## Abstract

A strongly coupled quantum dot-micropillar cavity system is studied under variation of the excitation power. The characteristic double peak spectral shape of the emission with a vacuum Rabi splitting of 85 *µ*eV at low excitation transforms gradually into a single broad emission peak when the excitation power is increased. Modelling the experimental data by a recently published formalism [Laussy et al., Phys. Rev. Lett. 101, 083601 (2008)] yields a transition from strong coupling towards weak coupling which is mainly attributed to an excitation power driven decrease of the exciton-photon coupling constant.

© 2009 Optical Society of America

The recent success in observing strong coupling in solid state systems has triggered intensive research activities in this field of cavity quantum electrodynamics (cQED). The strong interest in studying the strong coupling regime on the single emitter-single photon level is explained by the fact that it represents a coherently coupled two level system which can be exploited in quantum information processing. For instance, a strongly coupled quantum dot-microcavity system can act as a quantum mechanical interface to couple a localized qubit to a propagating “flying qubit” [1]. Experimentally, strong coupling between single quantum dots (QDs) and high-Q microcavity modes has been studied in various configurations mainly in the limit of low excitation powers. For instance temperature tuning has been applied to demonstrate the strong coupling regime for micropillar cavities [2], photonic crystal cavities [3] and microdisks [4]. Very recently, also electrooptical tuning was applied to perform resonance tuning of a single QD exciton and the cavity mode of a high-Q microcavity in the strong coupling regime [5, 6]. Even though it is predicted theoretically [7, 8] that the particular excitation conditions should have strong implications on the coupling behaviour of QD-microcavities a profound experimental study of this system is still lacking. In particular, it is still unclear how a coupled QD-microcavity system resembles an atomic system with respect to non-linear effects at high excitation powers. This open question has been raised, e.g. in [9, 10], and will have strong implications with respect to the study of the non-linear cQED-effects in semiconductor systems such field quantization [11, 12, 13] and single quantum dot lasing [14].

From a theoretical point of view, different approaches have been chosen to study a coupled QD-microcavity system. Most of the theoretical work describes the mode splitting in resonance very well as a function of the coupling constant g and the linewidths of the excitonic transition *γx* and the photonic mode *γc*, respectively [15]. Recently, also the dephasing of the emitter has been taken into account to model the emission spectra of coupled QD-microcavity systems [16, 17]. On the other hand, in order to obtain a full understanding of the underlying physics it is important to consider the particular excitation conditions of the coupled system. For instance, it has been shown that the spectrum of a coupled QD microcavity system depends sensitively on the primary channel of excitation which can be either the exciton or the photon mode [7]. In fact, a primary excitation via the photon mode can lead to a mode splitting in resonance - a typical characteristic of the strong coupling regime [18, 10] - even though the system is in the weak coupling regime [7]. These intriguing characteristics have also been studied theoretically in a recent paper by Laussy et al. who chose the quantum dissipative master equation in order to describe the coupled QD-microcavity system [8]. Their approach can describe the data presented in [2] in very good agreement. Moreover, Laussy et al. performed an in-depth study of the coupling character as a function of the exciton and cavity pumping. Taking realistic parameters into account it was predicted that a transition from strong to weak coupling can be initiated by an enhanced pumping via the exciton channel.

In the present letter we follow the work by Laussy et al. and study the influence of the optical excitation power on the coupling characteristics of a strongly coupled QD-microcavity system. The system under study is based on self-assembled In_{0.3}Ga_{0.7}As QDs embedded in the active layer of a high-Q micropillar cavity. Due to their enlarged shape, these type of QDs are predestinated for the observation of pronounced cQED effects and, in addition, they fit well to the approximations made in [8], i.e. a bosonic description of the QD excitons. Varying the optical pump power under non-resonant excitation conditions we observe a gradual decrease of the mode-splitting when the pump power is increased. The corresponding photoluminescence emission spectra are well described by the model introduced by Laussy et al. and the vanishing of the mode splitting is partly attributed to a pump power dependent broadening of the exciton line and to a lowering of the coupling constant at high excitation powers.

The excitation power dependent optical studies were performed on QD-micropillar cavities optimized for the observation of strong coupling. As reported in Refs. [19, 20] the micropillars are based on a planar microcavity sample consisting of a one-*λ* GaAs cavity embedded between a lower and an upper distributed Bragg reflector (DBR). Each DBR of the microcavitiy consists of a stack of *λ*/4-thick AlAs/GaAs mirror pairs. For the present work we have chosen a structure with 23/27 mirror pairs in the upper/lower DBR. The active layer of the microcavity is centered in the GaAs spacer and consists of a low density (≈5×10^{9} cm^{-2}) layer of enlarged In_{0.3}Ga_{0.7}As QDs. Micropillars with a diameter of *d _{c}*=1.6

*µ*m were shaped out of this planar microcavity by means of electron beam lithography in combination with electron cyclotron resonance (ECR) plasma etching. This sample technology allows us to observe strong coupling with coupling constants g up to about 60

*µ*eV and vacuum Rabi splittings of ≈100

*µ*eV [21].

The emission spectra of a strongly coupled QD-micropillar system were studied as a function of excitation power using high spectral resolution micro photoluminescence (*µ*PL) spectroscopy at low temperature (10–40 K). Excited by a frequency doubled emission line of a Nd:YAG laser at 532 nm (laser spot size on the sample ≈3 *µ*m) the emission of the micropillar was collected by a microscope objective (NA=0.4) and dispersed by a 0.75 m spectrometer. In conjunction with a liquid nitrogen cooled Si charge coupled device this setup provides a spectral resolution of 20 *µ*eV.

The influence of incoherent pumping in the strong coupling regime was investigated for a QD-micropillar (*d _{c}*=1.6

*µ*m) with a quality (Q) factor of 11000 and a resonance energy of the fundamental cavity mode

*E*=1.3314 eV where Q depends on the cavity mode linewidth (FWHM)

_{c}*γ*:

_{c}*Q*=

*E*. The micropillar was selected by temperature tuning at low excitation power

_{c}/γ_{c}*P*=500 nW. The chosen micropillar shows the characteristic anticrossing behaviour in the temperature dependent emission spectra depicted in Fig. 1(a) when a nearby QD exciton X is tuned through resonance with the cavity mode C. From the corresponding energy dispersion of the emission modes shown in Fig. 1(b) we extract a vacuum Rabi splitting of 80

_{exc}*µ*eV at resonance (31.7 K). The associated emission linewidths are shown in the inset of Fig. 1(b). At resonance the coupled system has equal linewidths as a further signature of the coherent exciton-photon interaction.

Next, we study the emission spectra of the micropillar as a function of excitation power at resonance. Special care has been taken to compensate for possible excitation power induced heating by a slight change (ΔT<1 K) of the sample temperature. This allows us to maintain the resonance condition between X and C also for comparatively high excitation powers. Fig. 2 shows the normalized excitation power dependent emission spectra. At the lowest excitation power of 60 nW a well pronounced double-peak emission structure is observed. The corresponding vacuum Rabi splitting amounts to 85 *µ*eV. Increasing the excitation power leads to a change of the spectral shape of the resonantly coupled QD-micropillar system. First of all, the dip in between the double peak structure becomes less prominent with increasing excitation power until a flat-topped spectral shape is observed at *P _{exc}*=8

*µ*W. At still higher excitation powers a single broad emission feature evolves. Secondly the splitting between the peaks becomes smaller when the excitation power is increased and reaches a value of 17

*µ*eV at the

*P*=64

_{exc}*µ*W. The spectral change from the double-peak lineshape to a single broad line can be attributed to an excitation power driven transition from strong coupling towards weak coupling which will be studied in more detail in the following.

To verify this interpretation and to get further insight in the underlying physics we modelled the emission spectra depicted in Fig. 2 by the self-consistent analytical expression derived recently by Laussy et al. [8]. Their model describes the spectral features of a coupled QD-microcavity system in a steady state maintained by a continuous incoherent pumping which fits very well to our experimental conditions. For a detailed description of the model we refer to Ref. [8]. Essentially, based on the quantum dissipative master equation for the density matrix the model takes into account the coupling strength *g* as well as the decay rates *γ _{c}, γ_{x}* and the pumping rates

*P*of the exciton and the photon mode, respectively, to obtain the emission spectra of the coupled system. Adopting this description we fitted the data presented in Fig. 2 under variation of

_{x}, P_{c}*g, P*and

_{c}, P_{x}*γx*. The experimental value

*γc*=80

*µ*eV is used for all excitation powers. Very good agreement between experiment and theory is achieved by this approach which is demonstrated in Fig. 2 where the calculated spectra (solid lines) are plotted together with the experimental data. Insight into the underlying physics can be obtained from the excitation power dependence of different fit parameters and derived quantities. For instance, fitting yields that the exciton line experiences a power broadening and its linewidth increases from ≈40

*µ*eV at 60 nW to ≈90

*µ*eV at 64

*µ*W which is presented in Fig. 3(b). This broadening is attributed to enhanced spectral diffusion at high excitation powers [22, 23] and partly explains the vanishing of the mode splitting at high excitation powers in a sense that the splitting can not be resolved if the contributing lines are too broad. The associated excitonic linewidths are in good agreement with those obtained via

*γx*=2

*γ*-

_{mean}*γc*(cf. Fig. 3(b)). Here,

*γ*denotes the mean value of the linewidth when the double peak emission spectra plotted in Fig. 2 are approximated by two Lorentzians. We have chosen this approach which allows for a reliable extraction of the exciton linewidth (cf. inset of Fig. 1(b)) at the resonance temperature instead of determining

_{mean}*γx*under out of resonance conditions for a different sample temperature. The chosen method ensures that a possible temperature dependent change of

*γx*can be excluded [23].

We will now discuss the excitation power dependent data in terms of a power driven transition from strong to weak coupling. Following the work by Laussy et al. [8] regions of strong and weak coupling can be distinguished by plotting *P _{x}/g* as a function of

*γc/g*. According to the standard threshold condition for strong coupling

*g*>|(

*γc*-

*γx*)/4| the two regimes are separated by a vertical line at

*γc/g*=4+

*γx/g*. However, taking pumping effects into account a more general threshold condition

*g*>|Γ-| can be derived and the vacuum Rabi splitting Δ

*E*(cf. Fig. 3(a)) is given by

_{R}where Γ-≡(*γ _{c}*-

*P*-(

_{c}*γ*))/4 [8]. Fig. 4 shows a phase diagram where regions of strong coupling (weak coupling) are highlighted in blue (red) according to the threshold condition

_{x}-P_{x}*g*>|Γ-|. The particular limit between the two coupling regimes in Fig. 4 depends on the system parameters

*g*,

*γ*and

_{x}*P*. We have chosen values (

_{c}*g*=43

*µ*eV,

*γ*=37

_{x}*µ*eV and

*P*=10 µeV) determined for the lowest excitation power to illustrate the regions in the phase diagram associated with strong and weak coupling in the present QD-micropillar system. With the power dependent fit parameters

_{c}*g*and

*P*at hand, it is possible to associate the coupling character at the excitation powers chosen in experiment with different data points (solid circles) in the phase diagram of Fig. 4. At low excitation powers (

_{x}*P*<8

*µ*W, points not labelled individually in Fig. 4 for the sake of visibility) the coupled system is clearly in the strong coupling regime. On the other hand, the system evolves towards the weak coupling regime at excitation powers exceeding 8

*µ*W. Even though this tendency is obvious in the phase diagram it is necessary to note that

*g*as well as

*γ*change with

_{x}*P*in a way that the region associated with strong coupling extends to higher values of

_{exc}*γc/g*at high excitation powers (not shown). Interestingly, for

*P*>8

_{exc}*µ*W the data points yield in good approximation a linear dependence between

*P*and

_{x}/g*γc/g*, i.e.

*g*decreases linearly with increasing

*P*.

_{x}In order to explain qualitatively the decrease of the coupling constant *g* with increasing excitation power it is necessary to take into account that *g* depends on the oscillator strength (OS) *f* of the QD [15]

Here, *ε _{r}* and

*ε*

_{0}denote the dielectric constants of the cavity material and vacuum, respectively.

*m*

_{0}is the free electron mass and

*V*represents the mode volume of the cavity. The In

_{m}_{0.3}Ga

_{0.7}As QDs used in the present study feature a particular large OS of up to 50 due to their enhanced lateral extension of typically 30 nm in width and 50–100 nm in length [19]. The large OS of the In

_{0.3}Ga

_{0.7}As QDs is attributed to an increased area of the center of mass wave function and clearly exceeds the value of about 10 typical for standard, small size (diameter: 15–20 nm) InGaAs QDs with a higher In content [15]. This interpretation explains the large OS of 30 % InGaAs QDs in the low excitation limit when the occupation of the QD by multiexcitons can be neglected. However, at higher excitation levels multiparticle effects need to be taken into account. While bound biexcitonic complexes with different spin configuration form in standard InGaAs QDs [24], a different behaviour can be expected in 30 % InGaAs QDs with a size that significantly exceeds the exciton Bohr radius (11 nm). In fact, in excitation power dependent studies we could not observe any indication of a biexcitonic transition for the studied QD. Thus, the enhanced occupation of these QDs by two or even more excitons with different spin states at high excitation powers can explain qualitatively the drop of g for

*P*>4

_{exc}*µ*W in terms of a reduced center of mass area and a correspondingly lower OS of the individual excitons. This interpretation is consistent with theoretical predictions saying that for large dots the Pauli exclusion can be taken into account by a phase-space filling effect that screens the exciton-photon interaction [25, 13].

n conclusion we have studied the evolution of a strongly coupled QD-micropillar system under variation of the incoherent optical pump intensity. The spectral shape of its emission pattern experiences a transition from the characteristic double peak structure associated with a vacuum Rabi splitting of 85 *µ*eV to a single broad emission peak when the pump power is increased by three orders of magnitude from 60 nW to 64 *µ*W. Taking an excitation power dependent excitonic linewidth and coupling constant *g* as well as the particular pumping channels into account the emission spectra can be theoretically described with high accuracy. The decrease of *g* at high excitation powers is attributed to a reduction of the oscillator strength when multiparticle occupation of the QD becomes significant.

## Acknowledgment

The authors acknowledge fruitful discussion with V. D. Kulakovskii. This work was financially supported by the Deutsche Forschungsgemeinschaft via the Research Group “Quantum Optics in Semiconductor Nanostructures” and the State of Bavaria. We thank M. Emmerling and A. Wolf for technical assistance.

## References and links

**1. **J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network,” Phys. Rev. Lett. **78**, 3221–3224 (1997).
[CrossRef]

**2. **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]

**3. **T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature **432**, 200–203 (2004).
[CrossRef] [PubMed]

**4. **E. Peter, P. Senellart, D. Martrou, A. Lemaitre, J. Hours, J. M. Gerard, and J. Bloch, “Exciton-Photon Strong-Coupling Regime for a Single Quantum Dot Embedded in a Microcavity,” Phys. Rev. Lett. **95**, 067401-1-4 (2005).
[CrossRef]

**5. **C. Kistner, T. Heindel, C. Schneider, A. Rahimi-Iman, S. Reitzenstein, S. Höfling, and A. Forchel, “Demonstration of strong coupling via electro-optical tuning in high-quality QD-micropillar systems,” Opt. Express **16**, 15006–15012 (2008).
[CrossRef] [PubMed]

**6. **A. Laucht, F. Hofbauer, N. Hauke, J. Angele, S. Stobbe, M. Kaniber, G. Böhm, P. Lodahl, and M.-C. Amann, “Electrical control of spontaneous emission and strong coupling for a single quantum dot,” New J. Phys. **11**, 023034-1-11 (2009).
[CrossRef]

**7. **L. V. Keldysh, V. D. Kulakovskii, S. Reitzenstein, M. N. Makhonin, and A. Forchel, “Interference effects in the emission spectra of quantum dots in high-quality cavities,” JETP Lett. **84**, 494–499 (2006).
[CrossRef]

**8. **F. P. Laussy, E. del Valle, and C. Tejedor, “Strong Coupling of Quantum Dots in Microcavities,” Phys. Rev. Lett. **101**, 083601-1-4 (2008).
[CrossRef]

**9. **Y. Yamamoto, T. Tassone, and H. Cao, “Biexcitonic Effects in Microcavities,” in Semiconductor Cavity Quantum Electrodynamics, G. Höhler, ed. (Springer, Berlin/Heidelberg2000), pp. 4958.

**10. **G. Khitrova, H.M. Gibbs, M. Kira, S.W. Koch, and A. Scherer, “Vacuum Rabi splitting in semiconductors,” Nat. Phys. **2**, 81–90 (2006).
[CrossRef]

**11. **E. Jaynes and F. Cummings, “Comparison of Quantum and Semiclassical Radiation Theory with Application to the Beam Maser,” Proc. IEEE **51**, 89–109 (1963).
[CrossRef]

**12. **L. Schneebeli, M. Kira, and S.W. Koch, “Characterization of Strong Light-Matter Coupling in Semiconductor Quantum-Dot Microcavities via Photon-Statistics Spectroscopy,” Phys. Rev. Lett. **101**, 097401-1-4 (2008).
[CrossRef]

**13. **E. del Valle, F. P. Laussy, F. M. Souza, and I. A. Shelykh, “Optical spectra of a quantum dot in a microcavity in the nonlinear regime,” Phys. Rev. B **78**, 085304-1-11 (2008).
[CrossRef]

**14. **J.-M. Gérard, “Solid-State Cavity-Quantum Electrodynamics with Self-Assembled Quantum Dots,” in Single Quantum Dots, Topics Appl. Phys. **90**, P. Michler, ed. (Springer, Berlin/Heidelberg2003), pp. 269315.

**15. **L. Andreani, G. Panzarini, and J.-M. Gérard, “Strong-coupling regime for quantum boxes in pillar microcavities: Theory,” Phys. Rev. B **60**, 13276–13279 (1999).
[CrossRef]

**16. **M. G. Guoqiang Cui and Raymer, “Emission spectra and quantum efficiency of single-photon sources in the cavity-QED strong-coupling regime,” Phys. Rev. A **73**, 053807-1-14 (2006).

**17. **A. Naesby, T. Suhr, P. T. Kristensen, and J. Mørk, “Influence of pure dephasing on emission spectra from single photon sources,” Phys. Rev. A **78**, 045802-1-4 (2008).
[CrossRef]

**18. **K. J. Vahala, “Optical Microcavities,” Nature **424**, 839–847 (2003).
[CrossRef] [PubMed]

**19. **A. Löffler, J. P. Reithmaier, G. Sek, C. Hofmann, S. Reitzenstein, M. Kamp, and A. Forchel, “Semiconductor quantum dot microcavity pillars with high-quality factors and enlarged dot dimensions,” Appl. Phys. Lett. **86**, 111105-1-3 (2005).
[CrossRef]

**20. **S. Reitzenstein, C. Hofmann, A. Gorbunov, M. Strauß, S. H. Kwon, C. Schneider, A. Löffler, S. Höfling, M. Kamp, and A. Forchel, “AlAs/GaAs micropillar cavities with quality factors exceeding 150.000,” Appl. Phys. Lett. **90**, 251109-1-3 (2007).
[CrossRef]

**21. **S. Reitzenstein, A. Löffler, C. Hofmann, A. Kubanek, M. Kamp, J. P. Reithmaier, A. Forchel, V. D. Kulakovskii, L. V. Keldysh, I. V. Ponomarev, and T. L. Reinecke, “Coherent photonic coupling of semiconductor quantum dots,” Opt. Letters **31**, 1738–1740 (2006).
[CrossRef]

**22. **S. A. Empedocles, D. J. Norris, and M. G. Bawendi, “Photoluminescence Spectroscopy of Single CdSe Nanocrystallite Quantum Dots,” Phys. Rev. Lett. **77**, 3873–3876 (1996).
[CrossRef] [PubMed]

**23. **I. Favero, A. Berthelot, G. Cassabois, C. Voisin, C. Delalande, Ph. Roussignol, R. Ferreira, and J. M. Gérard, “Temperature dependence of the zero-phonon linewidth in quantum dots: An effect of the fluctuating environment,” Phys. Rev. B **75**, 073308-1-4 (2007).
[CrossRef]

**24. **M. Bayer, O. Stern, P. Hawrylak, S. Fafard, and A. Forchel, “Hidden symmetries in the energy levels of excitonic artificial atoms,” Nature **405**, 923–926 (2000).
[CrossRef] [PubMed]

**25. **A. Imamoglu, “Phase-space filling and stimulated scattering of composite bosons,” Phys. Rev. B **57**, R4195–R4197 (1998).
[CrossRef]