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Abstract
We investigate the transmission of probe fields in a coupled-cavity system with polaritons and propose a theoretical schema for realizing a polariton-based photonic transistor. When probe light passes through such a hybrid optomechanical device, its resonant point with Stokes or anti-Stokes scattered effects, intensity with amplification or attenuation effects, as well as group velocity with slow or fast light effects can be effectively controlled by another pump light. This controlling depends on the exciton-photon coupling and single-photon coupling. We also discover an asymmetric Fano resonance in transparency windows under the strong exciton-photon coupling, which is different from general symmetric optomechanically induced transparency. Our results open up exciting possibilities for designing photonic transistors, which may be useful for implementing polariton integrated circuits.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
An optical transistor is a device which uses photons as signal carriers to control the probe laser field by using another pump laser field [1,2]. In comparison to the electronic transistor, the optical transistor possesses much higher transfer rates [3], causes much less hardware heating and lower dissipations [4], as well as remains effective at the nanometer scale [5]. Therefore, the optical transistor is often heralded as the next step of quantum information processing [6]. However, there are some challenges for building such an optical transistor because photons rarely interact each other and photon-photon interactions are very weak even in nonlinear optical media [7]. To remedy this shortcoming, several schemes for mediating weak light beams are proposed, such as enhancing resonances in optical emitters [8–10] and achieving strong coupling between light and matter [11–13]. A tightly focused laser beam can be coherently modulated by another gating beam via individual optical emitters realized by embedding a quantum dot in a photonic crystal nanocavity [5], embedding dye molecules in organic crystalline matrices [2] and embedding nitrogen vacancies in suitable host materials [14]. Light-matter interactions have been presented in a cavity optomechanical system [15–19], where individual atoms, ultra-cold atoms, or Bose-Einstein condensates are coupled to photons. With such strong coupling interactions, the mechanical motion of matters can easily modulate the transmission of photons [20,21], create sidebands below (the Stokes light) and above (the anti-Stokes light) the drive frequency, as well as lead to the effect of electromagnetically induced transparency [22,23].
A cavity optomechanical system is essentially a hybrid physical field of optical resonators coupled to mechanical oscillators [24–33]. In such a system, the radiative pressure and dynamical backaction enable the optical control of mechanical oscillators, which also reacts on optical resonators. Recently, a new type of cavity optomechanical system with the strong light-matter coupling between photons and excitons have been reported in a GaAs/AlAs quantum-well microcavity experimentally [34]. Such light-matter coupling results in the emergence of new quasiparticles, i.e., exciton polaritons. This semiconductor optomechanical system has its own advantages over atomic optomechanical systems. Exciton polaritons can be condensed at room temperatures owing to extremely light effective mass [35–38], and possess fast switching times owing to extremely high speed [39]. These properties are provided by the photonic component. They also inherit strong nonlinear effects from the electronic component [40–42]. Exciton polaritons demonstrate unique propagation properties [43], controllable amplification and cascadability [44], as well as the quantum nonequilibrium and non-Hermitian nature [45–50]. In experiments, exciton polaritons can be realized at room temperature [35–38] and directly imaged in momentum and real spaces through the cavity photoluminescence [51,52]. Hence, exciton-polariton condensates have not only fascinations in fundamental investigations but also potential applications in functional polariton devices, as well as have aroused enormous attentions both in theories and experiments. Moreover, in such an optomechanical system based on exciton polaritons, the strong coupling effect between excitons and photons can modulate the transmission of photons. Therefore, a question easily arises: can the transmission of photons be modulated by the exciton-photon coupling in such an optomechanical system with cavity polaritons? If so, can this modulated effect be used to design a polariton-based photonic transistor? This would lie at the heart of exploring the transmission of photons since the photonic transistor is the basis of optical computation and communication [6].
In this paper, we investigate the transmission of probe fields in an optomechanical resonator device with strong exciton-photon couplings (Fig. 1). This strong optical-mechanical coupling has been experimentally realized in a vertical GaAs/AlAs microcavity [34]. Using the standard input-output relation [53], we calculate the probe transmission coefficient and analyze the transmission spectrum. The result indicates that the pump light can effectively control the transmission of the probe light in a cavity optomechanical system with strong exciton-photon couplings. The output probe light can possess either Stokes or anti-Stokes scattered properties, be either amplified or attenuated, and display as either slow or fast light effects by suitably adjusting the pump light. Thus, this cavity optomechanical device can work as a polariton-based transistor. Moreover, we reveal the mechanism for resonant transmissions by using single-photon and single-phonon excitations, while discover an asymmetric Fano resonance. Our results provide a theoretical schema for the engineering of an all-optical polariton-based transistor.
Fig. 1. (a) Sketch of an optomechanical device in the simultaneous presence of a pump laser and a probe laser. The probe beam transmitted from the device are monitored by a detector. Optical cavity and mechanical oscillator (MO) is formed by two distributed Bragg reflectors (DBR) containing with quantum wells (QWs) placed at the antinode of the cavity field. There is a strong coupling effect between cavity photons and excitons in QWs. (b) Energy-level scheme of the system with a single-particle excitation.
2. Model
Motivated by the experiment in Refs. [34,44], we consider an optomechanical resonator device as shown in Fig. 1, where optical cavity and mechanical oscillator is formed by two movable end distributed Bragg reflectors, containing with quantum wells placed at the antinode of the cavity field. The cavity is driven by a strong pump laser with frequency $\omega _{pu}$ and a weak probe laser with frequency $\omega _{pr}$. In the frame rotating at the pump laser frequency $\omega _{pu}$, i.e., $\hat {H}=U^{-1}\hat {H}U-{\rm i}\hbar U^{-1}(\partial U/\partial t)$, where $U=\exp [-{\rm i}\omega _{pu}t(\hat {c}^{\dagger}\hat {c}+\hat {a}^{\dagger}\hat {a})]$, one can obtain the generic Hamiltonian of the system [34,44,54,55]:
In order to deal with the mean response of optomechanical oscillators to probe laser field, we introduce the position-like operator of the Bogoliubov mode $\hat {Q}=(\hat {b}+\hat {b}^{\dagger})/\sqrt {2}$ and momentum-like operator $\hat {P}={\rm i}(\hat {b}^{\dagger}-\hat {b})/\sqrt {2}$, then consider nonlinear Heisenberg-Langevin equations $\partial _{t}\hat {O}=({\rm i}/\hbar )[\hat {H},\hat {O}]+\hat {\mathcal {N}}$, where $\hat {O}=\{\hat {Q},\hat {P},\hat {a},\hat {c}\}$, $\mathcal {N}$ is the corresponding input vacuum noise operator. Thus, upon the communication relations $[\hat {a},\hat {a}^{\dagger}]=1$, $[\hat {c},\hat {c}^{\dagger}]=1$, and $[\hat {Q},\hat {P}]={\rm i}$, the temporal evolutions of mechanical oscillator modes, quantum well exciton modes, and cavity photon modes can be obtained by following equations:
In the strong driving region, i.e., the probe laser intensity is much weaker than the pump laser intensity, the probe field can be treated as the perturbation of the steady state. Thus, we can linearize Eq. (2) at the first order sidebands by using following ansatz:
Upon inserting Eq. (3) into Eq. (2), one can acquire
To obtain the transmission spectrum of the probe field, we use standard input-output relation, i.e., $c_{\rm out}(t)=c_{\rm in}(t)-\sqrt {2\kappa _{c}}c(t)$, where $c_{\rm out}$ and $c_{\rm in}$ are output and input optical field operators, respectively, which is appropriate and available for open cavity system. Here, $c_{\rm in}$ is determined by pump and probe laser fields. Therefore, we can obtain the output field in the following expression:
The total probe transmission coefficient is defined as the amplitude ratio of output and input optical fields at the frequency of probe laser field, i.e.,
Especially, in the absence of pump fields or mechanical oscillators,
In this case, the probe transmission can not be controlled by adjusting the pump intensity. No matter how the Rabi coupling between photons and excitons change, the probe transmission spectrum still behaves as an absorption window and there is not a transparency window. Namely, the output probe light is attenuated and cannot be amplified. Polariton-based transistors have already been realized at room-temperature in the absence of phonons [56]. However, when considering phonons, a radiation pressure force can be generated as cavity photons act on photons, which can react on photons and enhance photon-photon interactions, then the photon transmission is modulated.
Generally, the transmission coefficient of probe laser beams can be effectively manipulated by the pump laser field, which also depends on the coupling effect of cavity photons, quantum well excitons and mechanical oscillators. The property of transmission spectrum can be described by the amplitude and the phase of transmission coefficient $T_{\rm pr}(\omega _{pr})$. The amplitude can reveal amplifications and attenuations of probe transmission, while the phase is related to slow and fast probe transmission dynamics. Here, we focus on the control of probe light by using pump field, and define the probe transmission amplitude $T=|T_{\rm pr}(\omega _{pr})|$ to indicate the amplification and the attenuation of probe light. Probe light through the optomechanical system are amplified for $T>1$ and attenuated for $T<1$. We also define the group delay $\tau =-{\rm d}\phi /{\rm d}\delta$ (where $\phi ={\rm Arg}[T_{pr}(w_{pr})]$) to identify the slow and fast light of probe light [23]. The output probe light demonstrates a slow light effect for $\tau >0$ and a fast light effect for $\tau <0$. Therefore, the probe transmission can be controlled by appropriately adjusting the pump field intensity, probe-pump detuning, Rabi coupling strength, single-photon optomechanical coupling strength, exciton mode detuning and cavity mode detuning.
3. Transmission spectrum
Transmission spectrum of probe fields can be used to describe the property of output optical fields, which can be effectively modulated by pump fields. In the absence of the coupling between cavity photons and quantum well excitons, i.e., the Rabi coupling strength $\Omega =0$, the output probe light is modulated by cavity photons and mechanical oscillators as shown in Fig. 2(a). In this case, the probe transmission coefficient $T_{\rm pr}(\omega _{pr})=-[\kappa _{c}+{\rm i}(\delta +\Delta _{c})]/[\kappa _{c}-{\rm i}(\delta +\Delta _{c})]$ without pump fields [see Eq. (8)], i.e., when $\varepsilon _{pu}=0$, we have $T=1.0$ which indicates the probe light can completely transmit the optomechanical cavity [see the black solid line of Figs. 2(a) and 3(a)]. When a pump field is driven on the cavity, two transmission resonant points appear at $\delta =\delta _{\rm Res}\approx \pm \omega _{m}$. This resonant transmission can be called optomechanically induced transparency (OMIT) [23], which can be regarded as a strict optomechanical analog of electromagnetically induced transparency (EIT), originating from a similar effective interaction Hamiltonian. In the transparency window with a red probe-pump detuning (i.e., $\delta _{\rm Res}\approx -\omega _{m}$), the radiation pressure force at the beat frequency $\delta _{\rm Res}\approx -\omega _{m}$ gives rise to the coherent oscillation of the mechanical mode, which induces the Stokes scattering of light from the strong control (pump) field. This Stokes light is a amplified probe field due to its intensity $T_{\rm Res}>1$, and the intensity of Stokes light can be effectively controlled by a pump field. As pump field intensity is enhanced, $T_{\rm Res}$ first increases then decreases [see Fig. 2(a1)]. At the transparent point with a blue probe-pump detuning (i.e., $\delta \approx \omega _{m}$), anti-Stokes light can occurs. A pump field can either attenuate or amplify the anti-Stokes light. When pump field intensity is enhanced, the intensity of attenuated anti-Stokes light ($T_{\rm Res}<1$) first decreases then increases, and when pump field intensity is further enhanced, the intensity of amplified anti-Stokes light ($T_{\rm Res}>1$) first increases then decreases [see Fig. 2(a2)].
Fig. 2. Transmission spectrum of probe fields. (a)-(b) The transmission amplitude $T$ and (c)-(d) the transmission phase $\phi$ of probe fields versus the probe-pump detuning $\delta$. Insets depict the resonant transmission amplitude $T_{\rm Res}$ versus pump intensity $\varepsilon _{pu}$ in red [(a1) and (b1)] and blue [(a2) and (b2)] probe-pump detuning resonant regions. Other parameters are $g=10\omega _{m}$, $\Delta _{a}=\omega _{m}$, $\Delta _{c}=\omega _{m}$, $\kappa _{a}=\omega _{m}$, $\kappa _{c}=10\omega _{m}$, and $\gamma _{m}=0.01\omega _{m}$.
Fig. 3. Controlling Stokes scattered light. (a)-(b) The transmission amplitude $T$ and (c)-(d) the transmission group delay $\tau$ versus probe-pump detuning $\delta$. Insets [(a1) and (b1)] depict the resonant transmission amplitude $T_{\rm Res}$ versus pump intensity $\varepsilon _{pu}$. (e) The resonant transmission amplitude $T_{\rm Res}$ and (f) the resonant transmission group delay $\tau _{\rm Res}$ versus pump intensity $\varepsilon _{pu}$. Other parameters are $g=10\omega _{m}$, $\Delta _{a}=\omega _{m}$, $\Delta _{c}=\omega _{m}$, $\kappa _{a}=\omega _{m}$, $\kappa _{c}=10\omega _{m}$, and $\gamma _{m}=0.01\omega _{m}$.
When considering the coupling between photons and excitons [i.e., $\Omega =3\omega _{m}$ and see Figs. 2(b)], an absorption window appears at $\delta =\delta _{\rm Res}=-\omega _{m}$ for $\varepsilon _{pu}=0$ (due to $\Delta _{a}=\Delta _{c}=\omega _{m}$), where the probe transmission is suppressed. This is rooted in that excitons can absorb photons then generating polaritons. When a pump field is driven on the cavity, the absorption window turns into a transparency window, and Stokes light appears. As pump field intensity $\varepsilon _{pu}$ is enhanced, Stokes light is transformed from attenuations ($T_{\rm Res}<1$) into amplifications ($T_{\rm Res}>1$), and $T_{\rm Res}$ always increases [Fig. 2(b1)]. On the other hand, when pump fields are present, anti-Stokes light arises at a blue probe-pump detuning resonant point (i.e., $\delta =\delta _{\rm Res}\approx \omega _{m}$). As $\varepsilon _{pu}$ is enhanced, anti-Stokes lights are also transformed from attenuations into amplifications, and $T_{\rm Res}$ first decreases then increases [Fig. 2(b2)].
Moreover, in each transparency and absorption window, there is a steep slope in the curve of the transmission phase $\phi$ on the probe-pump detuning $\delta$ [Figs. 2(c) and 2(d)]. This phenomenon can be used for controlling the group velocity of the probe light passing through the cavity while generating fast and slow light. The effect of slow light has attracted widely interests because it can provide a route to implement the storage of light [57].
The physical origin of transparency and absorption windows above can be explained by single-photon and single-phonon excitations. We consider a $\Lambda$-type four-level system formed by four states $|n_{a},n_{b},n_{c}\rangle$, $|n_{a}+1,n_{b},n_{c}\rangle$, $|n_{a},n_{b}+1,n_{c}\rangle$, and $|n_{a},n_{b},n_{c}+1\rangle$ [Fig. 1(b)], where, $n_{a}$, $n_{b}$, and $n_{c}$ denote the number states of excitons, phonons and photons respectively. One can clearly see that as long as two-photon resonance condition is met (i.e., $|\omega _{pr}-\omega _{pu}|\approx \omega _{m}$), the destructive interference of excitation pathways for the intracavity probe field can induce transparency and absorption windows (i.e., OMIT appears), where Stokes and anti-Stokes scattering of light from pump fields are induced with frequency $\omega _{s}\approx \omega _{pu}-\omega _{m}$ and $\omega _{as}\approx \omega _{pu}+\omega _{m}$, respectively.
Apparently, when a probe light passes through the optomechanical cavity with polariton, its scattering property, intensity, and group velocity is significantly affected by the pump field. This effect may provide a useful scheme to manipulate the transmission of probe lights by appropriately adjusting pump lights, which enables to realize an optical transistor based on cavity polaritons.
4. Controllable transmission of probe fields
4.1 Controlling Stokes scattered light
When we drive the system in the red-detuning region ($\delta <0$), Stokes scattered light appears around $\delta =\delta _{\rm Res}\approx -\omega _{m}$ [see Figs. 3(a) and 3(b)], owing to the coupling between the sideband-driven cavity and the mechanical mode. The intensity, group velocity and resonant excitation point of the Stokes scattered light can be controlled by adjusting pump fields, which also depends on Rabi coupling strengths and single-photon optomechanical coupling strengths.
For vanished and weak Rabi coupling strengths, pump fields can amplify Stokes light and make the resonant detuning $\delta _{\rm Res}$ shift red [Fig. 3(a)]. As the pump field intensity $\varepsilon _{pu}$ is enhanced, there is a peak at $\varepsilon _{pu}=\varepsilon _{pu}^{\rm c}$, where $T_{\rm Res}$ reach to a maximum value [see Figs. 3(a) and 3(a1) as well as black and red lines of Fig. 3(e)]. When $\varepsilon _{pu}<\varepsilon _{pu}^{\rm c}$, pump fields is beneficial to the amplification of Stokes light. At present, Stokes light demonstrates as a fast light effect due to group delay $\tau _{\rm Res}<0$. The higher the pump field intensity, the faster the group velocity [see Fig. 3(c) and black and red lines of Fig. 3(f)]. However, when $\varepsilon _{pu}>\varepsilon _{pu}^{\rm c}$, the increase of $\varepsilon _{pu}$ can restrain the amplification of Stokes light, and the Stokes light demonstrates as a slow light effect due to group delay $\tau _{\rm Res}>0$. The lower the pump field intensity, the slower the group velocity. As Rabi coupling strengths increase, the peak value of $T_{\rm Res}$ and the transition point from fast to slow light both shift towards the higher pump intensity [see black and red lines of Figs. 3(e) and 3(f)]. Nevertheless, as single-photon optomechanical coupling strengths increases, the peak value of $T_{\rm Res}$ and the transition point from fast to slow light both shift towards the lower pump intensity [see Figs. 4(a1) and 4(b1)].
Fig. 4. Resonant transmission in red probe-pump detuning regions. The resonant transmission amplitude $T_{\rm Res}$ [(a1)-(a3)], group delay $\tau _{\rm Res}$ [(b1)-(b3)], and probe-pump detuning $\delta _{\rm Res}$ [(c1)-(c3)] versus pump intensity $\varepsilon _{pu}$ for different single-photon optomechanical coupling strength $g$. The inset (d) depicts a Fano resonance. Other parameters are $\Delta _{a}=\omega _{m}$, $\Delta _{c}=\omega _{m}$, $\kappa _{a}=\omega _{m}$, $\kappa _{c}=10\omega _{m}$, and $\gamma _{m}=0.01\omega _{m}$.
For stronger Rabi coupling strengths, pump fields result in Stokes light from attenuations to amplifications. In this case, when pump field intensity $\varepsilon _{pu}$ is enhanced, $T_{\rm Res}$ increases [Figs. 3(b) and 3(e)], Stokes light possesses a fast light effect ($\tau _{\rm Res}<0$) and $\tau _{\rm Res}$ first decreases then increases [Figs. 3(d) and 3(f)], as well as $\delta _{\rm Res}$ is nearly at $-\omega _{m}$ [Figs. 3(b) and 3(d)]. However, optomechanical coupling strengths $g$ can lead to $T_{\rm Res}$ increase, and enough stronger $g$ can result in amplified Stokes light from original attenuated one [Fig. 4(a2)].
When Rabi coupling strengths are enough strong, Stokes light is attenuated. As shown in Fig. 3(e), $T_{\rm Res}$ downs to a valley value, where the transition from slow to fast light occurs [Fig. 3(f)]. The stronger the Rabi coupling strength, the higher the pump field intensity that the valley value of $T_{\rm Res}$ and the transition from slow to fast light simultaneously arise. However, $g$ lead to the valley value of $T_{\rm Res}$ and the transition from slow to fast light both shift towards the lower pump field intensity [Figs. 4(a3) and 4(b3)]. Especially, for a enough strong $g$ [e.g. $g/\omega _{m}=20$ and $50$, and see Figs. 4(a3) and 4(b3)], a new valley value of $T_{\rm Res}$ arises at $\varepsilon _{pu}^{\rm c1}$ ($\varepsilon _{pu}^{\rm c1}>\varepsilon _{pu}^{\rm c}$), where the Stokes light is translated from fast light to slow light. As $\Omega$ and $g$ are both sufficiently large, Stokes light can be amplified for the higher pump field intensity. In this case, the curves of $T_{\rm Res}$, $\tau _{\rm Res}$ and $\delta _{\rm Res}$ versus $\varepsilon _{pu}$ have a jump, owing to a transition from minimum resonances of the transmission spectrum to maximum one, where a asymmetric Fano resonance can be discovered [Fig. 4(d)], which is different from general symmetric OMIT. Fano resonances have been studied in various hybrid optomechanical systems [58]. In our system, Fano resonances occurs at the transition from absorption windows (attenuated light) to transparency windows (amplified light). Moreover, Rabi coupling strengths can prevent the resonant excitation point $\delta _{\rm Res}$ from red shift, and single-photon optomechanical coupling strengths can promote the red shift [Figs. 4(c1)–4(c3)].
4.2 Controlling anti-Stokes scattered light
Anti-Stokes scattered light can be generated when the system is driven in the blue-detuning region ($\delta >0$). Its intensity, group velocity and resonant excitation point are all controllable.
For a vanished and weak Rabi coupling (e.g., $\Omega =0$ and $\omega _{m}$), as the pump field intensity is enhanced, the intensity of anti-Stokes light $T_{\rm Res}$ has a valley value at $\varepsilon _{pu}=\varepsilon _{pu}^{\rm c1}$ and a peak value at $\varepsilon _{pu}=\varepsilon _{pu}^{\rm c2}$ [see Figs. 5(a) and 5(a1) as well as black and red lines of Fig. 5(e)]. When $\varepsilon _{pu}<\varepsilon _{pu}^{\rm c1}$, anti-Stokes light is attenuated, and it demonstrates a slow light effect due to $\tau _{\rm Res}>0$ [see black and red lines of Fig. 5(f)]. As $\varepsilon _{pu}$ is enhanced, $T_{\rm Res}$ decreases and $\tau _{\rm Res}$ increases. When $\varepsilon _{pu}^{\rm c1}<\varepsilon _{pu}<\varepsilon _{pu}^{\rm c2}$, $\varepsilon _{pu}$ leads to the amplified anti-Stokes light from attenuated one, where the anti-Stokes light has a fast light effect. In this case, as $\varepsilon _{pu}$ is enhanced, $T_{\rm Res}$ still increases [Fig. 5(e)] and $\tau _{\rm Res}$ first increases then decreases [Fig. 5(f)]. When $\varepsilon _{pu}>\varepsilon _{pu}^{\rm c2}$, anti-Stokes light is amplified, and it has a slow light effect. As $\varepsilon _{pu}$ is enhanced, $T_{\rm Res}$ and $\tau _{\rm Res}$ both decrease. For a strong Rabi couping, the peak value of $T_{\rm Res}$ disappears. That is, when $\varepsilon _{pu}>\varepsilon _{pu}^{\rm c1}$, $T_{\rm Res}$ still increases for enhancing $\varepsilon _{pu}$ [Figs. 5(b), 5(b1) and 5(e)], and anti-Stokes light is always the fast light [Figs. 5(d), and 5(f)].
Moreover, Rabi coupling strengths can increase $\varepsilon _{pu}^{\rm c1}$ and $\varepsilon _{pu}^{\rm c2}$, while prohibit the blue shift of resonant detuning $\delta _{\rm Res}$. However, single-photon optomechanical coupling strengths can decrease $\varepsilon _{pu}^{\rm c1}$ and $\varepsilon _{pu}^{\rm c2}$, while promote the blue shift of resonant excitation points. That is, single-photon optomechanical coupling strengths are beneficial to the amplification of anti-Stokes light and modulate the delay time. These phenomena are clearly depicted in Fig. 6. Similar the case of Stokes light, when $\Omega$ is stronger and $\varepsilon _{pu}>\varepsilon _{pu}^{\rm c1}$, the curves of $T_{\rm Res}$, $\tau _{\rm Res}$ and $\delta _{\rm Res}$ versus $\varepsilon _{pu}$ have a jump, where a Fano resonance can also be discovered [Fig. 6(d)].
5. Experimental implementation
Experimentally, an all-optical transistor based on cavity polaritons can be proposed by an optomechanical device with distributed Bragg reflector GaAs/AlAs vertical cavity [34,44] as shown in Fig. 1. Distributed Bragg reflectors are periodic sequences of bilayers with multiple pairs of alternated AlAs and GaAs layers, and contains with quantum wells placed at the antinode of the cavity field. Pump and probe laser beams are generated by picosecond pulses which passes through a polarizing beam splitter and are split into two paths [23,34]. Theses picosecond pulses at a wavelength of $870$ nm and a spectral width of $0.8$ nm are delivered from a mode-locked Ti:sapphire laser [23,34]. The intensity of pump and probe laser beams can be controlled by using a half-wave plate and an acousto-optic modulator.
In typical exciton-polariton experiments [35–38,50–52,59–62], quantum wells are GaAs thin layers with the order of $10$ nm thickness, exciton polaritons have a lifetime on the order of $100$ ps. All the experiments can be performed at room temperature [23,34,59–62]. Realistic parameters can be estimated by experiments [23,34,59–62]: $\omega _{m}/2\pi \sim 20 {\rm GHz}$, $\kappa _{c}=1/\tau$ with $\tau \sim 5 {\rm ps}$ being the cavity mode lifetime, $\gamma _{m}=1/\tau _{\gamma }$ with $\tau _{\gamma }\sim 60 {\rm ns}$ being the cavity mode lifetime, $\kappa _{a}=1/\tau _{exc}$ with $\tau _{exc}\sim 0.5 {\rm ns}$ being the cavity mode lifetime, $P_{pu}\sim 300 {\rm mW}$, $P_{pr}\sim 1 {\rm mW}$, $\lambda _{pu}=870 {\rm nm}$, $\Delta \lambda _{c}\sim 0.8 {\rm nm}$, $\Delta \lambda _{pr}\sim 2 {\rm nm}$, $\Delta _a\sim 1 {\rm THz}$, $\Omega /2\pi \sim 0.48 {\rm THz}$, and $g/2\pi \sim 4.8\times 10^7 {\rm Hz}$. Thus, our predicted transistor based on cavity polaritons can be easily realized with current experimental conditions.
6. Discussion and conclusion
Actually, OMIT and Autler-Townes splitting (ATS) are difficult to be discriminated, because they can both cause an absorptive medium transparent and are quantified by a transparency window in transmission spectrum. However, they are fundamentally distinct in underlying physics [63–66]. OMIT originates from the destructive interference of two quantum pathways. ATS is rooted in energy level splitting due to the strong control field. Moreover, OMIT and ATS can be discerned by the application of the Akaike Information Criterion in whispering-gallery microcavities [67]. The propagation of the probe field is primarily reciprocal in the ATS configuration and nonreciprocal in the EIT configuration due to breaking the time-reversal symmetry, which is experimentally observed by using warm rubidium atoms [68].
In summary, we have proposed a theoretical schema for realizing a polariton-based photonic transistor. The probe light can be effectively controlled by another pump light when it passes through a hybrid optomechanical device which contains cavity photons, excitons, and mechanical oscillators. The output probe light modulated by the pump light can possess either Stokes or anti-Stokes scattered properties, be either amplified or attenuated, and demonstrate as either slow or fast light effects owing to exciton-photon couplings and single-photon optomechanical couplings. Thus, such a hybrid optomechanical resonator device with strong exciton-photon couplings can work as a polariton-based transistor [69,70]. This engineering may have potential applications in implementing polariton integrated circuits [71], which have low-energy dissipation [4] and fast switching times [39] as well as could easily be realized at room temperature [35–38]. This cavity-coupled optomechanical system with exciton polaritons may be used to control transport, bound states, and resonant states of single photon by expanding to a one-dimensional coupled-cavity array and embedding two-level atomic systems [72,73]. It maybe behaves as a quantum Zeno switch if two-level atomic systems are driven by an external periodic field [74]. The optical nonreciprocity (i.e., optical unidirectional propagation) can also be designed by adding another optical mode [75].
Funding
National Natural Science Foundation of China (12104374, 12164042); Natural Science Foundation of Gansu Province (20JR5RA526); Innovation Capability Enhancement Project of Gansu higher education (2019A-014); Creation of science and technology of Northwest Normal University (No. NWNU-LKQN-18-33).
Disclosures
The authors declare no conflicts of interest.
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.
References
1. D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3(11), 807–812 (2007). [CrossRef]
2. J. Hwang, M. Pototschnig, R. Lettow, G. Zumofen, A. Renn, S. Götzinger, and V. Sandoghdar, “A single-molecule optical transistor,” Nature (London) 460(7251), 76–80 (2009). [CrossRef]
3. M. Nazarathy, Z. Zalevsky, A. Rudnitsky, B. Larom, A. Nevet, M. Orenstein, and B. Fischer, “All-optical linear reconfigurable logic with nonlinear phase erasure,” J. Opt. Soc. Am. A 26(8), A21 (2009). [CrossRef]
4. M. Soljačić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004). [CrossRef]
5. J. J. Li and K. D. Zhu, “A quantum optical transistor with a single quantum dot in a photonic crystal nanocavity,” Nanotechnology 22(5), 055202 (2011). [CrossRef]
6. D. A. B. Miller, “Are optical transistors the logical next step,” Nat. Photonics 4(1), 3–5 (2010). [CrossRef]
7. R. W. Boyd, Nonlinear Optics, (Academic, New York, 1992).
8. S. E. Harris, “Electromagnetically Induced Transparency,” Phys. Today 50(7), 36–42 (1997). [CrossRef]
9. M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75(2), 457–472 (2003). [CrossRef]
10. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]
11. L. M. Duan and H. J. Kimble, “Scalable Photonic Quantum Computation through Cavity-Assisted Interactions,” Phys. Rev. Lett. 92(12), 127902 (2004). [CrossRef]
12. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436(7047), 87–90 (2005). [CrossRef]
13. E. Waks and J. Vuckovic, “Dipole Induced Transparency in Drop-Filter Cavity-Waveguide Systems,” Phys. Rev. Lett. 96(15), 153601 (2006). [CrossRef]
14. X. Xu, B. Sun, P. R. Berman, D. G. Steel, A. S. Bracker, D. Gammon, and L. J. Sham, “Coherent population trapping of an electron spin in a single negatively charged quantum dot,” Nat. Phys. 4(9), 692–695 (2008). [CrossRef]
15. F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450(7167), 268–271 (2007). [CrossRef]
16. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450(7167), 272–276 (2007). [CrossRef]
17. R. Lopes, C. Eigen, N. Navon, D. Clément, R. P. Smith, and Z. Hadzibabic, “Quantum Depletion of a Homogeneous Bose-Einstein Condensate,” Phys. Rev. Lett. 119(19), 190404 (2017). [CrossRef]
18. M. A. Kristensen, M. B. Christensen, M. Gajdacz, M. Iglicki, K. Pawłowski, C. Klempt, J. F. Sherson, K. Rzążewski, A. J. Hilliard, and J. J. Arlt, “Observation of Atom Number Fluctuations in a Bose-Einstein Condensate,” Phys. Rev. Lett. 122(16), 163601 (2019). [CrossRef]
19. M. A. Khamehchi, K. Hossain, M. E. Mossman, Y. Zhang, T. Busch, M. M. Forbes, and P. Engels, “Negative-Mass Hydrodynamics in a Spin-Orbit-Coupled Bose-Einstein Condensate,” Phys. Rev. Lett. 118(15), 155301 (2017). [CrossRef]
20. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77(3), 033804 (2008). [CrossRef]
21. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of Ground State Cooling of a Mechanical Oscillator Using Dynamical Backaction,” Phys. Rev. Lett. 99(9), 093901 (2007). [CrossRef]
22. A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011). [CrossRef]
23. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010). [CrossRef]
24. I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics 3(4), 201–205 (2009). [CrossRef]
25. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321(5893), 1172–1176 (2008). [CrossRef]
26. F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009). [CrossRef]
27. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]
28. D. G. Lai, J. F. Huang, X. L. Yin, B. P. Hou, W. L. Li, D. Vitali, F. Nori, and J. Q. Liao, “Nonreciprocal ground-state cooling of multiple mechanical resonators,” Phys. Rev. A 102(1), 011502 (2020). [CrossRef]
29. D. G. Lai, J. Q. Liao, A. Miranowicz, and F. Nori, “Noise-Tolerant Optomechanical Entanglement via Synthetic Magnetism,” Phys. Rev. Lett. 129(6), 063602 (2022). [CrossRef]
30. D. G. Lai, F. Zou, B. P. Hou, Y. F. Xiao, and J. Q. Liao, “Simultaneous cooling of coupled mechanical resonators in cavity optomechanics,” Phys. Rev. A 98(2), 023860 (2018). [CrossRef]
31. D. G. Lai, W. Qin, A. Miranowicz, and F. Nori, “Efficient optomechanical refrigeration of two vibrations via an auxiliary feedback loop: Giant enhancement in mechanical susceptibilities and net cooling rates,” Phys. Rev. Res. 4(3), 033102 (2022). [CrossRef]
32. D. G. Lai, W. Qin, B. P. Hou, A. Miranowicz, and F. Nori, “Significant enhancement in refrigeration and entanglement in auxiliary-cavity-assisted optomechanical systems,” Phys. Rev. A 104(4), 043521 (2021). [CrossRef]
33. D. G. Lai, J. Huang, B. P. Hou, F. Nori, and J. Q. Liao, “Domino cooling of a coupled mechanical-resonator chain via cold-damping feedback,” Phys. Rev. A 103(6), 063509 (2021). [CrossRef]
34. A. Fainstein, N. D. Lanzillotti-Kimura, B. Jusserand, and B. Perrin, “Strong Optical-Mechanical Coupling in a Vertical GaAs/AlAs Microcavity for Subterahertz Phonons and Near-Infrared Light,” Phys. Rev. Lett. 110(3), 037403 (2013). [CrossRef]
35. S. Christopoulos, G. Baldassarri Höger von Högersthal, A. J. D. Grundy, P. G. Lagoudakis, A. V. Kavokin, J. J. Baumberg, G. Christmann, R. Butté, E. Feltin, J. F. Carlin, and N. Grandjean, “Room-Temperature Polariton Lasing in Semiconductor Microcavities,” Phys. Rev. Lett. 98(12), 126405 (2007). [CrossRef]
36. J. J. Baumberg, A. V. Kavokin, S. Christopoulos, A. J. D. Grundy, R. Butté, G. Christmann, D. D. Solnyshkov, G. Malpuech, G. B. H. von Högersthal, E. Feltin, J. F. Carlin, and N. Grandjean, “Spontaneous Polarization Buildup in a Room-Temperature Polariton Laser,” Phys. Rev. Lett. 101(13), 136409 (2008). [CrossRef]
37. J. D. Plumhof, T. Stöferle, L. Mai, U. Scherf, and R. F. Mahrt, “Room-temperature Bose-Einstein condensation of cavity exciton-polaritons in a polymer,” Nat. Mater. 13(3), 247–252 (2014). [CrossRef]
38. R. Su, S. Ghosh, J. Wang, S. Liu, C. Diederichs, T. C. H. Liew, and Q Xiong, “Observation of exciton polariton condensation in a perovskite lattice at room temperature,” Nat. Phys. 16(3), 301–306 (2020). [CrossRef]
39. M. De Giorgi, D. Ballarini, E. Cancellieri, F. M. Marchetti, M. H. Szymanska, C. Tejedor, R. Cingolani, E. Giacobino, A. Bramati, G. Gigli, and D. Sanvitto, “Control and Ultrafast Dynamics of a Two-Fluid Polariton Switch,” Phys. Rev. Lett. 109(26), 266407 (2012). [CrossRef]
40. D. Bajoni, E. Semenova, A. Lemaître, S. Bouchoule, E. Wertz, P. Senellart, S. Barbay, R. Kuszelewicz, and J. Bloch, “Optical Bistability in a GaAs-Based Polariton Diode,” Phys. Rev. Lett. 101(26), 266402 (2008). [CrossRef]
41. J. Y. Lien, Y. N. Chen, N. Ishida, H. B. Chen, C. C. Hwang, and F. Nori, “Multistability and condensation of exciton-polaritons below threshold,” Phys. Rev. B 91(2), 024511 (2015). [CrossRef]
42. Z. F. Yu, J. K. Xue, L. Zhuang, J. K. Zhao, and W. M. Liu, “Non-Hermitian spectrum and multistability in exciton-polariton condensates,” Phys. Rev. B 104(23), 235408 (2021). [CrossRef]
43. T. C. H. Liew, A. V. Kavokin, T. Ostatnicky, M. Kaliteevski, I. A. Shelykh, and R. A. Abram, “Exciton-polariton integrated circuits,” Phys. Rev. B 82(3), 033302 (2010). [CrossRef]
44. D. Ballarini, M. De Giorgi, E. Cancellieri, R. Houdré, E. Giacobino, R. Cingolani, A. Bramati, G. Gigli, and D. Sanvitto, “All-optical polariton transistor,” Nat. Commun. 4(1), 1778 (2013). [CrossRef]
45. H. Deng, H. Haug, and Y. Yamamoto, “Exciton-polariton Bose-Einstein condensation,” Rev. Mod. Phys. 82(2), 1489–1537 (2010). [CrossRef]
46. I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85(1), 299–366 (2013). [CrossRef]
47. T. Byrnes, N. Y. Kim, and Y. Yamamoto, “Exciton-polariton condensates,” Nat. Phys. 10(11), 803–813 (2014). [CrossRef]
48. C. E. Whittaker, E. Cancellieri, P. M. Walker, D. R. Gulevich, H. Schomerus, D. Vaitiekus, B. Royall, D. M. Whittaker, E. Clarke, I. V. Iorsh, I. A. Shelykh, M. S. Skolnick, and D. N. Krizhanovskii, “Exciton-polariton condensation,” Contemp. Phys. 52(2), 131–151 (2011). [CrossRef]
49. L. Salasnich, B. A. Malomed, and F. Toigo, “Emulation of lossless exciton-polariton condensates by dual-core optical waveguides: Stability, collective modes, and dark solitons,” Phys. Rev. E 90(4), 043202 (2014). [CrossRef]
50. J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M. Marchetti, M. H. Szymańska, R. André, J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and L. S. Dang, “Bose-Einstein condensation of exciton polaritons,” Nature (London) 443(7110), 409–414 (2006). [CrossRef]
51. T. Gao, E. Estrecho, K. Bliokh, T. Liew, M. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Höfling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature (London) 526(7574), 554–558 (2015). [CrossRef]
52. M. Pieczarka, E. Estrecho, M. Boozarjmehr, O. Bleu, M. Steger, K. West, L. N. Pfeiffer, D. W. Snoke, J. Levinsen, M. M. Parish, A. G. Truscott, and E. A. Ostrovskaya, “Observation of quantum depletion in a non-equilibrium exciton-polariton condensate,” Nat. Commun. 11(1), 429 (2020). [CrossRef]
53. M. Paternostro, G. De Chiara, and G. M. Palma, “Cold-Atom-Induced Control of an Optomechanical Device,” Phys. Rev. Lett. 104(24), 243602 (2010). [CrossRef]
54. O. Kyriienko, T. C. H. Liew, and I. A. Shelykh, “Optomechanics with Cavity Polaritons: Dissipative Coupling and Unconventional Bistability,” Phys. Rev. Lett. 112(7), 076402 (2014). [CrossRef]
55. B. Sarma and A. K. Sarma, “Ground-state cooling of micromechanical oscillators in the unresolved-sideband regime induced by a quantum well,” Phys. Rev. A 93(3), 033845 (2016). [CrossRef]
56. A. V. Zasedatelev, A. V. Baranikov, D. Urbonas, F. Scafirimuto, U. Scherf, T. Stöferle, R. F. Mahrt, and P. G. Lagoudakis, “A room-temperature organic polariton transistor,” Nat. Photonics 13(6), 378–383 (2019). [CrossRef]
57. M. D. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature 413(6853), 273–276 (2001). [CrossRef]
58. L. Zhou, Z. R. Gong, Y. X. Liu, C. P. Sun, and F. Nori, “Controllable Scattering of a Single Photon inside a One-Dimensional Resonator Waveguide,” Phys. Rev. Lett. 101(10), 100501 (2008). [CrossRef]
59. J. Tang, J. Zhang, Y. C. Lv, H. Wang, F. F. Xu, C. Zhang, L. X. Sun, J. N. Yao, and Y. S. Zhao, “Room temperature exciton-polariton Bose-Einstein condensation in organic single-crystal microribbon cavities,” Nat. Commun. 12(1), 3265 (2021). [CrossRef]
60. M. Wurdack, E. Estrecho, S. Todd, T. Yun, M. Pieczarka, S. K. Earl, J. A. Davis, C. Schneider, A. G. Truscott, and E. A. Ostrovskaya, “Motional narrowing, ballistic transport, and trapping of room-temperature exciton polaritons in an atomically-thin semiconductor,” Nat. Commun. 12(1), 5366 (2021). [CrossRef]
61. F. Chen, H. Li, H. Zhou, S. Luo, Z. Sun, Z. Y. Ye, F. H. Sun, J. W. Wang, Y. L. Zheng, X. F. Chen, H. L. Xu, H. X. Xu, T. Byrnes, Z. H. Chen, and J. Wu, “Optically Controlled Femtosecond Polariton Switch at Room Temperature,” Phys. Rev. Lett. 129(5), 057402 (2022). [CrossRef]
62. Y. Li, X. K. Ma, X. K. Zhai, M. N. Gao, H. T. Dai, S. Schumacher, and T. G. Gao, “Room-temperature polariton quantum fluids in halide perovskites,” Nat. Commun. 13(1), 3785 (2022). [CrossRef]
63. H. C. Sun, Y. X. Liu, H. Ian, J. Q. You, E. Il’ichev, and F. Nori, “Electromagnetically induced transparency and Autler-Townes splitting in superconducting flux quantum circuits,” Phys. Rev. A 89(6), 063822 (2014). [CrossRef]
64. H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: Photon transmission through an optomechanical device with a two-level system,” Phys. Rev. A 90(2), 023817 (2014). [CrossRef]
65. H. Ian, Y. X. Liu, and F. Nori, “Tunable electromagnetically induced transparency and absorption with dressed superconducting qubits,” Phys. Rev. A 81(6), 063823 (2010). [CrossRef]
66. X. Gu, S. N. Huai, F. Nori, and Y. X. Liu, “Polariton states in circuit QED for electromagnetically induced transparency,” Phys. Rev. A 93(6), 063827 (2016). [CrossRef]
67. B. Peng, S. K. Özdemir, W. Chen, F. Nori, and L. Yang, “What is and what is not electromagnetically induced transparency in whispering-gallery microcavities,” Nat. Commun. 5(1), 5082 (2014). [CrossRef]
68. H. Wu, Y. Ruan, Z. Li, M. X. Dong, M. Cai, J. Tang, L. Tang, H. Zhang, M. Xiao, and K. Xia, “Fundamental Distinction of Electromagnetically Induced Transparency and Autler-Townes Splitting in Breaking the Time-Reversal Symmetry,” Laser Photonics Rev. 16(9), 2100708 (2022). [CrossRef]
69. R. Johne, I. A. Shelykh, D. D. Solnyshkov, and G. Malpuech, “Polaritonic analogue of Datta and Das spin transistor,” Phys. Rev. B 81(12), 125327 (2010). [CrossRef]
70. T. Gao, P. S. Eldridge, T. C. H. Liew, S. I. Tsintzos, G. Stavrinidis, G. Deligeorgis, Z. Hatzopoulos, and P. G. Savvidis, “Polariton condensate transistor switch,” Phys. Rev. B 85(23), 235102 (2012). [CrossRef]
71. A. W. Elshaari, W. Pernice, K. Srinivasan, O. Benson, and V. Zwiller, “Hybrid integrated quantum photonic circuits,” Nat. Photonics 14(5), 285–298 (2020). [CrossRef]
72. J. Q. Liao, Z. R. Gong, L. Zhou, Y. X. Liu, C. P. Sun, and F. Nori, “Controlling the transport of single photons by tuning the frequency of either one or two cavities in an array of coupled cavities,” Phys. Rev. A 81(4), 042304 (2010). [CrossRef]
73. L. Zhou, H. Dong, Y. X. Liu, C. P. Sun, and F. Nori, “Quantum supercavity with atomic mirrors,” Phys. Rev. A 78(6), 063827 (2008). [CrossRef]
74. L. Zhou, S. Yang, Y. X. Liu, C. P. Sun, and F. Nori, “Quantum Zeno switch for single-photon coherent transport,” Phys. Rev. A 80(6), 062109 (2009). [CrossRef]
75. K. Y. Bliokh, F. J. Rodríguez-Fortu no, A. Y. Bekshaev, Y. S. Kivshar, and F. Nori, “Electric-current-induced unidirectional propagation of surface plasmon-polaritons,” Opt. Lett. 43(5), 963–966 (2018). [CrossRef]
