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

Journal of the Optical Society of America B

Quantized motion of atoms in a quadrupole magnetostatic trap

T. H. Bergeman, Patrick J. McNicholl, Jan B. Kycia, Harold J. Metcalf, Nandor L. Balazs

Published Nov 1, 1989• 70 citations
Publisher

Abstract

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.

DOI
10.1364/josab.6.002249

Related papers

Quantum propagation of neutral atoms in a magnetic quadrupole guide

Edward A. Hinds, Claudia Eberlein

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.

Physical Review A•Feb 17, 2000•34 citations

Rotating trapped Bose-Einstein condensates

Alexander L. Fetter, Rocio R. Jauregui, J. Récamier +1 more

I review free and trapped Bose‐Einstein condensates (BECs) [1, 2, 3, 4, 5]. In an isotropic harmonic potential V(r) = 12Mω2r2, the single‐particle ground state introduces a new intrinsic scale of length [the ground‐state size d = ℏ/(Mω)] and energy [the ground‐state energy E0 = 32ℏω]. When the trap rotates [6, 7, 8], there is a critical angular velocity for the appearance of one or a few quantized vortices. For more rapid rotation [9, 10, 11, 12, 13] the condensate contains a vortex array. The resulting centrifugal forces expand the condensate radially and shrink it axially; thus the condensate becomes effectively two‐dimensional. If the external rotation speed approaches the frequency of the radial harmonic confining potential, the condensate enters the “lowest‐Landau‐level” regime, and a simple description again becomes possible [14, 15]. Eventually, the system is predicted to make a quantum phase transition to a highly correlated state analogous to the fractional quantum Hall states of electrons in a strong magnetic field [16, 17, 18, 19, 20].

AIP conference proceedings•Jan 1, 2008•77 citations

Applications of rotational excitations in Bose-Einstein condensates

Anssi Collin

This thesis contains theoretical studies of weakly interacting, trapped Bose-Einstein condensed gases. The main emphasis is on the rotational properties of these systems. A typical Bose-Einstein condensate reacts to external rotation by forming singly quantized vortices. By the word ”typical” we refer to a condensate with repulsive interparticle interactions and with a frozen spin degree of freedom. However, there are some atomic species for which the low-energy interactions are attractive. Moreover, the tuning of different condensate properties is relatively easy. For example, there are ways to experimentally change the effective interaction from repulsive to attractive. For an attractively interacting condensate the vortex is not the only possible rotational state but the sample may also experience rotation through its center of mass. In the first part of the thesis, we analyze some properties of a rotating attractive Bose gas at the zero temperature limit. First, we note that in order to maintain energetical stability, the gas has to be rotated in an external trap which is steeper than a harmonic trap. We study the system in an anharmonic quadratic + quartic potential. The rotating ground state phase-space is determined with analytical and numerical methods. If the interactions are strong enough, the rotating ground state appears to be a state with the rotating center of mass. On the other hand, at the limit of weak interactions the system experiences rotation by forming a multiply quantized vortex. We also determine the spectrum of elementary excitations for these quantum many-particle states. We find that the broken symmetry of the center of mass rotating state induces an additional zero-energy (Goldstone) mode and that the small oscillations of the ground state exhibit the decoupling between the center of mass and relative motions. If the spin of the atoms is allowed to change freely, a rich variety of topological excitations in condensates can be studied. External rotation is one of the tools for examining the excitations in spinor condensates. In a spinless scalar condensate the appearance of a quantized vortex is manifested by the vortex line in which the condensate density vanishes. In the plane perpendicular to the vortex line, one speaks of the vortex core for the density minimum. We study the properties of slowly rotated F = 1 spinor condensates and show that for ferromagnetic interactions the ground state is a coreless vortex. While the individual spin components may carry ordinary vortices the overall density of the system is coreless. We also study a method to create a point-like monopole structure in a ferromagnetic condensate by suitably imprinting vortices and solitons for different spin components of the condensate.

Työväentutkimus Vuosikirja•Mar 1, 2006•0 citations

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

Igor Lesanovsky, Peter Schmelcher

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.

Physical Review A•Mar 23, 2005•13 citations

Collisionless motion of neutral particles in magnetostatic traps

E.L. Surkov, Jook T. M. Walraven, Georgy V. Shlyapnikov

We present an analytical and numerical analysis of collisionless motion of neutral particles in magnetostatic traps. We treat the example of an idealized Ioffe quadrupole trap, emphasizing that the essential features of coupling between the degrees of freedom are of general relevance for static traps. In common situations the coupling between axial and radial motion predominantly occurs in a regular way. The coupling is weak, in particular, when the frequency of axial oscillations ${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$ is much smaller than the radial frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{\ensuremath{\rho}}}$ and the particle energy is much lower than the Zeeman energy in the center of the trap. Then, under an adiabatic change of the potential parameters, the quantity ${\mathit{scrE}}_{\mathit{z}}$/${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$, where ${\mathit{scrE}}_{\mathit{z}}$ is a characteristic axial energy, is an approximate adiabatic invariant. If ${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$ is decreasing and ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$ is constant, then the characteristic radial energy ${\mathit{scrE}}_{\mathrm{\ensuremath{\rho}}}$ will remain constant while ${\mathit{scrE}}_{\mathit{z}}$ decreases proportionally to ${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$. This implies one-dimensional adiabatic cooling, which is an interesting option for gravitational studies of light atoms such as hydrogen and antihydrogen.

Physical Review A•Jun 1, 1994•27 citations

Metastable Bose-Einstein condensate in a linear potential

D. Naik, Sérgio Ricardo Muniz, Chandra Raman

We have created a Bose-Einstein condensate whose spin orientation is metastable. Condensates were transferred into a quadrupole magnetic trap, where Majorana transitions limited the lifetime to a few hundred milliseconds, about 30 times the trapping period. Atoms held in the trap frequently displayed a ring-shaped time-of-flight distribution. We speculate that such a ring could be either a quantized vortex or a feature of the Majorana loss dynamics in the quantum regime.

arXiv:cond-mat/0510165•Physical Review A•Nov 22, 2005•12 citations
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.
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 $(x,\ensuremath{-}y).$ 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.
I review free and trapped Bose‐Einstein condensates (BECs) [1, 2, 3, 4, 5]. In an isotropic harmonic potential V(r) = 12Mω2r2, the single‐particle ground state introduces a new intrinsic scale of length [the ground‐state size d = ℏ/(Mω)] and energy [the ground‐state energy E0 = 32ℏω]. When the trap rotates [6, 7, 8], there is a critical angular velocity for the appearance of one or a few quantized vortices. For more rapid rotation [9, 10, 11, 12, 13] the condensate contains a vortex array. The resulting centrifugal forces expand the condensate radially and shrink it axially; thus the condensate becomes effectively two‐dimensional. If the external rotation speed approaches the frequency of the radial harmonic confining potential, the condensate enters the “lowest‐Landau‐level” regime, and a simple description again becomes possible [14, 15]. Eventually, the system is predicted to make a quantum phase transition to a highly correlated state analogous to the fractional quantum Hall states of electrons in a strong magnetic field [16, 17, 18, 19, 20].
This thesis contains theoretical studies of weakly interacting, trapped Bose-Einstein condensed gases. The main emphasis is on the rotational properties of these systems. A typical Bose-Einstein condensate reacts to external rotation by forming singly quantized vortices. By the word ”typical” we refer to a condensate with repulsive interparticle interactions and with a frozen spin degree of freedom. However, there are some atomic species for which the low-energy interactions are attractive. Moreover, the tuning of different condensate properties is relatively easy. For example, there are ways to experimentally change the effective interaction from repulsive to attractive. For an attractively interacting condensate the vortex is not the only possible rotational state but the sample may also experience rotation through its center of mass. In the first part of the thesis, we analyze some properties of a rotating attractive Bose gas at the zero temperature limit. First, we note that in order to maintain energetical stability, the gas has to be rotated in an external trap which is steeper than a harmonic trap. We study the system in an anharmonic quadratic + quartic potential. The rotating ground state phase-space is determined with analytical and numerical methods. If the interactions are strong enough, the rotating ground state appears to be a state with the rotating center of mass. On the other hand, at the limit of weak interactions the system experiences rotation by forming a multiply quantized vortex. We also determine the spectrum of elementary excitations for these quantum many-particle states. We find that the broken symmetry of the center of mass rotating state induces an additional zero-energy (Goldstone) mode and that the small oscillations of the ground state exhibit the decoupling between the center of mass and relative motions. If the spin of the atoms is allowed to change freely, a rich variety of topological excitations in condensates can be studied. External rotation is one of the tools for examining the excitations in spinor condensates. In a spinless scalar condensate the appearance of a quantized vortex is manifested by the vortex line in which the condensate density vanishes. In the plane perpendicular to the vortex line, one speaks of the vortex core for the density minimum. We study the properties of slowly rotated F = 1 spinor condensates and show that for ferromagnetic interactions the ground state is a coreless vortex. While the individual spin components may carry ordinary vortices the overall density of the system is coreless. We also study a method to create a point-like monopole structure in a ferromagnetic condensate by suitably imprinting vortices and solitons for different spin components of the condensate.
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 present an analytical and numerical analysis of collisionless motion of neutral particles in magnetostatic traps. We treat the example of an idealized Ioffe quadrupole trap, emphasizing that the essential features of coupling between the degrees of freedom are of general relevance for static traps. In common situations the coupling between axial and radial motion predominantly occurs in a regular way. The coupling is weak, in particular, when the frequency of axial oscillations ${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$ is much smaller than the radial frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{\ensuremath{\rho}}}$ and the particle energy is much lower than the Zeeman energy in the center of the trap. Then, under an adiabatic change of the potential parameters, the quantity scrEz{\mathit{scrE}}_{\mathit{z}}scrEz​/${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$, where scrEz{\mathit{scrE}}_{\mathit{z}}scrEz​ is a characteristic axial energy, is an approximate adiabatic invariant. If ${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$ is decreasing and ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$ is constant, then the characteristic radial energy ${\mathit{scrE}}_{\mathrm{\ensuremath{\rho}}}$ will remain constant while scrEz{\mathit{scrE}}_{\mathit{z}}scrEz​ decreases proportionally to ${\mathrm{\ensuremath{\Omega}}}_{\mathit{z}}$. This implies one-dimensional adiabatic cooling, which is an interesting option for gravitational studies of light atoms such as hydrogen and antihydrogen.