Hybrid Quantum-Classical Modeling of Electrically Driven Quantum Light Sources

Part of the Springer Theses book series (Springer Theses)


This chapter describes a comprehensive modeling approach for the simulation of quantum dot-based electrically driven quantum light sources, by coupling the van Roosbroeck system to a Markovian quantum master equation in Lindblad form. The Lindblad master equation for a quantum system weakly coupled to a carrier reservoir is derived under the additional constraint of a charge conserving system-reservoir interaction. Subsequently, a new hybrid quantum-classical model system is constructed, that allows for a self-consistent description of the dynamics of the QD-photon system and the carrier transport in realistic semiconductor device geometries. Thereby, the quantum optical figures of merit and the spatially resolved current flow can be calculated out of one box. The hybrid model is thoroughly investigated with respect to its thermodynamic properties and is shown to obey fundamental principles of (non-)equilibrium thermodynamics. Finally, the hybrid model is discussed from the viewpoint of the GENERIC formalism for thermodynamically consistent modeling of multi-physics problems.


  1. 1.
    Alicki R (1976) On the detailed balance condition for non-Hamiltonian systems. Rep Math Phys 10(2):249–258. Scholar
  2. 2.
    Alicki R (1977) The Markov master equations and the Fermi golden rule. Int J Theor Phys 16(5):351–355. Scholar
  3. 3.
    Baer N, Gartner P, Jahnke F (2004) Coulomb effects in semiconductor quantum dots. Eur Phys J B 42(2):231–237. Scholar
  4. 4.
    Baer N, Gies C, Wiersig J, Jahnke F (2006) Luminescence of a semiconductor quantum dot system. Eur Phys J B 50(3):411–418. Scholar
  5. 5.
    Benisty H (1995) Reduced electron-phonon relaxation rates in quantum-box systems: Theoretical analysis. Phys Rev B 51(19):13281. Scholar
  6. 6.
    Benisty H, Sotomayor-Torres CM, Weisbuch C (1991) Intrinsic mechanism for the poor luminescence properties of quantum-box systems. Phys Rev B 44(19):10945. Scholar
  7. 7.
    Bimberg D, Stock E, Lochmann A, Schliwa A, Tofflinger JA, Unrau W, Munnix M, Rodt S, Haisler VA, Toropov AI, Bakarov A, Kalagin AK (2009) Quantum dots for single-and entangled-photon emitters. IEEE Photon J 1(1):58–68. Scholar
  8. 8.
    Bockelmann U, Bastard G (1990) Phonon scattering and energy relaxation in two-, one-, and zero-dimensional electron gases. Phys Rev B 42(14):8947. Scholar
  9. 9.
    Bockelmann U, Egeler T (1992) Electron relaxation in quantum dots by means of Auger processes. Phys Rev B 46(23):15574. Scholar
  10. 10.
    Böckler C, Reitzenstein S, Kistner C, Debusmann R, Löffler A, Kida T, Höfling S, Forchel A, Grenouillet L, Claudon J, Gérard JM (2008) Electrically driven high-Q quantum dot-micropillar cavities. Appl Phys Lett 92(9):091107. Scholar
  11. 11.
    Boretti A, Rosa L, Mackie A, Castelletto S (2015) Electrically driven quantum light sources. Adv Opt Mat 3(8):1012–1033. Scholar
  12. 12.
    Brasken M, Lindberg M, Sopanen M, Lipsanen H, Tulkki J (1998) Temperature dependence of carrier relaxation in strain-induced quantum dots. Phys Rev B 58(24):R15993. Scholar
  13. 13.
    Breuer HP, Petruccione F (2002) The theory of open quantum systems. Oxford University Press, Oxford. Scholar
  14. 14.
    Buckley S, Rivoire K, Vučković J (2012) Engineered quantum dot single-photon sources. Rep Prog Phys 75(12):126503. Scholar
  15. 15.
    Carmichael HJ (1993) An open systems approach to quantum optics. Lecture notes in physics, vol 18. Springer, BerlinCrossRefGoogle Scholar
  16. 16.
    Carmichael HJ (1999) Statistical methods in quantum optics 1-master equations and Fokker-Planck equations. Springer, Berlin. Scholar
  17. 17.
    Chow WW, Jahnke F (2013) On the physics of semiconductor quantum dots for applications in lasers and quantum optics. Prog Quantum Electron 37(3):109–184. Scholar
  18. 18.
    Dachner MR, Malić E, Richter M, Carmele A, Kabuss J, Wilms A, Kim JE, Hartmann G, Wolters J, Bandelow U, Knorr A (2010) Theory of carrier and photon dynamics in quantum dot light emitters. Phys Status Solidi B 247(4):809–828. Scholar
  19. 19.
    Dümcke R, Spohn H (1979) The proper form of the generator in the weak coupling limit. Z Phys B Condens Matter 34(4):419–422. Scholar
  20. 20.
    Englund D, Fattal D, Waks E, Solomon G, Zhang B, Nakaoka T, Arakawa Y, Yamamoto Y, Vučković J (2005) Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal. Phys Rev Lett 95(1):013904. Scholar
  21. 21.
    Ferreira R, Bastard G (1999) Phonon-assisted capture and intradot Auger relaxation in quantum dots. Appl Phys Lett 74(19):2818. Scholar
  22. 22.
    Ferreira R, Bastard G (2000) Carrier capture and intra-dot Auger relaxation in quantum dots. Phys E 7(3–4):342–345. Scholar
  23. 23.
    Ferreira R, Bastard G (2015) Capture and relaxation in self-assembled semiconductor quantum dots. Morgan & Claypool Publishers, San Rafael, CA, pp 2053–2571.
  24. 24.
    Gemmer J, Michel M, Mahler G (2004) Quantum thermodynamics. Lecture notes in physics, vol 657. Springer, BerlinCrossRefGoogle Scholar
  25. 25.
    Gies C, Florian M, Steinhoff A, Jahnke F (2017) Theory of quantum light sources and cavity-QED emitters based on semiconductor quantum dots. In: Michler P (ed) Quantum dots for quantum information technologies, Springer series in nano-optics and nanophotonics, chap. 1. Springer, Cham, pp 3–40. Scholar
  26. 26.
    Gioannini M, Cedola AP, Santo ND, Bertazzi F, Cappelluti F (2013) Simulation of quantum dot solar cells including carrier intersubband dynamics and transport. IEEE J Photovolt 3(4):1271–1278. Scholar
  27. 27.
    Gomis-Bresco J, Dommers S, Temnov VV, Woggon U, Lämmlin M, Bimberg D, Malić E, Richter M, Schöll E, Knorr A (2008) Impact of Coulomb scattering on the ultrafast gain recovery in InGaAs quantum dots. Phys Rev Lett 101(25):256803. Scholar
  28. 28.
    Gorini V, Kossakowski A, Sudarshan ECG (1976) Completely positive dynamical semigroups of N-level systems. J Math Phys 17(5):821–825. Scholar
  29. 29.
    Gready D, Eisenstein G (2013) Carrier dynamics and modulation capabilities of 1.55-\(\mu \)m quantum-dot lasers. IEEE J Sel Top Quant 19(4):1900307–1900307. Scholar
  30. 30.
    Grmela M, Öttinger HC (1997) Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys Rev E 56:6620–6632. Scholar
  31. 31.
    Grundmann M, Heitz R, Bimberg D, Sandmann JHH, Feldmann J (1997) Carrier dynamics in quantum dots: modeling with master equations for the transitions between micro-states. Phys Status Solidi B 203(1):121–132.<121::AID-PSSB121>3.0.CO;2-MADSCrossRefGoogle Scholar
  32. 32.
    Haag R, Hugenholtz NM, Winnink M (1967) On the equilibrium states in quantum statistical mechanics. Commun Math Phys 5(3):215–236. Scholar
  33. 33.
    Haken H (1993) Quantenfeldtheorie des Festkörpers. Springer, Wiesbaden. Scholar
  34. 34.
    Hameau S, Guldner Y, Verzelen O, Ferreira R, Bastard G, Zeman J, Lemaitre A, Gérard J (1999) Strong electron-phonon coupling regime in quantum dots: evidence for everlasting resonant polarons. Phys Rev Lett 83(20):4152. Scholar
  35. 35.
    Hargart F, Kessler CA, Schwarzbäck T, Koroknay E, Weidenfeld S, Jetter M, Michler P (2013) Electrically driven quantum dot single-photon source at 2 GHz excitation repetition rate with ultra-low emission time jitter. Appl Phys Lett 102(1):011126. Scholar
  36. 36.
    Haug H, Jauho AP (2008) Quantum kinetics in transport and optics of semiconductors. Springer series in solid-state sciences, vol 123, 2 edn. Springer, Berlin.
  37. 37.
    Haug H, Koch SW (2004) Quantum theory of the optical and electronic properties of semiconductors, 4 edn. World Scientific, Singapore.
  38. 38.
    Heindel T, Schneider C, Lermer M, Kwon SH, Braun T, Reitzenstein S, Höfling S, Kamp M, Forchel A (2010) Electrically driven quantum dot-micropillar single photon source with 34% overall efficiency. Appl Phys Lett 96(1):011107. Scholar
  39. 39.
    Inoshita T, Sakaki H (1996) Electron-phonon interaction and the so-called phonon bottleneck effect in semiconductor quantum dots. Phys. B 227(1–4):373–377. Scholar
  40. 40.
    Jordan R, Kinderlehrer D, Otto F (1997) Free energy and the Fokker-Planck equation. Phys. D 107(2–4):265–271. Scholar
  41. 41.
    Jordan R, Kinderlehrer D, Otto F (1998) The variational formulation of the Fokker-Planck equation. SIAM J Math Anal 29(1):1–17. Scholar
  42. 42.
    Kaer P, Gregersen N, Mørk J (2013) The role of phonon scattering in the indistinguishability of photons emitted from semiconductor cavity QED systems. New J Phys 15(3):035027. Scholar
  43. 43.
    Kaer P, Nielsen TR, Lodahl P, Jauho AP, Mørk J (2012) Microscopic theory of phonon-induced effects on semiconductor quantum dot decay dynamics in cavity QED. Phys Rev B 86(8):085302. Scholar
  44. 44.
    Kantner M, Bandelow U, Koprucki T, Schulze JH, Strittmatter A, Wünsche HJ (2016) Efficient current injection into single quantum dots through oxide-confined p-n-diodes. IEEE Trans Electron Devices 63(5):2036–2042. Scholar
  45. 45.
    Kantner M, Mielke A, Mittnenzweig M, Rotundo N (2019) Mathematical modeling of semiconductors: From quantum mechanics to devices. In: Hintermüller M, Rodrigues J (eds) Topics in Applied Analysis and Optimisation. Springer, Cham, pp 269–293. Scholar
  46. 46.
    Kantner M, Mittnenzweig M, Koprucki T (2017) Hybrid quantum-classical modeling of quantum dot devices. Phys Rev B 96(20):205301. Scholar
  47. 47.
    Kantner M, Mittnenzweig M, Koprucki T (2018) A hybrid quantum-classical modeling approach for electrically driven quantum light sources. In: Proceedings of the SPIE photonics west: physics and simulation of optoelectronic devices XXVI, vol 10526. SPIE, p 1052603.
  48. 48.
    Koprucki T, Wilms A, Knorr A, Bandelow U (2011) Modeling of quantum dot lasers with microscopic treatment of Coulomb effects. Opt Quantum Electron 42(11):777–783. Scholar
  49. 49.
    Kosloff R (2013) Quantum thermodynamics: a dynamical viewpoint. Entropy 15(6):2100–2128. Scholar
  50. 50.
    Kossakowski A, Frigerio A, Gorini V, Verri M (1977) Quantum detailed balance and KMS condition. Commun Math Phys 57(2):97–110. Scholar
  51. 51.
    Kurtze H, Seebeck J, Gartner P, Yakovlev DR, Reuter D, Wieck AD, Bayer M, Jahnke F (2009) Carrier relaxation dynamics in self-assembled semiconductor quantum dots. Phys Rev B 80:235319. Scholar
  52. 52.
    Lindblad G (1975) Completely positive maps and entropy inequalities. Commun Math Phys 40(2):147–151. Scholar
  53. 53.
    Lindblad G (1976) On the generators of quantum dynamical semigroups. Commun Math Phys 48(2):119–130. Scholar
  54. 54.
    Lodahl P, Mahmoodian S, Stobbe S (2015) Interfacing single photons and single quantum dots with photonic nanostructures. Rev Mod Phys 87(2):347–400. Scholar
  55. 55.
    Lodahl P, Van Driel AF, Nikolaev IS, Irman A, Overgaag K, Vanmaekelbergh D, Vos WL (2004) Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals. Nature 430(7000):654. Scholar
  56. 56.
    Lorke M, Nielsen TR, Seebeck J, Gartner P, Jahnke F (2006) Influence of carrier-carrier and carrier-phonon correlations on optical absorption and gain in quantum-dot systems. Phys Rev B 73(8):085324. Scholar
  57. 57.
    Lüdge K, Bormann MJP, Malić E, Hövel P, Kuntz M, Bimberg D, Knorr A, Schöll E (2008) Turn-on dynamics and modulation response in semiconductor quantum dot lasers. Phys Rev B 78(3):035316. Scholar
  58. 58.
    Lüdge K, Schöll E (2009) Quantum-dot lasers-desynchronized nonlinear dynamics of electrons and holes. IEEE J Quantum Electron 45(11):1396–1403. Scholar
  59. 59.
    Magnúsdóttir I, Bischoff S, Uskov AV, Mørk J (2003) Geometry dependence of Auger carrier capture rates into cone-shaped self-assembled quantum dots. Phys Rev B 67(20):205326. Scholar
  60. 60.
    Magnúsdóttir I, Uskov AV, Bischoff S, Tromborg B, Mørk J (2002) One- and two-phonon capture processes in quantum dots. J Appl Phys 92(10):5982. Scholar
  61. 61.
    Malić E, Bormann MJP, Hövel P, Kuntz M, Bimberg D, Knorr A, Schöll E (2007) Coulomb damped relaxation oscillations in semiconductor quantum dot lasers. IEEE J Sel Top Quant 13(5):1242–1248. Scholar
  62. 62.
    Mielke A (2011) A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24:1329–1346. Scholar
  63. 63.
    Mielke A (2013) Dissipative quantum mechanics using GENERIC. In: Johann A, Kruse HP, Rupp F, Schmitz S (eds) Recent trends in dynamical systems, no. 35 in Springer proceedings in mathematics & statistics, chap. 21. Springer, Basel, pp 555–585. Scholar
  64. 64.
    Mielke A (2015) On thermodynamical couplings of quantum mechanics and macroscopic systems. In: Exner P, König W, Neidhardt H (eds) Mathematical results in quantum mechanics. World Scientific, Singapore, pp 331–348.
  65. 65.
    Mittnenzweig M, Mielke A (2017) An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models. J Stat Phys 167(2):205–233. Scholar
  66. 66.
    Narvaez GA, Bester G, Zunger A (2005) Excitons, biexcitons, and trions in self-assembled (In, Ga)As/ GaAs quantum dots: Recombination energies, polarization, and radiative lifetimes versus dot height. Phys Rev B 72(24):245318. Scholar
  67. 67.
    Nielsen TR, Gartner P, Jahnke F (2004) Many-body theory of carrier capture and relaxation in semiconductor quantum-dot lasers. Phys Rev B 69:235314. Scholar
  68. 68.
    Öttinger HC (2010) Nonlinear thermodynamic quantum master equation: properties and examples. Phys Rev A 82(5):052119. Scholar
  69. 69.
    Öttinger HC (2011) The geometry and thermodynamics of dissipative quantum systems. Europhys Lett 94(1):10006. Scholar
  70. 70.
    Öttinger HC, Grmela M (1997) Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys Rev E 56(6):6633–6655. Scholar
  71. 71.
    Purcell EM, Torrey HC, Pound RV (1946) Resonance absorption by nuclear magnetic moments in a solid. Phys Rev 69(1–2):37–38. Scholar
  72. 72.
    Reitzenstein S (2012) Semiconductor quantum dot-microcavities for quantum optics in solid state. IEEE J Sel Top Quant 18(6):1733–1746. Scholar
  73. 73.
    Reitzenstein S, Heindel T, Kistner C, Albert F, Braun T, Hopfmann C, Mrowinski P, Lermer M, Schneider C, Höfling S, Kamp M, Forchel A (2011) Electrically driven quantum dot micropillar light sources. IEEE J Sel Top Quant 17(6):1670–1680. Scholar
  74. 74.
    Reitzenstein S, Heindel T, Kistner C, Rahimi-Iman A, Schneider C, Höfling S, Forchel A (2008) Low threshold electrically pumped quantum dot-micropillar lasers. Appl Phys Lett 93(6):061104. Scholar
  75. 75.
    Rivas A, Huelga SF (2012) Open quantum systems–an introduction. SpringerBriefs in physics. Springer, Berlin. Scholar
  76. 76.
    Roy-Choudhury K, Hughes S (2017) Theory of phonon dressed light-matter interactions and resonance fluorescence in quantum dot cavity systems. In: Michler P (ed) Quantum dots for quantum information technologies. Springer series in nano-optics and nanophotonics, chap. 2. Springer, Cham, pp 41–74. Scholar
  77. 77.
    Santori C, Fattal D, Yamamoto Y (2010) Single-photon devices and applications. Wiley, WeinheimGoogle Scholar
  78. 78.
    Schäfer W, Wegener M (2002) Semiconductor optics and transport phenomena. Springer, Berlin. Scholar
  79. 79.
    Schaller G (2011) Quantum equilibration under constraints and transport balance. Phys Rev E 83(3):031111. Scholar
  80. 80.
    Schaller G (2014) Open quantum systems far from equilibrium. Lecture notes in physics, vol 881. Springer, ChamCrossRefGoogle Scholar
  81. 81.
    Schaller G, Kießlich G, Brandes T (2009) Transport statistics of interacting double dot systems: Coherent and non-markovian effects. Phys Rev B 80(24):245107. Scholar
  82. 82.
    Schlehahn A, Thoma A, Munnelly P, Kamp M, Höfling S, Heindel T, Schneider C, Reitzenstein S (2016) An electrically driven cavity-enhanced source of indistinguishable photons with 61% overall efficiency. APL Photonics 1(1):011301. Scholar
  83. 83.
    Scully MO, Zubairy MS (1997) Quantum optics. Cambridge University Press, Cambridge. Scholar
  84. 84.
    Seebeck J, Nielsen TR, Gartner P, Jahnke F (2005) Polarons in semiconductor quantum dots and their role in the quantum kinetics of carrier relaxation. Phys Rev B 71(12):125327. Scholar
  85. 85.
    Spohn H (1978) Entropy production for quantum dynamical semigroups. J Math Phys 19(5):1227–1230. Scholar
  86. 86.
    Spohn H, Lebowitz JL (1978) Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. In: Rice SA (ed) Advances in chemical physics, chap. 2, vol 38. Wiley, pp 109–142. Scholar
  87. 87.
    Steiger S, Veprek RG, Witzigmann B (2008) Unified simulation of transport and luminescence in optoelectronic nanostructures. J Comput Electron 7(4):509–520. Scholar
  88. 88.
    Steinhoff A, Gartner P, Florian M, Jahnke F (2012) Treatment of carrier scattering in quantum dots beyond the Boltzmann equation. Phys Rev B 85:205144. Scholar
  89. 89.
    Steinhoff A, Kurtze H, Gartner P, Florian M, Reuter D, Wieck AD, Bayer M, Jahnke F (2013) Combined influence of Coulomb interaction and polarons on the carrier dynamics in InGaAs quantum dots. Phys Rev B 88(20):205309. Scholar
  90. 90.
    Strauf S, Jahnke F (2011) Single quantum dot nanolaser. Laser Photonics Rev 5(5):607–633. Scholar
  91. 91.
    Stuhler J (2015) Quantum optics route to market. Nat Phys 11(4):293–295. Scholar
  92. 92.
    Unrau W, Quandt D, Schulze JH, Heindel T, Germann TD, Hitzemann O, Strittmatter A, Reitzenstein S, Pohl UW, Bimberg D (2012) Electrically driven single photon source based on a site-controlled quantum dot with self-aligned current injection. Appl Phys Lett 101(21):211119. Scholar
  93. 93.
    Uskov AV, Adler F, Schweizer H, Pilkuhn MH (1997) Auger carrier relaxation in self-assembled quantum dots by collisions with two-dimensional carriers. J Appl Phys 81(12):7895–7899. Scholar
  94. 94.
    Uskov AV, Magnúsdóttir I, Tromborg B, Mørk J, Lang R (2001) Line broadening caused by Coulomb carrier-carrier correlations and dynamics of carrier capture and emission in quantum dots. Appl Phys Lett 79(11):1679–1681. Scholar
  95. 95.
    Uskov AV, McInerney J, Adler F, Schweizer H, Pilkuhn MH (1998) Auger carrier capture kinetics in self-assembled quantum dot structures. Appl Phys Lett 72(1):58. Scholar
  96. 96.
    Villani C (2009) Optimal transport: old and new. Grundlehren der mathematischen Wissenschaften. Springer, Berlin, vol 338. Scholar
  97. 97.
    Ward MB, Farrow T, See P, Yuan ZL, Karimov OZ, Bennett AJ, Shields AJ, Atkinson P, Cooper K, Ritchie DA (2007) Electrically driven telecommunication wavelength single-photon source. Appl Phys Lett 90(6):063512. Scholar
  98. 98.
    Weisskopf V, Wigner E (1930) Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie. Z Phys 63(1–2):54–73. Scholar
  99. 99.
    Wetzler R, Wacker A, Schöll E (2004) Coulomb scattering with remote continuum states in quantum dot devices. J Appl Phys 95(12):7966–7970. Scholar
  100. 100.
    Wilcox RM (1967) Exponential operators and parameter differentiation in quantum physics. J Math Phys 8(4):962–982. Scholar
  101. 101.
    Wilms A, Mathé P, Schulze F, Koprucki T, Knorr A, Bandelow U (2013) Influence of the carrier reservoir dimensionality on electron-electron scattering in quantum dot materials. Phys Rev B 88:235421. Scholar
  102. 102.
    Wiseman HM, Milburn GJ (2009) Quantum measurement and control. Cambridge University Press, Cambridge. Scholar
  103. 103.
    Yuan Z, Kardynal BE, Stevenson RM, Shields AJ, Lobo CJ, Cooper K, Beattie NS, Ritchie DA, Pepper M (2002) Electrically driven single-photon source. Science 295(5552):102–105. Scholar

Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and Stochastics (WIAS)Leibniz Institute in Forschungsverbund Berlin e. V.BerlinGermany

Personalised recommendations