- Open Access
Learning spectral density functions in open quantum systems
Phys. Rev. A 114, 022402 – Published 3 August, 2026
DOI: https://doi.org/10.1103/bqw4-317c
Abstract
Spectral density functions quantify how environmental modes couple to quantum systems and govern their open dynamics. Inferring such frequency-dependent functions from time-domain measurements is an ill-conditioned inverse problem. Here, we use exactly solvable spin-boson models with pure-dephasing and amplitude-damping channels to reconstruct spectral density functions from noisy simulated data. First, we introduce a parameter estimation approach based on machine learning regressors to infer Lorentzian and Ohmic-like spectral density parameters, quantifying robustness to noise. Second, we show that a cosine transform inversion yields a physics-consistent spectral prior estimation, which is refined by a constrained neural network enforcing positivity and correct asymptotic behavior. Our neural-network framework robustly reconstructs structured spectral densities by filtering simulated noisy signals and learning general functional dependencies.
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References (26)
- I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys. 89, 015001 (2017).
- H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, Oxford, 2002).
- A. O. Caldeira and A. J. Leggett, Path integral approach to quantum Brownian motion, Physica A 121, 587 (1983).
- A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Dynamics of the dissipative two-state system, Rev. Mod. Phys. 59, 1 (1987).
- U. Weiss, Quantum Dissipative Systems, 3rd ed. (World Scientific, Singapore, 2008).
- I. Wilson-Rae and A. Imamoğlu, Quantum dot cavity-QED in the presence of strong electron-phonon interactions, Phys. Rev. B 65, 235311 (2002).
- A. Alkauskas, B. B. Buckley, D. D. Awschalom, and C. G. Van de Walle, First-principles theory of the luminescence lineshape for the triplet transition in diamond NV centres, New J. Phys. 16, 073026 (2014).
- A. Norambuena, S. A. Reyes, J. Mejía-López, A. Gali, and J. R. Maze, Microscopic modeling of the effect of phonons on the optical properties of solid-state emitters, Phys. Rev. B 94, 134305 (2016).
- M. C. Cambria, A. Norambuena, H. T. Dinani, G. Thiering, A. Gardill, I. Kemeny, Y. Li, V. Lordi, A. Gali, J. R. Maze, and S. Kolkowitz, Temperature-dependent spin-lattice relaxation of the nitrogen-vacancy spin triplet in diamond, Phys. Rev. Lett. 130, 256903 (2023).
- F. J. González, D. Tancara, H. T. Dinani, R. Coto, and A. Norambuena, Engineering non-Markovianity from defect-phonon interactions, New J. Phys. 25, 043004 (2023).
- J. Barr, G. Zicari, A. Ferraro, and M. Paternostro, Spectral density classification for environment spectroscopy, Mach. Learn.: Sci. Technol. 5, 015043 (2024).
- J. Barr, S. Mukherjee, A. Ferraro, M. Paternostro, and G. Zicari, A machine learning based approach to the identification of spectral densities in quantum open systems, Eur. Phys. J. Spec. Top. 235, 2977 (2026).
- J. Barr, A. Ferraro, M. Paternostro, and G. Zicari, Machine learning-enhanced characterisation of structured spectral densities: Leveraging the reaction coordinate mapping, arXiv:2501.07485.
- F. Mascherpa, A. Smirne, S. F. Huelga, and M. B. Plenio, Open systems with error bounds: Spin-boson model with spectral density variations, Phys. Rev. Lett. 118, 100401 (2017).
- A. Norambuena, J. R. Maze, P. Rabl, and R. Coto, Quantifying phonon-induced non-Markovianity in color centers in diamond, Phys. Rev. A 101, 022110 (2020).
- P. Haikka, T. H. Johnson, and S. Maniscalco, Non-Markovianity of local dephasing channels and time-invariant discord, Phys. Rev. A 87, 010103(R) (2013).
- J. Łuczka, Spin in contact with thermostat: Exact reduced dynamics, Physica A 167, 919 (1990).
- T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113, 140401 (2014).
- G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Syst. 2, 303 (1989).
- K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4, 251 (1991).
- P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, Philadelphia, 1998).
- F. Peleteiro, J. V. S. K. Lima, P. M. Prado, F. F. Fanchini, and A. Norambuena, Learning spectral density functions in open quantum systems: Code and reproducibility data, GitHub repository, 2026, https://github.com/Ariel-Norambuena/learning-spectral-density-functions.
- C. Cortes and V. Vapnik, Support-vector networks, Mach. Learn. 20, 273 (1995).
- L. Breiman, Random forests, Mach. Learn. 45, 5 (2001).
- J. H. Friedman, Greedy function approximation: A gradient boosting machine, Ann. Stat. 29, 1189 (2001).
- T. Chen and C. Guestrin, XGBoost: A scalable tree boosting system, in Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (ACM, New York, 2016), pp. 785–794.
