- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Interpreting Liouvillian Eigenmodes: Quantized Orbits in Dissipative Systems

Ashlin V. Thomas1,2, Felix Fritzsch3, Masudul Haque4,3, and Shovan Dutta5,*

  • *Contact author: shovan.dutta@rri.res.in
PDF

Phys. Rev. Lett. 137, 200404 – Published 12 November, 2026

DOI: https://doi.org/10.1103/nxhj-g7gh

Abstract

In closed quantum dynamics, energy eigenstates represent physical excitations which, for systems with a classical limit, localize on quantized orbits in phase space. Such a picture is lacking in open systems: Liouvillians have distinct right and left eigenstates that decay with time and are difficult to interpret. We recover a physical correspondence by introducing a gauge-invariant quasiprobability measure combining right and left eigenstates that interprets a Liouvillian mode as a set of coherences, allowing one to visualize the modes as quasiprobability distributions in a “doubled” phase space. Using this measure we show that, remarkably, a damped oscillator retains its quantized “orbits” for a large class of linear and nonlinear loss, with measurable signatures. A thermal bath broadens these orbits, similar to energy levels, leading to a characteristic multistage relaxation. This provides an experimentally accessible open-system analog of the eigenstate-orbit correspondence underlying semiclassical quantization. For quadratic systems, our results yield dissipative analogs of invariant tori—a central concept in Hamiltonian systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (94)

  1. P. Wölfle, Quasiparticles in condensed matter systems, Rep. Prog. Phys. 81, 032501 (2018).
  2. M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature (London) 452, 854 (2008).
  3. L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016).
  4. R. Nandkishore and D. A. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  5. C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Weak ergodicity breaking from quantum many-body scars, Nat. Phys. 14, 745 (2018).
  6. J. B. Keller, Semiclassical mechanics, SIAM Rev. 27, 485 (1985).
  7. I. C. Percival, Semiclassical theory of bound states, Adv. Chem. Phys. 36, 1 (1977).
  8. M. V. Berry and K. E. Mount, Semiclassical approximations in wave mechanics, Rep. Prog. Phys. 35, 315 (1972).
  9. P. M. Harrington, E. J. Mueller, and K. W. Murch, Engineered dissipation for quantum information science, Nat. Rev. Phys. 4, 660 (2022).
  10. P. Szańkowski, Introduction to the theory of open quantum systems, SciPost Phys. Lect. Notes 68, 1 (2023).
  11. S. Dutta, An introduction to Markovian open quantum systems, arXiv:2510.26530.
  12. A. J. Daley, Quantum trajectories and open many-body quantum systems, Adv. Phys. 63, 77 (2014).
  13. L. M. Sieberer, M. Buchhold, and S. Diehl, Keldysh field theory for driven open quantum systems, Rep. Prog. Phys. 79, 096001 (2016).
  14. H. Weimer, A. Kshetrimayum, and R. Orús, Simulation methods for open quantum many-body systems, Rev. Mod. Phys. 93, 015008 (2021).
  15. R. Fazio, J. Keeling, L. Mazza, and M. Schirò, Many-body open quantum systems, SciPost Phys. Lect. Notes, 99 (2025).
  16. I. Carusotto, How to exploit driving and dissipation to stabilize and manipulate quantum many-body states, C.R. Phys. 26, 533 (2025).
  17. M. Stefanini, A. A. Ziolkowska, D. Budker, U. Poschinger, F. Schmidt-Kaler, A. Browaeys, A. Imamoglu, D. Chang, and J. Marino, Is Lindblad for me?, SciPost Phys. Lect. Notes 129 (2026).
  18. I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys. 89, 015001 (2017).
  19. W. Demtröder, Laser Spectroscopy 1: Basic Principles (Springer, Berlin, 2014), Chap. 3.
  20. M. Brune, J. Bernu, C. Guerlin, S. Deléglise, C. Sayrin, S. Gleyzes, S. Kuhr, I. Dotsenko, J. M. Raimond, and S. Haroche, Process tomography of field damping and measurement of Fock state lifetimes by quantum nondemolition photon counting in a cavity, Phys. Rev. Lett. 101, 240402 (2008).
  21. H. Wang, M. Hofheinz, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, A. N. Cleland, and J. M. Martinis, Measurement of the decay of Fock states in a superconducting quantum circuit, Phys. Rev. Lett. 101, 240401 (2008).
  22. S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, Reconstruction of non-classical cavity field states with snapshots of their decoherence, Nature (London) 455, 510 (2008).
  23. A. I. Lvovsky and M. G. Raymer, Continuous-variable optical quantum-state tomography, Rev. Mod. Phys. 81, 299 (2009).
  24. I. Yang, T. Agrenius, V. Usova, O. Romero-Isart, and G. Kirchmair, Hot Schrödinger cat states, Sci. Adv. 11, eadr4492 (2025).
  25. D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Resolving photon number states in a superconducting circuit, Nature (London) 445, 515 (2007).
  26. M. F. Gely, M. Kounalakis, C. Dickel, J. Dalle, R. Vatré, B. Baker, M. D. Jenkins, and G. A. Steele, Observation and stabilization of photonic Fock states in a hot radio-frequency resonator, Science 363, 1072 (2019).
  27. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys. (N. Y.) 17, 821 (1976).
  28. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  29. H. J. Carmichael, An Open Systems Approach to Quantum Optics (Springer, Berlin, 1993).
  30. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, Berlin, 1999).
  31. C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer, Berlin, 2000).
  32. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, New York, 2002).
  33. H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, Cambridge, England, 2009).
  34. T. Prosen and T. H. Seligman, Quantization over boson operator spaces, J. Phys. A 43, 392004 (2010).
  35. J. Richter, M. Haque, and L. Sá, Localization and delocalization of quantum trajectories in the Liouvillian spectrum, arXiv:2511.19700.
  36. J. Watrous, The Theory of Quantum Information (Cambridge University Press, Cambridge, England, 2018).
  37. C. B. Naves and J. Larson, Liouville-Fock-state lattices: From environment-induced frustration to synthetic lattice dynamics, Phys. Rev. Res. 7, 043233 (2025).
  38. P. Kirton, M. M. Roses, J. Keeling, and E. G. Dalla Torre, Introduction to the Dicke model: From equilibrium to nonequilibrium, and vice versa, Adv. Quantum Technol. 2, 1800043 (2019).
  39. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86, 1391 (2014).
  40. J. C. Louw, M. Kastner, and J. N. Kriel, Bosonic representation of a Lipkin-Meshkov-Glick model with Markovian dissipation, Phys. Rev. B 102, 094430 (2020).
  41. A. K. Forbes, P. D. Blocher, and I. H. Deutsch, Modeling local decoherence of a spin ensemble using a generalized Holstein–Primakoff mapping to a bosonic mode, Opt. Quantum 2, 310 (2024).
  42. K. Husimi, Some formal properties of the density matrix, Proc. Phys. Math. Soc. Jpn. 22, 264 (1940).
  43. P. D. Drummond and D. F. Walls, Quantum theory of optical bistability. I. Nonlinear polarisability model, J. Phys. A 13, 725 (1980).
  44. B. Buča and T. Prosen, A note on symmetry reductions of the Lindblad equation: Transport in constrained open spin chains, New J. Phys. 14, 073007 (2012).
  45. V. V. Albert and L. Jiang, Symmetries and conserved quantities in Lindblad master equations, Phys. Rev. A 89, 022118 (2014).
  46. B. Baumgartner and H. Narnhofer, Analysis of quantum semigroups with GKS–Lindblad generators: II. General, J. Phys. A 41, 395303 (2008).
  47. M. S. Zubairy and J. J. Yeh, Photon statistics in multiphoton absorption and emission processes, Phys. Rev. A 21, 1624 (1980).
  48. H.-J. Briegel and B.-G. Englert, Quantum optical master equations: The use of damping bases, Phys. Rev. A 47, 3311 (1993).
  49. R. Endo, K. Fujii, and T. Suzuki, General solution of the quantum damped harmonic oscillator, Int. J. Geom. Methods Mod. Phys. 05, 653 (2008).
  50. D. Honda, H. Nakazato, and M. Yoshida, Spectral resolution of the Liouvillian of the Lindblad master equation for a harmonic oscillator, J. Math. Phys. (N. Y.) 51, 072107 (2010).
  51. S. Dutta, S. Zhang, and M. Haque, Quantum origin of limit cycles, fixed points, and critical slowing down, Phys. Rev. Lett. 134, 050407 (2025).
  52. N. Lu, Effects of dissipation on photon statistics and the lifetime of a pure number state, Phys. Rev. A 40, 1707 (1989).
  53. V. V. Dodonov and S. S. Mizrahi, Competition between one- and two-photon absorption processes, J. Phys. A 30, 2915 (1997).
  54. K. V. Kheruntsyan, Wigner function for a driven anharmonic oscillator, J. Opt. B 1, 225 (1999).
  55. N. Bartolo, F. Minganti, W. Casteels, and C. Ciuti, Exact steady state of a Kerr resonator with one- and two-photon driving and dissipation: Controllable Wigner-function multimodality and dissipative phase transitions, Phys. Rev. A 94, 033841 (2016).
  56. O. Scarlatella, A. A. Clerk, and M. Schiro, Spectral functions and negative density of states of a driven-dissipative nonlinear quantum resonator, New J. Phys. 21, 043040 (2019).
  57. D. Roberts and A. A. Clerk, Driven-dissipative quantum Kerr resonators: New exact solutions, photon blockade and quantum bistability, Phys. Rev. X 10, 021022 (2020).
  58. M. Tokieda, Exact diagonalization of a non-quadratic bosonic Liouvillian with two-body loss, arXiv:2603.27480.
  59. P. D. Drummond and Z. Ficek, Quantum Squeezing (Springer, Berlin, 2004).
  60. M. S. Kim, F. A. M. de Oliveira, and P. L. Knight, Properties of squeezed number states and squeezed thermal states, Phys. Rev. A 40, 2494 (1989).
  61. M. Saraceno and A. M. O. d. Almeida, Representation of superoperators in double phase space, J. Phys. A 49, 145302 (2016).
  62. B. Koczor, R. Zeier, and S. J. Glaser, Continuous phase-space representations for finite-dimensional quantum states and their tomography, Phys. Rev. A 101, 022318 (2020).
  63. G. S. Agarwal, Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions, Phys. Rev. A 24, 2889 (1981).
  64. J. P. Dowling, G. S. Agarwal, and W. P. Schleich, Wigner distribution of a general angular-momentum state: Applications to a collection of two-level atoms, Phys. Rev. A 49, 4101 (1994).
  65. J. C. Várilly and J. Gracia-Bondía, The Moyal representation for spin, Ann. Phys. (N.Y.) 190, 107 (1989).
  66. T. Tilma, M. J. Everitt, J. H. Samson, W. J. Munro, and K. Nemoto, Wigner functions for arbitrary quantum systems, Phys. Rev. Lett. 117, 180401 (2016).
  67. J. Davis, M. Kumari, R. B. Mann, and S. Ghose, Wigner negativity in spin-j systems, Phys. Rev. Res. 3, 033134 (2021).
  68. Q.-W. Wang, Exact dynamical correlations of nonlocal operators in quadratic open fermion systems: a characteristic function approach, SciPost Phys. Core 5, 027 (2022).
  69. D. A. Trifonov, Pseudo-boson coherent and Fock states, in Trends in Differential Geometry, Complex Analysis and Mathematical Physics (World Scientific, Singapore, 2009).
  70. F. Bagarello, Pseudobosons, Riesz bases, and coherent states, J. Math. Phys. (N. Y.) 51, 023531 (2010).
  71. F. Bagarello, A concise review of pseudobosons, pseudofermions, and their relatives, Theor. Math. Phys. 193, 1680 (2017).
  72. F. Bagarello, Pseudo-Bosons and Their Coherent States (Springer, Berlin, 2022).
  73. A. R. F. Neto, A. M. O. de Almeida, and O. Brodier, Exact Markovian evolution of quantum systems with several degrees of freedom: Phase space representations, Phys. Rev. E 108, 044106 (2023).
  74. E. Engdahl, T. Maniv, and N. Moiseyev, Complex quasiprobability for atoms trapped on surfaces: A novel application of the complex coordinate method, J. Chem. Phys. 88, 5864 (1988).
  75. W. T. Strunz and I. C. Percival, Classical mechanics from quantum state diffusion—a phase-space approach, J. Phys. A 31, 1801 (1998).
  76. J. Dubois, U. Saalmann, and J. M. Rost, Semi-classical Lindblad master equation for spin dynamics, J. Phys. A 54, 235201 (2021).
  77. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer, Berlin, 1996).
  78. P. Gaspard, G. Nicolis, A. Provata, and S. Tasaki, Spectral signature of the pitchfork bifurcation: Liouville equation approach, Phys. Rev. E 51, 74 (1995).
  79. M. D. Chekroun, A. Tantet, H. A. Dijkstra, and J. D. Neelin, Ruelle–Pollicott resonances of stochastic systems in reduced state space. Part I: Theory, J. Stat. Phys. 179, 1366 (2020).
  80. J.-P. Eckmann and M. Hairer, Spectral properties of hypoelliptic operators, Commun. Math. Phys. 235, 233 (2003).
  81. F. Hérau, M. Hitrik, and J. Sjöstrand, Tunnel effect and symmetries for Kramers–Fokker–Planck type operators, J. Inst. Math. Jussieu 10, 567 (2011).
  82. D. Edmunds and D. Evans, Spectral Theory and Differential Operators (Oxford University Press, Oxford, New York, 2018).
  83. E. B. Davies, Non-self-adjoint differential operators, Bull. Lond. Math. Soc. 34, 513 (2002).
  84. T. Prosen, Third quantization: A general method to solve master equations for quadratic open Fermi systems, New J. Phys. 10, 043026 (2008).
  85. M. Wolinsky and H. J. Carmichael, Quantum noise in the parametric oscillator: From squeezed states to coherent-state superpositions, Phys. Rev. Lett. 60, 1836 (1988).
  86. A. V. Thomas, F. Fritzsch, M. Haque, and S. Dutta, Code and data for “Interpreting Liouvillian eigenmodes: Quantized orbits in dissipative systems,” Zenodo (2026), 10.5281/zenodo.21363660.
  87. S. Axler, Linear Algebra Done Right (Springer, Cham, 2022), Chap. 5C.
  88. J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics (Cambridge University Press, Cambridge, England, 2020).
  89. F. Olver, D. Lozier, R. Boisvert, and C. Clark, The NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, England, 2010).
  90. R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues (Springer, Berlin, 2010), Chap. 9.10.
  91. X. Jin and R. Wong, Uniform asymptotic expansions for Meixner polynomials, Constr. Approx. 14, 113 (1998).
  92. L. Weisner, Group-theoretic origin of certain generating function, Pac. J. Math. 5, 1033 (1955).
  93. X. S. Wang and R. Wong, Uniform asymptotics of the Meixner polynomials, Sec. V, arXiv:0811.2624.
  94. W.-Y. Qiu and R. Wong, Global asymptotic expansions of the Laguerre polynomials—a Riemann–Hilbert approach, Numerical Algorithms 49, 331 (2008).

Outline

Information

Phys. Rev. Lett. 137, 200404– Published 12 November, 2026
  • Received 20 July 2026
  • Accepted 22 October 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation