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Topology by dissipation in atomic quantum wires

Nature Physics volume 7, pages 971–977 (2011)Cite this article

Abstract

Robust edge states and non-Abelian excitations are the trademark of topological states of matter, with promising applications such as ‘topologically protected’ quantum memory and computing. So far, topological phases have been exclusively discussed in a Hamiltonian context. Here we show that such phases and the associated topological protection and phenomena also emerge in open quantum systems with engineered dissipation. The specific system studied here is a quantum wire of spinless atomic fermions in an optical lattice coupled to a bath. The key feature of the dissipative dynamics described by a Lindblad master equation is the existence of Majorana edge modes, representing a non-local decoherence-free subspace. The isolation of the edge states is enforced by a dissipative gap in the p-wave paired bulk of the wire. We describe dissipative non-Abelian braiding operations within the Majorana subspace, and illustrate the insensitivity to imperfections. Topological protection is granted by a non-trivial winding number of the system density matrix.

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Figure 1: Schematic set-up for the dissipative Majorana quantum wire.
Figure 2: Microscopic implementation scheme for the Majorana Liouvillian equations (2), (3).
Figure 3: Characterization of the topological stationary state for a quantum wire with 50 lattice sites without (full circles) and with (open circles) static disorder.
Figure 4: Dissipative braiding.
Figure 5: Visualization of the topological invariant ν for chirally symmetric mixed states.

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References

  1. Kane, C. L. & Mele, E. J. Z 2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).

    Article  ADS  Google Scholar 

  2. Bernevig, B. A., Hughes, T. L. & Zhang, S-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

    Article  ADS  Google Scholar 

  3. Koenig, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).

    Article  ADS  Google Scholar 

  4. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Sarma, S. D. Non-abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  5. Fu, L. & Kane, C. L. Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008).

    Article  ADS  Google Scholar 

  6. Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article  ADS  Google Scholar 

  7. Kitaev, A. Y. Unpaired Majorana fermions in quantum wires. Phys. Usp. 44, 131–136 (2001).

    Article  ADS  Google Scholar 

  8. Sau, J. D., Lutchyn, R. M., Tewari, S. & Sarma, S. D. Generic new platform for topological quantum computation using semiconductor heterostructures. Phys. Rev. Lett. 104, 040502 (2010).

    Article  ADS  Google Scholar 

  9. Akhmerov, A. R., Nilsson, J. & Beenakker, C. W. J. Electrically detected interferometry of Majorana fermions in a topological insulator. Phys. Rev. Lett. 102, 216404 (2009).

    Article  ADS  Google Scholar 

  10. Alicea, J., Oreg, Y., Refael, G., von Oppen, F. & Fisher, M. P. A. Non-abelian statistics and topological quantum information processing in 1D wire networks. Nature Phys. 7, 412–417 (2011).

    Article  ADS  Google Scholar 

  11. Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267–10297 (2000).

    Article  ADS  Google Scholar 

  12. Ivanov, D. A. Non-abelian statistics of half-quantum vortices in p-wave superconductors. Phys. Rev. Lett. 86, 268–271 (2001).

    Article  ADS  Google Scholar 

  13. Lindner, N. H., Refael, G. & Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nature Phys. 7, 490–495 (2011).

    Article  ADS  Google Scholar 

  14. Kitagawa, T., Berg, E., Rudner, M. & Demler, E. Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010).

    Article  ADS  Google Scholar 

  15. Weimer, H., Müller, M., Lesanovsky, I., Zoller, P. & Büchler, H. P. A Rydberg quantum simulator. Nature Phys. 6, 382–388 (2010).

    Article  ADS  Google Scholar 

  16. Barreiro, J. T. et al. An open-system quantum simulator with trapped ions. Nature 470, 486–491 (2011).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  18. Stanescu, T. D., Galitski, V., Vaishnav, J. Y., Clark, C. W. & Sarma, S. D. Topological insulators and metals in atomic optical lattices. Phys. Rev. A 79, 053639 (2009).

    Article  ADS  Google Scholar 

  19. Goldman, N. et al. Realistic time-reversal invariant topological insulators with neutral atoms. Phys. Rev. Lett. 105, 255302 (2010).

    Article  ADS  Google Scholar 

  20. Bermudez, A., Goldman, N., Kubasiak, A., Lewenstein, M. & Martin-Delgado, M. A. Topological phase transitions in the non-Abelian honeycomb lattice. New J. Phys. 12, 033041 (2010).

    Article  ADS  Google Scholar 

  21. Jiang, L. et al. Majorana fermions in equilibrium and driven cold atom quantum wires. Phys. Rev. Lett. 106, 220402 (2011).

    Article  ADS  Google Scholar 

  22. Diehl, S. et al. Quantum states and phases in driven open quantum systems with cold atoms. Nature Phys. 4, 878–883 (2008).

    Article  ADS  Google Scholar 

  23. Kraus, B. et al. Preparation of entangled states by quantum Markov processes. Phys. Rev. A 78, 042307 (2008).

    Article  ADS  Google Scholar 

  24. Verstraete, F., Wolf, M. M. & Cirac, J. I. Quantum computation, quantum state engineering, and quantum phase transitions driven by dissipation. Nature Phys. 5, 633–636 (2009).

    Article  ADS  Google Scholar 

  25. Diehl, S., Yi, W., Daley, A. J. & Zoller, P. Dissipation-induced d-wave pairing of fermionic atoms in an optical lattice. Phys. Rev. Lett. 105, 227001 (2010).

    Article  ADS  Google Scholar 

  26. Eisert, J. & Prosen, T. Noise-driven quantum criticality. Preprint at http://arxiv.org/abs/1012.5013 (2010).

  27. Prosen, T. Third quantization: A general method to solve master equations for quadratic open Fermi systems. New J. Phys. 10, 043026 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  28. Prosen, T. Spectral theorem for the Lindblad equation for quadratic open fermionic systems. J. Stat. Mech. P07020 (2010).

  29. Lidar, D. A., Chuang, I. L. & Whaley, K. B. Decoherence free subspaces for quantum computation. Phys. Rev. Lett. 81, 2594–2597 (1998).

    Article  ADS  Google Scholar 

  30. Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. A 392, 45–57 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  31. Simon, B. Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys. Rev. Lett. 51, 2167–2170 (1983).

    Article  ADS  MathSciNet  Google Scholar 

  32. Wilczek, F. & Zee, A. Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett. 52, 2111–2114 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  33. Pachos, J., Zanardi, P. & Rasetti, M. Non-abelian Berry connections for quantum computation. Phys. Rev. A 61, 010305 (1999).

    Article  MathSciNet  Google Scholar 

  34. Carollo, A., Fuentes-Guridi, I., Franca Santos, M. & Vedral, V. Geometric phase in open systems. Phys. Rev. Lett. 90, 160402 (2003).

    Article  ADS  MathSciNet  Google Scholar 

  35. Beige, A., Braun, D., Tregenna, B. & Knight, P. L. Quantum computing using dissipation to remain in a decoherence-free subspace. Phys. Rev. Lett. 85, 1762–1765 (2000).

    Article  ADS  Google Scholar 

  36. Bravyi, S. Universal quantum computation with the ν=5/2 fractional quantum Hall state. Phys. Rev. B 73, 042313 (2006).

    Article  ADS  Google Scholar 

  37. Freedman, M., Nayak, C. & Walker, K. Towards universal topological quantum computation in the ν=5/2 fractional quantum Hall state. Phys. Rev. B 73, 245307 (2006).

    Article  ADS  Google Scholar 

  38. Altland, A. & Zirnbauer, M. R. Nonstandard symmetry classes in mesoscopic normal–superconducting hybrid structures. Phys. Rev. B 55, 1142–1161 (1997).

    Article  ADS  Google Scholar 

  39. Ryu, S., Schnyder, A., Furusaki, A. & Ludwig, A. W. W. Topological insulators and superconductors: Ten-fold way and dimensional hierarchy. New J. Phys. 12, 065010 (2010).

    Article  ADS  Google Scholar 

  40. Rudner, M. S. & Levitov, L. S. Topological transition in a non-Hermitian quantum walk. Phys. Rev. Lett. 102, 065703 (2009).

    Article  ADS  Google Scholar 

  41. Diehl, S., Tomadin, A., Micheli, A., Fazio, R. & Zoller, P. Dynamical phase transitions and instabilities in open atomic many-body systems. Phys. Rev. Lett. 105, 015702 (2010).

    Article  ADS  Google Scholar 

  42. Gurarie, V. Single-particle Green’s functions and interacting topological insulators. Phys. Rev. B 83, 085426 (2011).

    Article  ADS  Google Scholar 

  43. Bakr, W. S., Gillen, J. I., Peng, A., Foelling, S. & Greiner, M. A quantum gas microscope for detecting single atoms in a Hubbard regime optical lattice. Nature 462, 74–77 (2009).

    Article  ADS  Google Scholar 

  44. Bakr, W. S. et al. Probing the superfluid-to-Mott insulator transition at the single-atom level. Science 329, 547–550 (2010).

    Article  ADS  Google Scholar 

  45. Sherson, J. F. et al. Single-atom resolved fluorescence imaging of an atomic Mott insulator. Nature 467, 68–72 (2010).

    Article  ADS  Google Scholar 

  46. Weitenberg, C. et al. Single-spin addressing in an atomic Mott insulator. Nature 471, 319–324 (2011).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank C. Bardyn, V. Gurarie, A. Imamoglu, C. Kraus and M. Troyer for helpful discussions. We acknowledge support by the Austrian Science Fund (FOQUS), the European Commission (AQUTE, NAMEQUAM), the Institut für Quanteninformation GmbH and a grant from the US Army Research Office with funding from the Defense Advanced Research Projects Agency OLE program.

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Authors and Affiliations

  1. Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria

    Sebastian Diehl, Enrique Rico, Mikhail A. Baranov & Peter Zoller

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

    Sebastian Diehl, Enrique Rico, Mikhail A. Baranov & Peter Zoller

  3. RRC ‘Kurchatov Institute’, Kurchatov Square 1, 123182 Moscow, Russia

    Mikhail A. Baranov

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Correspondence to Sebastian Diehl.

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Diehl, S., Rico, E., Baranov, M. et al. Topology by dissipation in atomic quantum wires. Nature Phys 7, 971–977 (2011). https://doi.org/10.1038/nphys2106

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