For ultracold bosonic atoms in a two-dimensional optical superlattice — a square optical lattice superimposed with a second lattice of twice the period, producing two inequivalent sublattices with different potential depths — the homogeneous extended Bose–Hubbard model with staggered on-site energies and sublattice-dependent on-site interactions supports a Mott-insulator, a density-wave, and a modulated-superfluid phase, with characteristic peaks in the time-of-flight momentum distribution and in second-order (noise) correlation maps proposed for their experimental identification. Hou's original calculation (Mod. Phys. Lett. B 23, 25 (2009)) omitted the shallow harmonic trap present in every experiment, noting only qualitatively that the trap makes the effective chemical potential position-dependent, allows spatial phase separation, and should reduce correlation peaks at the cloud edge. The open problem is to compute quantitatively, for this two-dimensional superlattice Bose gas, how a realistic trap profile modifies the homogeneous phase boundaries, the local order-parameter and density distributions, and the cloud-integrated time-of-flight and noise-correlation signals used to identify the phases.

Background

An optical lattice is a standing-wave light shift potential that traps ultracold atoms at micrometer-scale periodic sites; a two-dimensional optical superlattice is built by superimposing two standing waves whose wavelengths differ by a factor of two, which splits the square lattice into two inequivalent sublattices (conventionally A and B) with different potential depths. In the lowest-band tight-binding description the system is governed by the extended Bose–Hubbard model with staggered on-site energies: bosons hop between nearest-neighbor sites with amplitude tt, feel sublattice-dependent repulsive on-site interactions (UAU_A on sublattice A, UBU_B on sublattice B), and experience an energy offset Δ\Delta between sublattices A and B (in Hou's model there is no nearest-neighbor repulsion; the 'extended' character of the Hamiltonian comes from the sublattice structure — both the on-site interaction and the effective chemical potential alternate between the two sublattices). Depending on the ratios of tt, UAU_A, UBU_B, Δ\Delta, and the chemical potential μ\mu, the ground state is a Mott insulator (MI; integer filling with a gap and vanishing superfluid order parameter), a density wave (DW; commensurate but sublattice-modulated integer filling, e.g. occupancy alternating between the sublattices, breaking translational symmetry), or a modulated superfluid (MS; superfluid order coexisting with unequal sublattice densities). Hou (2009) solved this model numerically with the Gutzwiller approach — a time-independent product-over-sites mean-field ansatz in which each site's on-site state is variationally optimized and superfluid order is encoded in the condensate order parameter ⟨ψi⟩\langle \psi_i \rangle — and mapped the MI/DW/MS phase diagram of the homogeneous two-dimensional superlattice (the displayed diagram uses UB=0.5UAU_B = 0.5 U_A and Δ=0.2UA\Delta = 0.2 U_A, with the density-wave lobes attributed to the competition among the tunneling term and the two different sublattice on-site interactions). The connection to experiment relies on two probes: after sudden release and free expansion for a time TT, the time-of-flight (TOF) absorption image reflects the first-order correlation function in lattice momentum space, peaked at reciprocal-lattice vectors such as (0,0)(0,0) and (±2π,±2π)(\pm 2\pi, \pm 2\pi); and shot-to-shot fluctuations of the expanded cloud, measured by noise (Hanbury Brown–Twiss) correlations — the second-order correlator of density fluctuations, proposed as a many-body probe by Altman, Demler, and Lukin (2004) — reveal additional Bragg-like peaks at (±π,±π)(\pm \pi, \pm \pi) that diagnose the diagonal long-range order of the DW and MS phases. A crucial simplification in all of this is the omission of the confining potential: real gases sit in a shallow harmonic trap, whose slowly varying energy offset can be absorbed into a position-dependent effective chemical potential μi=μ−Ωri2\mu_i = \mu - \Omega r_i^2. Then, in the spirit of the local-density approximation, different cloud regions realize different points of the homogeneous phase diagram, producing spatial coexistence ('wedding-cake') shells; spatial integration over the cloud also reshapes the momentum-space correlation peaks. This trap interface was studied early in adjacent settings: Rey, Satija, and Clark (2006) computed Hanbury Brown–Twiss interferograms for bosons in a period-2 superlattice in a confining potential, showing Mott-domain shell structure in the noise signal; Scarola, Demler, and Das Sarma (2006) carefully included the trapping potential in noise-correlation theory for the extended Bose–Hubbard model with density-wave and supersolid order, finding the noise-correlation peak width scales inversely with the trapped Mott-domain size; Roth and co-workers mapped two-color-superlattice phase diagrams from experimental parameters, with the trap included in one dimension by Schmitt, Hild, and Roth (2010). Uniform-lattice theory for the superlattice itself was refined after Hou by generalized effective-potential Landau theory (Wang, Zhang, Eggert, and Pelster 2013; Lin and Liu 2018), by mean-field decoupling (Dhar, Singh, Pai, and Das 2011), and by Wei, Zhang, and Jiang (2016), who computed the square-superlattice phase diagram and analytic time-of-flight absorption pictures — all in the homogeneous system. What was never done, and what Hou's closing discussion addressed only qualitatively — asserting that the presence of a shallow harmonic trapping potential does not change the intrinsic features of the homogeneous analysis, while sketching its consequences (position-dependent effective chemical potential, possible phase separation, lower correlation peaks at the cloud edge) without any calculation — is the corresponding quantitative treatment of the trap for the two-dimensional superlattice: how the trap-induced spatial dependence of μ\mu modifies the computed phase boundaries, the local order-parameter distributions, and the cloud-integrated time-of-flight and noise-correlation signals by which the MS, DW, and MI phases are to be identified experimentally.

Problem Statement

For ultracold bosons in a two-dimensional optical superlattice described by the extended Bose–Hubbard model with staggered sublattice energy offsets and sublattice-dependent on-site repulsions (hopping tt, on-site repulsions UAU_A and UBU_B on sublattices A and B respectively, offset Δ\Delta, chemical potential μ\mu; no nearest-neighbor interaction), whose homogeneous Gutzwiller phase diagram contains Mott-insulator, density-wave, and modulated-superfluid phases with characteristic first-order (time-of-flight) and second-order (noise) correlation peaks in lattice momentum space: determine quantitatively how a realistic shallow harmonic trapping potential modifies this picture. Concretely, incorporate the trap-induced spatial dependence of the effective chemical potential μi=μ−Ωri2\mu_i = \mu - \Omega r_i^2 (with Ω\Omega the trap curvature) and compute (i) how the homogeneous phase boundaries are shifted, distorted, or replaced by spatial coexistence (shell) structure across the cloud; (ii) the local, position-dependent order-parameter and density distributions, including where MS, DW, and MI regions coexist and whether phase separation occurs; and (iii) the observable, cloud-integrated time-of-flight momentum distributions and noise-correlation maps, including how the diagnostic peaks at (0,0)(0,0), at the reciprocal-lattice vectors (e.g. (±2π,±2π)(\pm 2\pi, \pm 2\pi), in units of the inverse lattice constant), and at (±π,±π)(\pm\pi,\pm\pi) are reduced or spatially modulated (e.g. lower peak heights at the cloud edge than at the center), so that the proposed experimental identification of the phases can be confronted with trapped-gas data.

References

  1. Jing-Min Hou, "Quantum phases of ultracold bosonic atoms in a two-dimensional optical superlattice", Modern Physics Letters B 23, 25–33 (2009). DOI: 10.1142/S0217984909017820, https://doi.org/10.1142/S0217984909017820 (preprint: arXiv:0705.1888, https://arxiv.org/abs/0705.1888)
  2. Ana Maria Rey, Indubala I. Satija, Charles W. Clark, "Hanbury Brown–Twiss interferometry for fractional and integer Mott phases", New Journal of Physics 8, 155 (2006). DOI: 10.1088/1367-2630/8/8/155, https://doi.org/10.1088/1367-2630/8/8/155
  3. V. W. Scarola, E. Demler, S. Das Sarma, "Searching for a supersolid in cold-atom optical lattices", Physical Review A 73, 051601(R) (2006). DOI: 10.1103/PhysRevA.73.051601, https://doi.org/10.1103/PhysRevA.73.051601
  4. Felix Schmitt, Markus Hild, Robert Roth, "Ab initio phase diagram of ultracold 87Rb in a one-dimensional two-colour superlattice", Journal of Physics B: Atomic, Molecular and Optical Physics 43, 235301 (2010). DOI: 10.1088/0953-4075/43/23/235301, https://doi.org/10.1088/0953-4075/43/23/235301 (preprint: arXiv:1005.3129, https://arxiv.org/abs/1005.3129)
  5. Fan Wei, Jun Zhang, Ying Jiang, "Quantum phase diagram and time-of-flight absorption pictures of an ultracold Bose system in a square optical superlattice", EPL 113, 16004 (2016). DOI: 10.1209/0295-5075/113/16004, https://doi.org/10.1209/0295-5075/113/16004
  6. Yan-Ying Li, Jing-Min Hou, "Quantum phases of cold bosonic atoms in an optical lattice with inhomogeneous atomic interactions and coexistence of multi phases", Communications in Theoretical Physics 67, 421 (2017). DOI: 10.1088/0253-6102/67/4/421, https://doi.org/10.1088/0253-6102/67/4/421
  7. Kuldeep Suthar, Kwai-Kong Ng, "Staggered quantum phases of dipolar bosons at finite temperatures", Physical Review A 106, 063313 (2022). DOI: 10.1103/PhysRevA.106.063313, https://doi.org/10.1103/PhysRevA.106.063313
  8. Tao Wang, Xue-Feng Zhang, Sebastian Eggert, Axel Pelster, "Generalized effective-potential Landau theory for bosonic quadratic superlattices", Physical Review A 87, 063615 (2013). DOI: 10.1103/PhysRevA.87.063615, https://doi.org/10.1103/PhysRevA.87.063615
  9. Zhi Lin, Wanli Liu, "Analytic calculation of high-order corrections to quantum phase transitions of ultracold Bose gases in bipartite superlattices", Frontiers of Physics 13, 136402 (2018). DOI: 10.1007/s11467-018-0811-1, https://doi.org/10.1007/s11467-018-0811-1 (preprint: arXiv:1804.08465, https://arxiv.org/abs/1804.08465)
  10. Arya Dhar, Manpreet Singh, Ramesh V. Pai, B. P. Das, "Mean-field analysis of quantum phase transitions in a periodic optical superlattice", Physical Review A 84, 033631 (2011). DOI: 10.1103/PhysRevA.84.033631, https://doi.org/10.1103/PhysRevA.84.033631
  11. Felix Schmitt, Markus Hild, Robert Roth, "Phase diagram of bosons in two-color superlattices from experimental parameters", Physical Review A 80, 023621 (2009). DOI: 10.1103/PhysRevA.80.023621, https://doi.org/10.1103/PhysRevA.80.023621
  12. P. Buonsante, V. Penna, A. Vezzani, "Fractional-filling Mott domains in two dimensional optical superlattices", Physical Review A 72, 031602 (2005). DOI: 10.1103/PhysRevA.72.031602, https://doi.org/10.1103/PhysRevA.72.031602
  13. Ehud Altman, Eugene Demler, Mikhail D. Lukin, "Probing many-body states of ultracold atoms via noise correlations", Physical Review A 70, 013603 (2004). DOI: 10.1103/PhysRevA.70.013603, https://doi.org/10.1103/PhysRevA.70.013603
  14. R. Roth, K. Burnett, "Phase diagram of bosonic atoms in two-color superlattices", Physical Review A 68, 023604 (2003). DOI: 10.1103/PhysRevA.68.023604, https://doi.org/10.1103/PhysRevA.68.023604