## Abstract

This research is a study of the photon radiation from the bilayer graphene perturbed by the electromagnetic field. Theoretically, our simulation shows vividly the asymmetry property of such bilayer graphene resulting in the outstanding attribute of the photon emission profiles. The methods employed in our work are a tight-binding model in the many-body system and Fermi’s golden rule. In this work, we show the emission profiles in various kinds such as the single-photon emission (both in linear polarization and circular polarization) and the double photon emission. Additionally, in the case of double photon emission, we illustrate the degree of entanglement between photon pairs by analyzing the certain coincident rate involved indirectly in the emission profiles. The results demonstrate that the degree of entanglement is maximum when the photon pair’s direction is perpendicular to each other (especially, in the case that one of the photons emits in the direction being perpendicular to the bilayer graphene plane). We also discover that the direction of the maximum entangled photons depends on the polarization between the photon’s pairs.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Since the advent of producing graphene in the early twenty one century, there have been various investigations on the properties of graphene, especially electronic attributes both in single layer [1–4] and bilayer [5–7]. A technique referred to as chemical vapor deposition is an advancement for synthesising high quality both single layer and bilayer graphene [8–10]. Concerning the photon absorption in graphene, there are diverse considerations whether one photon absorption [11–13] or two photon absorption [14,15]. According to the researches [16,17], demonstrating the two photon absorption theoretically and experimentally in regard to the bilayer graphene. By such explorations, the researchers discover that, in the case of double photons, the absorption is giant. On the other hand, concerning the photon emission characteristic, the experiments [18–20] report the photoluminescence emission of twisted bilayer graphene. These experiments show the results from resonant 2-photon excitation and clarify the emission peaks at van Hove singularities. In the thorough investigation, the graphene perturbed by the photon pump induces the interaction between photons and electrons inside the layer [21,22]. An Interesting result of such an interaction is entanglement of the electron’s momentum [23]. In modern quantum communication and computation, the generation of high-quality entangled photon pairs has been a significant goal. So far, the most widely used entangled photon pairs have been generated from spontaneous parametric downconversion (SPDC) [24]. This process is a nonlinear instant optical process that converts one photon of higher energy (namely, a pump photon), into a pair of photons (namely, a signal photon, and an idler photon) of lower energy. However, SPSC is intrinsically probabilistic and thus relegated to a regime of low rates of pair generation. Some researchers, therefore, propose other methods for entanglement photon generation such as employing a solid-state source [25] or symmetric GaAs quantum dots [26].

Theoretically, it is also possible to generate entanglement photons with certain perturbed graphene. Considering the photon emission from the graphene which is applied by a strong magnetic field, entangled photons are generated since the strong magnetic field causes an energy quantization in graphene to be Landau’s level [27]. Likewise, the bilayer graphene applied by a particular electrical field possibly generates entangled photons as well. A research [28] describes the photon emission in consequence of dipole-dipole interaction. Through ascertaining theoretically the interaction inside the electrically perturbed bilayer graphene, we will likely derive the photon’s emission profiles (both single photon [29] and double photons) which possess the entanglement property. In order to examine entanglement of photons, we can estimate the degree of entanglement indirectly by measuring the specific coincident rate between the evolutionary system and the unentangled system. This coincident rate implies the inverse on degree of entanglement [30].

The theoretical method of our work emphasizes a tight-binding model and many-body system. This method is useful for considering the dirac fermions [31,32] which cause, in specific conditions, the symmetry breaking of fixed mass [33]. The works [34,35] show the examples of tight binding systems in cold atoms driven by a temporal varying electrical field and higher harmonics, hence resulting in Mott insulator state. Employing Fermi’s golden rule [36], we discern vividly the scattering of the system through annihilation and creation operators. With the Hamiltonian eigen equations, we obtain the spinors and the energy of electrons in the graphene. To calculate the probability of a certain photon’s emission, we consider the transition rate from an initial state to a specific final state, hence providing the trend of evolutional states. Then, we possibly investigate the degree of entanglement concerning emission photons by considering the coincident rate. Eventually according to our theoretical results, it demonstrates evidently fundamental conservation laws which are energy and momentum conservation.

## 2. Bilayer graphene

In AB-stacked bilayer graphene, low-energy excitations which are much smaller than the vertical interlayer hopping $\gamma _{1}$ can be described by an effective $2\times 2$ Hamiltonian. For the electron energies $\mid \epsilon \mid <<\gamma _{1}=0.39$ eV, the second quantized effective Hamiltonian is given as

Following [37,38] , the single effective Hamiltonian can written as

The quantized field $\hat {\Psi }(\vec {r})$ is define as

We incorporate the electromagnetic field through the minimal coupling $\hat {\vec {p}}\to \hat {\vec {p}}-\frac {e}{c}\vec {\mathbf {A}}$, treating the new vector potential term as a quantized perturbation $\mathbf {H}_{int}$ in the full Hamiltonian

The vector potential $\vec {\mathbf {A}}$ is given by the expansion

Here $\phi _{\gamma }$ is the azimuthal angle, and $\theta _{\gamma }$ is the polar angle.

The electron-photon interaction rate $\Gamma _{c\rightarrow {v}}$ to go from an eigenstate $|\psi _{i}\rangle$ of the unperturbed electronic Hamiltonian $\mathbf {H}_{0}$ to a given final state $|\psi _{f}\rangle$ can be calculated using the standard arguments of Fermi’s Golden Rule,

whereWe calculate the electron-photon interaction rate with the simplest, tree-level Feynman diagram, the vertex connecting a photon with two electrons by using Fermi’s golden rule where $|\Psi (t)\rangle$ is the exact state that evolves from the initial state by time evolution operator. We approximate this time-evolution by considering the first order of Dyson’s series. Since the second order magnitude is proportional to the energy of the electron’s transition between conduction band and valence band, and since we consider band structure’s region where such the energy from the bilayer graphene is significantly less than from the single layer (as shown in the insert graph of Fig. 2), and since the second order has an order of magnitude $(v_{F}/c)^{2}$ that is a small number $(v_{F}/c\sim 1/300)$, we properly can approximate the time evolution state by calculating the first order only. Moreover, in the theoretical detail of the bilayer graphene, the mere first order of Dyson series exhibits the results of both single photon emission and double photon emission, whereas the first order of Dyson series, in the case of the single layer graphene, displays the result from single photon emission only. Therefore, in our research, it is appropriate and enough to consider merely the first order. Still, for the thorough analysis by considering the second order as well, there is an investigation as explained in the work [39]. If we calculate the time evolution state from the first order of Dyson series only, then

Initially, we consider two processes that create a valence electron $|\psi _{f}\rangle =|\chi _{v}(\vec {k}_v)\rangle \otimes |k_{\gamma },\widehat {\varepsilon }_{j}\rangle$ (single photon process) and $|\psi _{f}\rangle =|\chi _{v}(\vec {k}_v)\rangle \otimes |k^{(1)}_{\gamma },\widehat {\varepsilon }_{i};k^{(2)}_{\gamma },\widehat {\varepsilon }_{j}\rangle$ (double photon process), while destroying a conduction electron $|\chi _{c}(\vec {k}_c)\rangle$ and i) creating photon $|k_{\gamma },\widehat {\varepsilon }_{j}\rangle = \hat {a}^{\dagger }_{k_{\gamma },\widehat {\varepsilon }_{j}} |\textbf {vacuum} \rangle$ or ii) creating two photon $|k^{(1)}_{\gamma },\widehat {\varepsilon }_{i};k^{(2)}_{\gamma },\widehat {\varepsilon }_{j}\rangle =\hat {a}^{\dagger }_{k^{(1)}_{\gamma },\widehat {\varepsilon }_{i}}\hat {a}^{\dagger }_{k^{(2)}_{\gamma },\widehat {\varepsilon }_{j}} |\textbf {vacuum} \rangle$. Then the rate of transition for single photon and for double photon process are respectively given as $i)$ single photon process

The photon momentum satisfies energy conservation and the momentum conservation conditions,

The calculation is straightforward but quite cumbersome. We can solve the photon momentum obeying Eq. (17) in four analytical solutions that are isotropic corresponding to the band gap between conduction band and valence band. The angular matrix element for $\hat {\Delta }^{(j)}_{1}$ single and $\hat {\Delta }^{(j,j^{\prime })}_{2}$ double photon emission can be defined respectively as

The magnitude of transition is given as

Owing to the two conservation laws that are energy conservation and momentum conservation in Eq. (17), the pair of photons that emit from the bilayer intrinsically have the entanglement property. We can calculate the degree of entanglement by analysing the coincident rate [29] between the final state and unentangled state. This coincident rate define as

## 3. Radiation profiles of single and double photon emission

From now on, we will show the simulations from the equations of the previous section. First, we plot the energy band structure of bilayer graphene. Then, we will show the simulation results of photon emission (from both single photon and double photons) for each polarization. All of our simulation, we determine the numerical parameters following [37].

Through Eq. 2, we can calculate eigenvalues from this matrix as shown in Eq. (3). Then we plot the eigenvalues where they illustrate the characteristic of band structure for bilayer graphene. The simulation is shown as Fig. 2. We observe the outstanding characteristic of this band structure that there are two different Dirac points. This result is ascribed by the two kinds of electron’s motions. The first kind is the motion in the same layer and the other is between the different layers. Comparing to the single layer graphene band structure as shown in inserted graph Fig. 2, we discern that there are two different regions in the bilayer related to the single layer. At the first region, the energy gaps (between conduction band and valence band) of the bilayer are less than the single layer. On the other hand, at the other region, the energy gaps of the bilayer are more than the single layer. For the first region, we see that there is the obvious asymmetric property along the $y$-axis whereas the other region graph is rather symmetry. In detail, we see that the critical momentum between these two regions is $350 k_0$ with $k_0 = (1 \, meV)/v_F\approx 1,519 \,nm^{-1}$.

Considering the low energy gap region as demonstrated in Fig. 3, we discern the asymmetric property vividly. This characteristic implies that if the electron’s movements are different directions (despite the same momentum’s magnitude), it will induce the different characteristics of the photon’s emission and absorption. In detail, due to asymmetric band structure, the interaction between the bilayer and electrons induces a different effect in each direction of electron’s movement. Accordingly, the characteristic of a photon’s emission and absorption depends significantly on the initial direction of the electron. For this reason, the characteristic of a photon’s emission from the bilayer is obviously asymmetric whereas an emission from a single layer is symmetric. We, however, notice that even though the bilayer’s band structure is asymmetric along the $x$ axis, the band structure is symmetric along $xy$ axis (with 120 degrees related to the $x$ axis). Therefore, the interaction from bilayer affects the electrons along this $xy$ axis in both directions equally. Considering the Fig. 1, we see that the transition of electrons, in the case of inter-layer, seem asymmetry along the $x$ axis but symmetry along the $xy$ axis (with with 120 degrees related to the $x$ axis) due to hexagonal structure of the graphenes. This particular structure results in such the characteristic of band structure for the bilayer graphene. Our theoretical results conform to the experiment works as investigated in [40,41]. These works report the consequence of femion transport between gaps of layers, leading to the symmetry breaking.

#### 3.1 Single photon emission

The simulation, as shown in Fig. 4(a), demonstrates the intensity of the linear polarized photon in every direction. This simulation is derived from the integral of transition rate in Eq. 15 where we integrate overall momentum space and energy ($\vec {k}_{v}$ and $\hbar c k_{\gamma }$). The result imply the intensity form the polarization conforming to Eq. 10. According to the band structure, the photon’s intensity, from the electron’s transition in the $+x$ direction, is monotonic; the more magnitude of momentum the more intensity of photon. As shown in Fig. 4(b) and Fig. 4(d), the lowest momentum in our sample is $5k_0$ resulting in the lowest photon’s intensity. On the contrary, the highest momentum in the sample is $45k_0$ leading to the highest photon’s intensity. However, in the case of the electron’s transition in the $-x$ direction, the trend of the photon’s intensity is non-monotonic. From the sample in Fig. 4(c) and Fig. 4(e), the highest intensity does not take place where the electron’s momentum is $-45k_0$ but $-20k_0$. Considering the 3D graph in Fig. 4(a), we observe that the intensity is symmetric along the $zx$ plane. Such symmetry is due to $\hat {\epsilon }_{1}$ that there is no $z$ element. Furthermore, from the 3D simulation, we discover that the photon’s intensity in the $y$ direction is zero since we determine the initial momentum of the electron that consists of the $x$ direction only. There is, owing to the momentum conservation law, no momentum in the $y$ direction transferring to photons. Moreover, we discern from the simulation that the intensity of photons is maximum in the $z$ direction since the polarization in this direction is parallel with the $xy$ plane that is the same plane of bilayer graphene. As a theoretical analysis, the intensity of photons corresponds to the dot product between polarization $\hat {\epsilon }_{j}$ and spin pauli matrix $\vec {\sigma }$ that consists of $x$ and $y$ components only. Therefore, if polarization is parallel to the $xy$ plane, the intensity from such polarization will be maximum conforming to the simulation. Additionally, the characteristic of a photon’s emission on the $xy$ plane resembles the dipole emission and the characteristic on the $zx$ plane resembles the monopole emission.

The simulation in Fig. 5 shows the result from the integral of transition rate in Eq. 15 with the other polarization ($\hat {\epsilon }_2$) conforming to Eq. 11. In common with the former polarization, the photon’s intensity from the electron’s transition in the $+x$ direction is monotonic and in the case of the opposite direction is non-monotonic. Considering $+x$ direction from the cross section simulation, we see obviously that the less the electron momentum magnitude the less the photon intensity. From our sample, the highest electron momentum magnitude ($45k_0$) results in the maximum photon intensity and the lowest electron momentum magnitude ($5k_0$) leads to the minimum photon intensity. On the other hand, for $-x$ direction simulation, the highest electron momentum magnitude ($-45k_0$) does not necessarily result in the maximum photon intensity. Based on our sample for $-x$ direction, the intensity from the electron momentum at $-20k_0$ is more than the intensity from the electron momentum at $-45k_0$ and $-35k_0$ . This asserts that the intensity from electrons in the $-x$ direction is non-monotonic. Additionally, in contrast to previous polarization, the photon’s emission, both in the $yx$ plane and $zx$ plane, is similar to the P-dipole emission.

Figure 6 and Fig. 7 demonstrate the characteristic of the emission profiles for circular polarization. This simulation originate from the the integral of transition rate in Eq. 15 where the polarization is the combination with different phases of $\hat {\epsilon }_{1}$ and $\hat {\epsilon }_{2}$. The result shows the asymmetric characteristic along the $y$-axis. Such asymmetry is likely owing to the asymmetric band structure. We, moreover, notice that the tone of the result is similar to the energy band gaps from the band structure. Accordingly, the emission’s intensity corresponds to the energy band gaps inferring that the energy from an electron’s transition transfers to emission photons. Considering the simulation, we see that in the high electron momentum magnitude (approximately more than $35k_0$ for $+x$ and $45k_0$ for $-x$ ), the photon intensity is also high. On the other hand, the intensity from the low electron momentum magnitude (less than $35k_0$ for $+x$ and $45k_0$ for $-x$ approximately) is relatively very low with the asymmetric property along the $y$-axis. In detail, similar to linear polarization, the property for $+x$ direction is monotonic in that the less the electron momentum, the less the photon intensity. From our sample in the $+x$ direction, the maximum photon intensity takes place from the highest electron momentum ($45k_0$) and the minimum photon intensity occurs from the lowest electron momentum ($5k_0$). On the contrary, for the $-x$ direction that is non-monotonic, the more momentum magnitude does not lead to the more photon intensity. From our sample, the photon intensity from the electron momentum at $-20k_0$ is more than the intensity from the electron momentum both $-35k_0$ and $-45k_0$. In addition, We observe that the overall $3D$ simulation of circular polarization in the bilayer graphene is rather similar to the single layer graphene.

#### 3.2 Double photon emission and degree of entanglement

Figure 8 and Fig. 9 demonstrate the double photon emission profiles according to circular polarization. The Fig. 8 shows the emission from double photons which possess the same polarization and the latter figure expresses the emission from double photons which possess the opposite polarization to each other. These simulations come from the integral of the double photon emisson’s transition rate in Eq. 16 where we integrate some of momentum and energy space ($\vec {k}_v$ and $\hbar c k^{(1)}_{\gamma }$). In order to analyse vividly, we select the maximum emission where the amplitude factors are optimal where $(|\vec {k}^{(1)}_{\gamma }|= |\vec {k}^{(2)}_{\gamma }|)$. According to theoretical details, these results derive from the complicated mixture of angular matrices from linear polarization in different phases. Hence, the overall 3-D graphs look like the perplexing combination of Fig. 4 and Fig. 5. With the mixture of angular matrices, there are certain conditions, resulting in the maximum emission, that involve the two photon angles and angles of electron’s momentum in conduction band and valence band.

Let consider the Fig. 8 that is the emissions from double photons in the same circular polarization. The cross sections that are Fig. 8(b) and Fig. 8(c) show the probability where the first photon’s momentum direction is perpendicular to the bilayer graphene plane ($\phi ^{(1)}_\gamma = 0$ and $\theta ^{(1)}_\gamma = 0$) and the second photon’s momentum direction is parallel to the plane ($\theta ^{(2)}_\gamma = \pi /2$). On the other hand, Fig. 8(d) and Fig. 8(e) demonstrate the probability where the photon’s momentum directions are $\phi ^{(1)}_\gamma = 0$, $\phi ^{(2)}_\gamma = \pi /2$ and $\theta ^{(1)}_\gamma = 0$. The graphs in the left hand side are from positive electron momentum ($+x$) and the graphs in the right hand side are from negative electron momentum ($-x$). Our results from the left hand side show that the emission is maximum where electron momentum magnitude is highest ($45k_0$) implying the monotonic tendency that the more momentum magnitude the more emission intensity. On the contrary, according to results from the right hand side, the higher momentum magnitude in our sample does not always lead to the more emission intensity. As shown in Fig. 8(c) and Fig. 8(e), the intensity from the momentum at $-20k_0$ is more than the intensity from the momentum at both $-35k_0$ and $-45k_0$. Still, the intensity from $-35k_0$ is less than the intensity from $-45k_0$.

The Fig. 9 expresses the emission from double photons which have opposite circular polarization to each other. As shown in insert, the cross section that is Fig. 9(b) and Fig. 9(c) illustrate the intensity from double photon directions that are $\phi ^{(1)}_\gamma = 0$, $\theta ^{(1)}_\gamma = 0$ and $\theta ^{(2)}_\gamma = \pi /2$. Then the Fig. 9(d) and Fig. 9(e) are of the directions that are $\phi ^{(1)}_\gamma = 0$, $\phi ^{(2)}_\gamma = \pi /2$ and $\theta ^{(1)}_\gamma = 0$. According to our pattern, the left hand side results are derived from positive electron momentum ($+x$) and the right hand side results are from the opposite momentum direction ($-x$). We notice that the overall tendency of these results being from different circular polarization is rather similar to the results of the same polarization. The left hand side results show the monotonic tendency that the higher electron momentum magnitude the more the emission intensity. On the other hand, the right hand side results show the non-monotonic tendency. In summary, the overall tendency of double photon emission still resemble to the single photon emission

Following Eq. 22, such the coincident rate corresponds to the degree of entanglement since this coincident rate implies how much the particular state resembles the unentangled state. Hence the more this coincident rate, the less the degree of entanglement. Additionally, the method for deriving this coincident rate is rather similar to the transition rate with Fermi’s golden rule. We, therefore, can analyze the degree of entanglement by considering the transition rate shown indirectly by the emission profiles in the simulations. Comparing the coincident rate in Eq. 22 with the transition rate in Eqs. (15–16), we can infer that the more the emission intensity the less the degree of entanglement. On this ground, we can estimate the tendency of the entanglement’s degree by analysing the reciprocal of the emission profiles. For example, considering the Fig. 8 and Fig. 9 (the double emission profiles), if the first photon emit in the direction being perpendicular to the bilayer graphene plane, the second emission photon whose radiation’s direction parallel with the bilayer graphene plane will be the photon entangling with the first photon most. Still, even though both such the second photon in the Fig. 8 (whose polarization’s direction is same to the first photon) and such the second photon in the Fig. 9 (whose polarization’s direction is opposite to the first photon) have radiation’s direction paralleling with the bilayer graphene plane, the direction between these photon is perpendicular to each other.

## 4. Conclusion

The characteristic of the results in our research significantly originated from the asymmetry property in the band structure of the perturbed bilayer graphene. Such asymmetry property is confirmed by our simulation graphs and experiment works. According to the research [42] that considers quantum interference, the results of the two-photon absorption in the bilayer is significantly different from the results in the single layer. Through our analysing the eigenvalue of the energy in the spinor electron states, we can detect the unique characteristic of the bilayer graphene. Considering the photons emitting from the bilayer graphene, we discover that the intensity of photon emission is proportional to the energy from electron’s transition between conduction band and valence band. This tendency is caused by the energy conservation law that the energy from the transition of electrons transfers to the photons. In other words, the transition of electrons creates the emission of photons. Nevertheless, due to the asymmetry property in the bilayer graphene, the tendency of emission profile from the electron momentum in the particular direction differs from the tendency from the electron momentum in the opposite direction. The characteristic of emission profiles greatly depends on the angular matrices. In addition, by the essential conservation laws that are momentum conservation and energy conservation, the photon pairs emitting from the bilayer graphene possess the entanglement property. With the theoretical analysis between the coincident rate and transition rate of the photon emission, the degree of the entanglement can be detected indirectly by considering the inversion of the emission profiles. Based on our research, people can recognize theoretically the photon radiation profiles and the mechanism of the perturbed bilayer graphene. Besides, according to theoretical research [43,17], such bilayer graphene can be enhanced for optical absorption by microcavity. The work [44] also suggests an example to controlling the electronic structure of bilayer graphene by changing the Coulomb potential through selectively adjusting the carrier concentration in each layer. Concerning the optoelectronic application improvement, we can modify it by employing Floquet-Keldysh theory with dynamical mean field theory as demonstrated in [45,46] for semiconductors.

## Funding

Science Achievement Scholarship of Thailand; Kasetsart University Research and Development Institute.

## Acknowledgment

This research is supported by Science Achievement Scholarship of Thailand (SAST) and Kasetsart university research and development institute (KURDI), specialized center of rubber and polymer materials in agriculture and industry (RPM) and Faculty of Science, Kasetsart University for partial support.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

## 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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