Electromagnetically induced transparency and slow light in two-mode optomechanics

Cheng Jiang; Hongxiang Liu; Yuanshun Cui; Xiaowei Li; Guibin Chen; Bin Chen

doi:10.1364/OE.21.012165

1. Introduction

The emerging field of cavity optomechanics [1–4] studies the interaction between optical and mechanical modes via radiation pressure force, which enables to observe quantum mechanical behavior of macroscopic systems. Recent progress in fabrication and cooling techniques paves the way towards realizing strong coupling at the single-photon level in optomechanical systems [5–10] and cooling the nanomechanical resonators to their quantum ground state [11,12]. Moreover, the optical response of optomechanical systems is modified because of mechanical interactions, leading to the phenomenon of normal-mode splitting [13, 14] and electromagnetically induced transparency (EIT) [15–18]. In EIT [19] an opaque medium can be made transparent in the presence of a strong pump beam; the concomitant steep variation of the refractive index induces a drastic reduction in the group velocity of a probe beam, which can be used to slow and stop light [20, 21]. EIT has been first observed in atomic vapors [22] and recently in various solid state systems such as quantum wells [23], metamaterial [24] and nitrogen-vacancy centers [25]. In optomechanical systems, slowing and advancing of signals based on EIT have been observed both in optical [17, 18] and microwave domains [26]. Recently, the phenomena of electromechanically induced amplification [27] and absorption [28] with a blue-detuned pump field have also been presented in the circuit nano-electromechanical system consisted of a superconducting microwave resonator and a nanomechanical beam.

Most recently, two-mode optomechanics in which two optical modes are coupled to a mechanical mode have received a lot of research interest. Dobrindt et al.[29] have shown that the dual mode transducer leads to a dramatic reduction of the power to reach the standard quantum limit (SQL) for a high frequency resonator. Ludwig et al.[30] and Kómár et al.[31] have theoretically shown that quantum nonlinearities can be enhanced significantly in two-mode optomechanical systems, which can be used in optomechanical quantum information processing with photons and phonons [32]. However, the theoretical work of Dobrindt [29], Ludwig [30], and Kómár [31] require that the mechanical frequencies are nearly resonant to the optical level splitting, which is more demanding to realize experimentally. Recently, Qu and Agarwal [33] theoretically showed that double cavity optomechanical systems can be used both as memory elements as well as for the transduction of optical fields. Hill et al.[34] and Dong et al.[35] have experimentally demonstrated coherent wavelength conversion of optical photons between two different optical wavelengths in optomechanical crystal nanocavity and silica resonator, respectively. In the present paper, we investigate the optical response of the two-mode optomechanical system in the simultaneous presence of two strong pump beams and a weak probe beam. When the two cavities are pumped on their red sidebands (i.e., one mechanical frequency, ω_m, below cavity resonances, ω₁ and ω₂), respectively, a transparency window appears in the probe transmission spectrum.

2. Model and theory

We consider an optomechanical system as shown in Fig. 1, where two optical cavity modes a_k(k = 1, 2) are coupled to a common mechanical mode b. The left cavity is driven by a strong pump beam E_L with frequency ω_L and a weak probe beam E_p with frequency ω_p simultaneously, and the right cavity is only driven by a strong pump beam E_R with frequency ω_R. In a rotating frame at the pump frequency ω_L and ω_R, the Hamiltonian of the two-mode optomechanical system reads as follows [34]:

\begin{array}{l} H = & \sum_{k = 1, 2} \bar{h} Δ_{k} a_{k}^{†} a_{k} + \bar{h} ω_{m} b^{†} - b - \sum_{k = 1, 2} \bar{h} g_{k} a_{k}^{†} a_{k} (b^{†} + b) + i \bar{h} \sqrt{κ_{e, 1}} E_{L} (a_{1}^{†} - a_{1}) \\ + i \bar{h} \sqrt{κ_{e, 2}} E_{R} (a_{2}^{†} - a_{2}) + i \bar{h} \sqrt{κ_{e, 1}} E_{p} (a_{1}^{†} e^{- i δ t} - a_{1} e^{i δ t}) . \end{array}

The first term describes the energy of the two optical cavity modes with resonance frequency ω_k(k = 1, 2), where

a_{k}^{†}

(a_k) is the creation (annihilation) operator of each cavity mode. Δ₁ = ω₁ − ω_L and Δ₂ = ω₂ − ω_R are the corresponding cavity-pump field detunings. The second term gives the energy of the mechanical mode with creation (annihilation) operator b^† (b), resonance frequency ω_m and effective mass m. The third term is the radiation pressure coupling rate

g_{k} = (ω_{k} / L_{k}) \sqrt{\bar{h} / (2 m ω_{m})}

, where L_k is an effective length that depends on the cavity geometry. The last three terms represent the input fields, where E_L, E_R, and E_p are related to the power of the applied laser fields by

| E_{L} | = \sqrt{2 P_{L} κ_{1} / \bar{h} ω_{L}}

,

| E_{R} | = \sqrt{2 P_{R} κ_{2} / \bar{h} ω_{R}}

, and

| E_{p} | = \sqrt{2 P_{p} κ_{1} / \bar{h} ω_{p}}

(κ_k the linewidth of the kth cavity mode), respectively. The total cavity linewidth κ_k = κ_i,k + κ_e,k, where κ_e,k is the cavity decay rate due to coupling to an external photonic waveguide, as presented in the realistic two-mode optomechanical nanocavity [34]. δ = ω_p − ω_L is the detuning between the probe filed and the left pump field.

Fig. 1 Schematic of a two-mode optomechanical system where two optical cavity modes, a₁ and a₂, are coupled to the same mechanical mode b. The left cavity is driven by a strong pump beam E_L in the simultaneous presence of a weak probe beam E_p while the right cavity is only driven by a pump beam E_R.

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Applying the Heisenberg equations of motion for operators a₁, a₂, and Q which is defined as Q = b^† + b and introducing the corresponding damping and noise terms [36], we derive the quantum Langevin equations as follows:

{\dot{a}}_{1} = - i (Δ_{1} - g_{1} Q) a_{1} - κ_{1} a_{1} + \sqrt{κ_{e, 1}} (E_{L} + E_{p} e^{- i δ t}) + \sqrt{2 κ_{1}} a_{in, 1},

{\dot{a}}_{2} = - i (Δ_{2} - g_{2} Q) a_{2} - κ_{2} a_{2} + \sqrt{κ_{e, 2}} E_{R} + \sqrt{2 κ_{2}} a_{in, 2},

\ddot{Q} + γ_{m} \dot{Q} + ω_{m}^{2} Q = 2 g_{1} ω_{m} a_{1}^{†} a_{1} + 2 g_{2} ω_{m} a_{2}^{†} a_{2} + ξ,

where a_in,1 and a_in,2 are the input vacuum noise operators with zero mean value, ξ is the Brownian stochastic force with zero mean value [36].

Following standard methods from quantum optics, we derive the steady-state solution to Eqs. (2)–(4) by setting all the time derivatives to zero. They are given by

a_{s, 1} = \frac{\sqrt{κ_{e, 1}} E_{L}}{κ_{1} + i Δ_{1}^{'}}, a_{s, 2} = \frac{\sqrt{κ_{e, 2}} E_{R}}{κ_{2} + i Δ_{2}^{'}}, Q_{s} = \frac{2}{ω_{m}} (g_{1} {| a_{s, 1} |}^{2} + g_{2} {| a_{s, 2} |}^{2}),

where Δ′₁ = Δ₁ − g₁Q_s and Δ′₂ = Δ₂ − g₂Q_s are the effective cavity detunings including radiation pressure effects. We can rewrite each Heisenberg operator of Eqs. (2)–(4) as the sum of its steady-state mean value and a small fluctuation with zero mean value,

a_{1} = a_{s, 1} + δ a_{1}, a_{2} = a_{s, 2} + δ a_{2}, Q = Q_{s} + δ Q .

Inserting these equations into the Langevin equations Eqs. (2)–(4) and assuming |a_s,1| ≫ 1 and |a_s,2| ≫ 1, one can safely neglect the nonlinear terms

δ a_{1}^{†} δ a_{1}

,

δ a_{2}^{†} δ a_{2}

, δa₁δQ, and δa₂δQ. Since the drives are weak, but classical coherent fields, we will identify all operators with their expectation values, and drop the quantum and thermal noise terms [16]. Then the linearized Langevin equations can be written as:

〈 δ {\dot{a}}_{1} 〉 = - (κ_{1} + i Δ_{1}) 〈 δ a_{1} 〉 + i g_{1} Q_{s} 〈 δ a_{1} 〉 + i g_{1} a_{s, 1} 〈 δ Q 〉 + \sqrt{κ_{e, 1}} E_{p} e^{- i δ t},

〈 δ {\dot{a}}_{2} 〉 = - (κ_{2} + i Δ_{2}) 〈 δ a_{2} 〉 + i g_{2} Q_{s} 〈 δ a_{2} 〉 + i g_{2} a_{s, 2} 〈 δ Q 〉,

〈 δ \ddot{Q} 〉 + γ_{m} 〈 δ \dot{Q} 〉 + ω_{m}^{2} 〈 δ Q 〉 = 2 ω_{m} g_{1} a_{s, 1} (〈 δ a_{1} 〉 + 〈 δ a_{1}^{†} 〉) + 2 ω_{m} g_{2} a_{s, 2} (〈 δ a_{2} 〉 + 〈 δ a_{2}^{†} 〉) .

In order to solve equations (7)–(9), we make the ansatz [37] 〈δa₁〉 = a₁₊e^−iδt + a₁₋e^iδt, 〈δa₂〉 = a₂₊e^−iδt + a₂₋e^iδt, and 〈δQ〉 = Q₊e^−iδt + Q₋e^iδt. Upon substituting the above ansatz into Eqs. (7)–(9), we derive the following solution

a_{1 +} = \frac{\sqrt{κ_{e, 1}} E_{p}}{κ_{1} + i Δ_{1}^{'} - i δ} - \frac{1}{d (δ)} \frac{i g_{1}^{2} n_{1} \sqrt{κ_{e, 1}} E_{p}}{{(κ_{1} + i Δ_{1}^{'} - i δ)}^{2}},

where

d (δ) = \sum_{k = 1, 2} \frac{2 Δ_{k}^{'} g_{k}^{2} n_{k}}{{(κ_{k} - i δ)}^{2} + Δ_{k}^{' 2}} - \frac{ω_{m}^{2} - δ^{2} - i δ γ_{m}}{ω_{m}},

and n_k = |a_s,k|². Here n_k, approximately equal to the number of pump photons in each cavity, is determined by the following coupled equations

n_{1} = \frac{κ_{e, 1} E_{L}^{2}}{κ_{1}^{2} + {[Δ_{1} - 2 g_{1} / ω_{m} (g_{1} n_{1} - g_{2} n_{2})]}^{2}},

n_{2} = \frac{κ_{e, 2} E_{R}^{2}}{κ_{2}^{2} + {[Δ_{2} - 2 g_{2} / ω_{m} (g_{1} n_{1} - g_{2} n_{2})]}^{2}} .

The output field can be obtained by employing the standard input-output theory [38] $a_{out} (t) = a_{in} (t) - \sqrt{κ_{e}} a (t)$ , where a_out(t) is the output field operator. Considering the output field of the left cavity, we have

〈 a_{out} (t) 〉 = (E_{L} - \sqrt{κ_{e, 1}} a_{s, 1}) e^{- i ω_{L} t} + (E_{p} - \sqrt{κ_{e, 1}} a_{1 +}) e^{- i (δ + ω_{L}) t} - \sqrt{κ_{e, 1}} a_{1 -} e^{i (δ - ω_{L}) t}

The transmission of the probe field, defined by the ratio of the output and input field amplitudes at the probe frequency, is then given by

\begin{array}{l} t (ω_{p}) & = & \frac{E_{p} - \sqrt{κ_{e, 1}} a_{1 +}}{E_{p}} \\ = & 1 - [\frac{κ_{e, 1}}{κ_{1} + i Δ_{1}^{'} - i δ} - \frac{1}{d (δ)} \frac{i g_{1}^{2} n_{1} κ_{e, 1}}{{(κ_{1} + i Δ_{1}^{'} - i δ)}^{2}}] . \end{array}

The rapid phase dispersion ϕ = arg[t(ω_p)] of the transmitted probe laser beam leads to a group delay τ_g expressed as

τ_{g} = {\frac{d ϕ}{d ω_{p}} |}_{ω_{p} = ω_{1}} .

Note that, if E_R = 0 and g₂ = 0, the Eqs (10)–(16) lead to the well-known results for the single mode cavity optomechanical system, where electromagnetically induced transparency and slow light effect have been observed experimentally [16, 17]. In what follows, we will investigate theoretically this phenomenon in the two-mode optomechanics we consider here.

3. Results and discussion

To illustrate the numerical results, we choose a realistic two-mode cavity optomechanical system to calculate the transmission spectrum of the probe field. The parameters used are [34]: ω₁ = 2π × 205.3 THz, ω₂ = 2π × 194.1 THz, κ₁ = 2π × 520 MHz, κ₂ = 1.73 GHz, κ_e,1 = 0.2κ₁, κ_e,2 = 0.42κ₂, g₁ = 2π × 960 kHz, g₂ = 2π × 430 kHz, ω_m = 2π × 4 GHz, Q_m = 87 × 10³, where Q_m is the quality factor of the nanomechanical resonator, and the damping rate γ_m is given by $\frac{ω_{m}}{Q_{m}}$ . We can see that ω_m > κ₁ and ω_m > κ₂, therefore the system operates in the resolved-sideband regime also termed good-cavity limit which is a prerequisite for the ground state cooling of a mechanical resonator [39].

Characterization of the optomechanical cavity can be performed by using two strong pump beams combined with a weak probe beam. With both pump beams detuned a mechanical frequency to the red of their respective cavity modes (Δ₁ = Δ₂ = ω_m), a weak probe beam is then swept across the left cavity mode. The resulting transmission spectra of the probe beam as a function of the probe-cavity detuning Δ_p = ω_p − ω₁ are plotted in Fig. 2, where P_L = 0, 0.1, 1 and 10 μW, respectively, while the power of the right pump beam P_R is kept equal to 0.1 μW. When P_L = 0 μW, there is a transmission dip in the center of the probe transmission spectrum, as shown in Fig. 2(a). However, as P_L = 0.1 μW, the broad cavity resonance splits into two dips and a narrow transparency window appears when the probe beam is resonant with the cavity frequency. As the left pump power increases, and hence effective coupling strength $G_{1} = g_{1} \sqrt{n_{1}}$ , increase further, so does the probe transmission at the cavity resonance. The width of the transparency window also increases and is given by the modified mechanical damping rate $γ_{m}^{eff} \approx γ_{m} (1 + \frac{4 g_{1}^{2} n_{1}}{κ_{1} γ_{m}} + \frac{4 g_{2}^{2} n_{2}}{κ_{2} γ_{m}})$ [16, 34, 39]. This mechanically mediated electromagnetically induced transparency can be understood as a result of radiation pressure force oscillating at the beat frequency δ = ω_p − ω_L between the pump beam and the probe beam. If this driving force is close to the mechanical resonance frequency ω_m, the vibrational mode is excited coherently, resulting in Stokes and anti-Stokes scattering of light from the strong pump field. If the cavity is driven on its red sideband, the highly off-resonant Stokes scattering is suppressed and only the anti-Stokes scattering builds up within the cavity. However, when the probe beam is resonant with the cavity, destructive interference with the anti-Stokes field suppresses its build-up and hence a transparency window appears in the probe transmission spectrum. These processes are captured by the linearized Langevin Eqs. (7)–(9). In the resolved sideband regime (κ₁, κ₂ < ω_m), when the pump beam detuning Δ′₁ = Δ′₂ ≈ ω_m, the lower sideband can be neglected, i.e., a₁₋ ≈ 0 and a₂₋ ≈ 0 [16]. The solution for the left intracavity field reads

a_{1 +} \frac{\sqrt{κ_{e, 1}} E_{p}}{- i x + κ_{1} + \frac{g_{1}^{2} n_{1} / 2}{- i x + γ_{m} / 2 + \frac{g_{2}^{2} n_{2} / 2}{- i x + κ_{2}}}},

where x(= δ − ω_m) represents the detuning of the probe frequency to the cavity frequency. When g₂ = 0, Eq. (17) leads to the result for the single cavity [16]. This solution has a form well known from the response of an EIT medium to a probe field [19]. The role of the control laser’s Rabi frequency in an atomic system is taken by the parametrically enhanced optomechanical coupling rate

G_{1} = g_{1} \sqrt{n_{1}}

and

G_{2} = g_{2} \sqrt{n_{2}}

, and the two-photon resonance condition is given by Δ′₁ = Δ′₂ = ω_m.

Fig. 2 Probe transmission as a function of the probe-cavity detuning Δ_p = ω_p − ω₁ for left pump power P_L equals to 0, 0.1, 1, and 10 μW, respectively. The right pump power is kept equal to 0.1 μW. Both the cavities are pumped on their respective red sidebands, i.e., Δ₁ = ω_m and Δ₂ = ω_m. Other parameters used are ω₁ = 2π × 205.3 THz, ω₂ = 2π × 194.1 THz, κ₁ = 2π × 520 MHz, κ₂ = 1.73 GHz, κ_e,1 = 0.2κ₁, κ_e,2 = 0.42κ₂, g₁ = 2π × 960 kHz, g₂ = 2π × 430 kHz, ω_m = 2π × 4 GHz, Q_m = 87 × 10³.

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Much as in atomic EIT, this effect causes an extremely steep dispersion for the transmitted probe photons, leading to a group delay. Fig. 3 shows the magnitude and phase dispersion of the probe transmission as a function of probe-cavity detuning Δ_p with Δ₁ = Δ₂ = ω_m for P_L = 10 μW and P_R = 0.1 μW. It can be seen clearly that there is a transparency window combined with a steep positive phase dispersion at the cavity resonance, which will result in a tunable group delay of the transmitted probe beam. In addition, the delay in transmission is directly related to the advance on reflection through the bare cavity transmission contrast. To verify this, we plot the corresponding transmission group delay $τ_{g}^{(T)}$ of the probe beam versus the left pump power P_L with Δ₁ = Δ₂ = ω_m for P_R = 0.1 μW in Fig. 4(a). As can be seen from the figure, the maximum transmission delay is $τ_{g}^{(T)} \approx 4.5 ns$ . However, when the right pump beam beam is turned off, the transmission delay can be significantly increased, with a maximum delay 4 μs. Therefore, the group delay in the two-mode optomechanical system is smaller than the delay in the single optical cavity system. The physical origin can be explained as follows. When the two cavities are pumped on their respective red sidebands, the mechanical oscillation is damped by both the left and right cavity mode, with the total damping rate $γ_{m}^{eff} \approx γ_{m} (1 + \frac{4 g_{1}^{2} n_{1}}{κ_{1} γ_{m}} + \frac{4 g_{2}^{2} n_{2}}{κ_{2} γ_{m}})$ . The width of the transparency window is larger than the one in the single cavity where g₂ = 0, and the phase gradient in the probe transmission in the left cavity gets smaller. Therefore, the pump on the right cavity leads to the reduction of the group delay of the transmitted probe beam. However, much more flexible controllability of EIT and slow light effect in two-mode optomechanics can find potential applications in quantum memory [33] and coherent optical wavelength conversion [34], which is useful in communication system. In Fig. 4(c), we consider the effect of the external decay rate κ_e,1 on the group delay. If the external decay dominates the decay of the cavity, κ_e,1 = 0.6κ for example, the maximum transmission group delay can be increased further. Moreover, the group delay of the reflected probe beam as a function of the power of the left pump beam is plotted in Fig. 4(d) where the delay is negative, which represents the advancing of light pulses. Therefore, we can tune the group delay and advance of the probe beam by controlling the power of the pump beam.

Fig. 3 (a) Magnitude and (b) phase of the transmitted probe beam versus probe-cavity detuning Δ_p for P_L = 10 μW and P_R = 0.1 μW. Other parameters are the same with figure 2.

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Fig. 4 Group delay τ_g of the (a)–(c) transmitted (d) reflected probe beam as a function of the left pump power P_L with Δ₁ = ω_m and Δ₂ = ω_m considering the effects of κ_e,1 and P_R. Other parameters are ω₁ = 2π × 205.3 THz, ω₂ = 2π × 194.1 THz, κ₁ = 2π × 520 MHz, κ₂ = 1.73 GHz, κ_e,2 = 0.42κ₂, g₁ = 2π × 960 kHz, g₂ = 2π × 430 kHz, ω_m = 2π × 4 GHz, Q_m = 87 × 10³.

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4. Conclusion

In conclusion, we have demonstrated both electromagnetically induced transparency and amplification in a two-mode cavity optomechanics consisted of two optical cavity modes coupled to a common mechanical mode under different driving conditions. Destructive interference between the probe beam and the anti-Stokes field leads to a transparency window in the probe transmission spectrum in conjunction with a steep positive phase dispersion, giving rise to the corresponding slow light effect. Our theoretical results show an optically tunable delay of 4 μs of the transmitted probe beam.

Acknowledgments

The authors gratefully acknowledge support from National Natural Science Foundation of China (Grant Nos. 11074088 and 11174101) and Jiangsu Natural Science Foundation (Grant No. BK2011411).

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Electromagnetically induced transparency and slow light in two-mode optomechanics

Abstract

1. Introduction

2. Model and theory

3. Results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (17)

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