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

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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).
[CrossRef]

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

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

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

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).
[CrossRef]

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

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef]

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

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).
[CrossRef]

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).
[CrossRef] [PubMed]

1999

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).
[CrossRef]

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).
[CrossRef]

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

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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

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).
[CrossRef]

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).
[CrossRef]

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

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).
[CrossRef]

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).
[CrossRef] [PubMed]

1997

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

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).
[CrossRef]

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

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

1996

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

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

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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

1995

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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

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).
[CrossRef] [PubMed]

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

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).
[CrossRef] [PubMed]

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).
[CrossRef]

1994

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).
[CrossRef] [PubMed]

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

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).
[CrossRef] [PubMed]

1993

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

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

1992

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

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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

1990

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).
[CrossRef] [PubMed]

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

1989

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).
[CrossRef] [PubMed]

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

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).
[CrossRef]

1988

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

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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)
[CrossRef] [PubMed]

1987

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

1986

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

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

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

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).
[CrossRef] [PubMed]

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

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

1985

1983

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).
[CrossRef]

1982

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

1980

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

1979

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

1978

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).
[CrossRef]

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

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

1975

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

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

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

1950

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

Balykin, V. I.

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

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

Birkl, G.

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

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).
[CrossRef]

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).
[CrossRef] [PubMed]

Blatt, R.

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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)
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

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

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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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D. J. Wineland, J. Dalibard, and C. Cohen-Tannoudji, “Sisyphus cooling of a bound atom,” J. Opt. Soc. Am. B 9, 32–42 (1992).
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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).
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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).
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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).
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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)
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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).
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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).
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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).
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W. M. Itano and D. J. Wineland, “Laser cooling of ions stored in harmonic and Penning traps,” Phys. Rev. A 25, 35–54 (1982).
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D. J. Wineland and W. M. Itano, “Laser cooling of atoms,” Phys. Rev. A 20, 1521–1540 (1979).
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D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of bound resonant absorbers,” Phys. Rev. Lett. 40, 1639–1642 (1978).
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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).
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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).
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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).
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G. Birkl, J. A. Yeazell, R. Rückerl, and H. Walther, “Polarization gradient cooling of trapped ions,” Europhys. Lett. 27, 197–202 (1994).
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S. M. Yoo and J. Javanainen, “Polarization gradient cooling of a trapped ion,” Phys. Rev. A 48, R30–R33 (1993).
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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).
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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).
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Ya. B. Zel’dovich, “Cooling with the aid of high-frequency energy,” JETP Lett. 19, 74–75 (1974).

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

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Appl. Phys. B

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Eur. Phys. J. D

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

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).
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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).
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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).
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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).
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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).
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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).
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D. J. Wineland and W. M. Itano, “Laser cooling of atoms,” Phys. Rev. A 20, 1521–1540 (1979).
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W. M. Itano and D. J. Wineland, “Laser cooling of ions stored in harmonic and Penning traps,” Phys. Rev. A 25, 35–54 (1982).
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W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, “Localized visible Ba+ mono-ion oscillator,” Phys. Rev. A 22, 1137–1140 (1980).
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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).
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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).
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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).
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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).
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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).
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[CrossRef] [PubMed]

Phys. Today

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[CrossRef]

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

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).
[CrossRef]

Other

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.

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).

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.

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Figures (6)

Fig. 1
Fig. 1

Illustration of strong confinement (left) and weak confinement (right) in the Lamb–Dicke regime. The Lorentz curve is the atomic resonance of width Γ. The vertical bars display the relative absorption probabilities of the laser and its sidebands, |Fnm|2 for m=n, n±1, at ωL, ωLν. The laser spectrum is displayed in the reference frame of the ion, with the laser detuned for optimum cooling in both cases, i.e., ωL=ω0-ν and ωL=ω0-Γ/2 for strong and weak confinement, respectively.

Fig. 2
Fig. 2

(a) Cooling (|n|n-1) and heating (|n|n+1) transitions starting from state |g, n, at lowest order in η. The relative probabilities of the individual processes are indicated for cos θ=1. (b) Illustration of the rate coefficients for cooling and heating, after Ref. 14.

Fig. 3
Fig. 3

Levels and transition wavelengths relevant for resolved-sideband cooling in various ion species: (a) With a single 198Hg+ ion, Doppler cooling is performed on the transition near 194 nm. For sideband cooling, laser light near 281 and 398 nm is used. The red sideband of the S1/2D5/2 transition is indicated with a dashed horizontal line. (b) For Doppler cooling of 40Ca+ ions, the dipole transitions near 397 and 866 nm are employed. Resolved-sideband cooling is performed on the quadrupole transition near 729 nm while the dipole transition near 854 nm is used to increase the cooling rate. (c) For resolved-sideband cooling of 115In+ ions, the intercombination line near 231 nm is used. (d) Levels relevant for optical cooling of 9Be+ ions. Transitions D1, D2, and D3 are used for Doppler cooling and optical pumping. Raman sideband cooling is performed with laser pulses alternately driving the transitions R1, R2, and D1, D3. All wavelengths are near 313 nm; the hyperfine splitting of both P states is 1.250 GHz.  

Fig. 4
Fig. 4

Illustration of the normal modes of a two-ion string. Left, the common vibration is shown for the axial and breathing modes at frequencies νax and νb=νax3, for which the motion is purely along the zˆ direction of the trap. Right, radial and rocking modes at frequencies νrad and νr=νrad2-νax2 are depicted.

Fig. 5
Fig. 5

(a) Levels and transitions of the EIT-cooling scheme. The coupling laser drives |r|e, the cooling laser |g|e. The inset shows schematically the absorption rate on |g|e when the atom is strongly excited above resonance on |r|e. 76 (b) Absorption of cooling laser around Δg=Δr (solid curve); dashed lines mark the probabilities of carrier (|n|n) and sideband (|n|n±1) transitions for the case Δg=Δr. (c) Dressed-state picture: the cooling laser excites resonantly transitions from |g, n to the narrow dressed state denoted |+, n-1, which preferentially decays into |g, n-1. δ  is the light shift (ac Stark shift) created by the coupling laser.

Fig. 6
Fig. 6

Schematic of cooling by detection of no fluorescence: (a) Simplified level scheme showing the slow cooling transition SD and the fast transition for state-selective fluorescence SP. The Ba+ ion is taken as an example. Blue-sideband excitation on SD is alternated with probing on SP. (b) Starting in some distribution over the motional states in D (gray circles), the higher-lying states are more likely to be deexcited (arrows). The |D, n+1|S, n Rabi frequencies are indicated. If the ion is in |D, n=0, it will remain dark.

Equations (9)

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Φ=Φ0(αx2+βy2+γz2)/r02
Hmech=p22M+12 Mν2z2=νaa+12.
H=Hmech-Δ|ee|+(Ω/2)×[|eg|exp(ikzz)+H.c.],
ωR=k2/2M.
tρ=[H, ρ]/i+Lρ,
A±=Ω2Γ η2[cos2 θ W(Δν)+αW(Δ)],
ddt n=-(A--A+)n+A+.
nt=n0exp[-(A--A+)t]+n¯{1-exp[-(A--A+)t]},
Pa(τ)=ng, n|ρ|g, nsin2(ηn+1Ωτ),

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