1. Introduction

Graphene, a monolayer sheet of carbon atoms, exhibits intriguing optical properties that arise from its massless Dirac dispersion of electrons [1-4]. To predict the optical properties of a material such as the optical conductivity for an out-of-plane deformation in graphene, an approach based on solutions of the Dirac equation in curved space has been used [5]. From the point of view of the novel applications of graphene, a uniform external magnetic field is applied to the system in the same direction of photon propagation, and the interaction causes a rotation of a plane of polarization after photon passing through the layer. The effect depends on the strength of the external magnetic field and the thickness of a material. The angle of rotation in a single layer of graphene is remarkable because of an enormous effect [6], which is different from other material with the same thickness. By replacing the magnetic field by a nonuniform strain field which induces a pseudomagnetic field in graphene, Faraday rotation angle $\pi /8$ for suitable terahertz waves can be achieved [7]. For the theoretical description, the famous solution to solve this problem always defines the Hamiltonian of the system in terms of interaction-free and interaction Hamiltonian. It is convenient to express Hamiltonian and eigenstates of the electron in graphene in a second quantization interacting with the classical electric field. This method is called the equation of motion (EOM) [8]. For another method, a linearized collisionless quantum kinetic equation was used to the matrix distribution function that encodes $\uparrow$ and ↓ spin-density contributions in its diagonal elements and transverse spin-density contributions in its off-diagonal elements [9]. In predictions of EOM [8], Faraday’s rotation angle

For theoretical second quantization approach to the field and material interaction, many theoretical [10–13] have mainly considered the scattering properties of a single plasmon interacting with emitters as two-level quantum dots (QDs) and a V-type three-level QD. These are demonstrated that the transmission and reflection of a single plasmon can be switched on or off by controlling the detuning and changing the inter-particle distances between the QDs. The coupling between metal nanowires and quantum emitters is similar to a coupling between the polarization of light and electron in the thin-film matter.

This paper is motivated by [11]. We have investigated the quantum state of the polarization of light after passing a medium which has never been derived before. The scope of the present paper is to apply this method to describe the Faraday rotation effect in the full quantization limit by considering photon propagation in the form of the right-going and left-going photon field coupling with the uniform strained graphene and analyze in detail an alternative scheme which obtains from a new kind of stationary states solution describing Faraday rotation effect. This paper is arranged as follows. In Sec. II, we show our method that provides full second quantization describing Faraday rotation effect in the form of the right-going and left-going photon field coupling with the matter. Moreover, we propose a time-independent state of a single photon with the metal discussing an interband transition. In the next section, we consider the right and left hand circularly transmission (reflectance) for the strained graphene. The giant Faraday rotation in a single layer of graphene and uniform strained graphene can be provided. Finally, conclusions are presented in Sec. IV.

 

Fig. 1 Show a single photon propagating in the z direction with polarization in the x direction interacts with an electron in the material. The polarization of a reflected photon and a transmitted photon is rotated in Faraday effect.

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2. The right-going and left-going photon-topological insulators interaction with a full second quantization approaches

In our model, we consider a single photon, propagating in the direction perpendicular to graphene. Our method, full second quantization method is discussed. Let us start with Schrödinger equation for the right and left going photon, propagating in the z direction with polarization in the η direction, ($\eta =x,y$ is a photon polarization index) and interacting with an electron in material as Fig. 1

Following the Bernevig-Hughes-Zhang model [14–16], a tight-binding Hamiltonian in the $\overrightarrow{q}$ space is given

For the uniform strain formulation, the effects of deforming the lattice are taken into account, leading to a modification of the hopping. Modification of the hopping energies between different sites will, in turn, lead to new terms in the original Hamiltonian in the tight-binding formulation. Here the sum is over all nearest-neighbor pairs $t\left(\left|{\overrightarrow{\delta }}_{ij}\right|\right)$ having the strained hopping energy due to strain. The length and direction of the three nearest-neighbor vector ${\overrightarrow{\delta }}_i$ transform under strain according to ${\overrightarrow{\delta }}_i^{\prime }\simeq \left(1+\epsilon \right)\cdot {\overrightarrow{\delta }}_{0i}$ where ${\overrightarrow{\delta }}_{0i}$ represents a nearest-neighbor vector in the non-deformed graphene plane, and ε is the strain tensor [17],

For Cartesian strain tensor, we assume the x axis is along graphene armchair direction and the y axis is along graphene’s zigzag, respectively. In reciprocal space, the variation of hopping amplitude with inter carbon distance is $t\left(\left|{\overrightarrow{\delta }}_{ij}\right|\right)=t_0\exp [-\beta \left(\left|{\overrightarrow{\delta }}_{ij}\right|/a-1\right)]$ with $a=1.42A^o$ being the unstained nearest-neighbor separation, $t_0=2.7eV$, and $\beta \approx 3$. Then the hopping perturbation $\delta t_i$ can be obtained by expanding $t\left(\left|{\overrightarrow{\delta }}_{ij}\right|\right)$ and keeping the first order,

When the strain tensor vanish, the two velocities reduce to ${\overrightarrow{v}}_1=\left(0,v_F\right)$, ${\overrightarrow{v}}_2=\left(\tau v_F,0\right)$ and ${\overrightarrow{A}}_{pd}=0$, Eq. (11) then becomes the Dirac Hamiltonian.

For the non-uniform strained graphene [7], the lattice distortion is $\left(u_x,u_y\right)=\left(xy/R,-x^2/2R\right)$; R is the bending radius of the displacement applied, and the strain tensor is related to the displacement fields by ${\epsilon }_{xx}=y/R$, ${\epsilon }_{yy}=0$ and ${\epsilon }_{xy}=0$. The presence of a strain-induced vertical magnetic field $\left({\overrightarrow{B}}_{pd}=c\beta \hslash \pi /aR\left(0,0,1\right)\right)$ cause the cyclotron motion of the electron. A single interband transition plays a role in setting the Hall conductivity, even for zero Fermi energy and the Hall conductivity of graphene ${\sigma }_{xy}$ can be obtained as Eq. (1) of [7]. In our work, we are interested in the uniform strain (${\epsilon }_{xx}$, ${\epsilon }_{yy}$ and ${\epsilon }_{xy}$ are constants) then the uniform pseudo vector potential appears in the material and produces a zero pseudo magnetic field $\left({\overrightarrow{B}}_{pd}=0\right)$. The uniform strain only changes Fermi line from the isotropic circle to an elliptical shape due to the applying strains [18].

Following the method proposed in [10–13], Hamiltonian of the quantized electromagnetic field, moving along the z direction is rewritten as

To investigate Schrödinger for the right and left going photon, we introduce a time-independent state of the system consisting of linearly polarized photon state and a transition state in the matter

Assuming $E_{k,\pm }^{\left(\eta \right)}=E_{k,R}^{\left(\eta \right)}\pm E_{k,L}^{\left(\eta \right)}$ the linearly plus and minus fields with the η direction and applying the time independent Schrödinger equation ${\mathbf{H}}_{sys}\left|\psi \left(\overrightarrow{q},{ϵ}_k\right)〉=n_k\right|\psi \left(\overrightarrow{q},{ϵ}_k\right)〉$, we obtain the equations of motion :

Then the combination of relationship in Eqs. (21)-(22) can be expressed in the form of Maxwell’s equation of electric field

By considering the one to one interaction and having an electron transition from the conduction band to a final state in the valence band, one-photon conductivity tensor for one valley and spin ${\sigma }_{\eta ,{\eta }^{\prime }}$ is obtained in the form

Eq. (26) shows the resonance preserved in the optical conductivity tensor as a function of each value of photon energy $\hslash {\omega }_k$ which is a resonance energy eigenvalues. This optical conductivity corresponds with the optical conductivity investigated by an effective quantum field theory graphene sheet with arbitrary one-dimensional strain field from a microscopic effective low energy Hamiltonian at zero temperature [21]. The distortion of the reciprocal lattice transforms the original Dirac cone into a distorted one with a directional-dependent Fermi velocity. The Fermi line deforms from the isotropic circle into a different ellipse between two valleys, K and ${\mathbf{K}}^{\prime }$. This is due to the fact that in the presence of the strain, we have ${ϵ}_{\mathbf{K},\lambda }\left(\overrightarrow{q}\right)\ne {ϵ}_{{\mathbf{K}}^{\prime },\lambda }\left(\overrightarrow{q}\right)$. So the conductivity tensor in Eq. (26) no satisfy the relation ${\sigma }_{\eta ,{\eta }^{\prime }}^{\left(\mathbf{K}\right)}\left(\overrightarrow{q}\right)=-{\sigma }_{\eta ,{\eta }^{\prime }}^{\left({\mathbf{K}}^{\prime }\right)}\left(\overrightarrow{q}\right)$ for inequivalent valleys. Faraday’s rotations demonstrated by our approach will not cancel each other after we consider both valleys.

From Maxwell’s equation for electric field Eq. (24) and the boundary conditions at graphene with the air interface, the continuity of the tangential components of the electric field at the surface of graphene $\left(z=0\right)$ is,

To simplify the problem without loss of the generality, one can assume the polarization state of incoming photon by setting $E_{k,+}^{\left(y,i\right)}=0$. The right (left) hand circularly transmission $t_{+}\left(t_{-}\right)$ and reflections $r_{+}\left(r_{-}\right)$ coefficients are respectively given as

Optical transition matrix elements Eq. (33) contain the effects of the minimal-coupling between the light and matter that describe the possible transitions allowed by the spin selection rules [22]. The conservation of the electron momentum is in a plan of the strained graphene, and the pseudo-spin flip creates a change of the spin angular momentum of light when the transition appears between the positive and negative energy bands. Since an initial and final momentum point in the same direction and have the same magnitude; i.e., the transitions are vertical. According to Eq. (33) the maximum coupling of the photon and the spinor have to along the direction satisfying condition ${\varphi }_q-{\varphi }_{q,\eta }^{\left(s,\tau \right)}=\pm \pi /2.$ Phase shifts ${\varphi }_{q,\eta }^{\left(s,\tau \right)}$ determined by Eq. (33) give the same value for K and ${\mathbf{K}}^{\prime }$ valleys corresponding with velocities in Eqs. (12)-(13). We will discuss this result in the next section.

We want to point out the connection between polarization ratio $t_y/t_x$ and the general velocity matrix elements between the conduction band and the valence band ${\mathrm{\Pi }}^y$ and ${\mathrm{\Pi }}^x$ as

For near resonance frequency $\hslash {\omega }_k\approx △{ϵ}_{cv}$, the polarization ratio come to $\left|t_y/t_x|\approx |{\mathrm{\Pi }}^y/{\mathrm{\Pi }}^x\right|$, corresponding with the experimentally observed PL polarization ratio and the quantum mechanical pseudospin degrees of freedom in graphene [23]. Transmittance $T\left(\omega \right)$ and reflectance $R\left(\omega \right)$ can be written as

Let us consider a different between a classical electromagnetic field and a second quantization approach. We start with the scattered state, which can be expressed as $\left|out〉={\displaystyle \sum }_n{\mathbf{S}}^{\left(n\right)}\right|in〉$, where ${\mathbf{S}}^{\left(n\right)}$ is the S-matrix operator of n incident photons, whose elements can be calculated using scattering theory. For example, the one-mode one-photon S-matrix ${\mathbf{S}}^{\left(1\right)}$ can be written as

In experiments, the statistics of scattered photons is predominately determined by measuring first-order correlation $g_{\eta ,{\eta }^{\prime }}^{\left(1\right)}\left(z_1,z_2\right)$ of the scattered fields. For the transmitted beam, it is defined as

For a classical electromagnetic field approach, the first-order correlation is given as

For a quantum phenomenon, we start with a non-coupling of a single photon initial state $|in〉$ then we can obtain the scattered state as $\left|out〉=\mathbf{S}\right|in〉$, which contains many-body effects of the photon in the system. Two possible outcomes exist, Eq. (38) can be expanded as

The first term of Eq. (40) corresponds with the classical electromagnetic field first-order correlation Eq. (39). It implies that a single photon is due to a transition of an electron from the initial state in the conduction band to a possible final state in the valence band. Thus we can obtain a classical electromagnetic field approach in Faraday effect from a very large number of the photon to be in the time-independent state Eq. (18) and summing a completely random transition of electrons in the material. Next, the second term contains many photon effects, which transport through strained graphene. This effects had been considered as photon-photon correlations and entanglement generation in a one-dimensional waveguide coupled to two qubits with an arbitrary spatial separation [11, 12].

3. Calculated results

In this section, we consider the giant Faraday rotation in graphene as discussed by the exact solution of the time-independent Schrödinger equation for the light-material interaction Eq. (18). Two cases such as non-magnetic field and uniform strained graphene is moderated, deformation $<20\%$. This allows the hopping energy around its non-deformed value t, and $\partial t/\partial a\simeq -5eV$. The reciprocal space is shifted from the traditional Dirac points. The nature of the contours of the strained band structure has been discussed in [17]. For non-uniform strain, the Landau level structure is described as one modified by a uniform effective magnetic field [21] and found that the field is strongest when the force is applied in the direction perpendicular to armchair orientation. For strongly deformed lattice [24], the effect of the hopping mechanism for the next-nearest neighbors leads to anti-symmetric properties of the energy spectrum around zero energy. Furthermore, the robust deformation causes unusual dispersion relation which is linear in one direction and quadratic in the other [25, 26].

3.1. Kerr rotation and Faraday rotation-dependent on phase’s spinor and energy level

For example cases, we consider linearly-polarized light interacts with graphene for the non-magnetic field, and uniform strained graphene, the polarization state of the light will change in two ways. The first way involves the rotation of the polarization axis, and the second is a change from linearly-polarized to elliptically-polarized light that is characterized by the ellipticity. The phase (amplitude) variations between the right and left-circularly-polarized light components is responsible for the rotation, denoted by transmission $t_{+}\left(t_{-}\right)$ and reflectance $r_{+}\left(r_{-}\right)$ coefficients. We show that these characteristics correspond to the phase of spinor and energy level. The phenomenological and analytical expressions in such cases can be found in the next studies.

 

Fig. 2 Reflectance R and Kerr rotation θ_K for single electron interacting with single photon by varying q ≡ (q_x, q_y) in non-strained graphene. A red line for electron with wave vector (1,7.53) ×10⁸ m⁻¹, blackline for (5,5.72) ×10⁸ m⁻¹, blue line for (1,7.53) ×10⁷ m⁻¹ and green line for (5,5.72) ×10⁷ m⁻¹ around the K valley (τ = 1). Insets in Fig. 2(b) show the plots of imaginary part which are corresponding to the critical conditions.

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Reflectance and Kerr rotation : Refer to Eqs. (31)-(32), real parts and imaginary parts of left handed and right handed circular polarization are the keys to explain the optical properties, i.e. reflectance and Kerr rotation. Let usconsider four different wave vectors around the K valley for graphene such as, ${\mathbf{q}}_1=\left(1,7.53\right)\times {10}^8$ m${ }^{-1}$, ${\mathbf{q}}_2=\left(5,5.72\right)\times {10}^8$ m${ }^{-1}$, ${\mathbf{q}}_3=\left(1,7.52\right)\times {10}^7$ m${ }^{-1}$ and ${\mathbf{q}}_4=\left(5,5.72\right)\times {10}^7$ m${ }^{-1}$ and the energies for first two wave vector is $0.5eV$ and another two wave vectors are $0.05eV$, respectively. The corresponding phases of electronic spinors ϕ_q for each energy level are 1.44 and 0.85 radian.

The phase of spinor ϕ_q controls the characteristic curve of reflectance and photon energy $\hslash {\omega }_k$. In Fig. 2(a), there are two resonant peaks of reflectance. The first one is from $\hslash \omega$ to 0 and another one point, photon energy is equal to $△{ϵ}_{cv}$. Focusing on the low energy level in Fig. 2(a), the reflectance seems to be constant (blue and green lines) because of the merging of two resonant peaks. In higher energy level, the peaks split apparently as in the red and black lines. However, it appears that the magnitude of reflectance is independent of the energy level. For examples, As increasing the phase of pseudo-spin state ϕ_q near $\pi /2$ for high energy level and low energy level, the reflectance R reaches the maximum, according to the relation of Eq. (33). The magnitude of the curve is determined primarily by the phase of the pseudo-spin state.

In Fig. 2(b), the plots of Kerr rotation depends on the phase of spinor ϕ_q and energy level ${ϵ}_c\left(\overrightarrow{q}\right)$. The behaviors are also similar to reflectance. As increasing photon energy, there is the merging of 2 barriers for low energy level and ϕ_q near $\pi /2$ (green line). The peaks split apparently as in the red and black lines for high energy level. Another characteristic, Kerr rotation angle changes rapidly by π when photon frequency has a value around the specific frequencies. At the specific frequencies, the phase of the right (left) handed circularly reflected-photon is the same as that phase of the incident photon. The frequencies that change the polarization of reflected photon are called the critical frequencies ${\omega }_c^{(\pm )}$. The critical frequencies can be found by the conditions $\text{Im}\left[r_{+}\right]=0$ and $\text{Im}\left[r_{-}\right]=0$ or argument of right-handed $\left(r_{+}\right)$ and left-handed $\left(r_{-}\right)$ circular polarization to be zero (${\theta }_K^{(\pm )}=0$) [see Eq. (31)] as

These are valid only for $\omega >0$. From Eq. (31) and Eq. (33), the responses peaks of reflectance is invariant under the parity transformation, changing sign (positive or negative) and keeping the same magnitude of the wavevector, i.e. $\left(q_x,q_y\right)\to \left(-q_x,-q_y\right)$. Because the $r_{+}\text{ and }{r}_{-}$ are interchangeable, the reflectance and the Kerr rotation is still unchanged (symmetric) under the parity transformation.

 

Fig. 3 Transmittance T and Faraday rotation θ_F for single electron interacting with single photon by varying q ≡ (q_x, q_y) in non-strained graphene. A red line for electron with wave vector (1,7.53) ×10⁸ m⁻¹, black line (5,5.72) ×10⁸ m⁻¹, blue line (1,7.53) ×10⁷ m⁻¹ and green line (5,5.72) ×10⁷ m⁻¹ around the K valley (τ = 1). Insets in Fig. 3(b) show the plots of imaginary part which are corresponding to the critical conditions.

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Transmittance and Faraday rotation : The behavior of the transmitted photon corresponds to the reflected photon. It quite completely transparent except for the near-resonant region which implies the minimal coupling between the light and material. The lowest transmittances correspond with $\hslash {\omega }_k$ to 0 and $△{ϵ}_{cv}$ because of the maximum reflectance. When the phase of spinor decreases from $\pi /2$, by comparison, the transmittances are not too low as shown in Fig. 3(a) which the black lines are above the red lines (also for green and blue lines). The minimum transmittances are close to each other and merge for low energy level consistent with the reflectance. As an electronic energy level increasing, the transmittance split apparently (the red and black lines). This characteristic is seen in a curve of Faraday rotation too as shown in Fig. 3(b). Transmittance and Faraday rotation is invariant under the parity transformation which is agreeable with reflectance and Kerr rotation. The reasons are that deformation as shear strain, uniaxial armchair strain, and zigzag strain give the ellipse Dirac cone changing in the opposite direction of reciprocal lattice parameters against the real lattice parameters.However, the ellipse Dirac cone is invariant under the parity transformation.

3.2. Kerr rotation and Faraday rotation-dependent on strain types

Phase’s spinnor: We consider strain tensors as ${\epsilon }_{is}$ tensile isotropic strain, ${\epsilon }_{ss}$ shear strain, ${\epsilon }_{as}$ armchair uniaxial strain, and ${\epsilon }_{zs}$ zigzag strain, respectively. Here ν is Poisson’s ratio and ε is a strength of the strain. Each strain tensors is given as

 

Fig. 4 Show that transmittance T and reflectance R as a function of the phase of spinor for q = (5,3)×10⁸ m⁻¹ around the K valley (τ = 1) and Poisson’s ratio ν = 0.3 by varying strain parameter in different strain types compared with non strain case (black line). Inset in Fig. 4(b) show that the phase shift ${\varphi }_{q,\eta }^{\left(s,\tau \right)},\eta =x,y$ for shear strain type as a function of stain parameter.

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We calculate transmittance T and reflectance R as a function of the phase’s spinor for each strain type comparing with a non-strain case, evidently in Fig. 4(a). These results show that transmittances increase from non-strained-graphene (ε = 0, black line) for tensile isotropic strain (blue line) and zigzag strain (red line) with a strength of the strain parameter $\epsilon =0.1$. In the opposite case, the transmittance for armchair uniaxial strain (green line) decreases from the black line. For shear strain, the orange line is almost overlapping with the black line. As expected for reflectance, this characteristic curve is in the opposite of transmittance curve, $\left(T+R=1\right)$, see in Fig. 4(b). We consider the effects of strain on the phase shift’s spinor. From Eqs. (12)-(13) for tensile isotropic strain, zigzag strain, and armchair uniaxial strain, the phase shift’s spinor ${\varphi }_{q,y\left(x\right)}^{\left(s,\tau \right)}$ is $0\left(\pi /2\right)$ for all strain parameter, respectively. For shear strain case, the phase shift’s spinor has a slight change from the value 0 to $-\pi /40$ and in opposite from the value $-\pi /2$ to $-\pi /2+\pi /20$ when $\epsilon =0.2$, for ${\varphi }_{q,y}^{\left(s,\tau \right)}$ and ${\varphi }_{q,x}^{\left(s,\tau \right)}$ respectively, showed inset in Fig. 4(b). So the maximum and minimum point of transmittance and reflectance for each strain type are near the phase’s spinor 0, $2\pi$ (maximum) and $\pi /2$ (minimum). These results give the same values for K and ${\mathbf{K}}^{\prime }$.

 

Fig. 5 Reflectance R and Kerr rotation θ_K (in radian) for q = (5,3)×10⁸ m⁻¹ around the K valley (τ = 1) by varying strain parameter in different strain types. Using Poisson’s ratio ν = 0.3 and ε = 0 (red), ε = 0.05 (black) and ε = 0.1 (blue). Inset in Figs. 5(a), 5(c), show a little shift to the left corresponding to the energy level changing with strain.

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Fig. 6 Transmittance T and Faraday rotation θ_F (in radian) for q = (5,3)×10⁸ m⁻¹ around the K valley (τ = 1) by varying strain parameter in different strain types. sing Poisson’s ratio ν = 0.3 and ε = 0 (red), ε = 0.05 (black) and ε = 0.1 (blue). Inset in Figs. 6(a) and 6(c), show a little shift to the left corresponding to the energy level changing with strain.

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Reflectance and transmittance in strained-graphene: As applying force to graphene layer, the effects of strain on the change of the bond lengths are displayed by reflectance, transmittance, Kerr and Faraday rotation. We show the results of the optical response of 4 strain types as armchair, zigzag, shear, and isotropic strain. We use Poisson’s ration $\nu =0.3$ for armchair and zigzag strain types. The selected-wave vector is $\mathbf{q}=\left(5,3\right)\times {10}^8$ m${ }^{-1}$ and selected-strain parameters are ε = 0, $\epsilon =0.05$, and $\epsilon =0.1$. Evidently in Figs. 5-6, the peaks shift to the left for all of the strain types. They split clearly for isotropic (Figs. 5(b) and 5(d)) and zigzag types (Figs. 5(d) and 5(h)) while the splitting is much less for shear (Figs. 5(a) and 5(c)) and armchair strain types (Figs. 5(c) and 5(g)). The maximum angles are the same but a shift to the left corresponding to the energy level changing with strain. Evidently, the corresponding energies $△{ϵ}_{cv}$ of zigzag strain are $0.384$, $0.346$, and $0.313eV$ and of armchair strain are $0.384$, $0.376$, and $0.371eV$, as seen in inset of Figs. 5(a) and 5(c) for reflectance and Figs. 6(a) and 6(c) for transmittance cases of each strain parameters, respectively. The reflectances and transmittances in ^{Figs. 5 - 6} show the similar results as that in [13] dealing with transition energy. For phases of the spinor, strain parameters have a little change on the phases of the spinor such as zigzag strain type as $0.540$, $0.648$, and $0.780$ radians, and in armchair strain types as $0.540$, $0.480$, $0.417$ radians for each parameter ε, respectively.

4. Discussion and conclusions

In this paper, we showed the alternative way to consider Kerr and Faraday rotation for two-dimensional materials, which is a wave function-like description of a Dirac fermion with a second quantization approaches and presented applications of the theory and technique. We treat the incoming photon field in the second quantized form, allowing this way to obtain the classical result [8] in the limit of a large number of photons, due to a completely random transition of electrons in graphene. For the summation of the overall transition of electrons, Eqs. (22) - (23) come to a diagonal matrix and get Faraday angle to be zero. In this result, it corresponds to the relation in Eq. (1) to calculate Faraday’s rotation angle. Therefore, the Faraday effect does not occur in these predictions. However, we show that a single interband transition plays a role in understanding a single photon Faraday effect. For examples, we demonstrate the method for the cases of isotropic, shear, armchair uniaxial and zigzag strains. The Faraday rotations in the infrared regime are generated and measurable Faraday rotation angles. Reflection and transmission coefficients dependent on strain parameters and types are shown. An extremum of reflectance and transmittance exist at near the phase of spinor $\pi /2$, and the resonance peaks are found at $\hslash {\omega }_k\approx △{ϵ}_{cv}$. The Kerr rotation switches suddenly by π when the imaginary part of the reflection coefficient equals to zero (critical frequencies ${\omega }_c^{(\pm )}$). These show that the optical properties can be tuned by using strain control. To manipulate the geometrical phase of the electron by using strain control, a single photon Faraday effect can be operated as a switch to control the transmittance of the photon inside the quantum network and as a variable phase of a single photon qubit or the photon’s polarization rotation by varying strain parameter. In another way, it can be investigated by observing the Kerr and Faraday rotation process to study the band structure and the phase of the spinor of two-dimensional materials.