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Abstract
The coupling of intense laser field and electric field serves as a new method to achieve the desired electronic states, optical absorption coefficients and refractive index changes of cubic quantum dot for the first time, to the best of our knowledge. The stationary Schrödinger equation was derived and calculated by means of the Kramers-Henneberger transformation, the non-perturbative Floquet method, and the finite difference method. The energy-level anticrossing is activated by multi-physical field to transform suitable quantum states, resulting in the multiple-polarization-selective absorption and refractive index changes. The results show that ultra-wideband frequency shift and resonance enhancement characteristics of optical absorption coefficients and refractive index changes strongly depend on the laser-dressed parameter, the amplitude of electric field, and the polarization directions of the intense laser field and electric field.
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1. Introduction
As the foundation of modern nanophotonics, nonlinear optical nanostructures have been accelerating the development of integrated nonlinear optics, nonlinear nano-sources of light, and ultrafast optics [1–4]. Quantum dots, as zero-dimensional semiconductor heterostructures, manifest considerable superiority in supporting pronounced optical nonlinearities due to the significant quantum confinement effect, low threshold power, and ultrafast time response over quantum wells and quantum wires [4–10]. Particularly, cubic quantum dots (CQD) have been subjected to extensive attention for their crucial functionalities in the optical frontier fields such as high-power semiconductor lasers [11], biological detection and therapies [12,13], and optical imaging applications [14]. Therefore, acquiring pronounced nonlinear optical effects in CQD has gradually become significant research.
The electric field [15,16], magnetic field [17], and non-resonant intense laser fields (ILF) [18] have long been exploited as external perturbations to stimulate quantum system in order to realize the goal-oriented adjustment of nonlinear optical properties. An electric field may arouse the polarization of carrier distribution and the energy shift of quantum states, which is beneficial for modulating the intensity of optoelectronic devices. Recently, it is found that the max values of the optical absorption coefficients (OAC) and refractive index changes (RIC) in hollow cylindrical quantum well can be enhanced significantly at the same frequency position by an electric field [19]. Differently, the illumination of ILF motivates the confinement potential to oscillate periodically with time, resulting in the mutation of “effective” potential symmetry and strengthening the local effect of quantum states, which reveals the capacity of high tunability in the intersubband transitions [20–22]. You et al. have investigated the OAC of CQD under a terahertz laser field [23]. However, we believe that their results had statistical errors that needed to be corrected. On the other hand, it has been confirmed that the polarization-selective absorption can be achieved by adopting the anticrossing behavior of energy levels caused by the electric field and ILF [19,20]. Interestingly, different with a single physical field, multi-physics filed coupling can not only retain the advantages of a single field, but also excite new optical features. For instance, Ozturk had found that the position and magnitude of the OAC in a graded quantum well strongly depend on the laser parameter and electric field [24]. In a recent paper, it is discovered that the electric field and ILF not only can be used as a switch to control the intersubband transitions in conducting bands, but also can enhance and restore the optical properties of the coaxial cylindrical quantum well [19]. However, the previous studies on multi-physical field excitation mainly focus on one and two-dimensional quantum structures, while the research on zero-dimensional quantum dots still lacks. The dual action of electric field and ILF on zero-dimensional CQD is expected to achieve the tailorable electronic and optical properties, which has great significance for meeting the growing demand to integrate more functionalities into a single optoelectronic circuit.
Here, we will investigate the influence of the electric field and ILF on the electronic quantum states, OAC and RIC of CQD. Multi-physical field stimulates the energy-level anticrossing to exchange, warp and localize wave functions in the excited state. By manipulating appropriate laser-dressed parameter, amplitude of electric field, and their polarization directions, the rearrangement of quantum transitions results in multiple-polarization-selective absorption and refractive index change, accompanied by an ultra-wideband red shift and resonance enhancement of OAC and RIC. This adjustment mechanism can also be applied to other quantum structures.
2. Theory
In our modeling, the conduction-band electrons are tightly bound in a cubical GaAs surrounded by a wider band gap AlxGa1-xAs host materials, as illustrated in Fig. 1. The confinement potential can be expressed as [25]:
Fig. 1. Three-dimensional schematic view of CQD under ILF with laser-dressed parameter a0 and electric field F; (a) F = 100 kV/cm, a0 = 0, φ1 = 0°; (b) F = 100 kV/cm, a0 = 0, φ1 = 45°; (c) F = 0, a0 = 3 nm, φ2 = 0°; (d) F = 0, a0 = 3 nm, φ2 = 45°.
We assume that the system is exposed to a static electric field F and a non-resonant long-wavelength ILF with frequency ${\omega _L}$. The ILF is selected to be linearly polarized. With the effective mass approximation, the time-dependent Schrödinger equation in this system can be described as:
By performing Fourier series expansion on both ${\rm{\tilde \Phi }}$ and $\tilde V$ in the framework of the non-perturbative Floquet method [28–32], Eq. (6) can be deduced to a set of coupled differential equations in coordinate space for the Floquet components of ${\rm{\tilde \Phi }}$. Under the high-frequency regime, coupled differential equations can be reduced to a single system [33]:
A 3D finite difference method is introduced to numerically calculate the Schrödinger equation. After applying the central difference approximation to the derivatives, the discrete Schrödinger equation can be defined as:
Based on the compact density matrix approach and the iterative method, the linear and third-order nonlinear OAC with two-energy level model are, respectively, given by [19,20]:
The total OAC is obtained as [19,20]:
Similarly, the linear and the third-order nonlinear RIC are, respectively, expressed as [19,20]:
The total RIC is given by [19,20]:
Here, σν denotes the electron density, µ expresses the permeability, nr is the refractive index, ɛr is the real part of the permittivity, ω is the incident photon angular frequency, and $I = 2{\varepsilon _0}{n_r}{|{\hat E} |^2}$ is the optical intensity of the incident electromagnetic wave that leads to the intersubband optical transition. ${M_{ij}} = \left|{\left\langle {{{\rm{\Psi }}_i}|{e{\mathbf{r}}} |{{\rm{\Psi }}_j}} \right\rangle } \right|$ is the dipole matrix element, and ${E_{ij}} = \hbar \omega = {E_i} - {E_j}$ is the energy interval with i, j = 1, 2. In this work, we only discuss 1→2 intersubband transition.
3. Results and discussions
In this section, the parameters used in our calculations are as follows [34,35]: V0 = 228 meV, m* = 0.067 m0 (m0 is the free electron mass), x = 0.3, L = 10 nm, σν = 5.0×1024 m-3, nr = 3.2, ɛ0 = 8.85×10−12 Fm-1, Γ0 = 1/0.2 ps, I = 0.1 MW/cm2, ${\theta _1} = {\theta _2} = 90^\circ $, ${\varphi _1} = {\varphi _2} = 0^\circ \mathop \to \limits^{yields} {{\rm{Q}}_1} = 0^\circ $, ${\varphi _1} = 0^\circ \& {\varphi _2} = 45^\circ \mathop \to \limits^{yields} {{\rm{Q}}_1} = 45^\circ $, ${\varphi _1} = 0^\circ $&${\varphi _2} = 90^\circ \mathop \to \limits^{yields} {{\rm{Q}}_1} = 90^\circ $, ${\varphi _1} = 45^\circ $&${\varphi _2} = 45^\circ \mathop \to \limits^{yields} {{\rm{Q}}_2} = 0^\circ $, ${\varphi _1} = 45^\circ \& {\varphi _2} = 90^\circ \mathop \to \limits^{yields} {{\rm{Q}}_2} = 45^\circ $, and ${\varphi _1} = 45^\circ \& {\varphi _2} = 135^\circ \mathop \to \limits^{yields} {{\rm{Q}}_2} = 90^\circ $.
3.1 Influences of electric field and ILF on the electronic states
Figure 2 presents the energy levels of ground state E1, the first-excited state E2, and the second-excited state E3 in CQD as a function of F in different a0, with a0 = 0, 3, 6, and 9 nm. One can find from the Fig. 2 that E1, E2, and E3 exhibit an reduction with the increasing F whatever the values of Q1, Q2, and a0 are. Meanwhile, the reduction rate of energy levels can be further changed by regulating a0, Q1 and Q2. The origin of these features can be traced back to the variation of the confinement potential $\bar V$ with F, a0, Q1 and Q2.
The projection of confinement potential $\bar V$ onto a horizontal plane XOY for several typical values of F and a0 is given in Fig. 1 and Fig. 3. It can be explicitly found from Fig. 1 that the electric field can induce $\bar V$ to decrease linearly toward F (see Fig. 1(a-b)). However, the overall profile of $\bar V$ is not greatly affected, resulting in an almost fixed energy interval E21. ILF radiation can reinforce the electronic confinement symmetrically along the polarization direction of ILF. Meanwhile, the greater the ILF amplitude, the stronger the $\bar V$ intensification (see Fig. 1(c-d)). E1 increases with the increase of a0, while E21 decreases. Differently, the combination of electric field and ILF can not only transform the symmetry type of $\bar V$ (see Fig. 3(c, f)), but also break the symmetry in a selected direction (see Fig. 3(a, b, d, e)). The ILF radiation reinforces the sensitivity of energy levels to electric field. With the increase of F, the difference of falling rate between the excited state energy levels leads to the crossing (accidental degeneracies) and anticrossing (repulsion) of intersubbands, as shown by the pink circles in Fig. 2. The energy level roles and electron states will be exchanged at the intersection [19,20]. As illustrated in Fig. 4, the wave functions between the first excited state ${{\rm{\Psi }}_2}$ and the second excited state ${{\rm{\Psi }}_3}$ exchange after the anticrossing of E2 and E3. More interestingly, the polarization directions of electric field and ILF can determine whether quantum states exchange or not. As reflected in Fig. 2, the anticrossings of intersubbands will disappear if ${{\rm{Q}}_1} = 90^\circ $ or ${{\rm{Q}}_2} = 45^\circ $.
Fig. 3. The projection of confinement potential $\bar V$ onto a horizontal plane XOY of CQD for several typical values of Q1 and Q2 with F = 100 kV/cm and a0 = 3 nm.
Fig. 4. The wave functions of the first three subbands ${{\rm{\Psi }}_1}$, ${{\rm{\Psi }}_2}$, and ${{\rm{\Psi }}_3}$.
3.2 Influences of electric field and ILF on the OAC of CQD
It can be known from Eqs. (12-14) that the position of absorption peak can be determined by the derivative $d{\alpha ^{({1,3} )}}/d\omega = 0$ to obtain $\hbar \omega = {E_{21}}$. Specially, the optimal polarization direction of incident light that maximizes the OAC and RIC is discussed in Fig. 5. Figure 5 shows the OAC and RIC as a function of incident photon energy with different values of F and a0 in the optimal polarization of incident light. In the absence of ILF, an increase of F causes only a slight enhancement of the OAC and RIC at the same frequency position, which results from the fixed E21 we mentioned above. Differently, the resonance peaks of the OAC and RIC exhibit significant redshift with the increase a0 in the absence of F. The maximum redshifted range is about 90 meV, which is consistent with the behavior of E21 described in Section 3.1. The resonant peaks of third-order OAC α(3) are greatly enhanced with the increasing a0, which will reduce the total OAC α(ω, I) to the order of about $1 \times {10^7}{m^{ - 1}}$. It's worth noting that the OAC and RIC of X-polarized light disappear with the apply of electric field or low laser-dressed parameters a0 (see the inset “×” in Fig. 5(b)). But in the region of a0 ≥ 6 nm, the OAC and RIC of Y-polarized light and Z-polarized light disappear, while the ones of X-polarized light occur (see “√”). This transform results from the exchange of quantum states caused by anticrossing of intersubbands, which affects the transition direction from the ground state to the first excited state.
Fig. 5. The OAC and RIC as a function of incident photon energy with different values of F and a0 in the optimal polarization of incident light; “√” and “×” represent the presence and absence of the OAC and RIC of X-, Y-, and Z-polarized light, respectively.
In Fig. 6, α(1), α(3) and α(ω, I) as functions of incident photon energy are presented with different values of F, a0, Q1, and Q2 in the optimal polarization of incident light. The peak position of OAC appears ultra-wideband redshift with increase of a0. the resonant peaks of α(ω, I) remain in the order of ${10^7}{m^{ - 1}}$, which is at least an order of magnitude higher than that in spherical [36], cylindrical [37], and disk-shaped [38] quantum dots. Moreover, a further red shift of OAC can be achieved by enlarging F, which is completely different from the case of the electric field alone. It can be explained that the ILF intensifies the sensitivity of confinement potential to electric field by destroying the symmetry of $\bar V$. Notably, with the raising of Q1 and Q2, the resonant peaks of α(3) in high a0 can be significantly improved, and can be bleached in the process. The physical reason is that the distortion and localization of the wave functions expand the overlap between ground state and first excited state wave functions, resulting in the enhancement of dipole transition element.
Fig. 6. The OAC as functions of incident photon energy with different values of F, a0, Q1, and Q2 in the optimal polarization of incident light.
Figure 7 details the OAC of X-polarized light, Y-polarized light, and Z-polarized light as functions of incident photon energy with different values of F, a0, Q1, and Q2, which can be summarized into Table 1. One can find from Table 1 that the colored areas indicate that the absorption direction of linearly polarized light will transform inversely. The reason for the transform in absorption direction is the same as mentioned above. It means that the coupling of the electric field and ILF can possibly be used to distinguish among incident light polarized along the X-, Y- and Z-directions. In addition, Fig. 7 manifests that under the same conditions, the resonant peaks of the OAC of X-polarized, Y-polarized, and Z-polarized light are of the same order of magnitude as those in the optimal polarization direction. Ideally, when there are two orthogonal polarized light absorption at the same time, their peak values are almost at the same level (for instance, see curves “q” and “D” in Fig. 7(b1, c1)). The existence of the double absorption of orthogonal-polarized light is of great significance for the application of orthogonally polarized dual-wavelength passively q-switched crystals [39–42]. Besides, the double absorption of orthogonal-polarized light can be modulated into a single polarization absorption by exchanging excited state wave functions with energy-level anticrossing. To our knowledge, it is the first time that multiple polarization-selective absorption can be customized with the frame of multi-physics filed coupling.
Fig. 7. The OAC of X-polarized light (a1, a2), Y-polarized light (b1, b2), and Z-polarized light (c1, c2) as functions of incident photon energy.
Table 1. The existence form of the OAC of X-polarized light, Y-polarized light, and Z-polarized light
3.3 Influences of electric field and ILF on the RIC of CQD
Similarly, the Δn(1)/nr, Δn(3)/nr, and Δn/nr as functions of incident photon energy are described in Fig. 8 with different values of F, a0, Q1, and Q2 in the optimal polarization of incident light. The Δn(1)/nr, Δn(3)/nr, and Δn/nr reveal the ultra-wideband redshift and resonance enhancement characteristics under multi-physical field effect. The physical reason is similar to the explanation for the OAC. The maximum of total Δn/nr reaches 20, which is about one order of magnitude higher than that of spherical [36], cylindrical [37], and disk-shaped [38] quantum dots. According to Eqs. (15-17), the existence form of refractive index change of X-, Y-, and Z-polarized light is the same as that of OAC, which means that the multi-physics filed coupling can also control the multiple polarization-selective refractive index change. Therefore, by adopting the appropriate F, a0, Q1 and Q2, the RIC and phase of the incident light can be modulated at multiaspect as expected, which plays an important role in the manufacture of multi-polarization tunable optical phase modulator.
Fig. 8. The RIC as functions of incident photon energy with different values of F, a0, Q1, and Q2 in the optimal polarization of incident light.
4. Conclusion
To summarize, we have studied the influences of the ILF and electric field on the electronic states, OAC and RIC of CQD, and the results are compared with those of a electric field or ILF alone. The research provides an unambiguous picture for the evolution of energy-level anticrossing to exchange, warp and localize wave functions in the excited state manipulated by multi-physics filed coupling. By adjusting appropriate laser-dressed parameter, amplitude of electric field, and their polarization directions, multiple-polarization-selective absorption and refractive index change of CQD can be realized, accompanied by an ultra-wideband red shift and resonance enhancement of OAC and RIC. Our results not only enrich the understanding of interactions between intense laser field and electric field on low-dimensional nanostructure, but also provide an effective method to modulate nonlinear optical effects in the multiple polarization directions of incident light. We expect that our findings will be conducive to the development of optoelectronic devices such as optical switches, ultra-wideband photo-detectors and modulators.
Funding
National Natural Science Foundation of China (61935010, 62175091, 62175093, 11974146, 51872307, 51972149); Basic and Applied Basic Research Foundation of Guangdong Province (2020A1515110001); Special Project for Research and Development in Key areas of Guangdong Province (2020B090922006); Guangzhou Municipal Science and Technology Project (201904010294, 202102020949); Science and Technology Planning Project of Guangdong Province (2018B010114002).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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