Abstract

Trapped and laser-cooled ions are increasingly used for a variety of modern high-precision experiments, for frequency standard applications, and for quantum information processing. Therefore laser cooling of trapped ions is reviewed, the current state of the art is reported, and several new cooling techniques are outlined. The principles of ion trapping and the basic concepts of laser cooling for trapped atoms are introduced. The underlying physical mechanisms are presented, and basic experiments are briefly sketched. Particular attention is paid to recent progress by elucidating several milestone experiments. In addition, a number of special cooling techniques pertaining to trapped ions are reviewed; open questions and future research lines are indicated.

© 2003 Optical Society of America

PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Kastler, “Quelques suggestions concernant la production optique et la détection optique d’une inégalité de population des niveaux de quantification spatiale des atomes,” J. Phys. 11, 255–265 (1950).
  2. Ya. B. Zel’dovich, “Cooling with the aid of high-frequency energy,” JETP Lett. 19, 74–75 (1974).
  3. T. W. Hänsch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68–69 (1975).
  4. D. Wineland and H. Dehmelt, “Proposed 1014δν/ν laser fluorescence spectroscopy on Tl+ mono-ion oscillator III (side band cooling),” Bull. Am. Phys. Soc. 20, 637 (1975).
  5. W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, “Optical-sideband cooling of visible atom cloud confined in parabolic well,” Phys. Rev. Lett. 41, 233–236 (1978).
  6. D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1639–1642 (1978).
  7. D. J. Wineland and W. M. Itano, “Laser cooling of atoms,” Phys. Rev. A 20, 1521–1540 (1979).
  8. W. M. Itano and D. J. Wineland, “Laser cooling of ions stored in harmonic and Penning traps,” Phys. Rev. A 25, 35–54 (1982).
  9. W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, “Localized visible Ba+ mono-ion oscillator,” Phys. Rev. A 22, 1137–1140 (1980).
  10. D. J. Wineland, W. M. Itano, and R. S. van Dyck, “High resolution spectroscopy of stored ions,” Adv. At. Mol. Phys. 19, 135–186 (1983).
  11. W. Nagourney, J. Sandberg, and H. Dehmelt, “Shelved optical electron amplifier: observation of quantum jumps,” Phys. Rev. Lett. 56, 2797–2799 (1986).
  12. Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, “Observation of quantum jumps,” Phys. Rev. Lett. 57, 1696–1698 (1986).
  13. J. C. Bergquist, R. Hulet, W. M. Itano, and D. J. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).
  14. S. Stenholm, “Semiclassical theory of laser cooling,” Rev. Mod. Phys. 58, 699–739 (1986), and references therein.
  15. F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62, 403–406 (1989).
  16. Special Issue of J. Opt. Soc. Am. B 6, (1989).
  17. J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46, 2668–2681 (1992).
  18. J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Dark squeezed states of the motion of a trapped ion,” Phys. Rev. Lett. 70, 556–559 (1993).
  19. J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Preparation of Fock states by observation of quantum jumps in an ion trap,” Phys. Rev. Lett. 70, 762–765 (1993).
  20. C. A. Blockley, D. F. Walls, and H. Risken, “Quantum collapses and revivals in a quantized trap,” Europhys. Lett. 17, 509–514 (1992).
  21. D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, “Generation of nonclassical motional states of a trapped atom,” Phys. Rev. Lett. 76, 1796–1799 (1996).
  22. M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, “Quantum Rabi oscillation: a direct test of field quantization in a cavity,” Phys. Rev. Lett. 76, 1800–1803 (1996).
  23. J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Nonclassical states of motion in ion traps,” Adv. Atom., Mol., Opt. Phys. 37, 237–296 (1996).
  24. J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995).
  25. C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).
  26. Ch. Roos, Th. Zeiger, H. Rohde, H. C. Nägerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, “Quantum state engineering on an optical transition and decoherence in a Paul trap,” Phys. Rev. Lett. 83, 4713–4716 (1999).
  27. Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631–3634 (1998).
  28. C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).
  29. B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Cooling the collective motion of trapped ions to initialize a quantum register,” Phys. Rev. Lett. 81, 1525–1528 (1998).
  30. H. Rohde, S. T. Gulde, C. F. Roos, P. A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Sympathetic ground state cooling and coherent manipulation with two-ion-crystals,” J. Opt. B 3, S34–S41 (2001).
  31. G. Morigi, J. Eschner, and C. H. Keitel, “Ground state laser cooling using electromagnetically induced transparency,” Phys. Rev. Lett. 85, 4458–4461 (2000).
  32. C. F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Experimental demonstration of ground state laser cooling with electromagnetically induced transparency,” Phys. Rev. Lett. 85, 5547–5550 (2000).
  33. P. K. Gosh, Ion Traps (Clarendon, Oxford, UK, 1995).
  34. P. Kienle, “Sunshine by cooling,” Naturwissenschaften 88, 313–321 (2001), and references therein.
  35. J. Prestage, G. J. Dick, and L. Maleki, “New ion trap for frequency standard applications,” J. Appl. Phys. 66, 1013–1017 (1989).
  36. G. R. Janik and L. Maleki, “Simple analytic potentials for linear ion traps,” J. Appl. Phys. 67, 6050–6055 (1990).
  37. M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Ionic crystals in a linear Paul trap,” Phys. Rev. A 45, 6493–6501 (1992).
  38. M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Linear trap for high accuracy spectroscopy of stored ions,” J. Mod. Opt. 39, 233–242 (1992).
  39. A. Steane, “The ion trap quantum information processor,” Appl. Phys. B 64, 623–642 (1997).
  40. D. V. F. James, “Quantum dynamics of cold trapped ions, with application to quantum computation,” Appl. Phys. B 66, 181–190 (1998).
  41. In the absence of applied fields the trapped ion may still couple to the environment: Heating and decoherence of quantum states of the motion has been observed, 42 the origin of which is partly still unclear. Apparently this effect is critically determined by the trap characteristics, and the rate can be much slower than the rate of laser cooling.43 Furthermore, the charge of the ion couples to blackbody radiation. This process is also negligible on the time scales considered here. In the discussion, we ignore these effects.
  42. Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).
  43. D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, “Experimental issues in coherent quantum-state manipulation of trapped atomic ions,” J. Res. Natl. Inst. Stand. Technol. 103, 259–328 (1998).
  44. S. Stenholm, “Redistribution of molecular velocities by optical processes,” Appl. Phys. 15, 287–296 (1978).
  45. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions (Wiley, New York, 1992).
  46. When the natural linewidth is comparable with or smaller than the laser linewidth, the rate Γ has to be replaced by the spectroscopic linewidth obtained experimentally.
  47. D. J. Wineland, J. Dalibard, and C. Cohen-Tannoudji, “Sisyphus cooling of a bound atom,” J. Opt. Soc. Am. B 9, 32–42 (1992).
  48. J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Laser cooling of trapped ions with polarization gradients,” Phys. Rev. A 48, 1434–1445 (1993).
  49. D. J. Heinzen and D. J. Wineland, “Quantum-limited cooling and detection of radio-frequency oscillations by laser-cooled ions,” Phys. Rev. A 42, 2977–2994 (1990).
  50. I. Marzoli, J. I. Cirac, R. Blatt, and P. Zoller, “Laser cooling of trapped three-level ions: designing two-level systems for sideband cooling,” Phys. Rev. A 49, 2771–2779 (1994).
  51. G. Morigi, J. I. Cirac, M. Lewenstein, and P. Zoller, “Ground-state laser cooling beyond the Lamb–Dicke limit,” Europhys. Lett. 39, 13–18 (1997).
  52. L. Santos and M. Lewenstein, “Dynamical cooling of trapped gases: one-atom problem,” Phys. Rev. A 59, 613–619 (1999).
  53. E. Peik, J. Abel, Th. Becker, J. von Zanthier, and H. Walther, “Sideband cooling of ions in radio-frequency traps,” Phys. Rev. A 60, 439–449 (1999).
  54. In experiments the distance among the ions is usually of several wavelengths, and the only relevant interaction is the Coulomb repulsion. For distances of the order of the wavelength, the dipole–dipole interaction must be taken into account. Its effect in connection with laser cooling has been investigated in A. W. Vogt, J. I. Cirac, and P. Zoller, “Collective laser cooling of two trapped ions,” Phys. Rev. A 53, 950–968 (1995).
  55. I. Waki, S. Kassner, G. Birkl, and H. Walther, “Observation of ordered structures of laser-cooled ions in a quadrupole storage ring,” Phys. Rev. Lett. 68, 2007–2010 (1992).
  56. G. Birkl, S. Kassner, and H. Walther, “Multiple-shell struc-tures of laser-cooled 24Mg+ ions in a quadrupole storage ring,” Nature 357, 310–313 (1992).
  57. J. Javanainen, “Light-pressure cooling of a crystal,” Phys. Rev. Lett. 56, 1798–1801 (1986).
  58. J. Javanainen, “Laser cooling of trapped-ion clusters,” J. Opt. Soc. Am. B 5, 73–81 (1988).
  59. G. Morigi, J. Eschner, J. I. Cirac, and P. Zoller, “Laser cooling of two trapped ions: sideband cooling beyond the Lamb–Dicke limit,” Phys. Rev. A 59, 3797–3808 (1999).
  60. G. Morigi and J. Eschner, “Doppler cooling of a Coulomb crystal,” Phys. Rev. A 64, 063407 (2001).
  61. D. Kielpinski, B. E. King, C. J. Myatt, C. A. Sackett, Q. A. Turchette, W. M. Itano, C. Monroe, and D. J. Wineland, “Sympathetic cooling of trapped ions for quantum logic,” Phys. Rev. A 61, 032310 (2000).
  62. G. Morigi and H. Walther, “Two-species Coulomb chains for quantum information,” Eur. Phys. J. D 13, 261–269 (2001).
  63. S. L. Gilbert, J. J. Bollinger, and D. J. Wineland, “Shell-structure phase of magnetically confined strong coupled plasmas,” Phys. Rev. Lett. 60, 2022–2025 (1988)
  64. 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).
  65. A discussion of crystallization in these mesoscopic structures is beyond the scope of this work. The interested reader can find a review in D. H. Dubin and T. M. O’Neil, “Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium state),” Rev. Mod. Phys. 71, 87–172 (1999).
  66. F. Diedrich, E. Peik, J. M. Chen, W. Quint, and H. Walther, “Observation of a phase transition of stored laser-cooled ions,” Phys. Rev. Lett. 59, 2931–2934 (1987).
  67. D. J. Wineland, J. C. Bergquist, W. M. Itano, J. J. Bollinger, and C. H. Manney, “Atomic-ion Coulomb clusters in an ion trap,” Phys. Rev. Lett. 59, 2935–2938 (1987).
  68. R. Blümel, J. M. Chen, E. Peik, W. Quint, W. Schleich, Y. R. Shen, and H. Walther, “Phase transition of stored laser-cooled ions,” Nature 334, 309–313 (1988).
  69. D. S. Hall and G. Gabrielse, “Electron cooling of protons in a nested Penning trap,” Phys. Rev. Lett. 77, 1962–1965 (1996).
  70. W. Alt, M. Block, P. Seibert, and G. Werth, “Spatial separation of atomic states in a laser-cooled ion crystal,” Phys. Rev. A 58, R23–R25 (1998).
  71. P. Bowe, L. Hornekær, C. Brodersen, M. Drewsen, J. S. Hangst, and J. P. Schiffer, “Sympathetic crystallization of trapped ions,” Phys. Rev. Lett. 82, 2071–2074 (1999).
  72. H. C. Nägerl, D. Leibfried, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Coherent excitation of normal modes in a string of Ca+ ions,” Opt. Express 3, 89–96 (1998).
  73. C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).
  74. G. Morigi, H. Baldauf, W. Lange, and H. Walther, “Raman sideband cooling in the presence of multiple decay channels,” Opt. Commun. 187, 171–177 (2001).
  75. M. Lindberg and J. Javanainen, “Temperature of laser-cooled trapped three-level ion,” J. Opt. Soc. Am. B 3, 1008–1017 (1986).
  76. D. Reiß, A. Lindner, and R. Blatt, “Cooling of trapped multilevel ions: a numerical analysis,” Phys. Rev. A 54, 5133–5140 (1996).
  77. While we give the original explanation here, a more recent study shows that EIT cooling works without restrictions on the intensities see the two lasers, as long as the Lamb–Dicke limit applies; see G. Morigi, “Cooling atomic motion with quantum interference,” Phys. Rev. A (to be published); preprint available at http://arXiv.org/abs/quant-ph/0211043.
  78. E. Arimondo, “Coherent population trapping in laser spectroscopy,” in Progress in Optics XXXV, pp. 257–354, E. Wolf, ed. (North Holland, Amsterdam, 1996).
  79. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997), and references therein.
  80. F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73, 807–814 (2001).
  81. J. Dalibard and C. Cohen-Tannoudji, “Laser cooling below the Doppler limit by polarization gradients: simple theoretical models,” J. Opt. Soc. Am. B 6, 2023–2045 (1989).
  82. J. Dalibard and C. Cohen-Tannoudji, “Dressed-atom approach to atomic motion in laser light: the dipole force revisited,” J. Opt. Soc. Am. B 2, 1707–1720 (1985).
  83. S. M. Yoo and J. Javanainen, “Polarization gradient cooling of a trapped ion,” Phys. Rev. A 48, R30–R33 (1993).
  84. G. Birkl, J. A. Yeazell, R. Rückerl, and H. Walther, “Polarization gradient cooling of trapped ions,” Europhys. Lett. 27, 197–202 (1994).
  85. H. Katori, S. Schlipf, and H. Walther, “Anomalous dynamics of a single ion in an optical lattice,” Phys. Rev. Lett. 79, 2221–2224 (1997).
  86. S. Schlipf, H. Katori, L. Perotti, and H. Walther, “Diffusion of a single ion in a one-dimensional optical lattice,” Opt. Express 3, 97–103 (1998).
  87. J. Eschner, B. Appasamy, and P. E. Toschek, “Stochastic cooling of a trapped ion by null detection of its fluorescence,” Phys. Rev. Lett. 74, 2435–2438 (1995).
  88. B. Appasamy, Y. Stalgies, and P. E. Toschek, “Measurement-induced vibrational dynamics of a trapped ion,” Phys. Rev. Lett. 80, 2805–2808 (1998).
  89. T. Binnewies, U. Sterr, J. Helmcke, and F. Riehle, “Cooling by Maxwell’s demon: preparation of single-velocity atoms for matter-wave interferometry,” Phys. Rev. A 62, 011601 (2000).
  90. V. I. Balykin and V. S. Letokhov, “Informational cooling of neutral atoms,” Phys. Rev. A 64, 063410 (2001).
  91. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).
  92. R. Dum, P. Marte, T. Pellizzari, and P. Zoller, “Laser cooling to a single quantum state in a trap,” Phys. Rev. Lett. 73, 2829–2832 (1994).
  93. Ch. Raab, J. Eschner, J. Bolle, H. Oberst, F. Schmidt-Kaler, and R. Blatt, “Motional sidebands and direct measurement of the cooling rate in the resonance fluorescence of a single trapped ion,” Phys. Rev. Lett. 85, 538–541 (2000).
  94. R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Sub-dekahertz ultraviolet spectroscopy of 199Hg+,” Phys. Rev. Lett. 85, 2462–2465 (2000).
  95. R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).
  96. J. L. Clark and G. Rumbles, “Laser cooling in the condensed phase by frequency up-conversion,” Phys. Rev. Lett. 76, 2037–2040 (1996).
  97. C. W. Hoyt, M. Sheik-Bahae, R. I. Epstein, B. C. Edwards, and J. E. Anderson, “Observation of anti-Stokes fluorescence cooling in thulium-doped glass,” Phys. Rev. Lett. 85, 3600–3603 (2000).

2001 (7)

H. Rohde, S. T. Gulde, C. F. Roos, P. A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Sympathetic ground state cooling and coherent manipulation with two-ion-crystals,” J. Opt. B 3, S34–S41 (2001).

P. Kienle, “Sunshine by cooling,” Naturwissenschaften 88, 313–321 (2001), and references therein.

G. Morigi and J. Eschner, “Doppler cooling of a Coulomb crystal,” Phys. Rev. A 64, 063407 (2001).

G. Morigi and H. Walther, “Two-species Coulomb chains for quantum information,” Eur. Phys. J. D 13, 261–269 (2001).

G. Morigi, H. Baldauf, W. Lange, and H. Walther, “Raman sideband cooling in the presence of multiple decay channels,” Opt. Commun. 187, 171–177 (2001).

F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73, 807–814 (2001).

V. I. Balykin and V. S. Letokhov, “Informational cooling of neutral atoms,” Phys. Rev. A 64, 063410 (2001).

2000 (9)

T. Binnewies, U. Sterr, J. Helmcke, and F. Riehle, “Cooling by Maxwell’s demon: preparation of single-velocity atoms for matter-wave interferometry,” Phys. Rev. A 62, 011601 (2000).

Ch. Raab, J. Eschner, J. Bolle, H. Oberst, F. Schmidt-Kaler, and R. Blatt, “Motional sidebands and direct measurement of the cooling rate in the resonance fluorescence of a single trapped ion,” Phys. Rev. Lett. 85, 538–541 (2000).

R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Sub-dekahertz ultraviolet spectroscopy of 199Hg+,” Phys. Rev. Lett. 85, 2462–2465 (2000).

C. W. Hoyt, M. Sheik-Bahae, R. I. Epstein, B. C. Edwards, and J. E. Anderson, “Observation of anti-Stokes fluorescence cooling in thulium-doped glass,” Phys. Rev. Lett. 85, 3600–3603 (2000).

D. Kielpinski, B. E. King, C. J. Myatt, C. A. Sackett, Q. A. Turchette, W. M. Itano, C. Monroe, and D. J. Wineland, “Sympathetic cooling of trapped ions for quantum logic,” Phys. Rev. A 61, 032310 (2000).

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

G. Morigi, J. Eschner, and C. H. Keitel, “Ground state laser cooling using electromagnetically induced transparency,” Phys. Rev. Lett. 85, 4458–4461 (2000).

C. F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Experimental demonstration of ground state laser cooling with electromagnetically induced transparency,” Phys. Rev. Lett. 85, 5547–5550 (2000).

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).

1999 (6)

Ch. Roos, Th. Zeiger, H. Rohde, H. C. Nägerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, “Quantum state engineering on an optical transition and decoherence in a Paul trap,” Phys. Rev. Lett. 83, 4713–4716 (1999).

L. Santos and M. Lewenstein, “Dynamical cooling of trapped gases: one-atom problem,” Phys. Rev. A 59, 613–619 (1999).

E. Peik, J. Abel, Th. Becker, J. von Zanthier, and H. Walther, “Sideband cooling of ions in radio-frequency traps,” Phys. Rev. A 60, 439–449 (1999).

G. Morigi, J. Eschner, J. I. Cirac, and P. Zoller, “Laser cooling of two trapped ions: sideband cooling beyond the Lamb–Dicke limit,” Phys. Rev. A 59, 3797–3808 (1999).

A discussion of crystallization in these mesoscopic structures is beyond the scope of this work. The interested reader can find a review in D. H. Dubin and T. M. O’Neil, “Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium state),” Rev. Mod. Phys. 71, 87–172 (1999).

P. Bowe, L. Hornekær, C. Brodersen, M. Drewsen, J. S. Hangst, and J. P. Schiffer, “Sympathetic crystallization of trapped ions,” Phys. Rev. Lett. 82, 2071–2074 (1999).

1998 (8)

H. C. Nägerl, D. Leibfried, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Coherent excitation of normal modes in a string of Ca+ ions,” Opt. Express 3, 89–96 (1998).

W. Alt, M. Block, P. Seibert, and G. Werth, “Spatial separation of atomic states in a laser-cooled ion crystal,” Phys. Rev. A 58, R23–R25 (1998).

D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, “Experimental issues in coherent quantum-state manipulation of trapped atomic ions,” J. Res. Natl. Inst. Stand. Technol. 103, 259–328 (1998).

D. V. F. James, “Quantum dynamics of cold trapped ions, with application to quantum computation,” Appl. Phys. B 66, 181–190 (1998).

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631–3634 (1998).

B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Cooling the collective motion of trapped ions to initialize a quantum register,” Phys. Rev. Lett. 81, 1525–1528 (1998).

S. Schlipf, H. Katori, L. Perotti, and H. Walther, “Diffusion of a single ion in a one-dimensional optical lattice,” Opt. Express 3, 97–103 (1998).

B. Appasamy, Y. Stalgies, and P. E. Toschek, “Measurement-induced vibrational dynamics of a trapped ion,” Phys. Rev. Lett. 80, 2805–2808 (1998).

1997 (4)

H. Katori, S. Schlipf, and H. Walther, “Anomalous dynamics of a single ion in an optical lattice,” Phys. Rev. Lett. 79, 2221–2224 (1997).

A. Steane, “The ion trap quantum information processor,” Appl. Phys. B 64, 623–642 (1997).

G. Morigi, J. I. Cirac, M. Lewenstein, and P. Zoller, “Ground-state laser cooling beyond the Lamb–Dicke limit,” Europhys. Lett. 39, 13–18 (1997).

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997), and references therein.

1996 (6)

D. Reiß, A. Lindner, and R. Blatt, “Cooling of trapped multilevel ions: a numerical analysis,” Phys. Rev. A 54, 5133–5140 (1996).

D. S. Hall and G. Gabrielse, “Electron cooling of protons in a nested Penning trap,” Phys. Rev. Lett. 77, 1962–1965 (1996).

D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, “Generation of nonclassical motional states of a trapped atom,” Phys. Rev. Lett. 76, 1796–1799 (1996).

M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, “Quantum Rabi oscillation: a direct test of field quantization in a cavity,” Phys. Rev. Lett. 76, 1800–1803 (1996).

J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Nonclassical states of motion in ion traps,” Adv. Atom., Mol., Opt. Phys. 37, 237–296 (1996).

J. L. Clark and G. Rumbles, “Laser cooling in the condensed phase by frequency up-conversion,” Phys. Rev. Lett. 76, 2037–2040 (1996).

1995 (7)

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).

J. Eschner, B. Appasamy, and P. E. Toschek, “Stochastic cooling of a trapped ion by null detection of its fluorescence,” Phys. Rev. Lett. 74, 2435–2438 (1995).

J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

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).

In experiments the distance among the ions is usually of several wavelengths, and the only relevant interaction is the Coulomb repulsion. For distances of the order of the wavelength, the dipole–dipole interaction must be taken into account. Its effect in connection with laser cooling has been investigated in A. W. Vogt, J. I. Cirac, and P. Zoller, “Collective laser cooling of two trapped ions,” Phys. Rev. A 53, 950–968 (1995).

1994 (3)

I. Marzoli, J. I. Cirac, R. Blatt, and P. Zoller, “Laser cooling of trapped three-level ions: designing two-level systems for sideband cooling,” Phys. Rev. A 49, 2771–2779 (1994).

G. Birkl, J. A. Yeazell, R. Rückerl, and H. Walther, “Polarization gradient cooling of trapped ions,” Europhys. Lett. 27, 197–202 (1994).

R. Dum, P. Marte, T. Pellizzari, and P. Zoller, “Laser cooling to a single quantum state in a trap,” Phys. Rev. Lett. 73, 2829–2832 (1994).

1993 (4)

S. M. Yoo and J. Javanainen, “Polarization gradient cooling of a trapped ion,” Phys. Rev. A 48, R30–R33 (1993).

J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Laser cooling of trapped ions with polarization gradients,” Phys. Rev. A 48, 1434–1445 (1993).

J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Dark squeezed states of the motion of a trapped ion,” Phys. Rev. Lett. 70, 556–559 (1993).

J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Preparation of Fock states by observation of quantum jumps in an ion trap,” Phys. Rev. Lett. 70, 762–765 (1993).

1992 (7)

C. A. Blockley, D. F. Walls, and H. Risken, “Quantum collapses and revivals in a quantized trap,” Europhys. Lett. 17, 509–514 (1992).

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Ionic crystals in a linear Paul trap,” Phys. Rev. A 45, 6493–6501 (1992).

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Linear trap for high accuracy spectroscopy of stored ions,” J. Mod. Opt. 39, 233–242 (1992).

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46, 2668–2681 (1992).

D. J. Wineland, J. Dalibard, and C. Cohen-Tannoudji, “Sisyphus cooling of a bound atom,” J. Opt. Soc. Am. B 9, 32–42 (1992).

I. Waki, S. Kassner, G. Birkl, and H. Walther, “Observation of ordered structures of laser-cooled ions in a quadrupole storage ring,” Phys. Rev. Lett. 68, 2007–2010 (1992).

G. Birkl, S. Kassner, and H. Walther, “Multiple-shell struc-tures of laser-cooled 24Mg+ ions in a quadrupole storage ring,” Nature 357, 310–313 (1992).

1990 (2)

D. J. Heinzen and D. J. Wineland, “Quantum-limited cooling and detection of radio-frequency oscillations by laser-cooled ions,” Phys. Rev. A 42, 2977–2994 (1990).

G. R. Janik and L. Maleki, “Simple analytic potentials for linear ion traps,” J. Appl. Phys. 67, 6050–6055 (1990).

1989 (3)

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62, 403–406 (1989).

J. Prestage, G. J. Dick, and L. Maleki, “New ion trap for frequency standard applications,” J. Appl. Phys. 66, 1013–1017 (1989).

J. Dalibard and C. Cohen-Tannoudji, “Laser cooling below the Doppler limit by polarization gradients: simple theoretical models,” J. Opt. Soc. Am. B 6, 2023–2045 (1989).

1988 (4)

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).

J. Javanainen, “Laser cooling of trapped-ion clusters,” J. Opt. Soc. Am. B 5, 73–81 (1988).

R. Blümel, J. M. Chen, E. Peik, W. Quint, W. Schleich, Y. R. Shen, and H. Walther, “Phase transition of stored laser-cooled ions,” Nature 334, 309–313 (1988).

S. L. Gilbert, J. J. Bollinger, and D. J. Wineland, “Shell-structure phase of magnetically confined strong coupled plasmas,” Phys. Rev. Lett. 60, 2022–2025 (1988)

1987 (2)

F. Diedrich, E. Peik, J. M. Chen, W. Quint, and H. Walther, “Observation of a phase transition of stored laser-cooled ions,” Phys. Rev. Lett. 59, 2931–2934 (1987).

D. J. Wineland, J. C. Bergquist, W. M. Itano, J. J. Bollinger, and C. H. Manney, “Atomic-ion Coulomb clusters in an ion trap,” Phys. Rev. Lett. 59, 2935–2938 (1987).

1986 (6)

M. Lindberg and J. Javanainen, “Temperature of laser-cooled trapped three-level ion,” J. Opt. Soc. Am. B 3, 1008–1017 (1986).

J. Javanainen, “Light-pressure cooling of a crystal,” Phys. Rev. Lett. 56, 1798–1801 (1986).

W. Nagourney, J. Sandberg, and H. Dehmelt, “Shelved optical electron amplifier: observation of quantum jumps,” Phys. Rev. Lett. 56, 2797–2799 (1986).

Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, “Observation of quantum jumps,” Phys. Rev. Lett. 57, 1696–1698 (1986).

J. C. Bergquist, R. Hulet, W. M. Itano, and D. J. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).

S. Stenholm, “Semiclassical theory of laser cooling,” Rev. Mod. Phys. 58, 699–739 (1986), and references therein.

1985 (1)

1983 (1)

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

1982 (1)

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

1980 (1)

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

1979 (1)

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

1978 (3)

W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, “Optical-sideband cooling of visible atom cloud confined in parabolic well,” Phys. Rev. Lett. 41, 233–236 (1978).

D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1639–1642 (1978).

S. Stenholm, “Redistribution of molecular velocities by optical processes,” Appl. Phys. 15, 287–296 (1978).

1975 (2)

T. W. Hänsch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68–69 (1975).

D. Wineland and H. Dehmelt, “Proposed 1014δν/ν laser fluorescence spectroscopy on Tl+ mono-ion oscillator III (side band cooling),” Bull. Am. Phys. Soc. 20, 637 (1975).

1974 (1)

Ya. B. Zel’dovich, “Cooling with the aid of high-frequency energy,” JETP Lett. 19, 74–75 (1974).

1950 (1)

A. Kastler, “Quelques suggestions concernant la production optique et la détection optique d’une inégalité de population des niveaux de quantification spatiale des atomes,” J. Phys. 11, 255–265 (1950).

Abel, J.

E. Peik, J. Abel, Th. Becker, J. von Zanthier, and H. Walther, “Sideband cooling of ions in radio-frequency traps,” Phys. Rev. A 60, 439–449 (1999).

Alt, W.

W. Alt, M. Block, P. Seibert, and G. Werth, “Spatial separation of atomic states in a laser-cooled ion crystal,” Phys. Rev. A 58, R23–R25 (1998).

Anderson, J. E.

C. W. Hoyt, M. Sheik-Bahae, R. I. Epstein, B. C. Edwards, and J. E. Anderson, “Observation of anti-Stokes fluorescence cooling in thulium-doped glass,” Phys. Rev. Lett. 85, 3600–3603 (2000).

Appasamy, B.

B. Appasamy, Y. Stalgies, and P. E. Toschek, “Measurement-induced vibrational dynamics of a trapped ion,” Phys. Rev. Lett. 80, 2805–2808 (1998).

J. Eschner, B. Appasamy, and P. E. Toschek, “Stochastic cooling of a trapped ion by null detection of its fluorescence,” Phys. Rev. Lett. 74, 2435–2438 (1995).

Arimondo, E.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).

Aspect, A.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).

Baldauf, H.

G. Morigi, H. Baldauf, W. Lange, and H. Walther, “Raman sideband cooling in the presence of multiple decay channels,” Opt. Commun. 187, 171–177 (2001).

Balykin, V. I.

V. I. Balykin and V. S. Letokhov, “Informational cooling of neutral atoms,” Phys. Rev. A 64, 063410 (2001).

Barton, P. A.

H. Rohde, S. T. Gulde, C. F. Roos, P. A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Sympathetic ground state cooling and coherent manipulation with two-ion-crystals,” J. Opt. B 3, S34–S41 (2001).

Beall, J. A.

R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Sub-dekahertz ultraviolet spectroscopy of 199Hg+,” Phys. Rev. Lett. 85, 2462–2465 (2000).

Becker, Th.

E. Peik, J. Abel, Th. Becker, J. von Zanthier, and H. Walther, “Sideband cooling of ions in radio-frequency traps,” Phys. Rev. A 60, 439–449 (1999).

Bergquist, J. C.

R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Sub-dekahertz ultraviolet spectroscopy of 199Hg+,” Phys. Rev. Lett. 85, 2462–2465 (2000).

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Ionic crystals in a linear Paul trap,” Phys. Rev. A 45, 6493–6501 (1992).

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Linear trap for high accuracy spectroscopy of stored ions,” J. Mod. Opt. 39, 233–242 (1992).

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62, 403–406 (1989).

D. J. Wineland, J. C. Bergquist, W. M. Itano, J. J. Bollinger, and C. H. Manney, “Atomic-ion Coulomb clusters in an ion trap,” Phys. Rev. Lett. 59, 2935–2938 (1987).

J. C. Bergquist, R. Hulet, W. M. Itano, and D. J. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).

Binnewies, T.

T. Binnewies, U. Sterr, J. Helmcke, and F. Riehle, “Cooling by Maxwell’s demon: preparation of single-velocity atoms for matter-wave interferometry,” Phys. Rev. A 62, 011601 (2000).

Birkl, G.

G. Birkl, J. A. Yeazell, R. Rückerl, and H. Walther, “Polarization gradient cooling of trapped ions,” Europhys. Lett. 27, 197–202 (1994).

I. Waki, S. Kassner, G. Birkl, and H. Walther, “Observation of ordered structures of laser-cooled ions in a quadrupole storage ring,” Phys. Rev. Lett. 68, 2007–2010 (1992).

G. Birkl, S. Kassner, and H. Walther, “Multiple-shell struc-tures of laser-cooled 24Mg+ ions in a quadrupole storage ring,” Nature 357, 310–313 (1992).

Blatt, R.

F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73, 807–814 (2001).

H. Rohde, S. T. Gulde, C. F. Roos, P. A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Sympathetic ground state cooling and coherent manipulation with two-ion-crystals,” J. Opt. B 3, S34–S41 (2001).

C. F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Experimental demonstration of ground state laser cooling with electromagnetically induced transparency,” Phys. Rev. Lett. 85, 5547–5550 (2000).

Ch. Raab, J. Eschner, J. Bolle, H. Oberst, F. Schmidt-Kaler, and R. Blatt, “Motional sidebands and direct measurement of the cooling rate in the resonance fluorescence of a single trapped ion,” Phys. Rev. Lett. 85, 538–541 (2000).

Ch. Roos, Th. Zeiger, H. Rohde, H. C. Nägerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, “Quantum state engineering on an optical transition and decoherence in a Paul trap,” Phys. Rev. Lett. 83, 4713–4716 (1999).

H. C. Nägerl, D. Leibfried, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Coherent excitation of normal modes in a string of Ca+ ions,” Opt. Express 3, 89–96 (1998).

D. Reiß, A. Lindner, and R. Blatt, “Cooling of trapped multilevel ions: a numerical analysis,” Phys. Rev. A 54, 5133–5140 (1996).

J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Nonclassical states of motion in ion traps,” Adv. Atom., Mol., Opt. Phys. 37, 237–296 (1996).

I. Marzoli, J. I. Cirac, R. Blatt, and P. Zoller, “Laser cooling of trapped three-level ions: designing two-level systems for sideband cooling,” Phys. Rev. A 49, 2771–2779 (1994).

J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Laser cooling of trapped ions with polarization gradients,” Phys. Rev. A 48, 1434–1445 (1993).

J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Dark squeezed states of the motion of a trapped ion,” Phys. Rev. Lett. 70, 556–559 (1993).

J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Preparation of Fock states by observation of quantum jumps in an ion trap,” Phys. Rev. Lett. 70, 762–765 (1993).

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46, 2668–2681 (1992).

Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, “Observation of quantum jumps,” Phys. Rev. Lett. 57, 1696–1698 (1986).

Block, M.

W. Alt, M. Block, P. Seibert, and G. Werth, “Spatial separation of atomic states in a laser-cooled ion crystal,” Phys. Rev. A 58, R23–R25 (1998).

Blockley, C. A.

C. A. Blockley, D. F. Walls, and H. Risken, “Quantum collapses and revivals in a quantized trap,” Europhys. Lett. 17, 509–514 (1992).

Blümel, R.

R. Blümel, J. M. Chen, E. Peik, W. Quint, W. Schleich, Y. R. Shen, and H. Walther, “Phase transition of stored laser-cooled ions,” Nature 334, 309–313 (1988).

Bolle, J.

Ch. Raab, J. Eschner, J. Bolle, H. Oberst, F. Schmidt-Kaler, and R. Blatt, “Motional sidebands and direct measurement of the cooling rate in the resonance fluorescence of a single trapped ion,” Phys. Rev. Lett. 85, 538–541 (2000).

Bollinger, J. J.

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).

S. L. Gilbert, J. J. Bollinger, and D. J. Wineland, “Shell-structure phase of magnetically confined strong coupled plasmas,” Phys. Rev. Lett. 60, 2022–2025 (1988)

D. J. Wineland, J. C. Bergquist, W. M. Itano, J. J. Bollinger, and C. H. Manney, “Atomic-ion Coulomb clusters in an ion trap,” Phys. Rev. Lett. 59, 2935–2938 (1987).

Bowe, P.

P. Bowe, L. Hornekær, C. Brodersen, M. Drewsen, J. S. Hangst, and J. P. Schiffer, “Sympathetic crystallization of trapped ions,” Phys. Rev. Lett. 82, 2071–2074 (1999).

Brodersen, C.

P. Bowe, L. Hornekær, C. Brodersen, M. Drewsen, J. S. Hangst, and J. P. Schiffer, “Sympathetic crystallization of trapped ions,” Phys. Rev. Lett. 82, 2071–2074 (1999).

Brune, M.

M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, “Quantum Rabi oscillation: a direct test of field quantization in a cavity,” Phys. Rev. Lett. 76, 1800–1803 (1996).

Buchwald, M. I.

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).

Chen, J. M.

R. Blümel, J. M. Chen, E. Peik, W. Quint, W. Schleich, Y. R. Shen, and H. Walther, “Phase transition of stored laser-cooled ions,” Nature 334, 309–313 (1988).

F. Diedrich, E. Peik, J. M. Chen, W. Quint, and H. Walther, “Observation of a phase transition of stored laser-cooled ions,” Phys. Rev. Lett. 59, 2931–2934 (1987).

Cirac, J. I.

G. Morigi, J. Eschner, J. I. Cirac, and P. Zoller, “Laser cooling of two trapped ions: sideband cooling beyond the Lamb–Dicke limit,” Phys. Rev. A 59, 3797–3808 (1999).

G. Morigi, J. I. Cirac, M. Lewenstein, and P. Zoller, “Ground-state laser cooling beyond the Lamb–Dicke limit,” Europhys. Lett. 39, 13–18 (1997).

J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Nonclassical states of motion in ion traps,” Adv. Atom., Mol., Opt. Phys. 37, 237–296 (1996).

J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995).

In experiments the distance among the ions is usually of several wavelengths, and the only relevant interaction is the Coulomb repulsion. For distances of the order of the wavelength, the dipole–dipole interaction must be taken into account. Its effect in connection with laser cooling has been investigated in A. W. Vogt, J. I. Cirac, and P. Zoller, “Collective laser cooling of two trapped ions,” Phys. Rev. A 53, 950–968 (1995).

I. Marzoli, J. I. Cirac, R. Blatt, and P. Zoller, “Laser cooling of trapped three-level ions: designing two-level systems for sideband cooling,” Phys. Rev. A 49, 2771–2779 (1994).

J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Laser cooling of trapped ions with polarization gradients,” Phys. Rev. A 48, 1434–1445 (1993).

J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Preparation of Fock states by observation of quantum jumps in an ion trap,” Phys. Rev. Lett. 70, 762–765 (1993).

J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Dark squeezed states of the motion of a trapped ion,” Phys. Rev. Lett. 70, 556–559 (1993).

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46, 2668–2681 (1992).

Clark, J. L.

J. L. Clark and G. Rumbles, “Laser cooling in the condensed phase by frequency up-conversion,” Phys. Rev. Lett. 76, 2037–2040 (1996).

Cohen-Tannoudji, C.

Dalibard, J.

Dehmelt, H.

W. Nagourney, J. Sandberg, and H. Dehmelt, “Shelved optical electron amplifier: observation of quantum jumps,” Phys. Rev. Lett. 56, 2797–2799 (1986).

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

W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, “Optical-sideband cooling of visible atom cloud confined in parabolic well,” Phys. Rev. Lett. 41, 233–236 (1978).

D. Wineland and H. Dehmelt, “Proposed 1014δν/ν laser fluorescence spectroscopy on Tl+ mono-ion oscillator III (side band cooling),” Bull. Am. Phys. Soc. 20, 637 (1975).

Dick, G. J.

J. Prestage, G. J. Dick, and L. Maleki, “New ion trap for frequency standard applications,” J. Appl. Phys. 66, 1013–1017 (1989).

Diedrich, F.

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62, 403–406 (1989).

F. Diedrich, E. Peik, J. M. Chen, W. Quint, and H. Walther, “Observation of a phase transition of stored laser-cooled ions,” Phys. Rev. Lett. 59, 2931–2934 (1987).

Drewsen, M.

P. Bowe, L. Hornekær, C. Brodersen, M. Drewsen, J. S. Hangst, and J. P. Schiffer, “Sympathetic crystallization of trapped ions,” Phys. Rev. Lett. 82, 2071–2074 (1999).

Dreyer, J.

M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, “Quantum Rabi oscillation: a direct test of field quantization in a cavity,” Phys. Rev. Lett. 76, 1800–1803 (1996).

Drullinger, R. E.

D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1639–1642 (1978).

Dubin, D. H.

A discussion of crystallization in these mesoscopic structures is beyond the scope of this work. The interested reader can find a review in D. H. Dubin and T. M. O’Neil, “Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium state),” Rev. Mod. Phys. 71, 87–172 (1999).

Dum, R.

R. Dum, P. Marte, T. Pellizzari, and P. Zoller, “Laser cooling to a single quantum state in a trap,” Phys. Rev. Lett. 73, 2829–2832 (1994).

Edwards, B. C.

C. W. Hoyt, M. Sheik-Bahae, R. I. Epstein, B. C. Edwards, and J. E. Anderson, “Observation of anti-Stokes fluorescence cooling in thulium-doped glass,” Phys. Rev. Lett. 85, 3600–3603 (2000).

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).

Epstein, R. I.

C. W. Hoyt, M. Sheik-Bahae, R. I. Epstein, B. C. Edwards, and J. E. Anderson, “Observation of anti-Stokes fluorescence cooling in thulium-doped glass,” Phys. Rev. Lett. 85, 3600–3603 (2000).

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).

Eschner, J.

F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73, 807–814 (2001).

G. Morigi and J. Eschner, “Doppler cooling of a Coulomb crystal,” Phys. Rev. A 64, 063407 (2001).

H. Rohde, S. T. Gulde, C. F. Roos, P. A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Sympathetic ground state cooling and coherent manipulation with two-ion-crystals,” J. Opt. B 3, S34–S41 (2001).

G. Morigi, J. Eschner, and C. H. Keitel, “Ground state laser cooling using electromagnetically induced transparency,” Phys. Rev. Lett. 85, 4458–4461 (2000).

C. F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Experimental demonstration of ground state laser cooling with electromagnetically induced transparency,” Phys. Rev. Lett. 85, 5547–5550 (2000).

Ch. Raab, J. Eschner, J. Bolle, H. Oberst, F. Schmidt-Kaler, and R. Blatt, “Motional sidebands and direct measurement of the cooling rate in the resonance fluorescence of a single trapped ion,” Phys. Rev. Lett. 85, 538–541 (2000).

Ch. Roos, Th. Zeiger, H. Rohde, H. C. Nägerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, “Quantum state engineering on an optical transition and decoherence in a Paul trap,” Phys. Rev. Lett. 83, 4713–4716 (1999).

G. Morigi, J. Eschner, J. I. Cirac, and P. Zoller, “Laser cooling of two trapped ions: sideband cooling beyond the Lamb–Dicke limit,” Phys. Rev. A 59, 3797–3808 (1999).

H. C. Nägerl, D. Leibfried, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Coherent excitation of normal modes in a string of Ca+ ions,” Opt. Express 3, 89–96 (1998).

J. Eschner, B. Appasamy, and P. E. Toschek, “Stochastic cooling of a trapped ion by null detection of its fluorescence,” Phys. Rev. Lett. 74, 2435–2438 (1995).

Gabrielse, G.

D. S. Hall and G. Gabrielse, “Electron cooling of protons in a nested Penning trap,” Phys. Rev. Lett. 77, 1962–1965 (1996).

Gilbert, S. L.

S. L. Gilbert, J. J. Bollinger, and D. J. Wineland, “Shell-structure phase of magnetically confined strong coupled plasmas,” Phys. Rev. Lett. 60, 2022–2025 (1988)

Gilligan, J. M.

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Linear trap for high accuracy spectroscopy of stored ions,” J. Mod. Opt. 39, 233–242 (1992).

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Ionic crystals in a linear Paul trap,” Phys. Rev. A 45, 6493–6501 (1992).

Gosnell, T. R.

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).

Gulde, S. T.

H. Rohde, S. T. Gulde, C. F. Roos, P. A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Sympathetic ground state cooling and coherent manipulation with two-ion-crystals,” J. Opt. B 3, S34–S41 (2001).

Hagley, E.

M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, “Quantum Rabi oscillation: a direct test of field quantization in a cavity,” Phys. Rev. Lett. 76, 1800–1803 (1996).

Hall, D. S.

D. S. Hall and G. Gabrielse, “Electron cooling of protons in a nested Penning trap,” Phys. Rev. Lett. 77, 1962–1965 (1996).

Hangst, J. S.

P. Bowe, L. Hornekær, C. Brodersen, M. Drewsen, J. S. Hangst, and J. P. Schiffer, “Sympathetic crystallization of trapped ions,” Phys. Rev. Lett. 82, 2071–2074 (1999).

Hänsch, T. W.

T. W. Hänsch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68–69 (1975).

Haroche, S.

M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, “Quantum Rabi oscillation: a direct test of field quantization in a cavity,” Phys. Rev. Lett. 76, 1800–1803 (1996).

Harris, S. E.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997), and references therein.

Heinzen, D. J.

D. J. Heinzen and D. J. Wineland, “Quantum-limited cooling and detection of radio-frequency oscillations by laser-cooled ions,” Phys. Rev. A 42, 2977–2994 (1990).

Helmcke, J.

T. Binnewies, U. Sterr, J. Helmcke, and F. Riehle, “Cooling by Maxwell’s demon: preparation of single-velocity atoms for matter-wave interferometry,” Phys. Rev. A 62, 011601 (2000).

Hohenstatt, M.

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

W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, “Optical-sideband cooling of visible atom cloud confined in parabolic well,” Phys. Rev. Lett. 41, 233–236 (1978).

Hornekær, L.

P. Bowe, L. Hornekær, C. Brodersen, M. Drewsen, J. S. Hangst, and J. P. Schiffer, “Sympathetic crystallization of trapped ions,” Phys. Rev. Lett. 82, 2071–2074 (1999).

Hoyt, C. W.

C. W. Hoyt, M. Sheik-Bahae, R. I. Epstein, B. C. Edwards, and J. E. Anderson, “Observation of anti-Stokes fluorescence cooling in thulium-doped glass,” Phys. Rev. Lett. 85, 3600–3603 (2000).

Hulet, R.

J. C. Bergquist, R. Hulet, W. M. Itano, and D. J. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).

Itano, W. M.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).

D. Kielpinski, B. E. King, C. J. Myatt, C. A. Sackett, Q. A. Turchette, W. M. Itano, C. Monroe, and D. J. Wineland, “Sympathetic cooling of trapped ions for quantum logic,” Phys. Rev. A 61, 032310 (2000).

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Sub-dekahertz ultraviolet spectroscopy of 199Hg+,” Phys. Rev. Lett. 85, 2462–2465 (2000).

D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, “Experimental issues in coherent quantum-state manipulation of trapped atomic ions,” J. Res. Natl. Inst. Stand. Technol. 103, 259–328 (1998).

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631–3634 (1998).

B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Cooling the collective motion of trapped ions to initialize a quantum register,” Phys. Rev. Lett. 81, 1525–1528 (1998).

D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, “Generation of nonclassical motional states of a trapped atom,” Phys. Rev. Lett. 76, 1796–1799 (1996).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Linear trap for high accuracy spectroscopy of stored ions,” J. Mod. Opt. 39, 233–242 (1992).

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Ionic crystals in a linear Paul trap,” Phys. Rev. A 45, 6493–6501 (1992).

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62, 403–406 (1989).

D. J. Wineland, J. C. Bergquist, W. M. Itano, J. J. Bollinger, and C. H. Manney, “Atomic-ion Coulomb clusters in an ion trap,” Phys. Rev. Lett. 59, 2935–2938 (1987).

J. C. Bergquist, R. Hulet, W. M. Itano, and D. J. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).

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

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

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

James, D. V. F.

D. V. F. James, “Quantum dynamics of cold trapped ions, with application to quantum computation,” Appl. Phys. B 66, 181–190 (1998).

Janik, G. R.

G. R. Janik and L. Maleki, “Simple analytic potentials for linear ion traps,” J. Appl. Phys. 67, 6050–6055 (1990).

Javanainen, J.

S. M. Yoo and J. Javanainen, “Polarization gradient cooling of a trapped ion,” Phys. Rev. A 48, R30–R33 (1993).

J. Javanainen, “Laser cooling of trapped-ion clusters,” J. Opt. Soc. Am. B 5, 73–81 (1988).

J. Javanainen, “Light-pressure cooling of a crystal,” Phys. Rev. Lett. 56, 1798–1801 (1986).

M. Lindberg and J. Javanainen, “Temperature of laser-cooled trapped three-level ion,” J. Opt. Soc. Am. B 3, 1008–1017 (1986).

Jefferts, S. R.

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

Jelenkovic, B.

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).

Kaiser, R.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).

Kassner, S.

I. Waki, S. Kassner, G. Birkl, and H. Walther, “Observation of ordered structures of laser-cooled ions in a quadrupole storage ring,” Phys. Rev. Lett. 68, 2007–2010 (1992).

G. Birkl, S. Kassner, and H. Walther, “Multiple-shell struc-tures of laser-cooled 24Mg+ ions in a quadrupole storage ring,” Nature 357, 310–313 (1992).

Kastler, A.

A. Kastler, “Quelques suggestions concernant la production optique et la détection optique d’une inégalité de population des niveaux de quantification spatiale des atomes,” J. Phys. 11, 255–265 (1950).

Katori, H.

S. Schlipf, H. Katori, L. Perotti, and H. Walther, “Diffusion of a single ion in a one-dimensional optical lattice,” Opt. Express 3, 97–103 (1998).

H. Katori, S. Schlipf, and H. Walther, “Anomalous dynamics of a single ion in an optical lattice,” Phys. Rev. Lett. 79, 2221–2224 (1997).

Keitel, C. H.

G. Morigi, J. Eschner, and C. H. Keitel, “Ground state laser cooling using electromagnetically induced transparency,” Phys. Rev. Lett. 85, 4458–4461 (2000).

Kielpinski, D.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).

D. Kielpinski, B. E. King, C. J. Myatt, C. A. Sackett, Q. A. Turchette, W. M. Itano, C. Monroe, and D. J. Wineland, “Sympathetic cooling of trapped ions for quantum logic,” Phys. Rev. A 61, 032310 (2000).

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

Kienle, P.

P. Kienle, “Sunshine by cooling,” Naturwissenschaften 88, 313–321 (2001), and references therein.

King, B. E.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

D. Kielpinski, B. E. King, C. J. Myatt, C. A. Sackett, Q. A. Turchette, W. M. Itano, C. Monroe, and D. J. Wineland, “Sympathetic cooling of trapped ions for quantum logic,” Phys. Rev. A 61, 032310 (2000).

D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, “Experimental issues in coherent quantum-state manipulation of trapped atomic ions,” J. Res. Natl. Inst. Stand. Technol. 103, 259–328 (1998).

B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Cooling the collective motion of trapped ions to initialize a quantum register,” Phys. Rev. Lett. 81, 1525–1528 (1998).

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631–3634 (1998).

D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, “Generation of nonclassical motional states of a trapped atom,” Phys. Rev. Lett. 76, 1796–1799 (1996).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

Lange, W.

G. Morigi, H. Baldauf, W. Lange, and H. Walther, “Raman sideband cooling in the presence of multiple decay channels,” Opt. Commun. 187, 171–177 (2001).

Langer, C.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).

Leibfried, D.

H. Rohde, S. T. Gulde, C. F. Roos, P. A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Sympathetic ground state cooling and coherent manipulation with two-ion-crystals,” J. Opt. B 3, S34–S41 (2001).

F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73, 807–814 (2001).

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

C. F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Experimental demonstration of ground state laser cooling with electromagnetically induced transparency,” Phys. Rev. Lett. 85, 5547–5550 (2000).

Ch. Roos, Th. Zeiger, H. Rohde, H. C. Nägerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, “Quantum state engineering on an optical transition and decoherence in a Paul trap,” Phys. Rev. Lett. 83, 4713–4716 (1999).

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631–3634 (1998).

B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Cooling the collective motion of trapped ions to initialize a quantum register,” Phys. Rev. Lett. 81, 1525–1528 (1998).

D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, “Experimental issues in coherent quantum-state manipulation of trapped atomic ions,” J. Res. Natl. Inst. Stand. Technol. 103, 259–328 (1998).

H. C. Nägerl, D. Leibfried, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Coherent excitation of normal modes in a string of Ca+ ions,” Opt. Express 3, 89–96 (1998).

Letokhov, V. S.

V. I. Balykin and V. S. Letokhov, “Informational cooling of neutral atoms,” Phys. Rev. A 64, 063410 (2001).

Lewenstein, M.

L. Santos and M. Lewenstein, “Dynamical cooling of trapped gases: one-atom problem,” Phys. Rev. A 59, 613–619 (1999).

G. Morigi, J. I. Cirac, M. Lewenstein, and P. Zoller, “Ground-state laser cooling beyond the Lamb–Dicke limit,” Europhys. Lett. 39, 13–18 (1997).

Lindberg, M.

Lindner, A.

D. Reiß, A. Lindner, and R. Blatt, “Cooling of trapped multilevel ions: a numerical analysis,” Phys. Rev. A 54, 5133–5140 (1996).

Maali, A.

M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, “Quantum Rabi oscillation: a direct test of field quantization in a cavity,” Phys. Rev. Lett. 76, 1800–1803 (1996).

Maleki, L.

G. R. Janik and L. Maleki, “Simple analytic potentials for linear ion traps,” J. Appl. Phys. 67, 6050–6055 (1990).

J. Prestage, G. J. Dick, and L. Maleki, “New ion trap for frequency standard applications,” J. Appl. Phys. 66, 1013–1017 (1989).

Manney, C. H.

D. J. Wineland, J. C. Bergquist, W. M. Itano, J. J. Bollinger, and C. H. Manney, “Atomic-ion Coulomb clusters in an ion trap,” Phys. Rev. Lett. 59, 2935–2938 (1987).

Marte, P.

R. Dum, P. Marte, T. Pellizzari, and P. Zoller, “Laser cooling to a single quantum state in a trap,” Phys. Rev. Lett. 73, 2829–2832 (1994).

Marzoli, I.

I. Marzoli, J. I. Cirac, R. Blatt, and P. Zoller, “Laser cooling of trapped three-level ions: designing two-level systems for sideband cooling,” Phys. Rev. A 49, 2771–2779 (1994).

Meekhof, D. M.

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, “Experimental issues in coherent quantum-state manipulation of trapped atomic ions,” J. Res. Natl. Inst. Stand. Technol. 103, 259–328 (1998).

D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, “Generation of nonclassical motional states of a trapped atom,” Phys. Rev. Lett. 76, 1796–1799 (1996).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

Meyer, V.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).

Monroe, C.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).

D. Kielpinski, B. E. King, C. J. Myatt, C. A. Sackett, Q. A. Turchette, W. M. Itano, C. Monroe, and D. J. Wineland, “Sympathetic cooling of trapped ions for quantum logic,” Phys. Rev. A 61, 032310 (2000).

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, “Experimental issues in coherent quantum-state manipulation of trapped atomic ions,” J. Res. Natl. Inst. Stand. Technol. 103, 259–328 (1998).

B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Cooling the collective motion of trapped ions to initialize a quantum register,” Phys. Rev. Lett. 81, 1525–1528 (1998).

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631–3634 (1998).

D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, “Generation of nonclassical motional states of a trapped atom,” Phys. Rev. Lett. 76, 1796–1799 (1996).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

Morigi, G.

G. Morigi, H. Baldauf, W. Lange, and H. Walther, “Raman sideband cooling in the presence of multiple decay channels,” Opt. Commun. 187, 171–177 (2001).

F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73, 807–814 (2001).

G. Morigi and H. Walther, “Two-species Coulomb chains for quantum information,” Eur. Phys. J. D 13, 261–269 (2001).

G. Morigi and J. Eschner, “Doppler cooling of a Coulomb crystal,” Phys. Rev. A 64, 063407 (2001).

G. Morigi, J. Eschner, and C. H. Keitel, “Ground state laser cooling using electromagnetically induced transparency,” Phys. Rev. Lett. 85, 4458–4461 (2000).

G. Morigi, J. Eschner, J. I. Cirac, and P. Zoller, “Laser cooling of two trapped ions: sideband cooling beyond the Lamb–Dicke limit,” Phys. Rev. A 59, 3797–3808 (1999).

G. Morigi, J. I. Cirac, M. Lewenstein, and P. Zoller, “Ground-state laser cooling beyond the Lamb–Dicke limit,” Europhys. Lett. 39, 13–18 (1997).

Mundt, A.

F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73, 807–814 (2001).

C. F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Experimental demonstration of ground state laser cooling with electromagnetically induced transparency,” Phys. Rev. Lett. 85, 5547–5550 (2000).

Mungan, C. E.

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).

Myatt, C. J.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).

D. Kielpinski, B. E. King, C. J. Myatt, C. A. Sackett, Q. A. Turchette, W. M. Itano, C. Monroe, and D. J. Wineland, “Sympathetic cooling of trapped ions for quantum logic,” Phys. Rev. A 61, 032310 (2000).

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631–3634 (1998).

B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Cooling the collective motion of trapped ions to initialize a quantum register,” Phys. Rev. Lett. 81, 1525–1528 (1998).

Nägerl, H. C.

Ch. Roos, Th. Zeiger, H. Rohde, H. C. Nägerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, “Quantum state engineering on an optical transition and decoherence in a Paul trap,” Phys. Rev. Lett. 83, 4713–4716 (1999).

H. C. Nägerl, D. Leibfried, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Coherent excitation of normal modes in a string of Ca+ ions,” Opt. Express 3, 89–96 (1998).

Nagourney, W.

W. Nagourney, J. Sandberg, and H. Dehmelt, “Shelved optical electron amplifier: observation of quantum jumps,” Phys. Rev. Lett. 56, 2797–2799 (1986).

Neuhauser, W.

Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, “Observation of quantum jumps,” Phys. Rev. Lett. 57, 1696–1698 (1986).

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

W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, “Optical-sideband cooling of visible atom cloud confined in parabolic well,” Phys. Rev. Lett. 41, 233–236 (1978).

O’Neil, T. M.

A discussion of crystallization in these mesoscopic structures is beyond the scope of this work. The interested reader can find a review in D. H. Dubin and T. M. O’Neil, “Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium state),” Rev. Mod. Phys. 71, 87–172 (1999).

Oberst, H.

Ch. Raab, J. Eschner, J. Bolle, H. Oberst, F. Schmidt-Kaler, and R. Blatt, “Motional sidebands and direct measurement of the cooling rate in the resonance fluorescence of a single trapped ion,” Phys. Rev. Lett. 85, 538–541 (2000).

Parkins, A. S.

J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Nonclassical states of motion in ion traps,” Adv. Atom., Mol., Opt. Phys. 37, 237–296 (1996).

J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Dark squeezed states of the motion of a trapped ion,” Phys. Rev. Lett. 70, 556–559 (1993).

J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Preparation of Fock states by observation of quantum jumps in an ion trap,” Phys. Rev. Lett. 70, 762–765 (1993).

J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Laser cooling of trapped ions with polarization gradients,” Phys. Rev. A 48, 1434–1445 (1993).

Peik, E.

E. Peik, J. Abel, Th. Becker, J. von Zanthier, and H. Walther, “Sideband cooling of ions in radio-frequency traps,” Phys. Rev. A 60, 439–449 (1999).

R. Blümel, J. M. Chen, E. Peik, W. Quint, W. Schleich, Y. R. Shen, and H. Walther, “Phase transition of stored laser-cooled ions,” Nature 334, 309–313 (1988).

F. Diedrich, E. Peik, J. M. Chen, W. Quint, and H. Walther, “Observation of a phase transition of stored laser-cooled ions,” Phys. Rev. Lett. 59, 2931–2934 (1987).

Pellizzari, T.

R. Dum, P. Marte, T. Pellizzari, and P. Zoller, “Laser cooling to a single quantum state in a trap,” Phys. Rev. Lett. 73, 2829–2832 (1994).

Perotti, L.

Phillips, W. D.

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46, 2668–2681 (1992).

Prestage, J.

J. Prestage, G. J. Dick, and L. Maleki, “New ion trap for frequency standard applications,” J. Appl. Phys. 66, 1013–1017 (1989).

Quint, W.

R. Blümel, J. M. Chen, E. Peik, W. Quint, W. Schleich, Y. R. Shen, and H. Walther, “Phase transition of stored laser-cooled ions,” Nature 334, 309–313 (1988).

F. Diedrich, E. Peik, J. M. Chen, W. Quint, and H. Walther, “Observation of a phase transition of stored laser-cooled ions,” Phys. Rev. Lett. 59, 2931–2934 (1987).

Raab, Ch.

Ch. Raab, J. Eschner, J. Bolle, H. Oberst, F. Schmidt-Kaler, and R. Blatt, “Motional sidebands and direct measurement of the cooling rate in the resonance fluorescence of a single trapped ion,” Phys. Rev. Lett. 85, 538–541 (2000).

Rafac, R. J.

R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Sub-dekahertz ultraviolet spectroscopy of 199Hg+,” Phys. Rev. Lett. 85, 2462–2465 (2000).

Raimond, J. M.

M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, “Quantum Rabi oscillation: a direct test of field quantization in a cavity,” Phys. Rev. Lett. 76, 1800–1803 (1996).

Raizen, M. G.

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Ionic crystals in a linear Paul trap,” Phys. Rev. A 45, 6493–6501 (1992).

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Linear trap for high accuracy spectroscopy of stored ions,” J. Mod. Opt. 39, 233–242 (1992).

Reiß, D.

D. Reiß, A. Lindner, and R. Blatt, “Cooling of trapped multilevel ions: a numerical analysis,” Phys. Rev. A 54, 5133–5140 (1996).

Riehle, F.

T. Binnewies, U. Sterr, J. Helmcke, and F. Riehle, “Cooling by Maxwell’s demon: preparation of single-velocity atoms for matter-wave interferometry,” Phys. Rev. A 62, 011601 (2000).

Risken, H.

C. A. Blockley, D. F. Walls, and H. Risken, “Quantum collapses and revivals in a quantized trap,” Europhys. Lett. 17, 509–514 (1992).

Rohde, H.

H. Rohde, S. T. Gulde, C. F. Roos, P. A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Sympathetic ground state cooling and coherent manipulation with two-ion-crystals,” J. Opt. B 3, S34–S41 (2001).

Ch. Roos, Th. Zeiger, H. Rohde, H. C. Nägerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, “Quantum state engineering on an optical transition and decoherence in a Paul trap,” Phys. Rev. Lett. 83, 4713–4716 (1999).

Roos, C. F.

H. Rohde, S. T. Gulde, C. F. Roos, P. A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Sympathetic ground state cooling and coherent manipulation with two-ion-crystals,” J. Opt. B 3, S34–S41 (2001).

F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73, 807–814 (2001).

C. F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Experimental demonstration of ground state laser cooling with electromagnetically induced transparency,” Phys. Rev. Lett. 85, 5547–5550 (2000).

Roos, Ch.

Ch. Roos, Th. Zeiger, H. Rohde, H. C. Nägerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, “Quantum state engineering on an optical transition and decoherence in a Paul trap,” Phys. Rev. Lett. 83, 4713–4716 (1999).

Rowe, M.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).

Rowe, M. A.

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

Rückerl, R.

G. Birkl, J. A. Yeazell, R. Rückerl, and H. Walther, “Polarization gradient cooling of trapped ions,” Europhys. Lett. 27, 197–202 (1994).

Rumbles, G.

J. L. Clark and G. Rumbles, “Laser cooling in the condensed phase by frequency up-conversion,” Phys. Rev. Lett. 76, 2037–2040 (1996).

Sackett, C. A.

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

D. Kielpinski, B. E. King, C. J. Myatt, C. A. Sackett, Q. A. Turchette, W. M. Itano, C. Monroe, and D. J. Wineland, “Sympathetic cooling of trapped ions for quantum logic,” Phys. Rev. A 61, 032310 (2000).

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).

Sandberg, J.

W. Nagourney, J. Sandberg, and H. Dehmelt, “Shelved optical electron amplifier: observation of quantum jumps,” Phys. Rev. Lett. 56, 2797–2799 (1986).

Santos, L.

L. Santos and M. Lewenstein, “Dynamical cooling of trapped gases: one-atom problem,” Phys. Rev. A 59, 613–619 (1999).

Sauter, Th.

Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, “Observation of quantum jumps,” Phys. Rev. Lett. 57, 1696–1698 (1986).

Schawlow, A. L.

T. W. Hänsch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68–69 (1975).

Schiffer, J. P.

P. Bowe, L. Hornekær, C. Brodersen, M. Drewsen, J. S. Hangst, and J. P. Schiffer, “Sympathetic crystallization of trapped ions,” Phys. Rev. Lett. 82, 2071–2074 (1999).

Schleich, W.

R. Blümel, J. M. Chen, E. Peik, W. Quint, W. Schleich, Y. R. Shen, and H. Walther, “Phase transition of stored laser-cooled ions,” Nature 334, 309–313 (1988).

Schlipf, S.

S. Schlipf, H. Katori, L. Perotti, and H. Walther, “Diffusion of a single ion in a one-dimensional optical lattice,” Opt. Express 3, 97–103 (1998).

H. Katori, S. Schlipf, and H. Walther, “Anomalous dynamics of a single ion in an optical lattice,” Phys. Rev. Lett. 79, 2221–2224 (1997).

Schmidt-Kaler, F.

F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73, 807–814 (2001).

H. Rohde, S. T. Gulde, C. F. Roos, P. A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Sympathetic ground state cooling and coherent manipulation with two-ion-crystals,” J. Opt. B 3, S34–S41 (2001).

C. F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Experimental demonstration of ground state laser cooling with electromagnetically induced transparency,” Phys. Rev. Lett. 85, 5547–5550 (2000).

Ch. Raab, J. Eschner, J. Bolle, H. Oberst, F. Schmidt-Kaler, and R. Blatt, “Motional sidebands and direct measurement of the cooling rate in the resonance fluorescence of a single trapped ion,” Phys. Rev. Lett. 85, 538–541 (2000).

Ch. Roos, Th. Zeiger, H. Rohde, H. C. Nägerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, “Quantum state engineering on an optical transition and decoherence in a Paul trap,” Phys. Rev. Lett. 83, 4713–4716 (1999).

H. C. Nägerl, D. Leibfried, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Coherent excitation of normal modes in a string of Ca+ ions,” Opt. Express 3, 89–96 (1998).

M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, “Quantum Rabi oscillation: a direct test of field quantization in a cavity,” Phys. Rev. Lett. 76, 1800–1803 (1996).

Seibert, P.

W. Alt, M. Block, P. Seibert, and G. Werth, “Spatial separation of atomic states in a laser-cooled ion crystal,” Phys. Rev. A 58, R23–R25 (1998).

Sheik-Bahae, M.

C. W. Hoyt, M. Sheik-Bahae, R. I. Epstein, B. C. Edwards, and J. E. Anderson, “Observation of anti-Stokes fluorescence cooling in thulium-doped glass,” Phys. Rev. Lett. 85, 3600–3603 (2000).

Shen, Y. R.

R. Blümel, J. M. Chen, E. Peik, W. Quint, W. Schleich, Y. R. Shen, and H. Walther, “Phase transition of stored laser-cooled ions,” Nature 334, 309–313 (1988).

Stalgies, Y.

B. Appasamy, Y. Stalgies, and P. E. Toschek, “Measurement-induced vibrational dynamics of a trapped ion,” Phys. Rev. Lett. 80, 2805–2808 (1998).

Steane, A.

A. Steane, “The ion trap quantum information processor,” Appl. Phys. B 64, 623–642 (1997).

Stenholm, S.

S. Stenholm, “Semiclassical theory of laser cooling,” Rev. Mod. Phys. 58, 699–739 (1986), and references therein.

S. Stenholm, “Redistribution of molecular velocities by optical processes,” Appl. Phys. 15, 287–296 (1978).

Sterr, U.

T. Binnewies, U. Sterr, J. Helmcke, and F. Riehle, “Cooling by Maxwell’s demon: preparation of single-velocity atoms for matter-wave interferometry,” Phys. Rev. A 62, 011601 (2000).

Tan, J. N.

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).

Toschek, P.

W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, “Optical-sideband cooling of visible atom cloud confined in parabolic well,” Phys. Rev. Lett. 41, 233–236 (1978).

Toschek, P. E.

B. Appasamy, Y. Stalgies, and P. E. Toschek, “Measurement-induced vibrational dynamics of a trapped ion,” Phys. Rev. Lett. 80, 2805–2808 (1998).

J. Eschner, B. Appasamy, and P. E. Toschek, “Stochastic cooling of a trapped ion by null detection of its fluorescence,” Phys. Rev. Lett. 74, 2435–2438 (1995).

Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, “Observation of quantum jumps,” Phys. Rev. Lett. 57, 1696–1698 (1986).

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

Turchette, Q. A.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

D. Kielpinski, B. E. King, C. J. Myatt, C. A. Sackett, Q. A. Turchette, W. M. Itano, C. Monroe, and D. J. Wineland, “Sympathetic cooling of trapped ions for quantum logic,” Phys. Rev. A 61, 032310 (2000).

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631–3634 (1998).

B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Cooling the collective motion of trapped ions to initialize a quantum register,” Phys. Rev. Lett. 81, 1525–1528 (1998).

van Dyck, R. S.

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

Vansteenkiste, N.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).

Vogt, A. W.

In experiments the distance among the ions is usually of several wavelengths, and the only relevant interaction is the Coulomb repulsion. For distances of the order of the wavelength, the dipole–dipole interaction must be taken into account. Its effect in connection with laser cooling has been investigated in A. W. Vogt, J. I. Cirac, and P. Zoller, “Collective laser cooling of two trapped ions,” Phys. Rev. A 53, 950–968 (1995).

von Zanthier, J.

E. Peik, J. Abel, Th. Becker, J. von Zanthier, and H. Walther, “Sideband cooling of ions in radio-frequency traps,” Phys. Rev. A 60, 439–449 (1999).

Waki, I.

I. Waki, S. Kassner, G. Birkl, and H. Walther, “Observation of ordered structures of laser-cooled ions in a quadrupole storage ring,” Phys. Rev. Lett. 68, 2007–2010 (1992).

Walls, D. F.

C. A. Blockley, D. F. Walls, and H. Risken, “Quantum collapses and revivals in a quantized trap,” Europhys. Lett. 17, 509–514 (1992).

Walls, F. L.

D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1639–1642 (1978).

Walther, H.

G. Morigi and H. Walther, “Two-species Coulomb chains for quantum information,” Eur. Phys. J. D 13, 261–269 (2001).

G. Morigi, H. Baldauf, W. Lange, and H. Walther, “Raman sideband cooling in the presence of multiple decay channels,” Opt. Commun. 187, 171–177 (2001).

E. Peik, J. Abel, Th. Becker, J. von Zanthier, and H. Walther, “Sideband cooling of ions in radio-frequency traps,” Phys. Rev. A 60, 439–449 (1999).

S. Schlipf, H. Katori, L. Perotti, and H. Walther, “Diffusion of a single ion in a one-dimensional optical lattice,” Opt. Express 3, 97–103 (1998).

H. Katori, S. Schlipf, and H. Walther, “Anomalous dynamics of a single ion in an optical lattice,” Phys. Rev. Lett. 79, 2221–2224 (1997).

G. Birkl, J. A. Yeazell, R. Rückerl, and H. Walther, “Polarization gradient cooling of trapped ions,” Europhys. Lett. 27, 197–202 (1994).

G. Birkl, S. Kassner, and H. Walther, “Multiple-shell struc-tures of laser-cooled 24Mg+ ions in a quadrupole storage ring,” Nature 357, 310–313 (1992).

I. Waki, S. Kassner, G. Birkl, and H. Walther, “Observation of ordered structures of laser-cooled ions in a quadrupole storage ring,” Phys. Rev. Lett. 68, 2007–2010 (1992).

R. Blümel, J. M. Chen, E. Peik, W. Quint, W. Schleich, Y. R. Shen, and H. Walther, “Phase transition of stored laser-cooled ions,” Nature 334, 309–313 (1988).

F. Diedrich, E. Peik, J. M. Chen, W. Quint, and H. Walther, “Observation of a phase transition of stored laser-cooled ions,” Phys. Rev. Lett. 59, 2931–2934 (1987).

Werth, G.

W. Alt, M. Block, P. Seibert, and G. Werth, “Spatial separation of atomic states in a laser-cooled ion crystal,” Phys. Rev. A 58, R23–R25 (1998).

Wineland, D.

D. Wineland and H. Dehmelt, “Proposed 1014δν/ν laser fluorescence spectroscopy on Tl+ mono-ion oscillator III (side band cooling),” Bull. Am. Phys. Soc. 20, 637 (1975).

Wineland, D. J.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).

D. Kielpinski, B. E. King, C. J. Myatt, C. A. Sackett, Q. A. Turchette, W. M. Itano, C. Monroe, and D. J. Wineland, “Sympathetic cooling of trapped ions for quantum logic,” Phys. Rev. A 61, 032310 (2000).

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Sub-dekahertz ultraviolet spectroscopy of 199Hg+,” Phys. Rev. Lett. 85, 2462–2465 (2000).

D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, “Experimental issues in coherent quantum-state manipulation of trapped atomic ions,” J. Res. Natl. Inst. Stand. Technol. 103, 259–328 (1998).

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631–3634 (1998).

B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Cooling the collective motion of trapped ions to initialize a quantum register,” Phys. Rev. Lett. 81, 1525–1528 (1998).

D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, “Generation of nonclassical motional states of a trapped atom,” Phys. Rev. Lett. 76, 1796–1799 (1996).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

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).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

D. J. Wineland, J. Dalibard, and C. Cohen-Tannoudji, “Sisyphus cooling of a bound atom,” J. Opt. Soc. Am. B 9, 32–42 (1992).

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Linear trap for high accuracy spectroscopy of stored ions,” J. Mod. Opt. 39, 233–242 (1992).

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Ionic crystals in a linear Paul trap,” Phys. Rev. A 45, 6493–6501 (1992).

D. J. Heinzen and D. J. Wineland, “Quantum-limited cooling and detection of radio-frequency oscillations by laser-cooled ions,” Phys. Rev. A 42, 2977–2994 (1990).

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62, 403–406 (1989).

S. L. Gilbert, J. J. Bollinger, and D. J. Wineland, “Shell-structure phase of magnetically confined strong coupled plasmas,” Phys. Rev. Lett. 60, 2022–2025 (1988)

D. J. Wineland, J. C. Bergquist, W. M. Itano, J. J. Bollinger, and C. H. Manney, “Atomic-ion Coulomb clusters in an ion trap,” Phys. Rev. Lett. 59, 2935–2938 (1987).

J. C. Bergquist, R. Hulet, W. M. Itano, and D. J. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).

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

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

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

D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1639–1642 (1978).

Wood, C. S.

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631–3634 (1998).

B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Cooling the collective motion of trapped ions to initialize a quantum register,” Phys. Rev. Lett. 81, 1525–1528 (1998).

Yeazell, J. A.

G. Birkl, J. A. Yeazell, R. Rückerl, and H. Walther, “Polarization gradient cooling of trapped ions,” Europhys. Lett. 27, 197–202 (1994).

Yoo, S. M.

S. M. Yoo and J. Javanainen, “Polarization gradient cooling of a trapped ion,” Phys. Rev. A 48, R30–R33 (1993).

Young, B. C.

R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Sub-dekahertz ultraviolet spectroscopy of 199Hg+,” Phys. Rev. Lett. 85, 2462–2465 (2000).

Zeiger, Th.

Ch. Roos, Th. Zeiger, H. Rohde, H. C. Nägerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, “Quantum state engineering on an optical transition and decoherence in a Paul trap,” Phys. Rev. Lett. 83, 4713–4716 (1999).

Zel’dovich, Ya. B.

Ya. B. Zel’dovich, “Cooling with the aid of high-frequency energy,” JETP Lett. 19, 74–75 (1974).

Zoller, P.

G. Morigi, J. Eschner, J. I. Cirac, and P. Zoller, “Laser cooling of two trapped ions: sideband cooling beyond the Lamb–Dicke limit,” Phys. Rev. A 59, 3797–3808 (1999).

G. Morigi, J. I. Cirac, M. Lewenstein, and P. Zoller, “Ground-state laser cooling beyond the Lamb–Dicke limit,” Europhys. Lett. 39, 13–18 (1997).

J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Nonclassical states of motion in ion traps,” Adv. Atom., Mol., Opt. Phys. 37, 237–296 (1996).

J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995).

In experiments the distance among the ions is usually of several wavelengths, and the only relevant interaction is the Coulomb repulsion. For distances of the order of the wavelength, the dipole–dipole interaction must be taken into account. Its effect in connection with laser cooling has been investigated in A. W. Vogt, J. I. Cirac, and P. Zoller, “Collective laser cooling of two trapped ions,” Phys. Rev. A 53, 950–968 (1995).

I. Marzoli, J. I. Cirac, R. Blatt, and P. Zoller, “Laser cooling of trapped three-level ions: designing two-level systems for sideband cooling,” Phys. Rev. A 49, 2771–2779 (1994).

R. Dum, P. Marte, T. Pellizzari, and P. Zoller, “Laser cooling to a single quantum state in a trap,” Phys. Rev. Lett. 73, 2829–2832 (1994).

J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Laser cooling of trapped ions with polarization gradients,” Phys. Rev. A 48, 1434–1445 (1993).

J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Preparation of Fock states by observation of quantum jumps in an ion trap,” Phys. Rev. Lett. 70, 762–765 (1993).

J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Dark squeezed states of the motion of a trapped ion,” Phys. Rev. Lett. 70, 556–559 (1993).

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46, 2668–2681 (1992).

Adv. At. Mol. Phys. (1)

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

Adv. Atom., Mol., Opt. Phys. (1)

J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Nonclassical states of motion in ion traps,” Adv. Atom., Mol., Opt. Phys. 37, 237–296 (1996).

Appl. Phys. (1)

S. Stenholm, “Redistribution of molecular velocities by optical processes,” Appl. Phys. 15, 287–296 (1978).

Appl. Phys. B (3)

A. Steane, “The ion trap quantum information processor,” Appl. Phys. B 64, 623–642 (1997).

D. V. F. James, “Quantum dynamics of cold trapped ions, with application to quantum computation,” Appl. Phys. B 66, 181–190 (1998).

F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D. Leibfried, A. Mundt, and R. Blatt, “Laser cooling with electromagnetically induced transparency: application to trapped samples of ions or neutral atoms,” Appl. Phys. B 73, 807–814 (2001).

Bull. Am. Phys. Soc. (1)

D. Wineland and H. Dehmelt, “Proposed 1014δν/ν laser fluorescence spectroscopy on Tl+ mono-ion oscillator III (side band cooling),” Bull. Am. Phys. Soc. 20, 637 (1975).

Eur. Phys. J. D (1)

G. Morigi and H. Walther, “Two-species Coulomb chains for quantum information,” Eur. Phys. J. D 13, 261–269 (2001).

Europhys. Lett. (3)

G. Morigi, J. I. Cirac, M. Lewenstein, and P. Zoller, “Ground-state laser cooling beyond the Lamb–Dicke limit,” Europhys. Lett. 39, 13–18 (1997).

C. A. Blockley, D. F. Walls, and H. Risken, “Quantum collapses and revivals in a quantized trap,” Europhys. Lett. 17, 509–514 (1992).

G. Birkl, J. A. Yeazell, R. Rückerl, and H. Walther, “Polarization gradient cooling of trapped ions,” Europhys. Lett. 27, 197–202 (1994).

J. Appl. Phys. (2)

J. Prestage, G. J. Dick, and L. Maleki, “New ion trap for frequency standard applications,” J. Appl. Phys. 66, 1013–1017 (1989).

G. R. Janik and L. Maleki, “Simple analytic potentials for linear ion traps,” J. Appl. Phys. 67, 6050–6055 (1990).

J. Mod. Opt. (1)

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Linear trap for high accuracy spectroscopy of stored ions,” J. Mod. Opt. 39, 233–242 (1992).

J. Opt. B (1)

H. Rohde, S. T. Gulde, C. F. Roos, P. A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, “Sympathetic ground state cooling and coherent manipulation with two-ion-crystals,” J. Opt. B 3, S34–S41 (2001).

J. Opt. Soc. Am. B (5)

J. Phys. (1)

A. Kastler, “Quelques suggestions concernant la production optique et la détection optique d’une inégalité de population des niveaux de quantification spatiale des atomes,” J. Phys. 11, 255–265 (1950).

J. Res. Natl. Inst. Stand. Technol. (1)

D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, “Experimental issues in coherent quantum-state manipulation of trapped atomic ions,” J. Res. Natl. Inst. Stand. Technol. 103, 259–328 (1998).

JETP Lett. (1)

Ya. B. Zel’dovich, “Cooling with the aid of high-frequency energy,” JETP Lett. 19, 74–75 (1974).

Nature (4)

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).

R. Blümel, J. M. Chen, E. Peik, W. Quint, W. Schleich, Y. R. Shen, and H. Walther, “Phase transition of stored laser-cooled ions,” Nature 334, 309–313 (1988).

G. Birkl, S. Kassner, and H. Walther, “Multiple-shell struc-tures of laser-cooled 24Mg+ ions in a quadrupole storage ring,” Nature 357, 310–313 (1992).

R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, “Observation of laser-induced fluorescent cooling of a solid,” Nature 377, 500–503 (1995).

Naturwissenschaften (1)

P. Kienle, “Sunshine by cooling,” Naturwissenschaften 88, 313–321 (2001), and references therein.

Opt. Commun. (2)

T. W. Hänsch and A. L. Schawlow, “Cooling of gases by laser radiation,” Opt. Commun. 13, 68–69 (1975).

G. Morigi, H. Baldauf, W. Lange, and H. Walther, “Raman sideband cooling in the presence of multiple decay channels,” Opt. Commun. 187, 171–177 (2001).

Opt. Express (2)

Phys. Rev. A (20)

T. Binnewies, U. Sterr, J. Helmcke, and F. Riehle, “Cooling by Maxwell’s demon: preparation of single-velocity atoms for matter-wave interferometry,” Phys. Rev. A 62, 011601 (2000).

V. I. Balykin and V. S. Letokhov, “Informational cooling of neutral atoms,” Phys. Rev. A 64, 063410 (2001).

S. M. Yoo and J. Javanainen, “Polarization gradient cooling of a trapped ion,” Phys. Rev. A 48, R30–R33 (1993).

J. I. Cirac, R. Blatt, P. Zoller, and W. D. Phillips, “Laser cooling of trapped ions in a standing wave,” Phys. Rev. A 46, 2668–2681 (1992).

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

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

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

M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Ionic crystals in a linear Paul trap,” Phys. Rev. A 45, 6493–6501 (1992).

W. Alt, M. Block, P. Seibert, and G. Werth, “Spatial separation of atomic states in a laser-cooled ion crystal,” Phys. Rev. A 58, R23–R25 (1998).

D. Reiß, A. Lindner, and R. Blatt, “Cooling of trapped multilevel ions: a numerical analysis,” Phys. Rev. A 54, 5133–5140 (1996).

Q. A. Turchette, D. Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, “Heating of trapped ions from the quantum ground state,” Phys. Rev. A 61, 063418 (2000).

J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Laser cooling of trapped ions with polarization gradients,” Phys. Rev. A 48, 1434–1445 (1993).

D. J. Heinzen and D. J. Wineland, “Quantum-limited cooling and detection of radio-frequency oscillations by laser-cooled ions,” Phys. Rev. A 42, 2977–2994 (1990).

I. Marzoli, J. I. Cirac, R. Blatt, and P. Zoller, “Laser cooling of trapped three-level ions: designing two-level systems for sideband cooling,” Phys. Rev. A 49, 2771–2779 (1994).

G. Morigi, J. Eschner, J. I. Cirac, and P. Zoller, “Laser cooling of two trapped ions: sideband cooling beyond the Lamb–Dicke limit,” Phys. Rev. A 59, 3797–3808 (1999).

G. Morigi and J. Eschner, “Doppler cooling of a Coulomb crystal,” Phys. Rev. A 64, 063407 (2001).

D. Kielpinski, B. E. King, C. J. Myatt, C. A. Sackett, Q. A. Turchette, W. M. Itano, C. Monroe, and D. J. Wineland, “Sympathetic cooling of trapped ions for quantum logic,” Phys. Rev. A 61, 032310 (2000).

L. Santos and M. Lewenstein, “Dynamical cooling of trapped gases: one-atom problem,” Phys. Rev. A 59, 613–619 (1999).

E. Peik, J. Abel, Th. Becker, J. von Zanthier, and H. Walther, “Sideband cooling of ions in radio-frequency traps,” Phys. Rev. A 60, 439–449 (1999).

In experiments the distance among the ions is usually of several wavelengths, and the only relevant interaction is the Coulomb repulsion. For distances of the order of the wavelength, the dipole–dipole interaction must be taken into account. Its effect in connection with laser cooling has been investigated in A. W. Vogt, J. I. Cirac, and P. Zoller, “Collective laser cooling of two trapped ions,” Phys. Rev. A 53, 950–968 (1995).

Phys. Rev. Lett. (35)

I. Waki, S. Kassner, G. Birkl, and H. Walther, “Observation of ordered structures of laser-cooled ions in a quadrupole storage ring,” Phys. Rev. Lett. 68, 2007–2010 (1992).

P. Bowe, L. Hornekær, C. Brodersen, M. Drewsen, J. S. Hangst, and J. P. Schiffer, “Sympathetic crystallization of trapped ions,” Phys. Rev. Lett. 82, 2071–2074 (1999).

D. S. Hall and G. Gabrielse, “Electron cooling of protons in a nested Penning trap,” Phys. Rev. Lett. 77, 1962–1965 (1996).

J. Javanainen, “Light-pressure cooling of a crystal,” Phys. Rev. Lett. 56, 1798–1801 (1986).

F. Diedrich, E. Peik, J. M. Chen, W. Quint, and H. Walther, “Observation of a phase transition of stored laser-cooled ions,” Phys. Rev. Lett. 59, 2931–2934 (1987).

D. J. Wineland, J. C. Bergquist, W. M. Itano, J. J. Bollinger, and C. H. Manney, “Atomic-ion Coulomb clusters in an ion trap,” Phys. Rev. Lett. 59, 2935–2938 (1987).

S. L. Gilbert, J. J. Bollinger, and D. J. Wineland, “Shell-structure phase of magnetically confined strong coupled plasmas,” Phys. Rev. Lett. 60, 2022–2025 (1988)

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).

B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Cooling the collective motion of trapped ions to initialize a quantum register,” Phys. Rev. Lett. 81, 1525–1528 (1998).

G. Morigi, J. Eschner, and C. H. Keitel, “Ground state laser cooling using electromagnetically induced transparency,” Phys. Rev. Lett. 85, 4458–4461 (2000).

C. F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler, J. Eschner, and R. Blatt, “Experimental demonstration of ground state laser cooling with electromagnetically induced transparency,” Phys. Rev. Lett. 85, 5547–5550 (2000).

J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

Ch. Roos, Th. Zeiger, H. Rohde, H. C. Nägerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, “Quantum state engineering on an optical transition and decoherence in a Paul trap,” Phys. Rev. Lett. 83, 4713–4716 (1999).

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81, 3631–3634 (1998).

F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to the zero-point energy of motion,” Phys. Rev. Lett. 62, 403–406 (1989).

J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “Dark squeezed states of the motion of a trapped ion,” Phys. Rev. Lett. 70, 556–559 (1993).

J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Preparation of Fock states by observation of quantum jumps in an ion trap,” Phys. Rev. Lett. 70, 762–765 (1993).

D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, “Generation of nonclassical motional states of a trapped atom,” Phys. Rev. Lett. 76, 1796–1799 (1996).

M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche, “Quantum Rabi oscillation: a direct test of field quantization in a cavity,” Phys. Rev. Lett. 76, 1800–1803 (1996).

W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, “Optical-sideband cooling of visible atom cloud confined in parabolic well,” Phys. Rev. Lett. 41, 233–236 (1978).

D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1639–1642 (1978).

W. Nagourney, J. Sandberg, and H. Dehmelt, “Shelved optical electron amplifier: observation of quantum jumps,” Phys. Rev. Lett. 56, 2797–2799 (1986).

Th. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, “Observation of quantum jumps,” Phys. Rev. Lett. 57, 1696–1698 (1986).

J. C. Bergquist, R. Hulet, W. M. Itano, and D. J. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).

C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, and D. J. Wineland, “Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy,” Phys. Rev. Lett. 75, 4011–4014 (1995).

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).

R. Dum, P. Marte, T. Pellizzari, and P. Zoller, “Laser cooling to a single quantum state in a trap,” Phys. Rev. Lett. 73, 2829–2832 (1994).

Ch. Raab, J. Eschner, J. Bolle, H. Oberst, F. Schmidt-Kaler, and R. Blatt, “Motional sidebands and direct measurement of the cooling rate in the resonance fluorescence of a single trapped ion,” Phys. Rev. Lett. 85, 538–541 (2000).

R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Sub-dekahertz ultraviolet spectroscopy of 199Hg+,” Phys. Rev. Lett. 85, 2462–2465 (2000).

J. Eschner, B. Appasamy, and P. E. Toschek, “Stochastic cooling of a trapped ion by null detection of its fluorescence,” Phys. Rev. Lett. 74, 2435–2438 (1995).

B. Appasamy, Y. Stalgies, and P. E. Toschek, “Measurement-induced vibrational dynamics of a trapped ion,” Phys. Rev. Lett. 80, 2805–2808 (1998).

H. Katori, S. Schlipf, and H. Walther, “Anomalous dynamics of a single ion in an optical lattice,” Phys. Rev. Lett. 79, 2221–2224 (1997).

J. L. Clark and G. Rumbles, “Laser cooling in the condensed phase by frequency up-conversion,” Phys. Rev. Lett. 76, 2037–2040 (1996).

C. W. Hoyt, M. Sheik-Bahae, R. I. Epstein, B. C. Edwards, and J. E. Anderson, “Observation of anti-Stokes fluorescence cooling in thulium-doped glass,” Phys. Rev. Lett. 85, 3600–3603 (2000).

Phys. Today (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997), and references therein.

Rev. Mod. Phys. (2)

S. Stenholm, “Semiclassical theory of laser cooling,” Rev. Mod. Phys. 58, 699–739 (1986), and references therein.

A discussion of crystallization in these mesoscopic structures is beyond the scope of this work. The interested reader can find a review in D. H. Dubin and T. M. O’Neil, “Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium state),” Rev. Mod. Phys. 71, 87–172 (1999).

Other (7)

While we give the original explanation here, a more recent study shows that EIT cooling works without restrictions on the intensities see the two lasers, as long as the Lamb–Dicke limit applies; see G. Morigi, “Cooling atomic motion with quantum interference,” Phys. Rev. A (to be published); preprint available at http://arXiv.org/abs/quant-ph/0211043.

E. Arimondo, “Coherent population trapping in laser spectroscopy,” in Progress in Optics XXXV, pp. 257–354, E. Wolf, ed. (North Holland, Amsterdam, 1996).

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions (Wiley, New York, 1992).

When the natural linewidth is comparable with or smaller than the laser linewidth, the rate Γ has to be replaced by the spectroscopic linewidth obtained experimentally.

Special Issue of J. Opt. Soc. Am. B 6, (1989).

P. K. Gosh, Ion Traps (Clarendon, Oxford, UK, 1995).

In the absence of applied fields the trapped ion may still couple to the environment: Heating and decoherence of quantum states of the motion has been observed, 42 the origin of which is partly still unclear. Apparently this effect is critically determined by the trap characteristics, and the rate can be much slower than the rate of laser cooling.43 Furthermore, the charge of the ion couples to blackbody radiation. This process is also negligible on the time scales considered here. In the discussion, we ignore these effects.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Metrics