Bosonic atoms with spin-dependent contact interactions, trapped in state-dependent optical superlattices, realize through state-changing collisions an effective Bose-Hubbard model in which single atoms cannot hop but pairs hop coherently: H=∑⟨i,j⟩[U:(ni+nj)2:+Vninj−t ci†2cj2]H=\sum_{\langle i,j\rangle}[U:(n_i+n_j)^2:+V n_i n_j-t\,c_i^{\dagger 2}c_j^2]. The original proposal established three phases — a Mott insulator, a charge-density wave, and a pair superfluid — using Gutzwiller mean-field theory and matrix-product-state simulations on chains of at most 40 sites, but explicitly acknowledged that the size of those simulations left the asymptotic behavior of the two-particle correlator undetermined (numerical fits suggest an algebraic decay with a nonuniversal exponent around α<0.7\alpha<0.7 depending on U0U_0 and μ\mu) and that the Gutzwiller ansatz does not accurately reproduce the location of the phase transitions. This problem asks for a controlled, thermodynamic-limit characterization of this pair-hopping model — its quantitative phase diagram and the asymptotic decay of the pair correlator. Substantial progress has been made on related models, but this core problem remains open.

Background

An optical lattice is a periodic light-shift potential for ultracold neutral atoms, which under suitable conditions realizes the Bose-Hubbard model: bosons with on-site interaction and a tunnelling amplitude JJ between neighboring wells. In its standard form this model shows a superfluid-to-Mott-insulator transition — the Mott insulator has a fixed integer number of atoms per site and a gap to particle or hole excitations, while the superfluid has gapless phase excitations. A pair superfluid (PSF) is a variant of order in which bosons bind into pairs that delocalize coherently while single bosons remain localized: the single-particle correlator CΔ1=⟨ci†ci+Δ⟩C^1_\Delta=\langle c_i^\dagger c_{i+\Delta}\rangle decays exponentially, while the two-particle correlator CΔ2=⟨ci†2ci+Δ2⟩C^2_\Delta=\langle c_i^{\dagger 2}c_{i+\Delta}^2\rangle retains quasi-long-range order. In one dimension (1D), continuous symmetries cannot be broken at zero temperature, so superfluid order appears as algebraic (power-law) decay of the correlators, described by Tomonaga-Luttinger liquid (TLL) theory, in which the decay exponent is a nonuniversal function of the interactions. A charge-density wave (CDW) is an insulating state with a periodic modulation of the density, here an alternating pattern of 0 and 1 atoms per site. Correlated (or density-dependent) hopping is a lattice transport term in which the motion of a particle depends on the occupation of the sites involved, of the form ci†cjc_i^\dagger c_j multiplied by density operators, or — for pair hopping — the transfer of two atoms at once, ci†2cj2c_i^{\dagger 2}c_j^2. Eckholt and García-Ripoll (Phys. Rev. A 77, 063603 (2008)) proposed engineering such terms with two-state bosonic atoms in state-dependent optical superlattices, in which the two internal states feel lattices displaced relative to each other: a collision that converts a pair of atoms between internal states then forces a coordinated jump of the pair to a different site. For dressed states with asymmetric interactions, Hk=U02:(n↑+n↓)2:+U1n↑n↓H_k=\frac{U_0}{2}:(n_\uparrow+n_\downarrow)^2:+U_1 n_\uparrow n_\downarrow, the approximation U0,U1≪2JU_0,U_1\ll 2J (interactions weak compared to the band gap 2J2J of the superlattice double wells) yields the effective Hamiltonian H=∑⟨i,j⟩[U:(ni+nj)2:+Vninj−t ci†2cj2],H=\sum_{\langle i,j\rangle}\left[U:(n_i+n_j)^2:+V n_i n_j-t\,c_i^{\dagger 2}c_j^2\right], with U=(2U0+U1)/16U=(2U_0+U_1)/16, V=−U1/8V=-U_1/8 and t=U1/16t=U_1/16 for one superlattice geometry. Here U≥0U\ge 0 penalizes site occupation, VV drives neighboring-site correlations, and tt is the pair hopping; single-particle hopping is absent by construction. Unlike pairing schemes based on perturbative second-order tunnelling or Feshbach-resonance attraction, this mechanism is a direct strong process, so the pair binding energy can be comparable to the Mott-insulator interaction scale (of order 1 kHz). The authors mapped the phase diagram in the space of interaction asymmetry and chemical potential μ\mu using two variational methods: the Gutzwiller ansatz (a product of on-site wavefunctions), which is cheap but — as they state — 'does not reproduce accurately neither the location of phase transitions nor the behavior of correlators'; and matrix product states (MPS, a 1D variational wavefunction class whose accuracy is controlled by the bond dimension DD), for chains of up to L=40L=40 sites with D=40D=40. Both methods found Mott insulators with nˉ=1,2,3\bar n=1,2,3 atoms per site, a low-density CDW, and a PSF with vanishing CΔ1C^1_\Delta and slowly decaying CΔ2C^2_\Delta; numerical fits suggested algebraic decay CΔ2∼Δ−αC^2_\Delta\sim\Delta^{-\alpha} with a nonuniversal exponent below α≈0.7\alpha\approx 0.7 (one fitted example α≈0.32\alpha\approx 0.32). Detection was proposed via photoassociation (laser-induced conversion of atom pairs into molecules), which converts the pair correlator into molecular long-range order visible as interference fringes in time-of-flight expansion. The companion paper (New J. Phys. 11, 093028 (2009)) generalized the derivation to two classes of correlated hopping and repeated the numerics with an infinite-size MPS algorithm (iTEBD, bond dimensions up to 64), confirming algebraic decay of CΔ2C^2_\Delta with α\alpha between about 0.5 and 0.6, and stating that all phases are connected by second-order quantum phase transitions. In the broader context, pair superfluidity and pair supersolidity of lattice bosons with pair-hopping terms have since been studied in mixtures, Feshbach-pairing models, extended Bose-Hubbard chains, and triangular- and square-lattice models, establishing that pair hopping generically stabilizes paired condensates; but a controlled, thermodynamic-limit characterization of the specific collision-induced model — the parameter-resolved pair-correlator exponent and the quantitative phase boundaries — remains missing.

Problem Statement

Establish a controlled, thermodynamic-limit characterization of the collision-induced pair-hopping Bose-Hubbard model realized by two-state bosonic atoms in state-dependent optical superlattices, H=∑⟨i,j⟩[U:(ni+nj)2:+Vninj−t ci†2cj2]H=\sum_{\langle i,j\rangle}[U:(n_i+n_j)^2:+V n_i n_j-t\,c_i^{\dagger 2}c_j^2] with no single-particle hopping, the chemical potential μ\mu entering grand-canonically through the free energy F=H−μNF=H-\mu N and U1U_1 serving as the unit of energy (so that U0U_0 and U0/U1U_0/U_1 denote the same interaction asymmetry), replacing the original qualitative L≤40L\le 40 phase diagram by quantitative statements about the same model. The source explicitly acknowledges two jointly linked open issues: (i) determine the true asymptotic long-distance behavior of the two-particle correlator CΔ2=⟨ci†2ci+Δ2⟩C^2_\Delta=\langle c_i^{\dagger 2}c_{i+\Delta}^2\rangle in the pair-superfluid phase, in particular the thermodynamic-limit decay exponent as a function of the interaction asymmetry U0U_0 and chemical potential μ\mu; and (ii) determine the quantitative locations of the phase boundaries between the Mott insulators, the charge-density wave, and the pair superfluid, beyond the known inaccuracy of Gutzwiller mean field and the finite-size and finite-entanglement limitations of the original MPS calculations. The sources additionally discuss higher-dimensional generalizations of the scheme and photoassociation-based detection of the pair superfluid; these situate the model and its experimental relevance, but the source-acknowledged open issues are the two above.

References

  1. María Eckholt and Juan José García-Ripoll, Pair condensation of bosonic atoms induced by optical lattices, Phys. Rev. A 77, 063603 (2008). DOI: 10.1103/PhysRevA.77.063603, https://doi.org/10.1103/PhysRevA.77.063603 (arXiv:0707.0383, https://arxiv.org/abs/0707.0383)
  2. María Eckholt and Juan José García-Ripoll, Correlated hopping of bosonic atoms induced by optical lattices, New J. Phys. 11, 093028 (2009). DOI: 10.1088/1367-2630/11/9/093028, https://doi.org/10.1088/1367-2630/11/9/093028 (arXiv:0908.2383, https://arxiv.org/abs/0908.2383)
  3. Anzi Hu, L. Mathey, Ippei Danshita, Eite Tiesinga, Carl J. Williams, Charles W. Clark, Counterflow and paired superfluidity in one-dimensional Bose mixtures in optical lattices, Phys. Rev. A 80, 023619 (2009). DOI: 10.1103/PhysRevA.80.023619, https://doi.org/10.1103/PhysRevA.80.023619 (arXiv:0906.2150, https://arxiv.org/abs/0906.2150)
  4. M. J. Bhaseen, S. Ejima, F. H. L. Essler, H. Fehske, M. Hohenadler, B. D. Simons, Discrete Symmetry Breaking Transitions Between Paired Superfluids, Phys. Rev. A 85, 033636 (2012). DOI: 10.1103/PhysRevA.85.033636, https://doi.org/10.1103/PhysRevA.85.033636 (arXiv:1111.6778, https://arxiv.org/abs/1111.6778)
  5. Tomasz Sowiński, Ravindra W. Chhajlany, Omjyoti Dutta, Luca Tagliacozzo, Maciej Lewenstein, Criticality in the Bose-Hubbard model with three-body repulsion, Phys. Rev. A 92, 043615 (2015). DOI: 10.1103/PhysRevA.92.043615, https://doi.org/10.1103/PhysRevA.92.043615 (arXiv:1304.4835, https://arxiv.org/abs/1304.4835)
  6. Dirk-Sören Lühmann, Twisted superfluid phase in the extended one-dimensional Bose-Hubbard model, Phys. Rev. A 94, 011603 (2016). DOI: 10.1103/PhysRevA.94.011603, https://doi.org/10.1103/PhysRevA.94.011603 (arXiv:1605.05170, https://arxiv.org/abs/1605.05170)
  7. Wanzhou Zhang, Ruoxi Yin, Yancheng Wang, Pair supersolid with atom-pair hopping on the state-dependent triangular lattice, Phys. Rev. B 88, 174515 (2013). DOI: 10.1103/PhysRevB.88.174515, https://doi.org/10.1103/PhysRevB.88.174515 (preprint arXiv:1208.6506, https://arxiv.org/abs/1208.6506, titled 'Pair supersolid of the extended Bose-Hubbard model with atom-pair hopping on the triangular lattice' and authored by Wanzhou Zhang, Yancheng Wang, Wenan Guo)
  8. Yancheng Wang, Wanzhou Zhang, Hui Shao, Wenan Guo, Extended Bose-Hubbard model with pair hopping on the triangular lattice, Chin. Phys. B 22, 096702 (2013). DOI: 10.1088/1674-1056/22/9/096702, https://doi.org/10.1088/1674-1056/22/9/096702 (arXiv:1302.1376, https://arxiv.org/abs/1302.1376)
  9. Zhi Lin, Chenrong Liu, Yan Chen, Novel quantum phases of two-component bosons with pair hopping in synthetic dimension, Phys. Rev. Lett. 125, 245301 (2020). DOI: 10.1103/PhysRevLett.125.245301, https://doi.org/10.1103/PhysRevLett.125.245301 (arXiv:2004.00223, https://arxiv.org/abs/2004.00223)
  10. Julia Stasińska, Omjyoti Dutta, Luca Barbiero, Maciej Lewenstein, Ravindra W. Chhajlany, Clustered Superfluids in the One Dimensional Bose-Hubbard model with extended correlated hopping, Phys. Rev. B 103, 134513 (2021). DOI: 10.1103/PhysRevB.103.134513, https://doi.org/10.1103/PhysRevB.103.134513 (arXiv:2010.14459, https://arxiv.org/abs/2010.14459)
  11. Manali Malakar, Sudip Sinha, S. Sinha, Formation of paired phases of bosons and their excitations in a square lattice, Phys. Rev. B 108, 054518 (2023). DOI: 10.1103/PhysRevB.108.054518, https://doi.org/10.1103/PhysRevB.108.054518 (arXiv:2305.15307, https://arxiv.org/abs/2305.15307)
  12. C. Menotti and S. Stringari, Detection of pair-superfluidity for bosonic mixtures in optical lattices, Phys. Rev. A 81, 045604 (2010). DOI: 10.1103/PhysRevA.81.045604, https://doi.org/10.1103/PhysRevA.81.045604
  13. Nitya Cuzzuol, Luca Barbiero, Arianna Montorsi, Nonlocal order parameter of pair superfluids, SciPost Phys. Core 8, 020 (2025). DOI: 10.21468/SciPostPhysCore.8.1.020, https://doi.org/10.21468/SciPostPhysCore.8.1.020 (arXiv:2404.15972, https://arxiv.org/abs/2404.15972)
  14. Nitya Cuzzuol and Arianna Montorsi, Fundamental role of nonlocal orders in 1D extended Bose–Hubbard model, Chaos 34, 063116 (2024). DOI: 10.1063/5.0206798, https://doi.org/10.1063/5.0206798 (arXiv:2405.03336, https://arxiv.org/abs/2405.03336)
  15. Alexander Schuckert, Stefan Kühn, Kevin C. Smith, Eleanor Crane, Steven M. Girvin, Constrained many-body phases in a Z2\mathbb{Z}_2-Higgs lattice gauge theory, arXiv:2503.03828 (2025), https://arxiv.org/abs/2503.03828
  16. Domantas Burba, Giedrius Žlabys, Dzmitry Viarbitski, Thomas Busch, Gediminas Juzeliūnas, Chiral and pair superfluidity in triangular ladder produced by state-dependent Kronig-Penney lattice, arXiv:2603.04498 (2026), https://arxiv.org/abs/2603.04498
  17. A. Krzywicka and T. P. Polak, Coexistence of two kinds of superfluidity at finite temperatures in optical lattices, Ann. Phys. 443, 168973 (2022). DOI: 10.1016/j.aop.2022.168973, https://doi.org/10.1016/j.aop.2022.168973 (arXiv:2105.04788, https://arxiv.org/abs/2105.04788)