Abstract

In axially symmetric atomic-particle traps, the angular momentum of the particles about the symmetry axis is conserved in the absence of external torques. Changes in this angular momentum owing to laser scattering are discussed.

© 1985 Optical Society of America

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  1. A. Ashkin, “Applications of laser radiation pressure,” Science 210, 1081–1088 (1980).
    [CrossRef] [PubMed]
  2. W. D. Phillips, ed., Laser-Cooled and Trapped Atoms, Nat. Bur. Stand. (U.S.) Spec. Publ. 653, (1983); Prog. Quantum Electron. 8, 115–259 (1984).
  3. D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. D. Prestage, “Spectroscopy of stored atomic ions,” in Atomic Physics 9, R. S. Van Dyck, E. N. Fortson eds. (World Scientific, Singapore, 1985), pp. 3–27.
  4. D. J. Wineland, W. M. Itano, “Laser cooling of atoms,” Phys. Rev. A 20, 1521–1540 (1979).
    [CrossRef]
  5. W. M. Itano, D. J. Wineland, “Laser cooling of ions stored in harmonic and Penning traps,” Phys. Rev. A 25, 35–54 (1982).
    [CrossRef]
  6. H. G. Dehmelt, “Radiofrequency spectroscopy of stored ions. I and II,” Adv. At. Mol. Phys. 3, 53–72 (1967); Adv. At. Mol. Phys. 5, 109–154 (1969).
    [CrossRef]
  7. D. J. Wineland, W. M. Itano, R. S. Van Dyck, “High resolution spectroscopy of stored ions,” Adv. At. Mol. Phys. 19, 135–186 (1983).
    [CrossRef]
  8. A quantum mechanical treatment of rf trapping is given in R. J. Cook, D. G. Shankland, A. L. Wells, “Quantum theory of particle motion in a rapidly oscillating field,” Phys. Rev. A 31, 564–567 (1985).
    [CrossRef] [PubMed]
  9. E. Fischer, “Die dreidimensionale Stabilisierung von Ladungsträgern in einem Vierpolfeld,” Z. Phys. 156, 1–26 (1959).
    [CrossRef]
  10. F. G. Major, H. G. Dehmelt, “Exchange-collision technique for the rf spectroscopy of stored ions,” Phys. Rev. 170, 91–107 (1968).
    [CrossRef]
  11. H. A. Schuessler, E. N. Fortson, H. G. Dehmelt, “Hyperfine structure of the ground state of 3He+ by the ion-storage exchange-collision technique,” Phys. Rev. 187, 5–38 (1969).
    [CrossRef]
  12. D. J. Wineland, R. E. Drullinger, J. C. Bergquist, W. M. Itano, “Laser induced magnetron compression (expansion) of ions stored in a Penning trap,” Bull. Am. Phys. Soc. 24, 1185 (1979).
  13. W. D. White, J. H. Malmberg, C. F. Driscoll, “Resistive wall destabilization of diocotron waves,” Phys. Rev. Lett. 49, 1822–1826 (1982).
    [CrossRef]
  14. D. L. Eggleston, T. M. O’Neil, J. H. Malmberg, “Collective enhancement of radial transport in a nonneutral plasma,” Phys. Rev. Lett. 53, 982–984 (1984).
    [CrossRef]
  15. J. D. Crawford, T. M. O’Neil, J. H. Malmberg, “Effect of nonlinear collective processes on the confinement of a pure electron plasma,” Phys. Rev. Lett. 54, 697–700 (1985).
    [CrossRef] [PubMed]
  16. The potential for a uniformly charged ellipsoid of revolution can be found in W. D. MacMillan, The Theory of the Potential (Dover, New York, 1958), pp. 17 and 45; O. D. Kellog, Foundation of Potential Theory (Ungar, New York, 1929), p. 195.
  17. L. D. Landau, E. M. Lifshitz, Statistical Physics (Addison-Wesley, Reading, Mass., 1974).
  18. J. H. Malmberg, T. M. O’Neil, “Pure electron plasma, liquid, and crystal,” Phys. Rev. Lett. 39, 1333–1336 (1977).
    [CrossRef]
  19. S. A. Prasad, T. M. O’Neil, “Finite length thermal equilibria of a pure electron plasma column,” Phys. Fluids 22, 278–281 (1979).
    [CrossRef]
  20. T. M. O’Neil, C. F. Driscoll, “Transport to thermal equilibrium of a pure electron plasma,” Phys. Fluids 22, 266–277 (1979).
    [CrossRef]
  21. L. S. Cutler, R. P. Giffard, M. D. McGuire, “Mercury-199 trapped ion frequency standard: recent theoretical progress and experimental results,” in Proceedings of the 37th Annual Symposium on Frequency Control, 1983. (Copies available from Systematics General Corporation, Brinley Plaza, Rt. 38, Wall Township, N.J. 07719), pp. 32–36; “Thermalization of 199Hg ion macromotion by a light background gas in an rf quadrupole trap,” Appl. Phys. B 36, 137–142 (1985).
  22. J. S. deGrassie, J. H. Malmberg, “Waves and transport in the pure electron plasma,” Phys. Fluids 23, 63–81 (1980).
    [CrossRef]
  23. J. J. Bollinger, D. J. Wineland, “Strongly coupled nonneutral ion plasma,” Phys. Rev. Lett. 53, 348–351 (1984).
    [CrossRef]
  24. R. D. Knight, M. H. Prior, “Laser scanning measurements of the density distribution of confined 6Li+ ions,” J. Appl. Phys. 50, 3044–3049 (1979).
    [CrossRef]
  25. H. Schaaf, V. Schmeling, G. Werth, “Trapped ion density distribution in the presence of He-buffer gas,” Appl. Phys. 25, 249–251 (1981).
    [CrossRef]
  26. J. B. Jeffries, S. E. Barlow, G. H. Dunn, “Theory of space-charge shift of ion cyclotron resonance frequencies,” Int. J. Mass Spectrom. Ion Phys. 54, 169–187 (1983).
    [CrossRef]
  27. D. A. Church, H. G. Dehmelt, “Radiative cooling of an electrodynamically contained proton gas,” J. Appl. Phys. 40, 3421–3424 (1969).
    [CrossRef]
  28. H. G. Dehmelt, “Stored ion spectroscopy,” in Advances in Laser Spectroscopy, F. T. Arecchi, F. Strumia, H. Walther, eds. (Plenum, New York, 1983), pp. 153–187.
    [CrossRef]
  29. D. J. Wineland, “Spectroscopy of stored ions,” in Precision Measurement and Fundamental Constants II, B. N. Taylor, W. D. Phillips, eds., Natl. Bur. Stand. (U.S.) Spec. Publ.617(1984), p. 83–92.
  30. J. T. Davies, J. M. Vaughan, “A new tabulation of the Voigt profile,” Astrophys. J. 137, 1302–1305 (1963).
    [CrossRef]
  31. H. G. Dehmelt, F. L. Walls, “Bolometric technique for the rf spectroscopy of stored ions,” Phys. Rev. Lett. 21, 127–131 (1968).
    [CrossRef]
  32. E. W. McDaniel, Collision Phenomena in Ionized Gases (Wiley, New York, 1964), pp. 426–482.
  33. J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U. Daniel, G. Leuchs, “Energy and radiative lifetime of the 5d96s2 2D5/2state in Hg ii by Doppler-free two-photon laser spectroscopy,” Phys. Rev. Lett. (to be published).
  34. L. M. Chanin, M. A. Biondi, “Mobilities of mercury ions in helium, neon and argon,” Phys. Rev. 107, 1219–1221 (1957).
    [CrossRef]
  35. S. Ichimaru, “Strongly coupled plasmas: high density classical plasmas and degenerate electron liquids,” Rev. Mod. Phys. 54, 1017–1059 (1982).
    [CrossRef]
  36. W. Neuhauser, M. Hohenstatt, P. E. Toschek, H. G. Dehmelt, “Localized visible Ba+ mono-ion oscillator,” Phys. Rev. A 22, 1137–1140 (1980).
    [CrossRef]
  37. R. E. Drullinger, D. J. Wineland, J. C. Bergquist, “High-resolution optical spectra of laser cooled ions,” Appl. Phys. 22, 365–368 (1980).
    [CrossRef]
  38. J. J. Bollinger, J. D. Prestage, W. M. Itano, D. J. Wineland, “Laser cooled atomic frequency standard,” Phys. Rev. Lett. 54, 1000–1003 (1985).
    [CrossRef] [PubMed]
  39. C. F. Driscoll, J. H. Malmberg, “Length dependent containment of a pure electron-plasma column,” Phys. Rev. Lett. 50, 167–170 (1983).
    [CrossRef]
  40. L. S. Brown, G. Gabrielse, “Precision spectroscopy of a charged particle in an imperfect Penning trap,” Phys. Rev. A 25, 2423–2425 (1982).
    [CrossRef]
  41. M. D. McGuire, E. N. Fortson, “Penning-trap technique for studying electron-atom collisions at low energy,” Phys. Rev. Lett. 33, 737–739 (1974).
    [CrossRef]
  42. G. Graff, F. G. Major, R. W. H. Roeder, G. Werth, “Method for measuring the cyclotron and spin resonance of free electrons,” Phys. Rev. Lett. 21, 340–342 (1968).
    [CrossRef]
  43. G. Graff, M. Holzscheiter, “Method for trapped ion polarization and polarization detection,” Phys. Lett. 79A, 380–382 (1980).

1985

A quantum mechanical treatment of rf trapping is given in R. J. Cook, D. G. Shankland, A. L. Wells, “Quantum theory of particle motion in a rapidly oscillating field,” Phys. Rev. A 31, 564–567 (1985).
[CrossRef] [PubMed]

J. D. Crawford, T. M. O’Neil, J. H. Malmberg, “Effect of nonlinear collective processes on the confinement of a pure electron plasma,” Phys. Rev. Lett. 54, 697–700 (1985).
[CrossRef] [PubMed]

J. J. Bollinger, J. D. Prestage, W. M. Itano, D. J. Wineland, “Laser cooled atomic frequency standard,” Phys. Rev. Lett. 54, 1000–1003 (1985).
[CrossRef] [PubMed]

1984

D. L. Eggleston, T. M. O’Neil, J. H. Malmberg, “Collective enhancement of radial transport in a nonneutral plasma,” Phys. Rev. Lett. 53, 982–984 (1984).
[CrossRef]

J. J. Bollinger, D. J. Wineland, “Strongly coupled nonneutral ion plasma,” Phys. Rev. Lett. 53, 348–351 (1984).
[CrossRef]

1983

J. B. Jeffries, S. E. Barlow, G. H. Dunn, “Theory of space-charge shift of ion cyclotron resonance frequencies,” Int. J. Mass Spectrom. Ion Phys. 54, 169–187 (1983).
[CrossRef]

C. F. Driscoll, J. H. Malmberg, “Length dependent containment of a pure electron-plasma column,” Phys. Rev. Lett. 50, 167–170 (1983).
[CrossRef]

D. J. Wineland, W. M. Itano, R. S. Van Dyck, “High resolution spectroscopy of stored ions,” Adv. At. Mol. Phys. 19, 135–186 (1983).
[CrossRef]

W. D. Phillips, ed., Laser-Cooled and Trapped Atoms, Nat. Bur. Stand. (U.S.) Spec. Publ. 653, (1983); Prog. Quantum Electron. 8, 115–259 (1984).

1982

W. M. Itano, D. J. Wineland, “Laser cooling of ions stored in harmonic and Penning traps,” Phys. Rev. A 25, 35–54 (1982).
[CrossRef]

W. D. White, J. H. Malmberg, C. F. Driscoll, “Resistive wall destabilization of diocotron waves,” Phys. Rev. Lett. 49, 1822–1826 (1982).
[CrossRef]

L. S. Brown, G. Gabrielse, “Precision spectroscopy of a charged particle in an imperfect Penning trap,” Phys. Rev. A 25, 2423–2425 (1982).
[CrossRef]

S. Ichimaru, “Strongly coupled plasmas: high density classical plasmas and degenerate electron liquids,” Rev. Mod. Phys. 54, 1017–1059 (1982).
[CrossRef]

1981

H. Schaaf, V. Schmeling, G. Werth, “Trapped ion density distribution in the presence of He-buffer gas,” Appl. Phys. 25, 249–251 (1981).
[CrossRef]

1980

G. Graff, M. Holzscheiter, “Method for trapped ion polarization and polarization detection,” Phys. Lett. 79A, 380–382 (1980).

W. Neuhauser, M. Hohenstatt, P. E. Toschek, H. G. Dehmelt, “Localized visible Ba+ mono-ion oscillator,” Phys. Rev. A 22, 1137–1140 (1980).
[CrossRef]

R. E. Drullinger, D. J. Wineland, J. C. Bergquist, “High-resolution optical spectra of laser cooled ions,” Appl. Phys. 22, 365–368 (1980).
[CrossRef]

J. S. deGrassie, J. H. Malmberg, “Waves and transport in the pure electron plasma,” Phys. Fluids 23, 63–81 (1980).
[CrossRef]

A. Ashkin, “Applications of laser radiation pressure,” Science 210, 1081–1088 (1980).
[CrossRef] [PubMed]

1979

D. J. Wineland, W. M. Itano, “Laser cooling of atoms,” Phys. Rev. A 20, 1521–1540 (1979).
[CrossRef]

D. J. Wineland, R. E. Drullinger, J. C. Bergquist, W. M. Itano, “Laser induced magnetron compression (expansion) of ions stored in a Penning trap,” Bull. Am. Phys. Soc. 24, 1185 (1979).

S. A. Prasad, T. M. O’Neil, “Finite length thermal equilibria of a pure electron plasma column,” Phys. Fluids 22, 278–281 (1979).
[CrossRef]

T. M. O’Neil, C. F. Driscoll, “Transport to thermal equilibrium of a pure electron plasma,” Phys. Fluids 22, 266–277 (1979).
[CrossRef]

R. D. Knight, M. H. Prior, “Laser scanning measurements of the density distribution of confined 6Li+ ions,” J. Appl. Phys. 50, 3044–3049 (1979).
[CrossRef]

1977

J. H. Malmberg, T. M. O’Neil, “Pure electron plasma, liquid, and crystal,” Phys. Rev. Lett. 39, 1333–1336 (1977).
[CrossRef]

1974

M. D. McGuire, E. N. Fortson, “Penning-trap technique for studying electron-atom collisions at low energy,” Phys. Rev. Lett. 33, 737–739 (1974).
[CrossRef]

1969

D. A. Church, H. G. Dehmelt, “Radiative cooling of an electrodynamically contained proton gas,” J. Appl. Phys. 40, 3421–3424 (1969).
[CrossRef]

H. A. Schuessler, E. N. Fortson, H. G. Dehmelt, “Hyperfine structure of the ground state of 3He+ by the ion-storage exchange-collision technique,” Phys. Rev. 187, 5–38 (1969).
[CrossRef]

1968

F. G. Major, H. G. Dehmelt, “Exchange-collision technique for the rf spectroscopy of stored ions,” Phys. Rev. 170, 91–107 (1968).
[CrossRef]

H. G. Dehmelt, F. L. Walls, “Bolometric technique for the rf spectroscopy of stored ions,” Phys. Rev. Lett. 21, 127–131 (1968).
[CrossRef]

G. Graff, F. G. Major, R. W. H. Roeder, G. Werth, “Method for measuring the cyclotron and spin resonance of free electrons,” Phys. Rev. Lett. 21, 340–342 (1968).
[CrossRef]

1967

H. G. Dehmelt, “Radiofrequency spectroscopy of stored ions. I and II,” Adv. At. Mol. Phys. 3, 53–72 (1967); Adv. At. Mol. Phys. 5, 109–154 (1969).
[CrossRef]

1963

J. T. Davies, J. M. Vaughan, “A new tabulation of the Voigt profile,” Astrophys. J. 137, 1302–1305 (1963).
[CrossRef]

1959

E. Fischer, “Die dreidimensionale Stabilisierung von Ladungsträgern in einem Vierpolfeld,” Z. Phys. 156, 1–26 (1959).
[CrossRef]

1957

L. M. Chanin, M. A. Biondi, “Mobilities of mercury ions in helium, neon and argon,” Phys. Rev. 107, 1219–1221 (1957).
[CrossRef]

Ashkin, A.

A. Ashkin, “Applications of laser radiation pressure,” Science 210, 1081–1088 (1980).
[CrossRef] [PubMed]

Barlow, S. E.

J. B. Jeffries, S. E. Barlow, G. H. Dunn, “Theory of space-charge shift of ion cyclotron resonance frequencies,” Int. J. Mass Spectrom. Ion Phys. 54, 169–187 (1983).
[CrossRef]

Bergquist, J. C.

R. E. Drullinger, D. J. Wineland, J. C. Bergquist, “High-resolution optical spectra of laser cooled ions,” Appl. Phys. 22, 365–368 (1980).
[CrossRef]

D. J. Wineland, R. E. Drullinger, J. C. Bergquist, W. M. Itano, “Laser induced magnetron compression (expansion) of ions stored in a Penning trap,” Bull. Am. Phys. Soc. 24, 1185 (1979).

D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. D. Prestage, “Spectroscopy of stored atomic ions,” in Atomic Physics 9, R. S. Van Dyck, E. N. Fortson eds. (World Scientific, Singapore, 1985), pp. 3–27.

J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U. Daniel, G. Leuchs, “Energy and radiative lifetime of the 5d96s2 2D5/2state in Hg ii by Doppler-free two-photon laser spectroscopy,” Phys. Rev. Lett. (to be published).

Biondi, M. A.

L. M. Chanin, M. A. Biondi, “Mobilities of mercury ions in helium, neon and argon,” Phys. Rev. 107, 1219–1221 (1957).
[CrossRef]

Bollinger, J. J.

J. J. Bollinger, J. D. Prestage, W. M. Itano, D. J. Wineland, “Laser cooled atomic frequency standard,” Phys. Rev. Lett. 54, 1000–1003 (1985).
[CrossRef] [PubMed]

J. J. Bollinger, D. J. Wineland, “Strongly coupled nonneutral ion plasma,” Phys. Rev. Lett. 53, 348–351 (1984).
[CrossRef]

D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. D. Prestage, “Spectroscopy of stored atomic ions,” in Atomic Physics 9, R. S. Van Dyck, E. N. Fortson eds. (World Scientific, Singapore, 1985), pp. 3–27.

Brown, L. S.

L. S. Brown, G. Gabrielse, “Precision spectroscopy of a charged particle in an imperfect Penning trap,” Phys. Rev. A 25, 2423–2425 (1982).
[CrossRef]

Chanin, L. M.

L. M. Chanin, M. A. Biondi, “Mobilities of mercury ions in helium, neon and argon,” Phys. Rev. 107, 1219–1221 (1957).
[CrossRef]

Church, D. A.

D. A. Church, H. G. Dehmelt, “Radiative cooling of an electrodynamically contained proton gas,” J. Appl. Phys. 40, 3421–3424 (1969).
[CrossRef]

Cook, R. J.

A quantum mechanical treatment of rf trapping is given in R. J. Cook, D. G. Shankland, A. L. Wells, “Quantum theory of particle motion in a rapidly oscillating field,” Phys. Rev. A 31, 564–567 (1985).
[CrossRef] [PubMed]

Crawford, J. D.

J. D. Crawford, T. M. O’Neil, J. H. Malmberg, “Effect of nonlinear collective processes on the confinement of a pure electron plasma,” Phys. Rev. Lett. 54, 697–700 (1985).
[CrossRef] [PubMed]

Cutler, L. S.

L. S. Cutler, R. P. Giffard, M. D. McGuire, “Mercury-199 trapped ion frequency standard: recent theoretical progress and experimental results,” in Proceedings of the 37th Annual Symposium on Frequency Control, 1983. (Copies available from Systematics General Corporation, Brinley Plaza, Rt. 38, Wall Township, N.J. 07719), pp. 32–36; “Thermalization of 199Hg ion macromotion by a light background gas in an rf quadrupole trap,” Appl. Phys. B 36, 137–142 (1985).

Daniel, H.-U.

J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U. Daniel, G. Leuchs, “Energy and radiative lifetime of the 5d96s2 2D5/2state in Hg ii by Doppler-free two-photon laser spectroscopy,” Phys. Rev. Lett. (to be published).

Davies, J. T.

J. T. Davies, J. M. Vaughan, “A new tabulation of the Voigt profile,” Astrophys. J. 137, 1302–1305 (1963).
[CrossRef]

deGrassie, J. S.

J. S. deGrassie, J. H. Malmberg, “Waves and transport in the pure electron plasma,” Phys. Fluids 23, 63–81 (1980).
[CrossRef]

Dehmelt, H. G.

W. Neuhauser, M. Hohenstatt, P. E. Toschek, H. G. Dehmelt, “Localized visible Ba+ mono-ion oscillator,” Phys. Rev. A 22, 1137–1140 (1980).
[CrossRef]

D. A. Church, H. G. Dehmelt, “Radiative cooling of an electrodynamically contained proton gas,” J. Appl. Phys. 40, 3421–3424 (1969).
[CrossRef]

H. A. Schuessler, E. N. Fortson, H. G. Dehmelt, “Hyperfine structure of the ground state of 3He+ by the ion-storage exchange-collision technique,” Phys. Rev. 187, 5–38 (1969).
[CrossRef]

F. G. Major, H. G. Dehmelt, “Exchange-collision technique for the rf spectroscopy of stored ions,” Phys. Rev. 170, 91–107 (1968).
[CrossRef]

H. G. Dehmelt, F. L. Walls, “Bolometric technique for the rf spectroscopy of stored ions,” Phys. Rev. Lett. 21, 127–131 (1968).
[CrossRef]

H. G. Dehmelt, “Radiofrequency spectroscopy of stored ions. I and II,” Adv. At. Mol. Phys. 3, 53–72 (1967); Adv. At. Mol. Phys. 5, 109–154 (1969).
[CrossRef]

H. G. Dehmelt, “Stored ion spectroscopy,” in Advances in Laser Spectroscopy, F. T. Arecchi, F. Strumia, H. Walther, eds. (Plenum, New York, 1983), pp. 153–187.
[CrossRef]

Driscoll, C. F.

C. F. Driscoll, J. H. Malmberg, “Length dependent containment of a pure electron-plasma column,” Phys. Rev. Lett. 50, 167–170 (1983).
[CrossRef]

W. D. White, J. H. Malmberg, C. F. Driscoll, “Resistive wall destabilization of diocotron waves,” Phys. Rev. Lett. 49, 1822–1826 (1982).
[CrossRef]

T. M. O’Neil, C. F. Driscoll, “Transport to thermal equilibrium of a pure electron plasma,” Phys. Fluids 22, 266–277 (1979).
[CrossRef]

Drullinger, R. E.

R. E. Drullinger, D. J. Wineland, J. C. Bergquist, “High-resolution optical spectra of laser cooled ions,” Appl. Phys. 22, 365–368 (1980).
[CrossRef]

D. J. Wineland, R. E. Drullinger, J. C. Bergquist, W. M. Itano, “Laser induced magnetron compression (expansion) of ions stored in a Penning trap,” Bull. Am. Phys. Soc. 24, 1185 (1979).

Dunn, G. H.

J. B. Jeffries, S. E. Barlow, G. H. Dunn, “Theory of space-charge shift of ion cyclotron resonance frequencies,” Int. J. Mass Spectrom. Ion Phys. 54, 169–187 (1983).
[CrossRef]

Eggleston, D. L.

D. L. Eggleston, T. M. O’Neil, J. H. Malmberg, “Collective enhancement of radial transport in a nonneutral plasma,” Phys. Rev. Lett. 53, 982–984 (1984).
[CrossRef]

Fischer, E.

E. Fischer, “Die dreidimensionale Stabilisierung von Ladungsträgern in einem Vierpolfeld,” Z. Phys. 156, 1–26 (1959).
[CrossRef]

Fortson, E. N.

M. D. McGuire, E. N. Fortson, “Penning-trap technique for studying electron-atom collisions at low energy,” Phys. Rev. Lett. 33, 737–739 (1974).
[CrossRef]

H. A. Schuessler, E. N. Fortson, H. G. Dehmelt, “Hyperfine structure of the ground state of 3He+ by the ion-storage exchange-collision technique,” Phys. Rev. 187, 5–38 (1969).
[CrossRef]

Gabrielse, G.

L. S. Brown, G. Gabrielse, “Precision spectroscopy of a charged particle in an imperfect Penning trap,” Phys. Rev. A 25, 2423–2425 (1982).
[CrossRef]

Giffard, R. P.

L. S. Cutler, R. P. Giffard, M. D. McGuire, “Mercury-199 trapped ion frequency standard: recent theoretical progress and experimental results,” in Proceedings of the 37th Annual Symposium on Frequency Control, 1983. (Copies available from Systematics General Corporation, Brinley Plaza, Rt. 38, Wall Township, N.J. 07719), pp. 32–36; “Thermalization of 199Hg ion macromotion by a light background gas in an rf quadrupole trap,” Appl. Phys. B 36, 137–142 (1985).

Graff, G.

G. Graff, M. Holzscheiter, “Method for trapped ion polarization and polarization detection,” Phys. Lett. 79A, 380–382 (1980).

G. Graff, F. G. Major, R. W. H. Roeder, G. Werth, “Method for measuring the cyclotron and spin resonance of free electrons,” Phys. Rev. Lett. 21, 340–342 (1968).
[CrossRef]

Hemmati, H.

J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U. Daniel, G. Leuchs, “Energy and radiative lifetime of the 5d96s2 2D5/2state in Hg ii by Doppler-free two-photon laser spectroscopy,” Phys. Rev. Lett. (to be published).

Hohenstatt, M.

W. Neuhauser, M. Hohenstatt, P. E. Toschek, H. G. Dehmelt, “Localized visible Ba+ mono-ion oscillator,” Phys. Rev. A 22, 1137–1140 (1980).
[CrossRef]

Holzscheiter, M.

G. Graff, M. Holzscheiter, “Method for trapped ion polarization and polarization detection,” Phys. Lett. 79A, 380–382 (1980).

Ichimaru, S.

S. Ichimaru, “Strongly coupled plasmas: high density classical plasmas and degenerate electron liquids,” Rev. Mod. Phys. 54, 1017–1059 (1982).
[CrossRef]

Itano, W. M.

J. J. Bollinger, J. D. Prestage, W. M. Itano, D. J. Wineland, “Laser cooled atomic frequency standard,” Phys. Rev. Lett. 54, 1000–1003 (1985).
[CrossRef] [PubMed]

D. J. Wineland, W. M. Itano, R. S. Van Dyck, “High resolution spectroscopy of stored ions,” Adv. At. Mol. Phys. 19, 135–186 (1983).
[CrossRef]

W. M. Itano, D. J. Wineland, “Laser cooling of ions stored in harmonic and Penning traps,” Phys. Rev. A 25, 35–54 (1982).
[CrossRef]

D. J. Wineland, W. M. Itano, “Laser cooling of atoms,” Phys. Rev. A 20, 1521–1540 (1979).
[CrossRef]

D. J. Wineland, R. E. Drullinger, J. C. Bergquist, W. M. Itano, “Laser induced magnetron compression (expansion) of ions stored in a Penning trap,” Bull. Am. Phys. Soc. 24, 1185 (1979).

D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. D. Prestage, “Spectroscopy of stored atomic ions,” in Atomic Physics 9, R. S. Van Dyck, E. N. Fortson eds. (World Scientific, Singapore, 1985), pp. 3–27.

J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U. Daniel, G. Leuchs, “Energy and radiative lifetime of the 5d96s2 2D5/2state in Hg ii by Doppler-free two-photon laser spectroscopy,” Phys. Rev. Lett. (to be published).

Jeffries, J. B.

J. B. Jeffries, S. E. Barlow, G. H. Dunn, “Theory of space-charge shift of ion cyclotron resonance frequencies,” Int. J. Mass Spectrom. Ion Phys. 54, 169–187 (1983).
[CrossRef]

Knight, R. D.

R. D. Knight, M. H. Prior, “Laser scanning measurements of the density distribution of confined 6Li+ ions,” J. Appl. Phys. 50, 3044–3049 (1979).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Statistical Physics (Addison-Wesley, Reading, Mass., 1974).

Leuchs, G.

J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U. Daniel, G. Leuchs, “Energy and radiative lifetime of the 5d96s2 2D5/2state in Hg ii by Doppler-free two-photon laser spectroscopy,” Phys. Rev. Lett. (to be published).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Statistical Physics (Addison-Wesley, Reading, Mass., 1974).

MacMillan, W. D.

The potential for a uniformly charged ellipsoid of revolution can be found in W. D. MacMillan, The Theory of the Potential (Dover, New York, 1958), pp. 17 and 45; O. D. Kellog, Foundation of Potential Theory (Ungar, New York, 1929), p. 195.

Major, F. G.

F. G. Major, H. G. Dehmelt, “Exchange-collision technique for the rf spectroscopy of stored ions,” Phys. Rev. 170, 91–107 (1968).
[CrossRef]

G. Graff, F. G. Major, R. W. H. Roeder, G. Werth, “Method for measuring the cyclotron and spin resonance of free electrons,” Phys. Rev. Lett. 21, 340–342 (1968).
[CrossRef]

Malmberg, J. H.

J. D. Crawford, T. M. O’Neil, J. H. Malmberg, “Effect of nonlinear collective processes on the confinement of a pure electron plasma,” Phys. Rev. Lett. 54, 697–700 (1985).
[CrossRef] [PubMed]

D. L. Eggleston, T. M. O’Neil, J. H. Malmberg, “Collective enhancement of radial transport in a nonneutral plasma,” Phys. Rev. Lett. 53, 982–984 (1984).
[CrossRef]

C. F. Driscoll, J. H. Malmberg, “Length dependent containment of a pure electron-plasma column,” Phys. Rev. Lett. 50, 167–170 (1983).
[CrossRef]

W. D. White, J. H. Malmberg, C. F. Driscoll, “Resistive wall destabilization of diocotron waves,” Phys. Rev. Lett. 49, 1822–1826 (1982).
[CrossRef]

J. S. deGrassie, J. H. Malmberg, “Waves and transport in the pure electron plasma,” Phys. Fluids 23, 63–81 (1980).
[CrossRef]

J. H. Malmberg, T. M. O’Neil, “Pure electron plasma, liquid, and crystal,” Phys. Rev. Lett. 39, 1333–1336 (1977).
[CrossRef]

McDaniel, E. W.

E. W. McDaniel, Collision Phenomena in Ionized Gases (Wiley, New York, 1964), pp. 426–482.

McGuire, M. D.

M. D. McGuire, E. N. Fortson, “Penning-trap technique for studying electron-atom collisions at low energy,” Phys. Rev. Lett. 33, 737–739 (1974).
[CrossRef]

L. S. Cutler, R. P. Giffard, M. D. McGuire, “Mercury-199 trapped ion frequency standard: recent theoretical progress and experimental results,” in Proceedings of the 37th Annual Symposium on Frequency Control, 1983. (Copies available from Systematics General Corporation, Brinley Plaza, Rt. 38, Wall Township, N.J. 07719), pp. 32–36; “Thermalization of 199Hg ion macromotion by a light background gas in an rf quadrupole trap,” Appl. Phys. B 36, 137–142 (1985).

Neuhauser, W.

W. Neuhauser, M. Hohenstatt, P. E. Toschek, H. G. Dehmelt, “Localized visible Ba+ mono-ion oscillator,” Phys. Rev. A 22, 1137–1140 (1980).
[CrossRef]

O’Neil, T. M.

J. D. Crawford, T. M. O’Neil, J. H. Malmberg, “Effect of nonlinear collective processes on the confinement of a pure electron plasma,” Phys. Rev. Lett. 54, 697–700 (1985).
[CrossRef] [PubMed]

D. L. Eggleston, T. M. O’Neil, J. H. Malmberg, “Collective enhancement of radial transport in a nonneutral plasma,” Phys. Rev. Lett. 53, 982–984 (1984).
[CrossRef]

S. A. Prasad, T. M. O’Neil, “Finite length thermal equilibria of a pure electron plasma column,” Phys. Fluids 22, 278–281 (1979).
[CrossRef]

T. M. O’Neil, C. F. Driscoll, “Transport to thermal equilibrium of a pure electron plasma,” Phys. Fluids 22, 266–277 (1979).
[CrossRef]

J. H. Malmberg, T. M. O’Neil, “Pure electron plasma, liquid, and crystal,” Phys. Rev. Lett. 39, 1333–1336 (1977).
[CrossRef]

Prasad, S. A.

S. A. Prasad, T. M. O’Neil, “Finite length thermal equilibria of a pure electron plasma column,” Phys. Fluids 22, 278–281 (1979).
[CrossRef]

Prestage, J. D.

J. J. Bollinger, J. D. Prestage, W. M. Itano, D. J. Wineland, “Laser cooled atomic frequency standard,” Phys. Rev. Lett. 54, 1000–1003 (1985).
[CrossRef] [PubMed]

D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. D. Prestage, “Spectroscopy of stored atomic ions,” in Atomic Physics 9, R. S. Van Dyck, E. N. Fortson eds. (World Scientific, Singapore, 1985), pp. 3–27.

Prior, M. H.

R. D. Knight, M. H. Prior, “Laser scanning measurements of the density distribution of confined 6Li+ ions,” J. Appl. Phys. 50, 3044–3049 (1979).
[CrossRef]

Roeder, R. W. H.

G. Graff, F. G. Major, R. W. H. Roeder, G. Werth, “Method for measuring the cyclotron and spin resonance of free electrons,” Phys. Rev. Lett. 21, 340–342 (1968).
[CrossRef]

Schaaf, H.

H. Schaaf, V. Schmeling, G. Werth, “Trapped ion density distribution in the presence of He-buffer gas,” Appl. Phys. 25, 249–251 (1981).
[CrossRef]

Schmeling, V.

H. Schaaf, V. Schmeling, G. Werth, “Trapped ion density distribution in the presence of He-buffer gas,” Appl. Phys. 25, 249–251 (1981).
[CrossRef]

Schuessler, H. A.

H. A. Schuessler, E. N. Fortson, H. G. Dehmelt, “Hyperfine structure of the ground state of 3He+ by the ion-storage exchange-collision technique,” Phys. Rev. 187, 5–38 (1969).
[CrossRef]

Shankland, D. G.

A quantum mechanical treatment of rf trapping is given in R. J. Cook, D. G. Shankland, A. L. Wells, “Quantum theory of particle motion in a rapidly oscillating field,” Phys. Rev. A 31, 564–567 (1985).
[CrossRef] [PubMed]

Toschek, P. E.

W. Neuhauser, M. Hohenstatt, P. E. Toschek, H. G. Dehmelt, “Localized visible Ba+ mono-ion oscillator,” Phys. Rev. A 22, 1137–1140 (1980).
[CrossRef]

Van Dyck, R. S.

D. J. Wineland, W. M. Itano, R. S. Van Dyck, “High resolution spectroscopy of stored ions,” Adv. At. Mol. Phys. 19, 135–186 (1983).
[CrossRef]

Vaughan, J. M.

J. T. Davies, J. M. Vaughan, “A new tabulation of the Voigt profile,” Astrophys. J. 137, 1302–1305 (1963).
[CrossRef]

Walls, F. L.

H. G. Dehmelt, F. L. Walls, “Bolometric technique for the rf spectroscopy of stored ions,” Phys. Rev. Lett. 21, 127–131 (1968).
[CrossRef]

Wells, A. L.

A quantum mechanical treatment of rf trapping is given in R. J. Cook, D. G. Shankland, A. L. Wells, “Quantum theory of particle motion in a rapidly oscillating field,” Phys. Rev. A 31, 564–567 (1985).
[CrossRef] [PubMed]

Werth, G.

H. Schaaf, V. Schmeling, G. Werth, “Trapped ion density distribution in the presence of He-buffer gas,” Appl. Phys. 25, 249–251 (1981).
[CrossRef]

G. Graff, F. G. Major, R. W. H. Roeder, G. Werth, “Method for measuring the cyclotron and spin resonance of free electrons,” Phys. Rev. Lett. 21, 340–342 (1968).
[CrossRef]

White, W. D.

W. D. White, J. H. Malmberg, C. F. Driscoll, “Resistive wall destabilization of diocotron waves,” Phys. Rev. Lett. 49, 1822–1826 (1982).
[CrossRef]

Wineland, D. J.

J. J. Bollinger, J. D. Prestage, W. M. Itano, D. J. Wineland, “Laser cooled atomic frequency standard,” Phys. Rev. Lett. 54, 1000–1003 (1985).
[CrossRef] [PubMed]

J. J. Bollinger, D. J. Wineland, “Strongly coupled nonneutral ion plasma,” Phys. Rev. Lett. 53, 348–351 (1984).
[CrossRef]

D. J. Wineland, W. M. Itano, R. S. Van Dyck, “High resolution spectroscopy of stored ions,” Adv. At. Mol. Phys. 19, 135–186 (1983).
[CrossRef]

W. M. Itano, D. J. Wineland, “Laser cooling of ions stored in harmonic and Penning traps,” Phys. Rev. A 25, 35–54 (1982).
[CrossRef]

R. E. Drullinger, D. J. Wineland, J. C. Bergquist, “High-resolution optical spectra of laser cooled ions,” Appl. Phys. 22, 365–368 (1980).
[CrossRef]

D. J. Wineland, W. M. Itano, “Laser cooling of atoms,” Phys. Rev. A 20, 1521–1540 (1979).
[CrossRef]

D. J. Wineland, R. E. Drullinger, J. C. Bergquist, W. M. Itano, “Laser induced magnetron compression (expansion) of ions stored in a Penning trap,” Bull. Am. Phys. Soc. 24, 1185 (1979).

D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. D. Prestage, “Spectroscopy of stored atomic ions,” in Atomic Physics 9, R. S. Van Dyck, E. N. Fortson eds. (World Scientific, Singapore, 1985), pp. 3–27.

D. J. Wineland, “Spectroscopy of stored ions,” in Precision Measurement and Fundamental Constants II, B. N. Taylor, W. D. Phillips, eds., Natl. Bur. Stand. (U.S.) Spec. Publ.617(1984), p. 83–92.

J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U. Daniel, G. Leuchs, “Energy and radiative lifetime of the 5d96s2 2D5/2state in Hg ii by Doppler-free two-photon laser spectroscopy,” Phys. Rev. Lett. (to be published).

Adv. At. Mol. Phys.

H. G. Dehmelt, “Radiofrequency spectroscopy of stored ions. I and II,” Adv. At. Mol. Phys. 3, 53–72 (1967); Adv. At. Mol. Phys. 5, 109–154 (1969).
[CrossRef]

D. J. Wineland, W. M. Itano, R. S. Van Dyck, “High resolution spectroscopy of stored ions,” Adv. At. Mol. Phys. 19, 135–186 (1983).
[CrossRef]

Appl. Phys.

H. Schaaf, V. Schmeling, G. Werth, “Trapped ion density distribution in the presence of He-buffer gas,” Appl. Phys. 25, 249–251 (1981).
[CrossRef]

R. E. Drullinger, D. J. Wineland, J. C. Bergquist, “High-resolution optical spectra of laser cooled ions,” Appl. Phys. 22, 365–368 (1980).
[CrossRef]

Astrophys. J.

J. T. Davies, J. M. Vaughan, “A new tabulation of the Voigt profile,” Astrophys. J. 137, 1302–1305 (1963).
[CrossRef]

Bull. Am. Phys. Soc.

D. J. Wineland, R. E. Drullinger, J. C. Bergquist, W. M. Itano, “Laser induced magnetron compression (expansion) of ions stored in a Penning trap,” Bull. Am. Phys. Soc. 24, 1185 (1979).

Int. J. Mass Spectrom. Ion Phys.

J. B. Jeffries, S. E. Barlow, G. H. Dunn, “Theory of space-charge shift of ion cyclotron resonance frequencies,” Int. J. Mass Spectrom. Ion Phys. 54, 169–187 (1983).
[CrossRef]

J. Appl. Phys.

D. A. Church, H. G. Dehmelt, “Radiative cooling of an electrodynamically contained proton gas,” J. Appl. Phys. 40, 3421–3424 (1969).
[CrossRef]

R. D. Knight, M. H. Prior, “Laser scanning measurements of the density distribution of confined 6Li+ ions,” J. Appl. Phys. 50, 3044–3049 (1979).
[CrossRef]

Laser-Cooled and Trapped Atoms

W. D. Phillips, ed., Laser-Cooled and Trapped Atoms, Nat. Bur. Stand. (U.S.) Spec. Publ. 653, (1983); Prog. Quantum Electron. 8, 115–259 (1984).

Phys. Fluids

S. A. Prasad, T. M. O’Neil, “Finite length thermal equilibria of a pure electron plasma column,” Phys. Fluids 22, 278–281 (1979).
[CrossRef]

T. M. O’Neil, C. F. Driscoll, “Transport to thermal equilibrium of a pure electron plasma,” Phys. Fluids 22, 266–277 (1979).
[CrossRef]

J. S. deGrassie, J. H. Malmberg, “Waves and transport in the pure electron plasma,” Phys. Fluids 23, 63–81 (1980).
[CrossRef]

Phys. Lett.

G. Graff, M. Holzscheiter, “Method for trapped ion polarization and polarization detection,” Phys. Lett. 79A, 380–382 (1980).

Phys. Rev.

L. M. Chanin, M. A. Biondi, “Mobilities of mercury ions in helium, neon and argon,” Phys. Rev. 107, 1219–1221 (1957).
[CrossRef]

F. G. Major, H. G. Dehmelt, “Exchange-collision technique for the rf spectroscopy of stored ions,” Phys. Rev. 170, 91–107 (1968).
[CrossRef]

H. A. Schuessler, E. N. Fortson, H. G. Dehmelt, “Hyperfine structure of the ground state of 3He+ by the ion-storage exchange-collision technique,” Phys. Rev. 187, 5–38 (1969).
[CrossRef]

Phys. Rev. A

D. J. Wineland, W. M. Itano, “Laser cooling of atoms,” Phys. Rev. A 20, 1521–1540 (1979).
[CrossRef]

W. M. Itano, D. J. Wineland, “Laser cooling of ions stored in harmonic and Penning traps,” Phys. Rev. A 25, 35–54 (1982).
[CrossRef]

A quantum mechanical treatment of rf trapping is given in R. J. Cook, D. G. Shankland, A. L. Wells, “Quantum theory of particle motion in a rapidly oscillating field,” Phys. Rev. A 31, 564–567 (1985).
[CrossRef] [PubMed]

W. Neuhauser, M. Hohenstatt, P. E. Toschek, H. G. Dehmelt, “Localized visible Ba+ mono-ion oscillator,” Phys. Rev. A 22, 1137–1140 (1980).
[CrossRef]

L. S. Brown, G. Gabrielse, “Precision spectroscopy of a charged particle in an imperfect Penning trap,” Phys. Rev. A 25, 2423–2425 (1982).
[CrossRef]

Phys. Rev. Lett.

M. D. McGuire, E. N. Fortson, “Penning-trap technique for studying electron-atom collisions at low energy,” Phys. Rev. Lett. 33, 737–739 (1974).
[CrossRef]

G. Graff, F. G. Major, R. W. H. Roeder, G. Werth, “Method for measuring the cyclotron and spin resonance of free electrons,” Phys. Rev. Lett. 21, 340–342 (1968).
[CrossRef]

J. J. Bollinger, J. D. Prestage, W. M. Itano, D. J. Wineland, “Laser cooled atomic frequency standard,” Phys. Rev. Lett. 54, 1000–1003 (1985).
[CrossRef] [PubMed]

C. F. Driscoll, J. H. Malmberg, “Length dependent containment of a pure electron-plasma column,” Phys. Rev. Lett. 50, 167–170 (1983).
[CrossRef]

H. G. Dehmelt, F. L. Walls, “Bolometric technique for the rf spectroscopy of stored ions,” Phys. Rev. Lett. 21, 127–131 (1968).
[CrossRef]

J. J. Bollinger, D. J. Wineland, “Strongly coupled nonneutral ion plasma,” Phys. Rev. Lett. 53, 348–351 (1984).
[CrossRef]

J. H. Malmberg, T. M. O’Neil, “Pure electron plasma, liquid, and crystal,” Phys. Rev. Lett. 39, 1333–1336 (1977).
[CrossRef]

W. D. White, J. H. Malmberg, C. F. Driscoll, “Resistive wall destabilization of diocotron waves,” Phys. Rev. Lett. 49, 1822–1826 (1982).
[CrossRef]

D. L. Eggleston, T. M. O’Neil, J. H. Malmberg, “Collective enhancement of radial transport in a nonneutral plasma,” Phys. Rev. Lett. 53, 982–984 (1984).
[CrossRef]

J. D. Crawford, T. M. O’Neil, J. H. Malmberg, “Effect of nonlinear collective processes on the confinement of a pure electron plasma,” Phys. Rev. Lett. 54, 697–700 (1985).
[CrossRef] [PubMed]

Rev. Mod. Phys.

S. Ichimaru, “Strongly coupled plasmas: high density classical plasmas and degenerate electron liquids,” Rev. Mod. Phys. 54, 1017–1059 (1982).
[CrossRef]

Science

A. Ashkin, “Applications of laser radiation pressure,” Science 210, 1081–1088 (1980).
[CrossRef] [PubMed]

Z. Phys.

E. Fischer, “Die dreidimensionale Stabilisierung von Ladungsträgern in einem Vierpolfeld,” Z. Phys. 156, 1–26 (1959).
[CrossRef]

Other

D. J. Wineland, W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. D. Prestage, “Spectroscopy of stored atomic ions,” in Atomic Physics 9, R. S. Van Dyck, E. N. Fortson eds. (World Scientific, Singapore, 1985), pp. 3–27.

The potential for a uniformly charged ellipsoid of revolution can be found in W. D. MacMillan, The Theory of the Potential (Dover, New York, 1958), pp. 17 and 45; O. D. Kellog, Foundation of Potential Theory (Ungar, New York, 1929), p. 195.

L. D. Landau, E. M. Lifshitz, Statistical Physics (Addison-Wesley, Reading, Mass., 1974).

E. W. McDaniel, Collision Phenomena in Ionized Gases (Wiley, New York, 1964), pp. 426–482.

J. C. Bergquist, D. J. Wineland, W. M. Itano, H. Hemmati, H.-U. Daniel, G. Leuchs, “Energy and radiative lifetime of the 5d96s2 2D5/2state in Hg ii by Doppler-free two-photon laser spectroscopy,” Phys. Rev. Lett. (to be published).

L. S. Cutler, R. P. Giffard, M. D. McGuire, “Mercury-199 trapped ion frequency standard: recent theoretical progress and experimental results,” in Proceedings of the 37th Annual Symposium on Frequency Control, 1983. (Copies available from Systematics General Corporation, Brinley Plaza, Rt. 38, Wall Township, N.J. 07719), pp. 32–36; “Thermalization of 199Hg ion macromotion by a light background gas in an rf quadrupole trap,” Appl. Phys. B 36, 137–142 (1985).

H. G. Dehmelt, “Stored ion spectroscopy,” in Advances in Laser Spectroscopy, F. T. Arecchi, F. Strumia, H. Walther, eds. (Plenum, New York, 1983), pp. 153–187.
[CrossRef]

D. J. Wineland, “Spectroscopy of stored ions,” in Precision Measurement and Fundamental Constants II, B. N. Taylor, W. D. Phillips, eds., Natl. Bur. Stand. (U.S.) Spec. Publ.617(1984), p. 83–92.

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Figures (3)

Fig. 1
Fig. 1

Schematic representation of trapped atomic particles. The trap (not shown) is assumed to have axial symmetry about the z axis. If a laser beam is directed parallel to the y axis but does not intersect the z axis, then atomic particle–photon scattering imparts angular momentum (Lz) to the sample, which can cause it to spin about the z axis.

Fig. 2
Fig. 2

Schematic representation of the electrode configuration for the Paul (rf) or Penning trap. Electrode surfaces are figures of revolution about the z axis and are equipotentials of ϕ(r, z) = A(r2 − 2z2) (cylindrical coordinates are used with the origin at the center of the trap). Typical dimensions are 2 z 0 = r 0 1 cm cm. Typical operating parameters are the following: for the Paul trap, V0 = 300 V, Ω/2π ≅ 1 MHz; for the Penning trap, U0 ≅ 1 V, B ≅ 1 T.

Fig. 3
Fig. 3

Laser-beam–ion-cloud interaction geometry. The laser beam is assumed to be in the z = 0 plane; the radius of the ion cloud is rcl. For the rf trap, laser scattering will tend to cause the cloud to spin in the direction shown. For the Penning trap, the direction of rotation shown (ω > 0) is appropriate for negative ions; for positive ions the rotation is reversed (ω < 0).

Equations (52)

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L z = i = 1 N l z i = i = 1 N ( x i × p i ) z ,
p i = m i v i + q i A ( x i ) / c ,
ϕ T ( r , z ) = α r 2 + β z 2 ,
α = q 0 V 0 2 / m Ω 2 ξ 4 - U 0 / ξ 2 ,
β = 4 q V 0 2 / m Ω 2 ξ 4 + 2 U 0 / ξ 2 .
f = n 0 ( m 2 π k B T ) 3 / 2 exp [ - ( H - ω l z ) / k B T ] ,
H = m v 2 / 2 + q ϕ ( x )
l z = m v θ r + q A θ ( x ) r / c
f ( x , v ) = n ( x ) ( m 2 π k B T ) 3 / 2 exp [ - ½ m ( v - ω r θ ^ ) 2 / k B T ] ,
n ( x ) = n 0 exp { - [ q ϕ ( r , z ) - ½ m ω ( q q Ω c + ω ) r 2 / k B T ] } .
1 r r ( r ϕ r ) + 2 ϕ z 2 = - 4 π q n 0 × exp { - [ q ϕ - ½ m ω ( q q Ω c + ω ) r 2 ] / k B T } - 4 π ρ ,
λ D 2 = k B T / 4 π n 0 q 2 .
ϕ = ϕ I + ϕ T + ϕ ind = m ω ( Ω c q / q + ω ) r 2 / 2 q .
ϕ I = [ m ω ( Ω c q / q + ω ) / 2 q - α ] r 2 - β z 2 - ϕ ind
= - π q n 0 ( a r 2 + b z 2 ) - ϕ ind ,
a = k p { 1 / [ 2 ( 1 - k p 2 ) ] - ln [ ( 1 + k p ) / ( 1 - k p ) ] / 4 k p } ,
b = k p { ln [ ( 1 + k p ) / ( 1 - k p ) ] / 2 k p - 1 } ,
a = k 0 [ sin - 1 ( k 0 ) / 2 k 0 - ( 1 - k 0 2 ) 1 / 2 / 2 ] ,
b = k 0 [ ( 1 - k 0 2 ) - 1 / 2 - sin - 1 ( k 0 ) / k 0 ] ,
n 0 ( ω , α , β ) = m 2 π q 2 [ - ω ( Ω c q / q + ω ) + q m ( 2 α + β ) ] .
n 0 ( max ) = m 2 π q 2 [ Ω c 2 / 4 + q m ( 2 α + β ) ] .
ω = - Ω c q / 2 q ± [ ( Ω c / 2 ) 2 + q ( 2 α + β - 2 π q n 0 ) / m ] 1 / 2 .
L z = n 0 V [ m r 2 ω + r q c A θ ( r ) ] d 3 x = 2 5 N m ( Ω c q / 2 q + ω ) r cl 2 ,
T L = d L z d t L = I ( x ) h ν σ ( k · v , T ) h λ d n ( x ) d 3 x ,
T L = d L z d t L = 4 π N ˙ L n 0 σ ( v y , T ) d ( r cl 2 - d 2 ) 1 / 2 / λ ,
d p d t = F = q E = q v d / K .
T bkg = N r d p / d t = - N q r 2 ω / K = - N 2 5 q K r cl 2 ω ,
ϕ T ( r , z ) = α r 2 + β z 2 + ( x 2 - y 2 ) .
x = x 1 cos ( ω x t + θ x ) , y = y 1 cos ( w y t + θ y ) ,             z = z 1 cos ( ω z t + θ z ) ,
ω x 2 = 2 q ( α + ) / m ,             ω y 2 = 2 q ( α - ) / m ,             ω z 2 = 2 q β / m .
T = d L z d t - L z / τ ,             τ = π / 2 ω x - ω y .
d L z d t Total = T L + T bkg + T = 0.
[ N ˙ s × 3.0 × 10 5 - n ( He ) × 5.3 × 10 - 2 - Δ ν x y × 2.3 × 10 10 ] = 0 ,
m v ˙ = - q ϕ + q ( v × B ) / c ,
x + i y = r 1 exp ( i ω 1 t ) + r 2 exp ( i ω 2 t ) ,
x + i y = r m exp ( i ω m t ) + r c exp ( i ω c t ) ,
ω m = - q q { Ω c / 2 - [ ( Ω c / 2 ) 2 + 2 q α / m ] 1 / 2 } ,
ω c = - q q { Ω c / 2 + [ ( Ω c / 2 ) 2 + 2 q α / m ] 1 / 2 } ,
E x y ( total ) = m ( q Ω c / 2 q + ω m ) ( ω m r m 2 - ω c r c 2 ) ,
ϕ = U 0 ξ 2 [ 2 z 2 - r 2 + 3 ( y 2 - z 2 ) sin 2 δ - 3 z y sin 2 δ ] .
E y = - ϕ y = 3 z U 0 sin 2 δ / ξ 2
x ¨ + 2 q ( α + ) x / m = q Ω c y ˙ / q ,
y ¨ + 2 q ( α - ) y / m = - q Ω c x ˙ / q .
M ( ω ) [ x 0 y 0 ] = 0 ,
M ( ω ) = [ - ω 2 + 2 q ( α + ) / m i ω q Ω c / q - i ω q Ω c / q - ω 2 + 2 q ( α - ) / m ] .
2 ( ω ± ) 2 = Ω c 2 + 4 q α / m ± ( Ω c 4 + 8 q α Ω c 2 / m + 16 q 2 2 / m 2 ) 1 / 2
2 ( ω ± ) 2 = Ω c 2 + 2 ω r 2 ± ( Ω c 4 + 4 ω r 2 Ω c 2 + 4 ω 4 ) 1 / 2
( y 0 x 0 ) ω = ± ω + = i U + , U + = q [ Ω c 2 - 2 ω 2 + ( Ω c 4 + 4 ω r 2 Ω c 2 + 4 ω 4 ) 1 / 2 ] / 2 ω + Ω c q ,
( y 0 x 0 ) ω = ± ω - = i U - , U - = q [ Ω c 2 - 2 ω 2 - ( Ω c 4 + 4 ω r 2 Ω c 2 + 4 ω 4 ) 1 / 2 ] / 2 ω - Ω c q .
x = r + cos ( ω + t + θ + ) + r - cos ( ω - t + θ - ) , y = - U + r + sin ( ω + t + θ + ) - U - r - sin ( ω - t + θ - ) .
q q U ± 1 - 2 ω 2 / [ Ω c 2 ± ( Ω c 4 + 4 Ω c 2 ω r 2 ) 1 / 2 ] .
L z ± = m r ± 2 q q Ω c 2 [ ( 1 + 4 ω r 2 / Ω c 2 + 4 ω 2 / Ω c 4 ) 1 / 2 + ( U ± 2 - 1 ) sin 2 ( ω ± t + θ ± ) + 2 ω 2 / Ω c 2 ] .

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