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# Probing site-resolved correlations in a spin system of ultracold molecules

## Abstract

Synthetic quantum systems with interacting constituents play an important role in quantum information processing and in explaining fundamental phenomena in many-body physics. Following impressive advances in cooling and trapping techniques, ensembles of ultracold polar molecules have emerged as a promising platform that combines several advantageous properties1,2,3,4,5,6,7,8,9,10,11. These include a large set of internal states with long coherence times12,13,14,15,16,17 and long-range, anisotropic interactions. These features could enable the exploration of intriguing phases of correlated quantum matter, such as topological superfluids18, quantum spin liquids19, fractional Chern insulators20 and quantum magnets21,22. Probing correlations in these phases is crucial to understanding their properties, necessitating the development of new experimental techniques. Here we use quantum gas microscopy23 to measure the site-resolved dynamics of quantum correlations of polar 23Na87Rb molecules confined in a two-dimensional optical lattice. By using two rotational states of the molecules, we realize a spin-1/2 system with dipolar interactions between particles, producing a quantum spin-exchange model21,22,24,25. We study the evolution of correlations during the thermalization process of an out-of-equilibrium spin system for both spatially isotropic and anisotropic interactions. Furthermore, we examine the correlation dynamics of a spin-anisotropic Heisenberg model engineered from the native spin-exchange model by using periodic microwave pulses26,27,28. These experiments push the frontier of probing and controlling interacting systems of ultracold molecules, with prospects for exploring new regimes of quantum matter and characterizing entangled states that are useful for quantum computation29,30 and metrology31.

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## Data availability

Source data can be found in the Harvard Dataverse58. All other supporting data are available from the corresponding author upon reasonable request.

## Code availability

The code used in this manuscript is available from the corresponding author upon reasonable request.

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## Acknowledgements

We thank E. Guardado-Sanchez and G. Zheng for experimental assistance. This work was supported by the NSF (grant no. 1912154) and the David and Lucile Packard Foundation (grant no. 2016-65128). L.C. was supported by the NSF Graduate Research Fellowship Program. D.A.H. and A.M. were supported in part by the NSF QLCI grant no. OMA-2120757.

## Author information

Authors

### Contributions

W.S.B. and D.A.H. conceived the study and supervised the experiment. L.C., J.S.R., R.R. and Z.Z.Y. performed the experiments and the data analysis. A.M. and S.C. performed the numerical calculations. All authors contributed to the manuscript.

### Corresponding author

Correspondence to Waseem S. Bakr.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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## Extended data figures and tables

### Extended Data Fig. 1 Microwave spectroscopy.

a, Molecular rotational and hyperfine spectrum measured at 60 G. Green, blue and orange lines are the theoretical predictions using molecular parameters in refs. 54,56 for microwave transitions from $$\left|\uparrow \right\rangle$$ to selected hyperfine states in the N = 1 manifold using π, σ, and σ+ polarization, respectively. The transition on the far right, indicated by the black arrow, is the $$\left|\uparrow \right\rangle$$ to $$\left|\downarrow \right\rangle$$ transition. b, Sample Rabi oscillation between $$\left|\uparrow \right\rangle$$ and $$\left|\downarrow \right\rangle$$. The extracted Rabi frequency from this measurement is 2π × 9.529(4) kHz. Error bars are s.e.m.

### Extended Data Fig. 2 Differential polarizabilities between $${\boldsymbol{| }}{\boldsymbol{\uparrow }}{\boldsymbol{\rangle }}$$ and $${\boldsymbol{| }}{\boldsymbol{\downarrow }}{\boldsymbol{\rangle }}$$ versus trapping light intensity.

a, In the isotropic configuration, B = 60 G, and the angle between the light’s electric field and the quantization axis is 0°. The intensity varies by ~4% over the cloud, denoted by the grey shading. b, In the anisotropic configuration, B = 4.1 G, and the angle is 90°.

### Extended Data Fig. 3 Ramsey fringe contrast as a function of time at varying lattice filling.

Fringe contrast shown for 1.0(2)% (blue circles), 3.3(2)% (green squares), and 8.4(3)% (orange diamonds) peak lattice fillings. Dashed lines represent exponential fits with 1/e times of 83(4) ms, 25(4) ms, and 11(2) ms respectively. Error bars are s.e.m.

### Extended Data Fig. 4 Spin-exchange coupling.

The values of V(a)/h calculated for the isotropic (a) and anisotropic (b) cases for different separations in x and y.

### Extended Data Fig. 5 Numerical simulation comparison between XYY and Floquet dynamics.

a, Comparing magnetization dynamics for different initial states between the exact XYY model (shaded bands) and a Floquet drive with a 5.8 μs π-pulse time (points). Red: $$\left|+X\right\rangle$$ initial state. Orange: $$\left|+Y\right\rangle$$ initial state. The dashed line indicates the demagnetized value with N = N0/2. b, Correlation dynamics compared between the exact XYY model (shaded bands) and a Floquet drive with a 5.8 μs π-pulse time (points). Top: nearest-neighbor correlations. Middle: next-nearest neighbor correlations. Bottom: next-next-nearest neighbor correlations. c, Comparing magnetization dynamics for different initial states between the exact XYY model (shaded bands) and a Floquet drive with a 58 μs π-pulse time (points). Red: $$\left|+X\right\rangle$$ initial state. Orange: $$\left|+Y\right\rangle$$ initial state. The dashed line indicates the demagnetized value with N = N0/2. d, Correlation dynamics compared between the exact XYY model (shaded bands) and a Floquet drive with a 58 μs π-pulse time (points). Top: nearest-neighbor correlations. Middle: next-nearest neighbor correlations. Bottom: next-next-nearest neighbor correlations.

### Extended Data Fig. 6 π-pulse fidelity.

a, Microwave pulse sequence to measure the error in the π-pulse time. An even number of π-pulses interspersed with hold times τ are used to rotate the spins from $$\left|\uparrow \right\rangle$$ to $$\left|\downarrow \right\rangle$$ and back. b, Fraction of molecules remaining in $$\left|\uparrow \right\rangle$$ versus number of π-pulses N. The dashed line marks N/N0 = 1 indicating perfect π-pulses. Error bars are s.e.m.

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Christakis, L., Rosenberg, J.S., Raj, R. et al. Probing site-resolved correlations in a spin system of ultracold molecules. Nature 614, 64–69 (2023). https://doi.org/10.1038/s41586-022-05558-4

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• DOI: https://doi.org/10.1038/s41586-022-05558-4