Applied Physics B

, Volume 114, Issue 1–2, pp 157–166 | Cite as

Theory and simulation of ion Coulomb crystal formation in a Penning trap

  • Martin Asprusten
  • Simon Worthington
  • Richard C. ThompsonEmail author


Ion Coulomb crystals (ICCs) are formed by laser-cooled ions in both radio-frequency and Penning traps. In radio-frequency traps, the crystals are generally stationary. In Penning traps, ICCs always rotate. The frequency of rotation is often set by an applied rotating wall drive that forces the crystal to rotate at the same frequency as the drive. In the absence of any applied rotating or oscillating fields, ICCs in a Penning trap can be in stable equilibrium with a range of rotation frequencies. The density and shape of the crystal adjust with the rotation frequency to ensure that equilibrium is reached. Here, we show that the parameters of the radial laser-cooling beam determine the rotation frequency of a small crystal in a Penning trap when no driving fields are present. We demonstrate, using an approximate theoretical treatment and realistic simulations, that the crystal rotation frequency is independent of the number of ions and the trap parameters, so long as the crystal radius remains smaller than the cooling laser beam waist. As the rotation frequency increases, the crystal eventually becomes a linear string, at which point it is no longer able to adjust its density. Instead, a small amplitude vibration in the zigzag mode of oscillation manifests itself as a rotation of the crystal at a fixed frequency that depends only on the applied trap potential.


Laser Beam Rotation Frequency Laboratory Frame Laser Cool Radial Plane 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was supported by the European Commission STREP PICC (FP7 2007–2013 grant number 249958). We gratefully acknowledge financial support towards networking activities from COST Action MP 1001—Ion Traps for Tomorrows Applications.


  1. 1.
    F.G. Major, G. Gheorghe, V.N. Werth, Charged Particle Traps. (Springer, Heidelberg, 2005)Google Scholar
  2. 2.
    G. Werth, V.N. Gheorghe, F.G. Major, Charged Particle Traps II: Applications. (Springer, Heidelberg, 2005)Google Scholar
  3. 3.
    B. Roth, P. Blythe, S. Schiller, Motional resonance coupling in cold multispecies Coulomb crystals. Phys. Rev. A 75(2), 023402 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    H.C. Nagerl, D. Leibfried, F. Schmidt-Kaler, J. Eschner, R. Blatt, Coherent excitation of normal modes in a string of Ca+ ions. Opt. Express 3(2), 89–96 (1998)ADSCrossRefGoogle Scholar
  5. 5.
    A. Dantan, M. Albert, J.P. Marler, J.P. Herskind, M. Drewsen, Large ion Coulomb crystals: a near-ideal medium for coupling optical cavity modes to matter. Phys. Rev. A 80(4), 041802 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    J.J. Bollinger, T.B. Mitchell, X.P. Huang, W.M. Itano, J.N. Tan, B.M. Jelenkovic, D.J. Wineland, Crystalline order in laser-cooled, non-neutral ion plasmas. Phys. Plasmas 7(1), 7–13 (2000)ADSCrossRefGoogle Scholar
  7. 7.
    W.L. Slattery, G.D. Doolen, H.E. Dewitt, N-dependence in the classical one-component plasma Monte–Carlo calculations. Phys. Rev. A 26(4), 2255–2258 (1982)ADSCrossRefGoogle Scholar
  8. 8.
    D.H.E. Dubin, T.M. O’Neil, Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium states). Rev. Modern Phys. 71(1), 87–172 (1999)ADSCrossRefGoogle Scholar
  9. 9.
    R. Blatt, C.F. Roos, Quantum simulations with trapped ions. Nat. Phys. 8(4), 277–284 (2012)CrossRefGoogle Scholar
  10. 10.
    M. Mielenz, J. Brox, S. Kahra, G. Leschhorn, M. Albert, T. Schaetz, H. Landa, B. Reznik, Trapping of topological-structural defects in Coulomb crystals. Phys. Rev. Lett. 110(13), 133004 (2013)ADSCrossRefGoogle Scholar
  11. 11.
    S. Ulm, J. Rossnagel, G. Jacob, C. Deguenther, S.T. Dawkins, U.G. Poschinger, R. Nigmatullin, A. Retzker, M.B. Plenio, F. Schmidt-Kaler, Observation of the Kibble–Zurek scaling law for defect formation in ion crystals. Nat. Commun. 4, 2290 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    K. Pyka, J. Keller, H.L. Partner, R. Nigmatullin, T. Burgermeister, D.-M. Meier, K. Kuhlmann, A. Retzker, M.B. Plenio, W.H. Zurek, del A. Campo, T.E. Mehlstaeubler, Observation of the Kibble–Zurek scaling law for defect formation in ion crystals. Nat. Commun. 4, 2290 (2013)CrossRefGoogle Scholar
  13. 13.
    K. Molhave, M. Drewsen, Formation of translationally cold MgH+ and MgD+ molecules in an ion trap. Phys. Rev. A 62(1), 011401 (2000)ADSCrossRefGoogle Scholar
  14. 14.
    J.N. Tan, J.J. Bollinger, B. Jelenkovic, D.J. Wineland, Long-range order in laser-cooled, atomic-ion Wigner crystals observed by Bragg scattering. Phys. Rev. Lett. 75(23), 4198–4201 (1995)ADSCrossRefGoogle Scholar
  15. 15.
    J.W. Britton, B.C. Sawyer, A.C. Keith, C.C.J. Wang, J.K. Freericks, H. Uys, M.J. Biercuk, J.J. Bollinger, Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins. Nature 484(7395), 489–492 (2012)ADSCrossRefGoogle Scholar
  16. 16.
    R.C. Thompson, D.C. Wilson, The motion of small numbers of ions in a Penning trap. Zeitschrift Fur Physik D-Atoms Mol. Clust. 42(4), 271–277 (1997)ADSCrossRefGoogle Scholar
  17. 17.
    W.M. Itano, D.J. Wineland, Laser cooling of ions stored in harmonic and Penning traps. Phys. Rev. A 25(1), 35–54 (1982)ADSCrossRefGoogle Scholar
  18. 18.
    R.C. Thompson, J. Papadimitriou, Simple model for the laser cooling of an ion in a Penning trap. J. Phys. B-At. Mol. Opt. Phys. 33(17), 3393–3405 (2000)ADSCrossRefGoogle Scholar
  19. 19.
    R.J. Hendricks, E.S. Phillips, D.M. Segal, R.C. Thompson, Laser cooling in the Penning trap: an analytical model for cooling rates in the presence of an axializing field. J. Phys. B-At. Mol. Opt. Phys. 41(3), 035301 (2008)ADSCrossRefGoogle Scholar
  20. 20.
    J.J. Bollinger, D.J. Heinzen, F.L. Moore, W.M. Itano, D.J. Wineland, D.H.E. Dubin, Electrostatic modes of ion-trap plasmas. Phys. Rev. A 48(1), 525–545 (1993)ADSCrossRefGoogle Scholar
  21. 21.
    D.F.A. Winters, M. Vogel, D.M. Segal, R.C. Thompson, Electronic detection of charged particle effects in a Penning trap. J. Phys. B-At. Mol. Opt. Phys. 39(14), 3131–3143 (2006)ADSCrossRefGoogle Scholar
  22. 22.
    W.M. Itano, L.R. Brewer, L.R. Larson, L.R. Wineland, Perpendicular laser cooling of a rotating ion plasma in a Penning trap. Phys. Rev. A 38(11), 5698–5706 (1988)ADSCrossRefGoogle Scholar
  23. 23.
    A. Retzker, R.C. Thompson, D.M. Segal, M.B. Plenio, Double well potentials and quantum phase transitions in ion traps. Phys. Rev. Lett. 101(26), 260504 (2008)ADSCrossRefGoogle Scholar
  24. 24.
    R. Rafac, J.P. Schiffer, J.S. Hangst, J.S. Dubin, D.J. Wales, Stable configurations of confined cold ionic systems. Proc. Natl. Acad. Sci. USA 88(2), 483–486 (1991)ADSCrossRefGoogle Scholar
  25. 25.
    S. Mavadia, J.F. Goodwin, G. Stutter, S. Bharadia, D.R. Crick, D.M. Segal, R.C. Thompson, Control of the conformations of ion Coulomb crystals in a Penning trap. Nat. Commun. 4:2571 (2013)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Martin Asprusten
    • 1
  • Simon Worthington
    • 1
  • Richard C. Thompson
    • 1
    Email author
  1. 1.QOLS Group, Department of PhysicsImperial College LondonLondonUK

Personalised recommendations