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Non-local propagation of correlations in quantum systems with long-range interactions


The maximum speed with which information can propagate in a quantum many-body system directly affects how quickly disparate parts of the system can become correlated1,2,3,4 and how difficult the system will be to describe numerically5. For systems with only short-range interactions, Lieb and Robinson derived a constant-velocity bound that limits correlations to within a linear effective ‘light cone’6. However, little is known about the propagation speed in systems with long-range interactions, because analytic solutions rarely exist and because the best long-range bound7 is too loose to accurately describe the relevant dynamical timescales for any known spin model. Here we apply a variable-range Ising spin chain Hamiltonian and a variable-range XY spin chain Hamiltonian to a far-from-equilibrium quantum many-body system and observe its time evolution. For several different interaction ranges, we determine the spatial and time-dependent correlations, extract the shape of the light cone and measure the velocity with which correlations propagate through the system. This work opens the possibility for studying a wide range of many-body dynamics in quantum systems that are otherwise intractable.

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Figure 1: Sketch of experimental protocol.
Figure 2: Measured quench dynamics in a long-range Ising model.
Figure 3: Measured quench dynamics in a long-range XY model.
Figure 4: Correlations and dynamics beyond the perturbative regime.


  1. Nachtergaele, B., Ogata, Y. & Sims, R. Propagation of correlations in quantum lattice systems. J. Stat. Phys. 124, 1 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  2. Schachenmayer, J., Lanyon, B., Roos, C. & Daley, A. Entanglement growth in quench dynamics with variable range interactions. Phys. Rev. X 3, 031015 (2013)

    CAS  Google Scholar 

  3. Hauke, P. & Tagliacozzo, L. Spread of correlations in long-range interacting quantum systems. Phys. Rev. Lett. 111, 207202 (2013)

    Article  ADS  CAS  Google Scholar 

  4. Hazzard, K. R. A., Manmana, S. R., Foss-Feig, M. & Rey, A. M. Far-from-equilibrium quantum magnetism with ultracold polar molecules. Phys. Rev. Lett. 110, 075301 (2013)

    Article  ADS  Google Scholar 

  5. Eisert, J., Cramer, M. & Plenio, M. Area laws for the entanglement entropy. Rev. Mod. Phys. 82, 277 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  6. Lieb, E. & Robinson, D. The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972)

    Article  ADS  MathSciNet  Google Scholar 

  7. Hastings, M. & Koma, T. Spectral gap and exponential decay of correlations. Commun. Math. Phys. 265, 781–804 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  8. Bravyi, S., Hastings, M. B. & Verstraete, F. Lieb-Robinson bounds and the generation of correlations and topological quantum order. Phys. Rev. Lett. 97, 050401 (2006)

    Article  ADS  CAS  Google Scholar 

  9. Nachtergaele, B. & Sims, R. Lieb-Robinson bound and the exponential clustering theorem. Commun. Math. Phys. 265, 119–130 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  10. Hastings, M. An area law for one-dimensional quantum systems. J. Stat. Mech. 2007, P08024 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Michalakis, S. Stability of the area law for the entropy of entanglement. Preprint at (2012)

  12. Rigol, M., Dunjko, V., Yurovsky, V. & Olshanii, M. Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons. Phys. Rev. Lett. 98, 050405 (2007)

    Article  ADS  Google Scholar 

  13. Calabrese, P. & Cardy, J. Time dependence of correlation functions following a quantum quench. Phys. Rev. Lett. 96, 136801 (2006)

    Article  ADS  Google Scholar 

  14. Gong, Z.-X. & Duan, L.-M. Prethermalization and dynamic phase transition in an isolated trapped ion spin chain. New J. Phys. 15, 113051 (2013)

    Article  ADS  Google Scholar 

  15. Bose, S. Quantum communication through spin chain dynamics: an introductory overview. Contemp. Phys. 48, 13–30 (2007)

    Article  ADS  CAS  Google Scholar 

  16. Cheneau, M. et al. Light-cone-like spreading of correlations in a quantum many-body system. Nature 481, 484–487 (2012)

    Article  ADS  CAS  Google Scholar 

  17. Eisert, J., van den Worm, M., Manmana, S. & Kastner, M. Breakdown of quasi-locality in long-range quantum lattice models. Phys. Rev. Lett. 111, 260401 (2013)

    Article  ADS  Google Scholar 

  18. van den Worm, M., Sawyer, B., Bollinger, J. & Kastner, M. Relaxation timescales and decay of correlations in a long-range interacting quantum simulator. New J. Phys. 15, 083007 (2013)

    Article  ADS  Google Scholar 

  19. Jünemann, J., Cadarso, A., Perez-Garcia, D., Bermudez, A. & Garcia-Ripoll, J. J. Lieb-Robinson bounds for spin-boson lattice models and trapped ions. Phys. Rev. Lett. 111, 230404 (2013)

    Article  ADS  Google Scholar 

  20. Gong, Z.-X., Foss-Feig, M., Michalakis, S. & Gorshkov, A. V. Persistence of locality in systems with power-law interactions. Preprint at (2014)

  21. Jurcevic, P. et al. Quasiparticle engineering and entanglement propagation in a quantum many-body system. Nature (this issue)

  22. Islam, R. et al. Emergence and frustration of magnetic order with variable-range interactions in a trapped ion quantum simulator. Science 340, 583–587 (2013)

    Article  ADS  CAS  Google Scholar 

  23. Richerme, P. et al. Quantum catalysis of magnetic phase transitions in a quantum simulator. Phys. Rev. Lett. 111, 100506 (2013)

    Article  ADS  CAS  Google Scholar 

  24. Olmschenk, S. et al. Manipulation and detection of a trapped Yb+ hyperfine qubit. Phys. Rev. A 76, 052314 (2007)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  27. Kim, K. et al. Entanglement and tunable spin-spin couplings between trapped ions using multiple transverse modes. Phys. Rev. Lett. 103, 120502 (2009)

    Article  ADS  CAS  Google Scholar 

  28. Foss-Feig, M., Hazzard, K. R. A., Bollinger, J. J. & Rey, A. M. Nonequilibrium dynamics of arbitrary-range ising models with decoherence: an exact analytic solution. Phys. Rev. A 87, 042101 (2013)

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  30. Wang, C.-C. J. & Freericks, J. K. Intrinsic phonon effects on analog quantum simulators with ultracold trapped ions. Phys. Rev. A 86, 032329 (2012)

    Article  ADS  Google Scholar 

  31. Shen, C. & Duan, L.-M. Correcting detection errors in quantum state engineering through data processing. New J. Phys. 14, 053053 (2012)

    Article  ADS  Google Scholar 

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We thank J. Preskill, A. M. Rey, K. Hazzard, A. Daley, J. Schachenmayer, M. Kastner, S. Manmana and L.-M. Duan for discussions. This work is supported by the US Army Research Office (ARO) Award W911NF0710576 with funds from the DARPA Optical Lattice Emulator Program, ARO award W911NF0410234 with funds from the IARPA MQCO Program, and the US NSF Physics Frontier Center at JQI. M.F.-F. thanks the NRC for support. S.M. acknowledges funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontier Center with the support of the Gordon and Betty Moore Foundation (through grant GBMF1250).

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Experiments and data analysis were performed by P.R., A.L., C.S., J.S. and C.M. Theoretical calculations were done by Z.-X.G., M.F.-F., S.M., and A.V.G. All authors contributed to the preparation of the manuscript.

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Correspondence to Philip Richerme.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 A long-time partial revival in the long-range Ising model.

a, Spatial correlations measured at long times after a global quench of an Ising model with α = 0.63. b, A small partial revival in correlation between sites 1 and 2 is evident, showing quantum coherence at long times. The black line shows the exact solution predicted from equation (4). Error bars, 1 s.d.

Extended Data Figure 2 Numeric calculation of XY model correlations.

Calculated spatial and time-dependent correlations for an N = 22-spin XY model with spin–spin couplings Jij ≈ J0/|i − j|1.19, found by numerically evolving the Schrödinger equation.

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Richerme, P., Gong, ZX., Lee, A. et al. Non-local propagation of correlations in quantum systems with long-range interactions. Nature 511, 198–201 (2014).

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