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.
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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.).
The authors declare no competing financial interests.
Extended data figures and tables
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.
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).
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.
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Fukuhara, T., Schauß, P., Endres, M. et al. Microscopic observation of magnon bound states and their dynamics. Nature 502, 76–79 (2013). https://doi.org/10.1038/nature12541
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