Skip to main content

Thank you for visiting 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.

  • Letter
  • Published:

Microscopic observation of magnon bound states and their dynamics


The existence of bound states of elementary spin waves (magnons) in one-dimensional quantum magnets was predicted almost 80 years ago1. Identifying signatures of magnon bound states has so far remained the subject of intense theoretical research2,3,4,5, and their detection has proved challenging for experiments. Ultracold atoms offer an ideal setting in which to find such bound states by tracking the spin dynamics with single-spin and single-site resolution6,7 following a local excitation8. Here we use in situ correlation measurements to observe two-magnon bound states directly in a one-dimensional Heisenberg spin chain comprising ultracold bosonic atoms in an optical lattice. We observe the quantum dynamics of free and bound magnon states through time-resolved measurements of two spin impurities. The increased effective mass of the compound magnon state results in slower spin dynamics as compared to single-magnon excitations. We also determine the decay time of bound magnons, which is probably limited by scattering on thermal fluctuations in the system. Our results provide a new way of studying fundamental properties of quantum magnets and, more generally, properties of interacting impurities in quantum many-body systems.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Schematic representation of magnon propagation.
Figure 2: Spatial correlations after dynamical evolution.
Figure 3: Spreading wavefront velocity of bound and free magnons.
Figure 4: Stability of the bound state.

Similar content being viewed by others


  1. Bethe, H. A. Zur Theorie der Metalle. Z. Phys. 71, 205–226 (1931)

    Article  ADS  CAS  Google Scholar 

  2. Caux, J.-S. & Maillet, J. M. Computation of dynamical correlation functions of Heisenberg chains in a magnetic field. Phys. Rev. Lett. 95, 077201 (2005)

    Article  ADS  Google Scholar 

  3. Pereira, R. G., White, S. R. & Affleck, I. Exact edge singularities and dynamical correlations in spin-1/2 chains. Phys. Rev. Lett. 100, 027206 (2008)

    Article  ADS  Google Scholar 

  4. Kohno, M. Dynamically dominant excitations of string solutions in the spin-1/2 antiferromagnetic Heisenberg chain in a magnetic field. Phys. Rev. Lett. 102, 037203 (2009)

    Article  ADS  Google Scholar 

  5. Imambekov, A., Schmidt, T. L. & Glazman, L. I. One-dimensional quantum liquids: beyond the Luttinger liquid paradigm. Rev. Mod. Phys. 84, 1253–1306 (2012)

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  8. Ganahl, M., Rabel, E., Essler, F. & Evertz, H. Observation of complex bound states in the spin-1/2 Heisenberg XXZ chain using local quantum quenches. Phys. Rev. Lett. 108, 077206 (2012)

    Article  ADS  Google Scholar 

  9. Wortis, M. Bound states of two spin waves in the Heisenberg ferromagnet. Phys. Rev. 132, 85–97 (1963)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  10. Takahashi, M. One-dimensional Heisenberg model at finite temperature. Prog. Theor. Phys. 46, 401–415 (1971)

    Article  ADS  Google Scholar 

  11. Hanus, J. Bound states in the Heisenberg ferromagnet. Phys. Rev. Lett. 11, 336–338 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  12. Fogedby, H. C. The spectrum of the continuous isotropic quantum Heisenberg chain: quantum solitons as magnon bound states. J. Phys. C 13, L195–L200 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  13. Schneider, T. Solitons and magnon bound states in ferromagnetic Heisenberg chains. Phys. Rev. B 24, 5327–5339 (1981)

    Article  ADS  CAS  Google Scholar 

  14. Schreiber, A. et al. A 2D quantum walk simulation of two-particle dynamics. Science 336, 55–58 (2012)

    Article  ADS  CAS  Google Scholar 

  15. Lahini, Y. et al. Quantum walk of two interacting bosons. Phys. Rev. A 86, 011603(R) (2012)

    Article  ADS  Google Scholar 

  16. Venegas-Andraca, S. E. Quantum walks: a comprehensive review. Quant. Inf. Proc. 11, 1015–1106 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  18. Subrahmanyam, V. Entanglement dynamics and quantum-state transport in spin chains. Phys. Rev. A 69, 034304 (2004)

    Article  ADS  Google Scholar 

  19. Batchelor, T. The Bethe ansatz after 75 years. Phys. Today 60, 36–40 (2007)

    Article  ADS  Google Scholar 

  20. Date, M. & Motokawa, M. Spin-cluster resonance in CoCl2·2H2O. Phys. Rev. Lett. 16, 1111–1114 (1966)

    Article  ADS  CAS  Google Scholar 

  21. Torrance, J. B. & Tinkham, M. Excitation of multiple-magnon bound states in CoCl2·2H2O. Phys. Rev. 187, 595–606 (1969)

    Article  ADS  CAS  Google Scholar 

  22. Hoogerbeets, R., van Duyneveldt, A. J., Phaff, A. C., Swüste, C. H. W. & de Jonge, W. J. M. Evidence for magnon bound-state excitations in the quantum chain system (C6H11NH3)CuCl3 . J. Phys. C 17, 2595–2608 (1984)

    Article  ADS  CAS  Google Scholar 

  23. Winkler, K. et al. Repulsively bound atom pairs in an optical lattice. Nature 441, 853–856 (2006)

    Article  ADS  CAS  Google Scholar 

  24. Fölling, S. et al. Direct observation of second-order atom tunnelling. Nature 448, 1029–1032 (2007)

    Article  ADS  Google Scholar 

  25. Kuklov, A. & Svistunov, B. Counterflow superfluidity of two-species ultracold atoms in a commensurate optical lattice. Phys. Rev. Lett. 90, 100401 (2003)

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  28. Fukuhara, T. et al. Quantum dynamics of a mobile spin impurity. Nature Phys. 9, 235–241 (2013)

    Article  ADS  CAS  Google Scholar 

  29. 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  CAS  Google Scholar 

  30. Nishida, Y., Kato, Y. & Batista, C. D. Efimov effect in quantum magnets. Nature Phys. 9, 93–97 (2013)

    Article  ADS  CAS  Google Scholar 

  31. García-Ripoll, J. J. & Cirac, J. I. Spin dynamics for bosons in an optical lattice. New J. Phys. 5, 76 (2003)

    Article  Google Scholar 

  32. Altman, E., Hofstetter, W., Demler, E. & Lukin, M. D. Phase diagram of two-component bosons on an optical lattice. New J. Phys. 5, 113 (2003)

    Article  ADS  Google Scholar 

  33. Pertot, D., Gadway, B. & Schneble, D. Collinear four-wave mixing of two-component matter waves. Phys. Rev. Lett. 104, 200402 (2010)

    Article  ADS  Google Scholar 

  34. Hoefer, M. A., Chang, J. J., Hamner, C. & Engels, P. Dark-dark solitons and modulational instability in miscible two-component Bose-Einstein condensates. Phys. Rev. A 84, 041605 (2011)

    Article  ADS  Google Scholar 

  35. Karbach, M. & Müller, G. Introduction to the Bethe ansatz I. Comput. Phys. 11, 36–43 (1997)

    Article  ADS  Google Scholar 

Download references


We thank H. G. Evertz, M. Haque, J.-S. Caux and W. Zwerger for discussions. We thank J. Zeiher for proofreading the manuscript. This work was supported by MPG, DFG, EU (NAMEQUAM, AQUTE, Marie Curie Fellowship to M.C.) and JSPS (Postdoctoral Fellowship for Research Abroad to T.F.).

Author information

Authors and Affiliations



All authors contributed extensively to the work presented in this paper.

Corresponding author

Correspondence to Takeshi Fukuhara.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Quantum state analysis through the Bethe Ansatz.

a, Overlap of the initial state with the bound (red circles) and free (black dots) magnon states calculated for N = 16 lattice sites35. For illustration, we show only the states with wavevectors within the interval k [0,π/alat]. The inset b shows the corresponding energy spectrum. c, Spin–spin correlation of the magnon bound states as a function of the spin separation d = |j − i| for different wavevectors k. For k = π/alat the wavefunction of the bound magnon state corresponds to tightly bound spins on neighbouring sites, giving the largest overlap with our initial state.

Extended Data Figure 2 Propagation of bound magnons.

a, Calculated probability distribution (black lines) together with the Bessel function fit (green lines) for different evolution times (40, 80 and 120 ms). The red vertical lines show the width extracted from the fit. b, Determination of the velocity. The red line is the extracted width from the Bessel function fits for different evolution times. The black line corresponds to the expected maximum velocity (Jexalat/2).

Extended Data Figure 3 Propagation of free magnons.

Main figure, the blue and red lines show the Gaussian centre c and the centre plus the width, c + s. Note that the centre moves more slowly than the maximum wavefront velocity. The black line, almost overlapping with the red line, corresponds to twice the expected single magnon maximum velocity (2Jexalat/). Inset, an example of the Gaussian fit (green line). The grey circles represent the calculated correlation function for the evolution time of 80 ms. The blue shading highlights the region used for the fit.

PowerPoint slides

Source data

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fukuhara, T., Schauß, P., Endres, M. et al. Microscopic observation of magnon bound states and their dynamics. Nature 502, 76–79 (2013).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


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