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Journal of Physics B Atomic Molecular and Optical Physics

Neutral two-body systems in inhomogeneous magnetic fields: the quadrupole configuration

Hannes Bock, Igor Lesanovsky, Peter Schmelcher

Published Mar 16, 2005• 10 citations
Publisher

Abstract

We investigate the theoretical foundations of neutral two-body systems exposed to an inhomogeneous magnetic field. Various representations for the Hamiltonian describing the coupled centre-of-mass and internal motion are derived. For the specific case of a magnetic quadrupole field we establish the continuous and discrete symmetries and show that the energy levels of the interacting system are two-fold degenerate. We exploit the symmetries in two alternative ways and derive corresponding effective equations of motion. The first approach eliminates two of the six spatial degrees of freedom and leads to an (infinite) set of coupled channel equations for the spin and spatial degrees of freedom. The second approach introduces the projection of the total angular momentum onto the symmetry axis of the quadrupole field as a canonical momentum, thereby eliminating the corresponding cyclic angle.

DOI
10.1088/0953-4075/38/7/010

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We investigate the theoretical foundations of neutral two-body systems exposed to an inhomogeneous magnetic field. Various representations for the Hamiltonian describing the coupled centre-of-mass and internal motion are derived. For the specific case of a magnetic quadrupole field we establish the continuous and discrete symmetries and show that the energy levels of the interacting system are two-fold degenerate. We exploit the symmetries in two alternative ways and derive corresponding effective equations of motion. The first approach eliminates two of the six spatial degrees of freedom and leads to an (infinite) set of coupled channel equations for the spin and spatial degrees of freedom. The second approach introduces the projection of the total angular momentum onto the symmetry axis of the quadrupole field as a canonical momentum, thereby eliminating the corresponding cyclic angle.
We study electronically excited atoms exposed to a magnetic quadrupole field. In order to describe the electron dynamics, a one-body approach is employed. Due to the inhomogeneity of the field, the spatial and spin degrees of freedom become coupled in a unique way. The underlying unitary and anti-unitary symmetries are discussed in detail leading to remarkable features such as a two-fold degeneracy of any energy level. We analyse the energy level structure throughout a wide range of field gradients. An investigation of the electronic spin properties is performed by studying the spatially dependent spin-polarization which exhibits a rich nodal structure. We compute wavelengths and strengths for electromagnetic transitions and provide the selection rules. A discussion of the so-called ellipsoidal states which exhibit unique properties such as large mean orbital angular momenta and spatial compactness is provided. The property of magnetic field-induced permanent electric dipole moments is analysed in detail. Wherever reasonable, the results obtained for the quadrupole field are compared to the homogeneous field case.
We investigate the electronic structure and properties of Rydberg atoms exposed to a magnetic quadrupole field. It is shown that the spatial as well as generalized time-reversal symmetries lead to a two-fold degeneracy of the electronic states in the presence of the external field. A delicate interplay between the Coulomb and magnetic interactions in the inhomogeneous field leads to an unusual weak field splitting of the energy levels as well as complex spatial patterns of the corresponding spin polarization density of individual Rydberg states. Remarkably, the magnetic quadrupole field induces a permanent electric-dipole moment of the atom.
We study the translational motions of homonuclear diatomic molecules prepared in their 3Σ electronic states, deeply bound vibrational states, and rotational states of well-defined parity. The trapping potential arises due to the interaction of the total spin of electrons and orbital angular momentum of nuclei with the trap's quadrupole magnetic field. The translational motion of a molecule is treated classically. We examine the Hamilton equations that govern the center-of-mass dynamics both numerically and analytically. Using data of a hydrogen molecule at the ground vibrational state, we present global dynamics using the Poincaré section method and various types of trajectories: periodic, quasiperiodic, and chaotic. We prove that the Hamiltonian system governing this motion is non-integrable. The particle's orbits are confined to a bound region of space that grows with energy, but for small energies (<1.8 K), the motion is restricted to a processing chamber (a few centimeters). Solutions of equations of motion occurring on the symmetry axis and the horizontal plane are expressed in terms of Jacobi elliptic functions.
Exact centre-of-mass separation is performed for the problem of a charged two-body system in a homogeneous magnetic field. Common eigenstates of three exact and approximate constants of motion are constructed in the creation and annihilation operator formalism. Their expression as an infinite series in coordinate representation is obtained using two real linear canonical transformations. An expansion of the wavefunction in this eigenstate basis provides an infinite system of differential equations for the relative motion. This system of equations differs from the infinite nucleus mass approximation by the existence of small terms which couple states with different values of the magnetic quantum number. The usual limitation of the number of electron Landau states leads to a finite number of coupled equations.
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.
The approximate constant of motion introduced previously for the description of a charged system in a homogeneous magnetic field is interpreted physically as the kinetic momentum of the collective motion. The algebra satisfied by this operator and by the exact constants of motion describes the behavior in a magnetic field of a single particle possessing the total charge of the system. With this algebra, we generalize the approximate constant of motion and obtain a family of operators depending on arbitrary parameters. Canonical transformations based on these new operators separate the Hamiltonian into collective, internal, and coupling terms. These terms take the same form for charged and neutral systems although the collective energy represents different physical behaviors. The coupling between the internal and collective motions is small for some choices of the parameters if at least one particle is much heavier than the other ones.