1. Introduction

Nonlinear optical nanostructures comprise the foundation of modern nano-photonics, offering fascinating functionalities and applications ranging from integrated nonlinear optics to all-optical signal processing, nonlinear nano-sources of light, ultrafast optical switching, frequency up-conversion and optical solitons [1–5]. Acquiring pronounced nonlinear optical effects in nanostructures is an important step for greatly advancing the functionalities and applications. Low dimensional semiconductor nanostructures, such as quantum dots (QDs), quantum wells (QWs) and quantum well wires (QWWs), manifest considerable superiority in supporting pronounced optical nonlinearities, due to quantum-confined carrier, low threshold power, ultrafast time response and small footprints of semiconductor nanostructures [4,6–9]. Furthermore, low dimensional semiconductor nanostructures can be utilized for building basic blocks in nonlinear micro-nanostructures such as nonlinear plasmonic metasurfaces [10,11], nonlinear dielectric metasurfaces [12], and other nonlinear plasmonic nanostructures [2]. Therefore, nonlinear optical effects in low dimensional semiconductor nanostructures have been extensively investigated.

In recent years, there has been a growing interest in optical and electronic properties of low dimensional semiconductor nanostructures under the illumination of a terahertz field. Under the illumination of a weak terahertz field, although the confining potential of low dimensional semiconductor nanostructures is little altered, intriguing and significant physical phenomena can be achieved by designing appropriate energy level structure, such as switching from optical bistability to multistability via terahertz signal radiation in quantum dot [13], modifying the electron population in dark and bright levels of quantum dot [13], and realizingcoherent population trapping [14]. Under the illumination of an intense terahertz laser field, optical and electronic properties of low dimensional semiconductor nanostructures have received wide interest, mainly due to the forming of the laser-dressed states and the rapid development of high-power tunable laser sources in the terahertz regime [15–17]. Some exotic and intriguing physical phenomena associated with laser-driven low dimensional semiconductor nanostructures have been theoretically anticipated and experimentally observed ranging from linear stage to nonlinear stage [18–28], such as unusual Aharonov-Bohm oscillations in quantum rings [18], terahertz resonant absorption and zero-resistance states in two-dimensional electronic gas [19,20], changes in the electron density of states in QWs and QWWs via the dynamical Franz-Keldysh effect [21,22], modulating nonlinear optical effect in low dimensional semiconductor nanostructures [23,24]. The strength of the laser-dressed effects on low dimensional semiconductor nanostructures can be characterized by the laser-dressed parameter (also see the theory section) [18]. The increase of the laser-dressed parameter is feasible through increasing the optical intensity or decreasing the non-resonant frequency of the driving field [18], where the latter can avoid the onset of material damage for very high optical intensity [25]. The dressing effects of an intense terahertz laser field on low dimensional semiconductor nanostructures may be significant for constructing novel reliable lasers, optical modulators and ultra-fast infrared detectors [25]. In addition, the electronic states for low dimensional semiconductor nanostructures of fixed sizes stay almost unchanged in the absence of external field, while the application of an intense terahertz laser field allows us to flexibly tailor confinement potential configuration, electronic quantum states and optical effects in semiconductor nanostructures of fixed sizes. Achieving the tailorable electronic and optical properties in low dimensional semiconductor nanostructures is also of significance for meeting the growing demand for the integration of more functionalities in single optoelectronic circuit.

Nonlinear optical effects associated with the laser-dressed electronic states have been investigated in low dimensional nanostructures [23,25,29–35] through manipulating the laser-dressed parameter. Different from QWs and QDs, QWWs show quantum confinement effect in transversal section and allow electrons, holes or photons to moving freely along the longitudinal axis with lengths up to several micro-meters. Such unique features of quantum well wires not only bridge the quantum to classical world, but also enable us to integrate nanoscale components for electrical or optoelectronic device applications [36]. As a consequence, laser-dressed semiconductor QWWs have fascinated the nonlinear optical community. Theoretical studies on nonlinear optical effects associated with the laser-dressed electronic states of single QWW have been carried out [32–34,37]. Two closely positioned QWWs can couple with each other, resulting from the quantum tunneling, which not only shows merits in acquiring the desired electronic and optical properties such as larger nonlinear rectification [38,39], and quantum asymmetry of switching [40], but also can be made as good candidates for constructing building blocks in the fields of quantum computing [40] and quantum interference [41]. At present, the dressing effect of an intense terahertz laser field on the laterally-coupled QWWs remains still lack (see Fig. 1). Here, we will investigate electronic quantum states and optical absorption coefficients (OACs) and refractive index changes (RICs) of the LCQWWs subject to an intense terahertz laser. The confinement potential pattern, electronic quantum states as well as linear and nonlinear OACs in the LCQWWs become strongly dependent on the value of the laser-dressed parameters, and therefore, an intense terahertz laser can be proposed as an important tool for manipulating the electronic and optical performances of the LCQWWs with fixed sizes. A blue shift followed by a red shift with the increase of the laser-dressed parameter can be found for linear and nonlinear OACs as wells RICs. In contrast to the case without intense terahertz laser field, the max values of the linear, third-order nonlinear and total OACs as well as RICs can be obviously enlarged at the same frequency position by manipulating appropriate laser-dressed parameter. In addition, experiments to test some of our predictions would be most interesting. The advanced quantum-wire manufacturing technology allows us to produce single quantum wire, double quantum wires and a array of quantum wires [38,42–44], which provides the possibility to prepare the model and measure the result in our work.

 

Fig. 1. (a) Transversal section of the LQCWWs in the (x, y) plane with the center in the origin (0,0), and (b) three-dimensional schematic view of the LQCWWs.

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2. Theory section

We consider conduction-band electron confined in the LCQWWs, as illustrated in Fig. 1, where $\rho$ is the wire radius, d is the gap between the two wires, the wire direction is chosen along the z-axis, and the GaAs wire material is embedded by a Al_xGa_1−xAs ($x = 0.4$) host materials. The conduction-band electron has a free motion along the z-direction, but is significantly affected by quantum confinement and tunneling effects in the (x, y) plane. The lateral confinement potential of the LCQWWs is given by

We assume the LCQWWs to be exposure to an intense terahertz laser field with monochromatic plane wave of frequency ${\omega _\textrm{d}}$. The laser beam is non-resonant with the LCQWWs and linearly polarized along the x-axis direction. Thus, the laser field does not affect the axial motion of the electron and only modifies its transverse motion. The conduction-band electron dynamic of the LCQWWs in the (x, y) plane can be described by the time-dependent Schrdinger equation:

Equation (3) was treated with a Fourier series expansion of both $\tilde{\Phi }\textrm{(}x,y,t\textrm{)}$ and $\tilde{V}\textrm{(}x,y,t\textrm{)}$ in the framework of the non-perturbative Floquet method [47], which results in a set of coupled differential equations in coordinate space for the Floquet components of $\tilde{\Phi }\textrm{(}x,y,t\textrm{)}$. In the high-frequency limit, the set of coupled differential equations reduces to single one [26]

In the presence of the external terahertz laser field, the probability density functions of the ground and first-excited states in the LCQWWs are presented for different radius $\rho$ is given in Figure S2 (see Supplement 1). From Fig. S2, we see that both $|{\psi _0}{|^2}$ and $|{\psi _1}{|^2}$ stretch towards two sides of the LCQWWs along the x-direction and exhibit a reduction in their magnitude with increasing radius. In addition, the electron cloud of $|{\psi _0}{|^2}\,(|{\psi _1}{|^2})$ for $x > 0$ shows small overlapping with that of $|{\psi _0}{|^2}$ for $x < 0$ under larger radius. The feature lies in the reduced quantum confinement for increasing radius. In the presence of the external terahertz laser field, the probability density functions of the ground and first-excited states in the LCQWWs are presented for different distance d is given in Figure S3 (see Supplement 1). From Fig. S3, we see that the behavior of $|{\psi _0}{|^2}\,(|{\psi _1}{|^2})$ in Fig. S3 is similar to that of $|{\psi _0}{|^2}\,(|{\psi _1}{|^2})$ in Fig. S2 with increasing d, which is associated with the reduction of the coupling between the two quantum wires for increasing d.

Based on the compact density matrix approach and the iterative procedure, linear and third-order nonlinear optical absorption coefficients for a quantum system with two-energy level model are, respectively, given by [9,48, 49]

The total optical absorption coefficient is given by

Linear and third-order nonlinear refractive index changes related to dispersion properties for a quantum system with two-energy level model are, respectively, given by [48]

The total refractive index change is given by

Here, μ is the permeability of the material, ${n_r}$ is the refractive index, ${\varepsilon _r}$ is the real part of the permittivity, ${\varGamma _0}$ is the relaxation rate, I is the incident optical intensity, ${\varepsilon _0}$ is the vacuum permittivity, ${\rho _s}$ is the electron number density, $\omega$ is the incident photon frequency, ${E_{\textrm{10}}} = {E_\textrm{1}} - E_\textrm{0}^{}$ is the energy level interval between the first-excited state energy level ${E_\textrm{1}}$ and the ground state energy level $E_\textrm{0}^{},\,{M_{\textrm{10}}} = \;\mathrm{\ < }{\psi _\textrm{1}}\textrm{|}x\textrm{|}{\psi _\textrm{0}}\mathrm{\ > }$ is the dipole transition matrix element between the ground state wave function ${\psi _0}$ and the first-excited state wave function ${\psi _1}$, and ${M_{ii}}\textrm{ = }\;\mathrm{\ < }{\psi _i}\textrm{|}x\textrm{|}{\psi _i}\mathrm{\ > }\;\textrm{(}i\textrm{ = 0, 1)}$.

3. Results and discussion

In this section, we will discuss the dressed-effect of the terahertz laser field on electronic and optical properties of the laterally-coupled quantum well wires through manipulating the laser-pressed parameter ${\alpha _0}$. The parameters adopted here are as follows [37]: ${V_\textrm{0}} = \textrm{0}\textrm{.35 eV,}\,{m^\ast } = \textrm{0}\textrm{.067}{m_\textrm{0}}\,({m_0}$ is the free electron mass), ${\rho _s} = \textrm{3} \times \textrm{1}{\textrm{0}^{\textrm{22}}}{\textrm{m}^{ - \textrm{3}}}$ and ${\Gamma _\textrm{0}} = \textrm{1}/\textrm{1}\textrm{.4}\;\textrm{ps}$.

In Fig. 2, we present $E_\textrm{0}^{},\,E_\textrm{1}^{}$ and ${E_{\textrm{10}}}$ of the TCQQWs as a function of ${\alpha _0}$. One finds from the figure that both $E_\textrm{0}^{}$ and $E_\textrm{1}^{}$ exhibit an increase for increasing ${\alpha _0}$, but ${E_{\textrm{10}}}$ features an increase followed by a decrease with the increment in ${\alpha _0}$, and the max value of ${E_{\textrm{10}}}$ is reached at ${\alpha _0}\textrm{ = }5.29\textrm{ nm}$. The origin for the features in Fig. 2 can be traced back to the variation of the laser-dressed potential $\bar{V}$ with ${\alpha _0}$.

 

Fig. 2. The ground state energy level $E_\textrm{0}^{}$, the first-excited state energy level $E_\textrm{1}^{}$ and the energy level interval $E_{\textrm{10}}^{}$ of the LCQQWs as a function of the laser-dressed parameters ${\alpha _0}$ with $\rho \textrm{ = 4 nm}$ and $d\textrm{ = 1 nm}$.

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A three-dimensional profile of the laser-dressed confinement potential $\bar{V}$ of the LCQWWs for several typical values of the laser-dressed parameter ${\alpha _0}$ is given in Fig. 3. For further showing the details for Fig. 3, the profile of ${\alpha _0}$ along the cross planes of $y \textrm{ = }0,\textrm{ 2 nm}$ is given in Fig. 4. It is observed from Fig. 3 that the shape of confining potential of the LCQWWs is not affected and exhibits two closely cylinder configuration in the absence of terahertz laser field (${\alpha _0} = 0$), while in the presence of intense terahertz laser field, the shape of confining potential of the LCQWWs is greatly tailored, which will alter the electron quantum states. The similar phenomenon was also observed in other quantum structures [18,50]. It is clearly observed from Figs. (3, 4) that increasing the laser-dressed parameter ${\alpha _0}$ induces a pronounced change in the laser-dressed confinement potential $\bar{V}$ of the LCQWWs. For single cylinder quantum well wire, increasing the laser-dressed parameter will stretch the upper part and the lower part of the potential well for terahertz laser field polarized along the x-axis [25]. For the LCQWWs, increasing the laser-dressed parameter not only exhibits the feature for single cylinder quantum well wire, but also makes the confining potential of LCQWWs more complicated along with new shape and characteristic such as the appearance of muti-well configuration from to double well configuration to quadruple well configuration. For the case of ${\alpha _0}\textrm{ = }2,\textrm{ 4 nm}$, the profile of $\bar{V}$ exhibits a double well configuration, while for the case of ${\alpha _0}\textrm{ = }6,\textrm{ 8 nm}$, the profile of $\bar{V}$ exhibits a triple (${\alpha _0}\textrm{ = 8 nm}$) or quadruple (${\alpha _0}\textrm{ = 6 nm}$) well configuration with shallow secondary wells in the lower part of $\bar{V}$. The continuous evolution of ${E_0},\,{E_1}$ and ${E_{10}}$ with ${\alpha _0}$ in Fig. 2 can be qualitatively interpreted as follows. With ${\alpha _0}$ increasing from 2 to 4 nm, the upper and lower parts in the well regime of $\bar{V}$ tend to be wider and narrower along x-direction, respectively. Physically, the broadening of the upper part in the well regime of $\bar{V}$ will reduce the electronic quantum confinement regime, while the reduction of the upper part in the well regime of $\bar{V}$ will enhance the electronic quantum confinement regime. With ${\alpha _0}$ increasing from 2 to 4 nm, the shrinking of the well size along x-direction in the lower part of $\bar{V}$ will increase $E_\textrm{0}^{},\,E_\textrm{1}^{}$, and $E_{\textrm{10}}^{}$, while the enlarging of the well size along x-direction in the upper part of $\bar{V}$ will decrease ${E_0},\,{E_1}$ and $E_{\textrm{10}}^{}$. Owing to both $E_\textrm{0}^{}$ and $E_\textrm{1}^{}$ located in the lower part of $\bar{V}$ (see Fig. 4), the former relative to the latter predominates over determining the increase of ${E_0},\,{E_1}$ and $E_{\textrm{10}}^{}$ with ${\alpha _0}$ varying from 2 to 4 nm, which also elucidates the increase of ${E_0},\,{E_1}$ and $E_{\textrm{10}}^{}$ for ${\alpha _0} \le \textrm{ }5.29\textrm{ nm}$ in Fig. 2. With ${\alpha _0}$ increased from 6 to 8 nm, both $E_\textrm{0}^{}$ and $E_\textrm{1}^{}$ shift toward higher energy regions, which is mainly associated with the two competing physical mechanisms: the raising in the bottom of $\bar{V}$ and the enlargement of the well size along x-direction in the upper part of $\bar{V}$. The raising in the bottom of $\bar{V}$ will bring about the increase of both $E_\textrm{0}^{}$ and $E_\textrm{1}^{}$, while the enlargement of the well size along x-direction in the upper part of $\bar{V}$ will reduce quantum confinement, leading to the decrease of both $E_\textrm{0}^{}$ and $E_\textrm{1}^{}$. It is apparent that the raising in the bottom of $\bar{V}$ predominates over determining the increase of both $E_\textrm{0}^{}$ and $E_\textrm{1}^{}$ with ${\alpha _0}$ varying from 6 to 8 nm, as a result of both $E_\textrm{0}^{}$ and $E_\textrm{1}^{}$ located in the upper part of $\bar{V}$ (see Fig. 4), which also determines the increase of both $E_\textrm{0}^{}$ and $E_\textrm{1}^{}$ for ${\alpha _0} > \textrm{ }5.29\textrm{ nm}$ in Fig. 2. However, the enlargement of the well size along x-direction in the upper part of $\bar{V}$ plays a significant role in determining the decrease of $E_{\textrm{10}}^{}$ with ${\alpha _0}$ varying from 6 to 8 nm, which also leads to the decrease of $E_{\textrm{10}}^{}$ for ${\alpha _0} > \textrm{ }5.29\textrm{ nm}$ in Fig. 2.

 

Fig. 3. Three-dimensional profile of the laser-dressed confinement potential $\bar{V}$ of the LCQQWs for different values of the laser-dressed parameter ${\alpha _0}$, where the color gradient plot of the confinement potential is presented in the bottom plane. The case for ${\alpha _0} = 0$ corresponds to the absence of terahertz laser field. The results are presented for $\rho \textrm{ = 4 nm}$ and $d\textrm{ = 1 nm}$.

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Fig. 4. Laser-dressed confinement potential along the cross planes of y = 0, 2 nm (solid lines, corresponding to the x-axis) and energy levels (dots, corresponding to the ${\alpha _0}\textrm{ - }$ axis) for different values of the laser-dressed parameter ${\alpha _0}$. The case for ${\alpha _0} = 0$ is denoted by the black dashed lines, which corresponds to the absence of terahertz laser field. The results are presented for $\rho \textrm{ = 4 nm}$ and $d\textrm{ = 1 nm}$.

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Laser manipulation on the electronic confinement potential of the LCQQWs offers the possibility to tune the quantum states in the LCQWWs, thereby acquiring the desired quantum states. In Fig. 5, the probability density functions of the ground and first-excited states in the LCQWWs are presented for different laser-dressed parameter ${\alpha _0}$. We see from Fig. 5 that in the presence of intense terahertz laser field ${\alpha _0} \ne 0$, the shape of the electron cloud (the probability density function) is distorted in contrast to the case for the absence of intense terahertz laser field (${\alpha _0} = 0$). For small values of the laser-dressed parameter such ${\alpha _0} = 2\textrm{ nm}$, the probability density function suffers only a slight distortion compared with the case for ${\alpha _0} = 0$, while for large values of the laser-dressed parameter such as ${\alpha _0} = 4, 6, 8~\textrm{nm}$, the probability density function demonstrates dramatic variation in contrast to the case for ${\alpha _0} = 0$. It is also observed from Fig. 5 that both $|{\psi _0}{|^2}$ and $|{\psi _1}{|^2}$ spread symmetrically towards two sides of the LCQWWs along the x- direction and exhibit a reduction in their magnitude with the increase of ${\alpha _0}$. This is mainly attributed to the fact that the enlargement of the well size along x-direction in the upper part of $\bar{V}$ and the increase of both ${E_0}$ and ${E_1}$ allows the electron to acquire the ability for entering into the barrier regime at both sides of the LCQWWs. It should be noted that the extension of both $|{\psi _0}{|^2}$ and $|{\psi _1}{|^2}$ toward both sides of the LCQWWs is inapparent for small ${\alpha _0}$, but is significant for large ${\alpha _0}$. Furthermore, for the ground state, the electron cloud of $|{\psi _0}{|^2}$ for $x > 0$ shows more overlapping with that of $|{\psi _0}{|^2}$ for $x < 0$ under larger ${\alpha _0}$, which can be elucidated as follows. With the increase of ${\alpha _0}$, $|{\psi _0}{|^2}$ can more easily penetrate into the center barrier. The physical origin can be largely tracked back to the fact that both the higher electron level and the lowering of the center barrier height allows the electron to penetrate into the center barrier more easily under larger ${\alpha _0}$, where the center barrier collapses into a shallow quantum well for larger ${\alpha _0}$ such as ${\alpha _0}\textrm{ = 8 nm}$. Therefore, one observes the transition from the two peaks exhibited in $|{\psi _0}{|^2}$ for ${\alpha _0}\textrm{ = }2,\textrm{ 4 nm}$ into the single peak exhibited in $|{\psi _0}{|^2}$ at the center barrier for ${\alpha _0}\textrm{ = }6,\textrm{ 8 nm}$ (see the highlight regimes in Fig. 5(a)). For the first-excited state, the electron cloud of $|{\psi _1}{|^2}$ for $x > 0$ shows smaller overlapping with that of $|{\psi _1}{|^2}$ for $x < 0$ larger ${\alpha _0}$. Therefore, two peaks shown in $|{\psi _1}{|^2}$ is still maintained and is increasingly separated from each other with increasing ${\alpha _0}$ (see the highlight regimes in Fig. 5(b)). In addition, to assure the robustness of the features for $|{\psi _0}{|^2}$ and $|{\psi _1}{|^2}$, for the case of $d\textrm{ = }2\textrm{ nm}$, the probability density functions of the ground and first-excited states in the LCQWWs for different laser-dressed parameter ${\alpha _0}$ is provided in Figure S4 (see Supplement 1). We can see from Fig. S4 and Fig. 5 that the behavior of $|{\psi _0}{|^2}\,(|{\psi _1}{|^2})$ in Fig. S4 is similar to that of $|{\psi _0}{|^2}\,(|{\psi _1}{|^2})$ in Fig. 5 in shape with the increase of ${\alpha _0}$. The only difference between them is that the coupling between the two well wires is further reduced for larger distance d in Fig. S4 relative to Fig. 5.

 

Fig. 5. The probability density function $|{\psi _\textrm{0}}{|^2}$ (a) of the ground state and the probability density function $|{\psi _\textrm{1}}{|^2}$ (b) of the first-excited states in the LCQWWs for four values of the laser-dressed parameter ${\alpha _0}$ with $\rho \textrm{ = 4 nm}$ and $d\textrm{ = 1 nm}$.

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In Fig. 6, the linear, third-order nonlinear and total OACs of the LCQWWs are presented for different values of the laser-dressed parameter ${\alpha _0}$. As can be seen from Fig. 6, the resonant peak positions of the linear, third-order nonlinear and total OACs exhibit a non-monotonic behavior from blue shift to red shift with the increase of ${\alpha _0}$. A blue shift followed by a red shift for the OACs can be interpreted using the variation of $E_{\textrm{10}}^{}$ with ${\alpha _0}$ (see Fig. 2), due to resonant enhancement of the OACs at $\hbar \omega \textrm{ = }E_{\textrm{10}}^{}$ [44]. The physical reason for the variation of $E_{\textrm{10}}^{}$ with ${\alpha _0}$ has been analyzed using (3, 4). In contrast to the case for ${\alpha _0}\textrm{ = }\;0$,the peak values of the linear, third-order nonlinear and total OACs can be obviously enlarged at the same frequency position by choosing appropriate laser-dressed parameter (here, around ${\alpha _0}\textrm{ = }\;7.5\;\textrm{nm}$). Moreover, the peak values of the linear OACs demonstrate an increase followed by a decrease for increasing ${\alpha _0}$, the origin for which is dependent on the variation of ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{2}}$ with ${\alpha _0}$ through analyzing Eq. (10) (see Fig. 7). The peak values of the third-order OACs exhibit a continuous increase for increasing ${\alpha _0}$, the origin for which is dependent on the variation of ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{4}}$ with ${\alpha _0}$ through analyzing Eq. (11) (see Fig. 7). The peak values of the total OACs exhibit a non-monotonic behavior from increase to decrease, depending on the variation of the sum of the linear and nonlinear OACs with ${\alpha _0}$. To assure the robustness of the results for the OACs, for the case of $d\textrm{ = }2\textrm{ nm}$, the linear, third-order nonlinear and total OACs of the LCQWWs for different values of the laser-dressed parameter ${\alpha _0}$ is provided in Figure S5 (see Supplement 1). It is observed that similar feature can be found in Fig. S5 and Fig. 6 with respect to the peak position and peak value of the OACs.

 

Fig. 6. Linear (black lines), third-order nonlinear (red lines) and total (blue lines) optical absorption coefficients of the LCQWWs as a function of incident photon energy $\omega$ for different values of the laser-dressed parameter ${\alpha _0}$ with $\rho \textrm{ = 4 nm,}\,d\textrm{ = 1 nm}$, and I = 0.01 MW/cm².

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Fig. 7. $\textrm{|}{M_{\textrm{10}}}\textrm{|}$ (a), ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{2}}$ and ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{4}}$ (b) of the LCQWWs as a function of the laser-dressed parameters ${\alpha _0}$ with $\rho \textrm{ = 4 nm}$ and $d\textrm{ = 1 nm}$.

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In Fig. 7(a), we plot $\textrm{|}{M_{\textrm{10}}}\textrm{|}$ of the LCQWWs as a function of ${\alpha _0}$. It can be seen from the figure that $\textrm{|}{M_{\textrm{10}}}\textrm{|}$ shows a continuous increase for increasing ${\alpha _0}$, the reason for which can be elucidated as follows. With the increment in ${\alpha _0}$, the ground and first-excited state wave-functions extend towards both sides of the LCQQWs (see Fig. 5). Therefore, the overlap between the ground and first-excited state wave functions is enlarged, making the dipole transition element be large. In Fig. 7(b), an increase of ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{2}}$ followed by its decrease for increasing ${\alpha _0}$ is observed, and the max value of ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{2}}$ is achieved at ${\alpha _0}\textrm{ = }5.99\textrm{ nm}$, which captures the evolution of the linear OACs with ${\alpha _0}$. The non-monotonic ${\alpha _0}$ dependence of ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{2}}$ in Fig. 7(b) is elucidated as follows. With the increment in ${\alpha _0}$ for ${\alpha _0}\mathrm{\ < }5.29\textrm{ nm}$, both ${E_{10}}$ and $\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{2}}$ get to be larger, thereby inducing the increase of ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{2}}$. However, with the increment in ${\alpha _0}$ for ${\alpha _0}\mathrm{\ > }5.29\textrm{ nm}$, there is a competition between the reduced ${E_{10}}$ and the increased $\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{2}}$. The increase of $\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{2}}$ takes responsible for determining the increase of ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{2}}$ for $\textrm{ }5.29\mathrm{\ nm\ < }{\alpha _0}\mathrm{\ < 5}\textrm{.99 nm}$, while the decrease of ${E_{10}}$ plays a significant role in the decrease of ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{2}}$ for $\textrm{ }{\alpha _0}\mathrm{\ > 5}\textrm{.99 nm}$. Figure 7(b) also shows a continuous increase of ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{4}}$ for increasing ${\alpha _0}$, which determines the evolution of the peak values of third-order nonlinear OACs with ${\alpha _0}$. With the increment in ${\alpha _0}$ for ${\alpha _0}\mathrm{\ < }5.29\textrm{ nm}$, the increased both ${E_{10}}$ and $\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{4}}$ results in the increase of ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{4}}$. However, with the increment in ${\alpha _0}$ for ${\alpha _0}\mathrm{\ > }5.29\textrm{ nm}$, the increased ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{4}}$ is dominating the increment in ${E_{\textrm{10}}}\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{4}}$.

In Fig. 8, the linear, third-order nonlinear and total RICs of the LCQWWs are presented for different values of the laser-dressed parameter ${\alpha _0}$.The figure shows that the linear, third-order nonlinear and total RICs exhibit a blue shift followed by red shift with the increase of ${\alpha _0}$, the reason for which has same cause with the OACs in Fig. 6. In contrast to the case for ${\alpha _0}\textrm{ = }\;0$, the max value of the linear, third-order nonlinear and total RICs can be enhanced at the same frequency position by choosing appropriate laser-dressed parameter (here, around 7.5 nm). In addition, the figure shows a slow increase of the linear and nonlinear RICs followed by an obvious increase with the increase of ${\alpha _0}$.which can be interpreted as follows. The linear and nonlinear RICs are proportional to $\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{2}}$ and $\textrm{|}{M_{\textrm{10}}}{\textrm{|}^\textrm{4}}$, respectively. We see from Fig. 7(a) that $\textrm{|}{M_{\textrm{10}}}\textrm{|}$ shows a slow increase followed by an obvious increase with the increase of ${\alpha _0}$.

 

Fig. 8. Linear (black lines), third-order nonlinear (red lines) and total (blue lines) refractive index changes of the LCQWWs as a function of incident photon energy $\omega$ for different values of the laser-dressed parameter ${\alpha _0}$ with $\rho \textrm{ = 4 nm,}\,d\textrm{ = 1 nm}$ and I = 0.01 MW/cm².

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4. Conclusion

To summarize, we have studied the influences of an intense terahertz laser field on the electronic and optical properties of the LCQQWs. Increasing the laser-dressed parameter induces dramatic change of the potential in the LCQQWs, thereby allowing us to manipulate the laser-dressed quantum states for achieving the expected quantum states. Our studies provide an unambiguous picture for the evolution of the energy levels and wave functions with the laser-dressed parameter. Moreover, as the laser-dressed parameter increases, the resonant peak positions of the linear and nonlinear optical absorption coefficients of LCQQWs exhibit a non-monotonic behavior from blue shift to red shift. The peak values of the linear optical absorption coefficients feature an enhancement followed by a reduction, while the peak values of the nonlinear optical absorption coefficients exhibit an enhancement with the increment in the laser-dressed field parameter, which is well interpreted using the transition matrix element and the energy level interval. In contrast to the case without intense terahertz laser field, the linear, third-order nonlinear and total optical absorption coefficients as well as refractive index changes can be obviously enhanced at the same frequency position by choosing appropriate laser-dressed parameter. Our findings not only enrich the understanding for the evolution of the laser-dressed quantum states, but also provides an effective tool for manipulating nonlinear optical effects in LCQWWs with fixed sizes. We expect that our results will contribute to the understanding of interactions between intense terahertz laser field and low-dimensional nanostructure, accompanied with the potential applications such as optical switches, infrared photo-detectors and infrared modulators.