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arXiv:cond-mat/0403466 · Physical Review A

Nonadiabatic effects in the dynamics of atoms confined in a cylindric time-orbiting-potential magnetic trap

Roberto Franzosi, Bruno Zambon, E. Arimondo

Published Nov 4, 2004• 9 citations
arXivHTMLPDF

Abstract

In a time-orbiting-potential magnetic trap the neutral atoms are confined by means of an inhomogeneous magnetic field superimposed onto an uniform rotating one. We perform an analytic study of the atomic motion by taking into account the nonadiabatic effects arising from the spin dynamics about the local magnetic field. Geometric like magnetic fields determined by Berry's phase appear within the quantum description. Application of the time-dependent variational principle on the original quantum equation leads to a set of dynamical evolution equations for the quantum average value of the position operator and spin variables. Within this approximation we derive the quantum-mechanical ground-state configuration matching the classical adiabatic solution and perform some numerical simulations.

DOI
10.1103/physreva.70.053603

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In a time-orbiting-potential magnetic trap the neutral atoms are confined by means of an inhomogeneous magnetic field superimposed onto an uniform rotating one. We perform an analytic study of the atomic motion by taking into account the nonadiabatic effects arising from the spin dynamics about the local magnetic field. Geometric like magnetic fields determined by Berry's phase appear within the quantum description. Application of the time-dependent variational principle on the original quantum equation leads to a set of dynamical evolution equations for the quantum average value of the position operator and spin variables. Within this approximation we derive the quantum-mechanical ground-state configuration matching the classical adiabatic solution and perform some numerical simulations.
Nonadiabatic motion of Bose-Einstein condensates of rubidium atoms arising from the dynamical nature of a time-orbiting-potential (TOP) trap was observed experimentally. The orbital micromotion of the condensate in velocity space at the frequency of the rotating bias field of the TOP was detected by a time-of-flight method. A dependence of the equilibrium position of the atoms on the sense of rotation of the bias field was observed. We have compared our experimental findings with numerical simulations. The nonadiabatic following of the atomic spin in the trap rotating magnetic field produces geometric forces acting on the trapped atoms.
The time-dependent Schr\"odinger equation for a spin particle in a rotating magnetic field is solved analytically by the cranking method, and the exact solutions are employed to study the nonadiabatic Berry's phase. An alternative expression for Berry's phase is given, which shows that Berry's phase is related to the expectation value of spin along the rotating axis and gives Berry's phase a physical explanation besides its gauge geometric interpretation. This expression also presents a simple algorithm for calculating the nonadiabatic Berry's phase for Hamiltonians that are nonlinear functions of the SU(2) generators. It is shown that nonadiabaticity alters the time evolution ray and in turn changes its Berry's phase. For the SU(2) dynamical group, the nonadiabatic effect on Berry's phase manifests itself as spin alignment (a phenomenon in nuclear physics), and spin-alignment quantization (observed recently in high-spin nuclear physics) is related to Berry's-phase quantization.
The classical model that describes the motion of an atom in a magnetic trap is solved in order to investigate the relationship between the failure of the usual adiabatic approximation assumption and the physical parameters of the trap. This allows to evaluate the effect that reversing of the bias field rotation produces on the vertical position of the atomic orbit, a displacement that is closely related to the adiabatic character of the trap motion. The present investigation has been motivated by a similar experimental test previously carried out in the actual magnetic time orbiting potential trap. We find that the non-adiabatic effects provided by the classical model are extremely small. Thus, we conclude that the theoretical explanation of the experimental measures, requires a quantum description of the dynamics in magnetic traps.
Dynamics of neutral atoms in nonuniform magnetic fields, typical of quadrupole magnetic traps, is considered by applying an accurate method for solving nonlinear systems of differential equations. This method is more general than the adiabatic approximation and, thus, permits one to check the limits of the latter and also to analyze nonadiabatic regimes of motion. An unusual nonadiabatic regime is found when atoms are confined from one side of the zzz axis but are not confined from another side. The lifetime of atoms in a trap in this semiconfining regime can be sufficiently long for accomplishing experiments with a cloud of such atoms. At low temperature, the cloud is ellipsoidal, being stretched in the axial direction and moving along the zzz axis. The possibility of employing the semiconfining regime for studying the relative motion of one component through another in a binary mixture of gases is discussed.
It is well known that any cyclic solution of a spin 1/2 neutral particle moving in an arbitrary magnetic field has a nonadiabatic geometric phase proportional to the solid angle subtended by the trace of the spin. For neutral particles with higher spin, this is true for cyclic solutions with special initial conditions. For more general cyclic solutions, however, this does not hold. As an example, we consider the most general solutions of such particles moving in a rotating magnetic field. If the parameters of the system are appropriately chosen, all solutions are cyclic. The nonadiabatic geometric phase and the solid angle are both calculated explicitly. It turns out that the nonadiabatic geometric phase contains an extra term in addition to the one proportional to the solid angle. The extra term vanishes automatically for spin 1/2. For higher spin, however, it depends on the initial condition. We also consider the valence electron of an alkaline atom. For cyclic solutions with special initial conditions in an arbitrary strong magnetic field, we prove that the nonadiabatic geometric phase is a linear combination of the two solid angles subtended by the traces of the orbit and spin angular momenta. For more general cyclic solutions in a strong rotating magnetic field, the nonadiabatic geometric phase also contains extra terms in addition to the linear combination.
Abstract I consider a quantum spinless charged particle moving in the xy plane under the action of a time-dependent magnetic field described by means of the linear vector potential in the ‘circular’ gauge A = B ( t )(− y , x )/2 and in the presence of the confining potential m g 2 x 2 + y 2 / 2 . This potential guarantees finite values of all second-order moments in the initial equilibrium (thermal) state. Time-dependent mean values of the energy and magnetic moment are expressed in terms of solutions to the classical equation of motion ε ̈ + Ω 2 ( t ) ε = 0 . Simple approximate results are found in the cases of the sudden jump of magnetic field, the parametric resonance, and the adiabatic evolution. They are compared with exact solutions in terms of the hypergeometric function for the specific function B ( t ) of the Epstein–Eckart type. A great amplification of the magnetic moment is discovered for the initial high-temperature states, both for the fast and slow monotonous variations of the magnetic field. Classical adiabatic invariants do not exist if the magnetic field slowly goes to zero or changes its sign.