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# Exploring dynamical phase transitions with cold atoms in an optical  cavity

## Abstract

Interactions between atoms and light in optical cavities provide a means of investigating collective (many-body) quantum physics in controlled environments. Such ensembles of atoms in cavities have been proposed for studying collective quantum spin models, where the atomic internal levels mimic a spin degree of freedom and interact through long-range interactions tunable by changing the cavity parameters1,2,3,4. Non-classical steady-state phases arising from the interplay between atom–light interactions and dissipation of light from the cavity have previously been investigated5,6,7,8,9,10,11. These systems also offer the opportunity to study dynamical phases of matter that are precluded from existence at equilibrium but can be stabilized by driving a system out of equilibrium12,13,14,15,16, as demonstrated by recent experiments17,18,19,20,21,22. These phases can also display universal behaviours akin to standard equilibrium phase transitions8,23,24. Here, we use an ensemble of about a million strontium-88 atoms in an optical cavity to simulate a collective Lipkin–Meshkov–Glick model25,26, an iconic model in quantum magnetism, and report the observation of distinct dynamical phases of matter in this system. Our system allows us to probe the dependence of dynamical phase transitions on system size, initial state and other parameters. These observations can be linked to similar dynamical phases in related systems, including the Josephson effect in superfluid helium27, or coupled atomic28 and solid-state polariton29 condensates. The system itself offers potential for generation of metrologically useful entangled states in optical transitions, which could permit quantum enhancement in state-of-the-art atomic clocks30,31.

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## Data availability

Data relevant to the figures and conclusions of this manuscript are available at https://doi.org/10.5061/dryad.mgqnk98w951.

## Code availability

The codes used in the analysis of experimental data and to carry out associated theoretical calculations are available from the corresponding authors upon reasonable request.

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## Acknowledgements

We acknowledge discussions with I. Spielman, M. Holland and A. Shankar. This work is supported by the Air Force Office of Scientific Research (AFOSR) grant FA9550-18-1-0319, by the Defense Advanced Research Projects Agency (DARPA) Extreme Sensing and ARO grant W911NF-16-1-0576, the ARO single investigator award W911NF-19-1-0210, the US National Science Foundation (NSF) PHY1820885, NSF JILA-PFC PHY-1734006 grants, and by the National Institute of Standards and Technology (NIST). J.R.K.C. acknowledges financial support from NSF GRFP.

## Author information

Authors

### Contributions

J.A.M., D.J.Y., J.R.K.C. and J.K.T. collected and analysed the experimental data. R.J.L.-S., D.B. and A.M.R. developed the theoretical model. All authors discussed the results and contributed to the preparation of the manuscript.

### Corresponding authors

Correspondence to Ana Maria Rey or James K. Thompson.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Murray Barrett, Maria Luisa Chiofalo and Farokh Mivehvar for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Experimental platform.

a, An optical cavity is driven by a 689-nm coherent field that establishes an intra-cavity field $${\varOmega }_{{\rm{p}}}{{\rm{e}}}^{{\rm{i}}{\omega }_{{\rm{p}}}t}$$, which is near resonance with the 1S0 to 3P1 transition in 88Sr. Inside the cavity, an ensemble of atoms is confined in a 1D optical lattice at 813 nm. Different lasers are applied for shelving excited-state atoms into long-lived metastable excited states, for freezing the system dynamics, for applying a radiation pressure force that pushes ground states in a direction transverse to the cavity axis, for optically pumping atoms from long lived metastable excited states back to the ground state, and for fluorescence imaging of atoms in the ground state. b, A typical fluorescence image captured on a CCD, showing the state-resolved imaging technique. The Ne excited state atoms that were shelved into 3P0,2 while the freeze/push beam was applied remain near the trapping region. The Ng ground-state atoms are pushed away from the trapping region. Based on their spatial location, the atoms assigned to be in the excited (ground) state are shown in false colour blue (orange). c, The relevant energy levels for 88Sr, the laser wavelengths and their functions. d, Experimental timing sequence and typical timescales.

### Extended Data Fig. 2 Probing many-body dynamics and mapping the phase boundary.

a, Oscillation period as function of the cavity detuning Δ for 2Ωp/(Ng) = 0.104(4), δ = 0 and atoms starting in $$|\downarrow \,\rangle$$. Blue points are experimental values, solid red line represents the mean-field prediction for the same drive and atom number, and the shaded red area represents typical experimental fluctuations on 2Ωp/(Ng). The period is extracted from sinusoidal fits to data as in Fig. 2a, after removing a linear term caused by the single-particle dephasing effects. The mean-field value (red solid line) is Tosc = 2π/() with the effective replacements due to inhomogeneous coupling as discussed in Methods. Measurements are taken in the dispersive limit where $$\varDelta \gg \sqrt{N}g$$. b, Critical detuning δc as function of the drive Δ for Δ/(2π) = ±50 MHz (red and blue points, respectively). We also plot the theoretical prediction for the phase boundary (equation (11)) with rescaled parameters, and predictions of the numerical model (solid lines) including uncertainty based on the typical fluctuations in Ω/(χN). Error bars are statistical (1σ).

## Supplementary information

### Supplementary Information

This Supplementary Information file contains the following sections: I, Effective spin model; II, Axial motion; III, Anomalous decoherence due to residual motion and technical noise; IV, Dynamical phase diagram; and V, Mapping between spin model and macroscopic self-trapping.

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Muniz, J.A., Barberena, D., Lewis-Swan, R.J. et al. Exploring dynamical phase transitions with cold atoms in an optical  cavity. Nature 580, 602–607 (2020). https://doi.org/10.1038/s41586-020-2224-x

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