Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Quantum simulation of dynamical maps with trapped ions

Abstract

Dynamical maps describe general transformations of the state of a physical system—their iteration interpreted as generating a discrete time evolution. Prime examples include classical nonlinear systems undergoing transitions to chaos. Quantum mechanical counterparts show intriguing phenomena such as dynamical localization on the single-particle level. Here we extend the concept of dynamical maps to a many-particle context, where the time evolution involves both coherent and dissipative elements: we experimentally explore the stroboscopic dynamics of a complex many-body spin model with a universal trapped ion quantum simulator. We generate long-range phase coherence of spin by an iteration of purely dissipative quantum maps and demonstrate the characteristics of competition between combined coherent and dissipative non-equilibrium evolution—the hallmark of a previously unobserved dynamical phase transition. We assess the influence of experimental errors in the quantum simulation and tackle this problem by developing an efficient error detection and reduction toolbox based on quantum feedback.

This is a preview of subscription content

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Figure 1: Competing dissipative and Hamiltonian dynamical maps in the spin or hard-core boson model.
Figure 2: Experimental procedure to implement open-system dynamical maps.
Figure 3: Experimental results of dissipatively induced delocalization through composite dynamical maps with 3+1 ions.
Figure 4: Experimental results for competing dissipative and coherent dynamics with 3+1 and 4+1 ions.
Figure 5: Experimental error detection and reduction techniques.

References

  1. Maze, J. R. et al. Nanoscale magnetic sensing with an individual electronic spin in diamond. Nature 455, 644–647 (2008).

    ADS  Article  Google Scholar 

  2. Ladd, T. D. et al. Quantum computers. Nature 464, 45–53 (2010).

    ADS  Article  Google Scholar 

  3. Wrachtrup, J. & Jelezko, F. Processing quantum information in diamond. J. Phys. Condens. Matter 18, 807–824 (2006).

    ADS  Article  Google Scholar 

  4. Clarke, J. & Wilhelm, F. K. Superconducting quantum bits. Nature 453, 1031–1042 (2008).

    ADS  Article  Google Scholar 

  5. O’Brien, J. L. Optical quantum computing. Science 318, 1567–1570 (2007).

    ADS  Article  Google Scholar 

  6. Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in few-electron quantum dots. Rev. Mod. Phys. 79, 1217–1265 (2007).

    ADS  Article  Google Scholar 

  7. Schneider, C., Porras, D. & Schätz, T. Experimental quantum simulations of many-body physics with trapped ions. Rep. Prog. Phys. 75 024401 (2012).

  8. Saffman, M., Walker, T. G. & Mølmer, K. Quantum information with Rydberg atoms. Rev. Mod. Phys. 82, 2313–2363 (2010).

    ADS  Article  Google Scholar 

  9. Bloch, I., Dalibard, J. & Nascimbène, S. Quantum simulations with ultracold quantum gases. Nature Phys. 8, 267–276 (2012).

    ADS  Article  Google Scholar 

  10. Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. Nature Phys. 8, 277–284 (2012).

    ADS  Article  Google Scholar 

  11. Aspuru-Guzik, A. & Walther, P. Photonic quantum simulators. Nature Phys. 8, 285–291 (2012).

    ADS  Article  Google Scholar 

  12. Houck, A. A., Türeci, H. E. & Koch, J. On-chip quantum simulation with superconducting circuits. Nature Phys. 8, 292–299 (2012).

    ADS  Article  Google Scholar 

  13. Bacon, D. et al. Universal simulation of Markovian quantum dynamics. Phys. Rev. A 64, 062302 (2001).

    ADS  Article  Google Scholar 

  14. Lloyd, S. & Viola, L. Engineering quantum dynamics. Phys. Rev. A 65, 010101 (2001).

    MathSciNet  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  16. Baggio, G., Ticozzi, F. & Viola, L. in 2012 IEEE 51st Annual Conference on Decision and Control (CDC) 1072–1077 (IEEE, 2012).

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

    MATH  Google Scholar 

  18. Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996).

    ADS  MathSciNet  Article  Google Scholar 

  19. Lanyon, B. P. et al. Universal digital quantum simulation with trapped ions. Science 334, 57–61 (2011).

    ADS  Article  Google Scholar 

  20. Zhang, J., Yung, M-H., Laflamme, R., Aspuru-Guzik, A. & Baugh, J. Digital quantum simulation of the statistical mechanics of a frustrated magnet. Nature Commun. 3, 880 (2012).

    ADS  Article  Google Scholar 

  21. Poyatos, J. F., Cirac, J. I. & Zoller, P. Quantum reservoir engineering with laser cooled trapped ions. Phys. Rev. Lett. 77, 4728–4731 (1996).

    ADS  Article  Google Scholar 

  22. Cho, J., Bose, S. & Kim, M. S. Optical pumping into many-body entanglement. Phys. Rev. Lett. 106, 020504 (2011).

    ADS  Article  Google Scholar 

  23. Kastoryano, M. J., Reiter, F. & Sørensen, A. S. Dissipative preparation of entanglement in optical cavities. Phys. Rev. Lett. 106, 090502 (2011).

    ADS  Article  Google Scholar 

  24. Krauter, H. et al. Entanglement generated by dissipation and steady state entanglement of two macroscopic objects. Phys. Rev. Lett. 107, 080503 (2011).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  27. Pastawski, F., Clemente, L. & Cirac, J. I. Quantum memories based on engineered dissipation. Phys. Rev. A 83, 012304 (2011).

    ADS  Article  Google Scholar 

  28. Kliesch, M., Barthel, T., Gogolin, C., Kastoryano, M. & Eisert, J. Dissipative quantum Church-Turing theorem. Phys. Rev. Lett. 107, 120501 (2011).

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  31. Diehl, S., Rico, E., Baranov, M. A. & Zoller, P. Topology by dissipation in atomic quantum wires. Nature Phys. 7, 971–977 (2011).

    ADS  Article  Google Scholar 

  32. Gardiner, C. W. & Zoller, P. Quantum Noise (Springer, 1999).

    MATH  Google Scholar 

  33. Reichl, L. E. The Transition to Chaos In Conservative Classical Systems: Quantum Manifestations (Springer, 1992).

    Book  Google Scholar 

  34. Chirikov, B. V. A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263–379 (1979).

    ADS  MathSciNet  Article  Google Scholar 

  35. Izrailev, F. M. Simple models of quantum chaos: spectrum and eigenfunctions. Phys. Rep. 196, 299–392 (1990).

    ADS  MathSciNet  Article  Google Scholar 

  36. Haake, F. Quantum Signatures of Chaos (Synergetics Series, Springer, 2010).

    Book  Google Scholar 

  37. Moore, F. L., Robinson, J. C., Bharucha, C. F., Sundaram, B. & Raizen, M. G. Atom optics realization of the quantum delta-kicked rotor. Phys. Rev. Lett. 75, 4598–4601 (1995).

    ADS  Article  Google Scholar 

  38. Ammann, H., Gray, R., Shvarchuck, I. & Christensen, N. Quantum delta-kicked rotor: Experimental observation of decoherence. Phys. Rev. Lett. 80, 4111–4115 (1998).

    ADS  Article  Google Scholar 

  39. d’Arcy, M. B., Godun, R. M., Oberthaler, M. K., Cassettari, D. & Summy, G. S. Quantum enhancement of momentum diffusion in the delta-kicked rotor. Phys. Rev. Lett. 87, 074102 (2001).

    ADS  Article  Google Scholar 

  40. Henderson, K., Kelkar, H., Li, T. C., Gutierrez-Medina, G. & Raizen, M. G. Bose–Einstein condensate driven by a kicked rotor in a finite box. Europhys. Lett. 75, 392 (2006).

    ADS  Article  Google Scholar 

  41. Porras, D. & Cirac, J. I. Effective quantum spin systems with trapped ions. Phys. Rev. Lett. 92, 207901 (2004).

    ADS  Article  Google Scholar 

  42. Friedenauer, A., Schmitz, H., Glueckert, J. T., Porras, D. & Schaetz, T. Simulating a quantum magnet with trapped ions. Nature Phys. 4, 757–761 (2008).

    ADS  Article  Google Scholar 

  43. Kim, K. et al. Quantum simulation of frustrated Ising spins with trapped ions. Nature 465, 590–593 (2010).

    ADS  Article  Google Scholar 

  44. Islam, R. et al. Onset of a quantum phase transition with a trapped ion quantum simulator. Nature Commun. 2, 377 (2011).

    ADS  Article  Google Scholar 

  45. Britton, J. W. et al. Engineered two-dimensional Ising interactions in a trapped-ion quantum simulator with hundreds of spins. Nature 484, 489–492 (2012).

    ADS  Article  Google Scholar 

  46. 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).

    ADS  Article  Google Scholar 

  47. Sachdev, S. Quantum Phase Transitions (Cambridge Univ. Press, 1999).

    MATH  Google Scholar 

  48. Mølmer, K. & Sørensen, A. Multiparticle entanglement of hot trapped ions. Phys. Rev. Lett. 82, 1835–1838 (1999).

    ADS  Article  Google Scholar 

  49. Kielpinski, D., Monroe, C. & Wineland, D. J. Architecture for a large-scale ion-trap quantum computer. Nature 417, 709–711 (2002).

    ADS  Article  Google Scholar 

  50. Sayrin, C. et al. Real-time quantum feedback prepares and stabilizes photon number states. Nature 477, 73–77 (2011).

    ADS  Article  Google Scholar 

Download references

Acknowledgements

We gratefully acknowledge support by the Austrian Science Fund (FWF), through the SFB FoQus (FWF Project No. F4002-N16 and F4016-N16) and the START grant Y 581-N16 (S.D.), by the European Commission (AQUTE), as well as the Institut für Quantenoptik und Quanteninformation GmbH. This research was funded by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the Army Research Office grant W911NF-10-1-0284. All statements of fact, opinion or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of IARPA, the ODNI or the US Government. M.M. acknowledges support by the CAM research consortium QUITEMAD S2009-ESP-1594, European Commission PICC: FP7 2007-2013, Grant No. 249958, and the Spanish MICINN grant FIS2009-10061.

Author information

Authors and Affiliations

Authors

Contributions

M.M., P.S., J.T.B. and S.D. developed the research, based on theoretical ideas conceived with P.Z.; P.S. and D.N. performed the experiments; P.S. and T.M. analysed the data; P.S., J.T.B., D.N., T.M., E.A.M., M.H. and R.B. contributed to the experimental set-up; P.S., M.M. and S.D wrote the manuscript, with revisions provided by J.T.B., P.Z. and R.B; all authors contributed to the discussion of the results and manuscript.

Corresponding authors

Correspondence to P. Zoller or R. Blatt.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 1937 kb)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Schindler, P., Müller, M., Nigg, D. et al. Quantum simulation of dynamical maps with trapped ions. Nature Phys 9, 361–367 (2013). https://doi.org/10.1038/nphys2630

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys2630

Further reading

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing