⇦Chinese Physics Letters, 2017, Vol. 34, No. 1, Article code 017303 Modulating the Lasing Performance of the Quantum Dot-Cavity System by Adding a Resonant Driving Field * Li-Guo Qin(秦利国)1,2, Qin Wang(王琴)1,2** Affiliations 1Institute of Signal Processing and Transmission, Nanjing University of Posts and Telecommunications, Nanjing 210003 2Key Lab of Broadband Wireless Communication and Sensor Network Technology (Ministry of Education), Nanjing University of Posts and Telecommunications, Nanjing 210003 Received 25 September 2016 *Supported by the National Natural Science Foundation of China under Grant Nos 11274178, 61475197 and 61590932, the Natural Science Foundation of the Jiangsu Higher Education Institutions under Grant No 15KJA120002, the outstanding Youth Project of Jiangsu Province under Grant No BK20150039, and the Priority Academic Program Development of Jiangsu Higher Education Institutions under Grant No YX002001.
**Corresponding author. Email: qinw@njupt.edu.cn Citation Text: Qin L G and Wang Q 2017 Chin. Phys. Lett. 34 017303 Abstract We propose a new scheme on modulating the lasing performance of a quantum dot-cavity system. Compared to the conventional above-band pump, in our new scheme an additional resonant driving field is applied on the quantum dot-cavity system. By employing the master equation theory and the Jaynes–Cummings model, we are able to study the interesting phenomenon of the coupling system. To compare the different behaviors between using our new scheme and the conventional method, we carry out investigation for both the "good system" and "more realistic system", characterizing several important parameters, such as the cavity population, exciton population and the second-order correlation function at zero time delay. Through numerical simulations, we demonstrate that for both the good system and more realistic system, their lasing regimes can be displaced into other regimes in the presence of a resonant driving field. DOI:10.1088/0256-307X/34/1/017303 PACS:73.21.La, 78.67.Hc, 42.50.-p, 42.50.Wk © 2017 Chinese Physics Society Article Text Nanolasers have attracted extensive interests from the scientific world due to their bright prospect of applications in many fields, such as single molecular spectroscopy, ultra-high density of data storage, nano-lithography and quantum circuit. To date, they has been investigated in different kinds of systems such as atoms, molecules, trapped ions, NV centers, nanowires or quantum dots.[1-6] Among them, single quantum-dot (QD) lasers are standing out owing to many inherent excellent characteristics. For example, they are compatible with the current matured semiconductor fabrication technique, and can be easily integrated into an optical or electrical circuit on a chip. Such a system usually consists of a single quantum dot and a coupled micro-cavity, and it has been theoretically and experimentally investigated in recent years.[7-23] However, most researches on generating nanolasers with the quantum dot-cavity system mainly focus on using above-band pump conditions.[15-23] As we know, the above-band pump does not act directly on the QD like the resonant pump, thus it has a poorer efficiency than the resonant pump. It may result in poor lasing performance with realistic systems, such as a high pump threshold or a narrow lasing scope. In this Letter, we propose a new scheme on improving the lasing performance of a quantum-dot system by introducing an additional resonant pump field. Our system consists of a single-mode microcavity and a two-energy-level quantum dot, as schematically shown in Fig. 1. In the new scheme, the quantum dot-cavity system is driven by two external lasers. One pumps on the above-band, and the other pumps on the resonance with the QD. Then the QD and the cavity are coupled together through excitation fields, where the cavity has a leakage rate of $\kappa$ and the QD has a spontaneous emission rate of $\gamma$. In the following we employ the Jaynes–Cummings model[30] and carry out investigation on the lasing performance of the quantum dot-cavity system under the existence of both a coherent and an incoherent driving field. In addition, here we reasonably assume that the cavity follows bosonic statistics and the QD obeys Fermi statistics. Then we obtain the total Hamiltonian of the system from the rotating wave approximation as follows ($\hbar$=1 is assumed): $$\begin{align} H=\,&\omega_{a}a^†a+\omega_{\sigma}\sigma^†\sigma+g(a^†\sigma\\ &+a\sigma^†)+{\it \Omega}(\sigma^†+\sigma),~~ \tag {1} \end{align} $$ where $a$ ($a^†$) is the annihilation (creation) operator of the cavity mode, $\sigma$ ($\sigma^†$) represents the lowering (raising) operator of the QD exciton, $†$ denotes conjugation, $g$ denotes the coupling strength between the cavity and the QD, $\omega_{a}$ and $\omega_{\sigma}$ are the frequencies of the cavity mode and the QD, respectively, ${\it \Omega}$ is the Rabi frequency of the field to resonantly pump the QD, and ${\it \Delta}=\omega_{a}-\omega_{\sigma}$ is the cavity detuning from QD, which can be tuned by changing the temperature or depositing gases. Then the evolution of the coupled system follows the master equation $$ \frac{d\rho}{dt}=-i[H,\rho]+{\cal L}(\rho),~~ \tag {2} $$ where $\rho$ is the density matrix of the coupled system, the Liouvillian operator ${\cal L}(\rho)$ represents the dissipative processes, which includes the incoherent pumping process, the spontaneous emission of the QD, as well as the cavity loss.
cpl-34-1-017303-fig1.png
Fig. 1. (Color online) The schematic diagram of the quantum dot-cavity system. The QD interplays with the cavity mode with the coupling factor $g$. The blue circle represents an electron and the black circle represents a hole. The wavy line represents the cavity field caused by $P_{a}$. Here $P_{a}$ and $P_{\rm \sigma}$ are the above-band pumps exerting on the cavity mode and the QD, respectively, ${\it \Omega}$ is the Rabi frequency of the resonant field for the QD, $\kappa$ is the decay rate of microcavity, and $\gamma$ represents the spontaneous emission rate of the QD.
We use a system-reservoir interaction to describe the incoherent excitation,[20,24] where the cavity and exciton are both pumped incoherently via the corresponding reservoirs. The superoperator ${\cal L}(\rho)$ in Eq. (2) can be denoted in the Lindblad form $$\begin{align} {\cal L}(\rho)=\,&\frac{\kappa}{2}(2a\rho a^†-a^†a\rho-\rho a^†a)\\ &+\frac{\gamma}{2}(2\sigma\rho\sigma^†-\sigma^†\sigma\rho- \rho\sigma^†\sigma)+\frac{P_{a}}{2}\\ &(2a^†\rho a-aa^†\rho-\rho aa^†)\\ &+\frac{P_{\sigma}}{2}(2\sigma^†\rho\sigma- \sigma\sigma^†\rho-\rho\sigma\sigma^†),~~ \tag {3} \end{align} $$ where $\kappa$ and $\gamma$ represent the decay rate of the cavity and QD individually, and $P_{a}$ and $P_{\sigma}$ correspond to the incoherent pump rate for the cavity and QD, respectively.[17] Considering the fact that $[a, a^†]=1$, $[\sigma, \sigma^†]_{+}=1$, and $\frac{\partial\langle A\rangle}{\partial t}={\rm Tr}(A\frac{d\rho}{dt})$,[25] we can derive $$\begin{alignat}{1} \frac{\partial \langle a\rangle}{\partial t}=\,& \Big(\frac{P_{a}-\kappa}{2}-i\omega_{a}\Big)\langle a\rangle-ig\langle\sigma\rangle, \\ \frac{\partial\langle\sigma\rangle}{\partial t}=\,& -ig\langle a\rangle-\Big(\frac{P_{\sigma}+\gamma}{2}+i\omega_{\sigma}\Big) \langle\sigma\rangle-i{\it \Omega},~~ \tag {4} \end{alignat} $$ and $$\begin{alignat}{1} \frac{\partial\langle a^†a\rangle}{\partial t} =\,& P_{a}+(P_{a}-\kappa)\langle a^†a\rangle\\ &-ig\langle a^†\sigma\rangle+ig\langle a\sigma^†\rangle, \\ \frac{\partial\langle\sigma^{+}\sigma\rangle}{\partial t} =\,& P_{\sigma}-(P_{\sigma}+\gamma)\langle\sigma^†\sigma\rangle+ig\langle a^†\sigma\rangle\\ &-ig\langle a\sigma^†\rangle-{\it \Omega}\langle\sigma^†\rangle+i{\it \Omega}\langle\sigma\rangle, \\ \end{alignat} $$ $$\begin{alignat}{1} \frac{\partial \langle a^†\sigma\rangle}{\partial t} =\,& -ig\langle a^†a\rangle+ig\langle\sigma^†\sigma\rangle+\Big(i{\it \Delta}\\ &+\frac{P_{a}-P_{\sigma}-\kappa-\gamma}{2}\Big) \langle a^†\sigma\rangle-i{\it \Omega}\langle a^†\rangle, \\ \frac{\partial \langle a\sigma^†\rangle}{\partial t} =\,& ig\langle a^†a\rangle-ig\langle\sigma^†\sigma\rangle+\Big(-i{\it \Delta}\\ &+\frac{P_{a}-P_{\sigma}-\kappa-\gamma}{2}\Big) \langle a\sigma^†\rangle+i{\it \Omega}\langle a\rangle.~~ \tag {5} \end{alignat} $$ Let us consider the steady-state (ss) system, where the above partial derivative equation equals to zero. By solving Eqs. (4) and (5), we can obtain $$\begin{align} \langle a^†a\rangle_{\rm ss}=\,& (-1)/[(-P_{a}+P_{\sigma}+\kappa+\gamma)(-4\\ &+(P_{a}-\kappa)(P_{\sigma }+\gamma))^{2}]\times[-P_{a}^{3}(P_{\sigma}+\gamma)^{2}\\ &+P_{a}^{2}(P_{\sigma}+\gamma) (8+(P_{\sigma}+\gamma)(P_{\sigma}+\gamma+2\kappa))\\ &-4(P_{\sigma}^{2}\kappa+4{\it \Omega}^{2}(\kappa+\gamma)+P_{\sigma}(4+4{\it \Omega}^{2}\\ &+\kappa\gamma))+P_{a}(-16+16{\it \Omega}^{2}-(P_{\sigma}+\gamma)(4\gamma\\ &+\kappa(8+(P_{\sigma}+\gamma)(P_{\sigma}+\gamma+\kappa))))],\\ \langle\sigma^†\sigma\rangle_{\rm ss}=\,&\frac{-4(P_{a}+P_{\sigma})} {(-4+(P_{a}-\kappa)^{2})(-P_{a}+P_{\sigma} +\kappa+\gamma)}\\ &+\frac{4{\it \Omega}^{2}(P_{a}-\kappa)^{2}}{(-4+(P_{a}-\kappa)(P_{\sigma}+\gamma))^{2}}\\ &+[(P_{a}-\kappa)(P_{a}^{2}P_{\sigma}+P_{\sigma}\kappa^{2} +4P_{a}\\ &-2P_{a}P_{\sigma}\kappa)]/[(-4\\ &+(P_{a}-\kappa)^{2})(-4+(P_{a}-\kappa) (P_{\sigma}+\gamma))],\\ \langle a^†\sigma\rangle_{\rm ss}=\,&(2i)/[(-P_{a}+P_{\sigma}+ \kappa+\gamma)(-4+(P_{a}\\ &-\kappa)(P_{\sigma}+\gamma))^{2}]\times [P_{a}^{2}(4{\it \Omega}^{2}+(P_{\sigma}\\ &+\gamma)(2P_{\sigma}+\gamma)+\kappa(P_{\sigma}^{2}\kappa+4{\it \Omega}^{2}(\kappa+\gamma)\\ &+P_{\sigma}(4+4{\it \Omega}^{2}+\kappa\gamma))-P_{a}(3P_{\sigma}^{2}\kappa\\ &+4(1+{\it \Omega}^{2})\gamma+4P_{\sigma}^{2}(2+{\it \Omega}^{2}\\ &+\kappa\gamma)+\kappa(8{\it \Omega}^{2}+\gamma^{2}))] .~~ \tag {6} \end{align} $$ The second-order correlation function at zero delay is generally defined as $$\begin{align} g^{2}(0)=\frac{\langle a^† a^†a a\rangle_{\rm ss}}{\langle a^†a\rangle ^{2} _{\rm ss}},~~ \tag {7} \end{align} $$ where the numerator can be acquired by solving another partial derivative equation $$\begin{alignat}{1} \frac{\partial\langle a^† a^†a a\rangle}{\partial t} =\,& 2(P_{a}-\kappa)\langle a^† a^†a a\rangle+4P_{a}\langle a^†a\rangle\\ &+2ig(-2\langle a^†a\rangle\langle a^†\sigma\rangle+\langle a^†\sigma\rangle).~~ \tag {8} \end{alignat} $$ With the above coupled equations, we can obtain the steady-state solutions for the photon population in the cavity ($n_{a}=\langle a^†a\rangle_{\rm ss}$) and the exciton population in the QD ($n_{\sigma}=\langle \sigma^†\sigma\rangle_{\rm ss}$), respectively. Moreover, the value of the second-order correlation function at zero delay ($g^{2}(0)$) can be used to characterize a light field, i.e., when $g^{2}(0)=1$, it shows lasing field; if $g^{2}(0) < 1$, it represents quantum field; when $g^{2}(0)>1$, it refers to the classical field.[26-29] In principle, if we can obtain the density matrix of the coupled system by solving the above master equation, then we can carry out calculation for any relevant observable, considering that here it is very complicated to give analytical solutions. In the following we will carry out numerical solution for the master equation, i.e., Eq. (2), with the Hamiltonian in Eq. (1) and the Lindblad operators in Eq. (3). We build the operators in a matrix form on a Fock basis combining the occupation numbers with both the cavity and the exciton modes. Here we set up to 70 photons in the basis, which is enough for convergence. By searching for the eigenvector under steady-state conditions, we can reconstruct the density matrix and further calculate all observable quantities with $\langle O \rangle _{\rm ss} ={\rm Tr}\{{\hat O}\rho_{\rm ss}\}$.
cpl-34-1-017303-fig2.png
Fig. 2. (Color online) The good system: (a) the variation of the photon number in the cavity ($n_{a}$) with $P_{\sigma}$ and ${\it \Omega}$, (b) the exciton population in the QD ($n_{\sigma}$) versus $P_{\sigma}$ and ${\it \Omega}$, and (c) $g^2(0)$ changing with $P_{\sigma}$. From top to bottom, the solid, dotted, dashed, dashed-dotted, dotted and dashed curves correspond to ${\it \Omega}=0g$, $0.5g$, $2g$, $4g$ and $8g$, respectively. (d) The value of $g^2(0)$ versus ${\it \Omega}$, setting $P_{\sigma}=0.001g$ and $P_{a}=0.01g$.
By referring to the good system and more realistic system defined in Refs. [10,18], we calculate the photon numbers in the cavity ($n_{a}$), the exciton population in the QD ($n_{\sigma}$), and the second-order correlation function at zero time delay for the field in the cavity ($g^{2}(0)$). The "good system" is an ideal system with the smaller decay rates $(\kappa,\gamma)$, while the "more realistic system" is a practical system with relatively higher values of $\kappa$ and $\gamma$. During the simulation, we set reasonable parameters for the good system: $\kappa=0.1g$, $\gamma=0.01g$,[10] while for the more realistic system we set $\kappa=0.29g$, $\gamma=0.42g$.[18] Here $P_{a}=0.01g$, and $P_{\sigma}$ varies among $10^{-3}g$–$10^{3}g$. The corresponding simulation results are shown in Figs. 2–4. In Figs. 2(a)–2(d), we plot out the variation of $n_{a}$, $n_{\sigma}$ and $g^{2}(0)$ individually for the good system. Firstly, set ${\it \Omega}=0$. We find from Fig. 2(a)–2(c) that, for small $P_{\sigma}$ ($10^{-3}g$–$10^{-1}g$), the incoherent pump is too weak to overcome the decays ($\kappa,\gamma$). The value of $g^{2}(0)$ is larger than 1 and exhibits almost thermal distribution. When increasing the value of $P_{\sigma}$, $g^{2}(0)$ gradually decreases and becomes close to 1 ($P_{\sigma}\in [1g,50g]$), representing the lasing regime. Herein, the stimulated emission overtakes the spontaneous emission under relatively larger exciton pumping. Meanwhile, the photon number in the cavity accumulates to a maximum, and the possibility to be in the excited state of the exciton is high ($\sim0.5$). If $P_{\sigma}$ continues increasing, the QD will become saturated and the coherent coupling between the QD and cavity will be impaired, which will stop the QD from populating the cavity, resulting in self-quenching in the cavity. Then the lasing phenomenon finally disappears. Next, we consider the cases of ${\it \Omega}=0.5g$, $2g$, $4g$ and $8g$ individually. We find from Fig. 2(c) that, when $P_{\sigma}$ is weak ($10^{-3}g$–$10^{-1}g$), the value of $g^{2}(0)$ can be dramatically modulated by varying the value of ${\it \Omega}$, and it can even reach 1 when the proper resonant driving field is applied, wherein it is ${\it \Omega}$ that overcomes the decay ($\kappa,\gamma$) rather than incoherent exciton pumping. Now the photon number in the cavity is quite high and the field in the cavity becomes the Poisson distribution, showing the lasing phenomenon, see Figs. 2(a)–2(c). Moreover, to illustrate the relationship between the resonant driving field and the lasing regime, we plot out Fig. 2(d), assuming to use weak incoherent pumping ($P_{\sigma}=0.001g$ and $P_{a}=0.01g$). It is found that the lasing regime is within ${\it \Omega}\in[1.5g, 7g]$ in this circumstance. However, if $P_{\sigma}$ has intermediate or large values, the resonant driving field ${\it \Omega}$ will result in even more dramatic modulating effects on $g^{2}(0)$. Let us use the curves of ${\it \Omega}=4g$ as an example. When $P_{\sigma}$ grows to intermediate value ($0.1g$–$1g$), the photon number in the cavity continues decreasing and $g^{2}(0)$ keeps increasing, making the system leave the lasing regime. This is attributed to the competition effect between the resonant driving field and the incoherent pumping field. However, if $P_{\sigma}$ goes on increasing ($\sim10g$), the photon number in the cavity will re-accumulate and the value of $g^{2}(0)$ will drop again and return to 1, redisplaying lasing phenomenon. This is due to the re-dominance of the exciton pump. However, if the exciton pump develops too high a value ($>60g$), it will again saturate the QD and impair the coupling between the QD and cavity. Then the lasing phenomenon finally disappears. In Figs. 3(a)–3(d), we plot the variations of $n_{a}$, $n_{\sigma}$, $g^{2}(0)$ individually for the more realistic system. As we can see, when ${\it \Omega}=0$, their variations show similar behavior as the case in the good system, except that the lasing regime is much narrower ($P_{\sigma}\in [5g, 7g]$). Because here the coupling between the QD and the cavity is weaker and the leakage of the cavity is larger, the coherence of the lasing is harder to maintain. When ${\it \Omega}\neq0$, the lasing regime can be sharply replaced into other regimes by adding a proper resonant pump field. For example, when ${\it \Omega}=2g$, we can observe the lasing effect at very low incoherent pumping power ($[10^{-3}g, 10^{-1}g]$). It is the resonant driving field that overtakes the decays, fills the cavity, and attributes to lasing. Similar to Fig. 2(d), Fig. 3(d) illustrates the variation of $g^{2}(0)$ with ${\it \Omega}$ under very weak incoherent pumping ($P_{\sigma}=0.001g$, $P_{a}=0.01g$). We can observe the lasing phenomenon within the range ${\it \Omega}\in[0.5g, 3g]$. Furthermore, the second lasing regime can be obtained at relatively larger incoherent pumping power. For instance, for the curve of ${\it \Omega}=2g$ in Fig. 3(c), the lasing effect reappears when raising the incoherent pumping to $8g$, and disappears again when $P_{\sigma}>20g$. These can be attributed to the similar reasons as in the good system described above.
cpl-34-1-017303-fig3.png
Fig. 3. (Color online) The more realistic system: (a) the photon number in the cavity ($n_{a}$) versus $P_{\sigma}$ and ${\it \Omega}$, (b) the exciton population in the QD ($n_{\sigma}$) versus $P_{\sigma}$ and ${\it \Omega}$, and (c) $g^2(0)$ changing with $P_{\sigma}$. From top to bottom, the solid, the dotted, the dashed, the dashed-dotted and the dotted and dashed curves, each corresponds to ${\it \Omega}=0g$, $0.5g$, $2g$, $4g$ and $8g$, respectively. (d) The value of $g^2(0)$ versus ${\it \Omega}$, setting $P_{\sigma}=0.001g$ and $P_{a}=0.01g$.
In Figs. 2 and 3, we have discussed the lasing performance when the QD is on the resonance with the cavity. However, the dot might have some detuning with the cavity in the practical system. Therefore it is necessary to consider the off-resonant case. Next we will discuss the pump power dependence of the lasing performance when the coupling system is off-resonant. Here we reasonably set ${\it \Delta}=0.1g$ and plot out the second-order correlation function at zero time delay for the field in the cavity for both the good system and the more realistic system, see Figs. 4(a) and 4(b), respectively. We find that the field in the cavity also shows drastic modulation effect by adding a resonant driving field ${\it \Omega}$ for both the good system and the more realistic system. When compared with the resonant cases in Figs. 2(c) and 3(c), the lasing regimes here have been replaced to even further regimes and the lasing ranges become much narrower with the increase of ${\it \Omega}$. Therefore, detuning between the QD and cavity might diminish the lasing performance in some sense. However, we can still observe lasing phenomenon by applying proper resonant and non-resonant driving fields.
cpl-34-1-017303-fig4.png
Fig. 4. (Color online) The variation of $g^2(0)$ with $P_{\sigma}$ for the good system and the more realistic system in the off-resonant case. We set ${\it \Delta}=0.1g$. Other parameters are the same as used in Figs. 2 and 3. From top to bottom, the solid, dotted, dashed, dashed-dotted and dotted and dashed curves correspond to ${\it \Omega}=0g$, $0.5g$, $2g$, $4g$ and $8g$, respectively.
In conclusion, we have proposed a new approach on modulating the lasing effect in the coupled quantum dot-cavity systems. In this approach, two external pumping lights are employed, one is on the above band, and the other is on the resonance with the QD. Moreover, we carry out investigation on both the good system and more realistic system, and carry out corresponding numerical simulations. Through numerical simulations, we demonstrate that both the cavity intensity and the field distribution in the cavity can be dramatically modulated, and the lasing regime can be replaced into other regimes by applying a proper resonant driving field. Moreover, two lasing regimes can be observed when continuously increasing the incoherent pumping field. In addition, this scheme can be easily implemented with the current technology. Therefore, it may play an important role in studying cavity QED and in the realization of high-quality micro-lasers in the near future. References All-Color Plasmonic Nanolasers with Ultralow Thresholds: Autotuning Mechanism for Single-Mode LasingA review of the ecotoxicological effects of nanowiresSpectra of 4 2 S 1/2 →3 2 D 5/2 Transitions of a Single Trapped 40 Ca+ IonInterference with a quantum dot single-photon source and a laser at telecom wavelengthTemperature-Dependent Photoluminescence Characteristics of InAs/GaAs Quantum Dots Directly Grown on Si SubstratesEfficient Entanglement Concentration Schemes for Separated Nitrogen-Vacancy Centers Coupled to Low-Q MicroresonatorsSingle-mode solid-state single photon source based on isolated quantum dots in pillar microcavitiesSingle quantum dot controlled lasing effects in high-Q micropillar cavitiesVacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavityVacuum Rabi splitting in semiconductorsStrong Coupling of Quantum Dots in MicrocavitiesElectron–Phonon Coupling in CdSe/CdS Core/Shell Quantum DotsSingle germanium quantum dot embedded in photonic crystal nanocavity for light emitter on silicon chipQuantum theory of the emission spectrum from quantum dots coupled to structured photonic reservoirs and acoustic phononsPhoton emission from a quantum dot-photonic crystal microcavity system: The role of pumping and cavity decay ratesLuminescence spectra of quantum dots in microcavities. I. BosonsLuminescence spectra of quantum dots in microcavities. II. FermionsLuminescence spectra of quantum dots in microcavities. III. Multiple quantum dotsIncoherent excitation of the Jaynes-Cummings systemNonlinear photoluminescence spectra from a quantum-dot–cavity system: Interplay of pump-induced stimulated emission and anharmonic cavity QEDCavity-QED assisted attraction between a cavity mode and an exciton mode in a planar photonic-crystal cavityMaster-equation model of a single-quantum-dot microsphere laserQuantum calculations on quantum dots in semiconductor microcavities. Part IPhotonic crystal nanocavity laser with a single quantum dot gainSingle photon sources with single semiconductor quantum dotsOn the spectroscopy of quantum dots in microcavitiesStudy of s-exciton and p-exciton pump on the lasing generation of biexciton quantum dot–cavity systemComparison of quantum and semiclassical radiation theories with application to the beam maser
[1] Lu Y J, Wang C Y, Kim J et al 2014 Nano Lett. 14 4381
[2] Kwak J I and An Y J 2015 Int. J. Environ. Sci. Technol. 12 1163
[3] Gong S J, Zhou F and Wu H Y 2015 Chin. Phys. Lett. 32 013201
[4] Felle M, Huwer J, Stevenson R M et al 2015 Appl. Phys. Lett. 107 131106
[5] Wang T, Liu H Y and Zhang J J 2016 Chin. Phys. Lett. 33 044207
[6] Si B, Wen J J, Cheng L Y et al 2014 Int. J. Theor. Phys. 53 80
[7] Moreau E, Robert I, Gerard J M et al 2001 Appl. Phys. Lett. 79 2865
[8] Reitzenstein S, Bockler C, Bazhenov A et al 2008 Opt. Express 16 4848
[9] Yoshie T, Scherer A, Hendrickson J et al 2004 Nature 432 200
[10] Khitrova G, Gibbs H M, Kira M et al 2006 Nat. Phys. 2 81
[11] Laussy F P, del Valle E and Tejedor C 2008 Phys. Rev. Lett. 101 083601
[12] Lin C, Gong K, Kelley D F et al 2015 ACS Nano 9 8131
[13] Zeng C, Ma Y J, Zhang Y et al 2015 Opt. Express 23 22250
[14] Roy-Choudhury K and Hughes S 2015 Phys. Rev. B 92 205406
[15]Chen X and Mi X W 2011 Acta Phys. Sin. 60 044202
[16] Khasraghi A M, Shojaei S, Vala A S and Kalafi M 2013 Superlattice. Microst. 55 98
[17] Laussy F P, del Valle E and Tejedor C 2009 Phys. Rev. B 79 235325
[18] del Valle E, Laussy F P and Tejedor C 2009 Phys. Rev. B 79 235326
[19] Laussy F P, Laucht A, del Valle E et al 2011 Phys. Rev. B 84 195313
[20] Tian L and Carmichael H J 1992 Quantum Opt. 4 131
[21] Yao P J, Pathak P K, Illes E and Hughes S 2010 Phys. Rev. B 81 033309
[22] Tawara T, Kamada H, Tanabe T et al 2010 Opt. Express 18 2719
[23] Benson O and Yamamoto Y 1999 Phys. Rev. A 59 4756
[24]Carmichael H J 1999 Statistical Methods in Quantum Optics (Berlin: Springer-Verlag)
[25] Tsukanov A V and Kateev I Y 2014 Russ. Micro. 43 315
[26] Nomura M, Kumagai N, Iwamoto S et al 2009 Opt. Express 17 15975
[27] Shan G C, Yin Z Q, Shek C H and Huang W 2014 Front. Phys. 9 170
[28] Laussy F P and del Valle E 2010 J. Phys.: Conf. Ser. 210 012018
[29] Guan H, Yao P J, Lu Y H et al 2015 J. Opt. 17 015001
[30] Jaynes E T and Cummings F W 1963 Proc. IEEE 51 89