Quantum simulations with trapped ions

Journal name:
Nature Physics
Volume:
8,
Pages:
277–284
Year published:
DOI:
doi:10.1038/nphys2252
Received
Accepted
Published online

Abstract

In the field of quantum simulation, methods and tools are explored for simulating the dynamics of a quantum system of interest with another system that is easier to control and measure. Systems of trapped atomic ions can be accurately controlled and manipulated, a large variety of interactions can be engineered with high precision and measurements of relevant observables can be obtained with nearly 100% efficiency. Here, we discuss prospects for quantum simulations using systems of trapped ions, and review the available set of quantum operations and first proof-of-principle experiments for both analog and digital quantum simulations with trapped ions.

At a glance

Figures

  1. Principles of quantum simulation.
    Figure 1: Principles of quantum simulation.

    The three main steps of a quantum simulator consist of preparing the input state, evolving it over a time t and carrying out measurements on the evolved state to extract the physical information of interest. The time evolution of the simulator is designed to match the time evolution of the model system to be simulated. In an analog simulator, this is achieved by matching the dynamics of the simulator with the time evolution governing the dynamics of the simulated model. In a digital simulator, the propagator describing the dynamical evolution is constructed from a series of quantum gates. In the illustration, the horizontal lines represent qubits or other elementary constituents of the overall quantum system; grey boxes represent quantum operations acting on the respective constituents covered by the boxes.

  2. Trapped-ion quantum system used for quantum-information processing.
    Figure 2: Trapped-ion quantum system used for quantum-information processing.

    a, A qubit is encoded in two internal states |gright fence,|eright fence of an ion confined in a harmonic potential. Qubit read-out is accomplished by coupling one of the qubit levels to a short-lived state |sright fence. Thus, observation of fluorescence (bright, dark) indicates population of the ground (|gright fence) or excited (|eright fence) state, respectively. The qubit is coherently manipulated by laser pulses coupling states |gright fence and |eright fence on either a single- or two-photon transition. b, Qubit–motion coupling. The qubit is excited either on the carrier transition (2) or on the sideband transitions (3), (4), which couple it to a vibrational mode. Bracketed numbers refer to equation numbers in the text giving the corresponding Hamiltonians. Excitation on the sideband transitions decreases or increases the vibrational quantum number n on excitation of the qubit. c, In an ion crystal, all measurements are made by (spatially resolved) fluorescence detection. The vibrational states can be measured by coupling the ion motion to the qubits using sideband transitions, thus mapping information about the ion motion onto qubits that are subsequently read out. Effective spin–spin interactions are realized by interactions mediated by one or several of the crystal’s vibrational modes.

  3. Magnetization data.
    Figure 3: Magnetization data.

    The magnetization (as derived from the populations of the individual two-level systems that represent the simulated spins, for details see ref. 57) for N=2 spins (circles) is contrasted with that of N=9 spins (diamonds), with representative error bars for the detection process57. The data deviate from unity at B/|J|=0 by ~ 20%, predominantly owing to decoherence from spontaneous emission in Raman transitions and further dephasing from Raman beam intensity fluctuations57. The theoretical time-evolution curves (solid line for N=2 and dashed line for N=9 spins) are calculated by averaging over 10,000 quantum trajectories. Reproduced from ref. 57.

  4. Quantum simulation of relativistic scattering (Klein tunnelling) for linear potentials.
    Figure 4: Quantum simulation of relativistic scattering (Klein tunnelling) for linear potentials.

    Particle–wave packets (filled curves), presented in units of the size of the ground-state wavefunction, Δ, are compared with ideal predictions (solid black lines) and predictions taking corrections to the Lamb–Dicke approximation into account (dashed black lines). In the first and last frames of each sequence the positive (green)- and negative (red)-energy components are reconstructed separately. The blue colour scale of these panels represents the measured expectation value of momentum. The axis on the right shows the potential energy in units of initial kinetic energy. a, Without a potential, the particle moves to the right with constant velocity. bd, For a shallow potential gradient (b) the particle is almost completely reflected, and for steeper gradients (c,d) part of the wave packet propagates into the repulsive potential through Klein tunnelling. Figure reproduced with permission from ref. 71, © 2011 APS.

  5. Quantum toolbox with a string of trapped ions.
    Figure 5: Quantum toolbox with a string of trapped ions.

     Laser beams interact with the ion(s) for a predetermined time, corresponding to phases θ in units of π. For this, the ions are either simultaneously (global beam) or individually (local beam) addressed to implement the toolbox operations O1(θ,j) (local), O2(θ) and O3(θ,φ)(global) and the entangling O4(θ,φ) (global). These operations are applied sequentially according to a given simulation (or computational) task.

  6. Digital simulations of a two-spin system interacting through Ising, XY and XYZ interaction plus a transverse field.
    Figure 6: Digital simulations of a two-spin system interacting through Ising, XY and XYZ interaction plus a transverse field.

    Dynamics of the initial state using a fixed digital resolution of π/16. Each panel shows how a single digital step is built from the elementary interactions: C=O2(π/16), D=O4(π/16,0), E=O4(π/16,π), F=O3(π/4,0). Lines show exact dynamics induced by the respective Hamiltonian, open symbols the ideal digitized dynamics. Filled symbols are measured data (blue diamond, ; red square, ; black circle, ). Figure adapted with permission from ref. 77, © 2011 AAAS.

  7. Digital simulations of four- and six-spin systems.
    Figure 7: Digital simulations of four- and six-spin systems.

    Dynamics of the initial state where all spins point up. a, Four-spin long-range Ising system. Each digital step is D.C=O4(π/16,0).O2(π/32). Error bars are smaller than the point size. b, Six-spin six-body interaction. F=O1(θ,1), 4D=O4(π/4,0). Lines, exact dynamics. Open symbols, ideal digitized. Filled symbols, data (blue square, P0; magenta diamond, P1; black circle, P2; green triangle, P3; red right triangle, P4; cyan down triangle, P5; orange left triangle, P6, where Piis the total probability of finding i spins pointing down). Figure reproduced with permission from ref. 77, © 2011 AAAS.

References

  1. Zoller, P. et al. Quantum information processing and communication. Strategic report on current status, visions and goals for research in Europe. Eur. Phys. J. D 36, 203228 (2005).
  2. Schmidt, P. O. et al. Spectroscopy using quantum logic. Science 309, 749752 (2005).
  3. Weihs, G., Jennewein, T., Simon, C., Weinfurter, H. & Zeilinger, A. Violation of Bell’s inequality under strict Einstein locality conditions. Phys. Rev. Lett. 81, 50395043 (1998).
  4. Rowe, M. A. et al. Experimental violation of a Bell’s inequality with efficient detection. Nature 409, 791794 (2001).
  5. Shor, P. W. Proc. 35th Annual Symposium on Foundations of Computer Science124134 (IEEE Computer Soc. Press, 1994).
  6. Feynman, R. Simulating physics with computers. Int. J. Theoret. Phys. 21, 467488 (1982).
  7. Lloyd, S. Universal quantum simulators. Science 273, 10731078 (1996).
  8. Buluta, I. & Nori, F. Quantum simulators. Science 326, 108111 (2009).
  9. Trotter, H. F. On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545551 (1959).
  10. Army Research Office ARDA Quantum Computation Roadmap (Los Alamos National Laboratory, 2005), available at http://qist.lanl.gov/qcomp_map.shtml.
  11. Vandersypen, L. M. K. & Chuang, I. L. NMR techniques for quantum control and computation. Rev. Mod. Phys. 76, 10371069 (2005).
  12. Brown, K. L., Munro, W. J. & Kendon, V. M. Using quantum computers for quantum simulation. Entropy 12, 22682307 (2010).
  13. Jaksch, D., Bruder, C., Cirac, J. I., Gardiner, C. W. & Zoller, P. Cold bosonic atoms in optical lattices. Phys. Rev. Lett. 81, 31083111 (1998).
  14. Lewenstein, M. et al. Ultracold atomic gases in optical lattices: Mimicking condensed matter physics and beyond. Adv. Phys. 56, 243379 (2007).
  15. Jane, E., Vidal, G., Dür, W., Zoller, P. & Cirac, J. I. Simulation of quantum dynamics with quantum optical systems. Quant. Inf. Comp. 3, 1537 (2003).
  16. Bloch, I., Dalibard, J. & Nascimbéne, S. Quantum simulations with ultracold quantum gases. Nature Phys. 8, 267276 (2012).
  17. Bakr, W. S., Gillen, J. I., Peng, A., Fölling, S. & Greiner, M. A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice. Nature 462, 7477 (2009).
  18. Sherson, J. F. et al. Single-atom-resolved fluorescence imaging of an atomic Mott insulator. Nature 467, 6872 (2010).
  19. O’Brien, J. L. Optical quantum computing. Science 318, 15671570 (2007).
  20. Lanyon, B. P. et al. Towards quantum chemistry on a quantum computer. Nature Chem. 2, 106111 (2010).
  21. Ma, X., Dakic, B., Naylor, W., Zeilinger, A. & Walther, P. Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems. Nature Phys. 7, 399405 (2011).
  22. Aspuru-Guzik, A. & Walther, P. Photonic quantum simulators. Nature Phys. 8, 285291 (2012).
  23. Cirac, J. I. & Zoller, P. Quantum computations with cold trapped ions. Phys. Rev. Lett. 74, 40914094 (1995).
  24. Schoelkopf, R. J. & Girvin, S. M. Wiring up quantum systems. Nature 451, 664669 (2008).
  25. Houck, A. A., Türeci, H. E. & Koch, J. On-chip quantum simulation with superconducting circuits. Nature Phys. 8, 292299 (2012).
  26. Hanson, R. & Awschalom, D. D. Coherent manipulation of single spins in semiconductors. Nature 453, 10431049 (2008).
  27. Blatt, R. & Wineland, D. Entangled states of trapped atomic ions. Nature 453, 10081015 (2008).
  28. Häffner, H., Roos, C. F. & Blatt, R. Quantum computing with trapped ions. Phys. Rep. 469, 155203 (2008).
  29. Nagourney, W., Sandberg, J. & Dehmelt, H. Shelved optical electron amplifier: Observation of quantum jumps. Phys. Rev. Lett. 56, 27972799 (1986).
  30. Myerson, A. H. et al. High-fidelity readout of trapped-ion qubits. Phys. Rev. Lett. 100, 200502 (2008).
  31. Leibfried, D., Blatt, R., Monroe, C. & Wineland, D. Quantum dynamics of single trapped ions. Rev. Mod. Phys. 75, 281324 (2003).
  32. Turchette, Q. A. et al. Heating of trapped ions from the quantum ground state. Phys. Rev. A 61, 063418 (2000).
  33. Deslauriers, L. et al. Scaling and suppression of anomalous heating in ion traps. Phys. Rev. Lett. 97, 103007 (2006).
  34. Jaynes, E. T. & Cummings, F. W. Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51, 89109 (1963).
  35. Blockley, C. A., Walls, D. F. & Risken, H. Quantum collapses and revivals in a quantized trap. Europhys. Lett. 17, 509514 (1992).
  36. Cirac, J. I., Blatt, R., Parkins, A. S. & Zoller, P. Quantum collapse and revival in the motion of a single trapped ion. Phys. Rev. A 49, 12021207 (1994).
  37. Wineland, D. J. et al. Trapped ion quantum simulator. Proc. Am. Math. Soc. T76, 147151 (1998).
  38. Meekhof, D. M., Monroe, C., King, B. E., Itano, W. M. & Wineland, D. J. Generation of nonclassical motional states of a trapped atom. Phys. Rev. Lett. 76, 17961799 (1996).
  39. Leibfried, D. et al. Experimental determination of the motional quantum state of a trapped atom. Phys. Rev. Lett. 77, 42814285 (1996).
  40. Zähringer, F. et al. Realization of a quantum walk with one and two trapped ions. Phys. Rev. Lett. 104, 100503 (2010).
  41. Raizen, M. C., Bergquist, J. C., Gilligan, J. M., Itano, W. M. & Wineland, D. J. Linear trap for high-accuracy spectroscopy of stored ions. J. Mod. Opt. 39, 233242 (1992).
  42. Schmidt-Kaler, F. et al. Realization of the Cirsac–Zoller controlled-NOT quantum gate. Nature 422, 408411 (2003).
  43. Sørensen, A. & Mølmer, K. Quantum computation with ions in thermal motion. Phys. Rev. Lett. 82, 19711974 (1999).
  44. Sackett, C. A. et al. Experimental entanglement of four particles. Nature 404, 256259 (2000).
  45. Leibfried, D. et al. Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate. Nature 422, 412415 (2003).
  46. Benhelm, J., Kirchmair, G., Roos, C. F. & Blatt, R. Towards fault-tolerant quantum computing with trapped ions. Nature Phys. 4, 463466 (2008).
  47. Kirchmair, G. et al. Deterministic entanglement of ions in thermal states of motion. New J. Phys. 11, 023002 (2009).
  48. Porras, D. & Cirac, J. I. Effective quantum spin systems with trapped ions. Phys. Rev. Lett. 92, 207901 (2004).
  49. Johanning, M., Varón, A. F. & Wunderlich, C. Quantum simulations with cold trapped ions. J. Phys. B 42, 154009 (2009).
  50. Troyer, M. & Wiese, U-J. Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations. Phys. Rev. Lett. 94, 170201 (2005).
  51. Sandvik, A. W. Ground states of a frustrated quantum spin chain with long-range interactions. Phys. Rev. Lett. 104, 137204 (2010).
  52. Mintert, F. & Wunderlich, C. Ion-trap quantum logic using long-wavelength radiation. Phys. Rev. Lett. 87, 257904 (2001).
  53. Friedenauer, H., Schmitz, H., Glueckert, J., Porras, D. & Schaetz, T. Simulating a quantum magnet with trapped ions. Nature Phys. 4, 757761 (2008).
  54. Kim, K. et al. Entanglement and tunable spin–spin couplings between trapped ions using multiple transverse modes. Phys. Rev. Lett. 103, 120502 (2009).
  55. Kim, K. et al. Quantum simulation of frustrated Ising spins with trapped ions. Nature 465, 590593 (2010).
  56. Edwards, E. E. et al. Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins. Phys. Rev. B 82, 060412 (2010).
  57. Islam, R. et al. Onset of a quantum phase transition with a trapped ion quantum simulator. Nature Commun. 2, 377 (2011).
  58. Kim, K. et al. Quantum simulation of the transverse Ising model with trapped ions. New J. Phys. 13, 105003 (2011).
  59. Schneider, C., Porras, D. & Schaetz, T. Experimental quantum simulations of many-body physics with trapped ions. Rep. Prog. Phys. 75, 024401 (2012).
  60. Lin, G-D., Monroe, C. & Duan, L-M. Sharp phase transitions in a small frustrated network of trapped ion spins. Phys. Rev. Lett. 106, 230402 (2011).
  61. Lamata, L., León, J., Schätz, T. & Solano, E. Dirac equation and quantum relativistic effects in a single trapped ion. Phys. Rev. Lett. 98, 253005 (2007).
  62. Alsing, P. M., Dowling, J. P. & Milburn, G. J. Ion trap simulations of quantum fields in an expanding universe. Phys. Rev. Lett. 94, 220401 (2005).
  63. Menicucci, N. C. & Milburn, G. J. Single trapped ion as a time-dependent harmonic oscillator. Phys. Rev. A 76, 052105 (2007).
  64. Horstmann, B., Reznik, B., Fagnocchi, S. & Cirac, J. I. Hawking radiation from an acoustic black hole on an ion ring. Phys. Rev. Lett. 104, 250403 (2010).
  65. Menicucci, N. C., Olson, S. J. & Milburn, G. J. Simulating quantum effects of cosmological expansion using a static ion trap. New J. Phys. 12, 095019 (2010).
  66. Schützhold, R. et al. Analogue of cosmological particle creation in an ion trap. Phys. Rev. Lett. 99, 201301 (2007).
  67. Gerritsma, R. et al. Quantum simulation of the Dirac equation. Nature 463, 6871 (2010).
  68. Casanova, J., Garcia-Ripoll, J. J., Gerritsma, R., Roos, C. F. & Solano, E. Klein tunneling and Dirac potentials in trapped ions. Phys. Rev. A 82, 020101 (2010).
  69. Schrödinger, E. Über die kräftefreie Bewegung in der relativistischen Quantenmechanik. Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. 24, 418428 (1930).
  70. Klein, O. Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac. Z. Phys. 53, 157165 (1929).
  71. Gerritsma, R. et al. Quantum simulation of the Klein paradox with trapped ions. Phys. Rev. Lett. 106, 060503 (2011).
  72. Lutterbach, L. & Davidovich, L. Method for direct measurement of the Wigner function in cavity QED and ion traps. Phys. Rev. Lett. 78, 25472550 (1997).
  73. Lougovski, P., Walther, H. & Solano, E. Instantaneous measurement of field quadrature moments and entanglement. Eur. Phys. J. D 38, 423426 (2006).
  74. Wallentowitz, S. & Vogel, W. Reconstruction of the quantum mechanical state of a trapped ion. Phys. Rev. Lett. 75, 29322935 (1995).
  75. Casanova, J. et al. Quantum simulation of quantum field theories in trapped ions. Phys. Rev. Lett. 107, 260501 (2011).
  76. Brown, K. R., Clark, R. J. & Chuang, I. L. Limitations of quantum simulation examined by simulating a pairing hamiltonian using nuclear magnetic resonance. Phys. Rev. Lett. 97, 050504 (2006).
  77. Lanyon, B. P. et al. Universal digital quantum simulations with trapped ions. Science 334, 5761 (2011).
  78. Nebendahl, V., Häffner, H. & Roos, C. F. Optimal control of entangling operations for trapped-ion quantum computing. Phys. Rev. A 79, 012312 (2009).
  79. Barreiro, J. T. et al. An open-system quantum simulator with trapped ions. Nature 470, 486491 (2011).
  80. Ospelkaus, C. et al. Trapped-ion quantum logic gates based on oscillating magnetic fields. Phys. Rev. Lett. 101, 090502 (2008).
  81. Johanning, M. et al. Individual addressing of trapped ions and coupling of motional and spin states using rf radiation. Phys. Rev. Lett. 102, 073004 (2009).
  82. Ospelkaus, C. et al. Microwave quantum logic gates for trapped ions. Nature 476, 181184 (2011).
  83. Khromova, A. et al. A designer spin-molecule implemented with trapped ions in a magnetic gradient. Preprint at http://arxiv.org/abs/1112.5302 (2011).
  84. Wang, S. X., Labaziewicz, J., Ge, Y., Shewmon, R. & Chuang, I. L. Individual addressing of ions using magnetic field gradients in a surface-electrode ion trap. Appl. Phys. Lett. 94, 094103 (2009).
  85. Welzel, J. et al. Designing spin–spin interactions with one and two dimensional ion crystals in planar micro traps. Eur. Phys. J. D 65, 285297 (2011).
  86. Dubin, D. H. E. Theory of structural phase transitions in a trapped Coulomb crystal. Phys. Rev. Lett. 71, 27532756 (1993).
  87. Fishman, S., De Chiara, G., Calarco, T. & Morigi, G. Structural phase transitions in low-dimensional ion crystals. Phys. Rev. B 77, 064111 (2008).
  88. Lin, G-D. et al. Large-scale quantum computation in an anharmonic linear ion trap. Europhys. Lett. 86, 60004 (2009).
  89. Schaetz, T., Friedenauer, A., Schmitz, H., Petersen, L. & Kahra, S. Towards (scalable) quantum simulations in ion traps. J. Mod. Opt. 54, 23172325 (2007).
  90. Chiaverini, J. & Lybarger, J, W. E. Laserless trapped-ion quantum simulations without spontaneous scattering using microtrap arrays. Phys. Rev. A 77, 022324 (2008).
  91. Clark, R. J., Lin, T., Brown, K. R. & Chuang, I. L. A two-dimensional lattice ion trap for quantum simulation. J. App. Phys. 105, 013114 (2009).
  92. Schmied, R., Wesenberg, J. H. & Leibfried, D. Optimal surface-electrode trap lattices for quantum simulation with trapped ions. Phys. Rev. Lett. 102, 233002 (2009).
  93. Kumph, M., Brownnutt, M. & Blatt, R. Two-dimensional arrays of radio-frequency ion traps with addressable interactions. New J. Phys. 13, 073043 (2011).
  94. Schmied, R., Wesenberg, J. H. & Leibfried, D. Quantum simulation of the hexagonal Kitaev model with trapped ions. New J. Phys. 13, 115011 (2011).
  95. Porras, D., Marquardt, F., von Delft, J. & Cirac, J. I. Mesoscopic spin-boson models of trapped ions. Phys. Rev. A 78, 010101 (2008).
  96. Porras, D. & Cirac, J. I. Bose–Einstein condensation and strong-correlation behavior of phonons in ion traps. Phys. Rev. Lett. 93, 263602 (2004).
  97. Porras, D. & Cirac, J. I. Quantum manipulation of trapped ions in two dimensional Coulomb crystals. Phys. Rev. Lett. 96, 250501 (2006).
  98. Bermudez, A., Schätz, T. & Porras, D. Synthetic gauge fields for vibrational excitations of trapped ions. Phys. Rev. Lett. 107, 150501 (2011).
  99. Brown, K. R. et al. Coupled quantized mechanical oscillators. Nature 471, 196199 (2011).
  100. Harlander, M., Lechner, R., Brownnutt, M., Blatt, R. & Hänsel, W. Trapped-ion antennae for the transmission of quantum information. Nature 471, 200203 (2011).
  101. Schmied, R., Roscilde, T., Murg, V., Porras, D. & Cirac, J. I. Quantum phases of trapped ions in an optical lattice. New. J. Phys. 10, 045017 (2008).
  102. Müller, M., Liang, L., Lesanovsky, I. & Zoller, P. Trapped Rydberg ions: From spin chains to fast quantum gates. New J. Phys. 10, 093009 (2008).
  103. Schmidt-Kaler, F. et al. Rydberg excitation of trapped cold ions: A detailed case study. New J. Phys. 13, 075014 (2011).
  104. Mitchell, T. B. et al. Direct observations of structural phase transitions in planar crystallized ion plasmas. Science 282, 12901293 (1998).
  105. Sawyer, B. C. et al. Spectroscopy and thermometry of drumhead modes in a mesoscopic trapped-ion crystal using entanglement. Preprint at http://arxiv.org/abs/1201.4415 (2012).
  106. Schneider, C., Enderlein, M., Huber, T. & Schaetz, T. Optical trapping of an ion. Nature Photon. 4, 772775 (2010).
  107. Pruttivarasin, T., Ramm, M., Talukdar, I., Kreuter, A. & Häffner, H. Trapped ions in optical lattices for probing oscillator chain models. New J. Phys. 13, 075012 (2011).
  108. Benassi, A., Vanossi, A. & Tosatti, E. Nanofriction in cold ion traps. Nature Commun. 2, 236 (2011).
  109. Zipkes, C., Palzer, S., Sias, C. & Köhl, M. A trapped single ion inside a Bose–Einstein condensate. Nature 464, 388391 (2010).
  110. Schmid, S., Härter, A. & Denschlag, J. H. Dynamics of a cold trapped ion in a Bose–Einstein condensate. Phys. Rev. Lett. 105, 133202 (2010).
  111. Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 17041707 (2005).
  112. Hauke, P., Cucchietti, F. M., Tagliacozzo, L., Lewenstein, M. & Deutsch, I. On the robustness of quantum simulators. Preprint at http://arxiv.org/abs/1109.6457 (2011).
  113. Lidar, D. A., Chuang, I. L. & Whaley, K. B. Decoherence-free subspaces for quantum computation. Phys. Rev. Lett. 81, 25942597 (1998).
  114. Viola, L., Knill, E. & Lloyd, S. Dynamical decoupling of open quantum systems. Phys. Rev. Lett. 82, 24172421 (1999).
  115. Bennett, C. H., DiVincenzo, D. P., Smolin, J. A. & Wootters, W. K. Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 38243851 (1996).

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  1. Institut für Experimentalphysik, Universität Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria

    • R. Blatt &
    • C. F. Roos
  2. Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria

    • R. Blatt &
    • C. F. Roos

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