Lan and Lobo (Phys. Rev. A 90, 033627 (2014)) proposed an ultracold-atom quantum simulator of an electron-phonon system in which heavy atoms pinned at unit filling in a deep, species-selective optical lattice play the role of ions, while untrapped spin-polarized light fermions form an electronic band from overlapping inter-species dimers when the s-wave scattering length is tuned to the lattice spacing (a∼da\sim d). The paper computed the heavy-atom phonon dynamics and the Peierls instability but explicitly left open its central many-body question: whether the phonon-mediated interaction between the light atoms in this a∼da\sim d giant-orbital regime binds Cooper pairs of light atoms, producing a paired superfluid — the analog of BCS superconductivity in the simulator — and what the resulting phase is. This question remains unanswered in the literature to date: the closest follow-ups treat lattice Peierls/Su-Schrieffer-Heeger models or a qualitatively different Bose-condensate background, and none computes pairing for this continuum few-body-based setup. The problem is to derive the effective light-atom interaction mediated by the heavy-ion dynamics and to determine the pairing instability, its competition with Peierls charge-density-wave order, and the character of the paired phase.

Background

In standard optical-lattice physics, atoms trapped in a standing-wave light field are described by a Hubbard model, which requires the inter-atomic scattering length aa (the effective interaction range set by a Feshbach resonance, a magnetically tunable scattering resonance between two atomic species or states) to be much smaller than the lattice spacing dd, so that only the lowest band participates. Lan and Lobo (2014) proposed going beyond this paradigm: confining a heavy atomic species (e.g. 40^{40}K) in a deep optical lattice at unit filling — one heavy atom per site, with negligible tunneling between sites — while a second, light species of spin-polarized fermions (e.g. 6^{6}Li) is untrapped, sees no lattice, and interacts with the heavy atoms through an s-wave contact interaction. When aa is tuned by a Feshbach resonance to be on the order of dd, the two-body bound states (dimers) centered on neighboring sites overlap, forming 'giant orbitals' that extend over the whole lattice and constitute an electronic band for the light atoms; the deep lattice stabilizes these molecular states against three-body recombination, which destroys them in free space. The heavy atoms, oscillating in their sites, are the 'ions': without the light atoms their oscillators are independent (a flat optical phonon band), but interactions mediated by the light atoms produce a dispersive phonon branch. Treating the fast light atoms within the Born-Oppenheimer approximation (BOA, the adiabatic separation in which the light particles respond instantaneously to the slow heavy ones), valid when the light-atom band gap EgapE_{\mathrm{gap}} is open, the authors computed the mediated heavy-heavy interaction: at full filling of the light-atom band it favors a uniform displacement of all heavy atoms (the 'ferromagnetic' phase in the paper's pseudo-spin description of the site displacements), and at half filling it drives a Peierls instability — the spontaneous dimerization of the lattice that opens a gap ΔP\Delta_P at the Fermi wave vector, here of order 4Ed4E_d (with EdE_d the dimer energy scale) at displacement ∼0.2d\sim 0.2d — the analog of the charge-density-wave (CDW) instability of a one-dimensional metal. This realizes an electron-phonon simulator with Su-Schrieffer-Heeger (SSH)-type coupling, in which the light-atom hopping (band structure) rather than an on-site energy is modulated by the ion displacement. What the paper did not do is the essential superconductivity step of the electron-phonon analogy: the phonon-mediated interaction between the light atoms themselves was never computed in this a∼da\sim d regime, and whether it binds Cooper pairs (the Bardeen-Cooper-Schrieffer pairing of fermions induced by an attractive retarded interaction) was stated as intended future work. This is not a small extension: at a∼da\sim d the light atoms are genuinely few-body objects (overlapping dimers) not describable as Hubbard particles, the mediated interaction is expected to contain non-standard pair-hopping terms (depending on the total momentum of the interacting pair, as later emphasized in the lattice SSH context by Sous et al.), the pairing instability must be compared against the Peierls/CDW instability already present at commensurate fillings, and near unitarity (a→∞a\to\infty, where Egap→0E_{\mathrm{gap}}\to 0) the BOA itself breaks down. The problem thus asks for the many-body theory of the light fermions in this few-body-based simulator: the effective interaction mediated by the heavy-ion dynamics, and the fate of pairing across the BCS-BEC crossover of the light pairs (from weakly bound Cooper pairs to tightly bound bosonic dimers, whose condensation is a Bose-Einstein condensate of bipolarons, i.e. pairs of light atoms bound by phonon exchange).

Problem Statement

For the Lan-Lobo electron-phonon simulator — heavy atoms pinned at unit filling in a deep species-selective optical lattice, forming one-dimensional chains of sites with spacing dd (implemented experimentally as a two-dimensional array of one-dimensional tubes), with untrapped spin-polarized light fermions of mass mm interacting with the heavy atoms (mass MM) via an s-wave contact interaction with scattering length tuned to a∼da\sim d — address the following jointly required parts: (i) derive the effective interaction between light atoms that is mediated by the dynamics (onsite oscillations and Peierls distortions) of the heavy atoms; (ii) determine whether this mediated interaction produces Cooper pairing of the light atoms, that is, establish whether the paired (superfluid) state is the ground state of the light-atom band; (iii) compute the pairing scale and the BCS-BEC character of the pairs across the regime of a/da/d from weak coupling to the point where the BOA gap closes; and (iv) determine the outcome of the competition between pairing and the Peierls/CDW instability at half filling and at full filling. The answer should also identify the mechanism that fixes the outcome, namely the structure of the mediated coupling, including any pair-hopping/SSH-type components.

References

  1. Zhihao Lan and Carlos Lobo, 'Optical lattices with large scattering length: Using few-body physics to simulate an electron-phonon system', Physical Review A 90, 033627 (2014). DOI: 10.1103/PhysRevA.90.033627, https://doi.org/10.1103/PhysRevA.90.033627 (preprint arXiv:1310.4152, https://arxiv.org/abs/1310.4152)
  2. John Sous, Monodeep Chakraborty, Roman V. Krems, and Mona Berciu, 'Light Bipolarons Stabilized by Peierls Electron-Phonon Coupling', Physical Review Letters 121, 247001 (2018). DOI: 10.1103/PhysRevLett.121.247001, https://doi.org/10.1103/PhysRevLett.121.247001 (preprint arXiv:1805.06109, https://arxiv.org/abs/1805.06109)
  3. C. Zhang, J. Sous, D. R. Reichman, M. Berciu, A. J. Millis, N. V. Prokof'ev, and B. V. Svistunov, 'Bipolaronic High-Temperature Superconductivity', Physical Review X 13, 011010 (2023). DOI: 10.1103/PhysRevX.13.011010, https://doi.org/10.1103/PhysRevX.13.011010 (preprint arXiv:2203.07380, https://arxiv.org/abs/2203.07380)
  4. Axel Gagge, Th. K. Mavrogordatos, and Jonas Larson, 'Exploring phononlike interactions in one-dimensional Bose-Fermi mixtures', Physical Review Research 6, 013138 (2024). DOI: 10.1103/PhysRevResearch.6.013138, https://doi.org/10.1103/PhysRevResearch.6.013138 (preprint arXiv:2303.07083, https://arxiv.org/abs/2303.07083; the arXiv preprint version is titled 'Exploring phonon-like interactions in one-dimensional Bose-Fermi mixtures')
  5. Michał Tomza, Krzysztof Jachymski, Rene Gerritsma, Antonio Negretti, Tommaso Calarco, Zbigniew Idziaszek, and Paul S. Julienne, 'Cold hybrid ion-atom systems', Reviews of Modern Physics 91, 035001 (2019). DOI: 10.1103/RevModPhys.91.035001, https://doi.org/10.1103/RevModPhys.91.035001