## Abstract

Kerr-nonlinearity induced self-focusing or self-defocusing effect provides the opportunity for exploring fundamental phenomena related to the light-matter interactions. Here we show that the linear and nonlinear dispersion responses are significantly sensitive to both the detunings and the tunneling strengths of the indirect-excitonic (IX) states in an asymmetric triple quantum dot system. In particular, the nonlinear dispersion properties are dominated by the tunnel-enhanced cross-Kerr nonlinearity from one of the IX states. Meanwhile, by varying the detunings of other IX states, we reveal that the tunnel-enhanced cross-Kerr nonlinearity gives rise to the realization of the self-focusing and self-defocusing effects. Moreover, by taking into account the effect of the longitudinal-acoustic-phonon induced dephasing of the IX states, it is possible to modulate the height and position of the peak of the self-focusing or self-defocusing effect. Our results may have potential applications in nonlinear-optics and quantum-optics devices based on the tunnel-enhanced nonlinearities in this solid-state system.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Semiconductor quantum dot (QD) is of great interest for applications in quantum optics and nonlinear optics because it has discrete energy levels and exhibits optical properties similar to those in atomic systems [1–3]. A triple QD molecule, formed by the tunnel couplings among three individual dots, may provide additional tools and functions to study some interesting phenomena that do not appear in a single or double QD system [4–8]. It may be noted that carriers in the triple QD molecule are not strictly confined to single dot but may be delocalized over three dots. As such, the tunability of the excitonic energy spectrum in this QD system holds great promise for the exciton-based coherent control protocols as a feasible path for the modulation of the optical properties in the QD system [9]. Recently, there were many studies on the optical responses of the excitonic states in a QD molecule coupled by tunneling or pulsed laser fields, including the tunneling induced transparency, Autler-Townes splitting, slow light, and so on [2, 10–20].

It is well known that the optical Kerr effect, a third-order nonlinear process that includes the self-Kerr or cross-Kerr effect, can vary the refractive index of the medium and is directly proportional to the square of the field strength or the light intensity [21–23]. The refractive-index variation of a medium is responsible for the nonlinear optical effects, such as the self-focusing, self-defocusing, and modulational instability. Self-focusing or self-defocusing belongs to the self-action effects in which the sign of the third-order nonlinear susceptibility is positive or negative [24]. The Kerr-nonlinearity-induced self-focusing or self-defocusing effect was theoretically predicted [25–27] and experimentally demonstrated [28, 29] by studying the interaction of ruby lasers with glasses and liquids in the 1960s. The study of the self-focusing and self-defocusing effects enable many applications in laser physics, e.g., laser filamentation in transparent media [30, 31], self-compression of ultrashort laser pulses [32, 33], Kerr-lens modelocking [34], and other topics of the light-matter interactions.

In the present work, we harness a solid-state system (i.e., a triple QD molecule) to realize the above effects. Here the strong tunneling strength between any two nearest-neighbor dots plays the equivalent role of the strong optical control field, thus giving rise to the enhancement of the cross-Kerr nonlinearity. Moreover, it has been demonstrated that the indirect excitonic (IX) dephasing rates in the QD system can be effectively modulated by applying an electric field perpendicular to the growth direction, where the IX dephasing rate is comparable to the relaxation rate of the direct excitons [35] and it arises from the longitudinal-acoustic (LA) phonon-induced exciton relaxation [36, 37]. Therefore, in the present work, we show how the detuning and the tunneling strengths of the IX states affect the linear and nonlinear dispersion properties in an asymmetric triple QD system. In particular, the tunnel-enhanced cross-Kerr nonlinearity from one of the IX states is dominant due to the presence of the red-detuning rather than the blue-detuning in the other IX state. Moreover, the realization of self-focusing or self-defocusing effect induced by the tunnel-enhanced cross-Kerr nonlinearity is detuning-dependent. In addition, the LA phonon-induced dephasing of the IX states can be used to modulate the height and position of the peak of the self-focusing or self-defocusing effect.

The paper is organized as follows. Section 2 introduces the model of the triple QD. In Sec. 3, we study the linear dispersion response of a weak probe field in this triple QD system. In Sec. 4, the optical susceptibility is considered up to the third order. Then, we show the self-focusing and self-defocusing effects due to the tunnel-enhanced cross-Kerr nonlinearity, as well as the effects due to the LA phonon-induced dephasing of the indirect excitons. Finally, discussions and conclusions are given in Sec. 5.

## 2. Theoretical model of a triple QD system

Let us consider an asymmetric triple QD system coupled by tunneling through the thin barriers [see Fig. 1(a)]. It is composed of three aligned QDs with different sizes, similar to the case of the double QD with different sizes of the dots [11–14, 38–40]. Here the asymmetric triple QD system is suitably designed, so that by applying a gate voltage along the growth direction of the triple QD system [see see Fig. 1(b)], the conduction-band levels of the left and right dots can become resonant with that of the central dot [13, 38, 39]. Thus, even though the interdot separation is at nanoscale, the excitonic states consist of localized holes and delocalized electron states in the QD molecules [41]. The asymmetry of the triple QD system and the width of the barrier prevent the tunneling of holes [38] and thus we can just consider the electron tunneling between the left (right) and the central dots. Experimentally, this triple QD system can be produced on the GaAs (001) substrate by molecule beam epitaxy or in-situ atomic layer precise etching [41, 42], thus enabling a homogeneous triple QD aligned along the $[1\overline{1}0]$ direction. We also consider a weak probe laser field with frequency *ω _{p}* which is nearly resonant with the direct excitonic transition |0〉↔|1〉 (i.e., both the bound electron and hole are in the same QD) in the central dot, where |0〉 represents the excitonic vacuum state (i.e., no exciton in the triple QD) and |1〉 denotes the direct excitonic state formed by an applied pulsed laser field which excites an electron from the valence band to the conduction band, as shown in Fig. 1(c). The gate voltages can be used to control the electron to tunnel from the central dot to either the left or right dot, forming an indirect excitonic state |2〉 or |3〉, where the hole is in the central dot but the electron is in the left or right dot. The Hamiltonian for this triple QD system can be written as (we set

*ħ*= 1)

*σ*≡|

_{jk}*j*〉〈

*k*|, and Ω

*=*

_{p}*µ*

_{01}ℰ

*/2 denotes the Rabi frequency of the weak probe laser field, with ℰ*

_{p}*being the electric-field amplitude of the probe field and*

_{p}*µ*

_{01}the electric-dipole transition matrix element between states |0〉 and |1〉. T

_{12}(T

_{13}) is the tunneling coupling strength between the central dot and the left (right) dot, and

*ω*denotes the frequency differences between states |

_{jk}*j*〉 and |

*k*〉.

The time-dependent oscillatory terms in the Hamiltonian can be eliminated using a unitary transformation [43, 44] $U(t)=\mathrm{exp}[-i{\omega}_{p}({\displaystyle {\sum}_{j=1}^{3}{\sigma}_{jj}})t]$. Then, the Hamiltonian in the rotating frame is given by

_{1}=

*ω*

_{10}−

*ω*is the detuning of the probe field (i.e., the frequency difference between the probe field and

_{p}*ω*

_{10}), and Δ

_{2(3)}= Δ

_{1}+

*ω*

_{12(3)}, with

*ω*

_{12(3)}denoting the frequency difference between the indirect excitonic state |2〉(|3〉) and the direct excitonic state |1〉. Experimentally,

*ω*12(3) can be modulated by applying a gate field

*F*along the growth direction, leading to the energy-level shift Δ

*ω*

_{12(3)}=

*eFd*[13, 38], with

*d*being the width of the barrier between the right and central dots.

The dynamics of the system can be described by a master equation for the reduced density matrix of a triple QD system. By taking into account the energy relaxation rates Γ_{j}_{0} between | *j*〉 and |0〉 and the pure dephasing rates ${\gamma}_{j}^{\varphi}$ of the states | *j*〉 (*j* = 1, 2, and 3), the Lindblad master equation can be written, in the Born-Markov approximation, as

*ρ*

_{01}in the steady state is given by

*= Δ*

_{j}*+*

_{j}*i*Γ

*(*

_{j}*j*= 1, 2, 3), and Γ

*denotes the total decay rate of the energy-level |*

_{j}*j*〉, which is defined in the Appendix. For the considered triple QD system, we can obtain the linear optical susceptibility

*χ*(Δ

_{1}), which is proportional to the density matrix element

*ρ*

_{01}, i.e., where

*U*

_{01}= Γ

_{opt}|

*µ*

_{01}|

^{2}/(

*Vϵ*

_{0}

*ħ*), with

*µ*

_{01}/

*e*= 2.1 nm, Γ

_{opt}= 6 × 10

^{−3}describing the optical confinement factor,

*V*being the volume of a single QD, and

*ϵ*

_{0}the dielectric constant [37, 45–47]. Note that the linear optical susceptibility

*χ*(Δ

_{1}) is complex. The absorption profile and the refractive index of the medium can be determined from the imaginary and real parts of this linear susceptibility, respectively. Thus, the group velocity

*v*of the probe field, which is related to the linear optical susceptibility

_{g}*χ*(Δ

_{1}), is approximately equal to c/(1 + 2

*πω*dRe[

*χ*(Δ

_{1})]/dΔ

_{1}).

## 3. The linear dispersion properties

In Fig. 2, we show the linear dispersion response of a weak probe field in the triple QD system. The real part of the optical susceptibility, *χ*(Δ_{1})∝ *ρ*_{01}, is numerically obtained versus the detuning Δ_{1}. In the weak-tunneling regime, as shown in Figs. 2(a) and 2(b), we find that the sign of the group velocity of the probe field is positive when both the left and right dots are resonant with the central dot [see the black solid curves in Figs. 2(a) and 2(b)]. Here the probe field has a *normal linear dispersion* (i.e., the positive slope of the real part of the optical susceptibility dRe[*χ*(Δ_{1})]/dΔ_{1} *>* 0) and hence its propagation is subluminal, which can be termed as *slow light*. When the detuning of the right dot is blue-shifted from the resonance *ω*_{13} = Γ_{1}/5 [see the blue dashed curves in Figs. 2(a) and 2(b)], the normal dispersion slope increases, implying that the group velocity of the probe field becomes smaller at this moment. At the red-detuning *ω*_{13} = −Γ_{1}/5, we find that the normal dispersion slope further increases, which means that the group velocity of the probe field in the red-detuning case is even slower than the group velocity in the blue-detuning case [for clarity, see also the inset in Figs. 2(a) and 2(b)]. Also, it is interesting to note that, in comparison of the blue and red curves in Fig. 2(a) with those in Fig. 2(b), the group velocity of the probe field appears to be slower in certain condition when the tunneling strengths have T_{13} *>* T_{12}.

In the strong-tunneling regime, it is shown that compared with the results presented in Figs. 2(a) and 2(b), a similar conclusion can be made on the detuning-dependent normal dispersion properties of the probe field for different tunneling strengths [see Figs. 2(c) and 2(d)]. In addition, it should be noted that the phenomenon of the controllable slow light of the probe field is more obvious and distinguishable for relatively large detunings between the right and central dots.

## 4. Self-focusing and self-defocusing effects induced by the cross-Kerr nonlinearity

The Kerr-induced self-focusing effect was originally due to the optical Kerr effect, which is a third-order nonlinear process and produces an enhanced nonlinear component of the refractive index. It was demonstrated that an intense light beam propagating in a nonlinear medium may experience a self-focusing or self-defocusing effect [25], where the sign of the third-order nonlinear susceptibility is positive or negative. The spreading of an optical beam could be turned into the *spatial bright* (*dark*) *soliton*, as a result of a balance between the diffraction and the self-focusing (self-defocusing) effect. The tunable self-focusing and self-defocusing effects can also be used to determine the amplitude and group velocity of the spatial bright and dark solitons [26, 44, 48–51].

For the Λ-type three-level structure, it generally contains two ground states optically coupled to an excited state by a strong control field and a weak probe field, respectively. However, in the system of two coupled QDs, the control field was equivalently replaced by the electron tunneling between the conduction-band of the two QDs [11]. In doing so, it provides the possibility to generate tunneling-induced transparency and slow light phenomena by effectively regulating the external electric field. Furthermore, as demonstrated in the atomic system [52], the enhanced Kerr nonlinearities induced by the strong control field in the four-level *N*-type scheme are several orders of magnitude higher than that in the three-level ladder-type scheme. With regard to the triple QD system, there exist four effective excitonic states including two kinds of electron tunneling. The strong tunneling strengths, which have the equivalent effect as those of the strong light fields, are likely to cause the enhancement of the third-order cross-Kerr nonlinear susceptibility even for a weak probe field.

#### 4.1. The tunnel-enhanced cross-Kerr nonlinearity

In the triple QD system, by tuning the external electric field, it is possible to modulate the electron tunneling between the left (right) and central dots. The strong electron tunnelings in this triple QD system can play the equivalent role of the strong optical fields. Indeed, as shown in Eq. (5), the linear optical susceptibility *χ*(Δ_{1}) is nonlinear with respect to the electron tunneling strengths. Thus, the strong tunneling strength may give rise to the occurrence of cross-Kerr interactions even for the weak probe field in the triple QD system, with a suppressed self-Kerr interaction. To understand the effects of cross-Kerr nonlinearity (third-order dispersion effects) on the modulation of the group velocity of the probe field, we express the linear optical susceptibility by making a Taylor expansion with respect to the tunneling strengths ${\mathrm{T}}_{12}^{2}$ and ${\mathrm{T}}_{13}^{2}$ [22, 23],

The term *χ*^{(1)} in Eq. (7a) denotes the linear response part. The term *χ*^{CK−LD} (*χ*^{CK−RD}) in Eq. (7b) or (7c) corresponds to the effective third-order cross-Kerr nonlinear coefficient when considering the effect of the electron tunneling between the left (right) and center dots.

In Fig. 3, we show the detuning-dependent cross-Kerr nonlinearities Re[*χ*^{CK−LD}] and Re[*χ*^{CK−RD}] versus the probe detuning Δ_{1} for different tunneling strengths in the weak-tunneling regime. We can see that, for the on-resonance case with *ω*_{13} = 0, the cross-Kerr nonlinearity Re[*χ*^{CK−LD}] is dominant [for clarity, see the black solid curve of the inset in Fig. 3(a)] and shows an *anomalous third-order dispersion* (i.e., the negative slope of the real part of the third-order optical susceptibility dRe[*χ*^{CK−LD}(Δ_{1})]/dΔ_{1} *<* 0), when the tunneling strengths obey T_{12} *>* T_{13}. However, the cross-Kerr nonlinearity Re[*χ*^{CK−RD}] is dominant [see the black dashed curve in Fig. 3(b)] and has an *anomalous third-order dispersion*, when the tunneling strengths obey T_{12} *<* T_{13}.

As for the off-resonance case with *ω*_{13} ≠ 0, the tunnel-enhanced nonlinear interactions, as shown in the blue and red curves in Fig. 3(a), are dominated by the cross-Kerr nonlinearity Re[*χ*^{CK−LD}], which is compared to that of the cross-Kerr nonlinearity Re[*χ*^{CK−RD}] from the right dot [for clarity, see the blue and red curves in Fig. 3(b)]. It should be noted that, with a *normal third-order dispersion*, the enhancement of Re[*χ*^{CK−LD}] is one or two orders of magnitude larger than that of Re[*χ*^{CK−RD}]. Moreover, for the tunneling strengths with T_{12} *>* T_{13} or T_{12} *<* T_{13}, the enhancement of Re[*χ*^{CK−LD}] in the red-detuning case is larger than that in the blue-detuning [compare the red solid (dashed) curve with the blue solid (dashed) curve in Fig. 3(a)]. This detuning-dependent enhancement of the cross-Kerr nonlinearity Re[*χ*^{CK−LD}] also indirectly provides an evidence of the reason why the group velocity of the probe field in the red-or blue-detuning case is even slower, as shown in Figs. 2(a) and 2(b).

Figures 4(a) and 4(b) show that the tunnel-enhanced cross-Kerr nonlinearities Re[*χ*^{CK−LD}]and Re[*χ*^{CK−RD}] in the strong-tunneling regime are similar to the results presented in Figs. 3(a) and 3(b), respectively. Note that the enhancement of the cross-Kerr nonlinearity Re[*χ*^{CK−RD}], as shown in Fig. 4(b), is only twice larger in the on-resonance case (i.e., *ω*_{13} = 0) than that of the case shown in Fig. 3(b), but the enhancements of the cross-Kerr nonlinearity Re[*χ*^{CK−LD}], as shown in Fig. 4(a), are one or two orders of magnitude larger than that shown in Fig. 3(a) in the off-resonance case with *ω*_{13} ≠ 0.

#### 4.2. The self-focusing and self-defocusing effects

As shown in the inset of Fig. 3(b) (Fig. 4(b)), for the red-detuning case with *ω*_{13} = −Γ_{1}/5(*ω*_{13} = −2Γ_{1}), the cross-Kerr nonlinearity Re[*χ*^{CK−RD}] reveals *self-focusing effect*, while the *self-defocusing effect* appears for the blue-detunning case with *ω*_{13} = Γ_{1}/5(*ω*_{13} = 2Γ_{1}). Therein, the enhanced Re[*χ*^{CK−RD}] with T_{12} *<* T_{13} is larger than that with T_{12} *>* T_{13}, which can be indicated by the comparison of the red (blue) dashed curve with the red (blue) solid curve in the insets of Fig. 3(b) and Fig. 4(b). However, for the off-resonance detuning cases with *ω*_{13} ≠ 0, the enhancement of the cross-Kerr nonlinearity Re[*χ*^{CK−RD}] is much smaller than that of Re[*χ*^{CK−LD}] shown in Fig. 3(a) and Fig. 4(a), respectively.

For the purpose of obtaining the significantly tunnel-enhanced self-focusing or self-defocusing effect, we show in Fig. 5(a) the real part of the cross-Kerr nonlinearity *χ*^{CK−LD} versus the probe detuning Δ_{1} and the detuning *ω*_{13} of the right dot. For the relatively small blue-detuning *ω*_{13} = Γ_{1}/5 or red-detuning *ω*_{13} = −Γ_{1}/5, the cross-Kerr nonlinearity Re[*χ*^{CK−LD}] exhibits *anomalous third-order dispersion* [see the red dashed or solid curve in Fig. 5(b)]. For the relatively large value of blue-detuning *ω*_{13} = 3Γ^{1} or red-detuning *ω*_{13} = −3Γ_{1}, the cross-Kerr nonlinearity Re[*χ*^{CK−LD}] exhibits a *normal third-order dispersion* [see the blue dashed or solid curve in Fig. 5(e)]. It is interesting to see that, for the red-detuning decreasing from *ω*_{13} = −Γ_{1}/2 to *ω*_{13} = −Γ_{1}, the cross-Kerr nonlinearity Re[*χ*^{CK−LD}] evolves from *anomalous third-order dispersion* [see the green dashed curve in Fig. 5(c)] to *self-focusing effect* [see the black dashed curve in Fig. 5(d)]. Simultaneously, for the blue-detuning increasing from *ω*_{13} = Γ_{1}/2 to *ω*_{13} = Γ_{1}, the cross-Kerr nonlinearity Re[*χ*^{CK−LD}] evolves from *anomalous third-order dispersion* [see the green solid curve in Fig. 5(c)] to *self-defocusing effect* [see the black solid curve in Fig. 5(d)].

It is known that the IX energy can be adequately modulated by applying an electric field perpendicular to the growth direction. The time scale of the phonon-mediated relaxation between the discrete electronic states in the QD system is predicted to be comparable to that of the relaxation of the direct excitons [35]. Moreover, the LA phonon-induced exciton relaxation was supposed to be a primary mechanism for the dephasing of the indirect exciton in the QD system [36, 37].

In order to explore the effect of the LA phonon-induced dephasing of the indirect excitons on the self-focusing or self-defocusing effect, we plot in Fig. 6 the cross-Kerr nonlinearity Re[*χ*^{CK−LD}] versus the probe detuning Δ_{1} for different values of the LA phonon-induced dephasing rate of the indirect exciton. The black dashed curve in Fig. 6(a) or the black solid curve in Fig. 6(b) is the result that we have previously shown in Fig. 5(d). For a relatively large dephasing rate of the indirect exciton from the left dot (i.e., Γ_{2} = 10^{−2}Γ_{1}), the peak of the self-focusing or self-defocusing effect almost decreases by ten times [see the red dashed curve in the inset of Fig. 6(a) or the red solid curve in Fig. 6(b)]. However, for a relatively large dephasing rate of the indirect exciton from the right dot (i.e., Γ3 = 10^{−2}Γ_{1}), the peak of the self-focusing or self-defocusing effect remains nearly unchanged [for clarity, see the green dashed or solid curve in the inset of Fig. 6(a) or Fig. 6(b), respectively]. As the dephasing rate of the indirect exciton from the right dot further increases (i.e., Γ3 = 10^{−1}Γ_{1}), the peak of the self-focusing or self-defocusing effect only decreases ten percent [for clarity, see the blue dashed curve in the inset of Fig. 6(a) or the blue solid curve in the inset of Fig. 6(b)], where the peak position shifts to the blue-detuning (red-detuning) side for the self-focusing (self-defocusing) effect.

## 5. Conclusions

In summary, we have theoretically studied the linear and third-order nonlinear dispersion responses in an asymmetrical triple QD molecule system with four effective excitonic-state levels. It is shown that, in the case of the left dot being resonant with the central dot (i.e., the IX state in the left dot is on-resonance with *ω*_{12} = 0), the probe field presents a *normal dispersion*, which corresponds to the realization of *slow light*. By varying the detuning of the IX state in the right dot with *ω*_{13} ≠ 0, the group velocity of the probe field can be even slower in the red-detuning case than in the blue-detuning case. This can also be further slowed down when the tunneling strengths have T_{12} *<* T_{13}. As for the third-order nonlinear dispersion properties of the probe field, when *ω*_{12} = *ω*_{13} = 0, the tunnel-enhanced cross-Kerr nonlinearity of the left or right dot exhibits an *anomalous third-order dispersion* if the tunneling strengths obey T_{12} *>* T_{13} or T_{12} *<* T_{13}. For the off-resonant case with *ω*_{13} ≠ 0, the cross-Kerr nonlinearity of the left dot is one or two orders of magnitude larger than that of the right dot and its enhancement exhibits a *normal third-order dispersion*. Simultaneously, in spite of the tunneling strengths with T_{12} *>* T_{13} or T_{12} *<* T_{13}, the enhancements of the cross-Kerr nonlinearity of the left dot is larger when considering the red-detuning rather than the blue-detuning of the IX state in the right dot.

By increasing the detunings (either red-detuning or blue-detuning) of the IX state in the right dot, the cross-Kerr nonlinearity of the IX state in the left dot evolves from *anomalous third-order dispersion* into *normal third-order dispersion*. More interestingly, the self-focusing effect is fulfilled with an appropriate red-detuning of the IX state of the right dot, while the self-defocusing effect is accomplished in an adequate blue-detuning case. With an one order of magnitude increase of the LA phonon-induced dephasing of the IX state in the *left* dot, the peak of the self-focusing or self-defocusing effect almost decreases by more than ten times. With a two orders of magnitude increase of the LA phonon-induced dephasing of the IX state in the *right* dot, the peak of the self-focusing or self-defocusing effect only decreases ten percent.

Moreover, it is known that tunnel-coupled QD systems are proved to be promising candidates in many optical and electronic applications [1–3]. This is owing to the optical excitation combined with the electric-field-dependent tunneling interaction between the coupled QDs, which provides the feasibility for optical-electric field modulation. The coherent control of the electron tunneling in the coupled QD system has been directly observed by embedding QDs in the electric-field-tunable Schottky diodes [9, 38]. Also, the strong coherent tunneling between the coupled QDs can give rise to the formation of Autler-Townes doublet and triplet, and Mollow-like triplet [11, 13, 47]. All these phenomena were originally realized by considering the effect of the strong light fields in the atomic system. In the present work, the strong tunneling strengths among the triple QD system play the role to produce the enhancement of the third-order cross-Kerr nonlinearities even for the weak probe field.

In a cavity-qubit interaction system, when they are strongly coupled and the bare frequency of the cavity is far enough away from the frequency of the qubit, the resonance frequency of the cavity can be shifted compared with its bare frequency, depending on the state of the qubit. Thus, the measurement of the transmission or reflection of the cavity can, in turn, be used for reading out the state of the qubit or qutrit [53, 54]. This involves the quantum non-demolition measurement and is termed as the dispersive readout technique. As for the measurement of the linear and nonlinear responses when using a weak probe field in the triple QD system, it is possible to use dispersive readout technique by embedding the triple QD into a nanocavity, because the conditions of the cavity as mentioned above are feasible to fulfill in the QD-nanocavity system. Finally, our results pave the way to explore the experimental observation of the self-focusing and self-defocusing effects in the semiconductor QD molecule system and are further expected to have potential applications to all-optical devices in solid-state systems.

## Appendix. Derivation of the equations for density matrix elements

We here analytically derive the steady-state solution to the reduced density matrix elements that can be used to calculate the optical susceptibility of the probe field. From the Lindblad master equation in Eq. (3), we obtain the equations for density matrix elements,

*= Δ*

_{jk}*− Δ*

_{k}*, and ${\Gamma}_{jk}=({\Gamma}_{j0}+{\Gamma}_{k0})/2+{\gamma}_{j}^{\varphi}+{\gamma}_{k}^{\varphi}(j=1,2;k=2,3)$.*

_{j}With the time explicitly shown, the equations of *∂ _{t} ρ*

_{01},

*∂*

_{t}ρ_{02}, and

*∂*

_{t}ρ_{03}in Eq. (8) are given by

In the case of a weak probe field (Ω* _{p}* ≪ T

_{12}, T

_{13}) interacting with the four-level triple-QD system, one can assume that Ω

*[*

_{p}*ρ*

_{11}(

*t*)−

*ρ*

_{00}(

*t*)] ≈ Ω

*[*

_{p}*ρ*

_{11}(0)−

*ρ*

_{00}(0)], Ω

_{p}ρ_{02}(

*t*)≈ Ω

_{p}ρ_{02}(0), and Ω

_{p}ρ_{03}(

*t*)≈ Ω

_{p}ρ_{03}(0) in Eq. (9), respectively. Moreover, the system is assumed to be initially in the direct excitonic state |1〉, so

*ρ*

_{11}(0) = 1 and

*ρ*

_{12}(0) =

*ρ*

_{13}(0) =

*ρ*

_{00}(0) = 0. Then, Eq. (9) becomes

When the system approaches to the steady state, i.e., *∂ _{t} ρ*

_{01}=

*∂*

_{t}ρ_{02}=

*∂*

_{t}ρ_{03}= 0, we have

## Funding

Science Challenge Project (TZ2018003); National Key Research and Development Program of China (2016YFA0301200); NSFC (11774022, 11804074); NSAF (U1530401); Postdoctoral Science Foundation of China (2016M600905); .

## References

**1. **P. Michler, *Single Semiconductor Quantum Dots*(Springer, 2009). [CrossRef]

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

**3. **X. Q. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science **301**, 809–811 (2003). [CrossRef] [PubMed]

**4. **C. Y. Hsieh, Y. P. Shim, M. Korkusinski, and P. Hawrylak, “Physics of lateral triple quantumdot molecules with controlled electron numbers,” Rep. Prog. Phys. **75**, 114501 (2012). [CrossRef]

**5. **D. S. Saraga and D. Loss, “Spin-entangled currents created by a triple quantum dot,” Phys. Rev. Lett. **90**, 166803 (2003). [CrossRef] [PubMed]

**6. **K. G. Rasmussen, H. I. Jørgensen, T. Hayashi, P. E. Lindelof, and T. A. Fujisawa, “A triple quantum dot in a single-wall carbon nanotube,” Nano. Lett. **8**, 1055–1060 (2008). [CrossRef]

**7. **L. Gaudreau, G. Granger, A. Kam, G. C. Aers, S. A. Studenikin, P. Zawadzki, M. P. Ladrière, Z. R. Wasilewski, and A. S. Sachrajda, “Coherent control of three-spin states in a triple quantum dot,” Nat. Phys. **8**, 54–58 (2012). [CrossRef]

**8. **J. Y. Wang, S. Y. Huang, G. Y. Huang, D. Pan, J. H. Zhao, and H. Q. Xu, “Coherent transport in a linear triple quantum dot made from a pure-phase InAs nanowire,” Nano. Lett. **17**, 4158–4164 (2017). [CrossRef] [PubMed]

**9. **H. J. Krenner, S. Stufler, M. Sabathil, E. C. Clark, P. Ester, M. Bichler, G. Abstreiter, J. J. Finley, and A. Zrenner, “Recent advances in exciton-based quantum information processing in quantum dot nanostructures,” New J. Phys. **7**, 184 (2005). [CrossRef]

**10. **B. D. Gerardot, D. Brunner, P. A. Dalgarno, K. Karrai, A. Badolato, P. M. Petroff, and R. J. Warburton, “Dressed excitonic states and quantum interference in a three-level quantum dot ladder system,” New J. Phys. **11**, 013028 (2009). [CrossRef]

**11. **H. S. Borges, L. Sanz, J. M. Villas-Bôas, O. O. Diniz-Neto, and A. M. Alcalde, “Tunneling induced transparency and slow light in quantum dot molecules,” Phys. Rev. B **85**, 115425 (2012). [CrossRef]

**12. **J. Berney, M. T. Portella-Oberli, and B. Deveaud, “Dressed excitons within an incoherent electron gas: observation of a Mollow triplet and an Autler-Townes doublet,” Phys. Rev. B **77**, 121301 (2008). [CrossRef]

**13. **H. S. Borges, L. Sanz, J. M. Villas-Bôas, and A. M. Alcalde, “Quantum interference and control of the optical response in quantum dot molecules,” Appl. Phys. Lett. **103**, 222101 (2013). [CrossRef]

**14. **C. H. Yuan and K. D. Zhu, “Voltage-controlled slow light in asymmetry double quantum dots,” Appl. Phys. Lett. **89**, 052115 (2006). [CrossRef]

**15. **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**, 063601 (2017). [CrossRef] [PubMed]

**16. **P. L. Ardelt, M. Koller, T. Simmet, L. Hanschke, A. Bechtold, A. Regler, J. Wierzbowski, H. Riedl, J. J. Finley, and K. Müller, “Optical control of nonlinearly dressed states in an individual quantum dot,” Phys. Rev. B **93**, 165305 (2016). [CrossRef]

**17. **A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, “Fano resonances in nanoscale structures,” Rev. Mod. Phys. **82**, 2257–2298 (2010). [CrossRef]

**18. **D. Mogilevtsev, E. R. Gómez, S. B. Cavalcanti, and L. E. Oliveira, “Slow light in semiconductor quantum dots: effects of non-Markovianity and correlation of dephasing reservoirs,” Phys. Rev. B **92**, 235446 (2015). [CrossRef]

**19. **Y. H. Xu, W. X. Wang, Y. Q. Ge, H. Y. Guo, X. J. Zhang, S. Chen, Y. H. Deng, Z. G. Lu, and H. Zhang, “Stabilization of black phosphorous quantum dots in PMMA nanofiber film and broadband nonlinear optics and ultrafast photonics application,” Adv. Funct. Mater. **27**, 1702437 (2017). [CrossRef]

**20. **M. Sahrai, M. R. Mehmannavaz, and H. Sattari, “Optically controllable switch for light propagation based on triple coupled quantum dots,” Appl. Opt. **53**, 2375–2383 (2014). [CrossRef] [PubMed]

**21. **G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. **78**, 309–371 (2006). [CrossRef]

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

**23. **G. P. Agrawal, *Nonlinear Fiber Optics* (Academic, 2013).

**24. **R. W. Boyd, S. G. Lukishova, and Y. R. Shen, *Self-Focusing: Past and Present*(Springer, 2010).

**25. **P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. **15**, 1005–1008 (1965). [CrossRef]

**26. **R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. **13**, 479–482 (1964). [CrossRef]

**27. **N. F. Pilipetskii and A. R. Rustamov, “Observation of self-focusing of light in liquids,” JETP Lett. **2**, 55 (1965).

**28. **P. Lallemand and N. Bloembergen, “Self-focusing of laser beams and stimulated Raman gain in liquids,” Phys. Rev. Lett. **15**, 1010–1012 (1965). [CrossRef]

**29. **E. Garmire, R. Y. Chiao, and C. H. Townes, “Dynamics and characteristics of the self-trapping of intense light beams,” Phys. Rev. Lett. **16**, 347–349 (1966). [CrossRef]

**30. **J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. André, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science **301**, 61–64 (2003). [CrossRef] [PubMed]

**31. **A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. **441**, 47–189 (2007). [CrossRef]

**32. **G. Stibenz, N. Zhavoronkov, and G. Steinmeyer, “Self-compression of millijoule pulses to 7.8 fs duration in a white-light filament,” Opt. Lett. **31**, 274–276 (2006). [CrossRef] [PubMed]

**33. **A. Couairon, J. Biegert, C. P. Hauri, W. Kornelis, F. W. Helbing, U. Keller, and A. Mysyrowicz, “Self-compression of ultrashort laser pulses down to one optical cycle by filamentation,” J. Mod. Opt. **53**, 75–85 (2006). [CrossRef]

**34. **J. Herrmann, “Theory of Kerr-lens mode locking: Role of self-focusing and radially varying gain,” J. Opt. Soc. Am. B **11**, 498–512 (1994). [CrossRef]

**35. **U. Bockelmann and G. Bastard, “Phonon scattering and energy relaxation in two-, one-, and zero-dimensional electron gases,” Phys. Rev. B **42**, 8947–8951 (1990). [CrossRef]

**36. **B. Ohnesorge, M. Albrecht, J. Oshinowo, A. Forchel, and Y. Arakawa, “Rapid carrier relaxation in self-assembled InxGa1−xAs/GaAs quantum dots,” Phys. Rev. B **54**, 11532–11538 (1996). [CrossRef]

**37. **P. Borri, W. Langbein, U. Woggon, M. Schwab, M. Bayer, S. Fafard, Z. Wasilewski, and P. Hawrylak, “Exciton dephasing in quantum dot molecules,” Phys. Rev. Lett. **91**, 267401 (2003). [CrossRef]

**38. **A. S. Bracker, M. Scheibner, M. F. Doty, E. A. Stinaff, I. V. Ponomarev, J. C. Kim, L. J. Whitman, T. L. Reinecke, and D. Gammon, “Engineering electron and hole tunneling with asymmetric InAs quantum dot molecules,” Appl. Phys. Lett. **89**, 233110 (2006). [CrossRef]

**39. **J. M. Villas-Bôas, A. O. Govorov, and S. E. Ulloa, “Coherent control of tunneling in a quantum dot molecule,” Phys. Rev. B **69**, 125342 (2004). [CrossRef]

**40. **H. S. Borges and A. Alcalde, and S. E, Ulloa, “Exchange interaction and tunneling-induced transparency in coupled quantum dots,” Phys. Rev. B **90**, 205311 (2014). [CrossRef]

**41. **G. J. Beirne, C. Hermannstädter, L. Wang, A. Rastelli, O. G. Schmidt, and P. Michler, “Quantum light emission of two lateral tunnel-coupled (In, Ga)As/GaAs quantum dots controlled by a tunable static electric field,” Phys. Rev. Lett. **96**, 137401 (2006). [CrossRef]

**42. **B. Krause, T. H. Metzger, A. Rastelli, R. Songmuang, S. Kiravittaya, and O. G. Schmidt, “Shape, strain, and ordering of lateral InAs quantum dot molecules,” Phys. Rev. B **72**, 085339 (2005). [CrossRef]

**43. **Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. **85**, 623–653 (2013). [CrossRef]

**44. **X. Q. Luo, D. L. Wang, Z. Q. Zhang, J. W. Ding, and W. M. Liu, “Nonlinear optical behavior of a four-level quantum well with coupled relaxation of optical and longitudinal phonons,” Phys. Rev. A **84**, 033803 (2011). [CrossRef]

**45. **J. Kim, S. L. Chuang, P. C. Ku, and C. J. Chang-Hasnian, “Slow light using semiconductor quantum dots,” J. Phys: Condens. Matter **16**, S3727–S3735 (2004).

**46. **A. Tackeuchi, T. Kuroda, and K. Mase, “Dynamics of carrier tunneling between vertically aligned double quantum dots,” Phys. Rev. B **62**, 1568–1571 (2000). [CrossRef]

**47. **X. Q. Luo, Z. Z. Li, J. Jing, W. Xiong, T. F. Li, and T. Yu, “Spectral features of the tunneling-induced transparency and the Autler-Townes doublet and triplet in a triple quantum dot,” Sci. Rep. **8**, 3107 (2018). [CrossRef] [PubMed]

**48. **G. T. Adamashvili, C. Weber, A. Knorr, and N. T. Adamashvili, “Optical nonlinear waves in semiconductor quantum dots: solitons and breathers,” Phys. Rev. A **75**, 063808 (2007). [CrossRef]

**49. **W. X. Yang, A. X. Chen, R. K. Lee, and Y. Wu, “Matched slow optical soliton pairs via biexciton coherence in quantum dots,” Phys. Rev. A **84**, 013835 (2011). [CrossRef]

**50. **G. Gligorić, A. Maluckov, L. Hadžievski, G. Y. Slepyan, and B. A. Malomed, “Discrete solitons in an array of quantum dots,” Phys. Rev. B **88**, 155329 (2013). [CrossRef]

**51. **Z. G. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. **75**, 086401 (2012). [CrossRef] [PubMed]

**52. **H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. **21**, 1936–1938 (1996). [CrossRef] [PubMed]

**53. **Q. C. Liu, T. F. Li, X. Q. Luo, H. Zhao, W. Xiong, Y. S. Zhang, Z. Chen, J. S. Liu, W. Chen, F. Nori, J. S. Tsai, and J. Q. You, “Method for identifying electromagnetically induced transparency in a tunable circuit quantum electrodynamics system,” Phys. Rev. A **93**, 053838 (2016). [CrossRef]

**54. **A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Approaching unit visibility for control of a superconducting qubit with dispersive readout,” Phys. Rev. Lett. **95**, 060501 (2005). [CrossRef] [PubMed]