Abstract
Polar molecules confined in an optical lattice are a versatile platform to explore spin-motion dynamics based on strong, long-range dipolar interactions1,2. The precise tunability3 of Ising and spin-exchange interactions with both microwave and d.c. electric fields makes the molecular system particularly suitable for engineering complex many-body dynamics4,5,6. Here we used Floquet engineering7 to realize new quantum many-body systems of polar molecules. Using a spin encoded in the two lowest rotational states of ultracold 40K87Rb molecules, we mutually validated XXZ spin models tuned by a Floquet microwave pulse sequence against those tuned by a d.c. electric field through observations of Ramsey contrast dynamics. This validation sets the stage for the realization of Hamiltonians inaccessible with static fields. In particular, we observed two-axis twisting8 mean-field dynamics, generated by a Floquet-engineered XYZ model using itinerant molecules in two-dimensional layers. In the future, Floquet-engineered Hamiltonians could generate entangled states for molecule-based precision measurement9 or could take advantage of the rich molecular structure for quantum simulation of multi-level systems10,11.
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Acknowledgements
We thank N. Leitao, L. S. Martin and A. M. Rey for helpful discussions; K. P. Zamarski and F. Vietmeyer for technical contributions; and K. Kim and C. Luo for their comments on the manuscript. The work at JILA and Harvard was jointly supported by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator. Support is also acknowledged from the National Science Foundation (NSF) Quantum Leap Challenge Institutes (OMA-2016244), the NSF Physics Frontiers Centers (PHY-2317149), the Air Force Office of Scientific Research Multidisciplinary University Research Initiative, the Army Research Office Multidisciplinary University Research Initiative and the National Institute of Standards and Technology. H.G., H.Z. and M.D.L. acknowledge support from the Center for Ultracold Atoms, an NSF Physics Frontiers Center (PHY-2317134) and the NSF (PHY-2012023). C.M. acknowledges support from the Department of Defense through the National Defense Science and Engineering Graduate Fellowship. A.N.C. acknowledges support from the NSF Graduate Research Fellowship Program under grant number DGE 2040434.
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C.M., A.N.C., J.L., H.H. and J.Y. carried out the experiments and analysed the data. H.G., H.Z. and M.D.L. contributed ideas and methods for Floquet engineering. All authors discussed the results and contributed to the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Pulse sequences.
Pulse sequences for generating OAT (XY8; A) and TAT (XY8-TAT; B) dynamics are shown using the notation of ref. 7 Narrow (wide) rectangles represent \({\rm{\pi }}/2\) (\({\rm{\pi }}\)) pulses, red (blue) rectangles pulses about the ±X (±Y) axes, and pulses above (below) the line about the + (−) axes. The frame matrix representation shows which axis points along the +Z direction as a function of time, with yellow (green) blocks representing axes originally along the + (−) directions.
Extended Data Fig. 2 Imaging correction.
The measured value of \(\langle {S}_{Z}^{0}\rangle \) is plotted (blue circles) as a function of the value prepared by a microwave pulse. The dashed curve is a fit to the loss model (see Methods). By inverting the model, the data can be corrected for loss during imaging (orange crosses).
Extended Data Fig. 3 Optimizing Floquet pulse timing.
The fitted contrast decay rate Γ is plotted as a function of pulse spacing τ for molecules with initial densities around 1.8(5) × 107 cm−2 in a 3D lattice with parameters set to produce the XXX model. The green triangle is the electric field-tuned data with a KDD pulse sequence and the orange squares are the Floquet data with a DROID-R2D2 pulse sequence. The error bars on the plot are 1 s.e. from stretched exponential fits.
Extended Data Fig. 4 Number loss during TAT Floquet engineering.
The average molecule density is plotted as a function of time as the XY8-TAT pulse sequence is repeatedly applied. The solid curve shows an exponential fit to the data, with time constant 11.2(6) ms.
Extended Data Fig. 5 Single-particle contrast decay rates.
The single-particle contrast decay rate Γ0 is plotted as a function of χ electric field for electric field data (red circles and top axis) and Floquet data (blue squares) for (A) molecules pinned in a deep 3D lattice and (B) itinerant molecules in a 1D lattice. The error bars are 1 s.e. from a linear fit.
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Miller, C., Carroll, A.N., Lin, J. et al. Two-axis twisting using Floquet-engineered XYZ spin models with polar molecules. Nature 633, 332–337 (2024). https://doi.org/10.1038/s41586-024-07883-2
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DOI: https://doi.org/10.1038/s41586-024-07883-2
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