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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 ωpc. Epu (Epr) is the slowly varying envelope of the pump (probe) field, and ωpu (ωpr) 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 σ01, σ10 and σz. The typical physical situation is illustrated in Fig. 1(b).
Fig. 1 (a) Schematic diagram of a SQD embedded in a PC nanocavity. The system is driven by a strong pump laser and detected by a weak probe laser [37]. (b) The energy level scheme of an exciton in the SQD interacting with the photons in the PC nanocavity.
In the rotating frame, the total Hamiltonian of the system takes the form [38, 39]
where Δpu = ω10 ‒ ωpu is the exciton-pump field detuning, Δpc = ωpc ‒ ωpu is the nanocavity-pump field detuning, and δ = ωpr – ωpu is the probe-pump detuning. 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 = μEpu/h represents the effective Rabi frequency of the pump field, and μ represents the electric dipole moment. For simplicity, we set p = < σ01>, w = 2<σz> and 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]:
where Γ2 (Γ1) 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 = p0 + p1e–iδt + p–1 eiδt, w = w0 + w1e–iδt + w–1eiδt, and Λ = b0 + b1e–iδt + b–1eiδt [38], where p0, w0, and b0 is the solution of Eqs. (2) – (4) for the case in which only the pump field is present. Here |p0| >> |p1|, |p–1|, |w0| >> |w1|, |w–1|, |b0| >> |b1|, |b–1|. By substituting the above ansatz into Eqs. (2) – (4) and solving these equations, we can obtain the FWM signal defined as
whereThe steady-state population inversion of the exciton w0 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, Γ1 = 2Γ2 = 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/μEpr*h−1Γ2−1|, defined in Eq. (5), changes with the pumping intensity Ipu in three different cases including the weak coupling regime (g < Γ2, κ), the intermediate regime (g ~Γ2, κ), and the strong coupling regime (g > Γ2, 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 ωpr = ωpu 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 (g = 2 MHz), the FWM signal will be enhanced greatly as Ipu increases from 1 to 10 MHz2. As Ipu further increases to 100 MHz2, 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 (Ipu = 1000 MHz2), 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|.
Fig. 2 (a) The FWM signal |p−1/μEpr*h−1Γ2−1| as a function of the probe-pump detuning δ for Ipu = 1 MHz2 with and without the exciton-nanocavity coupling. Dependence of the FWM signal |p−1/μEpr*h−1Γ2−1| on the pumping intensity Ipu when g = 2 MHz (b), g = 6 MHz (c) and g = 30 MHz (d). Here Δpu = 0.
In the intermediate regime (g = 6 MHz), the FWM spectrum will change from double-peaked to triple-peaked by gradually increasing the pumping intensity Ipu [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 Ipu, and the peak value of the FWM signal increases linearly as Ipu 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 (Δpu = 0) vacuum Rabi splitting is 2g = 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 Δpu when Ipu = 1000 MHz2 and g = 30 MHz. In the strong coupling regime, as the pump field is exactly resonant with the exciton in the SQD (Δpu = 0), the FWM spectrum exhibits a double-peak structure whose symmetry axis is given by the line δ = 0. When the pump field is detuned from the exciton transition (Δpu ≠ 0), the scenario becomes completely different. As Δpu 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 Δpu, while the magnitudes of the other peaks decrease as Δpu increases. The FWM spectrum for Δpu = 30 MHz is the same as that in the case of Δpu = −30 MHz. Obviously, the FWM spectrum can be modified effectively via the off-resonant coupling between the SQD and the PC nanocavity.
Fig. 3 The FWM signal |p−1/μEpr*h−1Γ2−1| as a function of the probe-pump detuning δ when the exciton-pump field detuning Δpu is 0, 15 MHz, 30 MHz and −30 MHz. The parameters used here are g = 30 MHz and Ipu = 1000 MHz2.
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. ωpu = ωpr). Considering this, Fig. 4(a) shows how the FWM signal |p−1/μEpr*h−1Γ2−1| changes with the pumping intensity Ipu 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 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 Ipu increases. The results in Fig. 4(b) show that as Ipu increases, the population inversion w0 increases rapidly to a maximum value 0, then w0 almost keeps invariant despite the change of Ipu. 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, |p−1/μEpr*h−1Γ2−1| and w0 are both a single-value function of Ipu. This indicates that the bistable effect is always absent in the weak and intermediate coupling regimes.
Fig. 4 The FWM signal |p−1/μEpr*h−1Γ2−1| (a) and the population inversion w0 (b) as a function of the pumping intensity Ipu in the no, weak, and intermediate coupling regimes. The simulations are performed for Δpu = 0 and g = 0, 2, 6, and 30 MHz. (c) Dependence of OB on the excitation frequency (i.e. Δpu) in the strong coupling regime. The simulations are performed for g = 30 MHz and Δpu = 0, 15, and 30 MHz. (d) Optical hysteresis loop of the population difference w0 with the pumping intensity Ipu. Here Δpu = 0 and g = 30 MHz.
Figure 4(c) shows how the FWM signal |p−1/μEpr*h−1Γ2−1| changes with the pumping intensity Ipu for various excitation frequencies (i.e. Δpu) in the strong coupling regime (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 Δpu 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 Δpu = 30 MHz is only 15.9% of that for Δpu = 0. Moreover, the bistable effect to a certain extent will be suppressed as the pump field is detuned from the exciton resonance (|Δpu| > 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.
To further clarify this bistable behavior of the FWM signal, an optical hysteresis loop of the population difference w0 with the pumping intensity Ipu is plotted in Fig. 4(d). As the pumping intensity Ipu increases, the system firstly follows the lower (stable) branch and then jumps to the upper (stable) branch at Ipu = 6627.55 MHz2. With sweeping Ipu back, the system remains on the upper branch and then makes a transition to the lower branch at Ipu = 1156.32 MHz2. 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 [Ipu; g; different Δpu]. The boundaries between white and colored regions denote the bistable thresholds. The low and high bistable thresholds are denoted as Imin and Imax, 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 (Δpu = 0), for g < 9.13 MHz, the bistability is always absent despite the change of Ipu, while for g ≥ 9.13 MHz, the bistability may exist [Fig. 5(a)]. When the pump field is detuned from the exciton resonance (Δpu = 30 MHz), a larger critical value of 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 Δpu = 0, the critical conditions of OB are g = 9.13 MHz and Ipu = 91.5 MHz2. However, as Δpu increase to 30 MHz, the critical conditions of OB become g = 25.63 MHz and Ipu = 4188.1 MHz2. Obviously, the bistable thresholds of g and Ipu at off-resonant excitation (Δpu = 30 MHz) are both pushed to larger values compared to that at resonant excitation (Δpu = 0). Also, the bistable region for Δpu = 30 MHz becomes smaller than that for Δpu = 0 MHz. In a whole, the occurrence of OB depends strongly on the excitation frequency, pumping intensity and exciton-nanocavity coupling.
Fig. 5 Bistability phase diagrams of the FWM response of the SQD-nanocavity system in the parameter subspace (Ipu; g). (a) Δpu = 0; (b) Δpu = 30 MHz. The colored areas represent the subspace where the bistability exists. (c) Comparison of the results obtained in the above two bistability phase diagrams.
From the results presented in Figs. 4(c), we see that the bistable effect is strongly dependent on the excitation frequency (i.e. Δpu). To investigate this issue further, we plot the bistability phase diagram within a parameter subspace [Ipu; Δpu; 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 Δpu = 0 (i.e. ωpu = ω10). 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 ≤ Δpu ≤ 41.75 MHz. It is not difficult to find that the adjustable range of bistability at Δpu = 0 reaches a maximum value (i.e. 1156.3 MHz2 ≤ Ipu ≤ 6627.5 MHz2). The appearance of OB at a low pumping intensity means prospects of all-optical control using a weak pump light.
Fig. 6 Bistability phase diagram of the FWM response of the SQD-nanocavity system in the parameter subspace (Ipu; Δpu; g = 30 MHz). The colored area represents the subspace where the bistability exists.
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).
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