## Abstract

We perform a theoretical study of the bistable four-wave mixing (FWM) response in a coupled system comprised of a semiconductor quantum dot (SQD) and a photonic crystal (PC) nanocavity in which the SQD is embedded. It is shown that the shape of the FWM spectrum can switch among single-peaked, double-peaked, triple-peaked, and four-peaked arising from the vacuum Rabi splitting and the exciton-nanocavity coupling. Especially, we map out bistability phase diagrams within a parameter subspace of the system, and find that it is easy to turn on or off the bistable FWM response by only adjusting the excitation frequency or the pumping intensity. Our results offer a feasible means for measuring the SQD-PC nanocavity coupling strength and open a new avenue to design optical switches and memories.

© 2017 Optical Society of America

## 1. Introduction

Optical bistability (OB), as an important research branch in quantum optics, has inspired considerable investigation due to its potential to develop optical switches, logic gates, and memory elements [1–4]. So far, there are a number of various systems used to obtain OB such as atom-cavity systems [5–17], atomic systems [18, 19], metal-semiconductor hybrid nanostructures [20–25], optomechanical systems [26, 27], a superconducting qubit-microwave cavity hybrid system [28], etc. A typical system comprised of a two-level system and a cavity is often employed as a resource to achieve OB [29–32]. In an earlier article, absorptive OB has also been reported theoretically for a single two-level atom inside a resonant optical cavity [29]. Lü et al. investigated the bistability of the cavity field amplitude in a hybrid system consisting of a coherently driven two-level emitter strongly coupled to a high-quality microcavity which is embedded within a photonic crystal (PC), and explored the dependence of the threshold value and hysteresis loop on the photonic band gap of the PC, Rabi frequency of the driving field and dephasing processes [30]. Dumeige et al. showed that the optical property of a microcavity can be modified by inserting a driven, two-level system inside it, and demonstrated that optical absorptive or dispersive bistability can be combined with the population-oscillation-induced steep material dispersion to obtain a strong microcavity-quality-factor enhancement [31]. Recently, Xu et al. demonstrated that a single two-level atom in an asymmetric cavity can generate controllable OB in the Purcell regime and the bistable regime can be shifted by adjusting the asymmetric walls of the cavity [32].

As we all know, an optical cavity can modify the exciton-phonon interaction in a highly nonlinear fashion. Different from a micropillar cavity [33, 34], the PC nanocavity is a point defect embedded within periodic dielectric structures. Within a full band gap, photons will be completely localized in the vicinity of the defect. Nanoscale PC cavities may be promising due to its highly confined ultrasmall mode volume V and ultrahigh quality Q-factor [35]. Recently, Li et al. have investigated the OB in a hybrid system composed of a PC nanocavity, a single nitrogen-vacancy center embedded in the cavity, and a nearby photonic waveguide serving for in- and out-coupling of light into the PC nanocavity [36]. Despite these advantages of PC nanocavities, OB based on the PC nanocavity has not received much attention. Moreover, systematic investigation on the bistable four-wave mixing (FWM) response of a semiconductor quantum dot (SQD) coupled to a PC nanocavity has never been performed.

In this paper, we will study theoretically the variation of the FWM signal in the regime ranging from weak to strong coupling and map out bistability phase diagrams in the parameter space of a coupled system comprised of a SQD and a PC nanocavity.

## 2. Model and formalism

The system under consideration is a SQD embedded in a PC nanocavity in the simultaneous presence of a strong pump field and a weak probe field, as depicted schematically in Fig. 1(a). Herein, the pump field drives only one cavity mode with a frequency *ω _{pc}*.

*E*(

_{pu}*E*) is the slowly varying envelope of the pump (probe) field, and

_{pr}*ω*(

_{pu}*ω*) is the frequency of the pump (probe) field. The SQD can be modeled as a two-level system consisting of a ground state |0> and a single exciton state |1>. The exciton can be characterized by three pseudospin −1/2 operators

_{pr}*σ*

_{01},

*σ*

_{10}and

*σ*. The typical physical situation is illustrated in Fig. 1(b).

_{z}In the rotating frame, the total Hamiltonian of the system takes the form [38, 39]

*=*

_{pu}*ω*

_{10}‒

*ω*is the exciton-pump field detuning, Δ

_{pu}*=*

_{pc}*ω*‒

_{pc}*ω*is the nanocavity-pump field detuning, and

_{pu}*δ*=

*ω*–

_{pr}*ω*is the probe-pump detuning.

_{pu}*b*

^{+}and

*b*are the creation and annihilation operators for photons, respectively.

*g*denotes the coupling strength between the exciton in the SQD and the photons in the PC nanocavity, Ω =

*μE*/h represents the effective Rabi frequency of the pump field, and

_{pu}*μ*represents the electric dipole moment. For simplicity, we set

*p =*<

*σ*

_{01}>,

*w*= 2<

*σ*> and

_{z}*Λ*= <

*b*>.

Reducing the operators to their mean values and dropping the quantum and thermal noise terms, we can obtain the corresponding quantum Langevin equations [40]:

_{2}(Γ

_{1}) denotes the exciton dephasing rate (relaxation rate), and

*κ*refers to the decay rate of the cavity field.

In order to solve the above equations, we make the ansatz *p* = *p*_{0} + *p*_{1}*e*^{–}* ^{iδt}* +

*p*

_{–1}

*e*,

^{iδt}*w*=

*w*

_{0}+

*w*

_{1}

*e*

^{–}

*+*

^{iδt}*w*

_{–1}

*e*, and

^{iδt}*Λ*=

*b*

_{0}+

*b*

_{1}

*e*

^{–}

*+*

^{iδt}*b*

_{–1}

*e*[38], where

^{iδt}*p*

_{0},

*w*

_{0}, and

*b*

_{0}is the solution of Eqs. (2) – (4) for the case in which only the pump field is present. Here |

*p*

_{0}| >> |

*p*

_{1}|, |

*p*

_{–1}|, |

*w*

_{0}| >> |

*w*

_{1}|, |

*w*

_{–1}|, |

*b*

_{0}| >> |

*b*

_{1}|, |

*b*

_{–1}|. By substituting the above ansatz into Eqs. (2) – (4) and solving these equations, we can obtain the FWM signal defined as

The steady-state population inversion of the exciton *w*_{0} can be obtained by solving the following third-order equation

## 3. Numerical results and discussion

We start with a realistic system of an InAs/GaAs QD coupled to a PC nanocavity. We calculate the FWM signal and the exciton-population inversion in this system for the parameters [41]: *κ* = 8 MHz, Γ_{1} = 2Γ_{2} = 5.2 MHz.

As we all know, the exciton-phonon interaction plays a key role in the modification of optical properties of the QD system [42–47]. To reveal the impact of the exciton-photon interaction on the FWM response in a coupled SQD-PC nanocavity system, in Fig. 2 we show how |*p*_{−1}/*μE _{pr}**h

^{−1}Γ

_{2}

^{−1}|, defined in Eq. (5), changes with the pumping intensity

*I*in three different cases including the weak coupling regime (

_{pu}*g*< Γ

_{2},

*κ*), the intermediate regime (

*g*~Γ

_{2},

*κ*), and the strong coupling regime (

*g*> Γ

_{2},

*κ*). The results presented in Fig. 2(a) show that the full width at half maximum (FWHM) increases and the amplitude of the FWM peak decreases by turning the exciton-nanocavity coupling from off (

*g*= 0) to on (

*g*= 2 MHz). This peak located at

*ω*=

_{pr}*ω*can be ascribed to a usual optical absorptive behavior [48]. In fact, the SQD-PC nanocavity system in the no-coupling case can be modeled as a pure SQD system. Similar single-peaked FWM spectra have also been observed in individual InAs quantum dots [49]. As shown in Fig. 2(b), in the weak coupling regime (

_{pu}*g*= 2 MHz), the FWM signal will be enhanced greatly as

*I*increases from 1 to 10 MHz

_{pu}^{2}. As

*I*further increases to 100 MHz

_{pu}^{2}, however, the situation becomes quite distinct. The shape of the FWM spectrum will change from single-peaked to triple-peaked. The inset of Fig. 2(b) shows the origin of this triple-peaked structure which is attributed to a three-photon resonance process. Here the electron makes a transition from the lowest energy level |0,

*n*> to the highest energy level |1,

*n*+ 1> by simultaneous absorption of two pump photons and the emission of a photon [38, 39]. For a larger pumping intensity (

*I*= 1000 MHz

_{pu}^{2}), the amplitudes of these three peaks in the FWM spectrum are all reduced to lower values, and the peaks at two Rabi sidebands both move to the direction with a larger |

*δ*|.

In the intermediate regime (*g* = 6 MHz), the FWM spectrum will change from double-peaked to triple-peaked by gradually increasing the pumping intensity *I _{pu}* [Fig. 2(c)]. In the strong coupling regime (

*g*= 30 MHz), however, the scenario becomes rather different. The FWM spectrum is always double-peaked despite the increase in

*I*, and the peak value of the FWM signal increases linearly as

_{pu}*I*increases. These two peaks at Rabi sidebands are attributed to the vacuum Rabi splitting [50, 51]. Such double-peaked FWM spectra have also been observed experimentally in individual InAs QDs embedded in a low-Q asymmetrical GaAs/AlGaAs microcavity [52]. Here, the measured zero-detuning (Δ

_{pu}*= 0) vacuum Rabi splitting is 2*

_{pu}*g*= 60 MHz, which provides a new way to measure the SQD-nanocavity coupling strength and reveal the vacuum Rabi splitting of a SQD embedded in a PC nanocavity. These interesting results are plotted in Fig. 2(d). From the above discussion, we can draw a conclusion that the evolution of FWM signals depends strongly on a combined effect of the vacuum Rabi splitting and the exciton-nanocavity coupling.

In Fig. 3 we study the way that the FWM signal changes with the exciton-pump field detuning Δ* _{pu}* when

*I*= 1000 MHz

_{pu}^{2}and

*g*= 30 MHz. In the strong coupling regime, as the pump field is exactly resonant with the exciton in the SQD (Δ

*= 0), the FWM spectrum exhibits a double-peak structure whose symmetry axis is given by the line*

_{pu}*δ*= 0. When the pump field is detuned from the exciton transition (Δ

*≠ 0), the scenario becomes completely different. As Δ*

_{pu}*increases, two new peaks appear at the inside of two sideband peaks, and the shape of the FWM spectrum will change from two-peaked to four-peaked. This behavior may be ascribed to the off-resonant coupling between the SQD and the PC nanocavity. Such an off-resonant coupling character has been demonstrated in a coupled system composed of a single QD and a nanobeam PC cavity [53]. In addition, the magnitudes of the sideband peaks increase with an increase of Δ*

_{pu}*, while the magnitudes of the other peaks decrease as Δ*

_{pu}*increases. The FWM spectrum for Δ*

_{pu}*= 30 MHz is the same as that in the case of Δ*

_{pu}*= −30 MHz. Obviously, the FWM spectrum can be modified effectively via the off-resonant coupling between the SQD and the PC nanocavity.*

_{pu}To obtain a bistable FWM signal, we note that, the bistable effect can be registered only as the Eq. (6) has three real roots. The occurrence of OB in the FWM process is strongly correlated with the excitation frequency, pumping intensity, and exciton-nanocavity coupling strength. In the next calculations, we mainly study the OB of the FWM signal in the case of *δ* = 0 (i.e. *ω _{pu}* =

*ω*). Considering this, Fig. 4(a) shows how the FWM signal |

_{pr}*p*

_{−1}/

*μE**h

_{pr}^{−1}Γ

_{2}

^{−1}| changes with the pumping intensity

*I*for a given excitation frequency in the coupling regime ranging from weak to strong. The results show that as the pump field is exactly resonant with the exciton in the SQD, for the cases of

_{pu}*g*= 0 MHz,

*g*= 2 MHz, and

*g*= 6 MHz, the FWM signal first gradually becomes strong, reaching a maximum value, then declines to a stable value as

*I*increases. The results in Fig. 4(b) show that as

_{pu}*I*increases, the population inversion

_{pu}*w*

_{0}increases rapidly to a maximum value 0, then

*w*

_{0}almost keeps invariant despite the change of

*I*. This suggests that the enhanced pumping intensity will make the system much easier to saturation. In the weak and intermediate cases, the exciton-nanocavity coupling has a weak impact on the FWM response. In addition, |

_{pu}*p*

_{−1}/

*μE**h

_{pr}^{−1}Γ

_{2}

^{−1}| and

*w*

_{0}are both a single-value function of

*I*. This indicates that the bistable effect is always absent in the weak and intermediate coupling regimes.

_{pu}Figure 4(c) shows how the FWM signal |*p*_{−1}/*μE _{pr}**h

^{−1}Γ

_{2}

^{−1}| changes with the pumping intensity

*I*for various excitation frequencies (i.e. Δ

_{pu}*) in the strong coupling regime (*

_{pu}*g*= 30 MHz). As we expected, the bistable effect occurs, registering with an asymmetric “U-shaped” FWM curve. However, the hysteresis loops are difficult to be observed because the upper and lower branches of bistable curves almost overlap together. In order to reveal what exactly plotted in Fig. 4(c), the inset shows the magnification of these overlapped regions in Fig. 4(c) which exhibits a bistable exotic coiled curve. As Δ

*increases gradually, the bistable region becomes narrower and narrower and the corresponding bistable thresholds become larger and larger. In fact, the width of the bistable region for Δ*

_{pu}*= 30 MHz is only 15.9% of that for Δ*

_{pu}*= 0. Moreover, the bistable effect to a certain extent will be suppressed as the pump field is detuned from the exciton resonance (|Δ*

_{pu}*| > 0). Thus, based on the above discussion, we can draw a conclusion that strong exciton-photon coupling and the excitation frequency near the exciton resonance are beneficial to promote the occurrence of OB.*

_{pu}To further clarify this bistable behavior of the FWM signal, an optical hysteresis loop of the population difference *w*_{0} with the pumping intensity *I _{pu}* is plotted in Fig. 4(d). As the pumping intensity

*I*increases, the system firstly follows the lower (stable) branch and then jumps to the upper (stable) branch at

_{pu}*I*= 6627.55 MHz

_{pu}^{2}. With sweeping

*I*back, the system remains on the upper branch and then makes a transition to the lower branch at

_{pu}*I*= 1156.32 MHz

_{pu}^{2}. A hysteresis loop has been completed. The intermediate branch is unstable. It is also worth mentioning, the bistability can be revealed by detecting the optical hysteresis of the FWM signal.

The results presented in Figs. 4(a) and 4(c) show that the optical bistable effect may arise as the SQD couples strongly with the PC nanocavity. To clarify this further, Figs. 5(a) and 5(b) show the bistability phase diagrams within a parameter subspace [*I _{pu}*;

*g*; different Δ

*]. The boundaries between white and colored regions denote the bistable thresholds. The low and high bistable thresholds are denoted as*

_{pu}*I*and

_{min}*I*, respectively. In general, the bistability is removed because of a weak exciton-nanocavity coupling. When the pump field is exactly resonant with the exciton in the SQD (Δ

_{max}*= 0), for*

_{pu}*g*< 9.13 MHz, the bistability is always absent despite the change of

*I*, while for

_{pu}*g*≥ 9.13 MHz, the bistability may exist [Fig. 5(a)]. When the pump field is detuned from the exciton resonance (Δ

*= 30 MHz), a larger critical value of*

_{pu}*g*is required to attain the bistability. More precisely, the bistable effect possibly occurs only when g ≥ 25.63 MHz [Fig. 5(b)]. To further explore the physics of the bistability phase diagrams, we compare the results in the above two situations. The related results are plotted in Fig. 5(c). For Δ

*= 0, the critical conditions of OB are*

_{pu}*g*= 9.13 MHz and

*I*= 91.5 MHz

_{pu}^{2}. However, as Δ

*increase to 30 MHz, the critical conditions of OB become*

_{pu}*g*= 25.63 MHz and

*I*= 4188.1 MHz

_{pu}^{2}. Obviously, the bistable thresholds of

*g*and

*I*at off-resonant excitation (Δ

_{pu}*= 30 MHz) are both pushed to larger values compared to that at resonant excitation (Δ*

_{pu}*= 0). Also, the bistable region for Δ*

_{pu}*= 30 MHz becomes smaller than that for Δ*

_{pu}*= 0 MHz. In a whole, the occurrence of OB depends strongly on the excitation frequency, pumping intensity and exciton-nanocavity coupling.*

_{pu}From the results presented in Figs. 4(c), we see that the bistable effect is strongly dependent on the excitation frequency (i.e. Δ* _{pu}*). To investigate this issue further, we plot the bistability phase diagram within a parameter subspace [

*I*; Δ

_{pu}*;*

_{pu}*g*= 30 MHz]. As shown in Fig. 6, the bistable effect arises in the colored “triangle” region. This region exhibits a perfect symmetry with respect to the axis Δ

*= 0 (i.e.*

_{pu}*ω*=

_{pu}*ω*

_{10}). Similar bistability phase diagrams have also been observed by the group of Prof. Knoester in a SQD-metal nanoparticle heterodimer [54]. In the strong coupling regime (

*g*= 30 MHz), the bistability exists within a window of −41.75 MHz ≤ Δ

*≤ 41.75 MHz. It is not difficult to find that the adjustable range of bistability at Δ*

_{pu}*= 0 reaches a maximum value (i.e. 1156.3 MHz*

_{pu}^{2}≤

*I*≤ 6627.5 MHz

_{pu}^{2}). The appearance of OB at a low pumping intensity means prospects of all-optical control using a weak pump light.

As a concluding remark, we would like to point out two points: (1) A single SQD coupling to a PC cavity has emerged as a promising platform for realizing key components in quantum information processing. As the exciton in the SQD couples strongly with the photons in the PC cavity, a modification of the optical spectra of the QD or cavity can be observed [55–60]. However, our scheme is different from those works. In a strongly coupled system composed of a two-level SQD and a PC nanocavity, we find that the bistability can be revealed by detecting the optical hysteresis of the FWM signal. Especially, we map out bistability phase diagrams within a parameter subspace of the system. (2) The theoretical predictions are possibility to be demonstrated experimentally. Here, a realistic two-level InAs/GaAs QD has been selected as a research source. Also, the constructed nanoscale PC cavities are promising because of their highly confined ultrasmall mode volume V and ultrahigh quality Q-factor. Moreover, the experimental conditions can be easily realized because the PC nanocavity mode frequency strongly depends on the geometry of the PC and its material.

## 4. Summary

In summary, we have explored the variation of the FWM signal in all regimes from weak to intermediate to strong coupling in a coupled SQD-PC nanocavity system. Due to the vacuum Rabi splitting and exciton-nanocavity coupling, the shape of the FWM spectrum can switch among single-peaked, double-peaked, and triple-peaked as the pump field is resonant with the exciton transition, and the FWM spectrum will become four-peaked in the strong coupling regime under near-resonant excitation. Especially, we have mapped out bistability phase diagrams within the system’s parameter space, and found that the optical bistable effect emerges only as the SQD couples strongly with the nanocavity, and it is easy to turn on or off the bistable effect by only adjusting the excitation frequency or the pumping intensity. These results suggest that a coupled SQD-PC nanocavity system can act as a promising candidate for developing optical switches and memories. Finally, we hope that our results can be demonstrated experimentally in the near future.

## Funding and Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) under Grants Nos. 11404410, 11504105 and 11504434, the Hunan Provincial Natural Science Foundation of China under Grants Nos. 14JJ3116 and 2015JJ3174, the Foundation of Talent Introduction of Central South University of Forestry and Technology under Grant No.104-0260, and the Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province under Grant No. GD201403).

## References and links

**1. **H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry-Perot interferometer,” Phys. Rev. Lett. **36**(19), 1135–1138 (1976).

**2. **D. E. Grant and H. J. Kimble, “Optical bistability for two-level atoms in a standing-wave cavity,” Opt. Lett. **7**(8), 353–355 (1982). [PubMed]

**3. **B. Nagorny, T. Elsässer, and A. Hemmerich, “Collective atomic motion in an optical lattice formed inside a high finesse cavity,” Phys. Rev. Lett. **91**(15), 153003 (2003). [PubMed]

**4. **Z. P. Wang, S. Zhen, and B. Yu, “Controlling optical bistability of acceptor and donor quantum dots embedded in a nonlinear photonic crystal,” Laser Phys. Lett. **12**(4), 046004 (2015).

**5. **H. Wang, D. J. Goorskey, and M. Xiao, “Bistability and instability of three-level atoms inside an optical cavity,” Phys. Rev. A **65**(1), 011801 (2001).

**6. **H. Chang, H. Wu, C. Xie, and H. Wang, “Controlled Shift of optical bistability hysteresis curve and storage of optical signals in a four-level atomic system,” Phys. Rev. Lett. **93**(21), 213901 (2004). [PubMed]

**7. **Th. Elsässer, B. Nagorny, and A. Hemmerich, “Optical bistability and collective behavior of atoms trapped in a high-Q ring cavity,” Phys. Rev. A **69**(3), 033403 (2004).

**8. **A. Joshi, W. Yang, and M. Xiao, “Dynamical hysteresis in a three-level atomic system,” Opt. Lett. **30**(8), 905–907 (2005). [PubMed]

**9. **S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, “Cavity nonlinear optics at low photon numbers from collective atomic motion,” Phys. Rev. Lett. **99**(21), 213601 (2007). [PubMed]

**10. **A. Mitra and R. Vyas, “Entanglement and bistability in coupled quantum dots inside a driven cavity,” Phys. Rev. A **81**(1), 012329 (2010).

**11. **S. Yang, M. Alamri, J. Evers, and M. S. Zubairy, “Controllable optical switch using a Bose-Einstein condensate in an optical cavity,” Phys. Rev. A **83**(5), 053821 (2011).

**12. **J. Yuan, W. Feng, P. Li, X. Zhang, Y. Zhang, H. Zheng, and Y. Zhang, “Controllable vacuum Rabi splitting and optical bistability of multi-wave-mixing signal inside a ring cavity,” Phys. Rev. A **86**(6), 063820 (2012).

**13. **Z. Wang and B. Yu, “Switching from optical bistability to multistability in a coupled semiconductor double-quantum-dot nanostructure,” J. Opt. Soc. Am. B **30**(11), 2915–2920 (2013).

**14. **S. Safaei, Ö. E. Müstecaplıoğlu, and B. Tanatar, “Bistable behavior of a two-mode Bose-Einstein condensate in an optical cavity,” Laser Phys. **23**(3), 035501 (2013).

**15. **S. H. Asadpour and H. R. Soleimani, “Optical bistability in a three-level lambda molecule with permanent dipole moments,” J. Opt. Soc. Am. B **31**(12), 3123–3130 (2014).

**16. **S. Dutta and S. A. Rangwala, “All-optical switching in a continuously operated and strongly coupled atom-cavity system,” Appl. Phys. Lett. **110**(12), 121107 (2017).

**17. **A. Dalafi and M. H. Naderi, “Intrinsic cross-Kerr nonlinearity in an optical cavity containing an interacting Bose-Einstein condensate,” Phys. Rev. A **95**(4), 043601 (2017).

**18. **A. Joshi, W. Yang, and M. Xiao, “Dynamical hysteresis in a three-level atomic system,” Opt. Lett. **30**(8), 905–907 (2005). [PubMed]

**19. **F. Wang, X. Feng, and C. H. Oh, “Optical bistability and multistability via quantum coherence in chiral molecules,” Opt. Express **24**(13), 13702–13713 (2016). [PubMed]

**20. **R. D. Artuso and G. W. Bryant, “Optical response of strongly coupled quantum dot-metal nanoparticle systems: double peaked Fano structure and bistability,” Nano Lett. **8**(7), 2106–2111 (2008). [PubMed]

**21. **A. V. Malyshev, “Condition for resonant optical bistability,” Phys. Rev. A **86**(6), 065804 (2012).

**22. **J. B. Li, N. C. Kim, M. T. Cheng, L. Zhou, Z. H. Hao, and Q. Q. Wang, “Optical bistability and nonlinearity of coherently coupled exciton-plasmon systems,” Opt. Express **20**(2), 1856–1861 (2012). [PubMed]

**23. **J. B. Li, S. Liang, S. Xiao, M. D. He, N. C. Kim, L. Q. Chen, G. H. Wu, Y. X. Peng, X. Y. Luo, and Z. P. Guo, “Four-wave mixing signal enhancement and optical bistability of a hybrid metal nanoparticle-quantum dot molecule in a nanomechanical resonator,” Opt. Express **24**(3), 2360–2369 (2016). [PubMed]

**24. **B. S. Nugroho, A. A. Iskandar, V. A. Malyshev, and J. Knoester, “Instabilities in the optical response of a semiconductor quantum dot-metal nanoparticle heterodimer: self-oscillations and chaos,” J. Opt. **19**, 015004 (2017).

**25. **S. H. Asadpour and H. R. Soleimani, “Phase dependence of optical bistability and multistability in a four-level quantum system near a plasmonic nanostructure,” J. Appl. Phys. **119**(2), 023102 (2016).

**26. **B. Sarma and A. K. Sarma, “Controllable optical bistability in a hybrid optomechanical system,” J. Opt. Soc. Am. B **33**(7), 1335–1340 (2016).

**27. **S. H. Kazemi, S. Ghanbari, and M. Mahmoudi, “Controllable optical bistability in a cavity optomechanical system with a Bose-Einstein condensate,” Laser Phys. **26**, 055502 (2016).

**28. **T. K. Mavrogordatos, G. Tancredi, M. Elliott, M. J. Peterer, A. Patterson, J. Rahamim, P. J. Leek, E. Ginossar, and M. H. Szymańska, “Simultaneous bistability of a qubit and resonator in circuit quantum electrodynamics,” Phys. Rev. Lett. **118**(4), 040402 (2017). [PubMed]

**29. **C. M. Savage and H. J. Carmichael, “Single-atom optical bistability,” IEEE J. Quantum Electron. **24**(8), 1495–1498 (1988).

**30. **X. Guo and S. Lü, “Controllable optical bistability in photonic-crystal one-atom laser,” Phys. Rev. A **80**(4), 043826 (2009).

**31. **Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. L. Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A **85**(6), 063824 (2012).

**32. **X. Xia, J. Xu, and Y. Yang, “Controllable optical bistability of an asymmetric cavity containing a single two-level atom,” Phys. Rev. A **90**(4), 043857 (2014).

**33. **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**(7), 4848–4857 (2008). [PubMed]

**34. **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**(4), 304–308 (2010). [PubMed]

**35. **S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**(23), 2486–2489 (1987). [PubMed]

**36. **J. Li, R. Yu, C. Ding, and Y. Wu, “Optical bistability and four-wave mixing with a single nitrogen-vacancy center coupled to a photonic crystal nanocavity in the weak-coupling regime,” Opt. Express **22**(12), 15024–15038 (2014). [PubMed]

**37. **B. Gu, W. Ji, P. S. Patil, and S. M. Dharmaprakash, “Ultrafast optical nonlinearities and figures of merit in acceptor-substituted 3,4,5-trimethoxy chalcone derivatives: Structure-property relationships,” J. Appl. Phys. **103**(10), 103511 (2008).

**38. **R. W. Boyd, *Nonlinear Optics* (Academic Press, 2008).

**39. **J. J. Li and K. D. Zhu, “A quantum optical transistor with a single quantum dot in a photonic crystal nanocavity,” Nanotechnology **22**(5), 055202 (2011). [PubMed]

**40. **D. F. Walls and G. J. Milburn, *Quantum Optics* (Springer, 1994).

**41. **L. M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. **92**(12), 127902 (2004). [PubMed]

**42. **M. Glässl, A. M. Barth, and V. M. Axt, “Proposed robust and high-fidelity preparation of excitons and biexcitons in semiconductor quantum dots making active use of phonons,” Phys. Rev. Lett. **110**(14), 147401 (2013). [PubMed]

**43. **A. M. Barth, S. Lüker, A. Vagov, D. E. Reiter, T. Kuhn, and V. M. Axt, “Fast and selective phonon-assisted state preparation of a quantum dot by adiabatic undressing,” Phys. Rev. B **94**(4), 045306 (2016).

**44. **V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and factor in a photonic crystal waveguide,” Phys. Rev. B **75**(20), 205437 (2007).

**45. **S. Hughes and G. S. Agarwal, “Anisotropy-induced quantum interference and population trapping between orthogonal quantum dot exciton states in semiconductor cavity systems,” Phys. Rev. Lett. **118**(6), 063601 (2017). [PubMed]

**46. **J. M. Daniels, P. Machnikowski, and T. Kuhn, “Excitons in quantum dot molecules: Coulomb coupling, spin-orbit effects, and phonon-induced line broadening,” Phys. Rev. B **88**(20), 205307 (2013).

**47. **D. E. Reiter, T. Kuhn, M. Glässl, and V. M. Axt, “The role of phonons for exciton and biexciton generation in an optically driven quantum dot,” J. Phys. Condens. Matter **26**(42), 423203 (2014). [PubMed]

**48. **R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Harter, “Four-wave parametric interactions in a strongly driven two-level system,” Phys. Rev. A **24**(1), 411–423 (1981).

**49. **J. Kasprzak and W. Langbein, “Four-wave mixing from individual excitons: Intensity dependence and imaging,” Phys. Status Solidi, B Basic Res. **246**(4), 820–823 (2009).

**50. **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**(7014), 200–203 (2004). [PubMed]

**51. **Y. C. Yu, J. F. Liu, X. L. Zhuo, G. Chen, C. J. Jin, and X. H. Wang, “Vacuum Rabi splitting in a coupled system of single quantum dot and photonic crystal cavity: effect of local and propagation Green’s functions,” Opt. Express **21**(20), 23486–23497 (2013). [PubMed]

**52. **Q. Mermillod, D. Wigger, V. Delmonte, D. E. Reiter, C. Schneider, M. Kamp, S. Hofling, W. Langbein, T. Kuhn, G. Nogues, and J. Kasprzak, “Dynamics of excitons in individual InAs quantum dots revealed in four-wave mixing spectroscopy,” Optica **3**(4), 377–384 (2016).

**53. **A. Rundquist, A. Majumdar, and J. Vučković, “Off-resonant coupling between a single quantum dot and a nanobeam photonic crystal cavity,” Appl. Phys. Lett. **99**(25), 251907 (2011).

**54. **B. S. Nugroho, A. A. Iskandar, V. A. Malyshev, and J. Knoester, “Bistable optical response of a nanoparticle heterodimer: Mechanism, phase diagram, and switching time,” J. Chem. Phys. **139**(1), 014303 (2013). [PubMed]

**55. **K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature **445**(7130), 896–899 (2007). [PubMed]

**56. **M. R. Singh, C. Racknor, and D. Schindel, “Controlling the photoluminescence of acceptor and donor quantum dots embedded in a nonlinear photonic crystal,” Appl. Phys. Lett. **101**(5), 051115 (2012).

**57. **J. Liu, S. Ates, M. Lorke, J. Mørk, P. Lodahl, and S. Stobbe, “A comparison between experiment and theory on few-quantum-dot nanolasing in a photonic-crystal cavity,” Opt. Express **21**(23), 28507–28512 (2013). [PubMed]

**58. **P. Lodahl and S. Stobbe, “Solid-state quantum optics with quantum dots in photonic nanostructures,” Nanophotonics **2**(1), 39–55 (2013).

**59. **P. Lodahl, S. Mahmoodian, and S. Stobbe, “Interfacing single photons and single quantum dots with photonic nanostructures,” Rev. Mod. Phys. **87**(2), 347–400 (2015).

**60. **K. H. Madsen, T. B. Lehmann, and P. Lodahl, “Role of multilevel states on quantum-dot emission in photonic-crystal cavities,” Phys. Rev. B **94**(23), 235301 (2016).