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  • Letter
  • Open Access

Quantum correlations, mixed states, and bistability at the onset of lasing

Francesco Papoff1,*, Mark Anthony Carroll1, Gian Luca Lippi2, Gian-Luca Oppo1, and Giampaolo D'Alessandro3

  • *Contact author: f.papoff@strath.ac.uk
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Phys. Rev. A 111, L011501 – Published 7 January, 2025

DOI: https://doi.org/10.1103/PhysRevA.111.L011501

Abstract

We derive a model for a single-mode laser that includes all two-particle quantum correlations between photons and electrons. In contrast to the predictions of semiclassical models, we find that lasing takes place in the presence of quantum bistability between a nonlasing and a nonclassical coherent state. The coherent state is characterized by a central frequency and a finite linewidth and emerges with finite amplitude from a saddle-node bifurcation together with an unstable coherent state. Hence coherent emission in nanolasers originates through a mixing of lasing and nonlasing states. In the limit of a macrolaser with a large number of emitters and nonresonant modes, the laser threshold approaches the prediction of the semiclassical theory, but with the important difference that lasing can be achieved only in the presence of finite-size perturbations.

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References (46)

  1. J.-S. Tempel, I. Akimov, M. Aßmann, C. Schneider, S. Höfling, C. Kistner, S. Reitzenstein, L. Worschech, A. Forchel, and M. Bayer, Extrapolation of the intensity autocorrelation function of a quantum-dot micropillar laser into the thermal emission regime, J. Opt. Soc. Am. B 28, 1404 (2011).
  2. A. Koulas-Simos, J. Buchgeister, M. L. Drechsler, T. Zhang, L. K, G. Sinatkas, J. Xu, F. Lohof, Q. Kan, R. K. Zhang, F. Jahnke, C. Gies, W. W. Chow, C.-Z. Ning, and S. Reitzenstein, Quantum fluctuations and lineshape anomaly in a high-β silver-coated InP-based metallic nanolaser, Laser Photonics Rev. 16, 2200086 (2022).
  3. R.-M. Ma and R. F. Oulton, Applications of nanolasers, Nat. Nanotechnol. 14, 12 (2019).
  4. S. Kreinberg, W. W. Chow, J. Wolters, C. Schneider, C. Gies, F. Jahnke, S. Höfling, M. Kamp, and S. Reitzenstein, Emission from quantum-dot high-β microcavities: Transition from spontaneous emission to lasing and the effects of superradiant emitter coupling, Light: Sci. Appl. 6, e17030 (2017).
  5. J. Mork and K. Yvind, Squeezing of intensity noise in nanolasers and nanoLEDs with extreme dielectric confinement, Optica 7, 1641 (2020).
  6. M. Carroll, G. D'Alessandro, G. Lippi, G.-L. Oppo, and F. Papoff, Photon-number squeezing in nano- and microlasers, Appl. Phys. Lett. 119, 101102 (2021).
  7. C.-Z. Ning, What is laser threshold? IEEE J. Sel. Top. Quantum Electron. 19, 1503604 (2013).
  8. M. A. Carroll, G. D'Alessandro, G. L. Lippi, G.-L. Oppo, and F. Papoff, Thermal, quantum antibunching and lasing thresholds from single emitters to macroscopic devices, Phys. Rev. Lett. 126, 063902 (2021).
  9. N. B. Abraham and L. M. Narducci, Laser Physics and Laser Instabilities (World Scientific, Singapore, 1988).
  10. I. E. Protsenko and A. V. Uskov, Perturbation approach in Heisenberg equations for lasers, Phys. Rev. A 105, 053713 (2022).
  11. A. M. Yacomotti, Z. Denis, A. Biella, and C. Ciuti, Quantum density matrix theory for a laser without adiabatic elimination of the population inversion: Transition to lasing in the class-B limit, Laser Photonics Rev. 17, 2200377 (2023).
  12. A. A. Vyshnevyy, Gain-dependent Purcell enhancement, breakdown of Einstein's relations, and superradiance in nanolasers, Phys. Rev. B 105, 085116 (2022).
  13. J. Fricke, Transport equations including many-particle correlations for an arbitrary quantum system: A general formalism, Ann. Phys. 252, 479 (1996).
  14. W. Chow and F. Jahnke, On the physics of semiconductor quantum dots for applications in lasers and quantum optics, Prog. Quantum. Electron. 37, 109 (2013).
  15. W. Chow, F. Jahnke, and C. Gies, Emission properties of nanolasers during the transition to lasing, Light: Sci. Appl. 3, e201 (2014).
  16. C. Gies, J. Wiersig, M. Lorke, and F. Jahnke, Semiconductor model for quantum-dot-based microcavity lasers, Phys. Rev. A 75, 013803 (2007).
  17. M. A. Carroll, G. D'Alessandro, G. L. Lippi, G.-L. Oppo, and F. Papoff, Coherence buildup and laser thresholds from nanolasers to macroscopic lasers, Phys. Rev. A 107, 063710 (2023).
  18. N. Baer, C. Gies, J. Wiersig, and F. Jahnke, Luminescence of a semiconductor quantum dot system, Eur. Phys. J. B 50, 411 (2006).
  19. M. Kira and S. W. Koch, Semiconductor Quantum Optics (Cambridge University Press, Cambridge, UK, 2011).
  20. M. Sánchez-Barquilla, R. E. F. Silva, and J. Feist, The potential and global outlook of integrated photonics for quantum technologies, J. Chem. Phys. 152, 034108 (2020).
  21. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevA.111.L011501 for all the model equations used in this Letter, the details of the calculations of the first-order correlation, the laser threshold and frequency, and typical parameter values.
  22. H. A. Leymann, A. Foerster, and J. Wiersig, Expectation value based cluster expansion, Phys. Status Solidi C 10, 1242 (2013).
  23. H. A. M. Leymann, A. Foerster, and J. Wiersig, Expectation value based equation-of-motion approach for open quantum systems: A general formalism, Phys. Rev. B 89, 085308 (2014).
  24. C. B. Simmons, M. Thalakulam, N. Shaji, L. J. Klein, H. Qin, R. H. Blick, D. E. Savage, M. G. Lagally, S. N. Coppersmith, and M. A. Eriksson, Single-electron quantum dot in Si/SiGe with integrated charge sensing, Appl. Phys. Lett. 91, 213103 (2007).
  25. M. V. Ivanchenko, E. A. Kozinov, V. D. Volokitin, A. V. Liniov, I. B. Meyerov, and S. V. Denisov, Classical bifurcation diagrams by quantum means, Ann. Phys. 529, 1600402 (2017).
  26. 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).
  27. W. Casteels, R. Fazio, and C. Ciuti, Critical dynamical properties of a first-order dissipative phase transition, Phys. Rev. A 95, 012128 (2017).
  28. S. Olivares, Quantum optics in the phase space - A tutorial on Gaussian states, Eur. Phys. J. Spec. Top. 203, 3 (2012).
  29. M. A. Carroll, G. D'Alessandro, G. L. Lippi, G.-L. Oppo, and F. Papoff, Carroll et al. reply: Thermal, quantum antibunching and lasing thresholds from single emitters to macroscopic devices, Phys. Rev. Lett. 128, 029402 (2022).
  30. P. Fowler-Wright, K. B. Arnardóttir, P. Kirton, B. W. Lovett, and J. Keeling, Determining the validity of cumulant expansions for central spin models, Phys. Rev. Res. 5, 033148 (2023).
  31. M. Saldutti, Y. Yu, and J. Mórk, The onset of lasing in semiconductor nanolasers, Laser Photonics Rev. 18, 2300840 (2024).
  32. G. Habib, Predicting saddle-node bifurcations using transient dynamics: A model-free approach, Nonlinear Dynamics 111, 20579 (2023).
  33. H. Carmichael, Statistical Methods in Quantum Optics I (Springer, Heidelberg, 2002).
  34. S. Ates, C. Gies, S. M. Ulrich, J. Wiersig, S. Reitzenstein, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, Influence of the spontaneous optical emission factor β on the first-order coherence of a semiconductor microcavity laser, Phys. Rev. B 78, 155319 (2008).
  35. C. Henry, Phase noise in semiconductor lasers, J. Lightwave Technol. 4, 298 (1986).
  36. I. E. Protsenko, A. V. Uskov, E. C. André, J. Mørk, and M. Wubs, Quantum Langevin approach for superradiant nanolasers, New J. Phys. 23, 063010 (2021).
  37. A. S. Chirkin and A. V. Chipouline, Generalized expression for the natural width of the radiation spectrum of quantum oscillators, JETP Lett. 93, 114 (2011).
  38. G. Broggi and L. A. Lugiato, Transient noise-induced optical bistability, Phys. Rev. A 29, 2949 (1984).
  39. W. Lange, F. Mitschke, R. Deserno, and J. Mlynek, Study of fluctuations in transient optical bistability, Phys. Rev. A 32, 1271 (1985).
  40. B. Li, A. Das, S. Tserkis, P. Narang, P. K. Lam, and S. M. Assad, Quantum density matrix theory for a laser without adiabatic elimination of the population inversion: Transition to lasing in the class-B limit, Sci. Rep. 13, 11722 (2023).
  41. E. Pelucchi, G. Fagas, I. Aharonovich, D. Englund, E. Figueroa, Q. Gong, H. Hannes, J. Liu, C.-Y. Lu, N. Matsuda, J.-W. Pan, F. Schreck, F. Sciarrino, C. Silberhorn, J. Wang, and K. D. Jöns, The potential and global outlook of integrated photonics for quantum technologies, Nat. Rev. Phys. 4, 194 (2022).
  42. T. Wang, G. Puccioni, and G. Lippi, Dynamical buildup of lasing in mesoscale devices, Sci. Rep. 5, 15858 (2015).
  43. T. Wang, G. Wang, G. P. Puccioni, and G. L. Lippi, Exploration of VCSEL ultra-low biasing scheme for pulse generation, J. Opt. Soc. Am. B 36, 799 (2019).
  44. J. R. Tredicce, G. L. Lippi, P. Mandel, B. Charasse, A. Chevalier, and B. Picqué, Critical slowing down at a bifurcation, Am. J. Phys. 72, 799 (2004).
  45. M. Scheffer, S. R. Carpenter, T. M. Lenton, J. Bascompte, W. Brock, V. Dakos, J. van de Koppel, I. A. van de Leemput, S. A. Levin, E. H. van Nes, M. Pascual, and J. Vandermeer, Anticipating critical transitions, Science 338, 344 (2012).
  46. G. Zeng, J. Gao, L. Shekhtman, S. Guo, W. Lv, J. Wu, H. Liu, O. Levy, D. Li, Z. Gao, H. E. Stanley, and S. Havlin, Multiple metastable network states in urban traffic, Proc. Natl. Acad. Sci. USA 117, 17528 (2020).

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Phys. Rev. A 111, L011501– Published 7 January, 2025
  • Received 10 June 2024
  • Accepted 17 December 2024
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