We investigate theoretically the stationary states of two bosons in a one-dimensional optical lattice within the Bose-Hubbard model. Starting from a finite lattice with periodic boundary conditions, we effect a partial separation of the center-of-mass and relative motions of the two-atom lattice dimer in the lattice momentum representation, and carefully analyze the eigenstates of the relative motion. In the limit when the lattice becomes infinitely long, we find closed-form analytic expressions for both the bound state and the dissociated states of the lattice dimer. We outline the corresponding analysis in the position representation. The results are used to discuss three ways to detect the dimer: by measuring the momentum distribution of the atoms, by finding the size of the molecule with measurements of atom number correlations at two lattice sites, and by dissociating a bound state of the lattice dimer with a modulation of the lattice depth.Comment: This version is very close to the published version. 14 pages, 3 figure
We model two bosons in an optical lattice near a Feshbach or photoassociation resonance, focusing on the Bose-Hubbard model in one dimension. Whereas the usual atoms-only theory with a tunable scattering length yields one bound state for a molecular dimer for either an attractive or repulsive atom-atom interaction, for a sufficiently small direct background interaction between the atoms a two-channel atom-molecule theory may give two bound states that represent attractively and repulsively bound dimers occurring simultaneously. Such unusual molecular physics may be observable for an atom-molecule coupling strength comparable to the width of the dissociation continuum of the lattice dimer, which is the case, for instance, with narrow Feshbach resonances in Na, 87 Rb, and 133 Cs or low-intensity photoassociation in 174 Yb.
Within the Bose-Hubbard model we theoretically determine the stationary states of two distinguishable atoms in a one-dimensional optical lattice and compare with the case of two identical bosons. A heterodimer has odd-parity dissociated states that do not depend on the interactions between the atoms, and the lattice momenta of the two atomic species may have different averages even for a bound state of the dimer. We discuss methods to detect the dimer. The different distributions of the quasimomenta of the two species may be observed in suitable time-of-flight experiments. Also, an asymmetry in the line shape as a function of the modulation frequency may reveal the presence of the odd-parity dissociated states when a heterodimer is dissociated by modulating the depth of the optical lattice.
We study the states of one and two atoms in a rotating ring lattice in a Hubbard type tightbinding model. The model is developed carefully from basic principles in order to properly identify the physical observables. The one-particle ground state may be degenerate and represent a finite flow velocity depending on the parity of the number of lattice sites, the sign of the tunneling matrix element, and the rotation speed of the lattice. Variation of the rotation speed may be used to control one-atom states, and leads to peculiar behaviors such as wildly different phase and group velocities for an atom. Adiabatic variation of the rotation speed of the lattice may also be used to control the state of a two-atom lattice dimer. For instance, at a suitably chosen rotation speed both atoms are confined to the same lattice site.
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