Switching freely between superluminal and subluminal light propagation in a monolayer MoS<sub>2</sub> nanoresonator

Jian-Bo Li; Si Xiao; Shan Liang; Meng-Dong He; Nam-Chol Kim; Yongfeng Luo; Jian-Hua Luo; Li-Qun Chen

doi:10.1364/OE.25.013567

1. Introduction

In recent decades, a great deal of interest has been focused on how to advance or slow the propagation of light in the atomic vapors [1], solid-state materials [2], and nanomechanical oscillator systems [3] due to their potential applications in electronics and optoelectronics [4]. Fast and slow light effects can be observed by using coherent population oscillations [2,5], electromagnetically induced transparency [1,6], and stimulated Brillouin scattering [7]. Recently, Zhu et al. have proposed a tunable slow and fast light device based on a carbon nanotube resonator [8]. Javed Akram et al. have reported tunable fast and slow light effects of a transmitted probe field in a hybrid optomechanical system consisting of a high Q Fabry-Perot cavity, a mechanical resonator and a two-level atom, and proposed a tunable switch from superluminal to slow light is achievable in their model by simply adjusting the atomic detuning [9]. Then they have also studied Fano resonances and slow light in a nanocavity using Bose-Einstein condensate. The results showed that the slow light effect can be enhanced by the strength of the atom-atom interaction and its robustness against the condensate fluctuations using presently available technology [10]. Slow and fast light effects can be switched in the microwave regime using a coupled nanomechanical resonator (NR)-Cooper-pair box (CPB) system [11]. Very recently, Dumeige’s group has demonstrated that it is feasible to enhance the photon lifetime by several orders of magnitude by using slow-light effects in an active whispering-gallery-mode (WGM) microresonator [12]. Based on slow light in cavity optomechanics, an ultrasensitive mass sensing method has also been presented [13].

Compared with one-dimensional carbon nanotube with small bandgap, molybdenum disulfide (MoS₂) is an attractive semiconductor whose electronic structure depends on its layer number [14–20], which further promotes and extends the applications of MoS₂. As a new member of two-dimensional materials, layered MoS₂ with a large bandgap and surface-to-volume ratio possesses many excellent properties such as ultralow weight [15], exceptional strain limit [16], high elastic modulus [17], etc. Currently, significant research interest is devoted towards practical applications of such materials for transistors [18], photodetectors [20], small-signal amplifier [19], etc. Especially, nonlinear optical behaviors in MoS₂ nanosheets have been extensively studied [21–25]. Degenerate two-photon absorption [22], ultrafast saturable absorption [23], and harmonic effects [24] in MoS₂ nanosheets were reported one after another. Recent investigation has also shown that one can observe the dynamic self-diffraction in MoS₂ nanoflake solutions with laser [25]. Despite these significant impacts and immense applications, though, very few reports have been published on fast and slow light effects in MoS₂-based materials. Our main objective in this paper is to pave a new way to advance and slow light propagation using a suspended monolayer MoS₂ NR and open a new horizon for switching freely between fast (superluminal) and slow (subluminal) light propagation. The dependencies of the group velocity of the probe light on the excitation frequency, pumping intensity and exciton-phonon coupling strength will be studied and the absorption and dispersion spectra strongly correlated to fast and slow light effects will be discussed.

2. Model and formalism

We consider an optomechanical system based on monolayer MoS₂ suspended on a Si/SiO₂ substrate [26]. As schematically represented in Fig. 1(a), the system is driven by a strong pump field accompanying a weak probe field. E_pu (E_pr) is the slowly varying envelope of the pump (probe) field, and ω_pu (ω_pr) is the frequency of the pump (probe) field. In such structure, the lowest-energy resonance corresponds to the fundamental in-plane flexural mode with the frequency ω_n. The MoS₂ NR is assumed to be characterized by a 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 can be neglected as compared to the fundamental flexural mode [27]. Similar with a carbon nanotube, due to quantum confinement, the MoS₂ can act like as a QD when electrons are confined to a small region within a MoS₂ nanosheet [28]. The eigenmode of MoS₂ can be regarded as a quantum harmonic oscillator with two bosonic operators b⁺ and b (corresponding to the phonon creation and annihilation operators, respectively). The vibration Hamiltonian of the MoS₂ NR can be written as hω_nb⁺b. A localized exciton formed in the MoS₂ NR can be modeled as a two-level system consisting of a ground state |0> and single exciton state |1>. Such an exciton can be characterized by three pseudospin operators σ₀₁, σ₁₀ and σ_z. As the exciton interacts with two applied fields [29, 30], the physical situation is illustrated in Fig. 1(b).

Fig. 1 (a) Schematic representation of monolayer MoS₂ suspended on a Si/SiO₂ substrate by an optical pump-probe technique [31]. The system is driven by a strong pump laser and detected by a weak probe laser. (b) The level scheme of a localized exciton interacting with the phonons in the MoS₂ NR.

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In a rotating frame, the total Hamiltonian of the system in the simultaneous presence of two laser fields can be written as [27]

H = ℏ Δ_{p u} σ_{z} + ℏ ω_{n} b^{+} b + ℏ g ω_{n} σ_{z} (b^{+} + b) - μ E_{p u} (σ_{10} + σ_{01}) - μ (σ_{10} E_{p r} e^{- i δ t} + σ_{01} E_{p r}^{*} e^{i δ t}),

where Δ_pu = ω₁₀ ‒ ω_pu is the exciton-pump field detuning, and δ = ω_pr – ω_pu is the probe-pump frequency difference. σ_ij = |i><j| (i, j = 0, 1) is the transition operator between the states |i> and |j>. w = 2<σ_z> in which σ_z = σ₁₁ – σ₀₀ represents the population inversion of the exciton, g denotes the coupling strength between the exciton in the MoS₂ nanosheet and the phonons in the NR, and μ refers to the electric dipole moment. For simplicity, we set Ξ = < b⁺ + b> and p = <σ₀₁>.

By applying the Heisenberg equation of motion, we obtain the corresponding quantum Langevin equations as follows [32]:

\dot{p} = - (Γ_{2} + i Δ_{p u}) p - i g ω_{n} p Ξ - i Ω w - \frac{i μ}{ℏ} E_{p r} w e^{- i δ t},

\dot{w} = - Γ_{1} (w + 1) + 2 i Ω (p^{*} - p) + \frac{2 i μ}{ℏ} (p^{*} E_{p r} e^{- i δ t} - p E_{p r}^{*} e^{i δ t}),

\ddot{Ξ} + γ_{n} \dot{Ξ} + ω_{n}^{2} Ξ = - ω_{n}^{2} g w,

where Γ₁ (Γ₂) denotes the exciton relaxation rate (dephasing rate), Ω = μE_pu/h is the effective Rabi frequency of the pump field, and γ_n = ω_n/Q refers to the decay rate of the NR.

In order to solve Eqs. (2)-(4), we make the assumptions as follows: p = p₀ + p₁e^–^δt + p_–1 e^iδt, w = w₀ + w₁e^–^iδt + w_–1e^iδt and Ξ = Ξ₀ + Ξ₁e^–^iδt + Ξ_–1e^iδt [33], where p₀, w₀, Ξ₀ is the solution of Eqs. (2)-(4) for the case in which only the pump field is present. |p₀| >> |p₁|,|p_–1|; |w₀| >> |w₁|,|w_–1|; |Ξ₀| >> |Ξ₁|,|Ξ_–1|. Upon substituting the above ansatz into Eqs. (2)-(4), we can calculate the linear susceptibility defined as

χ^{(1)} = \frac{N μ p_{1}}{ε_{0} E_{p r}} = \frac{N μ^{2}}{ε_{0} ℏ Γ_{2}} \cdot χ_{0}^{(1)} = Π_{1} \cdot χ_{0}^{(1)},

χ_{0}^{(1)} = \frac{(C_{2} - C_{1}) C_{3} Γ_{2}}{(1 + C_{1} C_{4} - C_{2} C_{4})} w_{0} + \frac{p_{0}^{*} Γ_{2}}{(1 + C_{1} C_{4} - C_{2} C_{4}) Ω},

where Π₁ = Nμ²/(ε₀hΓ₂). N is the number density of MoS₂ nanosheets, and ε₀ is the dielectric constant of the surrounding medium.

p_{0}^{*} = i Ω w_{0} / [Γ_{2} - i (Δ_{p u} - g^{2} ω_{n} w_{0})]

,

C_{1} = - (δ + i Γ_{1}) / 2 Ω

,

C_{2} = i (g^{2} ω_{n} ζ p_{0}^{*} + Ω) / [Γ_{2} - i (Δ_{p u} + δ - g^{2} ω_{n} w_{0})]

,

C_{3} = - 1 / (g^{2} ω_{n} ζ p_{0} + Ω)

,

C_{4} = - [Γ_{2} + i (Δ_{p u} - δ - g^{2} ω_{n} w_{0})] / [i (g^{2} ω_{n} ζ p_{0} + Ω)]

,

ζ = ω_{n}^{2} / (δ^{2} + i γ_{n} δ - ω_{n}^{2})

.

The exciton-population inversion w₀ can be obtained from the following third-order equation

Γ_{1} (w_{0} + 1) [Γ_{2}^{2} + {(Δ_{p u} - g g_{0} w_{0})}^{2}] + 4 Ω^{2} Γ_{2} w_{0} = 0,

In our system, the group velocity of the probe light has the following form:

v_{g} = \frac{c}{n + ω_{p r} \cdot (d n / d ω_{p r})},

where n ≈1 + 2πχ⁽¹⁾(ω_pr) [34]. Having this, the Eq. (8) will become

\frac{c}{v_{g}} - 1 = 2 π \cdot Re {(χ^{(1)})}_{ω_{p r} = ω_{10}} + 2 π ω_{p r} \cdot Re {(\frac{d χ^{(1)}}{d ω_{p r}})}_{ω_{p r} = ω_{10}},

From Eq. (9) we can find that the dispersion is very steeply positive or negative when $Re {(χ^{(1)})}_{ω_{p r} = ω_{10}} = 0$ . Considering this, the effective group velocity index n_g can be written as

n_{g} = \frac{c}{v_{g}} - 1 = Π_{2} \cdot Γ_{2} Re {(\frac{d χ_{0}^{(1)}}{d ω_{p r}})}_{ω_{p r} = ω_{10}} .

where Π₂ = 2πω_exNμ²/(ε₀hΓ₂²).

3. Numerical results and discussion

In this scheme, we choose a realistic monolayer MoS₂ NR to illustrate the numerical results. All the parameters used here are accessible in experiment. We calculate the absorption and dispersion spectra, and the group velocity for the parameters [27, 35]: ω_n = 1GHz, Q = 2000, and Γ₂ = 0.25 GHz. In our calculations, we assume Γ₁ = 2Γ₂.

To study fast and slow light effects in a monolayer MoS₂ NR, in Fig. 2(a) we show the variation of the probe absorption spectrum Imχ⁽¹⁾ as the exciton-pump detuning Δ_pu is changed. For Δ_pu = ω_n, a deep dip appears in the absorption spectrum and the value of Imχ⁽¹⁾ is approximately equal to 0, suggesting the system almost becomes fully transparent to the probe light. This is rather expected, the system will behave like a three-level system in electromagnetically induced transparency [36]. For Δ_pu = −ω_n, a sharp absorption peak occurs at the top of the absorption spectrum. For Δ_pu = 0, however, the deep dip and sharp peak both vanish and only a weak absorption peak is left. As we all know, when the MoS₂ NR is subjected to two applied fields, a beat wave with a variable beat frequency δ = ω_pr – ω_pu will appear. When the beat frequency δ is close to the vibrational frequency ω_n, the MoS₂ NR will oscillate coherently, inducing the occurrence of the anti-Stokes (δ = –ω_n) and Stokes (δ = ω_n) scattering of light via the localized exciton. The corresponding dispersion spectra are shown in Fig. 2(b). As required by the linear Kramers-Kronig relation, the complementarity between the absorption and dispersion spectra is evident.

Fig. 2 The probe absorption Imχ⁽¹⁾ (a) and dispersion Reχ⁽¹⁾ (b) spectra (in units of Π₁) as a function of the probe-pump detuning δ for different exciton-pump detunings Δ_pu. (c) The energy levels of the exciton dressed with the phonon modes of the MoS₂ NR. Each energy level is split into a doublet with a separation Ω'. TP refers to the three-photon resonance, RL represents the Rayleigh resonance, and AC denotes the ac-Stark resonance.

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To understand these new features shown in the absorption and dispersion spectra, we depict a dressed-state picture. As the MoS₂ NR is excited by a strong pump field, the original energy levels of the exciton (|0>, |1>) will be dressed by the phonon modes in the MoS₂ NR. These two energy levels are split into four dressed states |0, n>, |0, n + 1>, |1, n> and |1, n + 1> [27,33], as shown in Fig. 2(c). The sharp peak in Fig. 2(a) is attributed to the three-photon resonance (TP) corresponding to a transition from the state |0, n> to |1, n + 1> by absorbing simultaneously two pump photons and emitting a photon with frequency ω_pu – Ω'. The middle peak is assigned to the Rayleigh resonance (RL) corresponding to a transition from |0, n> to |1, n>. The deep dip is induced by the ac-Stark effect corresponding to a usual absorption resonance. Additionally, we note that the dominant role of the resonance mechanisms is strongly corresponded to the excitation frequency. In the following sections, we mainly study the propagation properties of the probe light near the anti-Stokes and Stokes sidebands.

In Figs. 3 we study the way that the effective group velocity index n_g changes with the exciton-pump detuning Δ_pu when I_pu = 3.32 mW/cm² and 6.64 mW/cm². In Fig. 3(a), for the case of I_pu = 3.32 mW/cm², as Δ_pu increases to the anti-Stokes sideband (Δ_pu₀ = –ω_n), n_g will reach a maximum value 15831.7, suggesting the occurrence of an obvious slow light effect. Such large n_g implies that a monolayer MoS₂ NR would be an excellent candidate for slow light devices. As Δ_pu further increases to the Stokes sideband (Δ_pu₁ = ω_n), the peak value of n_g is only 240.8. In addition, n_g almost keeps invariant in the region between two sidebands. For I_pu = 6.64 mW/cm, however, the scenario becomes completely different. As Δ_pu increases, n_g rises monotonically, reaching a maximum value 549.5, then starts to decrease abruptly, reaching a minimum value –3133.2 with a further increase of Δ_pu. From Fig. 3(b) we find that it is easy to switch freely between fast and slow light propagation by only adjusting the frequency of the pump field. The results also demonstrate that the fast and slow light effects are highly sensitive to the excitation frequency.

Fig. 3 The group velocity index n_g (in units of Π₂) as a function of the exciton-pump detuning Δ_pu for I_pu = 3.32 mW/cm² (a) and I_pu = 6.64 mW/cm² (b). The insets show the magnification of these remarkable regions around Δ_pu = δ = ± ω_n.

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To better visualize the I_pu–dependence of fast and slow light effects, in Fig. 4 we plot n_g as a function of I_pu with and without the exciton-phonon coupling. As the exciton-phonon coupling is absent (g = 0), the slow light effect never occurs. Here we mainly consider the case of g = 0.08 and Δ_pu = –ω_n. As shown in Fig. 4(a), the switch is turned off when I_pu = 0. With the increase of I_pu, n_g first falls monotonously, reaching a minimum –5726 at I_pu = 2.74 mW/cm². This suggests that the fast light effect emerges. In other word, the fast light is switched on. By further increasing I_pu, n_g starts to increase abruptly, reaching a maximum 31536.6 at I_pu = 3.65 mW/cm². This suggests that the slow light effect appears and the system is switched from fast light to slow light. However, the switch is turned off again at I_pu = 4.48 mW/cm². Finally, n_g declines gradually to a stable value. These results show that the system can act as a two-channel continuous-tunable switch.

Fig. 4 The group velocity index n_g (in units of Π₂) as a function of the pumping intensity I_pu for Δ_pu = –ω_n (a), Δ_pu = 0 (b), and Δ_pu = ω_n (c) when the coupling between the exciton and the phonons turns on or off.

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The curves in Fig. 4(b) show that a fast light effect appears. The adjustable range of the fast light in MoS₂ NR is a litter larger than that in a carbon nanotube resonator [8]. Also, g modification does not change the nature of the curves, which implies that the exciton-phonon coupling has a weak effect on the fast light effect as the pump field is resonant with the exciton (Δ_pu = 0). Similar results in Fig. 4(c) have also been obtained in the carbon nanotube resonator, and the achieved slowdown of the group velocity in the MoS₂ nanoresonator is a bit higher than in the carbon nanotube resonator [8]. The results in Fig. 4 demonstrate that the fast and slow light effects depend strongly on the exciton-phonon coupling strength g.

To clarify why the group velocity index changes from negative to positive values and put some insight into the nature of the fast and slow light effects, we further explore the impact of the pumping intensity and exciton-phonon coupling strength on the absorption and dispersion spectra. In Fig. 5 we show how the absorption and dispersion spectra change with the pumping intensity when Δ_pu = −ω_n. Figure 5(a) shows the results of calculations of Imχ⁽¹⁾ as a function of the pumping intensity. The magnification of the region around the anti-Stokes sideband is shown in Fig. 5(b). Note that the system supports a critical value of I_c (I_c = 3.65mW/cm²) such that when I_pu < I_c, the magnitude of the absorption peak increases and the width of this peak decreases with an increase of I_pu. When I_pu = I_c, the magnitude reaches a maximum value and the width reaches a minimum value. After this value the absorption spectrum changes suddenly from a sharp peak to a dip. As required by the linear Kramers-Kronig relation, the dispersion spectrum Reχ⁽¹⁾ exhibits a perfect complementary behavior of the absorption spectrum Imχ⁽¹⁾. The corresponding dispersion spectra are shown in Fig. 5(c). Figure 5(d) shows the magnification of the most remarkable region in Fig. 5(c). Please note that the curves in Figs. 5(b) and 5(d) are in correlation. For example, for I_pu = 3.65mW/cm² the magnitude of the absorption peak has a maximum value and the slope of the dispersion spectrum at $Re {(χ^{(1)})}_{ω_{p r} = ω_{10}} = 0$ reaches a minimum value. That is, the dispersion spectrum displays a very steep negative slope. By combining Fig. 4(a) with Fig. 5, we can draw a conclusion that a prominent slow light effect arises accompanying with a very sharp absorption peak and a large dispersion with a very steep negative sloop.

Fig. 5 The absorption Imχ⁽¹⁾ (a) and dispersion Reχ⁽¹⁾ (c) spectra as a function of the probe-pump detuning δ for a different pumping intensity I_pu. Figures 5(b) and 5(d) show the magnification of the absorption and dispersion spectra around the anti-Stokes sideband (δ = –ω_n), respectively. The other parameters used are g = 0.08 and Δ_pu = –ω_n.

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Based on the above results, it is significant to further explore the dependence of the probe absorption and dispersion spectra near the anti-Stokes sideband (δ = –ω_n) on the exciton-phonon coupling strength so as to achieve a better understanding about the fast and slow light effects. Herein, the probe absorption Imχ⁽¹⁾ and dispersion Reχ⁽¹⁾, versus the probe-pump detuning δ is plotted in Fig. 6 for Δ_pu = –ω_n, against various values of the exciton-phonon coupling strength g. Here we mainly discuss that the amplification of the most remarkable regions in Figs. 6(a) and 6(c). When the coupling between the exciton and the phonons is absent (g = 0), the curve of the absorption spectrum keeps smooth, while when the coupling turns on (g > 0), it becomes very significant. In this case, by increasing g, the width of the absorption peak decreases and the magnitude increases until g approaches a critical value g_c (g_c = 0.08). When g passes this value, the sharp peak disappears and an attractive dip appears. The features presented in the dispersion spectra are similar to that in Fig. 5(d). This provides a feasible way to measure the exciton-phonon coupling strength g. The physics behind the slow light effect can be understood as follows: as the phonons in the MoS₂ NR interact with the exciton, quantum interference between two applied fields and MoS₂ NR via the exciton will occur, inducing the emergence of the slow light.

Fig. 6 The absorption Imχ⁽¹⁾ (a) and dispersion Reχ⁽¹⁾ (c) spectra as a function of the probe-pump detuning δ for a different coupling strength g. Figures 6(b) and 6(d) show the magnification of the absorption and dispersion spectra around the anti-Stokes sideband (δ = –ω_n), respectively. The other parameters used are Δ_pu = –ω_n and I_pu = 3.65 mW/cm².

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In this paper, a monolayer MoS₂ NR is proposed to advance or slow the propagation of the light, which may lead to some significant improvements in the field of fast and slow light devices. As we all know, carbon nanotube, as an attractive functional material with low power consumption and high hardness, can be used as a good candidate for fast and slow light devices [8]. Compared with carbon nanotube resonator, the proposed MoS₂ NR with a wider bandgap and larger surface-to-volume ratio has a greater advantage than carbon nanotube resonator in term of adjustment of the speed of light. By numerical calculation, it was found that the maximal group velocity index of the slow light in the monolayer MoS₂ NR is nearly 150 times larger than that in the carbon nanotube resonator under the condition with the same number density. Specifically, it is easy to switch freely between fast and slow light propagation by only adjusting the pumping intensity in the monolayer MoS₂ NR. In other word, such a system can act as a two-channel continuous-tunable switch.

4. Summary

In summary, we studied how to switch freely between fast (superluminal) and slow (subluminal) light propagation in a monolayer MoS₂ NR. Our results showed that when the exciton-pump detuning approaches the anti-Stokes sideband, the fast and slow light effects are highly sensitive to the excitation frequency. In addition, we found that these two effects can be switched freely by only adjusting the frequency or intensity of the pump field and strong exciton-phonon coupling has a great impact on the slow light effect. Especially, the slow light effect will appear along with a large dispersion with a very steep negative slope and a sharp absorption peak. Finally, we hope that our results can be demonstrated experimentally in the near future.

Funding

National Natural Science Foundation of China (NSFC) under Grants Nos. 11404410, 11504105 and 11504434; Hunan Provincial Natural Science Foundation of China under Grants Nos. 14JJ3116 and 2015JJ3174; Foundation of Talent Introduction of Central South University of Forestry and Technology under Grant No.104-0260.

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Switching freely between superluminal and subluminal light propagation in a monolayer MoS₂ nanoresonator

Abstract

Corrections

1. Introduction

2. Model and formalism

3. Numerical results and discussion

4. Summary

Funding

References and links

Cited By

Figures (6)

Equations (10)

Optics Express