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Physical Review A

Quantum propagation of neutral atoms in a magnetic quadrupole guide

Edward A. Hinds, Claudia Eberlein

Published Feb 17, 2000• 34 citations
Publisher

Abstract

We consider the quantized motion of neutral atoms at very low temperature in a two-dimensional magnetic quadrupole structure formed, for example, by four current-carrying wires along the z direction. The magnetic field $\mathbf{B}$ in the guide is proportional to the vector $(x,\ensuremath{-}y).$ We show that this field can be used to make a single-mode atomic de Broglie waveguide which has bound states of low angular momentum, even though the field at the center of the guide goes to zero. We investigate the spectrum and decay rate of the transverse modes for spin-1/2 and spin-1 atoms.

DOI
10.1103/physreva.61.033614

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We consider the quantized motion of neutral atoms at very low temperature in a two-dimensional magnetic quadrupole structure formed, for example, by four current-carrying wires along the z direction. The magnetic field B\mathbf{B}B in the guide is proportional to the vector (x,\ensuremath−y).(x,\ensuremath{-}y).(x,\ensuremath−y). We show that this field can be used to make a single-mode atomic de Broglie waveguide which has bound states of low angular momentum, even though the field at the center of the guide goes to zero. We investigate the spectrum and decay rate of the transverse modes for spin-1/2 and spin-1 atoms.
We consider a waveguide for cold neutral atoms with a magnetic moment proportional to their spin angular momentum. The waveguide consists of a thin wire carrying a constant current. For the spin-1 case, we find bound states for the two-dimensional part of the motion around the wire, and present numerical and approximate analytic results for these bound states. In an experiment, the bound states can decay due to various effects, and we calculate the time scales involved. \textcopyright{} 1996 The American Physical Society.
Atom waveguides are used to manipulate cold atoms in atom interferometers. The creation of atom interferometers using cold atoms in miniature magnetic waveguides is one of many goals of current atom chip research. To achieve a complete understanding of atom propagation in a complicated device such as a guided atom interferometer, a detailed understanding of the ground state and other nearby states is needed. The Frobenius series solutions for the bounded transverse modes of an atomic waveguide are presented here and arbitrary precision arithmetic is used to evaluate the series solutions without roundoff errors. The waveguide potential considered is an infinitely long quadrupole magnetic potential as used in various atom chip waveguides. The simplest case of a guided spin-12 particle is presented here. However, the basic series techniques can be extended to both higher order multipole potentials and higher spin particles, including atoms with hyperfine splitting. The low-field and the high-field seeking states together form the spectrum of the waveguide Hamiltonian. In the limit where the transverse dimension of the guide tends to infinity, the spectrum of the guide changes from a discrete set of low- and high-field seeking states to a continuum of high-field seeking states embedded with a discrete set of low-field seeking states. Although the low-field seeking states are not truly bound, the system is an approximate example of bound states in a continuum first discussed by von Neumann and Wigner. Depending on boundary conditions, the solutions form either a discrete set or a continuum of orthogonal waveguide modes. These are useful for further analysis of ideal waveguide behavior as well as the detailed perturbation studies necessary for analysis of atomic waveguide interferometers.
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.