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Invariant-based master equation applied to a driven qutrit coupled to a bath and a leaky cavity

Sagarika Basak1,2,*, A. Javadi1,2,3, and D. Blume1,2,†

  • 1Homer L. Dodge Department of Physics and Astronomy, The University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA
  • 2Center for Quantum Research and Technology, The University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA
  • 3School of Electrical and Computer Engineering, The University of Oklahoma, 110 W. Boyd Street, Norman, Oklahoma 73019, USA

  • *Contact author: basak.sagarika@ou.edu
  • †Contact author: doerte.blume-1@ou.edu
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Phys. Rev. B 114, 074108 – Published 31 August, 2026

DOI: https://doi.org/10.1103/lp44-ll2f

Abstract

We employ a generalized approach to the master equation for driven open N-level (N>2) quantum systems using Lewis–Riesenfeld invariants, which avoids the driving-strength restrictions inherent to conventional approaches. We show that the invariant-based master equation provides a unifying generalized framework, which reduces to the frequently employed laboratory-frame master equations and the less frequently employed rotating-frame master equation framework under appropriate simplifications. Extending the prototypical two-level system, we show that the inclusion of another state coupled to the ground state via reservoir-induced dephasing gives rise to qualitatively new dissipative behaviors that are, in general, not captured by standard approximations. We also apply the invariant-based master-equation framework to a driven quantum dot coupled to a leaky cavity, demonstrating the framework's ability to capture relevant dissipative dynamics without additional assumptions. Our work paves the way for quantum-control applications in the presence of dissipation.

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References (63)

  1. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2007).
  2. Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, SpringerBriefs in Physics (Springer, Berlin, 2012).
  3. D. Fernández de la Pradilla, E. Moreno, and J. Feist, Recovering an accurate Lindblad equation from the Bloch-Redfield equation for general open quantum systems, Phys. Rev. A 109, 062225 (2024).
  4. R. Dann, A. Levy, and R. Kosloff, Time-dependent Markovian quantum master equation, Phys. Rev. A 98, 052129 (2018).
  5. R. Hartmann and W. T. Strunz, Accuracy assessment of perturbative master equations: Embracing nonpositivity, Phys. Rev. A 101, 012103 (2020).
  6. A. D'Abbruzzo, V. Cavina, and V. Giovannetti, A time-dependent regularization of the Redfield equation, SciPost Phys. 15, 117 (2023).
  7. D. Tupkary, A. Dhar, M. Kulkarni, and A. Purkayastha, Fundamental limitations in Lindblad descriptions of systems weakly coupled to baths, Phys. Rev. A 105, 032208 (2022).
  8. D. Tupkary, A. Dhar, M. Kulkarni, and A. Purkayastha, Searching for Lindbladians obeying local conservation laws and showing thermalization, Phys. Rev. A 107, 062216 (2023).
  9. J. Jeske, D. J. Ing, M. B. Plenio, S. F. Huelga, and J. H. Cole, Bloch-Redfield equations for modeling light-harvesting complexes, J. Chem. Phys. 142, 064104 (2015).
  10. P. R. Eastham, P. Kirton, H. M. Cammack, B. W. Lovett, and J. Keeling, Bath-induced coherence and the secular approximation, Phys. Rev. A 94, 012110 (2016).
  11. G. McCauley, B. Cruikshank, D. I. Bondar, and K. Jacobs, Accurate Lindblad-form master equation for weakly damped quantum systems across all regimes, npj Quantum Inf. 6, 74 (2020).
  12. D. Davidović, Completely positive, simple, and possibly highly accurate approximation of the Redfield equation, Quantum 4, 326 (2020).
  13. D. Farina and V. Giovannetti, Open-quantum-system dynamics: Recovering positivity of the Redfield equation via the partial secular approximation, Phys. Rev. A 100, 012107 (2019).
  14. F. Nathan and M. S. Rudner, Universal Lindblad equation for open quantum systems, Phys. Rev. B 102, 115109 (2020).
  15. Z. C. Coleman and L. D. Carr, Exact analytical solution of the driven qutrit in an open quantum system: V and Λ configurations, J. Phys. B: At. Mol. Opt. Phys. 55, 065501 (2022).
  16. M. Boubakour, S. Endo, T. Fogarty, and T. Busch, Dynamical invariant based shortcut to equilibration in open quantum systems, Quantum Sci. Technol. 10, 025036 (2025).
  17. S. L. Wu, X. L. Huang, and X. X. Yi, Driven Markovian master equation based on the Lewis–Riesenfeld-invariant theory, Phys. Rev. A 106, 052217 (2022).
  18. S. Basak and D. Blume (unpublished).
  19. H. R. Lewis and W. B. Riesenfeld, An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field, J. Math. Phys. 10, 1458 (1969).
  20. X. Chen, E. Torrontegui, and J. G. Muga, Lewis–Riesenfeld invariants and transitionless quantum driving, Phys. Rev. A 83, 062116 (2011).
  21. G. Shavit, B. Horovitz, and M. Goldstein, Bridging between laboratory and rotating-frame master equations for open quantum systems, Phys. Rev. B 100, 195436 (2019).
  22. S. M. Ulrich, S. Ates, S. Reitzenstein, A. Löffler, A. Forchel, and P. Michler, Dephasing of triplet-sideband optical emission of a resonantly driven InAs/GaAs quantum dot inside a microcavity, Phys. Rev. Lett. 106, 247402 (2011).
  23. A. Muller, E. B. Flagg, P. Bianucci, X. Y. Wang, D. G. Deppe, W. Ma, J. Zhang, G. J. Salamo, M. Xiao, and C. K. Shih, Resonance fluorescence from a coherently driven semiconductor quantum dot in a cavity, Phys. Rev. Lett. 99, 187402 (2007).
  24. K. Konthasinghe, J. Walker, M. Peiris, C. K. Shih, Y. Yu, M. F. Li, J. F. He, L. J. Wang, H. Q. Ni, Z. C. Niu, and A. Muller, Coherent versus incoherent light scattering from a quantum dot, Phys. Rev. B 85, 235315 (2012).
  25. A. Ulhaq, S. Weiler, S. M. Ulrich, R. Roßbach, M. Jetter, and P. Michler, Cascaded single-photon emission from the Mollow triplet sidebands of a quantum dot, Nat. Photonics 6, 238 (2012).
  26. A. Nick Vamivakas, Y. Zhao, C.-Y. Lu, and M. Atatüre, Spin-resolved quantum-dot resonance fluorescence, Nat. Phys. 5, 198 (2009).
  27. E. B. Flagg, A. Muller, J. W. Robertson, S. Founta, D. G. Deppe, M. Xiao, W. Ma, G. J. Salamo, and C. K. Shih, Resonantly driven coherent oscillations in a solid-state quantum emitter, Nat. Phys. 5, 203 (2009).
  28. S. Ates, S. M. Ulrich, S. Reitzenstein, A. Löffler, A. Forchel, and P. Michler, Post-selected indistinguishable photons from the resonance fluorescence of a single quantum dot in a microcavity, Phys. Rev. Lett. 103, 167402 (2009).
  29. G. Wang, Y.-X. Liu, and P. Cappellaro, Observation of the high-order Mollow triplet by quantum mode control with concatenated continuous driving, Phys. Rev. A 103, 022415 (2021).
  30. B. L. Ng, C. H. Chow, and C. Kurtsiefer, Observation of the Mollow triplet from an optically confined single atom, Phys. Rev. A 106, 063719 (2022).
  31. F. Schuda, C. R. Stroud, and M. Hercher, Observation of the resonant Stark effect at optical frequencies, J. Phys. B: At. Mol. Phys. 7, L198 (1974).
  32. W. Hartig, W. Rasmussen, R. Schieder, and H. Walther, Study of the frequency distribution of the fluorescent light induced by monochromatic radiation, Z. Phys. A 278, 205 (1976).
  33. R. E. Grove, F. Y. Wu, and S. Ezekiel, Measurement of the spectrum of resonance fluorescence from a two-level atom in an intense monochromatic field, Phys. Rev. A 15, 227 (1977).
  34. L. Ortiz-Gutiérrez, R. C. Teixeira, A. Eloy, D. Ferreira Da Silva, R. Kaiser, R. Bachelard, and M. Fouché, Mollow triplet in cold atoms, New J. Phys. 21, 093019 (2019).
  35. V. Mosallanejad and W. Dou, Two-mode Floquet-Redfield quantum master equation approach for quantum transport, Phys. Rev. B 112, 174308 (2025).
  36. R. Alicki, D. Gelbwaser-Klimovsky, and G. Kurizki, Periodically driven quantum open systems: Tutorial, arXiv:1205.4552.
  37. A. Levy, R. Alicki, and R. Kosloff, Quantum refrigerators and the third law of thermodynamics, Phys. Rev. E 85, 061126 (2012).
  38. B. R. Mollow, Power spectrum of light scattered by two-level systems, Phys. Rev. 188, 1969 (1969).
  39. R. J. Glauber, The quantum theory of optical coherence, Phys. Rev. 130, 2529 (1963).
  40. D. A. Steck, Quantum and atom optics, https://steck.us/teaching (2024), online lecture notes, revision 0.13.4.
  41. M. Lax, Formal theory of quantum fluctuations from a driven state, Phys. Rev. 129, 2342 (1963).
  42. H.-T. Chen, T. E. Li, A. Nitzan, and J. E. Subotnik, Predictive semiclassical model for coherent and incoherent emission in the strong field regime: The Mollow triplet revisited, J. Phys. Chem. Lett. 10, 1331 (2019).
  43. K. Boos, S. K. Kim, T. Bracht, F. Sbresny, J. M. Kaspari, M. Cygorek, H. Riedl, F. W. Bopp, W. Rauhaus, C. Calcagno, J. J. Finley, D. E. Reiter, and K. Müller, Signatures of dynamically dressed states, Phys. Rev. Lett. 132, 053602 (2024).
  44. A. Stenquist, F. Zapata, E. Olofsson, Y. Liao, E. Svegborn, J. N. Bruhnke, C. Verdozzi, and J. M. Dahlström, Mollow-like triplets in ultrafast resonant absorption, Phys. Rev. Lett. 133, 063202 (2024).
  45. A. Ulhaq, S. Weiler, C. Roy, S. M. Ulrich, M. Jetter, S. Hughes, and P. Michler, Detuning-dependent Mollow triplet of a coherently-driven single quantum dot, Opt. Express 21, 4382 (2013).
  46. M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe, Electromagnetically induced transparency with single atoms in a cavity, Nature (London) 465, 755 (2010).
  47. X. Mi, J. Bai, D. Li, and H. Zhao, Coupling to a microdisk cavity containing a three-level quantum-dot with two orthogonal modes, Opt. Commun. 284, 2937 (2011).
  48. R. Dann, A. Tobalina, and R. Kosloff, Shortcut to equilibration of an open quantum system, Phys. Rev. Lett. 122, 250402 (2019).
  49. B. Mohan, R. Gangwar, T. Pandit, M. L. Bera, M. Lewenstein, and M. N. Bera, Coherent heat transfer leads to genuine quantum enhancement in the performances of continuous engines, Phys. Rev. Appl. 23, 044050 (2025).
  50. L. M. Cangemi, C. Bhadra, and A. Levy, Quantum engines and refrigerators, Phys. Rep. 1087, 1 (2024).
  51. P. Z. Zhao and L. Qiao, Dynamical decoupling protection for three-level systems, Phys. Rev. A 112, 032428 (2025).
  52. N. Jaseem, M. Hajdušek, V. Vedral, R. Fazio, L.-C. Kwek, and S. Vinjanampathy, Quantum synchronization in nanoscale heat engines, Phys. Rev. E 101, 020201(R) (2020).
  53. S. Basak, A. Javadi, and D. Blume, Data and code for “Invariant-based master equation applied to a driven qutrit coupled to a bath and a leaky cavity” [Dataset], In Physical Review B (Version 2.0.0), Zenodo, 2026, https://doi.org/10.5281/zenodo.21524114.
  54. Z.-y. Jin and J. Jing, Universal perspective on nonadiabatic quantum control, Phys. Rev. A 111, 012406 (2025).
  55. P. M. M. Paing and D. F. V. James, Conditions for time-independence of N-level systems under the rotating wave approximation (RWA) and dipole selection rules, J. Mod. Opt. 73, 350 (2026).
  56. S. Li, P. Shen, T. Chen, and Z.-Y. Xue, Noncyclic nonadiabatic holonomic quantum gates via shortcuts to adiabaticity, Front. Phys. 16, 51502 (2021).
  57. Z. Hua, G. Ying-Fang, and L. Jiu-Qing, Realization of adiabatic population transfer in a three-level system by using LR Hermitian invariants theory, Chin. Phys. 13, 865 (2004).
  58. Y.-H. Kang, Y.-H. Chen, B.-H. Huang, J. Song, and Y. Xia, Invariant-based pulse design for three-level systems without the rotating-wave approximation, Ann. Phys. 529, 1700004 (2017).
  59. J. Maisch, J. Grammel, N. Tran, M. Jetter, S. L. Portalupi, D. Hunger, and P. Michler, Investigation of Purcell enhancement of quantum dots emitting in the telecom O-band with an open fiber cavity, Phys. Rev. B 110, 165301 (2024).
  60. R. R. Puri, in Mathematical Methods of Quantum Optics, edited by W. T. Rhodes, Springer Series in Optical Sciences, Vol. 79 (Springer, Berlin, 2001).
  61. H. Bateman, Tables of Integral Transforms (McGraw-Hill, London, 1954).
  62. I. Martin, A. Shnirman, L. Tian, and P. Zoller, Ground-state cooling of mechanical resonators, Phys. Rev. B 69, 125339 (2004).
  63. Y. Yanay and A. A. Clerk, Reservoir engineering with localized dissipation: Dynamics and prethermalization, Phys. Rev. Res. 2, 023177 (2020).

Outline

Information

Phys. Rev. B 114, 074108– Published 31 August, 2026
  • Received 17 March 2026
  • Revised 24 June 2026
  • Accepted 27 July 2026
Article part of CHORUS
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