Abstract
One-dimensional systems exhibiting a continuous symmetry can host quantum phases of matter with true long-range order only in the presence of sufficiently long-range interactions1. In most physical systems, however, the interactions are short-ranged, hindering the emergence of such phases in one dimension. Here we use a one-dimensional trapped-ion quantum simulator to prepare states with long-range spin order that extends over the system size of up to 23 spins and is characteristic of the continuous symmetry-breaking phase of matter2,3. Our preparation relies on simultaneous control over an array of tightly focused individual addressing laser beams, generating long-range spin–spin interactions. We also observe a disordered phase with frustrated correlations. We further study the phases at different ranges of interaction and the out-of-equilibrium response to symmetry-breaking perturbations. This work opens an avenue to study new quantum phases and out-of-equilibrium dynamics in low-dimensional systems.
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Data availability
Data that support the findings of this study are available from the corresponding authors on reasonable request.
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Acknowledgements
This work is supported in part by the ARO through the IARPA LogiQ programme; the NSF STAQ, QLCI, RAISE-TAQS and QIS programmes; the AFOSR MURIs on Dissipation Engineering in Open Quantum Systems and on Quantum Verification Protocols; the ARO MURI on Modular Quantum Circuits; the DoE ASCR Quantum Testbed Pathfinder programme; NSF CAREER; AFOSR YIP; and the W. M. Keck Foundation. Support is also acknowledged from the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator.
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All authors contributed to the experimental design, construction and discussions, and wrote the manuscript. O.K. and L.F. collected the data, and L.F. analysed the results. C.H., Z.-X.G., M.M. and A.V.G. performed analytical and numerical calculation.
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C.M. is the chief scientist for IonQ, Inc. and has a personal financial interest in the company. The other authors declare no competing interests.
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Extended data figures and tables
Extended Data Fig. 1 Partial reconstruction of the interaction matrix in the second experimental configuration.
The experimentally reconstructed Jij matrix is shown for a second experimental configuration up to five nearest neighbors. The full modeled interaction is detailed in the Methods section and is shown in Extended Data Fig. 5. This matrix exhibits long-range interaction, which is nevertheless shorter than the interaction in Fig. 1c.
Extended Data Fig. 2 Decoherence rate matrix.
The measured decoherence rate matrix Γij is extracted from the reconstruction protocol for up to five nearest neighbors (see Methods). a, The measured relaxation accompanying the interaction matrix in the first configuration (Fig. 1c). b, The measured relaxation accompanying the interaction matrix in the second configuration (Extended Data Fig. 1).
Extended Data Fig. 3 Demonstration of the reconstruction protocol.
We measure the interactions between the i = -6 ion with its up to five-nearest neighbors, by turning on the a single pair of beams addressing two ions a time. Specifically (i, j) = (-6,-5) in a, (-6,-4) in b, (-6,-3) in c, and (-6,-2) in d and (-6,-1) in e, as indicated by a dark blue sphere. We fit the staggered magnetization \({m}_{s}=\frac{1}{2}\langle {\sigma }_{z}^{i}-{\sigma }_{z}^{j}\rangle \) to the function \(y=\cos (\pi {g}^{2}{J}_{ij}t){e}^{-{g}^{2}{\Gamma }_{ij}t}\) to extract the interaction strength Jij and the decoherence rate Γij. The interaction rate as a function of the inter-ion spacing is shown in f. The fitted Jij for i = − 6 (circles). The black line corresponds to the fit function in Eq. (5) with fitting parameters \({\alpha }^{{\prime} }=0.44,\,{\beta }^{{\prime} }=0.19\). The errorbars indicate one standard error of the mean. The exemplary data in this figure corresponds to the interaction matrix in Fig. 1c with a scaling factor g = 1.3; see Methods.
Extended Data Fig. 4 Average magnetization.
The measured regular and staggered magnetization along the x (a), y (b), and z (c) axes for the first configuration (with interaction matrix in Fig. 1c). The errorbars indicate one standard error of the mean.
Extended Data Fig. 5 Modeled interaction and measured correlations.
a, Numerically calculated spin-spin interaction based on Eq. (4) and a simple model of the trapping potential for the three experimental configurations, assuming a harmonic trap for the radial coordinates and a combination of quadratic and quartic potentials for the axial direction. The filled circles indicate the numerically calculated values with no free parameters. The solid lines are fits to the numerical results with a profile of \(J(l)=\bar{J}{e}^{-{\beta }^{{\prime} }(l-1)}{l}^{-{\alpha }^{{\prime} }}\). The fitted parameters are \({\alpha }^{{\prime} }=0.44,\,{\beta }^{{\prime} }=0.19\) for the first configuration (purple), \({\alpha }^{{\prime} }=1,\,{\beta }^{{\prime} }=0.19\) for the second configuration (black), and \({\alpha }^{{\prime} }=3.4,\,{\beta }^{{\prime} }=0\) for the third configuration (blue). Open squares are the experimental data in two out of the three experimental configurations, where the bars represent the spread of measured values of all pairs at a specific spacing l, namely one standard error of the mean. b, Measured spin correlation for the first (left) and the third (right) configuration for the state prepared at the end of the ramp. c, Comparison between the spatially averaged correlations CN(l) for N = 23 as a function of inter-spin distance l for the long-range (purple) and the short-range (blue) configurations.
Extended Data Fig. 6 Numerical simulation of the experiment in the first configuration.
a and b show the comparison of the correlation functions CN(l) between the experimental data (filled) and numerical results (unfilled) including modeled decoherence. In a, the system size is N = 7, and, in b, it is N = 11.
Extended Data Fig. 7 Numerical simulation of the quench dynamics.
We simulate the unitary quench evolution in Fig. 4 (i.e. without considering any decoherence processes). a, Measured spin-spin correlations \(\langle {\widehat{\sigma }}_{x}^{(i)}{\widehat{\sigma }}_{x}^{(j)}\rangle \) developed during the evolution by Eq. (3) for time τ. b, Average correlation as a function of time. The dots in different colors correspond to the correlation averaged within the corresponding colored contours shown in a (top left). The fast oscillations are due to the large but finite longitudinal magnetic field B.
Extended Data Fig. 8 Long-range correlation.
The correlation matrix \({C}_{ij}^{k}=\langle {\widehat{\sigma }}_{k}^{(i)}{\widehat{\sigma }}_{k}^{(j)}\rangle \), with (a) k = x or (b) k = y, measured in the x and y basis, respectively, for various system sizes of N = 7, 11, 15, 19, and 23. This data corresponds to the first experimental configuration (purple curve in Extended Data Fig. 5a) that leads to the correlations presented in Fig. 3 via the relation \({C}_{ij}=\frac{1}{2}({C}_{ij}^{x}+{C}_{ij}^{y})\).
Extended Data Fig. 9 Estimation of effective temperature.
Comparison of the correlations CN(l) between the measured state (Fig. 3b) (purple) and a numerically calculated thermal-state (black) for N = 11. The thermal state has an effective temperature \({k}_{{\rm{B}}}{\mathcal{T}}\approx 0.07{\bar{E}}_{{\rm{int}}}\), with \({\bar{E}}_{{\rm{int}}}=\frac{1}{N}{\sum }_{i\ne j}{J}_{i,j}\) denoting the average interaction energy. Here we consider the second experimental configuration presented as a black curve in Extended Data Fig. 5a.
Extended Data Fig. 10 Simulation of larger spin chains.
Numerical simulation of the order parameter \({\mathcal{M}}(N)\) of the CSB phase in the ground state of − H (s = 1) for spin chains of different sizes (N = 23 in red, 49 in magenta, and 89 in blue) as a function of the sideband detuning Δ. The sharp decrease in \({\mathcal{M}}\) at large Δ (short interaction range) compared to that at small Δ (large interaction range) indicates a phase transition from a disordered phase to the CSB phase.
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Feng, L., Katz, O., Haack, C. et al. Continuous symmetry breaking in a trapped-ion spin chain. Nature 623, 713–717 (2023). https://doi.org/10.1038/s41586-023-06656-7
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DOI: https://doi.org/10.1038/s41586-023-06656-7
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