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arXiv:cond-mat/0703573 · Physical Review A

Tunneling couplings in discrete lattices, single-particle band structure, and eigenstates of interacting atom pairs

Rune Piil, Klaus Mølmer

Published Aug 17, 2007• 46 citations
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Abstract

By adjusting the tunneling couplings over longer than nearest-neighbor distances, it is possible in discrete lattice models to reproduce the properties of the lowest energy band of a real, continuous periodic potential. We propose to include such terms in problems with interacting particles, and we show that they have significant consequences for scattering and bound states of atom pairs in periodic potentials.

DOI
10.1103/physreva.76.023607

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By adjusting the tunneling couplings over longer than nearest-neighbor distances, it is possible in discrete lattice models to reproduce the properties of the lowest energy band of a real, continuous periodic potential. We propose to include such terms in problems with interacting particles, and we show that they have significant consequences for scattering and bound states of atom pairs in periodic potentials.
So far we have discussed particles moving in space that is a continuum. In solidstate physics it is more useful to employ what is known as a tight-binding picture where an electron is assumed to be tied to a particular atom, occasionally hopping to a neighboring atom. In this picture, the electron’s position is a discrete quantity-it is either on one atom or on its neighbor and not anywhere in between. This tightbinding picture may be derived using appropriate basis functions. For our purposes, we postulate that the kinetic motion of the electron is brought about by hopping. Therefore the kinetic energy is c†iσc jσ, (9.1) where < i j > signifies that sites i and j are nearest neighbors. The negative sign implies that hopping lowers the energy of the system by an amount t. But there is a price to be paid for hopping, in the form of a potential energy. The potential energy is assumed to be short-ranged, namely it is present only if a site has two electrons so that they repel with some energy U . But Pauli’s exclusion principle forbids two electrons with the same spin from residing at the same site. Thus while hopping lowers energy, hopping onto a site already occupied by an electron is either forbidden or has an energy cost U . The potential energy may be written as V =U∑ i ni↑ni↓, (9.2) where niσ = c † iσciσ is the number of electrons in site i. These two put together, form the famous Hubbard model of condensed matter physics. It is mathematically well defined given that short distances have a lower bound, namely the lattice spacing. The phases of this model depend on the ratio U/t and also the number of electrons per site (and temperature, if present). One can have variants of this model as well. The extended Hubbard model includes interaction between the nearest neighbors. Then we have a term such as
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.
The quantum-mechanical transport properties of cold atoms in an optical lattice are described in terms of an analytically tractable model. The atoms are cooled down to the lowest energy band in the periodic optical potential, in the tight- binding limit. The transport mechanisms are coherent tunneling through potential barriers, and momentum recoil at optical pumping. Expressions are obtained for the time-dependent population distributions over the energy eigenstates and over localized states. These expressions are modified in the presence of a dark atomic state, that is uncoupled to the field.
We investigate tunneling properties of a bound pair of Fermi atoms in an optical lattice. Including a realistic cosine-shape optical lattice potential, we calculate the molecular wavefunction ψ. Starting from the weak-coupling regime, we show how ψ shrinks as one increases the strength of a pairing interaction between atoms. When the molecular size becomes comparable to the lattice constant in the intermediate coupling regime, we find from the spatial structure of ψ that the molecular tunneling between lattice sites is accompanied by virtual dissociation. This phenomenon is shown to be suppressed in the strong-coupling regime due to very strong binding energy. We point out that, while the molecular tunneling accompanied by virtual dissociation can be described by the Hubbard model, it cannot describe the molecular tunneling with no virtual dissociation in the strong-coupling regime.
Abstract. We study periodic well-to-well flopping of rubidium atoms in one-dimensional grey optical lattices using a nondestructive, real-time measurement technique and quantum Monte Carlo wavefunction simulations. The observed flopping rates as well as flopping rates extracted from exact band structure calculations can largely be reproduced using adiabatic models that employ the Born–Oppenheimer approximation. The adiabatic model is greatly improved by taking into account a gauge potential that is added to the usual adiabatic light-shift potential. The validity of the adiabatic model allows us to interpret the observed flopping phenomenon as periodic well-to-well tunnelling. At low intensities and in related far-off-resonant optical lattices the adiabatic model fails. There, a weak-coupling model becomes valid, which describes the well-to-well flopping as a Rabi oscillation between weakly coupled states, but not as a tunnel effect.