Abstract
Increasing the number of particles in a system often leads to qualitative changes in its properties, such as breaking of symmetries and the appearance of phase transitions. This renders a macroscopic system fundamentally different from its individual microscopic constituents. Lying between these extremes, mesoscopic systems exhibit microscopic fluctuations that influence behaviour on longer length scales, leading to critical phenomena and dynamics. Therefore, tracing the properties of well-controlled mesoscopic systems can help bridge the gap between an exact description of few-body microscopic systems and the emergent description of many-body systems. Here we explore the mesoscopic signatures of an optomechanical self-organization phase transition using arrays of cold atoms inside an optical cavity. By precisely engineering atom–cavity interactions, we reveal how critical behaviour depends on the atom number, identify characteristic dynamical behaviours in the self-organized regime and observe a finite optomechanical susceptibility at the critical point. These findings advance our understanding of particle-number- and time-resolved properties of phase transitions in mesoscopic systems.
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Data are available from the corresponding author upon reasonable request. Source data are provided with this paper.
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Code is available from the corresponding author upon reasonable request.
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Acknowledgements
We thank N. Song for assistance in the laboratory. We acknowledge support from the AFOSR (grant no. FA9550-1910328 (D.M.S.-K.)) and Young Investigator Prize grant no. 21RT0751 (A.A.-G.), from ARO through the MURI program (grant no. W911NF-20-1-0136 (D.M.S.-K.)), from DARPA (grant no. W911NF2010090 (D.M.S.-K.)), from the NSF (QLCI program through grant no. OMA-2016245 (D.M.S.-K.) and CAREER award no. 2047380 (A.A.-G.)), from the David and Lucile Packard Foundation (A.A.-G.), and from the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator (D.M.S.-K.). J.H. acknowledges support from the Department of Defense through the National Defense Science and Engineering Graduate (NDSEG) Fellowship Program. C.C.R. acknowledges support from the European Union’s Horizon Europe programme under the Marie-Skłodowska Curie Action LIME (grant no. 101105916).
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J.H., Y.-H.L., Z.Y. and T.X. contributed to building the experimental setup, performing the experiments and analysing the data. C.C.R., S.J.M., A.A.-G., D.M.S.-K. and J.H. contributed to the theoretical model. Z.Y. and D.M.S.-K. conceived the experiments. All authors contributed to the writing of the manuscript and discussed the results. All work was supervised by D.M.S.-K. and A.A.-G.
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Extended data
Extended Data Fig. 1 Schematic of the heterodyne detection system.
The pump and LO beams originate from separate fibres and are combined at a polarizing beamsplitter (PBS) with orthogonal polarizations. They are then both sent into a single-mode fibre, which ensures that they are mode-matched at the fibre output. Upon emerging from the fibre, they pass through a half-waveplate (labeled as λ/2) at 45∘ and are split at another PBS. Each output port of the last PBS is directed to a sensor on a balanced photodetector, which subtracts the signals on the two sensors. The output voltage of the detector is proportional to \({E}_{{\rm{LO}}}{E}_{{\rm{cav}}}\sin (\varDelta \omega t+\vartheta +\phi )\) where ELO is the LO electric field amplitude, Ecav is the cavity electric field amplitude, Δω = 2π × 20 MHz is the frequency difference between the pump and LO, ϕ is the phase of the LO relative to the pump, and ϑ is the phase of the cavity field relative to the pump.
Extended Data Fig. 2 Example time traces of the detected field in each of the three pumping frames.
a, Scatter plots of the complex cavity field before phase correction. The real (Re(\(\langle \hat{c}_\vartheta \rangle\))) and imaginary (Im(\(\langle \hat{c}_\vartheta \rangle\))) quadratures of the detected field with respect to the average phase of the cavity emission, ϑ, are plotted. Lighter points indicate earlier times and darker points indicate later times. From these data, the angle ϕ of the major axis in each frame is determined by PCA. b, Detected field after rotation by ϕ. Because PCA only determines the angle of the major axis up to a π rotation, in the antinode frames, we define the average projection of the field onto the axis defined by ϑ to be positive. The sign of the node frame is determined by comparing ϑnode to the angle linearly interpolated from ϑfirst frame and ϑlast frame and assuming that the two angles are within π/2 of each other. c, Time traces of cproj. Values of cproj are obtained by projecting the phase-corrected measurements of \(\hat{c}\) (shown in (b)) onto the ϑ-axis. The shaded dark purple regions indicate the shot noise level. Data shown in figure were taken with N = 18, Δpa = − 2π × 80 MHz, Δpc = − 2π × 2.15 MHz, and Ω = 2π × 63.9 MHz.
Extended Data Fig. 3 Dependence of the critical point on Δpa.
a, Data showing the bifurcation for 20 atoms at four values of Δpa. Markers represent the maxima of the fitted Boltzmann probability distribution of cproj. Shaded areas show the width of the distribution at half-maximum. Different pump-cavity detunings were chosen for each setting of Δpa to compensate for the different dispersive shifts on the cavity resonance frequency for 20 atoms placed at the nodes. The values used are Δpc = − 2π × {1.79, 1.9, 2.02, 2.26} MHz corresponding to Δpa = − 2π × {100, 80, 60, 40} MHz. These correspond approximately to Δpc(T) ≃ − 2π × 1.6 MHz when accounting for the shift that thermal atoms put on the cavity resonance frequency. b, Extracted Ωc (circles) exhibit an approximately linear dependence on Δpa. The dashed line shows a fit to equation (4), which gives a fitted temperature of T = 66 ± 8 μK. The shaded region shows the prediction of equation (4) for the independent temperature measurement. Error bars are smaller than the markers.
Extended Data Fig. 4 Interpolation of Ωc from Boltzmann fits.
a, Data showing the bifurcation for 20 atoms at Δpa = 2π × 80 MHz. Markers represent the maxima of the fitted Boltzmann probability distribution of cproj. Blue shaded areas show the width of the distribution at half-maximum. Bottom insets show histograms (light blue) and Boltzmann fits (dark blue) for values of Ω in the gray shaded region. b, By fitting experimentally obtained distributions, such as those shown in (a), to the Boltzmann distribution, we extract the parameter B at different values of Ω for various N. The N shown here correspond to the data shown in Fig. 2a of the main text. The value of Ωc (stars) is linearly interpolated from the two points directly on either side of the B = 0 line. Error bars on the fitted B values are smaller than the markers. Lines are guides to the eye.
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Supplementary Sections I–IV, Equations (1)–(34) and Fig. 1.
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Ho, J., Lu, YH., Xiang, T. et al. Optomechanical self-organization in a mesoscopic atom array. Nat. Phys. 21, 1071–1077 (2025). https://doi.org/10.1038/s41567-025-02916-7
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DOI: https://doi.org/10.1038/s41567-025-02916-7
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