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Physical Review A

Spectral properties and lifetimes of neutral fermions and bosons in a magnetic quadrupole trap

Igor Lesanovsky, Peter Schmelcher

Published Mar 23, 2005• 13 citations
Publisher

Abstract

We investigate the motion of neutral fermions and bosons in a three-dimensional magnetic quadrupole trap. Inspecting the underlying Hamiltonian a variety of symmetries is revealed which give rise to degeneracies of the resonance energies. Our numerical approach which involves the eigenvalue problem resulting from a Sturmian basis set together with the complex scaling method enables us to calculate several hundred resonance states. The distributions of the energies and decay widths of the resonances are analyzed for both spin-$\frac{1}{2}$ fermions and spin-1 bosons. We also investigate under what conditions quasibound states with long lifetimes can be achieved. An effective scalar Schr\"odinger equation describing such states is derived. The results are applied to the cases of the alkali-metal atoms $^{87}\mathrm{Rb}$ and $^{6}\mathrm{Li}$ trapped in a hyperfine ground state.

DOI
10.1103/physreva.71.032510

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We investigate the motion of neutral fermions and bosons in a three-dimensional magnetic quadrupole trap. Inspecting the underlying Hamiltonian a variety of symmetries is revealed which give rise to degeneracies of the resonance energies. Our numerical approach which involves the eigenvalue problem resulting from a Sturmian basis set together with the complex scaling method enables us to calculate several hundred resonance states. The distributions of the energies and decay widths of the resonances are analyzed for both spin-12\frac{1}{2}21​ fermions and spin-1 bosons. We also investigate under what conditions quasibound states with long lifetimes can be achieved. An effective scalar Schr"odinger equation describing such states is derived. The results are applied to the cases of the alkali-metal atoms 87Rb^{87}\mathrm{Rb}87Rb and 6Li^{6}\mathrm{Li}6Li trapped in a hyperfine ground state.
We investigate the resonant motion of neutral spin-12\frac{1}{2}21​-fermions in a magnetic guide. A wealth of unitary and antiunitary symmetries is revealed in particular giving rise to a twofold degeneracy of the energy levels. To compute the energies and decay widths of a large number of resonances the complex scaling method is employed. We discuss the dependence of the lifetimes on the angular momentum of the resonance states. In this context the existence of so-called quasibound states is shown. In order to approximately calculate the resonance energies of such states a radial Schr\"odinger equation is derived which improves the well-known adiabatic approximation. The effects of an additionally applied homogeneous Ioffe field on the resonance energies and decay widths are also considered. The results are applied to the case of the 6Li^{6}\mathrm{Li}6Li atom in the F=12F=\frac{1}{2}F=21​ hyperfine ground state.
We investigate the resonant quantum motion of spin-1 particles in a magnetic guide. A symmetry analysis is undertaken in order to reveal the complex symmetry properties of the system which lead to a twofold degeneracy in the energy-level spectrum. Lifetimes and energies of the resonance states are computed by employing the complex scaling method. By analyzing several hundreds of resonances we are able to make conclusions about the global properties of the resonance spectrum. The effect of an Ioffe field which is applied parallel to the guide is also discussed. We observe a global increase of the lifetimes with increasing Ioffe-field strength. For certain parameter regimes we find the ground-state resonance to exhibit a longer lifetime than the energetically neighbored excited states. The latter could have interesting implications on the time evolution of trapped ultracold atomic ensembles. Applications of our results to calculate the resonance energies andlifetimes of the trapped alkali-metal atoms 7Li^{7}\mathrm{Li}7Li and 87Rb^{87}\mathrm{Rb}87Rb are outlined.
Subject of this thesis is the investigation of the quantum dynamics of ultracold atoms in confined geometries. We discuss the behavior of ground state atoms inside a quadrupole magnetic field. Such atoms in enough weak magnetic fields can be approximately treated as neutral point-like particles. Complementary to the well-known positive energy resonances, we point out the existence of short-lived negative energy resonances. The latter originate from a fundamental symmetry of the underlying Hamiltonian. We drive a mapping of the two branches of the spectrum. Moreover, we analyze atomic hyperfine resonances in a magnetic quadrupole field. This corresponds to the case for which both the hyperfine and Zeeman interaction, are comparable, and should be taken into account. Finally we develop a general grid method for multichannel scattering of two atoms in a two-dimensional harmonic confienment. With our approach we analyze transverse excitations/deexcitations in the course of the collisional process (distinguishable or identical atoms) including all important partial waves and their couplings due to the broken spherical symmetry. Special attention is paid to suggest a non-trivial extension of the CIRs theory developed so far only for the single-mode regime and zero-energy limit.
We investigate resonances of spin-1 bosons in a three-dimensional magnetic quadrupole field. Complementary to the well-known positive energy resonances it is shown that there exist short-lived, i.e., broad, negative energy resonances. The latter are characterized by an atomic spin that is aligned antiparallel to the local magnetic field direction. In contrast to the positive energy resonances the lifetimes of the negative energy resonances decrease with increasing total magnetic quantum number. We derive a mapping of the two branches of the spectrum.
We consider quantized motion of neutral atoms cooled below the recoil limit in a quadrupole magnetostatic trap. Because of Majorana transitions to untrapped levels near the point of zero field at the trap center, all quantum levels have a nonzero decay rate. The Schrödinger equation associated with the potential gμB · S (S is the total atomic spin) takes the form of coupled equations in r when the spinor components are expanded in spherical harmonics. We integrate the multichannel problem numerically to obtain asymptotic phase shifts, resonance energies, and widths. For S = 1/2, the lowest levels have widths somewhat less than their spacing. Thus the trap quantum-level structure might possibly be observable if the atoms are sufficiently cold, namely, in the 0.1-μK regime for most atoms and attainable trap field gradients. The width decreases rapidly with increasing MJ, the angular momentum about the symmetry axis. Spectroscopic linewidths of a few hertz are possible if there is enough population in the lowest levels with a few MJ quanta. The decay rate of the lowest levels, however, is probably too rapid for studying Bose–Einstein condensation in such a trap.
It is shown that the energies and the lifetimes of unstable modes of quantum propagation of neutral atoms in a quadrupole magnetic guide can be calculated simply by diagonalizing the complex scaled Hamiltonian. The results reproduce those obtained for the same geometry, but by a different method, by E. A. Hinds and C. Eberlein [Phys. Rev. A 61, 033614 (2000)]. The destabilizing effect of a weak bias magnetic field on the l=0l=0l=0 propagation mode of s=1/2s=1/2s=1/2 atoms is illustrated.