Abstract
A class of systems exists in which dissipation, external drive and interactions compete and give rise to non-equilibrium phases that would not exist without the drive. There, phase transitions could occur without the breaking of any symmetry, yet with a local order parameter—in contrast to the Landau theory of phase transitions at equilibrium. One of the simplest driven–dissipative quantum systems consists of two-level atoms enclosed in a volume smaller than the wavelength of the atomic transition cubed, driven by a light field. The competition between collective coupling of the atoms to the driving field and their cooperative decay should lead to a transition between a phase where all the atomic dipoles are phase-locked and a phase governed by superradiant spontaneous emission. Here, we realize this model using a pencil-shaped cloud of laser-cooled atoms in free space, optically excited along its main axis, and observe the predicted phases. Our demonstration is promising in view of obtaining free-space superradiant lasers or observing new types of time crystal.
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Data availability
All data that support the plots within this paper and the Methods of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.
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Acknowledgements
We thank F. Robicheaux for stimulating conversations, and A.-M. Rey, J.K. Thompson, K. Moelmer, R.T. Sutherland, J. Marino, Bruno Laburthe-Tolra, D. Dreon and D. Clément for insightful discussions. We thank D. Goncalves-Romeu, L. Bombieri and D. Chang for insightful inputs on the role of the coherent dipole interactions. This project has received funding from the European Research Council (advanced grant no. 101018511, ATARAXIA), by the Agence National de la Recherche (ANR, project DEAR) and by the Région Ile-de-France in the framework of DIM SIRTEQ (projects DSHAPE and FSTOL).
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G.F. and A.G. carried out the experiments and analysed the data. G.F., I.F.-B. and A.B. conducted the theoretical analysis and simulations. All authors contributed to the writing of the manuscript.
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Extended data
Extended Data Fig. 1 Shift and width of the atomic transition as a function of the saturation parameter for N ≈ 3000.
Color (white) filled diamonds: shift (width). The dashed line is the single atom power broadening \(\sqrt{1+s/4}\). Error bars on Δ and Γ are from the fit. Error bars on s = I/Isat correspond to 10% shot-to-shot fluctuations evaluated from 1000 repetitions.
Extended Data Fig. 2
Steady state values of the collective dipole \({{{\mathbf{Im}}}}[\langle {\mathbf{S}}^{{\mathbf{-}}}\rangle ]\), the magnetiztion 〈sz〉, the effective Rabi frequency ΩEff and the superradiant emission rate γSR as a function of Ω/Γ, plotted for N = (2, 5, 10, 15) (red, blue, black, green).
Extended Data Fig. 3
Steady state values of ΩEff, and γSR as a function of N, for Ω/Γ = (1.1, 4.5, 11) (red dots, blue squares, black diamonds).
Extended Data Fig. 4 DDM and second order phase transition.
Comparison between the numerical solution of Eq. (21) for N = 20 (black line), and the analytical solution \(x=\sqrt{{\beta }^{2}-1}\) (red dashed line), showing the existence of a critical point for N → ∞.
Source data
Source Data Fig. 2
Data plotted in Fig. 2.
Source Data Fig. 3
Data plotted in Fig. 3.
Source Data Fig. 4
Data plotted in Fig. 4.
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Ferioli, G., Glicenstein, A., Ferrier-Barbut, I. et al. A non-equilibrium superradiant phase transition in free space. Nat. Phys. 19, 1345–1349 (2023). https://doi.org/10.1038/s41567-023-02064-w
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DOI: https://doi.org/10.1038/s41567-023-02064-w
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