Design method of low-threshold optical bistable switch based on double-layer graphene optical power system

Abstract

A low-threshold optical bistable switch design method based on a double-layer graphene light power system comprises the following steps that (1) the double-layer graphene light power system is established, the system is subjected to the combined action of pump light with the frequency being omega pu and probe light with the frequency being omega pr, and the amplitude of a pump field is Epu, and the amplitude of a probe field is Epr; the double-layer graphene of the double-layer graphene light power system is arranged in parallel, and an electric field vertically acts on a graphene film plane; 2) deducing a correlation quantity of the optical bistable characteristic; the definition formula of the dimensionless linear optical polarizability is shown as follows: 3) regulating and controlling the phonon-pumping field detuning amount delta n through the applied pumping field frequency; the phonon-exciton coupling strength g is controlled through the length of the graphene film; and the exciton-pumping detuning amount delta ex is regulated and controlled by adjusting the frequency of a pumping field. The invention provides a new method for designing low-threshold, single-channel and dual-channel optical bistable switches.

Description

Technical Field

[0001] The present invention relates to an optical bistable switch, and in particular to a design method for a low-threshold optical bistable switch based on a double-layer graphene optomechanical system. Background Art

[0002] Optical bistability, a typical nonlinear optical phenomenon, typically refers to the phenomenon in which one input state corresponds to two different output states in a nonlinear optical system. In recent years, optical bistability has been widely used in fields such as all-optical switches, optical transistors, and optical memories. As a novel two-dimensional material, graphene possesses a large nonlinear polarizability and a strong nonlinear response, making it considered an ideal material for fabricating low-threshold, tunable optical bistability devices. Currently, scholars both domestically and internationally have conducted a series of studies on the optical bistability of monolayer graphene.

[0003] Compared to single-layer graphene, bilayer graphene has higher carrier mobility and exhibits significant optical nonlinearity. This suggests that bilayer graphene has great potential for optoelectronic applications and microprocessors. However, little research has been reported on the optical bistability of bilayer graphene. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to overcome the deficiencies of the prior art and provide a low-threshold optical bistable switch design method based on a double-layer graphene optomechanical system.

[0005] To solve the above technical problems, the present invention proposes the following technical solutions: 1. A low-threshold optical bistable switch design method based on a double-layer graphene optomechanical system, comprising the following steps:

[0006] 1) Establish a double-layer graphene photomechanical system, which is subjected to a frequency of ω pu The pump light and frequency are ω pr The amplitudes of the pump field and the probe field are E pu and E pr ; The double-layer graphene of the double-layer graphene photomechanical system is set up in parallel, and the electric field Acting perpendicularly to the graphene film plane;

[0007] 2) Derivation of the relevant quantity of optical bistability; The electric field perpendicular to the graphene plane in the double-layer graphene optomechanical system in step 1) The axial symmetry of the double-layer graphene is broken, resulting in coupling between electrons and vibration modes; the two-level system of the double-layer graphene optomechanical system uses three pseudospin operators σ 01 , σ 10 and σ z to characterize;

[0008] Under the rotating wave approximation, the Hamiltonian of the system can be expressed as:

[0009]

[0010] is the pump field Rabi frequency, is the detection field Rabi frequency;

[0011] μ is the electric dipole moment of the exciton, δ=ω pr -ω pu is the probe-pump detuning, Δ n =ω n -ω pu is the G-mode phonon-pump field detuning, is the exciton-pump field detuning, ω n represents the vibration frequency of graphene;

[0012] The Hamiltonian representing the phonon energy, The Hamiltonian representing the energy of the electron-hole pair exciton, represents the phonon-exciton interaction Hamiltonian, The Hamiltonian representing the interaction between the pump light and the probe light and the phonon;

[0013] where b k + and b k denote the exciton creation and annihilation operators respectively, and the coupling strength of the G-mode phonon-electron state is g k ;

[0014] Let p = σ 01 ,ω ex k ≈ω ex ,∑ k g k b k =Ξ - ,∑ k g k b k + =Ξ + ;

[0015] Using Heisenberg's equations of motion: The following quantum Langevin equations can be obtained:

[0016]

[0017]

[0018]

[0019] Γ1 is the spontaneous emission rate of excitons, Γ2 is the dephasing rate of excitons, γ n is the electron-hole exciton relaxation rate, g is the center frequency ω of the phonon and exciton ex The coupling strength of ex =ω ex -ω pu ;

[0020] The effective first-order linear optical susceptibility is:

[0021]

[0022] (9) where ε0 is the dielectric constant of vacuum; the definition of the dimensionless linear optical polarizability is expressed as:

[0023]

[0024] Imx (1) and Reχ (1) denote the linear absorption coefficient and dispersion coefficient, respectively;

[0025] w0 is the exciton particle inversion number;

[0026] p0=iΩ pu w0(γ n +iΔ ex ) / [-(Γ2+iΔ n )(y n +iΔ ex )+g 2 w0],

[0027] p0 * =-iΩ pu w0(γ n -iΔ ex ) / [-(Γ2-iΔ n )(γ n -iΔ ex )+g 2 w0],

[0028] Ξ0=-ig 2 p0 / (γ u +iΔ ex ), p0 * =-iΩ pu w0(y n -iΔ ex ) / [-(Γ2-iA n )(γ n -iΔ ex )+g 2 w0],

[0029] Ξ0* = ig 2 p0 * / (γ n -iΔ ex ),

[0030] t1 = -i(Ω pu - Ξ0 * ) / {-Γ2 + i(Δ n + δ) + g 2 w0 / [γ n -i(δ + Δ ex )]},

[0031] t2 = (Γ1 - iδ) / {2g 2 p0 / [-γ n + i(δ + Δ ex ) - 2i(Ξ0 - Ω pu )},

[0032] t3 = -{g<00000​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​It is related to the phonon-exciton coupling strength g;

[0038] 3) Controlling the phonon-pump field detuning Δ by applying the pump field frequency n ;

[0039] The phonon-exciton coupling strength g is controlled by the length of the graphene film;

[0040] Tuning the exciton pump detuning Δ by adjusting the pump field frequency ex , usually the exciton frequency is constant.

[0041] In the above-mentioned low-threshold optical bistable switch design method based on the double-layer graphene optomechanical system, preferably, the exciton particle inversion number w0 is obtained by the third-order equation:

[0042]

[0043] In the above-mentioned low-threshold optical bistable switch design method based on the double-layer graphene optomechanical system, preferably, the parameter p1 is obtained by the following steps:

[0044] Using ansata processing: |p0|>>|p1|, |p -1 |,|w0|>>|w1|,|w -1 |,|Ξ0|>>|Ξ1|,|Ξ -1 |;

[0045] p(t)=p0+p1e -iδt +p -1 e iδt , (5)

[0046] w(t)=w0+w1e -iδt +w -1 e iδt , (6)

[0047] Ξ - (t) = Ξ0 + Ξ1e -iδt +Ξ -1 e iδt , (7)

[0048] Substituting formulas (5)-(7) into equations (2)-(4), we obtain the expression for parameter p1.

[0049] Compared with the prior art, the present invention has the advantages that: the present invention can control the phonon-pump field detuning amount Δ by applying the pump field frequency n The phonon-exciton coupling strength g is controlled by the length of the graphene film; the exciton-pump detuning Δ is regulated by adjusting the pump field frequency. exThus, an optical bistable switch is designed. The present invention provides a new method for designing low-threshold, single-channel and dual-channel optical bistable switches. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 Schematic diagram of the structure of the double-layer graphene photomechanical system in the present invention.

[0051] Figure 2 Schematic diagram of the energy levels of the double-layer graphene photomechanical system in the present invention.

[0052] Figure 3 When the phonon-exciton coupling strength g changes in Example 1, the linear absorption coefficient Im (1) With the probe-pump detuning δ pr Changing curve graph.

[0053] Figure 4 Graph showing the relationship between the distance between the two splitting peaks and the phonon-exciton coupling strength g in Example 1.

[0054] Figure 5 When Δ n =-0.2THz, 0THz and 0.2THz, the linear absorption coefficient Imχ (1) Graph showing the relationship with the phonon-exciton coupling strength g.

[0055] Figure 6 In the parameter space (Δ n ;g c ; Δ ex =0THz;Ω pu 2 =100THz 2 ) in the bistable phase diagram (where: g c0 Indicates the low threshold, g c1 indicates the high threshold).

[0056] Figure 7 is the pump field strength Ω in Example 1 pu 2 =100THz 2 50THz 2 , 100THz 2 When the linear absorption coefficient Imχ (1) Detuning amount Δ with exciton-pump field ex The changing relationship.

[0057] Figure 8 In the parameter space (Δ exc ;Ω pu 2 ; g = 0.8 THz; Δ n=0THz) in the bistable phase diagram (where: Δ ex0 Indicates the low threshold, Δ ex1 indicates the high threshold).

[0058] Figure 9 When the pump field strength Ω pu 2 =10THz 2 , 50THz 2 , 100THz 2 When the linear absorption coefficient Imχ (1) The relationship with the change of phonon-exciton coupling strength g.

[0059] Figure 10 In the parameter space (g c ;Ω pu 2 ; Δ ex =0THz;Δ n =0THz) in the bistable phase diagram (where: g c0 Indicates the low threshold, g c1 indicates the high threshold).

[0060] Legend DETAILED DESCRIPTION

[0061] In order to facilitate understanding of the present invention, the present invention will be described more comprehensively and meticulously below in conjunction with preferred embodiments, but the protection scope of the present invention is not limited to the following specific embodiments.

[0062] It should be noted that when an element is described as being "fixed, fixed, connected or communicated with" another element, it can be directly fixed, fixed, connected or communicated with the other element, or it can be indirectly fixed, fixed, connected or communicated with the other element through other intermediate connectors.

[0063] Unless otherwise defined, all technical terms used hereinafter have the same meanings as those generally understood by those skilled in the art. The technical terms used herein are only for the purpose of describing specific embodiments and are not intended to limit the scope of protection of the present invention.

[0064] In order to design a low-threshold optical bistable switch, the present invention establishes a double-layer graphene optical force system. The system is subjected to a frequency of ω pu The strong pump light and frequency are ω pr The combined effect of the weak probe light, where the amplitudes of the pump field and the probe field are E pu and E pr ,electric field Acting perpendicularly to the plane of the graphene film, such as Figure 1 As shown. The system is subjected to a frequency of ω puThe strong pump light E pu and frequency ω pr The weak detection light E pr The double-layer graphene optical force system can be equivalent to a discrete two-level state (G-mode phonon) modified by an electron cloud. The electric field perpendicular to the graphene plane The axial symmetry of the double-layer graphene is broken, resulting in coupling between electrons and vibration modes. The energy levels of the system are as follows: Figure 2 The two-level system can be represented by three pseudospin operators σ 01 , σ 10 and σ z To characterize.

[0065] Under the rotating wave approximation, the Hamiltonian of the system can be expressed as:

[0066]

[0067] Known: is the pump field Rabi frequency, is the Rabi frequency of the detection field. μ is the electric dipole moment of the exciton, δ=ω pr -ω pu is the probe-pump detuning, Δ n =ω n -ω pu is the G-mode phonon-pump field detuning, Δ ex k =ω ex k -ω pu is the exciton-pump field detuning, ω n Represents the vibration frequency of graphene. The Hamiltonian representing the phonon energy, The Hamiltonian representing the energy of the electron-hole pair exciton, represents the phonon-exciton interaction Hamiltonian, represents the Hamiltonian of the interaction between pump light and probe light and phonons. k + and b k denote the exciton creation and annihilation operators respectively, and the coupling strength of the G-mode phonon-electron state is g k Let p = σ 01 ,ω ex k ≈ω ex ,∑ k g k b k =Ξ - ,∑ k g k b k + =Ξ+ .

[0068] The commutation relation between operators is: [σ z ,σ 10 ]=σ 10 ,[σ z ,σ 01 ]=-σ 01 ,[σ 10 ,σ 01 ]=2σ z =w,[b k , b k + ]=1.

[0069] Using Heisenberg's equations of motion: The following quantum Langevin equations can be obtained:

[0070]

[0071]

[0072]

[0073] It is known that: Γ1 is the spontaneous emission rate of excitons, Γ2 is the dephasing rate of excitons, γ n is the electron-hole exciton relaxation rate, g is the center frequency ω of the phonon and exciton ex The coupling strength of Δ ex =ω ex -ω pu .

[0074] Using ansata processing: |p0|>>|p1|, |p -1 |,|w0|>>|w1|,|w -1 |,|Ξ0|>>|Ξ1|,|Ξ -1 |.

[0075] p(t)=p0+p1e -iδt +p -1 e iδt , (5)

[0076] w(t)=w0+w i e -iδt +w -1 e iδt , (6)

[0077] Ξ - (t) = Ξ0 + Ξ1e -iδt +Ξ -1 e iδt , (7)

[0078] Substituting formulas (5)-(7) into equations (2)-(4), we can obtain the expression of parameter p1:

[0079]

[0080] The effective first-order linear optical susceptibility is:

[0081]

[0082] Where ε0 is the dielectric constant of vacuum. The definition of dimensionless linear optical polarizability can be expressed as:

[0083]

[0084] The exciton population inversion w0 is obtained from the following third-order equation:

[0085]

[0086] Imx (1) and Reχ (1) Represent the linear absorption coefficient and dispersion coefficient respectively, and the relevant parameters are: p0=iΩ pu w0(y n +iΔ ex ) / [-(Γ2+iΔ n )(γ n +iΔ ex )+g 2 w0],Ξ0=-ig 2 p0 / (γ n +iΔ ex ), p0 * =-iΩ pu w0(γ n -iΔ ex ) / [-(Γ2-iΔ n )(y n -iΔ ex )+g 2 w0],Ξ0 * =ig 2 p0 * / (γ n -iΔ ex ), t1=-i(Ω pu -Ξ0 * ) / {-F2+i(Δ n +δ)+g 2 w0 / [γ n -i(δ+Δ ex )]}, t2=(Γ1-iδ) / {2g 2 p0 / [-γ n +i(δ+Δex )]-2i(Ξ0-Ω pu )}, t3=-{g 2 p0 * / [-y n +i(δ-Δ ex )]+i(Ξ0 * -Ω pu )} / {g 2 p0 / [-y n +i(δ+Δ ex )]-i(Ξ0-Ω pu )}, t5=-i / {g 2 p0 / [-γ n +i(δ+Δ ex )]-i(Ξ0-Ω pu )}, t6={-Γ2+i(δ-Δ n )-g 2 w0 / [-γ n +i(δ-Δ ex )]} / i(Ω pu -Ξ0), t8=-1 / (Ω pu -Ξ0).

[0087] From the linear absorption coefficient equation, it can be seen that the optical bistability is related to the phonon-pump field detuning Δ n , exciton-pump field detuning Δ ex , pump field strength Ω pu 2 It is related to the phonon-exciton coupling strength g. Phonon-pump field detuning Δ n The phonon-exciton coupling strength g can be controlled by the length of the graphene film. ex It can be controlled by adjusting the pump field frequency, and the exciton frequency is usually constant.

[0088] Example 1

[0089] The system of this embodiment is a double-layer graphene optomechanical system, and the specific parameters are as follows: phonon decay rate Γ1 = 2 THz, phonon dephasing rate Γ2 = Γ1 / 2 = 1 THz; exciton relaxation rate γ n =0.015THz; exciton electric dipole moment μ = 40D; frequency of G-mode phonon ω n =298.7THz.

[0090] First, under the action of a strong pump field, the linear absorption coefficient Imχ (1) The intrinsic correlation between the phonon-exciton coupling strength g. Select the pump field strength Ωpu 2 =100THz 2 , exciton-pump field detuning Δ ex =0THz, phonon-pump field detuning Δ n =0THz. Figure 3 As shown, the linear absorption line is about the central axis δ pr =0THz is symmetrically distributed, and the spectrum line is located at the detection-pump detuning amount δ pr =0THz peak will split. Figure 3 The middle illustration is a partial enlarged view of the two splitting peaks. In order to further explore the distance between the two splitting peaks and the size of the coupling factor, this embodiment draws Figure 4 ; Figure 4 Medium pump field strength Ω pu 2 =100THz 2 , exciton-pump field detuning Δ ex =0THz, phonon pump field detuning Δ n =0THz. When the phonon-exciton coupling strength g∈(0,0.63)THz, the distance between the two splitting peaks does not vary significantly with the coupling strength g, but rather exhibits a nonlinear relationship. When g∈[0.63,0.87]THz, an optical bistability effect occurs in the bilayer graphene optomechanical system. This effect can be used in the design of optical bistable switches, where the threshold of optical bistability is the switching threshold of the optical bistable switch. When g∈(0.87,2.00]THz, the distance between the two splitting peaks varies linearly with the coupling strength g, and the distance between the two splitting peaks is equal to 2g.

[0091] Next, this embodiment considers three different situations, namely ①Δ n =-0.2THz, ②Δ n =0THz、③Δ n =0.2THz, linear absorption coefficient Imχ (1) The relationship with the change of phonon-exciton coupling strength g. Figure 5 Display Δ n =-0.2THz and Δ n =0.2THz, the linear absorption spectra of the two cases are completely consistent. At the same time, the system shows optical bistability, and the low threshold of the bistability range (g c0 ) and high threshold (g c1 ) are all accompanied by |Δ n | decreases as | increases. Bistable phase Figure 6 Revealed the phonon-exciton coupling strength threshold g c Detuning Δ with the phonon-pump field n matching relationship; Figure 6 Medium pump field strength Ω pu2 =100THz 2 , exciton-pump field detuning Δ ex =0THz. When phonons and excitons are weakly coupled (g∈[0.63, 0.87]THz<Γ2=1THz) and the phonon-pump field detuning amount Δ n When the frequency range is [-3.5, 3.5] THz, the system shows optical bistability, and its interval width varies with |Δ n | value decreases and increases with Δ nA =0THz position reaches the maximum value of the bistable range g cA =0.87 THz. This shows that the threshold size and interval width of optical bistability can be controlled by the phonon-pump field detuning.

[0092] Figure 7 Shows that when Ω pu 2 =10THz 2 , 50THz 2 and 100THz 2 When the linear absorption coefficient Imχ (1) Exciton-pump detuning Δ ex In the following calculations, we select the phonon-exciton coupling strength g = 0.8 THz and the phonon-pump field detuning value Δ n =0THz. When Ω pu 2 =10THz 2 and Ω pu 2 =50THz 2 , the linear absorption line about the central axis Δ ex =0THz symmetry; when the pump field strength increases to Ω pu 2 =100THz 2 This symmetry disappears when Ω pu 2 =10THz 2 and Ω pu 2 =50THz 2 When Ω pu 2 =100THz 2 When the bistable phase Figure 8 It can be seen that when Ω pu 2 =5.78THz 2When the voltage is high, the double bistable interval begins to appear, and the width of the bistable interval increases with the increase of the pump field intensity. Such a low threshold also means the feasibility of developing a low-threshold bistable switch. Figure 8 The exciton-phonon coupling strength g = 0.8 THz, the phonon-pump field detuning Δ n =0THz. When the pump field strength increases to Ω pu 2 =83.24THz 2 When the double bistable interval becomes a single bistable interval, its bistable interval width reaches a maximum value of 0.042THz. pu 2 As the pump field intensity increases to Ω pu 2 =248.84THz 2 When , the bistability phenomenon disappears. It is worth noting that the optical bistability threshold Δ exc Always within the low threshold range of [-0.103, 0.103] and about the central axis Δ ex =0THz, showing a strictly symmetrical relationship. These results provide a new approach for developing low-threshold, single-channel and dual-channel optical bistable switches.

[0093] Considering the dependence of optical bistability on the phonon-exciton coupling strength g, Figure 9 It shows that when the pump field strength Ω pu 2 =10THz 2 , 50THz 2 and 100THz 2 When the linear absorption coefficient Imχ (1) The relationship between the phonon-exciton coupling strength g. At this time, the exciton-pump field detuning value Δ ex =0THz, phonon-pump field detuning Δ n =0THz. When the pump field strength Ω pu 2 =10THz 2 When the pump field strength Ω pu 2 =50THz 2 and Ω pu 2 =100THz 2 When the pump field intensity Ω is Ω, the system shows optical bistability, and the width of the bistability interval increases with the pump field intensity Ω. pu 2 According to the bistable phase Figure 10 It can be seen that the system reaches point E (Ω puE2 =14.88THz 2 , g cE =0.36THz), optical bistability begins to appear. Figure 10 Exciton-pump field detuning Δ ex = 0 THz, phonon-pump detuning Δ n =0THz. Bistable threshold g c With the pump field strength Ω pu 2 increases with the increase of g, and the high threshold g c1 The increase is significantly greater than the low threshold g c0 As the magnitude of the increase increases, the width of the hysteresis loop also gradually increases. This shows that by adjusting the pump field intensity, the bistability threshold and the width of the bistability interval can be effectively controlled. Therefore, the optical bistability characteristics can be controlled by changing the pump field intensity.

[0094] In the double-layer graphene optomechanical system of this embodiment, by changing the phonon-pump field detuning amount Δ n , exciton-pump field detuning Δ ex , pump field strength Ω pu 2 The magnitude of the phonon-exciton coupling strength g can be used to control the optical bistability of the system, thereby realizing a low-threshold optical bistability switch. The relevant conclusions are as follows: 1) When the phonon and exciton are weakly coupled (g∈[0.63, 0.87]THz<Γ2=1THz) and Ω pu 2 =100THz 2 When , the width of the optical bistability interval varies with the absolute value of the phonon-pump field detuning |Δ n 2) By changing the exciton-pump field detuning Δ ex The size of the optical bistability is regulated. When g=0.8THz, Δ n =0THz and Ω pu 2 =5.78THz 2 When the system begins to show two bistability intervals, the optical bistability threshold Δ exc Always within the range of [-0.103, 0.103] THz. 3) Under given conditions (ieΔ n =0THz, Δ ex =0THz), when the pump field strength Ω pu 2 =14.88THz 2 and the phonon-exciton coupling strength g c = 0.36THz, the system begins to show optical bistability effect, and its bistability threshold gc The embodiment provides a new method for developing low-threshold, single-channel and dual-channel optical bistable switches.

Claims

1. A low-threshold optical bistable switch design method based on a double-layer graphene optomechanical system, characterized by: The following steps are involved: 1) Establish a double-layer graphene photomechanical system, which is subjected to a frequency of ω pu The pump light and frequency are ω pr The amplitudes of the pump field and the probe field are E pu and E pr ; The double-layer graphene of the double-layer graphene photomechanical system is set up in parallel, and the electric field Acting perpendicularly to the graphene film plane; 2) Derivation of the relevant quantity of optical bistability; The electric field perpendicular to the graphene plane in the double-layer graphene optomechanical system in step 1) The axial symmetry of the double-layer graphene is broken, resulting in coupling between electrons and vibration modes; the two-level system of the double-layer graphene optomechanical system uses three pseudospin operators σ 01 , σ 10 and σ z to characterize; Under the rotating wave approximation, the Hamiltonian of the system can be expressed as: is the pump field Rabi frequency, is the detection field Rabi frequency; μ is the electric dipole moment of the exciton, δ=ω pr -ω pu is the probe-pump detuning, Δ n =ω n -ω pu is the G-mode phonon-pump field detuning, Δ ex k =ω ex k -ω pu is the exciton-pump field detuning, ω n represents the vibration frequency of graphene; The Hamiltonian representing the phonon energy, The Hamiltonian representing the energy of the electron-hole pair exciton, represents the phonon-exciton interaction Hamiltonian, The Hamiltonian representing the interaction between the pump light and the probe light and the phonon; where b k + and b k denote the exciton creation and annihilation operators respectively, and the coupling strength of the G-mode phonon-electron state is g k ; Let p = σ 01 , ω ex k ≈ ω ex , ∑ k g k b k = Ξ - , ∑ k g k b k + = Ξ + ; Using Heisenberg's equations of motion: The following quantum Langevin equations can be obtained: Γ1 is the spontaneous emission rate of excitons, Γ2 is the dephasing rate of excitons, γ n is the electron-hole exciton relaxation rate, g is the center frequency ω of the phonon and exciton ex The coupling strength of ex =ω ex -ω pu ; The effective first-order linear optical susceptibility is: (9) where ε0 is the dielectric constant of vacuum; the definition of the dimensionless linear optical polarizability is expressed as: Imx (1) and Reχ (1) denote the linear absorption coefficient and dispersion coefficient, respectively; w0 is the exciton particle inversion number; p0=iΩ pu w0(γ n +iΔ ex ) / [-(Γ2+iΔ n )(c n +iΔ ex )+g 2 w0], p0 * =-iΩ pu w0(γ n -iD ex ) / [-(Γ2-iΔ n )(c n -iD ex )+g 2 w0], Ξ0=-ig 2 p0 / (y n +iΔ ex ),p0 * =-iΩ pu w0(γ n -iD ex ) / [-(Γ2-iΔ n )(c n -iD ex )+g 2 w0], Ξ0 * =ig 2 p0 * / (y n -iD ex ), t1=-i(Ω pu -Ξ0 * ) / {-F2+i(Δ n +d)+g 2 w0 / [γ n -i(δ+δ ex )]}, t2=(Γ1-iδ) / {2g 2 p0 / [-γ n +i(δ+δ ex )]-2i(Ξ0-Ω pu )}, t3=-{g 2 p0 * / [-c n +i(δ-δ ex )]+i(Ξ0 * -Oh pu )} / {g 2 p0 / [-γ n +i(δ+δ ex )]-i(Ξ0-Ω pu )}, t5=-i / {g 2 p0 / [-y n +i(δ+δ ex )]-i(Ξ0-Ω pu )}, t6={-Γ2+i(δ-δ n )-g 2 w0 / [-γ n +i(δ-δ ex )]} / i(Ω pu -Ξ0), t8=-1 / (Ω pu -Ξ0); From the linear absorption coefficient equation, it can be seen that the optical bistability is related to the phonon pump field detuning Δ n , exciton pump field detuning Δ ex , pump field strength Ω pu 2 It is related to the phonon-exciton coupling strength g; 3) Controlling the phonon-pump field detuning Δ by applying the pump field frequency n ; The phonon-exciton coupling strength g is controlled by the length of the graphene film; Tuning the exciton-pump detuning Δ by adjusting the pump field frequency ex .

2. The low-threshold optical bistable switch design method based on a double-layer graphene optomechanical system according to claim 1, characterized in that: The exciton particle inversion number w0 is obtained by the third-order equation:

3. The low-threshold optical bistable switch design method based on a double-layer graphene optomechanical system according to claim 1, characterized in that: The parameter p1 is obtained by the following steps: Using ansata processing: |p0|>>|p1|, |p-1|, |w0|>>|w1|, |w -1 |,|Ξ0|>>|Ξ1|,|Ξ -1 |; p(t)=p0+p1e -iδt +p -1 e iδt , (5) w(t)=w0+w1e -iδt +in -1 e iδt , (6) X - (t)=Ξ0+Ξ1e -iδi +Ξ -1 e iδt , (7) Substituting formulas (5)-(7) into equations (2)-(4), we obtain the expression for parameter p1.

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