The 2004 realization of a Tonks–Girardeau gas of ultracold rubidium atoms in a two-dimensional optical lattice (Paredes et al., Nature 429, 277 (2004)) opened two experimental directions: (i) direct measurement of two-body correlations in the fermionized one-dimensional gas, and (ii) extension to two bosonic atomic species with tunable interactions to observe pairing and a BCS-like ('superconducting') regime, as proposed by Paredes and Cirac (Phys. Rev. Lett. 90, 150402 (2003)). Direction (i) is closed: Kinoshita, Wenger and Weiss (Phys. Rev. Lett. 95, 190406 (2005)) determined the local pair correlation function g(2)(0)g^{(2)}(0) of one-dimensional Bose gases from photoassociation rates over a wide range of coupling strengths, directly observing fermionization, in agreement with theory. The surviving open core is direction (ii): no experimental realization of interspecies pairing in a bosonic mixture in an optical lattice is known to date, while theory has mapped the pair-superfluid phases, their transition from two independent condensates, the weak-attraction BCS-like regime of extended pairs, and concrete detection protocols. The problem is to prepare such a two-species lattice gas with attractive interspecies interactions, drive and observe the transition to the paired (pair-superfluid; in one dimension, pair-Luttinger-liquid) state, and verify the pairing through correlation observables.

Background

An optical lattice is a periodic potential for neutral atoms created by interfering laser standing waves, with lattice spacing set by half the light wavelength; loading an ultracold Bose gas into a deep two-dimensional lattice freezes transverse motion and produces an array of independent one-dimensional (1D) 'tubes'. In a 1D Bose gas, repulsive contact interactions become dominant as the interaction parameter grows, and in the limit of infinitely strong repulsion the gas enters the Tonks–Girardeau (TG) regime, predicted by Girardeau in 1960: the bosonic many-body wavefunction vanishes whenever two particles coincide, so the bosons imitate fermions ('fermionization') in all local observables while keeping bosonic exchange symmetry in nonlocal ones such as the momentum distribution. Paredes et al. (Nature 429, 277 (2004)) realized this regime with ultracold 87Rb atoms in an array of 1D tubes formed by a two-dimensional optical lattice, adding a shallow axial lattice that increases the atoms' effective mass and thereby the effective ratio of interaction to tunneling energy; the measured time-of-flight momentum distributions agreed closely with a fermionization-based theory. The hallmark of fermionization in real space is the suppression of short-range density fluctuations, quantified by the local two-particle correlation function g(2)(0)g^{(2)}(0), which tends to zero in the TG limit; theory (Gangardt and Shlyapnikov, New J. Phys. 5, 379 (2003)) showed that this function can be read out from photoassociation rates — photoassociation is a light-induced process in which two colliding atoms are transferred to a bound molecular state, so its rate probes the short-distance pair wavefunction overlap. The same suppression also shows up indirectly as reduced three-body recombination, observed by Laburthe Tolra et al. (Phys. Rev. Lett. 92, 190401 (2004)). In 2004, direct two-body correlation measurements in the fermionized gas had not been performed. Looking beyond the single-species gas, Paredes and Cirac (Phys. Rev. Lett. 90, 150402 (2003)) proposed loading two bosonic species into an optical lattice and tuning the sign and strength of the inter-species scattering lengths (for example with Feshbach resonances — magnetically tunable scattering resonances that allow continuous control of the interaction strength): for attractive inter-species coupling the system should exhibit atomic pairing of bosons of different species, a bosonic analogue of Cooper pairing in superconductors, including a 'superconducting' regime with BCS-like extended pairs at weak attraction; in 1D lattices the paired state is a pair Luttinger liquid, a 1D quantum liquid with algebraic correlations carried by pairs rather than single particles. The relevant theoretical model is the two-species Bose–Hubbard Hamiltonian, the lattice analogue of interacting bosons with on-site intra- and inter-species interaction energies UU and UabU_{ab} and tunneling amplitude JJ. Subsequent theory established that the attractive two-species Bose–Hubbard model supports a pair superfluid (PSF) — a phase in which superfluid transport is carried exclusively by interspecies pairs, with no individual-species contribution — and proposed detection schemes: after switching the inter-species interaction from attractive to repulsive and releasing the gas, time-of-flight interference images and density correlations should reveal the disappearance of single-species interference fringes and the appearance of cross-species momentum correlations concentrated near opposite momenta. Two phenomena that an experiment must distinguish from genuine pairing are drag (Andreev–Bashkin) superfluidity — a miscible two-component superfluid in which the currents of the two components are coupled, each entraining the other, without any pairing of atoms — and the formation of tightly bound interspecies molecules. Cuzzuol, Barbiero and Montorsi (SciPost Phys. Core 8, 020 (2025)) added a further diagnostic: pair superfluids are characterized by a nonlocal 'odd-parity' order parameter, a pattern of alternating empty and occupied lattice sites read out from local density measurements, which discriminates them from ordinary atomic superfluids. Experiments with two bosonic species (spin states) in optical lattices exist, but so far with repulsive inter-species interactions, probing miscibility and many-body shifts rather than pairing (e.g. Gadway et al., Phys. Rev. Lett. 105, 045303 (2010)); and quantum-degenerate Bose–Bose mixtures with a broadly tunable interspecies interaction have been realized without a lattice, for instance the 162Dy–164Dy isotope mixture of Dürbeck et al. (Phys. Rev. A 114, L031301 (2026)) using a broad interspecies Feshbach resonance, though there the explored physics is the miscible–immiscible transition and dipolar supersolidity rather than interspecies pairing. No experiment has yet realized the attractive-regime paired state.

Problem Statement

Realize and observe the pairing of two bosonic atomic species with tunable interactions in an optical lattice, as proposed by Paredes and Cirac (Phys. Rev. Lett. 90, 150402 (2003)) and highlighted as an experimental direction by the Tonks–Girardeau-gas experiment of Paredes et al. (Nature 429, 277 (2004)). Concretely: (1) prepare a gas of two bosonic species in an optical lattice with inter-species interactions tuned from repulsive to attractive while keeping single-species coherence and controlled filling, without collapse or prohibitive three-body loss; (2) drive and observe the transition from two independent (or drag-coupled) superfluids to the interspecies-paired state — the bosonic analogue of the BCS 'superconducting' regime, i.e. a pair superfluid in which superfluid transport is carried by pairs, and, in one-dimensional lattices, the pair Luttinger liquid; and (3) verify the pairing through direct observables, resolving in particular the crossover between tightly bound on-site pairs and extended BCS-like pairs at weak attraction. The question is an experimental-physics one: can this paired regime of a bosonic mixture be created and unambiguously identified in the laboratory, distinguishing it from two independent condensates, molecule formation, and counterflow/drag phases?

References

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