time of the experiment (Fischer and Parish, Phys. Rev. B 90, 214503 (2014)).

Ries et al. (Phys. Rev. Lett. 114, 230401 (2015)) measured the in-situ pair momentum distribution of a strongly interacting quasi-2D $^6$Li gas (trap aspect ratio $\omega_z/\omega_r \approx 310$) by a magnetic-field projection technique, observed the emergence of a low-momentum pair condensate, and mapped the critical temperature $T_c/T_F$ across the crossover. Their interpretation assumed the strictly 2D description with $a_{2D}$, i.e. that the pairs' internal structure has negligible effect beyond the two-body sector. The paper states that this assumption is hardest to justify in the regime $E_B \approx \hbar\omega_z$ (near $\ln(k_Fa_{2D}) \approx 0.5$ in their system), that a theoretical treatment beyond the two-body sector there is 'extremely difficult due to strong interactions', and that experimental studies of these confinement effects would require even larger trap aspect ratios or smaller atom numbers than were feasible. The paper additionally reports that temperatures extracted from in-situ density-profile fits and from momentum-distribution fits disagree systematically at some magnetic fields for unclear reasons, and that the pair-momentum projection removes relative pair momentum information, so the Tan contact — the universal amplitude of the $k^{-4}$ high-momentum tail of the momentum distribution, which ties short-distance pair structure to thermodynamics — cannot be obtained from the measured pair momentum distribution. These gaps jointly limit how completely quasi-2D experiments can be mapped onto strictly 2D theoretical predictions.

Problem Statement

Two sub-questions are asked; both are required and both serve the single objective stated below.

(1) Determine quantitatively how the finite axial confinement and the three-dimensional internal structure of pairs modify the many-body observables of the quasi-2D Fermi gas across the BEC-BCS crossover — in particular the pair-condensation/superfluid transition temperature $T_c$ and the equation of state — beyond the two-body sector, in the strongly interacting regime where the dimer binding energy $E_B$ is comparable to the transverse confinement energy $\hbar\omega_z$ (around $\ln(k_Fa_{2D})\approx 0.5$ in the original experiment). This sub-question is resolved only by a controlled quantitative prediction of the confinement-induced modifications that states its regime of validity and is quantitatively compared with the strictly 2D prediction; a qualitative or order-of-magnitude statement does not resolve it.

(2) Establish experimentally realizable regimes or methods in which such confinement-induced modifications can be separated from many-body physics and quantitatively compared with strictly 2D predictions, together with an explicit criterion by which that separation is judged to be achieved. The parenthetical routes (larger trap aspect ratios or substantially smaller atom numbers) are illustrative and not exhaustive; naming them without such a criterion does not by itself resolve this sub-question.

Both sub-questions serve one objective: making the mapping of quasi-2D experiments onto 2D many-body theory quantitative rather than qualitative. Sub-question (1) — confinement effects beyond the two-body sector on $T_c$ and thermodynamics — is the load-bearing part; the experimental results of sub-question (2) are the means to test and calibrate it.

References

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