Abstract
The topological θ-angle is central to several gauge theories in condensed-matter and high-energy physics. For example, it is responsible for the strong CP problem in quantum chromodynamics and can emerge in effective theories of electrodynamics in topological insulators. Although analogue quantum simulators potentially offer a venue for realizing and controlling the θ-angle, doing so has hitherto remained an outstanding challenge. Here, we describe the experimental realization of a tunable topological θ-angle in a Bose–Hubbard gauge-theory quantum simulator, which was implemented through a tilted superlattice potential that induces an effective background electric field. We demonstrate the emerging physics through the direct observation of the confinement–deconfinement transition of (1 + 1)-dimensional quantum electrodynamics. Using an atomic-precision quantum gas microscope, we distinguish between the confined and deconfined phases by monitoring the real-time evolution of particle–antiparticle pairs. Our work provides a step forward in the realization of topological terms on modern quantum simulators.
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Data availability
Source data are provided within this paper. The data for the figures that support the findings of this study are also available from Zenodo at https://doi.org/10.5281/zenodo.13733573 (ref. 52).
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Acknowledgements
We thank H. Zhai for discussions. This work was supported by the NNSFC (Grant No. 12125409), the Innovation Programme for Quantum Science and Technology (Grant No. 2021ZD0302000) and an Anhui Provincial Major Science and Technology Project (Grant No. 202103a13010005). W.-Y.Z. acknowledges support from the Postdoctoral Fellowship Programme of CPSF (Grant No. GZC20241659). Y.C. acknowledges support from the NSFC (Grant Nos. 12204034 and 12374251) and Fundamental Research Funds for the Central Universities (Grant No. FRFTP-22-101A1). J.C.H. acknowledges funding by the Max Planck Society, the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2111 – 390814868, and the European Research Council (ERC) under the European Union’s Horizon Europe research and innovation program (Grant Agreement No. 101165667)—ERC Starting Grant QuSiGauge. B.Y. acknowledges the National Key R&D Programme of China (Grant No. 2022YFA1405800), the NNSFC (Grant No. 12274199) and a Guangdong Major Project of Basic and Applied Basic Research (Grant No. 2023B0303000011). W.Z. acknowledges support from the NSFC (Grant Nos. GG2030007011 and GG2030040453) and the Innovation Programme for Quantum Science and Technology (Grant No. 2021ZD0302004). P.H. acknowledges funding from the European Union through the HE research and innovation programme (Grant No. GA 101080086 NeQST) and the Next Generation EU, Mission 4 Component 2 (Grant No. CUP E53D23002240006), from the Italian Ministry of University and Research through Project FARE DAVNE (G R20PEX7Y3A) and Project DYNAMITE QUANTERA2_00056 ERANET, which were co-funded by the European Union H2020 (GA 101017733) and from the ICSC – Centro Nazionale di Ricerca in HPC, Big Data and Quantum Computing, the Provincia Autonoma di Trento and Q@TN. The views and opinions expressed are those of the authors only and do not necessarily reflect those of the granting authorities.
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Y.C., P.H., J.C.H., Z.-S.Y. and J.-W.P. conceived the research. Y.C., B.Y., P.H., W.Z. and J.C.H. developed the theory. W.-Y.Z., Z.-S.Y. and J.-W.P. designed the experiment. W.-Y.Z., Y.L., M.-G.H., H.-Y.W., T.-Y.W., Z.-H.Z., G.-X.S., Z.-Y.Z., Y.-G.Z. and H.S. conducted the experiments and collected the data. W.-Y.Z., Y.L. and Y.C. contributed to the data analysis. All authors contributed to writing the manuscript.
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Extended data
Extended Data Fig. 1 Overlaps between the target states and the deterministic configurations.
The numerical overlaps between the low-energy excited state \(\left\vert {\psi }_{t}\right\rangle\) and two deterministic states, namely \(\left\vert {\psi }_{d}\right\rangle =\left\vert 2020112020\right\rangle\) and \(\left\vert 1111201111\right\rangle\), are shown as solid blue and orange lines, respectively, for different values of \(\frac{m}{\kappa }\).
Extended Data Fig. 2 Procedures of initial state preparation in the positive mass region.
a. An illustration of the atom distribution of the prepared unity-filled Mott insulator state after staggered immersion cooling. Atoms in orange-shaded chains are then addressed by DMD light. b. An illustration of the atomic spin distribution after the site-dependent addressing. c. An illustration of the state after we merge every two atoms in \(\left\vert \downarrow \right\rangle\) state onto a single site along the y direction. d. The prepared copies of \(\left\vert 2020112020\right\rangle\) state along the y direction.
Extended Data Fig. 3 Procedures of initial state preparation in the negative mass region.
a. An illustration of the atom distribution of the prepared unity-filled Mott insulator state after staggered immersion cooling. Atoms in orange-shaded chains are then addressed by DMD light. b. An illustration of the atomic spin distribution after the site-dependent addressing. c. An illustration of the state after we adiabatically transfer every two atoms in the \(\left\vert \downarrow \right\rangle\) state onto a single site along the y direction. d. The prepared copies of \(\left\vert 1111201111\right\rangle\) state along the y direction.
Extended Data Fig. 4 Adiabatic passage.
a. Schematic total optical lattice potentials for the \(\left\vert \downarrow \right\rangle\) (solid blue line) atoms together with the relevant state-independent staggered potential δ and state-dependent tilt gradient Δ before and after the adiabatic passage process. Solid blue circles indicate an exemplary initial state of \(\left\vert \downarrow \right\rangle\) atoms in the optical lattice. The initial atom distribution \(\left\vert \downarrow ,\downarrow ,\downarrow ,\downarrow ,\downarrow ,\downarrow \right\rangle\) is transferred into \(\left\vert \downarrow \downarrow ,0,\downarrow \downarrow ,0,\downarrow \downarrow ,0\right\rangle\). b. The corresponding ramping protocols are employed in the adiabatic passage process. c. Schematic total optical lattice potentials for the \(\left\vert \downarrow \right\rangle\) (solid blue line) and \(\left\vert \uparrow \right\rangle\) atoms (solid orange line) together with the relevant state-independent staggered potential δ and state-dependent tilt gradient (Δ for \(\left\vert \downarrow \right\rangle\) atoms, and − 2Δ for \(\left\vert \uparrow \right\rangle\) atoms) before and after the adiabatic passage process. Solid circles indicate an exemplary initial state of \(\left\vert \downarrow \right\rangle\) (blue circles) and \(\left\vert \uparrow \right\rangle\) atoms (orange circles) in the optical lattice. The initial atom distribution \(\left\vert \downarrow ,\uparrow ,\downarrow ,\uparrow ,\downarrow ,\uparrow \right\rangle\) is kept unchanged. d. Numerical results of the time-resolved density profiles for \(\left\vert \uparrow \right\rangle\) state (left) and \(\left\vert \downarrow \right\rangle\) state (right), respectively. e. Probabilities of the atom distributions of \(\left\vert \downarrow ,\downarrow ,\downarrow ,\downarrow ,\downarrow ,\downarrow \right\rangle\) and \(\left\vert \downarrow \downarrow ,0,\downarrow \downarrow ,0,\downarrow \downarrow ,0\right\rangle\) during the adiabatic passage process. f. Numerical results of the time-resolved density profiles for \(\left\vert \uparrow \right\rangle\) state (left) and \(\left\vert \downarrow \right\rangle\) state (right), respectively. g. Probabilities of the atom distribution of \(\left\vert \downarrow ,\uparrow ,\downarrow ,\uparrow ,\downarrow ,\uparrow \right\rangle\) during the adiabatic passage process.
Extended Data Fig. 5 Numerical results in the positive mass region.
a. The time-resolved density profiles and two-point correlations at various χ/κ. The extracted time-resolved results of the observables b \({\mathcal{E}}(t)\), c \({\mathcal{M}}(t)\), and d \({\mathcal{E}}(t)/{\mathcal{M}}(t)\), at various χ/κ.
Extended Data Fig. 6 Numerical results in the negative mass region.
a. The time-resolved density profiles. b. The time-resolved two-point correlations. The extracted time-resolved results of the observables c \({\mathcal{E}}(t)\), d \({\mathcal{M}}(t)\), and e \({\mathcal{E}}(t)/{\mathcal{M}}(t)\).
Extended Data Fig. 7 Influence of disorder.
a. Numerical simulation results in the presence of disorders of different amplitudes d. b. The dark blue solid dots are the experimental results under χ = 0. The light blue solid dots are the experimental results under \(\frac{\chi }{\kappa }=0.1\). The dark-shaded region represents the numerical sampling result under χ = 0 with uniformly distributed random disorders in the interval (0, d = 0.24κ). The light-shaded region represents the numerical sampling result under \(\frac{\chi }{\kappa }=0.1\) with uniformly distributed random disorders in the interval (0, d = 0.24κ).
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Zhang, WY., Liu, Y., Cheng, Y. et al. Observation of microscopic confinement dynamics by a tunable topological θ-angle. Nat. Phys. 21, 155–160 (2025). https://doi.org/10.1038/s41567-024-02702-x
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DOI: https://doi.org/10.1038/s41567-024-02702-x
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