Abstract

We describe a technique and present results for imaging the modes of a laser-cooled plasma of 9Be+ ions in a Penning trap. The modes are excited by sinusoidally time-varying potentials applied to the trap electrodes. They are imaged by changes in the ion resonance fluorescence produced by Doppler shifts from the coherent ion velocities of the mode. For the geometry and conditions of this experiment, the mode frequencies and eigenfunctions have been calculated analytically. A comparison between theory and experiment for some of the azimuthally symmetric modes shows good agreement.

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References

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  1. Non-Neutral Plasma Physics II, eds. J. Fajans and D. H. E. Dubin (AIP, New York, 1995).
  2. J. H. Malmberg and T. M. O'Neil, ``Pure electron plasma, liquid, and crystal," Phys. Rev. Lett. 39, 1333-1336 (1977).
    [CrossRef]
  3. C. F. Driscoll, J. H. Malmberg, and K. S. Fine, ``Observation of transport to thermal equilibrium in pure electron plasmas," Phys. Rev. Lett. 60, 1290-1293 (1988).
    [CrossRef] [PubMed]
  4. L. R. Brewer, J. D. Prestage, J. J. Bollinger, W. M. Itano, D. J. Larson, and D. J. Wineland, ``Static properties of a non-neutral 9 Be + ion plasma," Phys. Rev. A 38, 859-873 (1988).
    [CrossRef] [PubMed]
  5. J. J. Bollinger, D. J. Wineland, and D. H. E. Dubin, ``Non-neutral ion plasmas and crystals, laser cooling, and atomic clocks," Phys. Plasmas 1, 1403-1414 (1994).
    [CrossRef]
  6. D. H. E. Dubin,``Theory of electrostatic uid modes in a cold spheroidal non-neutral plasma," Phys. Rev. Lett. 66, 2076-2079 (1991).
    [CrossRef] [PubMed]
  7. J. J. Bollinger, D. J. Heinzen, F. L. Moore, W. M. Itano, D. J. Wineland, and D. H. E. Dubin, ``Electrostatic modes of ion-trap plasmas," Phys. Rev. A 48, 525-545 (1993).
    [CrossRef] [PubMed]
  8. R. G. Greaves and C. M. Surko, ``Antimatter plasmas and antihydrogen," Phys. Plasmas 4, 1528-1543 (1997).
    [CrossRef]
  9. G. Gabrielse, X. Fei, L. A. Orozco, R. L. Tjoelker, J. Haas, H. Kalinowsky, T. A. Trainor, and W. Kells, ``Cooling and slowing of trapped antiprotons below 100 meV," Phys. Rev. Lett. 63, 1360 (1989).
    [CrossRef] [PubMed]
  10. M.H. Holzscheiter, X. Feng, T. Goldman, N.S.P. King, R. A. Lewis, M. M. Nieto, and G.A. Smith, ``Are antiprotons forever?," Phys. Lett. A 214, 279 (1996).
    [CrossRef]
  11. D. J. Heinzen, J. J. Bollinger, F. L. Moore, W. M. Itano, and D. J. Wineland, ``Rotational equilibria and low-order modes of a non-neutral ion plasma," Phys. Rev. Lett. 66, 2080-2083 (1991).
    [CrossRef] [PubMed]
  12. X.-P. Huang, J. J. Bollinger, T. B. Mitchell, and W. M. Itano, ``Phase-locked rotation of crystallized non-neutral plasmas by rotating electric fields," Phys. Rev. Lett. 80, 73-76 (1998).
    [CrossRef]
  13. D. H. E. Dubin and J. P. Schiffer, ``Normal modes of cold confined one-component plasmas," Phys. Rev. E 53, 5249-5267 (1996).
    [CrossRef]
  14. D. H. E. Dubin, ``Effects of correlations on the thermal equilibrium and normal modes of a non-neutral plasma," Phys. Rev. E 53, 5268-5290 (1996).
    [CrossRef]
  15. C. S. Weimer, J. J. Bollinger, F. L. Moore, and D. J. Wineland, ``Electrostatic modes as a diagnostic in Penning trap experiments," Phys. Rev. A 49, 3842-3853 (1994).
    [CrossRef] [PubMed]
  16. M. D. Tinkle, R. G. Greaves, and C. M. Surko, ``Low-order longitudinal modes of single-component plasmas," Phys. Plasmas 2, 2880-2894 (1995).
    [CrossRef]
  17. R. G. Greaves, M. D. Tinkle, and C. M. Surko, ``Modes of a pure ion plasma at the Brillouin limit," Phys. Rev. Lett. 74, 90-93 (1995).
    [CrossRef] [PubMed]
  18. C. F. Driscoll, ``Observation of an unstable m = 1 diocotron mode on a hollow electron column," Phys. Rev. Lett. 64, 1528-1543 (1990).
    [CrossRef]
  19. J. N. Tan, J. J. Bollinger, B. Jelenkovic, and D. J. Wineland, ``Long-range order in laser-cooled, atomic-ion Wigner crystals observed by Bragg scattering," Phys. Rev. Lett. 75, 4198-4201 (1995).
    [CrossRef] [PubMed]
  20. W. M. Itano, J. J. Bollinger, J. N. Tan, B. Jelenkovic, X.-P. Huang, and D. J. Wineland, ``Bragg diffraction from crystallized ion plasmas," Science 279, 686-689 (1998).
    [CrossRef] [PubMed]
  21. Here ! lm is the mode frequency in a frame rotating with the plasma. For the m = 0 modes discussed here this distinction is not necessary because their frequency is the same in either the laboratory or rotating frame.
  22. Information on the mode eigenfunction can be obtained from the side-view images even when there is a change in the phase-averaged ion uorescence. However, the images may no longer provide a linear measure of the mode axial velocity.
  23. R. C. Thompson, K. Dholakia, J-L. Hernandez-Pozos, G. Zs. K. Horvath, J. Rink, and D. M. Segal, ``Spectroscopy and quantum optics with ion traps," Phys. Scr. T72, 24-33 (1997).
    [CrossRef]

Other (23)

Non-Neutral Plasma Physics II, eds. J. Fajans and D. H. E. Dubin (AIP, New York, 1995).

J. H. Malmberg and T. M. O'Neil, ``Pure electron plasma, liquid, and crystal," Phys. Rev. Lett. 39, 1333-1336 (1977).
[CrossRef]

C. F. Driscoll, J. H. Malmberg, and K. S. Fine, ``Observation of transport to thermal equilibrium in pure electron plasmas," Phys. Rev. Lett. 60, 1290-1293 (1988).
[CrossRef] [PubMed]

L. R. Brewer, J. D. Prestage, J. J. Bollinger, W. M. Itano, D. J. Larson, and D. J. Wineland, ``Static properties of a non-neutral 9 Be + ion plasma," Phys. Rev. A 38, 859-873 (1988).
[CrossRef] [PubMed]

J. J. Bollinger, D. J. Wineland, and D. H. E. Dubin, ``Non-neutral ion plasmas and crystals, laser cooling, and atomic clocks," Phys. Plasmas 1, 1403-1414 (1994).
[CrossRef]

D. H. E. Dubin,``Theory of electrostatic uid modes in a cold spheroidal non-neutral plasma," Phys. Rev. Lett. 66, 2076-2079 (1991).
[CrossRef] [PubMed]

J. J. Bollinger, D. J. Heinzen, F. L. Moore, W. M. Itano, D. J. Wineland, and D. H. E. Dubin, ``Electrostatic modes of ion-trap plasmas," Phys. Rev. A 48, 525-545 (1993).
[CrossRef] [PubMed]

R. G. Greaves and C. M. Surko, ``Antimatter plasmas and antihydrogen," Phys. Plasmas 4, 1528-1543 (1997).
[CrossRef]

G. Gabrielse, X. Fei, L. A. Orozco, R. L. Tjoelker, J. Haas, H. Kalinowsky, T. A. Trainor, and W. Kells, ``Cooling and slowing of trapped antiprotons below 100 meV," Phys. Rev. Lett. 63, 1360 (1989).
[CrossRef] [PubMed]

M.H. Holzscheiter, X. Feng, T. Goldman, N.S.P. King, R. A. Lewis, M. M. Nieto, and G.A. Smith, ``Are antiprotons forever?," Phys. Lett. A 214, 279 (1996).
[CrossRef]

D. J. Heinzen, J. J. Bollinger, F. L. Moore, W. M. Itano, and D. J. Wineland, ``Rotational equilibria and low-order modes of a non-neutral ion plasma," Phys. Rev. Lett. 66, 2080-2083 (1991).
[CrossRef] [PubMed]

X.-P. Huang, J. J. Bollinger, T. B. Mitchell, and W. M. Itano, ``Phase-locked rotation of crystallized non-neutral plasmas by rotating electric fields," Phys. Rev. Lett. 80, 73-76 (1998).
[CrossRef]

D. H. E. Dubin and J. P. Schiffer, ``Normal modes of cold confined one-component plasmas," Phys. Rev. E 53, 5249-5267 (1996).
[CrossRef]

D. H. E. Dubin, ``Effects of correlations on the thermal equilibrium and normal modes of a non-neutral plasma," Phys. Rev. E 53, 5268-5290 (1996).
[CrossRef]

C. S. Weimer, J. J. Bollinger, F. L. Moore, and D. J. Wineland, ``Electrostatic modes as a diagnostic in Penning trap experiments," Phys. Rev. A 49, 3842-3853 (1994).
[CrossRef] [PubMed]

M. D. Tinkle, R. G. Greaves, and C. M. Surko, ``Low-order longitudinal modes of single-component plasmas," Phys. Plasmas 2, 2880-2894 (1995).
[CrossRef]

R. G. Greaves, M. D. Tinkle, and C. M. Surko, ``Modes of a pure ion plasma at the Brillouin limit," Phys. Rev. Lett. 74, 90-93 (1995).
[CrossRef] [PubMed]

C. F. Driscoll, ``Observation of an unstable m = 1 diocotron mode on a hollow electron column," Phys. Rev. Lett. 64, 1528-1543 (1990).
[CrossRef]

J. N. Tan, J. J. Bollinger, B. Jelenkovic, and D. J. Wineland, ``Long-range order in laser-cooled, atomic-ion Wigner crystals observed by Bragg scattering," Phys. Rev. Lett. 75, 4198-4201 (1995).
[CrossRef] [PubMed]

W. M. Itano, J. J. Bollinger, J. N. Tan, B. Jelenkovic, X.-P. Huang, and D. J. Wineland, ``Bragg diffraction from crystallized ion plasmas," Science 279, 686-689 (1998).
[CrossRef] [PubMed]

Here ! lm is the mode frequency in a frame rotating with the plasma. For the m = 0 modes discussed here this distinction is not necessary because their frequency is the same in either the laboratory or rotating frame.

Information on the mode eigenfunction can be obtained from the side-view images even when there is a change in the phase-averaged ion uorescence. However, the images may no longer provide a linear measure of the mode axial velocity.

R. C. Thompson, K. Dholakia, J-L. Hernandez-Pozos, G. Zs. K. Horvath, J. Rink, and D. M. Segal, ``Spectroscopy and quantum optics with ion traps," Phys. Scr. T72, 24-33 (1997).
[CrossRef]

Supplementary Material (3)

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» Media 3: MOV (81 KB)     

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Figures (5)

Figure 1.
Figure 1.

Sketch of the experimental apparatus. Modes were excited by applying in-phase or 180° out-of-phase sinusoidal potentials to the trap endcaps.

Figure 2.
Figure 2.

Plots of the frequencies of several magnetized plasma modes as a function of rotation frequency for Ω/2π=7.608 MHz and ωz /2π=1.13 MHz. The solid lines are the theoretical predictions and the symbols are experimental measurements. Only the highest frequency (9, 0) plasma mode and the second highest frequency (8, 0) plasma mode are plotted.

Figure 3.
Figure 3.

(a) Movie of sideview image data obtained on a plasma with ωr /2π= 1.00 MHz while driving a (2,0) mode at ω 2,0/2π=1.656 MHz. The magnetic field and axial laser beam point up. The ion cloud dimensions are 2z 0 = 0.76 mm and 2r 0 = 0.24 mm, and the density n 0 = 2.70 × 109 cm-3. Comparison of the amplitude (b) and phase (c) extracted from the (2,0) mode in (a) with the predictions of linear theory. The theory predictions are on the right. [Media 1]

Figure 4.
Figure 4.

(a) Movie of sideview image data obtained on the plasma of Fig. 3 with ωr /2π= 1.00 MHz while driving a (9,0) mode at ω 9,0/2π=2.952 MHz. Comparison of the amplitude (b) and phase (c) extracted from the (9, 0) mode in (a) with the predictions of linear theory. The theory predictions are on the right. [Media 2]

Figure 5.
Figure 5.

(a) Movie of sideview image data obtained on a plasma with ωr /2π= 638 kHz while driving with an even drive at 1.619 MHz. At this rotation frequency there is a crossing of the (2,0) mode and an (8,0) mode with a radial node. Comparison of the amplitude (b) and phase (c) extracted from the data in (a) with the predictions of linear theory. The predictions of both the (2, 0) and (8, 0) modes are given. For this plasma 2z 0 = 0.70 mm and 2r 0 = 0.29 mm. [Media 3]

Equations (2)

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ϕ T r z = m ω z 2 2 q ( z 2 r 2 2 ) .
Ψ lm P l m ( ξ ¯ 1 d ¯ ) P l m ( ξ ¯ 2 ) e i ( ω lm t ) .

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