1. Introduction
Atomically thin two-dimensional (2D) layered materials,such as graphene and monolayer molybdenum disulfide (MoS2),have attracted great interest recently due to their superior electrical-optical properties and potential applications[1-3]. The pristine graphene is a zero-bandgap nanomaterial and several applications requiring a bandgap will be limited. Compared with graphene being a semi-metal,MoS2 is a semiconductor with a thickness-dependent electronic structure[4,5]. With the decrease of the layer numbers of MoS2,the material undergoes a transition from an indirect bandgap (1.2 eV in bulk) to a direct bandgap semiconductor (1.8 eV in monolayer)[6]. Monolayer MoS2 may replace graphene to realize the applications in photodetection[7,8],photovoltaics[9],and MoS2-based transistors[10,11] due to this intrinsic property. In addition,the applications in sensors will benefit from another property of the large surface-to-volume ratio of monolayer MoS2,and chemical sensors based on multilayer and monolayer MoS2 have been demonstrated experimentally[12-14].
As atomically thin 2D layered nanomaterials,graphene and monolayer MoS2 could be considered as the ultimate nanomaterials to structure micromechanical resonators,and additionally,nano-optomechanical systems based on graphene have been obtained experimentally recently[15,16]. For MoS2,multilayer MoS2 nanomechanical resonator[17,18] and single layer MoS2 mechanical resonator[19] have also been realized experimentally,due to its properties of ultralow weight[17],exceptional strain limit[5],and high elastic modulus[20,21]. It is noticed that the applications such as transistors,sensors and nanomechanical systems based on monolayer/multilayer MoS2 mostly focused on the electronic regimes. However,MoS2-based devices and applications in the all-optical domain have not been undertaken yet.
Here,we present a nano-optomechanical system based on a plate-like circular monolayer MoS2 suspended on the Si/SiO2 substrate[17,19]. We investigate the probe absorption and dispersion spectra of the probe field induced by the coupling between the mechanical vibration and the exciton in the monolayer MoS2 resonator,and the phenomenon of phonon-induced transparency (PIT) is also demonstrated due to the exciton-resonator coupling. The transmission of the probe field can be manipulated by the optical pump power,and the probe field can be efficiently attenuated or amplified depending on the power of a second `gating' (pump) field,which indicates a promising candidate for a quantum optical transistor. The exciton-resonator coupling makes a great contribution to this coupled system which induces the significantly enhanced optical nonlinear effect. The optical Kerr effect and nonlinear absorption can be modulated via controlling the intensity of the pump laser. Parameters such as the quality factor of the resonator Q and the exciton-resonator coupling β affecting the optical Kerr effect are also studied.
2. System and method
Figure 1(a) gives the model based on monolayer MoS2[17,19] with the optical pump-probe technology. Figure 1(b) shows the structure of monolayer MoS2 consisting of a hexagonal plane of Mo atoms covalently bonded and sandwiched between two planes of S atoms. Both experiments[22,23] and theories[24,25] have studied the vibrational properties of bulk,few layer,and monolayer MoS2 recently,Although structural nonidealities and asymmetries of devices based on MoS2 nanostructures will occur and cause new mechanical resonances[26],for the sake of analytical simplicity we still take into account the model system as shown in Figure 1[17,19]. In this suspended structure,the lowest-energy resonance corresponds to the fundamental flexural mode and the resonator is assumed to be characterized by sufficiently high quality factor Q. Therefore the lifetime of the resonator is long enough. In this case,the new mechanical resonances caused by the edge effects and irregular shapes should be ignored compared to the fundamental flexural mode.
In the system,the lowest-energy resonance corresponds to the fundamental flexural mode,the resonator is assumed to be characterized by sufficiently high quality factor Q,and the lifetime of the resonator is long enough. The new mechanical resonances induced by the edge effects and irregular shapes can be ignored compared to the fundamental flexural mode. For vibrations of suspended MoS2 diaphragms,the fundamental mode resonance frequency can be derived as[27,28]
We introduce the bosonic annihilation (creation) operator a(a)+ to describe the eigenmode of the quantum harmonic oscillator,and HNR =
The pump-probe technology[30] affords an effective method to investigate light-matter interaction. Applying the scheme to the monolayer MoS2 resonator system,and HP-ex =
H=ℏΔcSz+ℏωma+a+ℏωmβSz(a++a)−μEc(S++S−)−μEs(S+e−iδt+S−eiδt), |
(1) |
where
Writing the Heisenberg equations of motion and adding dissipation of the corresponding damping and noise terms,the quantum Langevin equations can be derived as follows
˙Sz=−Γ1(Sz+1/2)+iΩc(S+−S−)+iμEsℏ(S+e−iδt−S−eiδt), |
(2) |
˙S−=−[i(Δc+ωmβN)+Γ2]S−+2iΩcSz−2iμSzEsℏe−iδt+⌢→τin(t), |
(3) |
¨N+γm˙N+ω2mN=−2ω2mβSz+⌢→ς(t), |
(4) |
in which the position operator is
⟨⌢→ς+(t)⌢→ς(t′)⟩=γmωm∫dω2πωe−iω(t−t′)[1+cothℏω2kBT], |
(5) |
where kB and T are the Boltzmann constant and the temperature of the reservoir.
Since the probe field is much weaker than the pump field,following the standard methods of quantum optics,the Heisenberg operator can be rewritten as the sum of its steady-state mean value and a small fluctuation with zero mean value:
Γ1(w0+1)[Γ22+(Δc−β2ωmw0)2]+4Ω2cw0Γ2=0. |
(6) |
Then keeping only the linear terms of the fluctuation operators,we make the ansatz[29]
χ(1)(ωs)={i(Γ2−iΔ2)[S∗0Π1−w0(Γ1−iδ)]Γ2−w0ΩcΠ2Γ2}×{[(Γ1−iδ)(Γ2−iΔ2)−iΩcΠ2](iΔ1+Γ2)+iΩcΠ1(Γ2−iΔ2)}−1, |
(7) |
where
χ(3)(ωs)=−[w0Ωc+iS0(Γ2−iΔ1)]Π3Γ32×({(Γ2−iΔ2)[(Γ1+iδ)(Γ2−iΔ1)−iΩcΠ4]+iΩcΠ3(Γ2−iΔ1)}Ω2c)−1. |
(8) |
The real and imaginary parts of
3. Results and discussions
The parameters of monolayer MoS2 resonator are shown as follows[28]: ω m≈ 1.2 GHz,Γ1 =0.5 GHz,Q=2000,and β =0.1. In Figure 2(a),the imaginary part (Imχ(1) and the real part (Reχ(1) of linear optical susceptibility as functions of probe-exciton detuning Δ s at Δc =0 describe the absorption and dispersion of the probe laser,respectively. There are two sharp peaks located at both sides of the spectra that just correspond to the vibrational frequency of the MoS2 resonator,and the middle parts indicate the absorption and dispersion of the exciton in the MoS2 resonator[31]. The phenomenon,for example,the absorption (i.e. the imaginary part Imχ(1) and the black curve in Figure 2(a) can be interpreted with a dressed-state picture. The two sharp peaks in the probe absorption spectrum represent the resonance amplification and absorption of the vibrational mode of the MoS2 resonator. The uncoupled two-level exciton (| g> and | e>) are dressed by the phonon mode n (n represents the number state of the phonon mode),and then the original eigenstates | g> and | e> are modified to form four dressed states,i.e. | g,n>,| g,n+1>,| e,n > and | e,n+1>. In Figure 2(a) (the black curve),the left sharp peak centered at -1.2 GHz in the absorption spectrum signifies the electron making a transition from two low dressed levels (| g,n> and | g,n+1> ) to two high dressed levels (| e,n> and | e,n+1> ). The progress simultaneously absorbs two pump photons and emits a photon at ω cω m,and therefore amplifies a wave at Δ s=-ω m. The right sharp peak located at 1.2 GHz indicates the transition from | g,n> to | e,n+1> corresponding to the usual excitonic absorption resonance induced by the ac-Stark effect. The middle parts shows the transition from | g,n> to | e,n> due to the MoS2 resonator-induced stimulated Rayleigh resonance.

In the case of red sideband Δ c =ω m,Imχ(1) and Reχ(1) of linear optical susceptibility exhibit zero absorption and a positive steep slope at Δ s=0,as shown in Figure 2(b). We also show the probe transmission of the system with and without the exciton-resonator coupling at Δ c =ω m. If there is no exciton-resonator coupling (β =0) in the system,the transmission spectrum of the probe field shows the usual Lorentzian line shape (see the black curve in Figure 2(c)). Once the exciton-resonator coupling has been taken into consideration,the transmission spectrum displays an analogous optomechanically-induced transparency[32] termed phonon-induced transparency (PIT) induced by the exciton-resonator coupling (the red curve in Figure 2(c)),which has been demonstrated in a coupled mechanical resonators system[33] and bilayer graphene nanoribbons[34]. Compared with the coupled mechanical resonators[33],the manufacturing process of the suspended monolayer MoS2 nanomechanical resonator system is simple and the frequency of the resonator can approach gigahertz (In Reference [32],the frequency of the mechanical mode is about kilohertz). The realistic monolayer MoS2 resonator can also exhibit a wide range of variations depending on its size-dependent parameters. Therefore,both the high and low frequency of the mechanical resonator are allowed in the scheme. The so-called PIT appear in bilayer graphene nanoribbons induced by the plasmon excitation due to the coupling with the infrared active γ-point optical phonon,and the function is more similar to that of the dark plasmon mode in the plasmon-induced transparency[34]. However,in the MoS2 resonator system,the phenomenon of PIT is induced by the exciton-resonator coupling,and PIT will disappear immediately without the exciton-resonator coupling. For the essential of PIT,PIT in bilayer graphene nanoribbons[34] is derived from coherent destructive interference of excitation pathways,while in our scheme,the phenomenon of PIT is due to mechanically-induced coherent population oscillation when the pump-probe detuning equals the frequency of MoS2 resonator[29]. Our demonstration opens an avenue for the exploration of slow light in this monolayer MoS2 resonator system.
The probe absorption and dispersion at the red sideband Δ c =ω m and the blue sideband Δ c =-ω m are also investigated,respectively. Figures 3(a) shows the probe absorption as a function of Δ s at Δ c =ω m (red curve) and Δ c =-ω m (green curve),respectively. The probe absorption experiences a significant drop,i.e. the probe fields with frequency ω s =ω e and ω s =-ω e,will present little absorption corresponding to the red and blue sideband. The insert in Figure 3(a) gives the origin of the new features,which shows the dressed states of exciton and the electron transitions induced by the pump light and probe light. When applying the pump and probe light on the MoS2 resonator system,a photon of the pump light corresponding to the detuning Δ c =ω m makes a transition from | g,n+1> to | e,n>,and emits two photons with frequency ω e. Another two transitions from | g,n> to | e,n> and | g,n+1> to | e,n+1> take place simultaneously with absorbing two ω e photons. The emission and absorption of ω e photons is almost at the same rate,and therefore the absorption is zero. The probe absorption at the case of Δ c =-ω m as shown in Figure 3(a) (the green curve) can be interpreted with the similar method. Figure 3(b) shows the dispersion of the probe light in the case of Δ c =ω m (red curve) and Δ c =-ω m (green curve),respectively. The steep positive slope and steep negative slope that appear at Δ s=0 indicate the potential of ultraslow light and the possibility of superluminal light based on the coupled monolayer MoS2 resonator system.

Switching Δ c from Δ c =ω m to Δ c =-ω m and increasing the pump field intensity,a series of transmission spectra of the probe field as a function of Δ s for several pump field powers are displayed as shown in Figure 4. If the pump field Ω c2=0,the negative transmission spectrum of the probe field shows the usual Lorentzian line shape. Turning on the pump field,the negative transmission of the probe field enhances significantly with increasing the pump power from Ω c2=0.002 GHz2 to Ω c2=0.006$ GHz2,termed as electromagnetically induced absorption (EIA)[33]. With further increase in the pump field power from Ω c2=0.008 GHz2 to Ω c2=0.01 GHz2,the system experiences the switch from EIA to parametric amplification (PA)[32] inducing a signal amplification. The monolayer MoS2 resonator system thus can act as a photonic transistor,where the probe (`source') field is manipulated by the pump (`gate') field. The exciton in the system behaves as an optical cavity. The pump field drives the vibration of the MoS2 resonator near its resonant frequency at
In Figure 5,we plot the optical Kerr coefficient Reχ(3) (black curve) and nonlinear absorption Imχ(3) (red curve) as functions of Δ s with Δ c =0 at β=0 and β=0.1,respectively. Fixing the pump laser on-resonance (Δ c =0) with the exciton frequency and scanning the probe laser,we obtain the large enhanced optical Kerr effect at
To explore the phenomenon above more carefully,we further show the optical Kerr coefficient Reχ(3) as functions of Δ s at the case of on-resonance (Δ c =0) under different parameters regime. Figure 6(a) shows that the optical Kerr coefficient peak increases monotonously with the quality factor Q of the MoS2 resonator,i.e. the long vibration lifetime of the resonator will induce observably Kerr coefficient. In Figure 6(b),the larger the coupling strength is,the higher the optical Kerr coefficient peak will be. Since the optical Kerr coefficient peak is caused by the exciton-resonator coupling,the peak will become more and more obvious when the coupling is made stronger. These results demonstrate that the coupling plays a key role in such a coupled system. Switching the pump-exciton detuning to the case of pump off-resonant Δ c =ω m,we demonstrate that the intensity of the pump laser will also affect the optical Kerr effect of the MoS2 resonator system. Figure 6(c) presents the optical Kerr coefficient as functions of Δ s at pump off-resonant Δ c =ω m with different Rabi frequencies of the pump laser,and the details are shown in Figure 6(d). We notice that the probe field experiences different optical Kerr coefficients with different pump laser intensities. Focusing on the detail,the optical Kerr effect will be weakened significantly via increasing the intensity of the pump laser. Therefore,the magnitude of optical Kerr effect can be tuned by controlling the pump intensity,which presents a method for modifying the nonlinear optical features of MoS2 resonator via coupling to the exciton in the system.

4. Conclusion
We have theoretically proposed a nano-optomechanical system based on monolayer MoS2 with optical pump-probe scheme. We investigate the coherent optical properties of this system,and the exciton-resonator coupling plays a key role which induces the phenomenon of phonon-induced transparency and significant nonlinear optical Kerr effect. The dispersion of the probe light in the case of the blue sideband may indicate the potential of ultraslow light and the possibility of superluminal light based on the coupled monolayer MoS2 resonator system. The phenomena of electromagnetically-induced absorption and parametric amplification are also demonstrated,and the monolayer MoS2 resonator system can work as an optical transistor,in which the pump field serves as a gate field which can effectively control the transmission of the probe field via regulating its power. Further,the high quality factor of the resonator and big exciton-resonator coupling strength will result in remarkable optical Kerr effect. By switching the pump-exciton detuning and manipulating the intensity of the pump laser,the optical Kerr effect of the system can be tuned. Our study,therefore,will indicate a new avenue for fabricating nanomechanical resonator systems based on layered nanomaterials and paves the way for various applications in nonlinear optics,quantum optics,and slow light devices.