1. Introduction

Electromagnetically induced grating (EIG) being closely related to electromagnetically induced transparency (EIT), has become an important and interesting research topic in quantum optics and nonlinear optics in recent years. When the EIT medium presents the spatial modulation induced by the standing-wave driving, the weak probe beam could be diffracted in the high-order direction [1]. However, the diffraction intensity is limited by the weak dispersion of the atoms on probe resonance, which could be enhanced for the case of probe detuning. And the two-dimensional EIG based on EIT has also been discussed [2,3]. It is shown that EIG has the potential applications in optical storage [4], quantum Talbot effect [5], optical bistability [6], and optical switching and routing [7] and so on. Therefore, some theoretical and experimental schemes for the preparation and realization of EIG are proposed in the various systems [8–22]. Strikingly, two approaches have been applied in the recent schemes in order to greatly improve the diffraction intensity. One is to enhance the dispersion and simultaneously inhibit the absorption to obtain the larger phase modulation accompanied by the high transmission. The so-called phase grating has been presented and received certain attention [14–18], where most of the light during the central region could be diffracted into other direction and over 30% diffraction intensity in the first-order direction could be reached. Another is gain-assisted diffraction grating, where the output probe field is amplified by using the Raman process, coherent and incoherent drivings and other ways to obtain the gain with large dispersion in the atomic system [19–22]. The results show that the diffraction intensity in the other direction is quite high, but the higher diffraction also occurs in the central direction, which may have an influence on the practical observation. Most recently, a great deal of attention has been devoted to the asymmetric diffraction grating including in one-dimensional and two-dimensional cases with periodic parity-time-symmetry and parity-time-antisymmetry [23–29].

On the other hand, semiconducting quantum dots have become a prominent platform for the quantum-optics and nonlinear-optics phenomena due to their discrete energy levels similar to natural atoms. Some inherent advantages are also involved in such a system. First, the large electric dipole moments are existent in the present system, which lead to the achievement of strong couplings with the quantum fields and other subsystems. Second, the quantum dot can be artificially structured with flexible designs and easily controlled for the experimental parameters. Last but not the least, the high nonlinear optical susceptibility could appear due to the size being on the order of several nanometers. For these reasons, many quantum optical effects have been investigated in the present system [30–35]. The EIT phenomenon is discussed via biexciton coherence in quantum dot system [33]. The gain without inversion and enhanced refractive index could be obtained in the transient regime for dephasing rates typical under room temperature and high excitation conditions [34]. Four-wave mixing with the high coefficient is also analyzed for a single bound-state quantum-dot [35]. The latest research reveals the generation of structured light in a four-level quantum-dot structure via four-wave mixing [36]. Furthermore, the hybrid quantum dot-metal nanoparticle system [37–43] may also provide additional functions to study some interesting phenomena, such as EIT, optical nonlinearity, bistability, modification of spontaneous emission spectra and so on. As the research moves along, the other systems with respect to quantum dot and cavity, quantum dot and resonator, quantum dots with dipole-dipole coupling have been well studied [44–48].

It is noted that the quantum-dot molecules (QDMs), which could be established via the tunnel couplings among two or more individual dots, play an important role in inducing the quantum coherence. In the QDM system an electron can pass through the potential barrier between quantum dots via the interdot tunneling, which could be controlled by the external static gate voltage [49,50]. Borges et al. proposed to use the electron tunneling of a quantum molecule to show the EIT effect [51], Autler-Townes doublets and a Mollow-like triplet [52,53], and coherent population trapping [54]. The formation of temporal optical dark and bright solitons has been discussed via combining the interdot tunneling and the external control optical field [55]. Luo et al. have shown that the tunnel-enhanced cross-Kerr nonlinearity gives rise to the realization of the self-focusing and self-defocusing effects in triple quantum dot [56]. The enhanced self-Kerr nonlinearity, fifth-order nonlinearity and cross-Kerr nonlinearity have been discussed via tunneling in multiple quantum dots [57]. Peng et al. have demonstrated the narrow tunneling-induced absorption for measuring interdot tunneling [58]. The dynamic propagation of a probe field is studied with the help of the tunneling-induced interference effect, in which tunneling has important effect on the probe group velocity [59]. Most recently, the interdot tunneling-dependent Kondo effect [60] and the tunneling-endowed exchange of optical vortices between two weak beams with different frequencies [61,62] are also discussed.

Here we use an artificial molecule model formed by two quantum dots to show that the interdot tunneling can play a critical role in generating the phase grating with the help of the weak standing-wave field. Importantly, the enhanced nonlinear modulation with vanishing absorption could be induced by the weak interdot tunneling, which allows the effective diffraction of a weak probe beam. The results show that high diffraction intensity could occur in the first-order and higher-order directions. In contrast to the schemes for enhancing nonlinearity, no additional coupling field is required. Moreover, the present grating is obtained under the condition of the weak tunneling and weak standing-wave driving, which may relax the constraint on the strong couplings. This paper is organized as follows. In section 2, we describe the system and derive the expressions of the linear and nonlinear susceptibility being closely relevant to the diffraction. In section 3, we discuss the nonlinear modulation with high transmission and its positive effect on the diffraction intensity. A summary is given in the last section.

2. Model and equation

Let us consider a QDM system consisting of two individual quantum dots, where two quantum dots are different in size and have different band structures, and they could be produced by self-assembly dot growth technology. In particular, a four-level system is formed due to the interdot tunneling and it is shown in Fig. 1(a). The ground state is denoted by $\left \vert 1\right \rangle$, and the other states $\left \vert 2\right \rangle$, $\left \vert 3\right \rangle$, $\left \vert 4\right \rangle$ are the excited states of the system. The tunneling of two quantum dots exists between state $\left \vert 2\right \rangle$ and state $\left \vert 3\right \rangle$. A control field with frequency $\omega _{c}$ is applied to the transition $\left \vert 3\right \rangle \leftrightarrow$ $\left \vert 4\right \rangle$ and a probe field with the frequency of $\omega _{p}$ is coupled to the states $\left \vert 1\right \rangle$ and $\left \vert 2\right \rangle$. Under the rotating-wave and the electric-dipole approximations, the Hamiltonian of the system could be written as ($\hbar =1$)

 

Fig. 1. (a) The four-level quantum dot system induced by interdot tunneling. The state $\left \vert 1\right \rangle$ is the ground state, and $\left \vert 2\right \rangle$, $\left \vert 3\right \rangle$ and $\left \vert 4\right \rangle$ are the excited states. A probe field with frequency $\omega _{p}$ and Rabi frequency $\Omega _{p}$ is coupled to the transition between states $\left \vert 1\right \rangle$ and $\left \vert 2\right \rangle$, A control field with frequency $\omega _{c}$ and Rabi frequency $\Omega _{c}$ interacts with the transition $\left \vert 3\right \rangle \leftrightarrow \left \vert 4\right \rangle$, and the tunneling of two quantum dots exists between state $\left \vert 2\right \rangle$ and state $\left \vert 3\right \rangle$. (b) The schematic diagram of control field and probe field propagating in quantum dots.

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According to the Schrödinger equation $\frac {d}{dt}\left \vert \psi _{I}\left ( t\right ) \right \rangle =-iH_{I}\left \vert \psi _{I}\left ( t\right ) \right \rangle$ and the superposition principle of the state $\left \vert \psi _{I}\left ( t\right ) \right \rangle =\overset {4}{\underset {j=1}{\sum }}a_{j}\left ( t\right ) \left \vert j\right \rangle$, we can obtain the evolution of the probability amplitudes over time $\dot {a}_{j}\left ( t\right ) =-i\sum _{k}H_{jk}a_{k}\left ( t\right )$. In consideration of the spontaneous emission of the system, the following dynamical equations can be obtained in the rotating frame

Next, we focus on the diffraction effect of the probe field based on the above linear and nonlinear polarization of the present system. From the Maxwell theory and under the slow amplitude approximation and the paraxial approximation, the propagation equation of the probe field in steady state is given by

3. Results and discussion

Now we study the diffraction properties of weak probe field based on the nonlinear modulation generated by tunnelling effect in double quantum dot system. Here we choose the realistic parameters for InAs self-assembled QDMs in the following discussion. As shown in Refs. [51,58,63], the interband dipole moment is $\left \vert \mu _{12}\right \vert /e=2.1$nm, the optical confinement factor is given by $\Gamma =6\times 10^{-3}$, and the QDM surface density is $2\times 10^{11}$cm$^{-2}$. And the weak tunneling strength and driving intensity are considered. Firstly, the dispersion and absorption spectra are under our consideration and they are shown in Fig. 2. Here all the physical parameters are scaled by the decay rate $\gamma _{2}$ and are chosen as $T_{e}=0.25,$ $\Omega _{0}=0.5$, $\gamma _{2}=\gamma _{4}=1,\gamma _{3}=0.0001,$ $\Delta _{c}=5$ (c,d) and $\Delta _{c}=25$ (e,f), where the graphs by the red dash-dot lines describe the cases of no tunneling coupling. Figures 2(a) and  2(b) show the linear susceptibility closely related with linear dispersion ($\operatorname {Re}\chi ^{(1)}$) and absorption ($\operatorname {Im}\chi ^{(1)}$) as a function of probe detuning, respectively. In the presence of tunneling, the transparent window appears at a central probe frequency ($\Delta _{p}=0$), and simultaneously the value of the dispersion is equal to zero for probe resonance. The nonlinear polarization responses generated by the tunneling coupling are shown in Figs. 2(c) and (e) and (d) and (f). When the driving detuning is small, the asymmetric feature is present for the nonlinear dispersion ($\operatorname {Re}\chi ^{(3)}$) shown in Fig. 2(c). At $\Delta _{p}=0$, the dispersion is greatly enhanced and its value is more than 3. But the large absorption is also existent with the value of $0.613$, as shown in Fig. 2(d), which will be not useful for the diffraction. Fortunately, the enhanced dispersion accompanied by near-vanishing absorption could be obtainable via increasing the driving detuning $\Delta _{c}$, which is found from Figs. 2(e) and 2(f) for the case of $\Delta _{c}=25$. Remarkably, the tunneling coupling between two quantum dots leads to the improvement of refractive index without absorption, which is very important for the fabrication of phase grating and the improvement of diffraction efficiency.

 

Fig. 2. The linear and nonlinear susceptibilities closely related with dispersion and absorption as a function of probe detuning, respectively. The corresponding parameters are given by $T_{e}=0.25,$ $\Omega _{0}=0.5$, $\gamma _{2}=\gamma _{4}=1,\gamma _{3}=0.0001,$ $\Delta _{c}=5$ (c,d) and $\Delta _{c}=25$ (e,f), where the graphs shown by the red dash-dot lines correspond to the case of no tunneling coupling.

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Next we focus on the phase $\Phi /\pi$ and amplitude $\left \vert T(x)\right \vert$ depending on the coordinate $x$ in the presence of tunneling coupling, which are shown in Fig. 3 with $\Delta _{c}=25$, $L=25z_{0},$ $\Delta _{p}=0$ and the other parameters being as same as those in Fig. (2). It is clear that the values of $\Phi /\pi$ and $|T(x)|$ approach to zero shown by the red dash-dot lines when the tunneling strength is $T_{e}=0$. That means that the probe light through the quantum dot system are totally absorbed, and it is impossible to realize the diffraction. Fortunately, combining the weak standing-wave driving with the tunneling coupling could increase the diffraction efficiency. For the case of weak tunneling coupling $T_{e}=0.25$, the phase modulation is large enough with the maximal value of $1.27\pi$, as shown in Fig. 3(a). At the same time, the absorption modulation is small which oscillates around an average transmissivity of 90%. That is to say, the phase grating with the high transmissivity could be obtained and the efficient diffraction will be present, which is mainly due to the enhanced nonlinearity with negligible absorption induced by the tunneling effect.

 

Fig. 3. The phase $\Phi /\pi$ (a)and amplitude $\left \vert T(x)\right \vert$ (b) depending on the coordinate $x$ in the presence of tunneling coupling with $\Delta _{c}=25$, $L=25z_{0},$ $\Delta _{p}=0$ and the other parameters being as same as those in Fig. (2).

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In order to clearly see the influence of the tunneling coupling on the diffraction efficiency, The diffraction intensity $I_{p}(\theta )$ vs $sin\theta$ is shown in Fig. 4, where the parameters are (a) $L=25z_{0}$ and (b) $L=35z_{0}$ and the rest parameters are same as those in Fig. 2 with $\Delta _{c}=25$, $R\;=\;5$, and $\Lambda / \lambda \mathrm{p}=4$. Without the tunneling coupling shown by red dash-dot line, we find that the value of $I_{p}(\theta )$ is close to $0$ and no diffraction pattern occurs. In sharp contrast, the diffraction intensity could be greatly increased in the first and higher order directions when the tunneling is involved ($T_{e}=0.25$). As shown in Fig. 4(a), The first-order diffraction intensity reached $0.265$ at $sin\theta =\pm 0.25$, which is close to that of an ideal sinusoidal phase grating. Meanwhile, the diffraction intensity in the zero-order direction is smaller. It is obvious that the QDMs act as the phase grating and can diffract the probe beam to other directions successfully. From Fig. 4(b) we could see that the higher diffraction intensity can be obtained in the second-order direction with the appropriate choice of the interaction length parameter. The resulting second-order diffraction efficiency of the grating is $16.4\%$ at $sin\theta =\pm 0.5$. The role of tunneling in the formation of phase gratings is to induce the great enhancement of refractive index with nearly-vanishing absorption, which are shown in Fig. 2(e). Therefore, the phase modulation is dominant in the Fraunhofer diffraction, which could improve the first-order and high-order diffraction efficiency.

 

Fig. 4. The diffraction intensity $I_{p}(\theta )$ vs $sin\theta$, where the parameters are (a) $L=25z_{0}$ and (b) $L=35z_{0}$ and the rest parameters are same as those in Fig. 2 with $\Delta _{c}=25$, $R\;=\;5$, and $\Lambda / \lambda \mathrm{p}=4$.

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Finally, we discuss the first-order diffraction intensity $I_{p}(\theta _{1})$. The variances of $I_{p}(\theta _{1})$ as a function of Rabi frequency $\ \Omega _{0}$, driving detuning $\Delta _{c}$, interaction length $\ L$ and tunneling strength $T_{e}$, are shown in Fig. 5, where the other parameters are the same as in Fig. 2. As shown in Fig. 5(a), the value of $I_{p}(\theta _{1})$ can be effectively modulated by changing $\Omega _{0}$. At the region of $\Omega _{0}$ $\approx 0.5$, the high first-order intensities is existent. When $\Omega _{0}$ continues to increase, the diffraction efficiency oscillates and then decreases. Similarly, we can see from Fig. 5(b) the diffraction intensity is the largest near $\Delta _{0}=28$, and then decreases. Figure 5(c) shows the relationship between the interaction length and the first-order diffraction intensity. It shows that there is an optimal value of $L=22.1z_{0}$, and the intensity of the first-order light increases significantly. When $L$ is large, the nonlinear absorption dominates and the diffraction efficiency decreases. As can be seen from Fig. 5(d), the diffraction intensity oscillates first, and then increases and finally decreases gradually with the increasing $T_{e}$. Near $T_{e}=0$, the diffraction intensity is almost $0$ and light can not pass through the quantum dots due to the existence of large absorption. In the region of $T_{e} \approx 0.27$, there is a high first-order diffraction intensity with the value 27.3%. The diffraction efficiency decreases as the tunneling continues to increase. These results allow us to say that with an appropriate choice of Rabi frequency $\Omega _{0}$, driving detuning $\Delta _{c}$, interaction length $\ L$, and tunneling strength $T_{e}$, we can optimize diffraction phenomena in artificial QDMs and obtain the perfect phase grating in the solid-state system. However, the electric field controlling the tunneling strength $T_{e}$ also causes the change of the energy of levels in the two quantum dots, and this situation, makes the tunneling manipulation technically complicated.

 

Fig. 5. The variances of $I_{p}(\theta _{1})$ as a function of Rabi frequency $\ \Omega _{0}$, driving detuning $\Delta _{c}$, interaction length $\ L$ and tunneling strength $T_{e}$.

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4. Conclusion

In conclusion, the diffraction properties of the quantum dot system have been studied via considering the tunneling coupling between two quantum dots. With the help of the weak standing-wave control field, the quantum dot system exhibits the periodic phase modulation with high transmission, which could result in the probe beams being diffracted into first-order and higher-order directions. The responsible mechanism is the tunneling-induced dispersion enhancement accompanied by the nearly-vanishing absorption for the case of probe resonance. In our scheme, the phase grating is obtained under the condition of weak tunneling and weak standing-wave driving, which can relax the restriction on the strong couplings. Furthermore, the diffraction intensities can be easily adjusted by changing the parameters such as the weak driving intensity, driving detuning, tunneling intensity and interaction length. The scheme we present has potential applications in optical switch, quantum information processing, development of new photonic devices and measurement of solid materials.