Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

American Journal of Physics

1D toy model for magnetic trapping

S. Gov, S. Shtrikman, H. Thomas

Published Apr 1, 2000• 10 citations
Publisher

Abstract

We study, both classically and quantum mechanically, the problem of a neutral particle with a spin angular momentum S, mass m, and magnetic moment μ, moving in one dimension in an inhomogeneous magnetic field given by B=B0ẑ+B⊥′xŷ. This problem serves for us as a toy model to study the trapping of neutral particles. We identify K≡[S2(B⊥′)2/μmB03], which is the ratio between the precessional frequency of the particle and its vibrational frequency, as the relevant parameter of the problem. Classically, we find that when μ is antiparallel to B, the particle is trapped provided that K<0.5. We also find that viscous friction, be it translational or precessional, destabilizes the system. Quantum mechanically, we study the problem of a spin S=ℏ/2 particle in the same field. Treating K as a small parameter for the perturbation from the adiabatic Hamiltonian, we find that the lifetime Tesc of the particle in its trapped ground state is Tesc=(Tvib/2π)(1/8πK)exp(2/K), where Tvib=2πmB0/μ(B⊥′)2 is the classical period of the particle when placed in the adiabatic potential V=μ|B|.

DOI
10.1119/1.19436

Related papers

Magnetic Trapping of Neutral Particles: A Study of a Physically Realistic Model

S. Gov, S. Shtrikman, H. Thomas

We study, both classically and quantum-mechanically, the problem of a neutral particle with spin S, mass m and magnetic moment mue, moving in two-dimensions in an inhomogeneous magnetic field given by B_x=B'*x; B_y=-B'*y; B_z=B; We identify K, the ratio between the precessional frequency of the particle and its vibration frequency, as the relevant parameter of the problem. Classicaly, we find that when the magnetic moment is antiparallel to B, the particle is trapped provided that K

arXiv:quant-ph/9812079•Dec 28, 1998•2 citations

Magnetic trapping of neutral particles: Classical and quantum-mechanical study of a Ioffe–Pritchard type trap

S. Gov, S. Shtrikman, H. Thomas

Recently, we developed a method for calculating the lifetime of a particle inside a magnetic trap with respect to spin flips, as a first step in our efforts to understand the quantum mechanics of magnetic traps. The one-dimensional toy model that was used in this study was physically unrealistic because the magnetic field was not curl free. Here, we study, both classically and quantum mechanically, the problem of a neutral particle with spin S, mass m, and magnetic moment μ, moving in three dimensions in an inhomogeneous magnetic field corresponding to traps of the Ioffe–Pritchard “clover-leaf” and “baseball” type. Defining by ωp, ωz, and ωr the precessional, the axial, and the lateral vibrational frequencies, respectively, of the particle in the adiabatic potential Veff =μ|B|, we find classically the region in the (ωr/ωp)−(ωz/ωp) plane where the particle is trapped. Quantum mechanically, we study the problem of a spin-one particle in the same field. Treating ωr/ωp and ωz/ωp as small parameters for the perturbation from the adiabatic Hamiltonian, we derive a closed-form expression for the transition rate 1/Tesc of the particle from its trapped ground state. In the extreme cases, the expression for 1/Tesc reduces to 4πωr exp(−2ωp/ωr) for ωp≫ωr≫ωz, to 82πωpωi exp(−2ωp/ωi) for ωp≫ωr=ωz≡ωi and to π/2 ωr(ωz/ωp)3/2exp(−2ωp/ωz) for ωp≫ωz≫ωr.

arXiv:quant-ph/9907044•Journal of Applied Physics•Apr 15, 2000•13 citations

Magnetic trapping of neutral particles: Classical and Quantum-mechanical study of a Ioffe-Pritchard type trap

Gov, S, Shtrikman, S, Thomas, H

Recently, we developed a method for calculating the lifetime of a particle inside a magnetic trap with respect to spin flips, as a first step in our efforts to understand the quantum-mechanics of magnetic traps. The 1D toy model that was used in this study was physically unrealistic because the magnetic field was not curl-free. Here, we study, both classically and quantum-mechanically, the problem of a neutral particle with spin S, mass m and magnetic moment mu, moving in 3D in an inhomogeneous magnetic field corresponding to traps of the Ioffe-Pritchard, `clover-leaf' and `baseball' type. Defining by omega_p, omega_z and omega_r the precessional, the axial and the lateral vibrational frequencies, respectively, of the particle in the adiabatic potential, we find classically the region in the $(ømega_{r}% (omega_r -- omega_z) plane where the particle is trapped. Quantum-mechanically, we study the problem of a spin-one particle in the same field. Treating omega_r / omega_p and omega_z / omega_p as small parameters for the perturbation from the adiabatic Hamiltonian, we derive a closed-form expression for the transition rate 1/T_{esc} of the particle from its trapped ground-state. We find that in the extreme cases, the expression for 1/T_{esc} is dominated by the largest of the two frequencies omega_r and omega_z.

•Jul 13, 1999•15 citations

Adiabatic motion of a neutral spinning particle in an inhomogeneous magnetic field

Robert G. Littlejohn, Stefan Weigert

The motion of a neutral particle with a magnetic moment in an inhomogeneous magnetic field is considered. This situation, occurring, for example, in a Stern-Gerlach experiment, is investigated from classical and semiclassical points of view. It is assumed that the magnetic field is strong or slowly varying in space, i.e., that adiabatic conditions hold. To the classical model, a systematic Lie-transform perturbation technique is applied up to second order in the adiabatic-expansion parameter. The averaged classical Hamiltonian contains not only terms representing fictitious electric and magnetic fields but also an additional velocity-dependent potential. The Hamiltonian of the quantum-mechanical system is diagonalized by means of a systematic WKB analysis for coupled wave equations up to second order in the adiabaticity parameter, which is coupled to Planck's constant. An exact term-by-term correspondence with the averaged classical Hamiltonian is established, thus confirming the relevance of the additional velocity-dependent second-order contribution.

Physical Review A•Aug 1, 1993•61 citations

Translational dynamics of diatomic molecule in magnetic quadrupole trap

Yurij Yaremko, Maria Przybylska, Andrzej J. Maciejewski

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.

arXiv:2603.03898•The Journal of Chemical Physics•Oct 15, 2025•0 citations

Trapping neutral particles endowed with a magnetic moment by an electromagnetic wave carrying orbital angular momentum: Semiclassical theory

Iwo Białynicki‐Birula, Tomasz Radożycki

The motion of a neutral atom endowed with a magnetic moment interacting with the magnetic field is determined from the Ehrenfest-like equations of motion. These equations for the average values of the translational and spin degrees of freedom are derived from the Schr\"odinger-Pauli wave equation, and they form a set of nine coupled nonlinear evolution equations. The numerical and analytic solutions of these equations are obtained for the combination of the rotating magnetic field of a wave carrying orbital angular momentum and a static magnetic field. The running wave traps the atom only in the transverse direction, while the standing wave traps the atom also in the direction of the beam.

arXiv:1605.08894•Physical Review A•Jun 2, 2016•5 citations
We study, both classically and quantum mechanically, the problem of a neutral particle with a spin angular momentum S, mass m, and magnetic moment μ, moving in one dimension in an inhomogeneous magnetic field given by B=B0ẑ+B⊥′xŷ. This problem serves for us as a toy model to study the trapping of neutral particles. We identify K≡[S2(B⊥′)2/μmB03], which is the ratio between the precessional frequency of the particle and its vibrational frequency, as the relevant parameter of the problem. Classically, we find that when μ is antiparallel to B, the particle is trapped provided that K<0.5. We also find that viscous friction, be it translational or precessional, destabilizes the system. Quantum mechanically, we study the problem of a spin S=ℏ/2 particle in the same field. Treating K as a small parameter for the perturbation from the adiabatic Hamiltonian, we find that the lifetime Tesc of the particle in its trapped ground state is Tesc=(Tvib/2π)(1/8πK)exp(2/K), where Tvib=2πmB0/μ(B⊥′)2 is the classical period of the particle when placed in the adiabatic potential V=μ|B|.
Recently, we developed a method for calculating the lifetime of a particle inside a magnetic trap with respect to spin flips, as a first step in our efforts to understand the quantum mechanics of magnetic traps. The one-dimensional toy model that was used in this study was physically unrealistic because the magnetic field was not curl free. Here, we study, both classically and quantum mechanically, the problem of a neutral particle with spin S, mass m, and magnetic moment μ, moving in three dimensions in an inhomogeneous magnetic field corresponding to traps of the Ioffe–Pritchard “clover-leaf” and “baseball” type. Defining by ωp, ωz, and ωr the precessional, the axial, and the lateral vibrational frequencies, respectively, of the particle in the adiabatic potential Veff =μ|B|, we find classically the region in the (ωr/ωp)−(ωz/ωp) plane where the particle is trapped. Quantum mechanically, we study the problem of a spin-one particle in the same field. Treating ωr/ωp and ωz/ωp as small parameters for the perturbation from the adiabatic Hamiltonian, we derive a closed-form expression for the transition rate 1/Tesc of the particle from its trapped ground state. In the extreme cases, the expression for 1/Tesc reduces to 4πωr exp(−2ωp/ωr) for ωp≫ωr≫ωz, to 82πωpωi exp(−2ωp/ωi) for ωp≫ωr=ωz≡ωi and to π/2 ωr(ωz/ωp)3/2exp(−2ωp/ωz) for ωp≫ωz≫ωr.
Recently, we developed a method for calculating the lifetime of a particle inside a magnetic trap with respect to spin flips, as a first step in our efforts to understand the quantum-mechanics of magnetic traps. The 1D toy model that was used in this study was physically unrealistic because the magnetic field was not curl-free. Here, we study, both classically and quantum-mechanically, the problem of a neutral particle with spin S, mass m and magnetic moment mu, moving in 3D in an inhomogeneous magnetic field corresponding to traps of the Ioffe-Pritchard, `clover-leaf' and `baseball' type. Defining by omega_p, omega_z and omega_r the precessional, the axial and the lateral vibrational frequencies, respectively, of the particle in the adiabatic potential, we find classically the region in the $(ømega_{r}% (omega_r -- omega_z) plane where the particle is trapped. Quantum-mechanically, we study the problem of a spin-one particle in the same field. Treating omega_r / omega_p and omega_z / omega_p as small parameters for the perturbation from the adiabatic Hamiltonian, we derive a closed-form expression for the transition rate 1/T_{esc} of the particle from its trapped ground-state. We find that in the extreme cases, the expression for 1/T_{esc} is dominated by the largest of the two frequencies omega_r and omega_z.
The motion of a neutral particle with a magnetic moment in an inhomogeneous magnetic field is considered. This situation, occurring, for example, in a Stern-Gerlach experiment, is investigated from classical and semiclassical points of view. It is assumed that the magnetic field is strong or slowly varying in space, i.e., that adiabatic conditions hold. To the classical model, a systematic Lie-transform perturbation technique is applied up to second order in the adiabatic-expansion parameter. The averaged classical Hamiltonian contains not only terms representing fictitious electric and magnetic fields but also an additional velocity-dependent potential. The Hamiltonian of the quantum-mechanical system is diagonalized by means of a systematic WKB analysis for coupled wave equations up to second order in the adiabaticity parameter, which is coupled to Planck's constant. An exact term-by-term correspondence with the averaged classical Hamiltonian is established, thus confirming the relevance of the additional velocity-dependent second-order contribution.
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.
The motion of a neutral atom endowed with a magnetic moment interacting with the magnetic field is determined from the Ehrenfest-like equations of motion. These equations for the average values of the translational and spin degrees of freedom are derived from the Schr\"odinger-Pauli wave equation, and they form a set of nine coupled nonlinear evolution equations. The numerical and analytic solutions of these equations are obtained for the combination of the rotating magnetic field of a wave carrying orbital angular momentum and a static magnetic field. The running wave traps the atom only in the transverse direction, while the standing wave traps the atom also in the direction of the beam.