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arXiv:quant-ph/0605147 · Physical Review A

Quasi-Hermitian pseudopotential for higher partial wave scattering

Iris Reichenbach, Andrew Silberfarb, R. Stock, Ivan Deutsch

Published Oct 31, 2006• 9 citations
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Abstract

We formulate a quasi-Hermitian delta-shell pseudopotential for higher partial wave scattering, and show that any such potential must have an energy-dependent regularization. The quasi-Hermiticity of the Hamiltonian leads to a complete set of biorthogonal wave functions that can be used as a basis to expand and diagonalize other two-body Hamiltonians. We demonstrate this procedure for the case of ultracold atoms in a polarization-gradient optical lattice, interacting pairwise when two atoms are transported together from separated lattice sites. Here the pseudopotential depends explicitly on the trapping potential. Additionally, we calculate the location of trap-induced resonances for higher partial waves, which occur when a molecular eigenstate is shifted to resonance with a trap eigenstate. We verify the accuracy of the pseudopotential approach using a toy model in which a square well acts as the true interaction potential, and see excellent agreement.

DOI
10.1103/physreva.74.042724

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We formulate a quasi-Hermitian delta-shell pseudopotential for higher partial wave scattering, and show that any such potential must have an energy-dependent regularization. The quasi-Hermiticity of the Hamiltonian leads to a complete set of biorthogonal wave functions that can be used as a basis to expand and diagonalize other two-body Hamiltonians. We demonstrate this procedure for the case of ultracold atoms in a polarization-gradient optical lattice, interacting pairwise when two atoms are transported together from separated lattice sites. Here the pseudopotential depends explicitly on the trapping potential. Additionally, we calculate the location of trap-induced resonances for higher partial waves, which occur when a molecular eigenstate is shifted to resonance with a trap eigenstate. We verify the accuracy of the pseudopotential approach using a toy model in which a square well acts as the true interaction potential, and see excellent agreement.
We derive a generalized zero-range pseudopotential applicable to all partial wave solutions to the Schrödinger equation based on a delta-shell potential in the limit that the shell radius approaches zero. This properly models all higher order multipole moments not accounted for with a monopolar delta function at the origin, as used in the familiar Fermi pseudopotential for s-wave scattering. By making the strength of the potential energy dependent, we derive self-consistent solutions for the entire energy spectrum of the realistic potential. We apply this to study two particles in an isotropic harmonic trap, interacting through a central potential, and derive analytic expressions for the energy eigenstates and eigenvalues.
We present a zero-range pseudopotential applicable for all partial wave interactions between neutral atoms. For p and d waves, we derive effective pseudopotentials, which are useful for problems involving anisotropic external potentials. Finally, we consider two nontrivial applications of the p-wave pseudopotential: we solve analytically the problem of two interacting spin-polarized fermions confined in a harmonic trap, and we analyze the scattering of p-wave interacting particles in a quasi-two-dimensional system.
We consider the feasibility of observing a trap-induced resonance [Stock et al., Phys. Rev. Lett. 91, 183201 (2003)] for the case of two 133Cs^{133}\mathrm{Cs}133Cs atoms, trapped in separated wells of a polarization-gradient optical lattice, and interacting through a multichannel scattering process. Due to the anomalously large scattering length of cesium dimers, a strong coupling can occur between vibrational states of the trap and a weakly bound molecular state that is made resonant by the ac Stark shift of the lattice. We calculate the energy spectrum of the two-atom system as a function of the distance between two potential wells by connecting the solutions of the Schr\"odinger equation for the short-range molecular potential to that of the long-range trap in a self-consistent manner. The short-range potential is treated through a multichannel pseudopotential, parametrized by the KKK matrix, calculated numerically for atoms in free space in a close-coupling approximation. This captures both the bound molecular spectrum as well as the energy-dependent scattering for all partial waves. We establish realistic operating conditions under which the trap-induced resonance could be observed and show that this strong and coherent interaction could be used as a basis for high-fidelity two-qubit quantum logic operations.
We derive pseudopotentials that describe the scattering between two particles in two dimensions for any partial wave mmm, whose scattering strength is parametrized in terms of the phase shift ${\ensuremath{\delta}}_{m}$. Using our m=0m=0m=0 pseudopotential, we develop a coupled-channel model with two-dimensional (2D) zero-range interactions, which describes the two-body physics across a Feshbach resonance. Our model predicts the scattering length, the binding energy, and the ``closed channel molecular fraction'' of two particles; these observables can be measured in experiments on ultracold quasi-2D atomic Bose and Fermi gases with present-day technology.
We describe a model for s-wave collisions between ground-state atoms in optical lattices, considering especially the limits of quasi-one- and two-dimensional axisymmetric harmonic confinement. When the atomic interactions are modeled by an s-wave Fermi pseudopotential, the relative motion energy eigenvalues can easily be obtained. The results show that except for a bound state, the trap eigenvalues are consistent with one- and two-dimensional scattering with renormalized scattering amplitudes. For absolute scattering lengths large compared with the tightest trap width, our model predicts a bound state of low energy and nearly-isotropic wave function extending on the order of the tightest trap width.
The zero-range potential approach is extended for the description of situations where two-body scattering is resonant in arbitrary partial waves. The formalism generalizes the Fermi pseudopotential which can be used only for sss-wave broad resonances. In a given channel, the interaction is described either in terms of a contact condition on the wave function or with a family of pseudopotentials. We show that it is necessary to introduce a regularized scalar product for wave functions obtained in the zero-range potential formalism (except for the Fermi pseudopotential). This metrics shows that the geometry of these Hilbert spaces depends crucially on the interaction.