In Ref. [1] , it is well established that the first complete realization of Bose-Einstein Condensation (BEC) in the history of science occurred on June 5, 1995, at MIT and the University of Colorado Boulder in the United States. In Ref. [2] , scientists discovered that silicon alloys Yb₂Si₂O₇ can successfully achieve the BEC phase. This finding represents a significant milestone in materials physics, highlighting the profound potential of silicon-based materials. It suggests that silicon, silicon alloys, and silicon topological superconductors possess either latent potential or demonstrated capability to achieve the BEC phase. The findings discussed in Refs. [3] - [25] demonstrate that the technology of Electromagnetically Induced Transparency (EIT) for light storage (quantum storage or mode locking) relies on laser light sources provided externally rather than generated internally by Yb₂Si₂O₇. By effectively applying the concept of half-value thickness from nuclear physics absorption models, it can be concluded that the ytterbium (Yb) atoms in a Bose-Einstein Condensate (BEC) may be considered absent. This also implies that single-crystal silicon ingots could be excluded at the initial stages of manufacturing. In other words, doping substrates with Yb atoms is unnecessary (as supported in Appendix A), and Yb can be eliminated entirely from the process.

This paper addresses a long-standing issue in the semiconductor industry: the challenges posed by 1-nanometer circuit nodes (quantum wires). Our work derives and verifies a solution to this critical problem. We have conducted extensive pioneering studies in this area and are now presenting them here. The focus of this paper is to explore whether silicon can achieve BEC and to conduct verification work related to this hypothesis.

Additionally, we have demonstrated that electron transport through 0.5-nanometer channels is fundamentally prohibited due to quantum mechanical scattering states. Specifically, electrons are scattered by the potential barriers within the channels. Detailed discussions and supporting evidence can be found in Appendices B and C.

We aim to ensure that these discussions align with experimental procedures and engineering realities. We confirm that this work is the same as the current BEC experiments. This is based on the properties of solid silicon crystals, which naturally play the role of a MOT, and the process of BEC laser cooling silicon is also the same as that of the current laser cooling Rb-87.

Based on the discussions in sections 2 and 3 previously, we have deduced that silicon can serve as one of the standard materials for a Fermi condensate. With appropriate transformations or technical manipulations—typically within the framework of BCS theory (e.g., effective applications of Cooper pairs)—a Fermi condensate can be sufficiently transformed into a Bose-Einstein condensate. In the manufacturing process, the intrinsic properties of silicon are preserved. Using quantum processing technologies, such as the so-called “light storage of EIT,” silicon can be advanced into a novel material known as a “silicon topological superconductor.” However, the fabrication of 0.5 nm channels is not feasible (it is explicitly prohibited). For further details on this specific case, refer to Appendices B and C.

After the completion of this paper, we would like to extend our gratitude to the exchange student from Switzerland, studying in the Department of Computer Engineering at National Yang Ming Chiao Tung University, Taiwan region, for providing valuable advice that contributed to this work.

Fundamentally, Yb₂Si₂O₇ contains 11 atoms but does not allow its 11th atom to be accommodated within 1-nanometer wires, assuming an average atomic size of 0.1 nm, i.e.,

$11\cdot \left(0.1\text{nm}\right)=1.1\text{nm}\ge 1\text{nm}$ (A.1)

Clearly, a matter of 0.1 nm cannot fit, indicating that one atom would be cut, leading to boundary effects. Such effects are not permissible in a 1-nanometer channel.

For silicon, with a lattice constant of $a=0.543\AA \approx \frac{1\text{nm}}{\text{18}\text{.4162}}$ and a face-centered cubic (F.C.C.) structure, silicon atoms can be fully accommodated without cutting. This configuration allows for a maximum of 10 silicon atoms to be arranged within 1-nanometer wires.

Restriction of Boundary Effect: Nanowires cannot permit non-integer numbers of atoms to be accommodated within a channel. As the size of the nanowire decreases, the ratio of surface atoms to the total number of atoms increases, making the “boundary effect” more pronounced [27] . At this scale, if an atom is cut, it leads to a violation of the boundary effect, which also contradicts the quantization effect. As noted in Ref. [27] , quantum size effects, quantum interference effects, and general quantization effects strongly support the arguments presented in this paper.

Remark.

Considering the atomic radius of silicon (111 pm), we next examine the constraints of a 1 nm channel. Specifically, this requires the exclusion of two silicon atoms to avoid boundary effects. Based on this, we initially obtain

$\begin{array}{l}\frac{1\text{nm}}{2\times 111\text{pm}+a}-2=\frac{{10}^{-9}\text{m}}{222\times {10}^{-13}\text{m}+0.543\AA }-2\approx 13.072-2=11.072,\\ \left[11.072\right]=11\end{array}$ (A.2)

Returning to Equation (A.1), it becomes evident that one additional silicon atom must also be excluded artificially. Consequently, the final result is:

$\frac{1\text{nm}}{2\times 111\text{pm}+a}-3=10.072,\left[10.072\right]=10$ (A.3)

This implies that only 10 silicon atoms can be accommodated within 1 nanowire. Such a reversal solution for Equation (A.1) indicates that Yb₂Si₂O₇ cannot be applied to 1 nanowire unless Yb atoms are not doped, leaving Si₂O₇.

However, even in this scenario, Yb₂Si₂O₇ still provides valuable insights (or contributes ideas) that can promote experimental advancements. These advancements may help transform silicon (Si) or Si₂O₇ into novel materials when subjected to BEC laser cooling, respectively.

Consider a special case: wave-numbers of electrons in 1D quantum wells of 0.5 nm-channels, namely 0.5 nm channel as a barrier is forbidden in physics. Consider a special case involving the wave numbers of electrons in one-dimensional (1D) quantum wells within 0.5 nm channels. Specifically:

$\kappa =\frac{2\pi }{\lambda }$ (B.1)

If the channel is contracted from 1 nm to 0.5 nm, the boundary conditions (B.C.) of the infinite potential wells change from $x=\pm L/2$ (for the 1 nm channel, used as the reference level) to $x=\pm L/4$ (for the 0.5 nm channel).With $\kappa =\pi /\lambda$ applied, this contraction produces the eigenvalues of potential energy as follows⁵:

$E=\frac{{\hslash }^2{\kappa }^2}{2m{L}^2}=\frac{{\hslash }^2}{2m}{\left(\frac{\pi /\lambda }{L/4}\right)}^2=16\frac{{\left(\pi \hslash \right)}^2}{2m}\frac{1}{{\left(\lambda L\right)}^2}$ in channels (1D-wells) (B.2)

Moreover,

$E=\frac{{\hslash }^2}{2m}{\left(\frac{4\pi }{L}\right)}^2\frac{1}{{\lambda }^2}$ (B.3)

Derive the above equation by varying $\lambda$ when an electron is input into the 0.5 nm channel from the tuning voltage source of integrated circuits

$\frac{\partial E}{\partial \lambda }=-\frac{{\hslash }^2}{m}{\left(\frac{4\pi }{L}\right)}^2\frac{1}{{\lambda }^3}<0$ with fixed $L=0.5\text{nm}$ (B.4)

And it is in the bound state due to the negative sign mentioned above. However, when taking the second derivative of the above equation by varying $\lambda$ as an electron is “quasi-standing” in the 0.5 nm channel space (regarding this part, it shall be seen in new sections (Appendix C)), so that

$\frac{{\partial }^{2}E}{\partial {\lambda }^2}=\frac{3{\hslash }^2}{m}{\left(\frac{4\pi }{L}\right)}^2\frac{1}{{\lambda }^4}\propto \frac{1}{{\lambda }^4},\Omega \equiv 4\pi$ (B.5)

With a constant $L=0.5\text{nm}$ retained and solid angles $\Omega \equiv 4\pi$ considered in a spherical coordinate system (Refer to Ref. [28] ), the actual result implies Rayleigh scattering in the case of a Fermi gas particle as the subject of interaction—provided we disregard the effects of the Pauli exclusion principle and spin (i.e., treating it as a classical particle for faster solutions). Note that the barrier, acting as the scattering center, is exactly half the width of the 1 nm channels (i.e., 0.5 nm channels). Since scattering states occur in 0.5 nm channels, we propose that it is unnecessary to produce these in physical engineering based on this section's deduction, thereby avoiding unnecessary waste. Notably, other solutions, such as destructive interference in quantum mechanics, can be explored to address this issue. Therefore with eigenfunctions:

$\begin{array}{l}{\psi }_n=\sqrt{\frac{2}{L/2}}\sin \left(\frac{n\pi x}{L/2}\right),x\ne 0,\\ {{\psi }^{″}}_n=-\sqrt{\frac{2}{L/2}}{\left(\frac{n\pi }{L/2}\right)}^2\sin \left(\frac{n\pi x}{L/2}\right),x\ne 0,\\ n=1,2,3,\cdots \end{array}$ (B.6)

The above equation implies that electrons disappear at the scattering center $\left(x=0\right)$. In other words, in the case of $x=0.05\text{nm}$,

$\begin{array}{l}{\psi }_n={\sqrt{\frac{2}{L/2}}\sin \left(\frac{n\pi x}{L/2}\right)|}_{x=L/2=0.5\text{nm}}=0,\\ n=1,2,3,\cdots \end{array}$ (B.7)

Thus, electrons are completely absent within the 0.5 nm channels $\left(x\le 0.5\text{nm}\right)$ when affected by circuit nodes. In this case, the complete quantum mechanical solution with significant allowed values is

$\begin{array}{l}{\psi }_n=\sqrt{\frac{2}{L/2}}\sin \left(\frac{n\pi x}{L/2}\right),x>0.5\text{nm}\\ {{\psi }^{″}}_n=-\sqrt{\frac{2}{L/2}}{\left(\frac{n\pi }{L/2}\right)}^2\sin \left(\frac{n\pi x}{L/2}\right),x>0.5\text{nm}\\ n=1,2,3,\cdots \end{array}\}$ (B.8)

For all space, whereas in industrial settings, the space in laboratories or fabrication facilities is considered complete.

Remark.

Note that if the scattering angle $\theta =\pi$ $\theta =\pi$ (with cylindrical coordinates chosen) possibly occurs in the 0.5 nm channel, this choice retains valid, since that

$\Omega =4\pi ={2\pi \left(1-\cos \theta \right)|}_{\theta =\pi }$ (B.9)

By applying the Heisenberg uncertainty principle, we obtain the lifetime of the quasi-standing state represented in terms of

$\Delta {\tau }_{\min }=\frac{\hslash }{2\Delta E}=\frac{\hslash m}{16}{\left(\frac{\lambda L}{\pi \hslash }\right)}^2\approx \text{3}\text{.4340}\times {\text{10}}^{\text{-36}}\sec \ll {10}^{-9}\sec$ (C.1)

Under the given conditions $L\approx \lambda$, the photon-electronic sensitive time (10⁻⁹ sec) cannot immediately match the lifetime of the system. Therefore, it can be considered that the electron is initially well-scattered, resulting in outcomes similar to those observed in other solutions. We have primarily derived energy $\lambda$ as a “twice,” caused by electron quantum tunneling. This tunneling process occurs at the boundaries of 0.5 nm channels (quantum wells), which serve as scattering centers at $r=0$.

It is important to note that the parameter L must approach the wavelengths of $\lambda$, making it necessary to achieve a Mie scattering pattern $\left(L\approx \lambda \right)$. Next, we examine f $\kappa$ for the matter-wave properties of the electron within the 0.5 nm channel. Here, $V_0$ is determined by the applied voltages in the integrated circuits, which can be manually adjusted to fine-tune the required wavelengths.

$\begin{array}{l}\kappa =\frac{2\pi }{\lambda },\\ \lambda =\frac{2\pi }{\kappa }=\frac{h}{p}=\frac{h}{\sqrt{2m_e{V}_0}}\equiv \frac{C}{\sqrt{V_0}}\end{array}$ (C.2)

When the standing wave patterns formed by electron subjects are established within the 0.5 nm channel, and $\lambda =\frac{2\pi }{\kappa }=0.5\text{nm}$ is tuned (i.e., fixed at specific points a and b):

$\begin{array}{l}\left|x_{ab}\right|=\left|x_b-x_a\right|=0.5\text{nm}\equiv L/2\\ L=1\text{nm}\end{array}\}$ (C.3)

When the wavelength is confined within the 0.5 nm channel, consider the relationship between the wavelengths and the particle size under Mie scattering conditions. Here, the scattering center (corresponding to the particle size, $L=1\text{nm}$) and the wavelengths of electron matter-waves (analogous to light wavelengths) are key factors. Associated with $\lambda =0.5\text{nm}$ and $L=1\text{nm}$, the relationship is expressed as:

$L\approx \lambda$ (C.4)

This clearly corresponds to the Mie scattering conditions outlined in Equation (C.1). Furthermore, by manually decreasing the value of $V_0$ (tuned to smaller values), a greater $\lambda$ would be applied. As a result, Mie scattering transitions into Rayleigh scattering, while maintaining a fixed $L$ (i.e., $L\gg \lambda$) to remain consistent with the prior discussions.

Associated with Equations (B.4) - (B.8) (excluding Equation (B.7)), it is evident that the results align with the sinc function. Based on Equation (B.4) and Equation (B.5), observation leads to the corresponding wavelengths of $2\lambda$ versus the phase angle $4\pi$, represented in figures of sinc functions. This observation is also consistent with Equation (B.1).

Rewriting Equation (B.4) such yields:

$\frac{\partial E}{\partial \lambda }=-\frac{{\hslash }^2}{m}{\left(\frac{4\pi }{\lambda L}\right)}^2\frac{1}{\lambda }<0$ with fixed $L=0.5\text{nm}$ (D.1)

Make the same procedure, so

$\frac{{\partial }^{2}E}{\partial {\lambda }^2}=\frac{3{\hslash }^2}{m}{\left(\frac{4\pi }{\lambda L}\right)}^2\frac{1}{{\lambda }^2}\propto \frac{1}{{\lambda }^4},\Omega \equiv 4\pi$ (D.2)

Referring to the second formulations of Equation (B.6) and Equation (B.8), we derive:

$\frac{{{\psi }^{″}}_n}{C}=-\sqrt{\frac{2}{L/2}}{C}^2\frac{\sin \left(C\right)}{C},x\ne 0$ (D.3)

And

$\frac{{{\psi }^{″}}_n}{C}=-\sqrt{\frac{2}{L/2}}{C}^2\frac{\sin \left(C\right)}{C},x>0.5\text{nm}$ (D.4)

where $C\equiv \frac{n\pi x}{L/2}$ is defined here.

It is evident that both Equation (D.3) and Equation (D.4) can be represented as sinc functions⁶.

$\frac{{{\psi }^{″}}_n}{C}=-\sqrt{\frac{2}{L/2}}{C}^2\sin c\left(C\right),x\ne 0$ (D.5)

And

$\frac{{{\psi }^{″}}_n}{C}=-\sqrt{\frac{2}{L/2}}{C}^2\sin c\left(C\right),x>0.5\text{nm}$ (D.6)

This implies that they serve as strong evidence precisely supporting the previous discussions (i.e., Equation (D.1) and Equation (D.2)). Specifically, through a step-by-step approach using mathematical physics, these results directly correspond to “Mie scattering.”

In this section, we briefly demonstrate how the general interference patterns of an incident wave degenerate into particle scattering patterns.

In case of a 0.5 nm channel, when electron matter-waves encounter a slit with a width of $a$, the diffraction patterns observed on the screen degenerate into an optical spot governed by the properties of sinc functions. This occurs due to the limiting condition with $C\to 0$ in Equation. (D.5) and Equation (D.6) (i.e., $a\approx \lambda$). In other words, when the wavelength $\lambda \cong \text{0}\text{.5}\text{nm}$ of the incident wave approaches or equals the slit width $a$, the wave-like behavior transitions into a particle-like spot. However, in earlier sections of this paper, nano-silicon particles were arranged within a 0.5 nm channel (as described in Equation (A.1), where up to five silicon atoms can be partially accommodated within this space). Thus, the slit width can be approximately equated to the lattice spacing, estimated between 0.1 nm and 0.5 nm. These values align in terms of magnitude, allowing us to conceptually replace the slits (i.e., lattice spacing) with the particles themselves. Through this reasoning, we conclude that wave-like behaviors degenerate into particle-like behaviors within the framework of quantum mechanics. This strongly supports the idea of specific scattering states, consistent with the results of the previous sections. These scattering states are well-established for a 0.5 nm channel. In summary, the work presented in this paper addresses related issues in a self-consistent and complete manner.

Using Equation (C.1), within a 1 nm channel, if one aims to confine electron gas within the wires while imposing infinite pressure (or momentum), the tunneling probability becomes zero. Thus, we derive condition

$\Delta {\theta }_{\min }=\underset{\Delta P\to \infty }{\lim }{\frac{\hslash }{2\Delta P\Delta x}|}_{\Delta x=1\text{nm}}=0$ with mode locking (F.1)

This implies that the phase retains synchronized for all electrons in silicon atoms. The technological approach undoubtedly relies on Bose-Einstein Condensate (BEC) laser cooling and Electromagnetically Induced Transparency (EIT) to achieve mode locking. To ensure Moore’s Law remains valid, the following must hold:

BEC silicon = BEC laser cooling Si + EIT (F.2)

In Kramers-Kronig relations, the dielectric constant can be written as

$\chi \left(\omega \right)={\chi }_1\left(\omega \right)+i{\chi }_2\left(\omega \right)$ (G.1)

where ${\chi }_1\left(\omega \right)$ is the real part, and ${\chi }_2\left(\omega \right)$ is the imaginary part (and ${\omega }^{\prime }\in \mathrm{Im}$).

Moreover,

${\chi }_2\left(\omega \right)=-\frac{2}{\pi }\Im {\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{\omega {\chi }_1\left({\omega }^{\prime }\right)}{{{\omega }^{\prime }}^2-{\omega }^2}\text{d}{\omega }^{\prime }}$ (G.2)

where $\Im$ means: taking imaginary parts. (The above integral must be convergence to a finite value.) In a BEC phase, obviously resonant $\omega$ keeps the same frequency, thus set $\omega \equiv 1$ [1/sec] fixed as a singular point in polar coordinates.

${\chi }_2\left(\omega \right)=\frac{2}{\pi }\Im {\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{{\chi }_1\left({\omega }^{\prime }\right)}{1-{{\omega }^{\prime }}^2}\text{d}{\omega }^{\prime }}$ (G.3)

Since that ${\omega }^{\prime }\in \mathrm{Im}$, so that

$\begin{array}{l}{\chi }_2\left(\omega \right)=\frac{2}{\pi }\Im {\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{{\chi }_1\left(i\omega \right)}{1-{\left(i\omega \right)}^2}\text{d}{\omega }^{\prime }},\\ {\chi }_2\left(\omega \right)=\frac{2}{\pi }\Im {\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{{\chi }_1\left(i\omega \right)}{{\omega }^2+1}\text{d}{\omega }^{\prime }}\end{array}$ (G.4)

Using the residue-method:

$\begin{array}{l}\frac{{\chi }_1\left(i\omega \right)}{{\omega }^2+1}=\frac{{\chi }_1\left(i\omega \right)}{2i}\left(\frac{1}{\omega -i}-\frac{1}{\omega +i}\right),\\ Re{s}_{\omega =i}f\left(\omega \right)=\frac{{\chi }_1\left(i\omega \right)}{2i}\end{array}$ (G.5)

Then due to the theorem of residues we have:

$\Im {\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{{\chi }_1\left(i\omega \right)}{{\omega }^2+1}\text{d}{\omega }^{\prime }}=\Im \left\{2\pi iRe{s}_{\omega =i}f\left(\omega \right)\right\}=\Im \left\{2\pi i\frac{{\chi }_1\left(i\omega \right)}{2i}\right\}=\pi {\chi }_1\left(\omega \right)$ (G.6)

Such that yields in Equation (G.4),

${\chi }_2\left(\omega \right)=\frac{2}{\pi }\Im {\displaystyle \underset{0}{\overset{\infty }{\int }}\frac{{\chi }_1\left(i\omega \right)}{{\omega }^2+1}\text{d}{\omega }^{\prime }}=2{\chi }_1\left(\omega \right)$ (G.7)

As far, therefore:

$\begin{array}{l}\chi \left(\omega \right)={\chi }_1\left(\omega \right)+i{\chi }_2\left(\omega \right),\\ \Im \chi \left(\omega \right)={\chi }_2\left(\omega \right)=2{\chi }_1\left(\omega \right),\\ \Re \chi \left(\omega \right)={\chi }_1\left(\omega \right),\\ {\displaystyle \underset{i=1}{\overset{2}{\sum }}{\chi }_i\left(\omega \right)}={\chi }_1\left(\omega \right)+2{\chi }_1\left(\omega \right)=3{\chi }_1\left(\omega \right),\\ {\displaystyle \underset{i=1}{\overset{2}{\sum }}{\epsilon }_{i,r}\left(\omega \right)}={\epsilon }_1\left(\omega \right)+2{\epsilon }_1\left(\omega \right)=3{\epsilon }_1\left(\omega \right)\equiv 3\epsilon {\left(\omega \right)}^{\prime }\end{array}$ (G.8)

Equation (G.8) complete fits the current solution way by using “high-κ dielectric” but the BEC laser plus EIT artificial promoted by this paper is much more advanced and affordable than it.

Notice that in Equation (G.6), the result of $\pi {\chi }_1\left(\omega \right)$ actually provides one that the Zak phase: ${\phi }_{Zk}=\pi$. Associated with this quantity and its potential optical resonance, both ensure the symmetry-protected topological (SPT) order for Equation (F.2), namely this paper’s highly-technology. In another respect, the axion-graphene system (Dr. ‘t Hooft condition 2D-graphenes with QFT-axions) reveals that the Berry curvature is zero (i.e., a plane) as shown in Equations (36)-(37) of Ref. [29] , i.e., the electrons running in 1 nm wires which represent as the nature of topological invariance (i.e., 3D topological insulators) undoubted. Moreover, the above deductions completely adhere Chern-Simon theory, since that due to Ref. [29] the Chern-numbers are 1,3,5 (gives Fractional Quantum Hall effect (FQHE) $\frac{1}{2},\frac{3}{2},\frac{5}{2}$ ) where $5/2$ denotes Dirac matters presented for Equation (F.2), and the technology provided by Equation (F.2) is a production better cheap than current graphene materials. —The big breakthrough.

Claims

Point 1: If a 0.5 nm channel is positioned at the center of scattering, it renders any manufacturing process impossible to directly implement. However, alternative approaches remain feasible. We emphasize that this paper does not endorse or support any commercial manufacturing without explicit authorization through proper purchasing agreements. Any commercial production of 1 nm chips based on the ideas presented in this paper must utilize materials developed through “silicon tomography” as outlined herein.

Point 2: While AI calculations may independently derive results similar to those presented in this paper, the intellectual priority belongs to this work. This precedence is established by our earlier efforts and contributions in advancing these findings.

Point 3: A possible issue and/or argument might be occurred at that the research done by this successful work whether if the first author’s mother-school attended it or not? The first author claims that the all work done by himself independently (of course the second author has attended) and confirms that this work without any request(ing) from National Chung Cheng University (in Taiwan region) nor obtaining any help (e.g. its equipment or instruments) from it, although the first author had ever learned there. And this point of claims is only limited in this paper.

¹In thermal equilibrium, the energy that the photon gas has, which reflects to the energy of the absorption material of silicon is $E=nhv=\frac{1}{2}\left(\gamma kT+hv\right)$ where $n>1$, $\gamma =3$. Via an approximation by large number we confirm that $T_{\gamma }\equiv \left(1+\epsilon \right)T_c$, $\epsilon \equiv 1/{10}^{\gamma }$.

²To simplify the discussion, we have adopted the use of absolute value notation to exclude complex numbers when analyzing the projection of one quantum state onto another.

³In the Fresnel approximation, supposed that a nano tunnel in 2nm, the widely-known expression as practical material of silicon, $\rho =\pm \sqrt{{\left(x_1-{x^{\prime }}_2\right)}^2+{\left(x_2-{x^{\prime }}_2\right)}^2}$, $r={\displaystyle \underset{i=1}{\overset{3}{\sum }}{x}_i}$ for the wavelengths of gamma-rays: ${\lambda }_{\gamma }\ll \rho \ll x_3$ and then $r\approx x_3+\frac{{\rho }^2}{2x_3}$, $\Omega =4\pi$ such that $r\approx 1+\frac{{\rho }^2}{2}$ and $r^{\prime }\approx 2+\frac{{\rho }^2}{4}$. Therefore $\frac{r^{\prime }}{2}=1+4\times \frac{{\rho }^2}{8}$ with ${\psi }_n=0$, this leads to corresponding: $\ln 2\to \ln 1$.

⁴In other words, it is not essential that the structures required such as Gate-SiO₂-Si Substrate to be manufactured in a semiconductor manufactures by this paper. (BEC Si plus EIT can be feasible.) Here given reasons: ${\epsilon }_r=\epsilon \left(\omega \right)/{\epsilon }_0$ and due to Equation (11), Equation (17), and Equation (18) $\omega$ in BEC is obviously coherent. Therefore, BEC resonant $\omega$ is regarded as contributing to imaginary parts of dielectric constants: ${\epsilon }_r\approx \frac{3\text{nm}}{1\text{nm}}\epsilon {\left(\omega \right)}^{\prime }=3\times \underset{{\text{SiO}}_{\text{2}}}{\underset{︸}{3.9}}\approx \underset{\text{Si}}{\underset{︸}{11.68}}$ which is compared with 3 nm (Gate-SiO₂-Si substrate) and 1nm tunnels. It also can be computed by Kramers–Kronig relations that we confirm in Appendix G.

⁵The electron Hamiltonian is given as $H=E+V\left(x\right)$, since it behaves as a free particle when affected, so that $H=E$.

⁶ $\sin c^2\left(c\right)$, exhibits extremely small differences in area relative to $\sin c\left(c\right)$, making it an effective approximation. Note that, evidently the sinc functions effectively represent quantum barriers.