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Anyonic interferometry and protected memories in atomic spin lattices

Nature Physics volume 4, pages 482–488 (2008)Cite this article

Abstract

Strongly correlated quantum systems can exhibit exotic behaviour called topological order which is characterized by non-local correlations that depend on the system topology. Such systems can exhibit remarkable phenomena such as quasiparticles with anyonic statistics and have been proposed as candidates for naturally error-free quantum computation. However, anyons have never been observed in nature directly. Here, we describe how to unambiguously detect and characterize such states in recently proposed spin–lattice realizations using ultracold atoms or molecules trapped in an optical lattice. We propose an experimentally feasible technique to access non-local degrees of freedom by carrying out global operations on trapped spins mediated by an optical cavity mode. We show how to reliably read and write topologically protected quantum memory using an atomic or photonic qubit. Furthermore, our technique can be used to probe statistics and dynamics of anyonic excitations.

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Figure 1: Generators for the encoded qubits.
Figure 2: Cavity-assisted controlled-string operation based on the single-photon approach.
Figure 3: Phase accumulation for the geometric phase gate approach (equation (7)).
Figure 4: Braiding operations.
Figure 5: Fringe contrast of anyonic interferometry as a function of time (in units of 〈(hex)2〉−1/2) for anyonic diffusion.

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References

  1. Wen, X.-G. Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons (Oxford Univ. Press, Oxford, 2004).

    Google Scholar 

  2. Einarsson, T. Fractional statistics on a torus. Phys. Rev. Lett. 64, 1995–1998 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  3. Das Sarma, S., Freedman, M., Nayak, C., Simon, S. H. & Stern, A. Non-abelian anyons and topological quantum computation. Preprint at <http://arxiv.org/abs/0707.1889> (2007).

  4. Camino, F. E., Zhou, W. & Goldman, V. J. Realization of a Laughlin quasiparticle interferometer: Observation of fractional statistics. Phys. Rev. B 72, 075342 (2005).

    Article  ADS  Google Scholar 

  5. Rosenow, B. & Halperin, B. I. Influence of interactions on flux and back-gate period of quantum Hall interferometers. Phys. Rev. Lett. 98, 106801 (2007).

    Article  ADS  Google Scholar 

  6. Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  7. Duan, L. M., Demler, E. & Lukin, M. D. Controlling spin exchange interactions of ultracold atoms in optical lattices. Phys. Rev. Lett. 91, 090402 (2003).

    Article  ADS  Google Scholar 

  8. Micheli, A., Brennen, G. K. & Zoller, P. A toolbox for lattice-spin models with polar molecules. Nature Phys. 2, 341–347 (2006).

    Article  ADS  Google Scholar 

  9. Brennen, G. K. & Pachos, J. K. Why should anyone care about computing with anyons? Proc. R. Soc. Lond. A 464, 1–24 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  10. Pachos, J. K. The wavefunction of an anyon. Ann. Phys. 322, 1254–1264 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  11. Zhang, C. W., Scarola, V. W., Tewari, S. & Das Sarma, S. Anyonic braiding in optical lattices. Proc. Natl Acad. Sci. USA 104, 18415–18420 (2007).

    Article  ADS  Google Scholar 

  12. Mabuchi, H. & Doherty, A. C. Cavity quantum electrodynamics: Coherence in context. Science 298, 1372–1377 (2002).

    Article  ADS  Google Scholar 

  13. Gupta, S., Moore, K. L., Murch, K. W. & Stamper-Kurn, D. M. Cavity nonlinear optics at low photon numbers from collective atomic motion. Phys. Rev. Lett. 99, 213601 (2007).

    Article  ADS  Google Scholar 

  14. Colombe, Y. et al. Strong atom-field coupling for Bose–Einstein condensates in an optical cavity on a chip. Nature 450, 272–276 (2007).

    Article  ADS  Google Scholar 

  15. Brennecke, F. et al. Cavity QED with a Bose–Einstein condensate. Nature 450, 268–271 (2007).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  17. Lu, C.-Y., Gao, W.-B., Guhne, O., Zhou, X.-Q. & Chen, Z.-B. Demonstration of fractional statistics of anyons in the Kitaev lattice-spin model. Preprint at <http://arxiv.org/abs/0710.0278> (2007).

  18. Pachos, J. K. et al. Revealing anyonic statistics with multiphoton entanglement. Preprint at <http://arxiv.org/abs/0710.0895> (2007).

  19. Han, Y. J., Raussendorf, R. & Duan, L. M. Scheme for demonstration of fractional statistics of anyons in an exactly solvable model. Phys. Rev. Lett. 98, 150404 (2007).

    Article  ADS  MathSciNet  Google Scholar 

  20. Briegel, H. J., Dur, W., Cirac, J. I. & Zoller, P. Quantum repeater: The role of imperfect local operations in quantum communication. Phys. Rev. Lett. 81, 5932–5935 (1998).

    Article  ADS  Google Scholar 

  21. Jiang, L., Taylor, J. M., Khaneja, N. & Lukin, M. D. Optimal approach to quantum communication algorithms using dynamics programming. Proc. Natl Acad. Sci. USA 104, 17291–17296 (2007).

    Article  ADS  Google Scholar 

  22. Kitaev, A. Y. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  23. Dennis, E., Kitaev, A., Landahl, A. & Preskill, J. Topological quantum memory. J. Math. Phys. 43, 4452–4505 (2002).

    Article  ADS  MathSciNet  Google Scholar 

  24. Trotzky, S. et al. Time-resolved observation and control of superexchange interactions with ultracold atoms in optical lattices. Science 319, 295–299 (2008).

    Article  ADS  Google Scholar 

  25. Scully, M. O. & Zubairy, M. S. Quantum Optics (Cambridge Univ. Press, Cambridge, 1997).

    Book  Google Scholar 

  26. Cho, J. Addressing individual atoms in optical lattices with standing-wave driving fields. Phys. Rev. Lett. 99, 020502 (2007).

    Article  ADS  Google Scholar 

  27. Gorshkov, A., Jiang, L., Greiner, M., Zoller, P. & Lukin, M. D. Coherent quantum optical control with sub-wavelength resolution. Phys. Rev. Lett. 100, 093005 (2008).

    Article  ADS  Google Scholar 

  28. Wang, X. & Zanardi, P. Simulation of many-body interactions by conditional geometric phases. Phys. Rev. A 65, 032327 (2002).

    Article  ADS  Google Scholar 

  29. Purcell, E. M. Spontaneous emission probabilities at radio frequencies. Phys. Rev. 69, 681 (1946).

    Article  Google Scholar 

  30. Michler, P. et al. A quantum dot single-photon turnstile device. Science 290, 2282–2285 (2000).

    Article  ADS  Google Scholar 

  31. Vandersypen, L. M. K. & Chuang, I. L. NMR techniques for quantum control and computation. Rev. Mod. Phys. 76, 1037–1069 (2004).

    Article  ADS  Google Scholar 

  32. Brown, K. R., Harrow, A. W. & Chuang, I. L. Arbitrarily accurate composite pulse sequences. Phys. Rev. A 70, 052318 (2004).

    Article  ADS  Google Scholar 

  33. Dusuel, S., Schmidt, K. P. & Vidal, J. Creation and manipulation of anyons in the Kitaev model. Preprint at <http://arxiv.org/abs/0802.0379> (2008).

  34. Duan, L. M., Blinov, B. B., Moehring, D. L. & Monroe, C. Scalable trapped ion quantum computation with a probabilistic ion-photon mapping. Quant. Inf. Comput. 4, 165–173 (2004).

    MathSciNet  MATH  Google Scholar 

  35. Lim, Y. L., Barrett, S. D., Beige, A., Kok, P. & Kwek, L. C. Repeat-until-success quantum computing using stationary and flying qubits. Phys. Rev. A 73, 012304 (2006).

    Article  ADS  Google Scholar 

  36. Benjamin, S. C., Browne, D. E., Fitzsimons, J. & Morton, J. J. L. Brokered graph-state quantum computation. New J. Phys. 8, 141 (2006).

    Article  ADS  Google Scholar 

  37. Jiang, L., Taylor, J. M., Sorensen, A. & Lukin, M. D. Distributed quantum computation based-on small quantum registers. Phys. Rev. A 76, 062323 (2007).

    Article  ADS  Google Scholar 

  38. Arovas, D., Schrieffer, J. R. & Wilczek, F. Fractional statistics and the quantum Hall effect. Phys. Rev. Lett. 53, 722–723 (1984).

    Article  ADS  Google Scholar 

  39. Freedman, M., Nayak, C. & Shtengel, K. Extended NMR model with ring exchange: A route to a non-abelian topological phase. Phys. Rev. Lett. 94, 066401 (2005).

    Article  ADS  Google Scholar 

  40. Doucot, B., Feigel’man, M. V., Ioffe, L. B. & Ioselevich, A. S. Protected qubits and Chern-Simons theories in Josephson junction arrays. Phys. Rev. B 71, 024505 (2005).

    Article  ADS  Google Scholar 

  41. Nielsen, M. A. & Chuang, I. Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge, 2000).

    MATH  Google Scholar 

  42. Levitt, M. H. Spin Dynamics: Basics of Nuclear Magnetic Resonance (Wiley, Chichester, 2001).

    Google Scholar 

  43. Bacon, D. Operator quantum error-correcting subsystems for self-correcting quantum memories. Phys. Rev. A 73, 012340 (2006).

    Article  ADS  Google Scholar 

  44. Milman, P. et al. Topologically decoherence-protected qubits with trapped ions. Phys. Rev. Lett. 99, 020503 (2007).

    Article  ADS  Google Scholar 

  45. Levin, M. A. & Wen, X. G. String-net condensation: A physical mechanism for topological phases. Phys. Rev. B 71, 045110 (2005).

    Article  ADS  Google Scholar 

  46. Hastings, M. B. & Wen, X.-G. Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance. Phys. Rev. B 72, 045141 (2005).

    Article  ADS  Google Scholar 

  47. Aguado, M., Brennen, G. K., Verstraete, F. & Cirac, J. I. Creation, manipulation and detection of anyons in optical lattices. Preprint at <http://arxiv.org/abs/0802.3163> (2008).

  48. Grier, D. G. A revolution in optical manipulation. Nature 424, 810–816 (2003).

    Article  ADS  Google Scholar 

  49. Porto, J. V., Rolston, S., Tolra, B. L., Williams, C. J. & Phillips, W. D. Quantum information with neutral atoms as qubits. Phil. Trans. R. Soc. Lond. A 361, 1417–1427 (2003).

    Article  ADS  Google Scholar 

  50. Bullock, S. S. & Brennen, G. K. Qudit surface codes and gauge theory with finite cyclic groups. J. Phys. A 40, 3481–3505 (2007).

    ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We gratefully acknowledge conversations with H. P. Buchler, S. Dusuel, M. Greiner, L. Ioffe, A. Peng, A. M. Rey and J. Vidal. Work at Harvard is supported by NSF, ARO-MURI, CUA, DARPA, AFOSR and the Packard Foundation. Work at Innsbruck is supported by the Austrian Science Foundation, the EU under grants OLAQUI, SCALA, and the Institute for Quantum Information.

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Authors and Affiliations

  1. Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA

    Liang Jiang, Alexey V. Gorshkov, Mohammad Hafezi, Eugene Demler & Mikhail D. Lukin

  2. Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria

    Gavin K. Brennen, Klemens Hammerer & Peter Zoller

  3. Institute for Quantum Optics and Quantum Information of the Austrian Academy of Science, 6020 Innsbruck, Austria

    Gavin K. Brennen, Klemens Hammerer & Peter Zoller

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Jiang, L., Brennen, G., Gorshkov, A. et al. Anyonic interferometry and protected memories in atomic spin lattices. Nature Phys 4, 482–488 (2008). https://doi.org/10.1038/nphys943

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