Dimensional crossover of bound complexes in a two-species Bose-Hubbard lattice: correlations and dynamics

We study the equilibrium and nonequilibrium formation of four-particle complexes in a balanced two-species Bose-Hubbard model with repulsive intra- and attractive inter-species interactions. Using exact diagonalization, we characterize the transition from weakly- to strongly-correlated dimer and tetramer states along the one- to two-dimensional crossover in coupled-chain geometries by combining local correlation signatures with global diagnostics such as the binding energy and interspecies entanglement entropy. We show that transverse connectivity between chains qualitatively reshapes the phase diagram, substantially enlarging the tetramer region and, in particular, stabilizing weakly bound tetramers when compared to the one-dimensional chains. By tuning the interchain hopping, we identify a transition from a degenerate manifold of spatially separated dimers to a localized tetramer ground state, driven by the lifting of one-dimensional configurational degeneracies and an associated kinetic-energy gain. Finally, we demonstrate interaction and geometric quench protocols to dynamically prepare these complexes with high fidelity. Our results provide a microscopic framework for engineering and probing few-body bosonic bound states in tunable lattice geometries.

💡 Research Summary

In this work the authors investigate the formation and manipulation of few‑body bound states in a balanced two‑species Bose‑Hubbard lattice with repulsive intra‑species interactions (U>0) and attractive inter‑species interactions (U_AB<0). They focus on the minimal system of four particles (N_A=N_B=2) and employ exact diagonalization in real space to obtain both static and dynamical properties.

The Hamiltonian includes nearest‑neighbour hopping along the x‑direction (J_x, set to unity) and, optionally, along the y‑direction (J_y) that couples parallel 1D chains into a quasi‑2D geometry. By varying the number of coupled chains and the transverse hopping J_y, the authors smoothly interpolate between a strictly one‑dimensional chain and a two‑dimensional square lattice.

Key observables are: (i) the one‑body density ρ^(1)(i,j;σ), (ii) the diagonal elements of the two‑body reduced density matrix ρ^(2)(i,j;i′,j′;σσ′) which directly reveal intra‑ and inter‑species correlations, (iii) the four‑particle binding energy E_b = E_AABB – 2E_AB, and (iv) the inter‑species von Neumann entanglement entropy S_N obtained from the Schmidt decomposition of the many‑body wavefunction. A negative E_b signals a stable bound complex, while the curvature of S_N as a function of U/J_x pinpoints the crossover between weakly‑correlated dimers, strongly‑correlated dimers, and tetramers.

In the pure 1D limit (J_y=0) the phase diagram in the (U_AB/U, U/J_x) plane shows three distinct regions: (a) weakly‑correlated dimers where AB pairs are spatially extended, (b) strongly‑correlated dimers with tightly bound AB pairs but no four‑particle clustering, and (c) tetramers where all four particles occupy the same site. The transition from dimers to tetramers is marked by a sharp drop in the binding energy and a peak in ∂S_N/∂(U/J_x).

Introducing transverse hopping J_y lifts the extensive degeneracy of spatially separated dimers that exists in the decoupled chains. As J_y increases, kinetic‑energy gain favors configurations where the two AB dimers co‑locate, leading to a ground‑state reordering from a manifold of degenerate dimer states to a unique localized tetramer. This dimensional crossover dramatically expands the tetramer region, especially a narrow corridor where tetramers remain stable even when the constituent AB pairs are only weakly bound—a regime absent in strictly 1D systems. The authors demonstrate that this regime can be identified by characteristic two‑body correlation patterns: inter‑species correlations become sharply peaked at zero separation while intra‑species correlations retain a broader profile.

Beyond equilibrium properties, the paper proposes two experimentally realistic protocols to dynamically prepare the bound complexes. (1) Interaction ramps: U_AB is varied in time from zero (no inter‑species attraction) to a target negative value, allowing adiabatic evolution from an unbound state to a dimer or tetramer. (2) Geometric quenches: J_y is suddenly switched on (or off), driving a rapid dimensional crossover. Time‑dependent Schrödinger evolution is solved with a fourth‑order Runge‑Kutta scheme, and the authors monitor the real‑time development of densities, binding energy, entanglement entropy, and the fidelity O(t)=|⟨Ψ_eq(H_f)|Ψ(t)⟩| with the final ground state. For sufficiently slow ramps, the fidelity approaches unity, indicating near‑adiabatic preparation of the desired phase. Even for sudden quenches, the system exhibits coherent oscillations that can be damped by modest adjustments of J_y, achieving fidelities above 0.9.

The study thus provides a comprehensive microscopic framework for (i) diagnosing few‑body bound states using a combination of local correlation functions, global energetic criteria, and entanglement measures; (ii) understanding how lattice geometry and transverse connectivity reshape the stability and internal structure of dimers and tetramers; and (iii) designing concrete dynamical protocols that are compatible with current experimental capabilities such as site‑resolved imaging, deterministic few‑particle preparation, and tunable lattice depths. The results open pathways for exploring richer few‑body physics—such as higher‑order multimers, impurity‑mediated binding, and lattice‑induced Efimov‑like states—in controllable quantum simulators.