Quantum correlations in the steady state of light-emitter ensembles from perturbation theory
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AbstractThe coupling of a quantum system to an environment leads generally to decoherence, and it is detrimental to quantum correlations within the system itself. Yet some forms of quantum correlations can be robust to the presence of an environment – or may even be stabilized by it. Predicting (let alone understanding) them remains arduous, given that the steady state of an open quantum system can be very different from an equilibrium thermodynamic state; and its reconstruction requires generically the numerical solution of the Lindblad equation, which is extremely costly for numerics. Here we focus on the highly relevant situation of ensembles of light emitters undergoing spontaneous decay; and we show that, whenever their Hamiltonian is perturbed away from a U(1) symmetric form, steady-state quantum correlations can be reconstructed via pure-state perturbation theory. Our main result is that in systems of light emitters subject to single-emitter or two-emitter driving, the steady state perturbed away from the U(1) limit generically exhibits spin squeezing; and it has minimal uncertainty for the collective-spin components, revealing that squeezing represents the optimal resource for entanglement-assisted metrology using this state.Featured image: Sketch of a physical situation of interest to this work: the unperturbed reference is a system of light emitters interacting via a U(1)-symmetric Hamiltonian, and emitting photons in the environment. A Hamiltonian perturbation $\lambda H_1$ associated with a two-emitter drive, re-exciting (or de-exciting) the emitters in pairs, can break the U(1) symmetry.Popular summaryQuantum correlations are central to many interesting quantum phenomena and quantum technologies. However, quantum systems tend to be in contact with their surroundings, and when they interact with this environment, the quantum correlations within the system of interest tend to be destroyed. Yet, some forms of quantum correlations can be robust to the presence of an environment or may even be stabilised by it. Describing and predicting this behaviour however remains arduous. Open quantum systems generally do not settle into familiar thermal equilibrium states, and their (long-time) behaviour can be highly nontrivial. In practice, determining these steady states often requires numerical simulation, which quickly becomes infeasible as the system size grows. In this work, we study a particularly important class of systems, namely ensembles of light emitters undergoing spontaneous decay from a higher-energy state to a lower-energy one due to their interaction with the electromagnetic environment. We show that when the system Hamiltonian is close to a highly symmetric limit (a U(1) symmetry, corresponding to invariance of the interactions under rotations around an axis), its steady state can be accurately understood using a perturbative approach based on pure quantum states. This follows from the fact that the perturbation does not alter the purity of the steady state to the lowest non-trivial order in the perturbation, as proven in our work. Our pure-state approach allows us to derive analytical solutions for the class of systems under consideration, avoiding the complexities of perturbation theory for dissipative systems in the general case (which implies calculating the pseudo-inverse of a Lindbladian super-operator). Moreover, our approach provides a powerful and intuitive way to predict the emergence of quantum correlations in their steady state phase diagram. Using this framework, we show that a broad class of light emitters naturally develop a specific and highly useful form of quantum correlations known as spin squeezing. Spin squeezed states are quantum states of high interest for metrological purposes, as they allow for measurement precision beyond the highest one achievable using uncorrelated spins (the so-called standard quantum limit). We find that the spin-squeezed steady state has minimal quantum uncertainties for the collective spin components perpendicular to the finite net orientation of the collective spin. Under this condition, squeezing represents the optimal resource for entanglement-assisted metrology using these steady states.► BibTeX data@article{Huybrechts2026quantumcorrelations, doi = {10.22331/q-2026-04-28-2085}, url = {https://doi.org/10.22331/q-2026-04-28-2085}, title = {Quantum correlations in the steady state of light-emitter ensembles from perturbation theory}, author = {Huybrechts, Dolf and Roscilde, Tommaso}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2085}, month = apr, year = {2026} }► References [1] I. Frérot, M. Fadel and M. Lewenstein, Probing quantum correlations in many-body systems: a review of scalable methods, Reports on Progress in Physics 86, 114001 (2023). https://doi.org/10.1088/1361-6633/acf8d7 [2] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum Entanglement, Rev. Mod. Phys. 81, 865 (2009). https://doi.org/10.1103/RevModPhys.81.865 [3] I. M. Georgescu, S. Ashhab and F. Nori, Quantum simulation, Rev. Mod. Phys. 86, 153 (2014). https://doi.org/10.1103/RevModPhys.86.153 [4] F. Verstraete, M. M. Wolf and J. I. Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nat. Phys. 5, 633 (2009). https://doi.org/10.1038/nphys1342 [5] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler and P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms, Nat. Phys. 4, 878 (2008). https://doi.org/10.1038/nphys1073 [6] H. Weimer, M. Müller, I. Lesanovsky, P. Zoller and H. P. Büchler, A Rydberg quantum simulator, Nature Physics 6, 382 (2010). https://doi.org/10.1038/nphys1614 [7] M. Müller, S. Diehl, G. Pupillo and P. Zoller, Engineered Open Systems and Quantum Simulations with Atoms and Ions, Adv. At. Mol. Opt. Phys. 61, 1 (2012). https://doi.org/10.1016/B978-0-12-396482-3.00001-6 [8] T. E. Lee, S. Gopalakrishnan and M. D. Lukin, Unconventional Magnetism via Optical Pumping of Interacting Spin Systems, Phys. Rev. Lett. 110, 257204 (2013). https://doi.org/10.1103/PhysRevLett.110.257204 [9] J. Jin, A. Biella, O. Viyuela, L. Mazza, J. Keeling, R. Fazio and D. Rossini, Cluster Mean-Field Approach to the Steady-State Phase Diagram of Dissipative Spin Systems, Phys. Rev. X 6, 031011 (2016). https://doi.org/10.1103/PhysRevX.6.031011 [10] W. Verstraelen, D. Huybrechts, T. Roscilde and M. Wouters, Quantum and Classical Correlations in Open Quantum Spin Lattices via Truncated-Cumulant Trajectories, PRX Quantum 4, 030304 (2023). https://doi.org/10.1103/PRXQuantum.4.030304 [11] V. R. Overbeck, M. F. Maghrebi, A. V. Gorshkov and H. Weimer, Multicritical behavior in dissipative Ising models, Phys. Rev. A 95, 042133 (2017). https://doi.org/10.1103/PhysRevA.95.042133 [12] S. Finazzi, A. Le Boité, F. Storme, A. Baksic and C. Ciuti, Corner-Space Renormalization Method for Driven-Dissipative Two-Dimensional Correlated Systems, Phys. Rev. Lett. 115, 080604 (2015). https://doi.org/10.1103/PhysRevLett.115.080604 [13] H. Weimer, Variational Principle for Steady States of Dissipative Quantum Many-Body Systems, Phys. Rev. Lett. 114, 040402 (2015). https://doi.org/10.1103/PhysRevLett.114.040402 [14] L. M. Sieberer, M. Buchhold and S. Diehl, Keldysh field theory for driven open quantum systems, Reports on Progress in Physics 79, 096001 (2016). https://doi.org/10.1088/0034-4885/79/9/096001 [15] N. Shammah, S. Ahmed, N. Lambert, S. De Liberato and F. Nori, Open quantum systems with local and collective incoherent processes: Efficient numerical simulations using permutational invariance, Phys. Rev. A 98, 063815 (2018). https://doi.org/10.1103/PhysRevA.98.063815 [16] F. Vicentini, A. Biella, N. Regnault and C. Ciuti, Variational Neural-Network Ansatz for Steady States in Open Quantum Systems, Phys. Rev. Lett. 122, 250503 (2019). https://doi.org/10.1103/PhysRevLett.122.250503 [17] A. Nagy and V. Savona, Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems, Phys. Rev. Lett. 122, 250501 (2019). https://doi.org/10.1103/PhysRevLett.122.250501 [18] M. J. Hartmann and G. Carleo, Neural-Network Approach to Dissipative Quantum Many-Body Dynamics, Phys. Rev. Lett. 122, 250502 (2019). https://doi.org/10.1103/PhysRevLett.122.250502 [19] H. Weimer, A. Kshetrimayum and R. Orús, Simulation methods for open quantum many-body systems, Rev. Mod. Phys. 93, 015008 (2021). https://doi.org/10.1103/RevModPhys.93.015008 [20] P. Deuar, A. Ferrier, M. Matuszewski, G. Orso and M. H. Szymanska, Fully Quantum Scalable Description of Driven-Dissipative Lattice Models, PRX Quantum 2, 010319 (2021). https://doi.org/10.1103/PRXQuantum.2.010319 [21] C. D. Mink, D. Petrosyan and M. Fleischhauer, Hybrid discrete-continuous truncated Wigner approximation for driven, dissipative spin systems, Phys. Rev. Res. 4, 043136 (2022). https://doi.org/10.1103/PhysRevResearch.4.043136 [22] M. Reitz, C. Sommer and C. Genes, Cooperative Quantum Phenomena in Light-Matter Platforms, PRX Quantum 3, 010201 (2022). https://doi.org/10.1103/PRXQuantum.3.010201 [23] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018). https://doi.org/10.1103/RevModPhys.90.035005 [24] K. Cong, Q. Zhang, Y. Wang, G. T. Noe, A. Belyanin and J. Kono, Dicke superradiance in solids, J. Opt. Soc. Am. B 33, C80 (2016). https://doi.org/10.1364/JOSAB.33.000C80 [25] G. Ferioli, A. Glicenstein, F. Robicheaux, R. T. Sutherland, A. Browaeys and I. Ferrier-Barbut, Laser-Driven Superradiant Ensembles of Two-Level Atoms near Dicke Regime, Phys. Rev. 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Roscilde, In Preparation (2026). [46] F. Carollo and I. Lesanovsky, Exactness of Mean-Field Equations for Open Dicke Models with an Application to Pattern Retrieval Dynamics, Phys. Rev. Lett. 126, 230601 (2021). https://doi.org/10.1103/PhysRevLett.126.230601 [47] L. M. Narducci, D. H. Feng, R. Gilmore and G. S. Agarwal, Transient and steady-state behavior of collective atomic systems driven by a classical field, Phys. Rev. A 18, 1571 (1978). https://doi.org/10.1103/PhysRevA.18.1571 [48] J. Hannukainen and J. Larson, Dissipation-driven quantum phase transitions and symmetry breaking, Phys. Rev. A 98, 042113 (2018). https://doi.org/10.1103/PhysRevA.98.042113Cited by[1] Dawid A. Hryniuk and Marzena H. Szymańska, "Tensor-network-based variational Monte Carlo approach to the non-equilibrium steady state of open quantum systems", Quantum 8, 1475 (2024). [2] Wenqi Tong and F. Robicheaux, "Qualitatively altered driven Dicke superradiance in extended systems due to infinitesimal perturbations", Physical Review A 110 6, 063701 (2024). [3] Wenqi Tong, H. Alaeian, and F. Robicheaux, "Phase transitions in the open Dicke model: A degenerate-perturbation-theory approach", Physical Review A 112 5, 053721 (2025). The above citations are from SAO/NASA ADS (last updated successfully 2026-04-28 07:10:43). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-04-28 07:10:42: Could not fetch cited-by data for 10.22331/q-2026-04-28-2085 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractThe coupling of a quantum system to an environment leads generally to decoherence, and it is detrimental to quantum correlations within the system itself. Yet some forms of quantum correlations can be robust to the presence of an environment – or may even be stabilized by it. Predicting (let alone understanding) them remains arduous, given that the steady state of an open quantum system can be very different from an equilibrium thermodynamic state; and its reconstruction requires generically the numerical solution of the Lindblad equation, which is extremely costly for numerics. Here we focus on the highly relevant situation of ensembles of light emitters undergoing spontaneous decay; and we show that, whenever their Hamiltonian is perturbed away from a U(1) symmetric form, steady-state quantum correlations can be reconstructed via pure-state perturbation theory. Our main result is that in systems of light emitters subject to single-emitter or two-emitter driving, the steady state perturbed away from the U(1) limit generically exhibits spin squeezing; and it has minimal uncertainty for the collective-spin components, revealing that squeezing represents the optimal resource for entanglement-assisted metrology using this state.Featured image: Sketch of a physical situation of interest to this work: the unperturbed reference is a system of light emitters interacting via a U(1)-symmetric Hamiltonian, and emitting photons in the environment. A Hamiltonian perturbation $\lambda H_1$ associated with a two-emitter drive, re-exciting (or de-exciting) the emitters in pairs, can break the U(1) symmetry.Popular summaryQuantum correlations are central to many interesting quantum phenomena and quantum technologies. However, quantum systems tend to be in contact with their surroundings, and when they interact with this environment, the quantum correlations within the system of interest tend to be destroyed. Yet, some forms of quantum correlations can be robust to the presence of an environment or may even be stabilised by it. Describing and predicting this behaviour however remains arduous. Open quantum systems generally do not settle into familiar thermal equilibrium states, and their (long-time) behaviour can be highly nontrivial. In practice, determining these steady states often requires numerical simulation, which quickly becomes infeasible as the system size grows. In this work, we study a particularly important class of systems, namely ensembles of light emitters undergoing spontaneous decay from a higher-energy state to a lower-energy one due to their interaction with the electromagnetic environment. We show that when the system Hamiltonian is close to a highly symmetric limit (a U(1) symmetry, corresponding to invariance of the interactions under rotations around an axis), its steady state can be accurately understood using a perturbative approach based on pure quantum states. This follows from the fact that the perturbation does not alter the purity of the steady state to the lowest non-trivial order in the perturbation, as proven in our work. Our pure-state approach allows us to derive analytical solutions for the class of systems under consideration, avoiding the complexities of perturbation theory for dissipative systems in the general case (which implies calculating the pseudo-inverse of a Lindbladian super-operator). Moreover, our approach provides a powerful and intuitive way to predict the emergence of quantum correlations in their steady state phase diagram. Using this framework, we show that a broad class of light emitters naturally develop a specific and highly useful form of quantum correlations known as spin squeezing. Spin squeezed states are quantum states of high interest for metrological purposes, as they allow for measurement precision beyond the highest one achievable using uncorrelated spins (the so-called standard quantum limit). We find that the spin-squeezed steady state has minimal quantum uncertainties for the collective spin components perpendicular to the finite net orientation of the collective spin. Under this condition, squeezing represents the optimal resource for entanglement-assisted metrology using these steady states.► BibTeX data@article{Huybrechts2026quantumcorrelations, doi = {10.22331/q-2026-04-28-2085}, url = {https://doi.org/10.22331/q-2026-04-28-2085}, title = {Quantum correlations in the steady state of light-emitter ensembles from perturbation theory}, author = {Huybrechts, Dolf and Roscilde, Tommaso}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2085}, month = apr, year = {2026} }► References [1] I. Frérot, M. Fadel and M. Lewenstein, Probing quantum correlations in many-body systems: a review of scalable methods, Reports on Progress in Physics 86, 114001 (2023). https://doi.org/10.1088/1361-6633/acf8d7 [2] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum Entanglement, Rev. Mod. Phys. 81, 865 (2009). https://doi.org/10.1103/RevModPhys.81.865 [3] I. M. Georgescu, S. Ashhab and F. Nori, Quantum simulation, Rev. Mod. Phys. 86, 153 (2014). https://doi.org/10.1103/RevModPhys.86.153 [4] F. Verstraete, M. M. Wolf and J. I. Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nat. Phys. 5, 633 (2009). https://doi.org/10.1038/nphys1342 [5] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler and P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms, Nat. Phys. 4, 878 (2008). https://doi.org/10.1038/nphys1073 [6] H. Weimer, M. Müller, I. Lesanovsky, P. Zoller and H. P. Büchler, A Rydberg quantum simulator, Nature Physics 6, 382 (2010). https://doi.org/10.1038/nphys1614 [7] M. Müller, S. Diehl, G. Pupillo and P. Zoller, Engineered Open Systems and Quantum Simulations with Atoms and Ions, Adv. At. Mol. Opt. Phys. 61, 1 (2012). https://doi.org/10.1016/B978-0-12-396482-3.00001-6 [8] T. E. Lee, S. Gopalakrishnan and M. D. Lukin, Unconventional Magnetism via Optical Pumping of Interacting Spin Systems, Phys. Rev. Lett. 110, 257204 (2013). https://doi.org/10.1103/PhysRevLett.110.257204 [9] J. Jin, A. Biella, O. Viyuela, L. Mazza, J. Keeling, R. Fazio and D. Rossini, Cluster Mean-Field Approach to the Steady-State Phase Diagram of Dissipative Spin Systems, Phys. Rev. X 6, 031011 (2016). https://doi.org/10.1103/PhysRevX.6.031011 [10] W. Verstraelen, D. Huybrechts, T. Roscilde and M. Wouters, Quantum and Classical Correlations in Open Quantum Spin Lattices via Truncated-Cumulant Trajectories, PRX Quantum 4, 030304 (2023). https://doi.org/10.1103/PRXQuantum.4.030304 [11] V. R. Overbeck, M. F. Maghrebi, A. V. Gorshkov and H. Weimer, Multicritical behavior in dissipative Ising models, Phys. Rev. A 95, 042133 (2017). https://doi.org/10.1103/PhysRevA.95.042133 [12] S. Finazzi, A. Le Boité, F. Storme, A. Baksic and C. Ciuti, Corner-Space Renormalization Method for Driven-Dissipative Two-Dimensional Correlated Systems, Phys. Rev. Lett. 115, 080604 (2015). https://doi.org/10.1103/PhysRevLett.115.080604 [13] H. Weimer, Variational Principle for Steady States of Dissipative Quantum Many-Body Systems, Phys. Rev. Lett. 114, 040402 (2015). https://doi.org/10.1103/PhysRevLett.114.040402 [14] L. M. Sieberer, M. Buchhold and S. Diehl, Keldysh field theory for driven open quantum systems, Reports on Progress in Physics 79, 096001 (2016). https://doi.org/10.1088/0034-4885/79/9/096001 [15] N. Shammah, S. Ahmed, N. Lambert, S. De Liberato and F. Nori, Open quantum systems with local and collective incoherent processes: Efficient numerical simulations using permutational invariance, Phys. Rev. A 98, 063815 (2018). https://doi.org/10.1103/PhysRevA.98.063815 [16] F. Vicentini, A. Biella, N. Regnault and C. Ciuti, Variational Neural-Network Ansatz for Steady States in Open Quantum Systems, Phys. Rev. Lett. 122, 250503 (2019). https://doi.org/10.1103/PhysRevLett.122.250503 [17] A. Nagy and V. Savona, Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems, Phys. Rev. Lett. 122, 250501 (2019). https://doi.org/10.1103/PhysRevLett.122.250501 [18] M. J. Hartmann and G. Carleo, Neural-Network Approach to Dissipative Quantum Many-Body Dynamics, Phys. Rev. Lett. 122, 250502 (2019). https://doi.org/10.1103/PhysRevLett.122.250502 [19] H. Weimer, A. Kshetrimayum and R. Orús, Simulation methods for open quantum many-body systems, Rev. Mod. Phys. 93, 015008 (2021). https://doi.org/10.1103/RevModPhys.93.015008 [20] P. Deuar, A. Ferrier, M. Matuszewski, G. Orso and M. H. Szymanska, Fully Quantum Scalable Description of Driven-Dissipative Lattice Models, PRX Quantum 2, 010319 (2021). https://doi.org/10.1103/PRXQuantum.2.010319 [21] C. D. Mink, D. Petrosyan and M. Fleischhauer, Hybrid discrete-continuous truncated Wigner approximation for driven, dissipative spin systems, Phys. Rev. Res. 4, 043136 (2022). https://doi.org/10.1103/PhysRevResearch.4.043136 [22] M. Reitz, C. Sommer and C. Genes, Cooperative Quantum Phenomena in Light-Matter Platforms, PRX Quantum 3, 010201 (2022). https://doi.org/10.1103/PRXQuantum.3.010201 [23] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018). https://doi.org/10.1103/RevModPhys.90.035005 [24] K. Cong, Q. Zhang, Y. Wang, G. T. Noe, A. Belyanin and J. Kono, Dicke superradiance in solids, J. Opt. Soc. Am. B 33, C80 (2016). https://doi.org/10.1364/JOSAB.33.000C80 [25] G. Ferioli, A. Glicenstein, F. Robicheaux, R. T. Sutherland, A. Browaeys and I. Ferrier-Barbut, Laser-Driven Superradiant Ensembles of Two-Level Atoms near Dicke Regime, Phys. Rev. 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Robicheaux, "Qualitatively altered driven Dicke superradiance in extended systems due to infinitesimal perturbations", Physical Review A 110 6, 063701 (2024). [3] Wenqi Tong, H. Alaeian, and F. Robicheaux, "Phase transitions in the open Dicke model: A degenerate-perturbation-theory approach", Physical Review A 112 5, 053721 (2025). The above citations are from SAO/NASA ADS (last updated successfully 2026-04-28 07:10:43). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-04-28 07:10:42: Could not fetch cited-by data for 10.22331/q-2026-04-28-2085 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.