Two-state bosonic atoms in state-dependent optical superlattices with asymmetric spin-dependent on-site interactions realize, in the perturbative regime U0,U1≪2JU_0,U_1\ll 2J, an effective Bose-Hubbard model with correlated pair hopping and density-assisted hopping, whose one-dimensional ground states include a pair superfluid (PSF) - a phase with quasi-long-range order of bound atomic pairs but no single-particle superfluidity (Eckholt and García-Ripoll, Phys. Rev. A 77, 063603 (2008)). The qualitative phase diagram was established with Gutzwiller mean-field theory and matrix-product-state simulations on chains of at most L=40L=40 sites, leaving the quantitative critical characterization open: the thermodynamic-limit decay exponent of the pair correlator and its parameter dependence, the universality classes of the quantum phase transitions bounding the PSF phase, accurate phase boundaries, and the stability of pair condensation beyond one dimension and beyond the derivation regime.

Background

Ultracold bosonic atoms trapped in optical lattices - periodic potentials made from standing-wave laser light - realize the Bose-Hubbard model, in which atoms hop between neighboring sites and interact on site; depending on the ratio of interaction to hopping, the ground state is either a superfluid or a Mott insulator (MI), a commensurately filled, gapped state with a fixed number of atoms per site. Eckholt and García-Ripoll (2008) analyzed two-state bosonic atoms loaded into a state-dependent optical superlattice (a lattice with a doubled unit cell, whose potential differs for the two internal atomic states) with asymmetric on-site interactions Hk=(U0/2):(n↑+n↓)2:+U1n↑n↓H_k=(U_0/2):(n_\uparrow+n_\downarrow)^2:+U_1 n_\uparrow n_\downarrow, where nσn_\sigma counts atoms in internal state σ\sigma. They showed that when the interaction asymmetry is small compared with the band splitting 2J2J of the double wells formed by the superlattice (U0,U1≪2JU_0,U_1\ll 2J), second-order tunnelling yields an effective low-band Hamiltonian with correlated (pair) hopping,

H=∑⟨i,j⟩[U : ⁣(ni+nj)2 ⁣:+V ninj−t ci†2cj 2],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; here cic_i annihilates a boson on site ii, ni=ci†cin_i=c_i^\dagger c_i, :⋯::\cdots: denotes normal ordering, ⟨i,j⟩\langle i,j\rangle runs over neighboring sites, and the pair-hopping term implicitly includes its Hermitian conjugate (written explicitly in the 2009 follow-up) so that HH is Hermitian. The distinctive term is −t ci†2cj 2-t\,c_i^{\dagger 2}c_j^{\,2}: to this order, single atoms cannot hop - only bound pairs tunnel. Using Gutzwiller mean-field theory (a site-factorized variational approximation, useful for qualitative trends but unreliable for correlations and transition locations) and matrix-product-state (MPS) calculations - a variational method for one-dimensional quantum many-body ground states, here limited to chains of L≤40L\le 40 sites and bond dimension D=40D=40 - the authors mapped out a phase diagram containing the MI, density-wave (DW) phases with alternating site occupations, and a pair superfluid (PSF): a gapless phase with quasi-long-range pair order, diagnosed by the pair correlator CΔ2=⟨ci†2ci+Δ 2⟩C^2_\Delta=\langle c_i^{\dagger 2}c_{i+\Delta}^{\,2}\rangle decaying algebraically (as a power law) with distance Δ\Delta while single-particle correlations decay faster. Because continuous symmetries cannot be spontaneously broken in one dimension, such algebraic "quasi-long-range" order is the 1D signature of superfluidity (Luttinger-liquid behavior, the generic algebraically ordered 1D quantum fluid). The observed power-law decay carried a fitted exponent that varied with parameters (an example fit α≈0.32\alpha\approx 0.32, with fitted nonuniversal values up to α≲0.7\alpha\lesssim 0.7), but the accessible system sizes were too small to establish the thermodynamic-limit exponent, its dependence on U0,U1U_0,U_1 and the chemical potential, or the universality classes (the sets of critical exponents and scaling laws shared by all systems in a given transition class) of the MI–PSF and DW–PSF boundaries. For detection, the authors proposed photoassociation - using light to convert atom pairs into molecules, whose interference pattern would reveal pair coherence. Their open questions also included the behavior when the derivation regime U0,U1≪2JU_0,U_1\ll 2J fails and the stability of pair condensation in spatial dimension d>1d>1 (where true long-range pair order, rather than algebraic order, becomes possible); on the experimental side, the robustness of the proposed detection signatures against losses, finite photoassociation efficiency, and noise likewise remains open.

Problem Statement

Determine the complete critical characterization of the pair superfluid (PSF) of the effective correlated-hopping Hamiltonian

H=∑⟨i,j⟩[U : ⁣(ni+nj)2 ⁣:+V ninj−t ci†2cj 2]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]

on a one-dimensional lattice with nearest-neighbor pairs ⟨i,j⟩\langle i,j\rangle, where cic_i annihilates a boson on site ii, ni=ci†cin_i=c_i^\dagger c_i, :⋯::\cdots: denotes normal ordering, the pair-hopping term ci†2cj 2c_i^{\dagger 2}c_j^{\,2} implicitly includes its Hermitian conjugate so that HH is Hermitian, and U=(2U0+U1)/16U=(2U_0+U_1)/16, V=−U1/8V=-U_1/8, t=U1/16t=U_1/16. This is the model derived by Eckholt and García-Ripoll for bosonic atoms in state-dependent optical superlattices, together with its two-asymmetry-channel generalization from the 2009 follow-up, in which pair hopping t∝2g↑↓−g↑↑−g↓↓t\propto 2g_{\uparrow\downarrow}-g_{\uparrow\uparrow}-g_{\downarrow\downarrow} and correlated hopping j∝g↑↑−g↓↓j\propto g_{\uparrow\uparrow}-g_{\downarrow\downarrow} arise from the two independent asymmetries of the interactions gσσ′g_{\sigma\sigma'} between the internal atomic states σ,σ′\sigma,\sigma'. Concretely, establish with control over finite-size and finite-entanglement effects: (1) the thermodynamic-limit asymptotic behavior of the pair correlator CΔ2=⟨ci†2ci+Δ 2⟩C^2_\Delta=\langle c_i^{\dagger 2}c_{i+\Delta}^{\,2}\rangle - whether it decays algebraically throughout the PSF phase, and the decay exponent as a function of the model parameters (U,V,tU,V,t and the chemical potential, together with the correlated hopping jj for the 2009 two-channel generalization; on the microscopic realization, U0,U1U_0,U_1 fix U,V,tU,V,t through the relations above, so that V=−2tV=-2t); (2) the universality classes (the sets of critical exponents and scaling laws shared by all systems at the same transition) of the quantum phase transitions bounding the PSF phase, including the Mott-insulator (MI)–PSF and density-wave (DW)–PSF boundaries and any first-order segments; (3) quantitatively accurate phase boundaries of the PSF phase, beyond Gutzwiller mean-field theory and finite-bond-dimension matrix-product-state (MPS) calculations on short chains; and (4) whether pair condensation survives in spatial dimension d>1d>1 - true long-range pair order and the nature of its transitions - and how the phase structure changes when the perturbative derivation condition U0,U1≪2JU_0,U_1\ll 2J breaks down. These four questions are facets of one objective: a controlled phase diagram and critical characterization of the interaction-induced pair condensate of this model. (Experimental identification of the pair condensate against losses, finite photoassociation efficiency, and noise is a closely related but distinct open direction, related to the detection proposals of the source work; it is recorded here for context and is not part of this problem's completion boundary.)

References

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