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arXiv:2106.07586 · Physical Review A

Scattering of two particles in a one-dimensional lattice

Seth T. Rittenhouse, P. Giannakeas, Nirav P. Mehta

Published Aug 6, 2021• 0 citations
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Abstract

This study concerns the two-body scattering of particles in a one-dimensional periodic potential. A convenient ansatz allows for the separation of center-of-mass and relative motion, leading to a discrete Schr\"odinger equation in the relative motion that resembles a tight-binding model. A lattice Green's function is used to develop the Lippmann-Schwinger equation, and ultimately derive a multiband scattering $K$ matrix which is described in detail in the two-band approximation. Two distinct scattering lengths are defined according to the limits of zero relative quasimomentum at the top and bottom edges of the two-body collision band. Scattering resonances occur in the collision band when the energy is coincident with a bound state attached to another higher or lower band. Notably, repulsive on-site interactions in an energetically closed lower band lead to collision resonances in an excited band.

DOI
10.1103/physreva.104.023308

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This study concerns the two-body scattering of particles in a one-dimensional periodic potential. A convenient ansatz allows for the separation of center-of-mass and relative motion, leading to a discrete Schr"odinger equation in the relative motion that resembles a tight-binding model. A lattice Green's function is used to develop the Lippmann-Schwinger equation, and ultimately derive a multiband scattering KKK matrix which is described in detail in the two-band approximation. Two distinct scattering lengths are defined according to the limits of zero relative quasimomentum at the top and bottom edges of the two-body collision band. Scattering resonances occur in the collision band when the energy is coincident with a bound state attached to another higher or lower band. Notably, repulsive on-site interactions in an energetically closed lower band lead to collision resonances in an excited band.
We study quasi-one-dimensional scattering of one and two particles with short-range interactions on a discrete lattice model in two dimensions. One of the directions is tightly confined by an arbitrary trapping potential. We obtain the collisional properties of these systems both at finite and zero Bloch quasimomenta, considering as well finite sizes and transversal traps that support a continuum of states. This is made straightforward by using the exact ansatz for the quasi-one-dimensional states from the beginning. In the more interesting case of genuine two-particle scattering, we find that more than one confinement-induced resonances appear due to the nonseparability of the center-of-mass and relative coordinates on the lattice. This is done by solving its corresponding Lippmann-Schwinger-like equation. We characterize the effective one-dimensional interaction and compare it with a model that includes only the effect of the dominant, broadest resonance, which amounts to a single-pole approximation for the interaction coupling constant.
We study the effective Bloch-wave scattering of a spinless Fermi gas in one-dimensional (1D) optical lattices. By tuning the odd-wave scattering length, we find multiple resonances of Bloch waves scattering at the bottom (and the top) of the lowest band, beyond which an attractive (and a repulsive) two-body bound state starts to emerge. These resonances exhibit comparable widths in the deep lattice limit, and the finite interaction range plays an essential role in determining their locations. Based on exact two-body solutions, we construct an effective microscopic model for the low-energy scattering of fermions. The model can reproduce not only the scattering amplitudes of Bloch waves at the lowest-band bottom or top, but also the attractive or repulsive bound states within a reasonably large energy range below or above the band. These results lay the foundation for quantum simulating topological states in cold Fermi gases confined in 1D optical lattices.
In this work, a system including two neutral atoms confined to an external one-dimensional Morse potential was modelled. The problem can be relevant to cold atom physics, where neutral atoms may be effectively confined in radially tight tubes formed by optical lattices. The atom-atom interaction was considered as a nonlocal separable potential. Analytical expressions for wave-function as well as transition matrix were derived. The contributions of bound states and resonances in the complex energy plane were calculated. For numerical computations, the bound states in a system of argon gas confined in graphite were considered. Since the most important quantity in the low energy quantum scattering problems is “scattering length,” considering various values of Morse parameters, the behavior of this parameter was described versus the reduced energy.
We consider the two-body problem in a periodic potential, and study the bound-state dispersion of a spin-$\ensuremath{\uparrow}$ fermion that is interacting with a spin-$\ensuremath{\downarrow}$ fermion through a short-range attractive interaction. Based on a variational approach, we obtain the exact solution of the dispersion in the form of a set of self-consistency equations, and apply it to tight-binding Hamiltonians with on-site interactions. We pay special attention to the bipartite lattices with a two-point basis that exhibit time-reversal symmetry, and show that the lowest-energy bound states disperse quadratically with momentum, whose effective-mass tensor is partially controlled by the quantum metric tensor of the underlying Bloch states. In particular, we apply our theory to the Mielke checkerboard lattice, and study the special role played by the interband processes in producing a finite effective mass for the bound states in a nonisolated flat band.
We study the low-energy two-body scattering in optical lattices with higher-band effects included in an effective potential, using a renormalization group approach. The approach captures the most dominating higher-band effects as well as all multiple scattering processes in the lowest band. For an arbitrary negative free space scattering length (a(s)), a resonance of low-energy scattering occurs as the lattice potential depths reach a critical value v(c); these resonances, with continuously tunable positions v(c) and widths W, can be mainly driven either by intraband or both intra- and interband effects depending on the magnitude of a(s). We have also studied the scattering amplitudes and the formation of molecules when interband effects are dominating, and discussed an intimate relation between molecules for negative a(s) and repulsively bound states pioneered by Winkler et al. [Nature (London) 441, 853 (2006)].