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Full Counting Statistics and First-Passage Times in Quantum Markovian Processes: Ensemble Relations, Metastability, and Fluctuation Theorems

Paul Menczel1,*, Christian Flindt2,1, Fredrik Brange2, Franco Nori1,3, and Clemens Gneiting1

  • *Contact author: paul@menczel.net
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PRX Quantum 7, 010304 – Published 8 January, 2026

DOI: https://doi.org/10.1103/4hc1-l8w4

Abstract

We develop a comprehensive framework for characterizing fluctuations in quantum transport and nonequilibrium thermodynamics using two complementary approaches: full counting statistics and first-passage times. Focusing on open quantum systems governed by Markovian Lindblad dynamics, we derive general ensemble relations that connect the two approaches at all times, and we clarify how the steady states reached at long times relate to those reached at large jump counts. In regimes of metastability, long-lived intermediate states cause violations of experimentally testable cumulant relations, as we discuss. We also formulate a fluctuation theorem governing the probability of rare fluctuations in the first-passage time distributions based on results from full counting statistics. Our results apply to general integer-valued trajectory observables that do not necessarily increase monotonically in time. Three illustrative applications, a two-state emitter, a driven qubit, and a variant of the Su-Schrieffer-Heeger model, highlight the physical implications of our results and provide guidelines for practical calculations. Our framework provides a complete picture of first-passage time statistics in Markovian quantum systems, encompassing multiple earlier results, and it has direct implications for current experiments in quantum optics, superconducting circuits, and nanoscale heat engines.

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Popular Summary

Open quantum systems, such as electrons tunneling through a nanoscale device, are inherently random: the tunneling of electrons corresponds to quantum jumps in the device, which happen stochastically. To analyze the behavior of such systems, physicists typically count how many quantum jumps occur within a fixed amount of time. However, one can flip the question: how long does it take for a specific number of jumps to occur? These first-passage times offer a fresh perspective, acting like a stopwatch rather than a counter. In this study, we develop a comprehensive framework that unifies the two viewpoints, providing new insights into the fluctuations of quantum systems and novel techniques for probing these fluctuations in experiments.

As our main result, we derive general relations that translate the mathematics of the “how many” approach directly into the “how long” picture and vice versa. This correspondence acts as a dictionary, allowing researchers to switch between methods depending on which is easier to calculate. We discover that violations of simple relations between these statistics act as a signature of metastability, where the system gets temporarily stuck in long-lived states. We also formulate a fluctuation theorem that predicts the likelihood of rare events, such as a current flowing in reverse.

Our findings provide a versatile toolbox for analyzing nonequilibrium dynamics in emerging technologies, from superconducting circuits to nanoscale heat engines. Looking ahead, our results lay the foundations for the exploration of more complicated quantum systems, for example ones with memory, potentially revealing new thermodynamic principles.

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References (146)

  1. P. Hänggi and H. Thomas, Stochastic processes: Time evolution, symmetries and linear response, Phys. Rep. 88, 207 (1982).
  2. C. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer, Berlin, 2004).
  3. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 2011).
  4. M. H. Devoret and R. J. Schoelkopf, Superconducting Circuits for Quantum Information: An Outlook, Science 339, 1169 (2013).
  5. J. P. Pekola, Towards quantum thermodynamics in electronic circuits, Nat. Phys. 11, 118 (2015).
  6. X. Gu, A. F. Kockum, A. Miranowicz, Y.-X. Liu, and F. Nori, Microwave photonics with superconducting quantum circuits, Phys. Rep. 718–719, 1 (2017).
  7. A. F. Kockum and F. Nori, in Fundamentals and Frontiers of the Josephson Effect, edited by F. Tafuri (Springer International Publishing, Cham, 2019).
  8. M. A. Kastner, The single-electron transistor, Rev. Mod. Phys. 64, 849 (1992).
  9. J. V. Koski, T. Sagawa, O.-P. Saira, Y. Yoon, A. Kutvonen, P. Solinas, M. Möttönen, T. Ala-Nissila, and J. P. Pekola, Distribution of entropy production in a single-electron box, Nat. Phys. 9, 644 (2013).
  10. S. Gustavsson, R. Leturcq, B. Simovič, R. Schleser, T. Ihn, P. Studerus, K. Ensslin, D. C. Driscoll, and A. C. Gossard, Counting statistics of single electron transport in a quantum dot, Phys. Rev. Lett. 96, 076605 (2006).
  11. J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler, and K. Singer, A single-atom heat engine, Science 352, 325 (2016).
  12. M. Josefsson, A. Svilans, A. M. Burke, E. A. Hoffmann, S. Fahlvik, C. Thelander, M. Leijnse, and H. Linke, A quantum-dot heat engine operating close to the thermodynamic efficiency limits, Nat. Nanotechnol. 13, 920 (2018).
  13. J. Klatzow, J. N. Becker, P. M. Ledingham, C. Weinzetl, K. T. Kaczmarek, D. J. Saunders, J. Nunn, I. A. Walmsley, R. Uzdin, and E. Poem, Experimental demonstration of quantum effects in the operation of microscopic heat engines, Phys. Rev. Lett. 122, 110601 (2019).
  14. J. P. S. Peterson, T. B. Batalhão, M. Herrera, A. M. Souza, R. S. Sarthour, I. S. Oliveira, and R. M. Serra, Experimental characterization of a spin quantum heat engine, Phys. Rev. Lett. 123, 240601 (2019).
  15. A. Guthrie, C. D. Satrya, Y.-C. Chang, P. Menczel, F. Nori, and J. P. Pekola, Cooper-pair box coupled to two resonators: An architecture for a quantum refrigerator, Phys. Rev. Appl. 17, 064022 (2022).
  16. T. Dittrich, P. Hänggi, G.-L. Ingold, B. Kramer, G. Schön, and W. Zwerger, Quantum Transport and Dissipation (Wiley-VCH, Weinheim, 1998).
  17. Y. V. Nazarov and Y. M. Blanter, Quantum Transport: Introduction to Nanoscience (Cambridge University, Cambridge, 2009).
  18. D. D’Alessandro, Introduction to Quantum Control and Dynamics (CRC, New York, 2007).
  19. J. Zhang, Y.-x. Liu, R.-B. Wu, K. Jacobs, and F. Nori, Quantum feedback: Theory, experiments, and applications, Phys. Rep. 679, 1 (2017).
  20. F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Eds.), Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, Fundamental Theories of Physics, Vol. 195 (Springer International Publishing, Cham, 2018).
  21. G. Kurizki and A. G. Kofman, Thermodynamics and Control of Open Quantum Systems (Cambridge University, Cambridge, 2022).
  22. M. B. Plenio and P. L. Knight, The quantum-jump approach to dissipative dynamics in quantum optics, Rev. Mod. Phys. 70, 101 (1998).
  23. C. Gneiting, A. V. Rozhkov, and F. Nori, Jump-time unraveling of Markovian open quantum systems, Phys. Rev. A 104, 062212 (2021).
  24. G. T. Landi, M. J. Kewming, M. T. Mitchison, and P. P. Potts, Current fluctuations in open quantum systems: Bridging the gap between quantum continuous measurements and full counting statistics, PRX Quantum 5, 020201 (2024).
  25. S. Singh, P. Menczel, D. S. Golubev, I. M. Khaymovich, J. T. Peltonen, C. Flindt, K. Saito, É. Roldán, and J. P. Pekola, Universal first-passage-time distribution of non-Gaussian currents, Phys. Rev. Lett. 122, 230602 (2019).
  26. Y. M. Blanter and M. Büttiker, Shot noise in mesoscopic conductors, Phys. Rep. 336, 1 (2000).
  27. Y. V. Nazarov (Ed.), Quantum Noise in Mesoscopic Physics (Springer, Dordrecht, 2003).
  28. H. Touchette, The large deviation approach to statistical mechanics, Phys. Rep. 478, 1 (2009).
  29. C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690 (1997).
  30. G. E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60, 2721 (1999).
  31. U. Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theorem, Phys. Rev. Lett. 95, 040602 (2005).
  32. M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys. 81, 1665 (2009).
  33. M. Campisi, P. Hänggi, and P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and applications, Rev. Mod. Phys. 83, 771 (2011).
  34. S. Redner, A Guide to First-Passage Processes (Cambridge University, Cambridge, 2001).
  35. R. Metzler, S. Redner, and G. Oshanin, First-Passage Phenomena and Their Applications (World Scientific, Singapore, 2014).
  36. G. Manzano, R. Fazio, and É. Roldán, Quantum martingale theory and entropy production, Phys. Rev. Lett. 122, 220602 (2019).
  37. S. L. Rudge and D. S. Kosov, Fluctuating-time and full counting statistics for quantum transport in a system with internal telegraphic noise, Phys. Rev. B 100, 235430 (2019).
  38. S. L. Rudge and D. S. Kosov, Nonrenewal statistics in quantum transport from the perspective of first-passage and waiting time distributions, Phys. Rev. B 99, 115426 (2019).
  39. S. L. Rudge and D. S. Kosov, Counting quantum jumps: A summary and comparison of fixed-time and fluctuating-time statistics in electron transport, J. Chem. Phys. 151, 034107 (2019).
  40. T. Van Vu and K. Saito, Thermodynamics of precision in Markovian open quantum dynamics, Phys. Rev. Lett. 128, 140602 (2022).
  41. Y. Hasegawa, Thermodynamic uncertainty relation for quantum first-passage processes, Phys. Rev. E 105, 044127 (2022).
  42. X. He, P. Pakkiam, A. A. Gangat, M. J. Kewming, G. J. Milburn, and A. Fedorov, Effect of measurement backaction on quantum clock precision studied with a superconducting circuit, Phys. Rev. Appl. 20, 034038 (2023).
  43. M. J. Kewming, A. Kiely, S. Campbell, and G. T. Landi, First passage times for continuous quantum measurement currents, Phys. Rev. A 109, L050202 (2024).
  44. F. Liu, S. Xia, and S. Su, Semi-Markov processes in open quantum systems. III. Large deviations of first-passage-time statistics, Phys. Rev. E 110, 064145 (2024).
  45. G. Bakewell-Smith, F. Girotti, M. Guţă, and J. P. Garrahan, Bounds on fluctuations of first passage times for counting observables in classical and quantum Markov processes, J. Stat. Phys. 192, 126 (2025).
  46. A. A. Budini, R. M. Turner, and J. P. Garrahan, Fluctuating observation time ensembles in the thermodynamics of trajectories, J. Stat. Mech. 2014, P03012 (2014).
  47. T. R. Gingrich and J. M. Horowitz, Fundamental bounds on first passage time fluctuations for currents, Phys. Rev. Lett. 119, 170601 (2017).
  48. A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett. 114, 158101 (2015).
  49. T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Dissipation bounds all steady-state current fluctuations, Phys. Rev. Lett. 116, 120601 (2016).
  50. U. Seifert, Stochastic thermodynamics: From principles to the cost of precision, Phys. A 504, 176 (2018).
  51. J. M. Horowitz and T. R. Gingrich, Thermodynamic uncertainty relations constrain non-equilibrium fluctuations, Nat. Phys. 16, 15 (2020).
  52. F. Meier, Y. Minoguchi, S. Sundelin, T. J. G. Apollaro, P. Erker, S. Gasparinetti, and M. Huber, Precision is not limited by the second law of thermodynamics, Nat. Phys. 21, 1147 (2025).
  53. K. Macieszczak, Ultimate kinetic uncertainty relation and optimal performance of stochastic clocks, arXiv:2407.09839 [cond-mat.stat-mech].
  54. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys. 17, 821 (1976).
  55. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  56. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, Oxford, 2002).
  57. D. A. Lidar, Lecture notes on the theory of open quantum systems, arXiv:1902.00967 [quant-ph].
  58. K. Macieszczak, M. Guţă, I. Lesanovsky, and J. P. Garrahan, Towards a theory of metastability in open quantum dynamics, Phys. Rev. Lett. 116, 240404 (2016).
  59. K. Macieszczak, D. C. Rose, I. Lesanovsky, and J. P. Garrahan, Theory of classical metastability in open quantum systems, Phys. Rev. Res. 3, 033047 (2021).
  60. W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42, 1698 (1979).
  61. C. Gneiting, A. Koottandavida, A. V. Rozhkov, and F. Nori, Unraveling the topology of dissipative quantum systems, Phys. Rev. Res. 4, 023036 (2022).
  62. C. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences (Springer, Berlin, 2010).
  63. I. Goldhirsch and Y. Gefen, Analytic method for calculating properties of random walks on networks, Phys. Rev. A 33, 2583 (1986).
  64. I. Goldhirsch and Y. Gefen, Biased random walk on networks, Phys. Rev. A 35, 1317 (1987).
  65. É. Roldán, I. Neri, M. Dörpinghaus, H. Meyr, and F. Jülicher, Decision making in the arrow of time, Phys. Rev. Lett. 115, 250602 (2015).
  66. K. Saito and A. Dhar, Waiting for rare entropic fluctuations, EPL 114, 50004 (2016).
  67. I. Neri, É. Roldán, and F. Jülicher, Statistics of infima and stopping times of entropy production and applications to active molecular processes, Phys. Rev. X 7, 011019 (2017).
  68. O. Artime, N. Khalil, R. Toral, and M. San Miguel, First-passage distributions for the one-dimensional Fokker-Planck equation, Phys. Rev. E 98, 042143 (2018).
  69. I. Neri, Second law of thermodynamics at stopping times, Phys. Rev. Lett. 124, 040601 (2020).
  70. G. Manzano, D. Subero, O. Maillet, R. Fazio, J. P. Pekola, and É. Roldán, Thermodynamics of gambling demons, Phys. Rev. Lett. 126, 080603 (2021).
  71. S. Chandrasekhar, Dynamical friction. II. The rate of escape of stars from clusters and the evidence for the operation of dynamical friction, Astrophys. J. 97, 263 (1943).
  72. H. C. Berg and E. M. Purcell, Physics of chemoreception, Biophys. J. 20, 193 (1977).
  73. A. Szabo, K. Schulten, and Z. Schulten, First passage time approach to diffusion controlled reactions, J. Chem. Phys. 72, 4350 (1980).
  74. R. Zwanzig, A. Szabo, and B. Bagchi, Levinthal’s paradox, Proc. Natl. Acad. Sci. 89, 20 (1992).
  75. O. Bénichou, C. Chevalier, J. Klafter, B. Meyer, and R. Voituriez, Geometry-controlled kinetics, Nat. Chem. 2, 472 (2010).
  76. S. N. Majumdar, in The Legacy of Albert Einstein (World Scientific, Singapore, 2006).
  77. J. Perelló, M. Gutiérrez-Roig, and J. Masoliver, Scaling properties and universality of first-passage-time probabilities in financial markets, Phys. Rev. E 84, 066110 (2011).
  78. J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Mod. Phys. 48, 571 (1976).
  79. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  80. J. Dalibard, Y. Castin, and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Phys. Rev. Lett. 68, 580 (1992).
  81. K. Mølmer, Y. Castin, and J. Dalibard, Monte Carlo wave-function method in quantum optics, J. Opt. Soc. Am. B 10, 524 (1993).
  82. C. Flindt, T. Novotný, and A.-P. Jauho, Full counting statistics of nano-electromechanical systems, EPL 69, 475 (2005).
  83. H. Carmichael, An Open Systems Approach to Quantum Optics, Lecture Notes in Physics, Vol. 18 (Springer, Berlin, 1993).
  84. P. Menczel, C. Flindt, and K. Brandner, Quantum jump approach to microscopic heat engines, Phys. Rev. Res. 2, 033449 (2020).
  85. L. S. Levitov, H. Lee, and G. B. Lesovik, Electron counting statistics and coherent states of electric current, J. Math. Phys. 37, 4845 (1996).
  86. D. A. Bagrets and Y. V. Nazarov, Full counting statistics of charge transfer in Coulomb blockade systems, Phys. Rev. B 67, 085316 (2003).
  87. F. Minganti, A. Miranowicz, R. W. Chhajlany, I. I. Arkhipov, and F. Nori, Hybrid-Liouvillian formalism connecting exceptional points of non-Hermitian Hamiltonians and Liouvillians via postselection of quantum trajectories, Phys. Rev. A 101, 062112 (2020).
  88. X.-K. Gu, L.-Z. Tan, F. Nori, and J. Q. You, Exploring dynamics of open quantum systems in naturally inaccessible regimes, arXiv:2503.06946 [quant-ph].
  89. M. Marcuzzi, M. Buchhold, S. Diehl, and I. Lesanovsky, Absorbing state phase transition with competing quantum and classical fluctuations, Phys. Rev. Lett. 116, 245701 (2016).
  90. F. Carollo, E. Gillman, H. Weimer, and I. Lesanovsky, Critical behavior of the quantum contact process in one dimension, Phys. Rev. Lett. 123, 100604 (2019).
  91. F. Carollo and I. Lesanovsky, Nonequilibrium dark space phase transition, Phys. Rev. Lett. 128, 040603 (2022).
  92. G. Perfetto, F. Carollo, J. P. Garrahan, and I. Lesanovsky, Reaction-limited quantum reaction-diffusion dynamics, Phys. Rev. Lett. 130, 210402 (2023).
  93. Formally, the integral is only convergent for parameters s with large enough real parts. However, the functions c¯n(s) can be analytically continued to holomorphic functions on the entire complex plane, except for some poles and branch cuts. Here, we always implicitly consider the analytic continuations of conditionally convergent integrals such as this one.
  94. G. T. Landi, Patterns in the jump-channel statistics of open quantum systems, arXiv:2305.07957 [quant-ph].
  95. J. P. Garrahan, Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables, Phys. Rev. E 95, 032134 (2017).
  96. K. Hiura and S.-i. Sasa, Kinetic uncertainty relation on first-passage time for accumulated current, Phys. Rev. E 103, L050103 (2021).
  97. A. Pal, S. Reuveni, and S. Rahav, Thermodynamic uncertainty relation for first-passage times on Markov chains, Phys. Rev. Res. 3, L032034 (2021).
  98. A. Raghu and I. Neri, Thermodynamic bounds and symmetries in first-passage problems of fluctuating currents, New J. Phys. 27, 124602 (2025).
  99. A. N. Jordan and E. V. Sukhorukov, Transport statistics of bistable systems, Phys. Rev. Lett. 93, 260604 (2004).
  100. G. Schaller, G. Kießlich, and T. Brandes, Counting statistics in multistable systems, Phys. Rev. B 81, 205305 (2010).
  101. T. Karzig and F. von Oppen, Signatures of critical full counting statistics in a quantum-dot chain, Phys. Rev. B 81, 045317 (2010).
  102. A. Deger, K. Brandner, and C. Flindt, Lee-Yang zeros and large-deviation statistics of a molecular zipper, Phys. Rev. E 97, 012115 (2018).
  103. J. K. Asbóth, L. Oroszlány, and A. Pályi, A Short Course on Topological Insulators, Lecture Notes in Physics, Vol. 919 (Springer International Publishing, Cham, 2016).
  104. F. Schmolke, Asymptotic fate of continuously monitored quantum systems, arXiv:2506.10873 [quant-ph].
  105. J. L. Lebowitz and H. Spohn, A Gallavotti–Cohen-type symmetry in the large deviation functional for stochastic dynamics, J. Stat. Phys. 95, 333 (1999).
  106. D. Andrieux and P. Gaspard, A fluctuation theorem for currents and non-linear response coefficients, J. Stat. Mech. 2007, P02006 (2007).
  107. D. Manzano and P. I. Hurtado, Symmetry and the thermodynamics of currents in open quantum systems, Phys. Rev. B 90, 125138 (2014).
  108. H. Spohn, Entropy production for quantum dynamical semigroups, J. Math. Phys. 19, 1227 (1978).
  109. T. Guérin, N. Levernier, O. Bénichou, and R. Voituriez, Mean first-passage times of non-Markovian random walkers in confinement, Nature 534, 356 (2016).
  110. T. Akimoto and K. Saito, Trace of anomalous diffusion in a biased quenched trap model, Phys. Rev. E 101, 042133 (2020).
  111. H. Scher and E. W. Montroll, Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B 12, 2455 (1975).
  112. F. Höfling and T. Franosch, Anomalous transport in the crowded world of biological cells, Rep. Prog. Phys. 76, 046602 (2013).
  113. 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).
  114. W.-M. Zhang, P.-Y. Lo, H.-N. Xiong, M. W.-Y. Tu, and F. Nori, General non-Markovian dynamics of open quantum systems, Phys. Rev. Lett. 109, 170402 (2012).
  115. H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Non-Markovian dynamics in open quantum systems, Rev. Mod. Phys. 88, 021002 (2016).
  116. I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Rev. Mod. Phys. 89, 015001 (2017).
  117. P. Talkner and P. Hänggi, Colloquium: Statistical mechanics and thermodynamics at strong coupling: Quantum and classical, Rev. Mod. Phys. 92, 041002 (2020).
  118. C. Flindt, T. Novotný, A. Braggio, M. Sassetti, and A.-P. Jauho, Counting statistics of non-Markovian quantum stochastic processes, Phys. Rev. Lett. 100, 150601 (2008).
  119. C. Flindt, T. Novotný, A. Braggio, and A.-P. Jauho, Counting statistics of transport through Coulomb blockade nanostructures: High-order cumulants and non-Markovian effects, Phys. Rev. B 82, 155407 (2010).
  120. W. D. Heiss, The physics of exceptional points, J. Phys. A: Math. Theor. 45, 444016 (2012).
  121. M.-A. Miri and A. Alù, Exceptional points in optics and photonics, Science 363, eaar7709 (2019).
  122. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  123. F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps, Phys. Rev. A 100, 062131 (2019).
  124. S. Khandelwal, N. Brunner, and G. Haack, Signatures of Liouvillian exceptional points in a quantum thermal machine, PRX Quantum 2, 040346 (2021).
  125. J.-D. Lin, P.-C. Kuo, N. Lambert, A. Miranowicz, F. Nori, and Y.-N. Chen, Non-Markovian quantum exceptional points, Nat. Commun. 16, 1289 (2025).
  126. J. P. Garrahan and I. Lesanovsky, Thermodynamics of quantum jump trajectories, Phys. Rev. Lett. 104, 160601 (2010).
  127. A. I. Pavlov, Y. Gefen, and A. Shnirman, Topological transitions in quantum jump dynamics: Hidden exceptional points, Phys. Rev. B 111, 104301 (2025).
  128. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).
  129. K. S. U. Kansanen, P. Portugal, C. Flindt, and P. Samuelsson, Photon counting statistics in Gaussian bosonic networks, SciPost Phys. 18, 116 (2025).
  130. S. Gleyzes, S. Kuhr, C. Guerlin, J. Bernu, S. Deléglise, U. Busk Hoff, M. Brune, J.-M. Raimond, and S. Haroche, Quantum jumps of light recording the birth and death of a photon in a cavity, Nature 446, 297 (2007).
  131. A. Kubanek, M. Koch, C. Sames, A. Ourjoumtsev, P. W. H. Pinkse, K. Murr, and G. Rempe, Photon-by-photon feedback control of a single-atom trajectory, Nature 462, 898 (2009).
  132. C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune, J.-M. Raimond, and S. Haroche, Real-time quantum feedback prepares and stabilizes photon number states, Nature 477, 73 (2011).
  133. R. Vijay, D. H. Slichter, and I. Siddiqi, Observation of quantum jumps in a superconducting artificial atom, Phys. Rev. Lett. 106, 110502 (2011).
  134. K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi, Observing single quantum trajectories of a superconducting quantum bit, Nature 502, 211 (2013).
  135. A. Delteil, W.-B. Gao, P. Fallahi, J. Miguel-Sanchez, and A. İmamoğlu, Observation of quantum jumps of a single quantum dot spin using submicrosecond single-shot optical readout, Phys. Rev. Lett. 112, 116802 (2014).
  136. A. Chantasri, M. E. Kimchi-Schwartz, N. Roch, I. Siddiqi, and A. N. Jordan, Quantum trajectories and their statistics for remotely entangled quantum bits, Phys. Rev. X 6, 041052 (2016).
  137. S. J. Weber, K. W. Murch, M. E. Kimchi-Schwartz, N. Roch, and I. Siddiqi, Quantum trajectories of superconducting qubits, C. R. Phys. 17, 766 (2016).
  138. A. Kurzmann, P. Stegmann, J. Kerski, R. Schott, A. Ludwig, A. D. Wieck, J. König, A. Lorke, and M. Geller, Optical detection of single-electron tunneling into a semiconductor quantum dot, Phys. Rev. Lett. 122, 247403 (2019).
  139. M. Heyl, Dynamical quantum phase transitions: A review, Rep. Prog. Phys. 81, 054001 (2018).
  140. J. R. Johansson, P. D. Nation, and F. Nori, QuTiP: An open-source Python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 183, 1760 (2012).
  141. J. R. Johansson, P. D. Nation, and F. Nori, QuTiP 2: A Python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 184, 1234 (2013).
  142. N. Lambert, E. Giguère, P. Menczel, B. Li, P. Hopf, G. Suárez, M. Gali, J. Lishman, R. Gadhvi, R. Agarwal, A. Galicia, N. Shammah, P. Nation, J. R. Johansson, S. Ahmed, S. Cross, A. Pitchford, and F. Nori, QuTiP 5: The quantum toolbox in Python, Phys. Rep. 1153, 1 (2026).
  143. B. Baumgartner and H. Narnhofer, Analysis of quantum semigroups with GKS–Lindblad generators: II. General, J. Phys. A 41, 395303 (2008).
  144. M. V. Keldysh, On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators, Russ. Math. Surv. 26, 15 (1971).
  145. R. Mennicken and M. Möller, Non-Self-Adjoint Boundary Eigenvalue Problems (Elsevier, Amsterdam, 2003).
  146. W.-J. Beyn, An integral method for solving nonlinear eigenvalue problems, Linear Algebra Appl. 436, 3839 (2012).

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PRX Quantum 7, 010304– Published 8 January, 2026
  • Received 3 September 2025
  • Revised 17 November 2025
  • Accepted 21 November 2025
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