Manipulation of Coherent Optical Propagation Based on Monolayer MoS2 Resonator

Atomically thin two-dimensional semiconductor nanomaterials have attained considerable attention currently. We here theoretically investigate the phenomena of slow and superluminal light based on the MoS2 resonator system driven by two-tone fields. Superluminal and ultraslow probe light without absorption can be obtained via manipulating the pump laser on- and off-resonant with the exciton frequency under different parameters regimes, respectively, of which the magnitude is larger than that in a carbon nanotube resonator. The bandwidth of the probe spectrum determined by the quality factor Q of MoS2 resonator is also presented. Furthermore, we also demonstrate the phenomenon of phonon induced transparency and show an optical transistor in the system. The all-optical device based on MoS2 resonator may indicate potential chip-scale applications in quantum information with the currently popular pump-probe technology.


Introduction
Slow light effect [1] in ultracold atoms induced by electromagnetically induced transparency (EIT) [2] has been derived with numerous important developments in optical physics [3] and has inspired various potential applications [4]. Besides the phenomenon of slow light, superluminal phenomena are also observed in an atomic system [5] and a crystal structure [6]. At present, several schemes for obtaining slow light have also been proposed and realized, including coherent population oscillations in solids [7], stimulated Brillouin and Raman scattering in optical fibers [8], and wave-mixing in nematic liquid crystals [9]. Recently, cavity optomechanics which studies the interaction between light and mechanical oscillation [10] has become a rapidly developing field and plays an important role in many fields of physics, including gravitational-wave detectors [11], cooling of mechanical oscillators [12], and mass sensing [13]. Furthermore, the manipulation of light propagation, such as the slow and superluminal light, has also been investigated both theoretically and experimentally in optomechanical systems [14,15], and one has been successfully observed an optically tunable delay of 50 ns with near-unity optical transparency, and superluminal light with a 1.4 s signal advance [15].
On the other hand, atomically thin two-dimensional (2D) layered materials, such as monolayer molybdenum disulfide (MoS 2 ) with enormous stiffness, low density, intrinsically small size, and excellent mechanical properties, have attracted great interest recently due to their superior electrical-optical properties and potential applications [16]. MoS 2 is a semiconductor with the thickness-dependent electronic structure [17,18], and with a decrease in the layer numbers of MoS 2 , the material undergoes a transition from an indirect bandgap (1.2 eV in bulk) to a direct bandgap semiconductor (1.8 eV in monolayer) [19]. Monolayer MoS 2 has potential applications in photodetection [20,21], photovoltaics [22], and MoS 2 -based transistors [23,24] due to this intrinsic property. Monolayer MoS 2 could be considered as the ultimate nanomaterials to structure micromechanical resonators, and additionally, multilayer MoS 2 nanomechanical resonator [25,26] and single-layer MoS 2 mechanical resonator [27] have also been realized experimentally. It is noticed that although these significant impacts and immense applications based on monolayer/multilayer MoS 2 system have been demonstrated, the coherent optical propagation in MoS 2 -based micromechanical resonator systems in all-optical domain has not been undertaken yet.
In this paper, we present a tunable slow-and fast-light device based on a platelike circular monolayer MoS 2 suspended on the Si/SiO 2 substrate [28] driven by two-tone fields. Due to the interaction of excition-phonon in the system, the probe laser will undergo a significant velocity change when applying a strong pump laser with a detuning under different parameters regimes, such as different exciton-resonator coupling and different resonance detuning. Adjusting the pump laser on-and off-resonance with the exciton frequency in the resonator, the probe laser group velocity undergoes fast light and slow light with little absorption. Furthermore, the linewidth of signal light spectrum is determined by the quality factor Q of MoS2 resonator. The larger the quality factor of MoS 2 resonator is, the narrower the probe spectrum bandwidth is. In addition, electromagnetically induced absorption (EIA) and parametric amplification (PA) are also demonstrated with manipulating the pump field intensity, which may serve as a quantum optical transistor. Figure 1 gives the model based on monolayer MoS 2 [28,29] with the optical pump-probe technology. Both experiments [30] and theories [31] have studied the vibrational properties of bulk, few layer, and monolayer MoS 2 recently. Although structural nonidealities and asymmetries of device based on MoS 2 nanostructures will occur and cause new mechanical resonances [32], for the sake of the analytical simplicity, we still take into account the model system as shown in Fig. 1 [25,27]. 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. So 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 neglected compared with the fundamental flexural mode.  [13] induce the coupling between the exciton in monolayer MoS 2 resonator and the flexural motion, and therefore the interaction Hamiltonian between the resonator and the two-level exciton is

System and methods
a coupling strength  . For the exciton-phonon interaction, symmetry-dependent exciton-phonon coupling in 2D and bulk MoS 2 observed by resonance Raman scattering has been investigated by Carvalho et al. [33]. They showed the 1g A phonon mode is enhanced by the A and B excitons, however, the enhancement decreases with a decreasing number of layers, due to the dependence of the lifetime of the intermediate excitonic states on the number of layers. Except the distinct exciton-phonon coupling can be explained considering the symmetries of the 1g A and 1 2 g E phonons associated with the A, B, and C excitons, the symmetry-dependent exciton-phonon coupling can be also discussed using group theory arguments [32]. In addition, the electron-phonon coupling is also investigated by Raman and resonance Raman spectra of MoS2 nanoparticles, and the electron-phonon coupling responsible for the strong-resonance conditions is identified through dynamic band calculations [34].
On the other hand, in the monolayer MoS 2 resonator system, there are multi-phonon modes exist in the system. Discussing all the phonon modes couple to the exciton is inefficient as the system size increases, and it is also unnecessary, because usually only a few modes or a signal mode within a small bandwidth is used for transmitting quantum states or quantum information processing. Therefore, we only consider the exciton and longitudinal optical phonons (LOPs) coupling in the system, and the optical propagation is investigated by the strong exciton-LOP interaction, in which the exciton behaves as an optical cavity and the phonon like a mechanical resonator analogy in a cavity optomechanics system.
The pump-probe technology [35] affords an effective method to investigate light-matter interaction. Applying the scheme to the monolayer MoS2 resonator system, indicates the Hamiltonian of the exciton coupled to the two fields, where  is the dipole moment of the exciton, and i E is the slowly varying envelope of the field. In a rotating frame of the pump field frequency c  , we obtain the total Hamiltonian of the system as [28,29] are the detuning of the pump frequency from the exciton frequency and probe frequency, respectively.
Writing the Heisenberg equations of motion and adding dissipation of the corresponding damping and noise terms, we obtain the following quantum Langevin equations as The resonator mode is affected by a Brownian stochastic force with zero mean value, and ( ) t  has the correlation function as [28] i ( ) where B k and T are the Boltzmann constant and the temperature of the reservoir, respectively.
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: The steady-state values are governed by the pump power and the small fluctuations by the probe power. In the steady state, disregarding the probe field, the time derivatives vanish, and the static solutions for the population inversion ( 0 ) of the exciton obey the following algebraic equation Keeping only the linear terms of the fluctuation operators, we make the ansatz [13] i i Solving the equation set and working to the lowest order in s E but to all orders in c E , we obtain the linear optical susceptibility as The imaginary and real parts of (1) ( ) s   indicate absorption and dispersion, respectively.
Based on the MoS 2 resonator system, we can determine the light group velocity as [36] / [

Results and discussion
The parameters of monolayer MoS 2 resonator are shown as follows [28,29]  , respectively. We find that the output probe light can be about 10 times faster than input probe light in vacuum simply via tuning the pump laser on the resonant with exciton frequency in MoS 2 resonator system at 0.07   as shown in Fig. 2(c). However, when increasing the exciton-resonator coupling strength to 0.4

 
, the evolution of group velocity index of probe laser g n becomes more complicated as shown in Fig. 2(d), and there is a saltus with an increase in the Rabi frequency displays an analogous optomechanically-induced transparency [38] and termed as phonon-induced transparency (PIT) induced by the exciton-resonator coupling, and the phenomenon has been demonstrated in the coupled mechanical resonators system [39] and bilayer graphene nanoribbons [40]. Obviously, with an increase in the exciton-resonator coupling strength, the absorption spectrum splits into two peaks, which is analogous to the Rabi splitting of two-level systems in quantum optics and has a zero absorption point at 0 s   . Compared with the coupled mechanical resonators [39], the manufacturing process of the suspended monolayer MoS 2 nanomechanical resonator system is simple, and the frequency of the resonator can approach to gigahertz (In [32], the frequency of the mechanical mode is about kilohertz). The realistic monolayer MoS 2 resonator can also exhibit a wide range of variations depending on its size-dependent parameters. Therefore, both the high and low frequencies of the mechanical resonator are allowed in the scheme. The so-called PIT appears 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 [40]. However, in the MoS 2 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 [40] 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 MoS 2 resonator [13]. Our demonstration opens an avenue for the exploration of slow light in this monolayer MoS 2 resonator system. , respectively. That is, the output signal pulse will be 200 and 156 times slower than the input light with a single MoS 2 resonator under two different coupling strengths. Compared with Fig. 3(c), the curve is steeper in Fig. 3(d) due to the stronger exciton-resonator coupling in the system. The physical origin of this result is the coupling between exciton and MoS 2 resonator vibration, which makes quantum interference between the MoS 2 resonator and the two optical fields via the exciton as Similar results in Fig. 3(c) have also been obtained in the carbon nanotube resonator, and the achieved slowdown of the group velocity in the MoS 2 nanoresonator is a bit higher than in the carbon nanotube resonator [41].

Rabi frequency
Imχ (1) (ωs)(a.u.) Reχ (1) (ωs)(a.u.) From this figure, we can demonstrate that the width of the signal spectrum decreases as the quality factor Q increases. Therefore, the larger the MoS 2 resonator quality factor is, the narrower of the probe spectrum width is. As a result, the MoS 2 resonator with high quality factor is beneficial to the transparency window. Due to the high quality factor (or short decay rate) of MoS 2 resonator, the slow light and fast light effect performed in MoS 2 is obviously better than that in other quantum systems such as quantum dots carbon nanotube resonator [41].  Switching the pump-exciton detuning to the blue sideband c m     and increasing the pump field intensity, the probe transmission displays a deeper dip as shown in Fig. 6(a). In Fig. 6(a), the negative transmission of the probe laser with an increase in the pump intensity is the so-called electromagnetically induced absorption (EIA) [42]. However, with further increasing the pump laser (GHz) c  intensity, the system switches from EIA to parametric amplification (PA) resulting in the probe laser amplification as shown in Fig. 6(b), which has been demonstrated in the conventional optomechanical system [42]. The elliptical inset in Fig. 6 shows that there exists a turning point around 2 2 0.008(GHz) c   which switches the probe transmission from EIA to PA. Therefore, the MoS 2 resonator system can not only switch the weak probe laser from off to on, but also serve as a quantum optomechanical transistor due to the probe amplifier effect. Turning off the pump laser, the weak probe laser displays the transmission spectrum owning to the usual exciton absorption resonance as shown in Fig. 6(c). However, turning on the pump laser and fixing the pump-exciton detuning c m     , the dip switches to a transmission peak immediately as shown in Fig. 6(d). This amplification comes from the quantum interference between the phonons and the beat of the two optical fields via the exciton in the MoS 2 resonator. Due to dressing with the phonon modes, the original two levels of exciton in the MoS 2 resonator system split into several metastable levels. When applying a strong pump laser to the system, the electrons can transit between the metastable levels, which induces the constructive interference, and eventually amplifies the weak probe laser.

Conclusions
We have theoretically investigated the coherent optical propagate based on monolayer MoS 2 nano-optomechanical system driven by two-tone fields. The dispersion of the probe light together with zero absorption indicates the potential for ultraslow light and the possibility of superluminal light based on the coupled monolayer MoS 2 resonator system with manipulating pump laser onand off-resonant with exciton frequency in the resonator system. In addition, the bandwidth of the signal spectrum determined by the quality factor of monolayer MoS 2 resonator system is also demonstrated. Further, electromagnetically induced absorption and parametric amplification are demonstrated by the exciton-resonator coupling in the system, which may suggest a quantum optical transistor. Our study, therefore, will indicate a new avenue for fabricating nanomechanical resonator systems based on layered nanomaterials and such a tunable slow and fast light device based on graphene may have potential applications in optical networks and engineering in nanometer scale.