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Power-law distributions in nonequilibrium open quantum systems

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Power-law distributions in nonequilibrium open quantum systems

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AbstractPower-law probability distributions are widely used to model extreme statistical events in complex systems, with applications to a vast array of natural phenomena ranging from earthquakes to stock market crashes to pandemics. We show that analogous heavy tails arise naturally in open quantum systems with nonlinear dissipation. Introducing a prototypical family of quantum dynamical models, we analytically prove the emergence of power-law tails in the steady state energy distribution, originating from an amplification of quantum noise whose microscopic fluctuations grow with energy. Moreover, our analysis suggests a general mechanism for heavy-tail statistics in the nonequilibrium steady states of open quantum systems: Nonlinear dissipation generically induces multiplicative quantum noise, enforced by the constraints of quantum mechanics, which is responsible for the heavy-tail behavior. This is supported by numerical simulations of a general class of nonlinear dynamics known as quantum LiÃ©nard systems. Remarkably, even when the corresponding classical system is stable, we find power-law tails in both steady-state populations and coherences, which occur for typical parameters without fine-tuning. This phenomenon can potentially be harnessed to develop extreme photon sources for novel applications in light-matter interaction and sensing.Featured image: The steady-state density matrix elements decay as a power law with increasing excitation number, indicating the presence of rare but large fluctuations. The inset shows the corresponding Wigner distribution in the steady state.► BibTeX data@article{Mok2026powerlaw, doi = {10.22331/q-2026-04-08-2054}, url = {https://doi.org/10.22331/q-2026-04-08-2054}, title = {Power-law distributions in nonequilibrium open quantum systems}, author = {Mok, Wai-Keong}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2054}, month = apr, year = {2026} }► References [1] MEJ Newman. Power laws, pareto distributions and zipf's law. Contemp. Phys., 46 (5): 323–351, 2005. 10.1080/00107510500052444. URL https://doi.org/10.1080/00107510500052444. https://doi.org/10.1080/00107510500052444 [2] Pablo A. Marquet, Renato A. Quiñones, Sebastian Abades, Fabio Labra, Marcelo Tognelli, Matias Arim, and Marcelo Rivadeneira. Scaling and power-laws in ecological systems. J. Exp. Biol., 208 (9): 1749–1769, 05 2005. ISSN 0022-0949. 10.1242/jeb.01588. URL https://doi.org/10.1242/jeb.01588. https://doi.org/10.1242/jeb.01588 [3] Alvaro Corral and Alvaro González. Power law size distributions in geoscience revisited.

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The Quantum Theory of Nonlinear Optics.

Cambridge University Press, 2014. 10.1017/CBO9780511783616. https://doi.org/10.1017/CBO9780511783616 [59] Neill Lambert, Eric Gigu`ere, Paul Menczel, Boxi Li, Patrick Hopf, Gerardo Su'arez, Marc Gali, Jake Lishman, Rushiraj Gadhvi, Rochisha Agarwal, Asier Galicia, Nathan Shammah, Paul Nation, J. R. Johansson, Shahnawaz Ahmed, Simon Cross, Alexander Pitchford, and Franco Nori. Qutip 5: The quantum toolbox in Python. Physics Reports, 1153: 1–62, 2026. ISSN 0370-1573. 10.1016/j.physrep.2025.10.001. URL https://www.sciencedirect.com/science/article/pii/S0370157325002704. https://doi.org/10.1016/j.physrep.2025.10.001 https://www.sciencedirect.com/science/article/pii/S0370157325002704Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-08 08:41:11: Could not fetch cited-by data for 10.22331/q-2026-04-08-2054 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-08 08:41:11: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractPower-law probability distributions are widely used to model extreme statistical events in complex systems, with applications to a vast array of natural phenomena ranging from earthquakes to stock market crashes to pandemics. We show that analogous heavy tails arise naturally in open quantum systems with nonlinear dissipation. Introducing a prototypical family of quantum dynamical models, we analytically prove the emergence of power-law tails in the steady state energy distribution, originating from an amplification of quantum noise whose microscopic fluctuations grow with energy. Moreover, our analysis suggests a general mechanism for heavy-tail statistics in the nonequilibrium steady states of open quantum systems: Nonlinear dissipation generically induces multiplicative quantum noise, enforced by the constraints of quantum mechanics, which is responsible for the heavy-tail behavior. This is supported by numerical simulations of a general class of nonlinear dynamics known as quantum LiÃ©nard systems. Remarkably, even when the corresponding classical system is stable, we find power-law tails in both steady-state populations and coherences, which occur for typical parameters without fine-tuning. This phenomenon can potentially be harnessed to develop extreme photon sources for novel applications in light-matter interaction and sensing.Featured image: The steady-state density matrix elements decay as a power law with increasing excitation number, indicating the presence of rare but large fluctuations. The inset shows the corresponding Wigner distribution in the steady state.► BibTeX data@article{Mok2026powerlaw, doi = {10.22331/q-2026-04-08-2054}, url = {https://doi.org/10.22331/q-2026-04-08-2054}, title = {Power-law distributions in nonequilibrium open quantum systems}, author = {Mok, Wai-Keong}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2054}, month = apr, year = {2026} }► References [1] MEJ Newman. Power laws, pareto distributions and zipf's law. Contemp. Phys., 46 (5): 323–351, 2005. 10.1080/00107510500052444. URL https://doi.org/10.1080/00107510500052444. https://doi.org/10.1080/00107510500052444 [2] Pablo A. Marquet, Renato A. Quiñones, Sebastian Abades, Fabio Labra, Marcelo Tognelli, Matias Arim, and Marcelo Rivadeneira. Scaling and power-laws in ecological systems. J. Exp. Biol., 208 (9): 1749–1769, 05 2005. ISSN 0022-0949. 10.1242/jeb.01588. URL https://doi.org/10.1242/jeb.01588. https://doi.org/10.1242/jeb.01588 [3] Alvaro Corral and Alvaro González. Power law size distributions in geoscience revisited.

The Quantum Theory of Nonlinear Optics.

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Source: Quantum Journal

Website: https://quantum-journal.org/feed/

Power-law distributions in nonequilibrium open quantum systems

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