1. Introduction

Cavity quantum electrodynamics (cQED) have attracted a lot of attention due to the fundamental physical interest and extensive practical application, for example, the non-classical behavior can be investigated, such as anti-bunching, entanglement, quantum decoherence and so on, it can also be applied to quantum information, quantum cryptography, etc. First experiments on cQED were performed in atomic systems using dipole traps to precisely localize an atom in the antinode of the cavity mode [1]. Although remarkable achievements have been made with all of these advancements, the unfixed property of atomic system and high-finesse cavity impose an extreme technical condition which limits its further commercialization. From practical perspective, semiconductor QD-cavity system provide an attractive alternative. Beneficial from semiconductor technology advancement, the extreme high quality factor and ultrasmall mode volume has been realized in various cavity, such as micropillar [2], microdisk [3], Noda cavity [4] and so on, besides that, single semiconductor QDs have many unique attributes beyond the other single emitters, such as large exciton dipole moments, integrability with compact semiconductor cavity systems, [5–7], fixed in position and compatible with electronics and telecom components [8].

Over the last decade, substantial progress has been made on the semiconductor self-assembled QD coupled to a nanoscale optical cavity mode. In 2004, Yoshie et al. and Reithmaier et al. experimentally realized vacuum Rabi splitting in the solid state system: a single QD strongly coupled to semiconductor micropillar and photonic crystal nanocavity respectively [2, 9]. Subsequently, Hennessy et al. precisely located the QD at the field antinode of single cavity mode by atomic force microscopy metrology, and validated the antibunching of generated photon stream in the strong coupling regime. All of these work pave the way to realize lasing oscillation with a single QD and a monolithic cavity of single mode. Almost at the same time, Xie et al. investigate the influence of a Single QD State on the Characteristics of a Microdisk Laser [10], which however is not a real single QD, actually, many random QDs are embedded in the microdisk, those nontarget QDs will inevitably disturb the properties of emission from QD-cavity system. Soon after that, a similar experiment is realized in micropillar cavity by A. Forchel’s group [11]. By further reducing the mode volume and decreasing the density of QDs (about 0.4 QD per cavity in average), Nomura et al. firstly realized the lasing oscillation from the real single QD in a photonic crystal cavity [12, 13].

On the theoretical side, due to atomic-like properties of QDs, the atomic model (two levels or four levels) [13–16] is usually used to simulate single QD lasers, in which two levels associated with light emission is described by the Jaynes-Cummings Hamiltonian. Atomic model has successfully explained a bunch of QD-cavity phenomena, such as strong coupling, nonlinear, exciton attraction and so on. However, beyond atomic model, QDs have their own peculiar properties: larger volume, minor energy separation, electrical neutrality and so on. Recently, Ritter et al.[17] used two-electrons multi-configuration QD model to investigate emission properties of a single QD laser and this method was also used in a similar recent work by Gies et al.[18]. Alternatively, exact methods (for very few electrons) or other approximate approaches are often reported to calculate the energy spectrum in a variety of work. For example, by single-site Hubbard, Hartree-Fock and exact methods, Rontani et al. discussed the range of validity of approximate theoretical scheme and found that the dimensionality of the dots has a crucial impact on the accuracy of the predicted addition spectra [19].

Presently, the investigations on the single QD laser are usually focused on the resonant case of QD-cavity system. However, the off-resonant case is not well studied theoretically. In this work, the lasing oscillation based on off-resonant QD-cavity system are investigated detailedly through two-electrons QD model. Firstly, cavity emission spectra at selected pump rates and photon statistics is investigated in the presence of exciton-cavity detunings. Then through gradually increasing the pump rate, we show the typical lasing signature with and without detuning, include the spectral transition from multiple peaks to single peak, antibunching to Poissonian distribution and approximately linear increase of photon number. To uncover the effect of exchange energy on the statistical properties of emission, the mean photon number and the second order correlation versus pump rate is investigated for different exchange energy. Finally, we change the pump dependencies by fixing the relaxation rates and exchange energy, and investigate the properties of photon statistics with and without detuning.

2. Theoretical model

The system under investigation consists of a single QD with a deep confinement potential allowing for two confined shells for electrons and holes, which is coupled to an optical cavity, see Fig. 1. Six possible configurations connected with the transition processes arise from different occupations of these single-particle states with two carriers. We assume Coulomb interaction between configurations with opposite spin carriers is smaller than the line broadening, and, for simplicity, only one-spin subsystem is taken into consideration in the system [17], then the QD with two confined shells can be characterized by a two electrons four level model (the right side of Fig. 1). Furthermore, the following assumptions are imposed on the open dot-cavity system: the cavity is single-mode with a decay rate κ; the coupling rate between the target QD exciton (s-exciton) and cavity is denoted as g; the QD-cavity system is driven incoherently by exciton pump P (γ₄₁) which pumps the QD level |1〉 → |4〉. Thus the dynamics of the whole system can be governed by a graceful master equation for the reduced density operator ρ of the system, which can be derived using a Born-Markov and dipole approximation. In the interaction picture it reads

 

Fig. 1 Schematic illustration of semiconductor QD-microcavity system with QD’s energy levels.

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3. Simulation results and discussions

In this section, we investigate the influence of the detuning on the statistical properties for various pump rates and study the emission spectra at certain pump rates as a function of detuning by directly solving the master equation of Eq. (1) for the reduced density operator. Firstly, the dependence of emission spectra on pump rate is studied [see Fig. 2(a) and 2(b ∼ d) corresponds to the resonant and non-resonant case respectively]. For resonant case, a well pronounced multi-peak emission structure is observed at low pump power [blue line in Fig. 2(a)]. The weak side peaks (at about Δω = ±1.4meV) and the two weak peaks (at about Δω = −0.06, 0.14meV) are mainly ascribed to the next step of the Jaynes-Cummings ladder. The additional peak from the sp-biexciton transitions is positioned at twice the Coulomb exchange energy E_sp[17]. However, with increasing pump rate, the main peaks of spectra show a transition from multi-peak to a singlet which is partly attributed to a pump power dependent broadening of the exciton line, and weak peaks induced by the next step of the Jaynes-Cummings ladder gradually fade out, and the whole system goes into weak coupling regime. In the meanwhile, the “sub-peak” is shifted to right but still positioned at the left side of the main peak as pump rate increases, since Coulomb exchange energy decreases with elevated pump rate. While for detuning case [Fig. 2(b∼d)], it is clear to see that the main peak of the spectra always shows singlet induced by the cavity mode, and there appears a peak at the right side of the main peak which is due to exciton mode. With gradually increasing detuning the “sub-peak” of spectra will even go across the main peak from left to right, and as depicted in Fig. 2(c) it is nearly overlapped the main peak at Δ = 4meV, and it is completely pulled to the right side of the main peak for Δ = 6meV, owing to the complex combined effect of detuning factor and pump rate.

 

Fig. 2 Photon emission spectra at selected pump rates. (a) P = 0.1, 0.15, 0.3, 1/ps from bottom to up for resonant case and (b–d) for non-resonant case at Δ = 2, 4, 6meV. For all curves, the decay rates of cavity is κ = 0.1/ps, the light-matter coupling constant g = 0.9/ps, the spontaneous emission rates γ₂₃ = γ₁₄ = 0.01/ps, and the pure dephasing rate is Γ_pd = 0.035 meV. And Δω = ω − ω_c in all the plots.

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Next, the pump rate is further increased and lasing oscillation is studied. Figure 3(a) depicts that the mean photon number of cavity at steady state varies with pump rate for different cavity-s-exciton detuning. At weak excitation, the mean photon number is monotonously decreased with increasing detuning. However, the situation becomes more complex when the pump rate is further increased, for example, compared with the other detunings, the mean photon number for detuning Δ = 5meV is maximum in the medium pumping region (about 0.1 ∼ 1/ps), while the mean photon number for detuning Δ = 2meV is maximum in strong pumping region (approximately > 1/ps). Generally, the smaller detuning the larger the mean photon number, and it is important to note that the detuning here is the “dressed” detuning. By directly solving the master equation, we find that the exchange energy is not monotonously changed with the pump rate and the detuning, and so evidently the complex appearance of the mean photon number is mainly from complex dependance of exchange energy on the pump and detuning. In addition, the mean photon number is approximately linearly increased with the pump rate in the strong pumping region, as we all know, the approximately linear increase of mean photon number usually means the generation of coherent light, to confirm it, we investigate the dependance of the second order correlation at zero delay g⁽²⁾(0) on the pump rate [shown in Fig. 3(b)]. For all the cavity-s-exciton detunings, g⁽²⁾(0) gradually approaches to 1 but at different speed when the pump rate is continuously increased. For detuning Δ = 2meV, g⁽²⁾(0) approaches to unity at about P = 2.4/ps, while for detuning Δ = 5meV, g⁽²⁾(0) approaches to unity only when P > 1000/ps, which is coincident with the variation of the mean photon number. However, it is not sufficient to judge coherent light only by g⁽²⁾(0) = 1, instead the normalized second-order correlation function g⁽²⁾(τ) = 1 means coherent light, for example it is not a coherent light if γ₁₂ = γ₃₄ = 0.05 meV and E_sp = 3.3 meV in our model [the grey line in inset of Fig. 3(c)] though g⁽²⁾(0) = 1. Figure 3(c) shows g⁽²⁾(τ) for different detuning with P = 1000/ps, the normalized second-order correlation function g⁽²⁾(τ) is approximately stabilized at unity which indicates the output light is coherent, the tiny difference in g⁽²⁾(τ) among various detuning indicates the different laser threshold. Furthermore, Fig. 3(d) shows that the laser threshold varies as a function of detuning, here the threshold definition is proposed by Gunnar Björk et al. where the mean photon number in the mode at threshold is unity [25], and the optimum (smallest) laser threshold is attained at 0.197/ps when the QD is blue detuned with respect to the cavity by 4.36meV. It is worth to clarify that the detuning case is the “bare” detuning, and in fact, it is exactly resonant with the dressed energy level dressed by exchange energy, to confirm it, the emission spectrum for the detuning Δ = 4.36meV is calculated, and we find the “sub-peak” is completely overlapped with the main peak. Actually, the “sub-peak” in Fig. 2(c) is already very close to the main peak when detuning Δ = 4meV. When the detuning is deviated from the optimum value, the threshold will grow rapidly because the energy exchange between the QD and cavity is very difficult when the “dressed” detuning is far larger than coupling constant g. In order to compare the different definitions on the laser threshold, two more essential laser threshold definitions are taken into consideration: the first threshold definition is that the stimulated emission simply equals the spontaneous emission [26], and the second one is that the net stimulated emission (that is, the stimulated emission minus the stimulated absorption) equals the spontaneous emission [26, 27]. Based on a direct solution of the master equation, we calculate the stimulated emission, absorption emission and spontaneous emission. We find that, the laser threshold by the first definition almost overlap with the one of the unity mean photon number, and the threshold by the second definition is slightly larger than the first one. In addition, to observe the effect of the pure dephasing, the photon statistics without pure dephasing is investigated, we find that the lasing threshold is a little larger than the case with pure dephasing processes, and the mean photon number shows a little increase, moreover, the presence of pure dephasing will slightly accelerate the lasing speed which is coincident with [28] though the optimum laser threshold remains at the same detuning.

 

Fig. 3 (a) Steady-state mean photon number 〈n〉 vs. pump rate P. (b) Second-order photon correlation function g⁽²⁾(0) versus pump rate P. (c) Normalized second-order correlation function g⁽²⁾(τ) versus the delay time τ at P = 1000/ps. The inset is for E_sp = 3.3 meV, γ₁₂ = γ₃₄ = 0.05 meV in our model without considering the pure dephasing processes. (d) The laser threshold versus detuning between QD and cavity, the inset is the enlarged plot of the laser threshold where the red rectangle dot indicates the optimum value of laser threshold. For Fig. 3(a–c): the black lines correspond to resonant case; for the green lines, Δ = 2meV; in comparison, Δ = 1meV (Δ = −1meV) is used for the solid (dashed) purple lines and Δ = 5meV for the blue lines. Other parameters are the same as Fig. 2.

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The dependence of the emission spectra on the detuning is shown in Fig. 4 for weak and strong pump respectively. For a weak pump rate P = 0.1/ps [Fig. 4(a)], the emission spectra show a multipeak structure which is coincident with g⁽²⁾(0) ≠ 1, i.e. non-lasing, and the characteristic anti-crossing behavior between the cavity line and exciton line is observed in the detuning dependent emission spectra. The weak peaks (at about Δω = ±1.4, −0.06, 0.14meV for resonant case, see the black line in Fig. 4(a)) due to the next step of the Jaynes-Cummings ladder gradually fade out with increased detuning and disappear at Δ = ±2meV. Unusually, the “sub-peak” from the sp-biexciton transitions is pulled to right along with blue detuning and left with red detuning, comparing with the resonant case under the same pump power, this pulling phenomenon depicts that the detuning factor will pronouncedly influence the sp-biexciton transitions and blue detuning of QD-cavity in some way enhances the coupling efficiency between QD and cavity mode. To bring evidences from experiments, we reproduce the PL emission with detuning Δλ = −0.03nm in [7], the same parameters as the experiment are adopted in our simulation except the coupling constant g = 0.3meV. From the inset of Fig. 4(a), we can see that the lineshape of the emission qualitatively agree with the experiment result: the two side peaks is mainly ascribed to the Rabi-splitting in the strong coupling regime, and the additional middle peak is due to the nonlinear effect of the Jaynes-Cummings ladder. It is noted that the horizontal ordinate in our manuscript is corresponding to frequency, and that in the reference wavelength, so the position of the peaks is inversed to the experiment result. For a high pump rate P = 500/ps as described in Fig. 4(b), the spectra becomes singlet and the linewidth shows significant reduction compared to the case of the weak pump rate, which indicates that strong coupling regime fades out and the whole system goes into weak coupling regime, especially, the resonant case doesn’t correspond to minimum linewidth and the dependence of linewidth on the detuning is not monotonic, which implies that “suitable detuning” will indeed facilitate lasing oscillation.

 

Fig. 4 Photon emission spectrum for selected pump rates. (a) P = 0.1/ps (semi-logarithmic plot). (b) P = 500/ps (linear plot). From bottom-to-top s-exciton-cavity detuning is chosen as Δ = (−5, −2, −1, 0, 1, 2, 5)meV in both subfigure. Lines are vertically displaced. The inset in (a) is the simulation results of Fig. 3(b) in [7] when Δλ = −0.03nm, the same parameters as the experiment are used in our model except the coupling constant g = 0.3meV. For all curves, the other parameters are the same as Fig. 2.

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To investigate the influence of exchange energy on the photon statistics, the mean photon number and the second order correlation function varying as a function of pump rate is studied for different exchange energy, the results are shown in Fig. 5. It is found that, under weak excitation, the mean photon number increases approximately linearly along with pump rate and keeps invariant for various E_sp. However, at strong pump rate, the mean photon number is declined with the increase of exchange energy. It is known that the biexciton emission will play a more and more important role in the emission process with the increase of pump rate, on the other hand, the emission of biexciton will be gradually out of resonance with the cavity mode due to the exchange energy, so the photon number shows a reduction with the increase of exchange energy. In this sense, the exchange energy is actually detrimental to the lasing oscillation, which is manifested by the second order correlation function. Just as we see from Fig. 5(b), for E_sp = 0, g⁽²⁾(0) approaches to 1 more quickly than E_sp = 2.5, 5 meV. Fortunately, the effect of exchange energy can be offset by cavity-s-exciton detuning, for example, just as we discussed above, for E_sp = 2.32 meV at P = 0.197/ps, the detuning Δ = 4.36meV will complete offset the exchange energy and the “sub-peak” will complete overlap the main peak.

 

Fig. 5 (a) Steady-state mean photon number 〈n〉 vs. pump rate P. (b) Second-order photon correlation function g⁽²⁾(0) versus pump rate P. The black solid lines correspond to E_sp = 0meV. For the red dashed lines E_sp = 2.5 meV. In comparison, E_sp = 5 meV is used for the blue dash-dotted lines. For all curves, the other parameters are set as Fig. 2 except Δ = 0.

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Finally, the properties of photon statistics is investigated with the fixed relaxation rates and exchange energy, the results are described in Fig. 6. For all the detuning cases, the mean photon number increases approximately linearly with the pump power, and then arrive at a steady value at high pump rate as depicted in Fig. 6(a). However, as described in the inset of Fig. 6(a), the mean photon number at steady value will be dependent on the detuning, the maximum is 6.9 which is achieved at Δ = 4meV. The steady value is because the electrons will stack at the 4th level rather than relax to the third level once the pump rate P is larger than relaxation rate γ₃₄ (the fixed relaxation rate γ₃₄ will limit the transition), while the detuning, at which the maximum mean photon number at steady value is achieved, is related to the exchange energy. In like wise, the second order correlation function will be stable when the pump rate exceeds the rate γ₃₄. It means the system will never generate coherent light if the emission is still not coherent when the pumping rate is less than γ₃₄. In the meanwhile, the results manifest that an suitable detuning will facilitate the generation of coherent light, for example the emission for Δ = 4meV is coherent as the red line in Fig. 6(b) shows.

 

Fig. 6 (a) Steady-state mean photon number 〈n〉 vs. pump rate P. The inset is the mean photon number at steady value versus detuning, and the red rectangle dot indicates the maximum value at Δ = 4meV. (b) Second-order photon correlation function g⁽²⁾(0) versus pump rate P. The black lines correspond to resonant case; the solid/dashed purple lines is for Δ = 1meV/ − 1meV, the green lines Δ = 2meV, the red lines Δ = 4meV and the blue lines Δ = 5meV. Other parameters are the same as Fig. 2 except that γ₁₂ = γ₃₄ = 1 meV, E_sp = 2 meV.

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4. Conclusion

In summary, we have investigated photon statistical and spectral properties of non-resonant single QD-cavity system through two-electrons QD model. Within the weak pumping regime, the nonlinear process and higher order Jaynes-Cummings ladder is obviously inhibited by detun-ing just as expected, unusually, with increasing detuning, the “sub-peak” of spectra will cross the main peak from left to right due to the interplay between exchange energy and cavity-s-exciton detuning. When the pump rate is increased continuously, the typical lasing signature is observed, specially, we find that a certain suitable “bare” detuning of s-exciton from cavity will facilitate the lasing oscillation and coherent light generation, accordingly, its linewidth of spectrum will be prominently narrowed, which are all owed to the presence of exchange energy. Compared to the weak pumping regime, the biexciton emission will be dominated and play a more important role in the strong pumping regime, on the other hand, the exchange energy will shift the emission of biexciton from cavity, so the exchange energy is inherently detrimental to the lasing oscillation. Fortunately, the exchange effect can be offset by the detuning, for example, the detuning Δ = 4.36meV will completely offset the energy shift of biexciton caused by E_sp = 2.32 meV, therefore the laser threshold will reach its minimum P_th = 0.197/ps which is much lower than the resonant case. Finally, we would like to emphasize that the cavity resonance is only detuned from the s-exciton but exactly on-resonant with the QD configuration dressed by exchange energy.