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.
This is a preview of subscription content, access via your institution
Relevant articles
Open Access articles citing this article.
-
Quantum metrology with boundary time crystals
Communications Physics Open Access 17 October 2023
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 print issues and online access
$209.00 per year
only $17.42 per issue
Rent or buy this article
Prices vary by article type
from$1.95
to$39.95
Prices may be subject to local taxes which are calculated during checkout
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.
References
Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153–185 (2014).
Parmee, C. D. & Cooper, N. R. Phases of driven two-level systems with nonlocal dissipation. Phys. Rev. A 97, 053616 (2018).
Olmos, B., Yu, D. & Lesanovsky, I. Steady-state properties of a driven atomic ensemble with nonlocal dissipation. Phys. Rev. A 89, 023616 (2014).
Parmee, C. D. & Ruostekoski, J. Signatures of optical phase transitions in superradiant and subradiant atomic arrays. Commun. Phys. 3, 205 (2020).
Muniz, J. A. et al. Exploring dynamical phase transitions with cold atoms in an optical cavity. Nature 580, 602–607 (2020).
Dicke, R. H. Coherence in spontaneous radiation processes. Phys. Rev. 93, 99–110 (1954).
Agarwal, G. S., Brown, A. C., Narducci, L. M. & Vetri, G. Collective atomic effects in resonance fluorescence. Phys. Rev. A 15, 1613–1624 (1977).
Narducci, L. M., Feng, D. H., Gilmore, R. & Agarwal, G. S. Transient and steady-state behavior of collective atomic systems driven by a classical field. Phys. Rev. A 18, 1571–1576 (1978).
Carmichael, H. J. & Walls, D. F. Hysteresis in the spectrum for cooperative resonance fluorescence. J. Phys. B 10, L685–L691 (1977).
Walls, D. F., Drummond, P. D., Hassan, S. S. & Carmichael, H. J. Non-equilibrium phase transitions in cooperative atomic systems. Prog. Theor. Phys. Suppl. 64, 307–320 (1978).
Walls, D. F. Cooperative fluorescence from N coherently driven two-level atoms. J. Phys. B 13, 2001–2009 (1980).
Hannukainen, J. & Larson, J. Dissipation-driven quantum phase transitions and symmetry breaking. Phys. Rev. A 98, 042113 (2018).
Ritsch, H., Domokos, P., Brennecke, F. & Esslinger, T. Cold atoms in cavity-generated dynamical optical potentials. Rev. Mod. Phys. 85, 553–601 (2013).
Baumann, K., Guerlin, C., Brennecke, F. & Esslinger, T. Dicke quantum phase transition with a superfluid gas in an optical cavity. Nature 464, 1301–1306 (2010).
Klinder, J., Keßler, H., Bakhtiari, M. R., Thorwart, M. & Hemmerich, A. Observation of a superradiant Mott insulator in the Dicke–Hubbard model. Phys. Rev. Lett. 115, 230403 (2015).
Iemini, F. et al. Boundary time crystals. Phys. Rev. Lett. 121, 035301 (2018).
Keßler, H. et al. Observation of a dissipative time crystal. Phys. Rev. Lett. 127, 043602 (2021).
Meiser, D., Ye, J., Carlson, D. R. & Holland, M. J. Prospects for a millihertz-linewidth laser. Phys. Rev. Lett. 102, 163601 (2009).
Bohnet, J. G. et al. A steady-state superradiant laser with less than one intracavity photon. Nature 484, 78–81 (2012).
Norcia, M. A. & Thompson, J. K. Cold-strontium laser in the superradiant crossover regime. Phys. Rev. X 6, 011025 (2016).
Laske, T., Winter, H. & Hemmerich, A. Pulse delay time statistics in a superradiant laser with calcium atoms. Phys. Rev. Lett. 123, 103601 (2019).
Schäffer, S. A. et al. Lasing on a narrow transition in a cold thermal strontium ensemble. Phys. Rev. A 101, 013819 (2020).
Glicenstein, A. et al. Preparation of one-dimensional chains and dense cold atomic clouds with a high numerical aperture four-lens system. Phys. Rev. A 103, 043301 (2021).
Gross, M. & Haroche, S. Superradiance: an essay on the theory of collective spontaneous emission. Phys. Rep. 93, 301–396 (1982).
Allen, L. & Eberly, J. H. Optical Resonance and Two-Level Atoms (Dover, 1987).
Sutherland, R. T. & Robicheaux, F. Superradiance in inverted multilevel atomic clouds. Phys. Rev. A 95, 033839 (2017).
Tanji-Suzuki, H. et al. in Advances in Atomic, Molecular, and Optical Physics Vol. 60 (eds Arimondo, E., Berman, P. & Lin, C.) pp 201–237 (Academic, 2011).
Ferioli, G. et al. Laser-driven superradiant ensembles of two-level atoms near dicke regime. Phys. Rev. Lett. 127, 243602 (2021).
Loudon, R. The Quantum Theory of Light (Oxford Univ. Press, 2000).
Gold, D. C. et al. Spatial coherence of light in collective spontaneous emission. PRX Quantum 3, 010338 (2022).
Liedl, C. et al. Observation of superradiant bursts in waveguide QED. Preprint at https://arxiv.org/abs/2211.08940 (2022).
Somech, O. & Shahmoon, E. Quantum entangled states of a classically radiating macroscopic spin. Preprint at https://arxiv.org/abs/2204.05455 (2022).
Hassan, S., Bullough, R., Puri, R. & Lawande, S. Intensity fluctuations in a driven Dicke model. Physica A 103, 213–225 (1980).
Carmichael, H. J. Analytical and numerical results for the steady state in cooperative resonance fluorescence. J. Phys. B 13, 3551–3575 (1980).
Pellegrino, J. et al. Observation of suppression of light scattering induced by dipole–dipole interactions in a cold-atom ensemble. Phys. Rev. Lett. 113, 133602 (2014).
Glicenstein, A. et al. Collective shift in resonant light scattering by a one-dimensional atomic chain. Phys. Rev. Lett. 124, 253602 (2020).
Debnath, K., Zhang, Y. & Mølmer, K. Lasing in the superradiant crossover regime. Phys. Rev. A 98, 063837 (2018).
Zhang, Y., Zhang, Y.-X. & Mølmer, K. Monte-Carlo simulations of superradiant lasing. N. J. Phys. 20, 112001 (2018).
Jäger, S. B., Liu, H., Cooper, J., Nicholson, T. L. & Holland, M. J. Superradiant emission of a thermal atomic beam into an optical cavity. Phys. Rev. A 104, 033711 (2021).
Parmee, C. D. & Ruostekoski, J. Bistable optical transmission through arrays of atoms in free space. Phys. Rev. A 103, 033706 (2021).
Glicenstein, A., Ferioli, G., Browaeys, A. & Ferrier-Barbut, I. From superradiance to subradiance: exploring the many-body dicke ladder. Opt. Lett. 47, 1541–1544 (2022).
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).
Author information
Authors and Affiliations
Contributions
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.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41567-023-02064-w
This article is cited by
-
Quantum metrology with boundary time crystals
Communications Physics (2023)
