## The single quantum dot-laser: lasing and strong coupling in the high-excitation regime |

Optics Express, Vol. 19, Issue 15, pp. 14370-14388 (2011)

http://dx.doi.org/10.1364/OE.19.014370

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### Abstract

The emission properties of a single quantum dot in a microcavity are studied on the basis of a semiconductor model. As a function of the pump rate of the system we investigate the onset of stimulated emission, the possibility to realize stimulated emission in the strong-coupling regime, as well as the excitation-dependent changes of the photon statistics and the emission spectrum. The role of possible excited charged and multi-exciton states, the different sources of dephasing for various quantum-dot transitions, and the influence of background emission into the cavity mode are analyzed in detail. In the strong coupling regime, the emission spectrum can contain a line at the cavity resonance in addition to the vacuum doublet caused by off-resonant transitions of the same quantum dot. If strong coupling persists in the regime of stimulated emission, the emission spectrum near the cavity resonance additionally grows due to broadened contributions from higher rungs of the Jaynes-Cummings ladder.

© 2011 OSA

## 1. Introduction

1. P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science **290**, 2282–2285 (2000). [CrossRef] [PubMed]

2. M. Pelton, C. Santori, J. Vuckovic, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. **89**, 233602 (2002). [CrossRef] [PubMed]

3. A. Muller, W. Fang, J. Lawall, and G. S. Solomon, “Creating polarization-entangled photon pairs from a semiconductor quantum dot using the optical stark effect,” Phys. Rev. Lett. **103**, 217402 (2009). [CrossRef]

5. C. L. Salter, R. M. Stevenson, I. Farrer, C. A. Nicoll, D. A. Ritchi, and A. J. Shields, “An entangled-light-emitting diode,” Nature **465**, 594–597 (2010). [CrossRef] [PubMed]

10. J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Bestermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature **460**, 245–249 (2009). [CrossRef] [PubMed]

11. Z. G. Xie, S. Götzinger, W. Fang, H. Cao, and G. S. Solomon, “Influence of a single quantum dot state on the characteristics of a microdisk laser,” Phys. Rev. Lett. **98**, 117401 (2007). [CrossRef] [PubMed]

14. M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single-quantum-dot nanocavity system,” Nat. Phys. **6**, 279–283 (2010). [CrossRef]

13. M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Photonic crystal nanocavity laser with a single quantum dot gain,” Opt. Express **17**, 15975–15982 (2009). [CrossRef] [PubMed]

15. J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature **432**, 197–200 (2004). [CrossRef] [PubMed]

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*before*the onset of stimulated emission. Then the double-peak structure of the hybridized states merges into a single emission line due to excitation-induced dephasing as well as loss of oscillator strength for the QD transition. In this regime the behavior of the single-emitter is analogous to the polariton normal-mode coupling of spatially extended excitonic states [17

17. G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode-coupling semiconductor microcavities,” Rev. Mod. Phys. **71**, 1591–1639 (1999). [CrossRef]

14. M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single-quantum-dot nanocavity system,” Nat. Phys. **6**, 279–283 (2010). [CrossRef]

*in the presence of strong coupling*, clusters of lines appear close to the degenerate cavity and exciton resonances. Their appearance has been reported in the theoretical work [18

18. J. I. Cirac, H. Ritsch, and P. Zoller, “Two-level system interacting with a finite-bandwidth thermal cavity mode,” Phys. Rev. A **44**, 4541–4551 (1991). [CrossRef] [PubMed]

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20. J. Kasprzak, S. Reitzenstein, E. A. Muljarov, C. Kistner, C. Schneider, M. Strauss, S. Höfling, A. Forchel, and W. Langbein, “Up on the Jaynes-Cummings ladder of a quantum-dot/microcavity system,” Nat. Mater. **9**, 304–308 (2010). [CrossRef] [PubMed]

*g*(determined by the QD oscillator strength and spatial and spectral coupling of the QD to the cavity field) is required for the photon production rate to exceed the cavity losses. The maximum photon output does not only depend on the recombination rate, but also relies on fast feeding of carriers into the laser levels, a limit being set by saturation due to Pauli blocking. The fast carrier dynamics needed to drive the coupled QD-cavity system into stimulated emission also introduces dephasing that broadens the laser transition line. If the broadening becomes comparable to the light-matter coupling strength, strong coupling is no longer feasible.

19. E. del Valle, F. P. Laussy, and C. Tejedor, “Luminescence spectra of quantum dots in microcavities. ii.) Fermions,” Phys. Rev. B **79**, 235326 (2009). [CrossRef]

21. Y. Mu and C. M. Savage, “One-atom lasers,” Phys. Rev. A **46**, 5944–5954 (1992). [CrossRef] [PubMed]

22. A. Auffèves, D. Gerace, J.-M. Gérard, M. F. Santos, L. C. Andreani, and J.-P. Poizat, “Controlling the dynamics of a coupled atom-cavity system by pure dephasing,” Phys. Rev. B **81**, 245419 (2010). [CrossRef]

23. S. Ritter, P. Gartner, C. Gies, and F. Jahnke, “Emission properties and photon statisticsof a single quantum dot laser,” Opt. Express **18**, 9909–9921 (2010). [CrossRef] [PubMed]

13. M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Photonic crystal nanocavity laser with a single quantum dot gain,” Opt. Express **17**, 15975–15982 (2009). [CrossRef] [PubMed]

14. M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single-quantum-dot nanocavity system,” Nat. Phys. **6**, 279–283 (2010). [CrossRef]

## 2. Single-QD microcavity system

*ρ*(

*t*) in a state space defined by the confined QD states and the Fock states of the cavity mode. The electronic part is naturally limited due to the finite confinement potential for electrons and holes, which leads to a finite number of localized states to be occupied by carriers; for the cavity photons we consider a sufficiently large maximum photon number. The von-Neumann equation for the QD-photon density matrix

*ρ*(

*t*) has the form where the interaction picture has been used (

*h̄*= 1). The non-perturbative light-matter interaction between the QD-interband transitions and the quantized field of the microcavity mode is described by the Jaynes-Cummings (JC) Hamiltonian Here,

*a*

^{†},

*a*are fermionic creation and annihilation operators for electrons, and the index

*i*(

*j*) refers to valence (conduction)-band states, as the elementary processes in the JC Hamiltonian comply with the rotating wave approximation.

*b*

^{†}and

*b*are the bosonic operators for photons in the laser mode, and

*g*is the light-matter coupling strength for the respective electronic levels and the cavity mode.

_{i,j}*H*

_{Coul}between the QD carriers in the von-Neumann equation. Using the full Coulomb Hamiltonian, the numerical solution is then equivalent to a configuration interaction (CI) calculation since it describes the time-evolution of the system in terms of interacting states, which are also eigenstates of the diagonalized CI Hamiltonian. For details of the considered electronic states and the Coulomb interaction, see Appendix A.

24. T. S. Sosnowski, T. B. Norris, H. Jiang, J. Singh, K. Kamath, and P. Bhattacharya, “Rapid carrier relaxation in InGaAs/GaAs quantum dots characterized by differential transmission spectroscopy,” Phys. Rev. B **57**, R9423–R9426 (1998). [CrossRef]

25. H. Kurtze, J. Seebeck, P. Gartner, D. R. Yakovlev, D. Reuter, A. D. Wieck, M. Bayer, and F. Jahnke, “Carrier relaxation dynamics in self-assembled semiconductor quantum dots,” Phys. Rev. B **80**, 235319 (2009). [CrossRef]

26. U. Bockelmann and T. Egeler, “Electron relaxation in quantum dots by means of Auger processes,” Phys. Rev. B **46**, 15574 (1992). [CrossRef]

27. T. R. Nielsen, P. Gartner, and F. Jahnke, “Many-body theory of carrier capture and relaxation in semiconductor quantum-dot lasers,” Phys. Rev. B **69**, 235314 (2004). [CrossRef]

28. J. Seebeck, T. R. Nielsen, P. Gartner, and F. Jahnke, “Polarons in semiconductor quantum-dots and their role in the quantum kinetics of carrier relaxation,” Phys. Rev. B **71**, 125327 (2005). [CrossRef]

*γ*are intraband scattering rates between the levels

_{ij}*j*and

*i*. In a similar fashion, carrier scattering can lead to the capture of electrons and holes from the delocalized states into the QD states and vice versa. The corresponding Lindblad terms contain the scattering rates

*i*and

27. T. R. Nielsen, P. Gartner, and F. Jahnke, “Many-body theory of carrier capture and relaxation in semiconductor quantum-dot lasers,” Phys. Rev. B **69**, 235314 (2004). [CrossRef]

28. J. Seebeck, T. R. Nielsen, P. Gartner, and F. Jahnke, “Polarons in semiconductor quantum-dots and their role in the quantum kinetics of carrier relaxation,” Phys. Rev. B **71**, 125327 (2005). [CrossRef]

*j*and a hole in state

*i*at rate

*κ*are given by [29]

23. S. Ritter, P. Gartner, C. Gies, and F. Jahnke, “Emission properties and photon statisticsof a single quantum dot laser,” Opt. Express **18**, 9909–9921 (2010). [CrossRef] [PubMed]

30. K. Matsuda, K. Ikeda, T. Saiki, H. Saito, and K. Nishi, “Carrier-carrier interaction in single In_{0.5}Ga_{0.5}As quantum dots at room temperature investigated by near-field scanning optical microscope,” Appl. Phys. Lett. **83**, 2250–2252 (2003). [CrossRef]

31. C. Santori, G. S. Solomon, M. Pelton, and Y. Yamamoto, “Time-resolved spectroscopy of multiexcitonic decay in an InAs quantum dot,” Phys. Rev. B **65**, 073310 (2002). [CrossRef]

32. A. Laucht, M. Kaniber, A. Mohtashami, N. Hauke, M. Bichler, and J. J. Finley, “Temporal monitoring of non-resonant feeding of semiconductor nanocavity modes by quantum dot multiexciton transitions,” Phys. Rev. B **81**, 241302 (2010). [CrossRef]

**6**, 279–283 (2010). [CrossRef]

## 3. Configuration space in a single- and two-spin description

*p*-shell carriers and may be called ‘hot’ configurations. The highest multi-exciton state is the

*sp*-biexciton, which can be expected to behave similarly to the filled-dot configuration in the two-spin picture.

*p*-shell carriers), which is often the case. The line broadening due to dephasing can mask the small splittings especially at elevated excitation conditions. Furthermore, at high excitation, the system dynamics is mainly determined by the highest multi-exciton configuration, in which case the influence of the other configurations plays a smaller role. Still, the larger number of possible configurations in a two-spin description can modify the system dynamics. Also a detailed study of the emission characteristics of e.g. the charged exciton line [33

33. M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamođlu, “Quantum dot spectroscopy using cavity quantum electrodynamics,” Phys. Rev. Lett. **101**, 226808 (2008). [CrossRef] [PubMed]

*s*-exciton (top panel) or the highest multi-exciton state (bottom panel) is tuned to the cavity mode. The first is the preferred configuration in the weak excitation limit, while the latter dominates the emission at high excitation. We conclude that, for the considered situation, the single-spin model is able to capture the relevant physics with quantitative agreement. Deviations between both models result from the higher number of possible configurations in the two-spin model, as well as from different energetic positions of the contributing configurations due to the Coulomb interaction. For example, the smaller mean photon number in the top panel of Fig. 1 is the result of a larger detuning of the

*sp*-biexciton for the used parameters, while in the two-spin case the additional Coulomb interaction reduces the energetic separation between the cavity mode and the configuration corresponding to the filled QD. A detailed discussion of the underlying physics is given in the remainder of this manuscript, whereas more information about the two-spin calculation will be found in Ref. [34].

## 4. Laser emission from a single QD

**6**, 279–283 (2010). [CrossRef]

*μ*eV, corresponding to a Jaynes-Cummings coupling between the electron and hole ground states (

*s*-states) of

*g*= 0.15/ps. The typical cavity decay rate of

*κ*= 0.1/ps corresponds to a cavity-

*Q*= 20,000 in the red spectral range of the InGaAs QD emission. Solely driven by a single QD emitter, we are unable to reach the regime of stimulated emission with these parameters. Determining the requirements for stimulated emission in the multilevel-QD system is not straightforward, and a discussion of emission rates in terms of QD parameters is given in Appendix B.

*s*and

*p*, as well as one spin sub-system as discussed in Section 3. The system is pumped by injecting electrons from the continuum into the conduction-band

*p*-state and ejecting them out of the valence-band

*p*-state (hole capture). These two processes are modeled by the Lindblad terms of Eq. (4) with the capture rates

*p*-to-

*s*scattering is considered in both bands, with the corresponding rates

*g*= 0.3/ps, which we expect to be achievable now or in the near future. Independent calculations along the lines of [27

27. T. R. Nielsen, P. Gartner, and F. Jahnke, “Many-body theory of carrier capture and relaxation in semiconductor quantum-dot lasers,” Phys. Rev. B **69**, 235314 (2004). [CrossRef]

28. J. Seebeck, T. R. Nielsen, P. Gartner, and F. Jahnke, “Polarons in semiconductor quantum-dots and their role in the quantum kinetics of carrier relaxation,” Phys. Rev. B **71**, 125327 (2005). [CrossRef]

*g*and

*κ*values, we find that

*p*-to-

*s*intraband relaxation rates

*γ*

^{nl}= 0.01/ps. This set of parameters we label in the following as Set A. Using a constant carrier capture rate, we evolve Eqs. (1)–(4) for the density matrix elements of the initially empty system in time until steady state is reached.

**Mean photon number.**In the left column of Fig. 2 we show input/output curves for a SQD laser with the cavity being in resonance with the 1

*X*exciton, the

_{s}*X*biexciton transition, respectively. The contributions to the mean photon number from the 1

_{sp}*X*,

_{s}*X*configurations are shown separately as dashed, dotted and dash-dotted lines, respectively. At low excitation powers they exhibit different slopes, the one of the biexciton being twice that of the exciton, and the one of the charged exciton states in between. For the considered confinement situation these are the only bright configurations allowing for a recombination at the

_{sp}*s*shell.

*p*shell becomes more favorable for capture rates above 10

^{−3}/ps and then leads to a preference of the two charged exciton configurations. As a consequence, the mean photon number is higher in this regime if the cavity is tuned to the charged exciton transitions. At high excitation carriers get captured into the higher QD

*p*-states much faster than the optical recombination process, so that the

*sp*biexciton configuration is always dominant. Thus, tuning the biexciton transition into resonance with the cavity maximizes the photon output. Only in this case stimulated emission with 〈

*n*〉 > 1 can be reached for the chosen parameter set. At the same time, the realization of the other bright configurations becomes strongly suppressed, and this is one reason why quenching is observed in the cases of the cavity at the neutral or charged exciton resonances— dephasing introduced by the capture processes is the other.

*β*-factors close to unity, see e.g. the review article [35

35. S. Strauf and F. Jahnke, “Single quantum dot nanolaser,” Laser Photonics Rev. **5**, n/a. doi:(2011). [CrossRef]

*n*. As an example we consider the case where the biexciton is tuned to the cavity mode. For weak pumping (left in Fig. 4) the photon statistics

*p*deviates from a single-photon Fock state albeit the probability

_{n}*p*for n>1 is below that of either ideal thermal or coherent light with the same mean photon number.

_{n}*g*

^{(2)}exhibits bunching when the cavity is resonant with either the exciton or the charged exciton transition, while it does not if the biexciton is tuned to resonance. Looking at Fig. 2, we infer that bunching appears when more than one configuration contribute with comparable amounts to the total photon production. In other words, the bunching is a reflection of competing emission channels which are realized in the density matrix with comparable likelihoods.

**Spectra and linewidths.**The contribution of separate configurations is also reflected in the cavity emission spectrum

*S*, which is given by the Fourier transform

*ℱ*of the first-order two-time photon correlation function

*g*

^{(1)}(

*t,τ*) = 〈

*b*

^{†}(

*t*)

*b*(

*t*+

*τ*)〉 with respect to the delay time

*τ*The latter is obtained by invoking the quantum regression theorem, which requires a separate time evolution of the density operator in

*τ*with modified initial conditions obtained from the steady state solution of the

*t*-time evolution

*ρ*(

*t*) [36

_{ss}36. P. Zoller and C. Gardiner, *Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics*, 3rd ed. (Springer-Verlag, 2004). [PubMed]

*s*-exciton resonance, the charged exciton contributions split off by the Coulomb exchange energy, and the

*sp*-biexciton separated by twice the exchange energy from the

*s*exciton. The increased photonic density of states at the cavity mode leads to a significant enhancement of the resonant emission. The broadening of the spectral lines reflects the possible scattering processes due to carrier capture, relaxation and recombination, which differ for various excitation configurations. Deduced from the contribution of the Lindblad terms to the equation of motion for the relevant polarization, a simple rule can be derived: A given optical transition is dephased by those processes changing either its initial or its final configuration. For example the 1

*X*excitonic transition is dephased by the carrier capture processes, which act upon the initial and final configurations, but is not influenced by the relaxation as long as one neglects

_{s}*s*-to-

*p*scattering processes, which are unlikely at low temperatures. For the biexciton transition it is the other way round, only the

*p*-to-

*s*relaxation, but not the capture rates contribute. For the charged exciton transition both do. Hence, the neutral and the charged exciton contributions to the photon output are subject to quenching by means of excitation-induced dephasing (see Fig. 2), while the biexcitonic contribution is not.

*X*→ 0

_{s}*X*transition. At low capture rates the finite cavity lifetime is the main source of dephasing (

*s*-exciton transition is in close spectral vicinity to the cavity mode, both hybridize and exhibit the double-peak structure of the vacuum Rabi splitting— the well-known signature for strong coupling that is seen in the upper right panel of Fig. 2. With increasing capture of carriers into the QD

*p*-states, the capture process itself starts to contribute to the dephasing, resulting in broadening and finally the transition into weak coupling.

*s*exciton being on resonance: Already at low pump rates an inner peak emerges within the Rabi doublet of the

*s*exciton. It originates from the detuned resonances of the charged exciton and biexciton. These are too far detuned to hybridize with the cavity mode. As a consequence, each possesses

*two*spectral contributions, one at the

*s*-shell transition energy that is renormalized by the Coulomb interaction, and one at the bare cavity energy. In the spectra the bare cavity peak of the detuned configurations can be observed to emerge and gain oscillator strength as the charged exciton states and the biexciton state gain probability (see the upper right panel in Fig. 2). This feature has also been discussed in [37

37. S. Hughes and P. Yao, “Theory of quantum light emission from a strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express **17**, 3322–3330 (2009). [CrossRef] [PubMed]

38. M. Yamaguchi, T. Asano, K. Kojima, and S. Noda, “Quantum electrodynamics of a nanocavity coupled with exciton complexes in a quantum dot,” Phys. Rev. B **80**, 155326 (2009). [CrossRef]

## 5. Single QD with background contributions

*Q*mode is typically extended over a micrometer range in the emitter plane. Often additional emitters are present in this area that are also subjected to the pump process. While these emitters typically exhibit off-resonant emission with respect to the cavity mode, it is now widely accepted that even transitions detuned from the cavity mode by tens of meV can still contribute to cavity feeding.

**6**, 279–283 (2010). [CrossRef]

19. E. del Valle, F. P. Laussy, and C. Tejedor, “Luminescence spectra of quantum dots in microcavities. ii.) Fermions,” Phys. Rev. B **79**, 235326 (2009). [CrossRef]

39. A. Laucht, N. Hauke, J. M. Villas-Bôas, F. Hofbauer, G. Böhm, M. Kaniber, and J. J. Finley, “Dephasing of exciton polaritons in photoexcited ingaas quantum dots in gaas nanocavities,” Phys. Rev. Lett. **103**, 087405 (2009). [CrossRef] [PubMed]

41. F. P. Laussy, E. del Valle, and C. Tejedor, “Strong coupling of quantum dots in microcavities,” Phys. Rev. Lett. **101**, 083601 (2008). [CrossRef] [PubMed]

**6**, 279–283 (2010). [CrossRef]

*κ*the two corresponding Lindblad terms simulate the contact of the photon subsystem with a reservoir of a temperature determined by the ratio Γ/

*κ*=

*e*

^{−βh̄ω}. This treatment of the residual emission can only produce thermal, incoherent photons. Moreover, the situation with Γ >

*κ*leads to a pathological, exponentially increasing “negative-temperature” photon statistics.

42. M. Scully and W. Lamb, “Quantum theory of an optical maser. I. General theory,” Phys. Rev. **159**, 208–226 (1967). [CrossRef]

*S*. We use a value of

*S*= 0 limit of Eq. (9).

12. S. Reitzenstein, C. Böckler, A. Bazhenov, A. Gorbunov, A. Löffler, M. Kamp, V. D. Kulakovskii, and A. Forchel, “Single quantum dot controlled lasing effects in high-Q micropillar cavities,” Opt. Express **16**, 4848–4857 (2008). [CrossRef] [PubMed]

**6**, 279–283 (2010). [CrossRef]

*g*

^{(2)}= 1 indicating coherent light emission is reached, see Fig. 6. The photon production does not saturate but keeps increasing with increasing capture rates. Regarding the carrier system, the capture rates appear together with population factors that ensure that the Pauli principle is obeyed and each state is filled with one fermion only.

*g*

^{(2)}takes on a value of 2 instead. Note that a stable solution cannot be obtained over the whole excitation range, as Γ <

*κ*must be fulfilled. Clearly, both methods to account for background emission channels are not arbitrarily interchangeable, as they describe different physical situations in the way we have discussed above. Their impact on the photon statistics does not depend on parameters, but lies in the origin of the terms. If background emission accounts for a significant part of the photon production, no coherent light emission can be expected from the inverse cavity loss term.

## 6. Lasing in the strong coupling regime

**Strong coupling and lasing at the exciton transition.**Since the carrier capture rate determines the broadening of the 1

*X*exciton transition, which eventually destroys the strong coupling, as well as all the necessary supply of carriers for the laser transition, an optimal balance between the two counteracting effects is required. With an interband relaxation rate of 2/ps and a light-matter coupling strength

_{s}*g*= 1.8/ps, strong coupling in the presence of stimulated emission can be obtained. We refer to this as Parameter Set B.

*g*

^{(2)}(0) = 0.9 (not shown). Due to the increased probability of having several photons in the regime of stimulated emission, peaks from higher transitions between Jaynes-Cummings rungs appear, e.g. at 3.7meV, corresponding to the

**Strong coupling and lasing at the biexciton transition.**We now take the cavity to be in resonance with the

*sp*-biexciton, which, of all configurations, maximizes the photon production rate at high excitation. In Section 4 we have identified the intraband carrier relaxation processes to act as the dominant dephasing channel for this transition. In a QD in which these scattering rates are strongly reduced, signatures of strong coupling are expected to be seen. To demonstrate this, we consider a third Parameter Set C, where the relaxation rates are reduced to

*s*-states after capture into the

*p*-states is significantly lowered. In order to still obtain stimulated emission, an improved cavity

*Q*of 100, 000 (corresponding to

*κ*= 0.02/ps) is required, together with additional feeding of photons from background emitters parameterized with

*α*= 0.001 and

*S*= 3, cf. Section 5. The used light-matter coupling strength is

*g*= 0.3/ps like in Parameter Set A.

## 7. Conclusion

*Q*cavity mode, the choice of parameters being close to what is possible to realize experimentally to date. The presence of carriers in excited states modifies the emission energy at the

*s*-shell due to the Coulomb interaction, and we have discussed cases of particular interest. Choosing the highest multi-exciton configuration (in our particular QD the

*sp*-biexciton) to be in resonance with the cavity mode maximizes the photon output from the device. The main reason for this lies in the fact that it is the dominating configuration at high excitation densities. Capture and relaxation processes are typically faster than the radiative interband recombination, so that the dot fills up before emission takes place. At the same time, higher multi-exciton configurations are subject to strong dephasing due to fast carrier relaxation, which prohibits strong coupling to be observed at these resonances. Nevertheless, stimulated and coherent laser emission can be achieved in the weak coupling regime, characterized by Poissonian photon statistics,

*g*

^{(2)}≈ 1, and a mean photon number exceeding unity.

*s*-exciton transition, on the other hand, is not dephased by the relaxation but the capture processes. Thus, if resonant with the cavity mode, strong coupling becomes possible in the emission spectrum at low excitation densities for typical values of the light-matter coupling strength, but disappears as capture rates increase with pumping. The attainable photon output is significantly reduced due to quenching of the transition. At high excitation, only the off-resonant highest multi-exciton state contributes. Then the emission characteristics depends strongly on the detuning. Excitation-induced screening and dephasing can enhance the overlap in the presence of a high WL carrier density [23

23. S. Ritter, P. Gartner, C. Gies, and F. Jahnke, “Emission properties and photon statisticsof a single quantum dot laser,” Opt. Express **18**, 9909–9921 (2010). [CrossRef] [PubMed]

## Appendix A: Electronic states and solution of the von-Neumann equation

*p*and

*s*shell from the WL continuum are, for the electrons, 40 and 80meV, respectively, and 15 and 30meV for the holes.

*H*

_{Coul}in Eq. (1) within our subspace of QD states. For confinement potentials of typical self-assembled QDs it has been discussed in [45

45. N. Baer, P. Gartner, and F. Jahnke, “Coulomb effects in semiconductor quantum dots,” Eur. Phys. J. B **42**, 231–237 (2004). [CrossRef]

*λ*′ to the two middle states in the matrix elements

*W*of the screened Coulomb potential [46

46. C. Gies, M. Florian, P. Gartner, and F. Jahnke, “A semiconductor model for the single quantum dot laser,” Phys. Status Solidi B **248**, 879–882 (2011). [CrossRef]

*s*-exciton and the

*p*-exciton, and has the value of 2–3meV. This is one order of magnitude smaller than the energetic separation between the states involved (tens of meV’s – see above) so that the mixing of these configurations can be safely neglected. Throughout the paper (except in Section 3) we also restrict ourselves to the simplified case of equal envelopes for the electron and hole wave functions [47

47. P. Hawrylak, “Excitonic artificial atoms: Engineering optical properties of quantum dots,” Phys. Rev. B **60**, 5597 (1999). [CrossRef]

## Appendix B: Quantum-dot emission rates

*γ*. The value of

*γ*is determined for each transition (1

*X*→ 0

_{s}*X*,

*X*→ 1

_{sp}*X*) according to the discussion of the cavity emission spectra in Section 4.

_{p}*R*follows by multiplying 1/

_{X}*τ*with the occupation probability

_{sp}*f*of the considered exciton configuration:

_{X}*n*〉 = 1, electrons and holes need to be generated in the QD ground state with a rate of (at least) 0.1/ps (the assumed value for

*κ*). This adds up to a dephasing rate = 0.2/ps (neglecting the small

*g*= 0.15/ps one finds 1/

*τ*= 0.3/ps. Only if the occupation probability of the corresponding electronic configuration exceeds the value of 1/3, photon emission balances the cavity losses. At this point the population dynamics comes into play. For a carrier generation rate slower than that of the spontaneous emission, the carriers in the QD-ground state recombine faster than they get refilled by the stationary pump process. Hence the QD-exciton population remains small. If the pump rate becomes faster than the spontaneous recombination rate, the QD-exciton population starts to increase, but so does

_{sp}*γ*. This pump-induced dephasing, in turn, reduces the emission rate according to Eq. (11). If carriers start to populate also higher QD states, the probability of the excitonic configuration is reduced for the benefit of excited configurations. Note that adding confined carriers to a QD exciton leads to the formation of a higher multi-exciton state. Consequently the probability of the exciton configuration

*f*, and with it the exciton recombination rate

_{X}decreases*R*.

_{X}## References and links

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3. | A. Muller, W. Fang, J. Lawall, and G. S. Solomon, “Creating polarization-entangled photon pairs from a semiconductor quantum dot using the optical stark effect,” Phys. Rev. Lett. |

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5. | C. L. Salter, R. M. Stevenson, I. Farrer, C. A. Nicoll, D. A. Ritchi, and A. J. Shields, “An entangled-light-emitting diode,” Nature |

6. | J. Hendrickson, B. C. Richards, J. Sweet, S. Mosor, C. Christenson, D. Lam, G. Khitrova, H. M. Gibbs, T. Yoshie, A. Scherer, O. B. Shchekin, and D. G. Deppe, “Quantum dot photonic-crystal-slab nanocavities: quality factors and lasing,” Phys. Rev. B |

7. | S. Reitzenstein, A. Bazhenov, A. Gorbunov, C. Hofmann, S. Münch, A. Löffler, M. Kamp, J. P. Reithmaier, V. D. Kulakovskii, and A. Forchel, “Lasing in high-Q quantum-dot micropillar cavities,” Appl. Phys. Lett. |

8. | S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L. C. Andreani, E. L. Hu, P. M. Petroff, and D. Brouwmeester, “Self-tuned quantum dot gain in photonic crystal lasers,” Phys. Rev. Lett. |

9. | S. M. Ulrich, C. Gies, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, “Photon statistics of semiconductor microcavity lasers,” Phys. Rev. Lett. |

10. | J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Bestermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature |

11. | Z. G. Xie, S. Götzinger, W. Fang, H. Cao, and G. S. Solomon, “Influence of a single quantum dot state on the characteristics of a microdisk laser,” Phys. Rev. Lett. |

12. | S. Reitzenstein, C. Böckler, A. Bazhenov, A. Gorbunov, A. Löffler, M. Kamp, V. D. Kulakovskii, and A. Forchel, “Single quantum dot controlled lasing effects in high-Q micropillar cavities,” Opt. Express |

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18. | J. I. Cirac, H. Ritsch, and P. Zoller, “Two-level system interacting with a finite-bandwidth thermal cavity mode,” Phys. Rev. A |

19. | E. del Valle, F. P. Laussy, and C. Tejedor, “Luminescence spectra of quantum dots in microcavities. ii.) Fermions,” Phys. Rev. B |

20. | J. Kasprzak, S. Reitzenstein, E. A. Muljarov, C. Kistner, C. Schneider, M. Strauss, S. Höfling, A. Forchel, and W. Langbein, “Up on the Jaynes-Cummings ladder of a quantum-dot/microcavity system,” Nat. Mater. |

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22. | A. Auffèves, D. Gerace, J.-M. Gérard, M. F. Santos, L. C. Andreani, and J.-P. Poizat, “Controlling the dynamics of a coupled atom-cavity system by pure dephasing,” Phys. Rev. B |

23. | S. Ritter, P. Gartner, C. Gies, and F. Jahnke, “Emission properties and photon statisticsof a single quantum dot laser,” Opt. Express |

24. | T. S. Sosnowski, T. B. Norris, H. Jiang, J. Singh, K. Kamath, and P. Bhattacharya, “Rapid carrier relaxation in InGaAs/GaAs quantum dots characterized by differential transmission spectroscopy,” Phys. Rev. B |

25. | H. Kurtze, J. Seebeck, P. Gartner, D. R. Yakovlev, D. Reuter, A. D. Wieck, M. Bayer, and F. Jahnke, “Carrier relaxation dynamics in self-assembled semiconductor quantum dots,” Phys. Rev. B |

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27. | T. R. Nielsen, P. Gartner, and F. Jahnke, “Many-body theory of carrier capture and relaxation in semiconductor quantum-dot lasers,” Phys. Rev. B |

28. | J. Seebeck, T. R. Nielsen, P. Gartner, and F. Jahnke, “Polarons in semiconductor quantum-dots and their role in the quantum kinetics of carrier relaxation,” Phys. Rev. B |

29. | H. J. Carmichael, |

30. | K. Matsuda, K. Ikeda, T. Saiki, H. Saito, and K. Nishi, “Carrier-carrier interaction in single In |

31. | C. Santori, G. S. Solomon, M. Pelton, and Y. Yamamoto, “Time-resolved spectroscopy of multiexcitonic decay in an InAs quantum dot,” Phys. Rev. B |

32. | A. Laucht, M. Kaniber, A. Mohtashami, N. Hauke, M. Bichler, and J. J. Finley, “Temporal monitoring of non-resonant feeding of semiconductor nanocavity modes by quantum dot multiexciton transitions,” Phys. Rev. B |

33. | M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamođlu, “Quantum dot spectroscopy using cavity quantum electrodynamics,” Phys. Rev. Lett. |

34. | F. Jahnke, ed., |

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36. | P. Zoller and C. Gardiner, |

37. | S. Hughes and P. Yao, “Theory of quantum light emission from a strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express |

38. | M. Yamaguchi, T. Asano, K. Kojima, and S. Noda, “Quantum electrodynamics of a nanocavity coupled with exciton complexes in a quantum dot,” Phys. Rev. B |

39. | A. Laucht, N. Hauke, J. M. Villas-Bôas, F. Hofbauer, G. Böhm, M. Kaniber, and J. J. Finley, “Dephasing of exciton polaritons in photoexcited ingaas quantum dots in gaas nanocavities,” Phys. Rev. Lett. |

40. | A. Laucht, J. M. Villas-Bôas, S. Stobbe, N. Hauke, F. Hofbauer, G. Böhm, P. Lodahl, M.-C. Amann, M. Kaniber, and J. J. Finley, “Mutual coupling of two semiconductor quantum dots via an optical nanocavity,” Phys. Rev. B |

41. | F. P. Laussy, E. del Valle, and C. Tejedor, “Strong coupling of quantum dots in microcavities,” Phys. Rev. Lett. |

42. | M. Scully and W. Lamb, “Quantum theory of an optical maser. I. General theory,” Phys. Rev. |

43. | S. Stenholm, “Quantum theory of electromagnetic fields interacting with atoms and molecules,” Phys. Rep. |

44. | D. Walls and G. Milburn, |

45. | N. Baer, P. Gartner, and F. Jahnke, “Coulomb effects in semiconductor quantum dots,” Eur. Phys. J. B |

46. | C. Gies, M. Florian, P. Gartner, and F. Jahnke, “A semiconductor model for the single quantum dot laser,” Phys. Status Solidi B |

47. | P. Hawrylak, “Excitonic artificial atoms: Engineering optical properties of quantum dots,” Phys. Rev. B |

**OCIS Codes**

(270.5290) Quantum optics : Photon statistics

(130.3990) Integrated optics : Micro-optical devices

(250.5590) Optoelectronics : Quantum-well, -wire and -dot devices

**ToC Category:**

Optoelectronics

**History**

Original Manuscript: May 12, 2011

Revised Manuscript: June 20, 2011

Manuscript Accepted: June 22, 2011

Published: July 12, 2011

**Citation**

Christopher Gies, Matthias Florian, Paul Gartner, and Frank Jahnke, "The single quantum dot-laser: lasing and strong coupling in the high-excitation regime," Opt. Express **19**, 14370-14388 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14370

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### References

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