Photonic negative differential transistor based on cavity polaritons

We theoretically provide a scheme for realizing a photonic negative differential transistor (NDT) by using a two-port asymmetric system with cavity exciton polaritons. In such a hybrid optomechanical system, the transmission of the probe light can be completely regulated by the pump field. Interestingly, the resonance transmission curve of probe light has a negative (positive) slope to the pump intensity, which depends on the coupling among excitons, photons and phonons. Therefore, the probe transmission exhibits the characteristic of negative (positive) differential transistors. The transmission spectrum of probe fields is modified by Stokes and anti-Stokes scattering effects, resulting in the output probe light be either attenuated or amplified. Moreover, we find that the transmission of pump fields has a bistability characteristic with appropriate parameters due to nonlinear effects. Our results open up exciting new possibilities for designing a photonic NDT, which may be applied to implement polariton integrated circuits.


Introduction
Photonic transistors are usually heralded as the next core device in the quantum information processing [1].They use photons as signal carriers to manipulate the transmission of a probe field (signal field) by another pump field (gate field) via nonlinear optical interactions [2].The photonic negative differential transistor (NDT) is an interesting and counterintuitive optical device, which describes the signal field decrease as the applied gate field increases [3].Analogous to a negative differential resistance [4], the NDT holds great potential for applications such as oscillators, amplifiers, and logic switches [5].Photons as signal carriers have more advantages than electrons, such as higher transfer speeds, lower power dissipations, less hardware heating, and avoiding crosstalk interference [6,7].However, there are some difficulties for realizing such a transistor due to weak photon-photon interactions [6].To circumvent this difficulty, the weak light beam is mediated by increasing resonances in optical emitters [8,9] and realizing strong light-matter couplings in cavity optomechanical systems [10][11][12][13].
A general cavity optomechanical system is composed of optical resonators and mechanical oscillators (MOs) [13][14][15][16][17][18][19][20][21][22][23][24][25][26].Optical and MOs can be coupled to each other via the radiation pressure which results from momentum transfer due to the reflection of a photon [23].When the cavity is driven by a pump laser beam, the circulating light in the cavity gives rise to a radiation pressure force which can deflect the mirror.The periodic deflection of mirrors (it can be regarded as movable mirrors) generates a MO which reacts on cavity modes, then changes the total cavity length and modifies the resonance frequency of the cavity [24].Thus, the interaction between optical and mechanical modes can be effectively modulated by pump laser fields via the radiation pressure [23][24][25][26][27][28][29].The hybrid optomechanical system presents abundant and fascinating phenomena, such as optomechanically induced transparency (OMIT) [27][28][29], optical bistability [14][15][16], parity-symmetry-breaking quantum phase transitions [18], and optomechanical entanglements [22].The system has also some potential applications such as phonon lasers [30], all-optical switches and transistors [31,32], single-photon routers [33], and force sensors [34].In particular, OMIT can be used for designing optical transistors and routers, the optical bistability provides a way for realizing a controllable optical switch, and the optomechanical entanglement can be applied to quantum network communication.In recent years, a new type of hybrid optomechanical system with the strong coupling between excitons and photons in a vertical quantum-well (QW) cavity has been realized theoretically and experimentally, respectively [35][36][37][38][39][40][41][42][43][44][45][46][47].Microcavity exciton polaritons are hybrid light-matter bosonic quasi-particles consisting of confined photons and bound excitons [35].They inherite remarkable properties from both their photonic part and their electronic component, such as extremely high speeds, very light effective masses, and strong nonlinear effects [36,39,40].In particular, exciton-polariton condensates can be achieved experimentally at room temperature [35,41,42], and directly observed in the real and momentum space via photoluminescence spectra [43].Thus, exciton polaritons have provided an ideal platform for studying novel phenomena in fundamental physics, such as non-Hermitian physics [44], non-equilibrium processes [42], and open systems [43].Exciton polaritons have also aroused a lot of attentions in applications of polariton devices, such as optical polariton transistors [36], ultrafast optical polarization switches [45], exciton-polariton lasers [46], and polaritonic light-emitting diodes [47].Moveover, the strong photon-exciton coupling in such a coupled-cavity system can regulate the transmission of probe fields, and lead to novel OMIT, which is useful for realizing the photonic NDT.However, the photonic NDT based on such a hybrid cavity system is rarely explored.
In this paper, we investigate OMIT in a two-port asymmetric cavity optomechanical system with exciton polaritons, driven by a strong pump field and a weak probe field, which can be realized experimentally based on a vertical GaAs/AlAs microcavity with QWs as shown in figure 1.Using the input-output relation, we calculate the transmission of the probe light and discuss the transmission spectrum.We find that the pump field can completely regulate the transmission of probe light, and the output probe light is amplified or attenuated by the pump field due to Stokes and anti-Stokes scattering effects, which also depends on the coupling among excitons, photons and phonons.We also discover the resonance transmission of the probe light presents a negative (positive) slope to the pump intensity, which indicates this hybrid optomechanical system with strong exciton-photon couplings can be regarded as a polariton-based NDT.Moreover, we find that the transmission of the pump light demonstrates as a bistability characteristic with appropriate parameters due to nonlinear effects.Therefore, our results have provided a theoretical schema for a polariton-based photonic NDT.

Model
We consider a two-port asymmetric cavity optomechanical system as shown in figure 1, where an optomechanical resonator is formed by a micropillar with movable distributed Bragg reflectors (DBRs).The device includes QWs located at the antinode of the cavity in which excitons are trapped.The cavity is driven by a strong pump laser field b pu and another weak probe laser field b pr along the cavity axis.The pump (probe) photon b pu (b pr ) is input at the left (right) side of the cavity, and the output pump (probe) photon C pu (C pr ) is monitored by a detector at the right (left) side of the cavity.In the rotating frame at the pump field frequency ω pu , the Hamiltonian of the system can be obtained as [26,[36][37][38]: where â (â † ), ê (ê † ), and ĉ (ĉ † ) denotes the annihilation (creation) operator for phonons (MO modes), QW excitons, and cavity photons with a resonance frequency ω m , ω e , and ω c , respectively.∆ e = ω pu − ω e and ∆ c = ω pu − ω c is the pump-exciton detuning and the pump-cavity detuning, respectively.The fourth term describes the photon-exciton coupling effect with a Rabi coupling strength Ω.The fifth term describes the photon-phonon coupling effect with a single-photon optomechanical coupling strength g.The last term describes driven fields including the pump light with the amplitude b pu = √ P pu /hω pu and the probe light with the amplitude b pr = √ P pr /hω pr , where ω pr is the probe light frequency, and P pu (P pr ) refers to the pump (probe) light power.δ = ω pr − ω pu is the probe-pump detuning.κ c1 and κ c2 is the decay rate from the left and right port of the cavity, respectively.The spatial symmetry of cavity decay rate is broken when The dynamical evolution of the system follows the quantum master equation: where ρ is the density matrix operator of the system.
,ĉ} is the Liouvillian superoperator, which describes dissipation processes of the system.γ k = {γ m , κ e , κ c }, where γ m is the damping rate of MO modes, κ e is the decay rate of exciton modes, and κ c = κ c1 + κ c2 is the average decay rate of cavity modes.The expectation value of the operator Ô (O ≡ ⟨ Ô⟩ = Tr( Ôρ)) can be calculated by using the master equation.We introduce the position-like operator Q = (â † + â)/ √ 2 of mechanical oscillator modes and the corresponding momentum-like operator and [ Q, P] = i, then, the open quantum system can be described by following quantum Langevin equations: where χin (t) = ω m P in (t), P in , ξ in , η in1 , and η in2 are corresponding Markovian noise operators with zero mean values and relevant delta correlation functions ⟨P in (t)P in (t Here, q in = {ξ in , η in1 , η in2 }, and e, c) is the equilibrium mean thermal phonon, exciton, and photon number with the Boltzmann constant k B and the environmental temperature T.Under the high frequency condition, i.e. hω q ≫ k B T, we have n q th ≈ 0. Thus, these corresponding noises can be neglected.In fact, the quantum Langevin equation of momentum and position operators of MO modes can also be derived from the quantum Langevin equation of the bosonic operator â and â † without rotating wave approximation [48].
When considering strong pump fields, i.e. |b pu | ≫ |b pr |, the probe field can be regarded as a perturbation.Therefore, one can solve equation (3) at the first order sidebands (the higher order sidebands are neglected) by using following ansatz: By inserting equation (4) into equation ( 3), we can obtain and where the pump intensity E pu = |b pu | 2 , the average intracavity photon number and σ = {e, c}.Generally, n 0 can be obtained by equation (5).Once n 0 is known, we can obtain other steady-state values, i.e.
, and e 0 = Ωc 0 /(∆ e + iκ e ).In the following discussion about the transmission of the probe light, we focus on the blue pump-cavity detuning region (i.e.∆ c = ω m > 0), where n 0 only has one steady-state value.
To obtain the optical transmission property of the probe light, we introduce an input-output relation, namely, c out (t) = c in (t) − √ 2κc(t) [49], where c in and c out are input and output optical field operators.Then, the expectation value of output fields from left and right mirrors can be obtained as: Firstly, we consider the transmission of the probe field, and focus on the total transmission coefficient of the probe field, which is defined as the ratio of the amplitude between output and input optical fields at the probe frequency, i.e.
In order to describe the amplification and attenuation of the probe light transmission, we introduce Then the probe light through this coupled-cavity system is attenuated for T < 1 and amplified for T > 1. Especially, in the absence of the pump field, where In this case, the transmission of probe photons is effectively modulated by the Rabi coupling between photons and excitons, and independent of the single-photon optomechanical coupling between photons and MOs.However, when a pump field driving on the cavity, the transmission of probe photons is controlled by the coupling effect among excitons, photons and MOs.

Transmission of the probe light in a symmetric cavity
The transmission of the probe light can be demonstrated by a transmission window as shown in figure 2. We first consider a symmetric cavity case (i.e.κ c1 = κ c2 ).Without the photon-exciton coupling, i.e.Ω = 0 (see figure 2(a)), the probe transmission ] for E pu = 0, then T ≈ 1 due to κ c ≫ {∆ c , δ} (see equation ( 9)), which demonstrates that the probe light completely transmits through the coupled-cavity system (see the black line of figure 2(a)).When a weak pump light driving the cavity, the radiation pressure force at a beat frequency δ = δ Res ≈ ±ω m causes the coherent oscillation of mechanical modes.Thus, the Stokes scattered light from the strong intracavity field is induced at the transmission resonant frequency δ Res ≈ −ω m , and the optomechanically induced amplification is produced due to probe photons absorping pump photons (the resonant transmission T Res > 1) (see the purple line of figure 2(c)).Meanwhile the anti-Stokes scattered light is induced at δ = δ Res ≈ ω m , the optomechanically induced absorption is produced due to pump photons absorping probe photons (T Res < 1) (see the purple line of figure 2(d)).Namely, in the transmission spectrum of probe fields, a transparency window appears in the red probe-pump detuning region (δ < 0), and an absorption window appears in the blue probe-pump detuning region (δ > 0).When the pump intensity E pu is increased, T Res is first enhanced then reduced for the Stokes scattered light (δ Res ≈ −ω m ) (figure 2(a1)), and the corresponding δ Res shifts red (figure 2(c)).For the anti-Stokes scattered light (δ Res ≈ ω m ), the pump light can either amplify or attenuate the probe light.When the pump intensity is increased, the absorption window becomes a transparency window, while the attenuated probe light (T Res < 1) is first reduced then enhanced, and the amplified probe light (T Res > 1) is first enhanced then reduced (figure 2(a2)).In this case, the corresponding δ Res shifts blue (figure 2(d)).
When the photon-exciton coupling is present (i.e.Ω = 3ω m ), as E pu = 0, an absorption window arises at δ = δ Res ≈ −ω m , where the transmission of probe fields is suppressed due to polaritons generated by QW excitons absorbing photons (see the black line of figure 2  The strong pump field can give rise to an energy transfer into the mechanical degree of freedom.This effect results in that the probe light can induce stimulated emission of cavity photons near a frequency ω pr = ω pu + δ Res ≈ ω pu ± ω m [28,29], and effectively leading to an amplification of the probe light.When excitons are present (i.e.Ω ̸ = 0), the strong Rabi coupling between photons and excitons makes |n c , n e + 1, n a ⟩ and |n c + 1, n e , n a ⟩ (|n c , n e + 1, n a + 1⟩ and |n c + 1, n e , n a + 1⟩) influence each other (figures 1(b) and (c)).Thus, the quantum interference condition of generating OMIT is changed, i.e. the frequency and amplitude of output probe light are modified.
Obviously, the transmission of probe fields can be effectively controlled by pump fields, which also depends on the photon-excition coupling and the photon-phonon coupling.As the pump intensity increases, the intensity of probe fields can be either enhanced or reduced.Namely, the probe transmission presents either a positive slope to the pump intensity (∂T Res /∂E pu > 0) or a negative slope (∂T Res /∂E pu < 0).The transmission with a positive (negative) slope to the pump intensity provides a way to design a positive (negative) differential transistor (PDT/NDT).

Optical negative differential transmission of Stokes scattered light
When we drive a two-port symmetric coupled-cavity system with polaritons in a red-detuning region (δ < 0), the Stokes scattered light arises at δ = δ Res ≈ −ω m (see red zones of figures 2(a) and (b)).The transmission of probe light (i.e. the Stokes scattered light) presents either a PDT or NDT as shown in figure 3. Figure 3(a) depicts the dependence of the probe resonant transmission T Res (i.e. the Stokes light intensity) on the pump intensity E pu .For vanished and weak exciton-photon couplings Ω, the probe light is always amplified by pump fields (T Res > 1) (see black and red lines of figure 3(a)).As E pu is increased, T Res reaches the first peak value at E pu = E c1 pu , i.e.T Res has a maximum value (see figure 3(a)).Namely, when pu , the Stokes scattered light intensity presents a positive slope to E pu (∂T Res /∂E pu > 0).Thus, the transmission of the Stokes light demonstrates as a PDT.As E pu is further increased, T Res downs to the first valley value at E pu = E c2 pu , i.e.T Res has a minimum value.Namely, when E c1 pu < E pu < E c2 pu , the Stokes light presents a negative slope to E pu (∂T Res /∂E pu < 0).Thus, the transmission of the Stokes light demonstrates as a NDT.Similarly, when E c2 pu < E pu < E c3 pu and E pu > E c3 pu , the transmission demonstrates as a PDT and NDT, respectively.Therefore, pump fields make the transmission of the Stokes light undergo a PDT-NDT-PDT-NDT transition for weak photon-excition couplings.However, for strong Rabi coupling strengths Ω, the Stokes light is attenated for weak pump fields, and an increase in the pump intensity can result in the Stokes light from attenuations to amplifications (see figure 3 Moreover, as the Rabi coupling strength Ω increases, critical values E ci pu (i = 1, 2, 3) for the transition between PDT and NDT tend to a bigger pump intensity.This is rooted in excitons absorbing much more photons and being converted into polaritons for strong Rabi coupling strengths Ω between excitons and photons.These photons should be provided by a stronger pump field, i.e. a bigger pump intensity.However, as the single-photon optomechanical coupling strength g increases, E ci pu (i = 1, 2, 3) shifts to a lesser pump intensity (see figures 3(b)-(d)).The single-photon optomechanical coupling strength can enhance the coherent oscillation of mechanical modes, and photons used to driving the coherent oscillation of mechanical modes are reduced.So the corresponding pump intensity decreases.

Optical negative differential transmission of anti-Stokes scattered light
The anti-Stokes scattered light (δ Res > 0) can be induced as the system is driven by pump fields in a blue-detuning region (see blue zones of figures 2(a) and (b)).The anti-Stokes scattered light also presents either a NDT or PDT as shown in figure 4. Figure 4(a) depicts the dependence of the anti-Stokes light intensity T Res on E pu .For a given Rabi coupling strength, as the pump intensity E pu increases, the attenuated probe light is reduced when E pu < E c1 pu , and T Res downs to the first valley value at E pu = E c1 pu (see figure 4(a)).Thus, the transmission of the anti-Stokes light demonstrates as a NDT when E pu < E c1 pu .As E pu is further increased, the attenuated probe light turns into the amplified probe light, and T Res reaches the first peak value at E pu = E c2 pu (figure 4(a)).Thus, the transmission of the anti-Stokes light demonstrates as a PDT when E c1 pu < E pu < E c2 pu .As E pu is sequentially increased, the second peak and valley values of T Res appear.Similarly, when pu and E pu > E c4 pu , the transmission demonstrates as a NDT, PDT and NDT, respectively.On the other hand, for weak pump fields, strong Rabi coupling strengths Ω can lead to a distinct attenuation of anti-Stokes light (see green and blue lines of figure 4(a)).
In a word, as the Rabi coupling strength Ω is increased, critical values E ci pu (i = 1, 2, 3, 4) for the transition between NDT and PDT tend to a bigger pump intensity.However, the single-photon optomechanical coupling strength g leads to E ci pu (i = 1, 2, 3, 4) shifting to a lesser pump intensity (see figures 4(b)-(d)).The coupling effect among excitons, photons and phonons results in the transmission of anti-Stokes light undergoing a transition between NDT and PDT.

Transmission of the probe light in an asymmetric cavity
The transmission of probe fields also depends on the asymmetry of the cavity (i.e.κ c1 ̸ = κ c2 ), which is demonstrated in figure 5.In the absence of exciton-photon couplings, for the Stokes light, when κ c1 /κ c2 is

increased, the critical pump intensity E c
pu where the Stokes light is transferred from attenuations to amplifications first reduces then enhances.Particularly, when κ c1 /κ c2 = 1 (i.e. a symmetric cavity), E c pu = 0, which indicates that the Stokes light is amplified for arbitrary E pu (see figure 5(a)).When considering exciton-photon couplings (figure 5(c)), E c pu first reduces then enhances for increasing κ c1 /κ c2 .On the other hand, the critical value E ci pu (i = 1, 2, 3) for a transition between PDT and NDT first reduces then remains almost unchanged as κ c1 /κ c2 is increased.Interestingly, the transmission of the attenuated Stokes light still demonstrates as a PDT, however, the transmission of the amplified Stokes light demonstrates as either a PDT or a NDT (figures 5(a) and (c)).
The transmission of the anti-Stokes light is similar to the case of Stokes light for changing κ c1 /κ c2 .However, as the pump intensity E pu = 0, the transmission of anti-Stokes light is still attenuated even for a symmetric cavity (i.e.κ c1 /κ c2 = 1), and the transmission of the attenuated anti-Stokes light displays as either a NDT or a PDT (figures 5(b) and (d)), which is different from the case of Stokes light.

Transmission of the pump light
We turn, now, to show the transmission of the pump field.According to the pump field input-output relationship, one can acquire the output pump field C pu = − √ 2κ c2 c 0 (see equation ( 7)), and the output pump intensity fulfills where E pu-out = |C pu | 2 and E pu-in = E pu = |b pu | 2 denotes the output and input intensity of pump fields, respectively.Apparently, the relationship between the input pump intensity E pu-in and the output pump intensity E pu-out can be described by the cubic equation of E pu-out .So for a given input pump field, there are three output pump fields with appropriate parameters.This phenomenon is known as optical bistability, which has been observed in many physical systems [14-16, 39, 40].In the absence of the photon-phonon coupling (i.e.g = 0), E pu-out linearly depends on E pu-in (see equation (10)), and the dependence of E pu-out on the pump-cavity detuning ∆ c demonstrates as a standard Lorentzian curve (see figure 6(a1)).When considering photon-phonon couplings (figures 6(a2) and (b)), the dependences of E pu-out on E pu-in and ∆ c both present a S-shaped curve.Namely, when E c1 pu-in < E pu-in < E c2 pu-in , E pu-out has multiple values for a given E pu-in , i.e. output pump fields are bistable.Out of this region (i.e.E pu-in < E c1 pu-in and E pu-in > E c2 pu-in ), E pu-out has only one value for a given E pu-in , i.e. output pump fields are monostable.Similarly, the output pump field presents bistability for ∆ c1 < ∆ c < ∆ c2 (figure 6(a2)).These phenomena are also clearly displayed in figures 6(c) and (d), where the phase diagram for optical bistability is depicted in ∆ c − E pu-in plane.
Moreover, the Rabi coupling strength Ω between excitons and photons can decrease output pump photons due to excitons absorbing photons (figures 6(a1) and (a2)), and makes bistability regions shift up to a bigger input pump intensity (figures 6(c) and (d)).However, strong single-photon optomechanical coupling strengths g between MOs and photons can make bistability regions shift down to a lesser input pump intensity (figures 6(c) and (d)), which because that g enhances the coherent oscillation of mechanical modes, and reduces input pump photons used to driving the coherent oscillation of mechanical modes.

Noise analysis
To describe the noise spectrum of the system, from equation (3), linearized quantum Langevin equations [i.e.α i = α 0 + δα (α i = c, e, Q, P)] can be expressed as a matrix form: where V = (δc, δc † , δe, δe † , δQ, δP) T , Γ pr t in = ( √ 2κ c2 b pr e −iδt , √ 2κ c2 b pr e iδt , 0, 0, 0, 0) T , By introducing the Fourier transform O(ω) = ´+∞ −∞ O(t)e iωt dt and using the input-output relation, the noise term in the frequency domain is obtained as f out (ω) = U(ω)f in (ω), where f out = (η out1 , η † out1 , η out2 , η † out2 , ξ out , ξ † out , 0, P out ) T and the coefficient matrix U(ω) = I − Γ T f (M − iωI) −1 Γ f .Then the output noise spectrum corresponding to an output probe field (i.e. a signal field) can be written as [50] S ηout1 (ω) = 1 2 , S e , and S m refers to the effect of the photon-thermal noise, exciton-thermal noise, and phonon-thermal noise, respectively, on the output spectrum S ηout1 (ω).Here we use the nonzero correlation function of input noise operators in the frequency domain, i.e. ⟨P in (ω)P in (ω The noise-to-signal ratio (NSR) can be defined as the ratio of the integral of the output noise spectrum S ηout1 (ω) and the integral of the output probe (signal) intensity S pr-out (δ) [50].Experimentally, the noise can be measured with a small bandwidth 2∆ω around ω = 0. Thus, NSR can be obtained as where S pr-out (δ) = E pr T with the probe transmission T = |T pr | 2 and the probe intensity Generally, the effect of thermal noises is usually ignored.However, in a usual optomechanical system, such a high frequency condition for the MO is hard to satisfy, i.e. n m th ̸ = 0.Even in this case, the noise-to-signal ratio NSR ≪ 0.1 in the working region of figure 2 as shown in figure 7. Therefore, it is applicable to ignore the effect of thermal noises in our designed photonic NDT.

Experimental implementation
In experiments, we can propose a photonic NDT based on a hybrid optomechanical system with strong exciton-photon couplings by using a vertical GaAs/AlAs microcavity as shown in figure 1.The vertical resonant microcavity is enclosed by two Al 0.18 Ga 0.82 As/AlAs DBRs (the movable left mirror connecting a MO) consisting of periodic sequences of bilayers with multiple pairs of alternated AlAs and GaAs layers, and contains with QWs placed inside the cavity field [36][37][38].QWs are used to trap excitons, which are constructed of the λ/2 GaAs thin layer.In a quasi-one-dimensional system (excitons can only move freely in one spatial dimension), the QW can be regarded as a quantum wire.The cavity with QWs works as an optomechanical resonator which enables phonon and exciton modes coupled to the optical mode.Moreover, GaAs materials have many advantages to construct a QW cavity.For instance, GaAs materials possess a large direct band gap which can enhance optical-mechanical couplings by optical resonance, the GaAs system is very attractive due to the perfect lattice matching of its compounds, and the molecular beam epitaxy technology to manufacture GaAs-based microcavities is mature [37].Picosecond light pulses delivering from a Ti:sapphire laser are split into two paths (pump and probe) when passing through a polarizing beam splitter [27,37].The power ratio of pump and probe laser beams can be controlled by using an acousto-optic modulator and a half-wave plate.A reliable estimation of cavity and QW performance parameters can be obtained by experiments [27,37,[51][52][53][54][55] as shown in table 1.
In a typical GaAs film-Al x Ga 1−x As substrate system, the exciton binding energy is defined as f /2a * 0 ] being the dimension of a solid [56,57].Here, ε 0 is the dielectric permittivity of vacuum, ε * is the dielectric constant of GaAs film-Al x Ga 1−x As substrate material, µ * is the effective exciton reduced mass, m e is the free-electron mass, R H is the Rydberg constant, h is the Planck constant, c is the light velocity constant, L * f is the effective film thickness, and a * 0 = (ε * /ε 0 )(m e /µ * )a H is the effective Bohr radius with the Bohr radius a H .In our system, E b ≈ 31.7 meV.Thus, our predicted polariton-based NDT can be easily realized at room temperature with current experimental conditions [27,37,[51][52][53][54][55].

Conclusion
In conclusion, we have theoretically provided a scheme for designing a photonic NDT in a two-port asymmetric coupled-cavity system with the coupling effect among photons, excitons and phonons.The pump field can effectively control the transmission of the probe light in such a hybrid cavity system with polaritons.The output probe light can be either amplified or attenuated by the pump field due to Stokes and anti-Stokes scattered effects.Interestingly, the resonance transmission curve of probe light has either a negative or positive slope to the pump intensity, which indicates that the transmission demonstrates as either a NDT or PDT.The NDT may be useful to realizing stable ultra-weak light (even single-photon) sources because it can provide a unusual negative feedback [3].Moreover, we find the transmission of the pump field has a bistability characteristic with appropriate parameters due to nonlinear effects.Thus, such a hybrid optomechanical system can work as a photonic NDT.However, it is not appropriate to discuss the unidirectional amplification [50,58], because the probe field is only transmitted along one direction in our system (i.e. from the right side to the left side).Our engineering may have potential applications in implementing polariton integrated circuits, such as all-optical switches and transistors, single-photon routers, and sensors, which has controllable cascade amplification [36] and all-optical logic operation [6] as well as could easily be realized at room temperature [41,42].

Figure 1 .
Figure 1.(a) Schematic diagram for an asymmetric optomechanical cavity coupled to quantum well (QW) excitons in the simultaneous presence of a strong pump laser bpu and another weak probe laser bpr along the cavity axis.The output probe laser beam Cpr and pump laser beam Cpu are monitored by respective detectors.The cavity decay rate of left and right sides is given by κc1 and κc2, respectively.An optomechanical resonator is formed by two distributed Bragg reflectors (DBRs), where the movable left mirror connects a mechanical oscillator (MO).The device contains with QWs located at the antinode of cavity fields.(b) and (c) Energy-level scheme of optomechanically induced transparency for Stokes (b) and anti-Stokes (c) scattered light with the state |nc, ne, na⟩, where nc,e,a is the excitation number of photons, excitons, and phonons, respectively.
(a)).Meanwhile, E pu leads to the transmission of the Stokes light translating between PDT and NDT (see figures 3(a) and (c)).

Figure 6 .
Figure 6.Transmission of pump fields in a symmetric cavity.The output pump intensity Epu-out versus the pump-cavity detuning ∆c (a1)-(a2) and the input pump intensity E pu-in (b) for different Rabi coupling strengths Ω. Phase diagram for optical bistability in ∆c − E pu-in plane ((c) and (d)) for different single-photon optomechanical coupling strengths g.Other parameters are ∆e = ωm, κc1 = κc2 = 10ωm, and κe = ωm.

Table 1 .
A summary of experimentally accessible parameters and normalized parameters used for numerical simulations.