Loss-induced transparency in optomechanics

H. Zhang; F. Saif; Y. Jiao; H. Jing

doi:10.1364/OE.26.025199

1. Introduction

Cavity optomechanics (COM) [1, 2], viewed as a new milestone [3] in the history of optics, has significantly extended fundamental studies and practical applications of coherent light-matter interactions in the past decade. A wide range of COM devices, such as solid-state resonators [4], gaseous cold atoms [5], or liquid droplets [6], has been created in experiments for diverse purposes. Important applications of COM devices [2] include quantum transducer [7–9], mechanical quantum squeezing [10], coherent mechanical lasing [11–14], and ultra-sensitive motion sensing [15–17]. Another intriguing example closely related to the present work is optomechanically induced transparency (OMIT) [18–26], as already demonstrated in a wide range of experimental systems, e.g., a microtoroid resonator [20], a crystal-nanobeam system [21], a microwave circuit [22], a membrane-in-the-middle cavity [23], a cascaded COM device [24], an optical cavity coupled to Bogoliubov mechanical modes [25], and a nonlinear Kerr resonator [26]. OMIT is generally viewed as an analog of electromagnetically induced transparency (EIT) well-known in atomic physics [27,28], i.e., arising due to destructive interference of two absorption channels of the probe photons (by the cavity or the phonon mode). Beyond this picture, novel effects have also been revealed, such as nonlinear OMIT [29–32], two-color OMIT [33, 34], nonreciprocal OMIT [35], and reversed OMIT [36]. Promising applications of OMIT devices are actively explored as well, such as optical memory [21,37], phononic engineering [38–41], and precision measurements [42,43], to mention a few examples.

In this work, we study the unexpected role of loss in OMIT with a compound COM system. We note that in comparison with single-cavity devices, coupled-cavity COM has several unique properties enabling more advantages in applications. For example, both the input light and its frequency sideband can be resonantly tuned to achieve efficient phonon cooling [44]. The inter-cavity coupling strength, strongly affecting the circulating power in the resonators, also provides a tunable parameter to realize e.g., phonon lasing [11–14], unconventional photon blockade [45], reversed OMIT [36], and highly-efficient optical control [46]. In particular, by coupling an active (e.g., Erbium ion-doped) resonator to a lossy one [47, 48], COM devices with an exceptional point (EP) [13,14,36,49–51], featuring non-Hermitian coalescence of both eigenvalues and eigenfunctions [52–54], can be created. In view of novel functionalities enabled by EP devices, e.g., single-mode lasing, nonreciprocal optical transmissions, or ultrahigh-sensitive sensing [55–57], this opens up a promising new route to engineer light-matter interactions and operate quantum devices at EPs.

Here we probe the EP features in OMIT, without using any active gain, but increasing the optical loss [12,58,59]. We note that in a recent experiment [58], by placing an external nanotip near a microresonator and thus increasing the optical loss, counterintuitive EP features, i.e., suppression and revival of lasing were demonstrated [58]. Similar EP features in optical transmissions, i.e., loss-induced transparency (LIT) were reported previously in a purely optical experiment [59]. Our purpose here is to show the LIT features in OMIT devices, i.e., loss-induced suppression and revival of optical transparency at the EP. In addition, we find that by increasing the optical loss, strong absorption regimes in conventional OMIT can become transparent, accompanying by a slow-to-fast light switch in the vicinity of the EP (for similar reversed-OMIT features, see also Ref. [36] in an active COM system). The unconventional role of loss on the higher-order OMIT sidebands [30–32] is also probed, which reveals a similar turning-point (TP) behavior as the linear OMIT transmission spectrum. In contrast to the reversed OMIT in active-passive-coupled resonators [36], our results here indicate a counterintuitive way to achieve coherent optical switch and communications with OMIT devices, without the need of any active gain or complicated materials.

2. Model and results

As shown in Fig. 1, we consider two whispering-gallery-mode (WGM) microtoroid resonators coupled through evanescent fields, with the tunable coupling strength J and the intrinsic optical loss γ₁ or γ₂, respectively [47, 58]. The external lights are input and output via tapered-fiber waveguides. As in Refs. [12,58,59], an additional optical loss γ_tip is induced on the right (i.e., purely optical) resonator by a chromium (Cr)-coated silica nanofiber tip, in order to see the role of increasing optical loss on such a compound COM system. As in the experiments, the cone-like-shape nanotip used here (see Fig. 1(a)) can be fabricated by wet etching a standard fiber taper [60], with a size in diameter ranging from tens of nanometers to several hundreds of nanometers [61]. When coupling to a resonator, the additional optical loss induced by the nanotip can be finely tuned by varying both the position and the effective size of the nanotip within the mode volume (for more details, see e.g., Refs. [60, 61]). The left resonator, supporting also a mechanical mode of frequency ω_m and an effective mass m , is driven by a strong red-detuned pump laser at frequency ω_L and a weak probe laser at frequency ω_P [20,21], with the optical field amplitudes

ε_{L} = \sqrt{2 γ_{c} P_{L} / ℏ ω_{L}}, ε_{P} = \sqrt{2 γ_{c} P_{in} / ℏ ω_{P}},

respectively, where for simplicity we take γ₁ = γ₂ = γ_c and P_L or P_in is the power of the pump or the probe light.

Fig. 1 (a) Schematic diagram of the compound COM system, with an additional optical loss γ_tip induced by a Cr-coated nanofiber tip on the right (i.e., purely optical) resonator [12,58,59]. (b) The frequency spectrum of the compound COM system, with the red line or the blue lines denoting the red sideband (Stokes process) or the blue sidebands (anti-Stokes process), respectively [20,21].

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In a frame rotating at frequency ω_L, the Hamiltonian of this compound COM system can be written at the simplest level as

\begin{array}{l} H = H_{0} + H_{int} + H_{dr}, \\ H_{0} = \frac{p^{2}}{2 m} + \frac{1}{2} m ω_{m}^{2} x^{2} + ℏ Δ_{L} (a_{1}^{†} a_{1} + a_{2}^{†} a_{2}), \\ H_{int} = - ℏ J (a_{1}^{†} a_{2} + a_{2}^{†} a_{1}) - ℏ g a_{1}^{†} a_{1} x, \\ H_{dr} = i ℏ ε_{L} (a_{1}^{†} - a_{1}) + i ℏ ε_{P} (a_{1}^{†} e^{- i ∊ t} - a_{1} e^{i ∊ t}), \end{array}

where a₁ (

a_{1}^{†}

) and a₂(

a_{2}^{†}

) are the optical bosonic annihilation (creation) operators, g denotes the COM coupling strength, x or p is the mechanical displacement or momentum operator, and the optical detunings are

Δ_{L} = ω_{c} - ω_{L}, ∊ = ω_{P} - ω_{L} .

where ω_c is the resonant frequency of the optical modes. Here for simplicity, we assume that the two optical modes are degenerate; nevertheless, this can be readily extended to a more generalized situation with non-degenerate optical modes. In fact, we note that in two very recent experiments, the emergence of EPs has been demonstrated in optomechanical or electromechanical systems with highly dissimilar elements [62,63].

The Heisenberg equations of motion of this compound system are

\begin{array}{l} \ddot{x} = - Γ_{m} \dot{x} - ω_{m}^{2} x + \frac{ℏ g}{m} a_{1}^{†} a_{1}, \\ {\dot{a}}_{1} = (- i Δ_{L} - γ_{1} + i g x) a_{1} + i J a_{2} + ε_{L} + ε_{P} e^{- i ∊ t}, \\ {\dot{a}}_{2} = (- i Δ_{L} - γ_{2} + γ_{tip}) a_{2} + i J a_{1}, \end{array}

where Γ_m is the mechanical loss rate. For ε_P ≪ ε_L, we can take the probe light as a perturbation, the dynamical variables can be expressed as a_i = a_i,s + δa_i (i =1,2) and x = x_s + δx, where the steady-state solutions of the system, by setting all the derivatives of the variables as zero, are easily obtained as

\begin{array}{l} x_{s} = \frac{ℏ g}{m ω_{m}^{2}} {| a_{1, s} |}^{2}, \\ a_{1, s} = \frac{ε_{L} (i Δ_{L} + γ_{2} + γ_{tip})}{(i Δ_{L} + γ_{1} - i g x_{s}) (i Δ_{L} + γ_{2} + γ_{tip}) + J^{2}}, \\ a_{2, s} = \frac{i J ε_{L}}{(i Δ_{L} + γ_{1} - i g x_{s}) (i Δ_{L} + γ_{2} + γ_{tip}) + J^{2}} . \end{array}

For comparisons, we first consider the purely optical case [59] by ignoring the COM coupling. In this special case, by using the input-output relation [64] $a_{1}^{out} = a_{1}^{in} - \sqrt{2 γ_{1}} a_{1}$ , we can derive the optical transmission rate as

T = {| \frac{a_{1}^{out}}{a_{1}^{in}} |}^{2} = {| 1 - \frac{2 γ_{1} (i Δ_{2} + γ_{2} + γ_{tip})}{(i Δ_{1} + γ_{1}) (i Δ_{2} + γ_{2} + γ_{tip}) + J^{2}} |}^{2},

where Δ_i = ω_P − ω_i (i = 1, 2) is the detuning between the probe and the cavity mode. For simplicity, here we take Δ₁ = Δ₂ = Δ_P. As shown in Fig. 2(a), the LIT feature can be clearly seen at the resonance (i.e., Δ_P = 0) in the transmission spectrum, that is, the transmission rate firstly decreases and then increases by increasing the tip loss γ_tip [59]. The TP position turns out to be

γ_{tip}^{TP} = - (i Δ_{2} + γ_{2}) + \frac{(i Δ_{1} + γ_{1}) J^{2}}{Δ_{1}^{2} + γ_{1}^{2}},

which, for the parameter values chosen here, corresponds to

γ_{tip}^{TP} / γ_{c} = 3

(illustrated in Fig. 2(b)). Interestingly, we also note that by increasing γ_tip, the strong-absorption regimes in the conventional transmission spectrum (at Δ_P = ±11 MHz) become transparent, see Fig. 2(b), which is not reported in Ref. [59]. This phenomenon is induced by the reduction of interference caused by the tip loss. And from our numerical estimation,

γ_{tip}^{TP}

is ∼ 0 for Δ_P = ±11 MHz, i.e., by reversing the tip loss to an active gain, it is possible to reverse the dip in the EIT spectrum to a peak as already observed in the recent reversed EIT experiment performed by T. Oishi and M. Tomita [65]. LIT is generally viewed as the evidence of the EP emergence in this lossy system [59], or the existence of hidden parity-time symmetry (under a suitable mathematical transformation) [66]. The eigenfrequencies of this coupled optical system are

ω_{\pm} = \frac{1}{2} [(ω_{1} + ω_{2}) - i (γ_{1} + γ_{2} + γ_{tip})] \pm \frac{1}{2} \sqrt{{[(ω_{1} - ω_{2}) + i (γ_{2} + γ_{tip} - γ_{1})]}^{2} + 4 J^{2}} .

For ω₁ = ω₂ = ω_c, the EP condition is simplified as

γ_{tip}^{TP} = γ_{1} - γ_{2} + 2 J

, or for the parameter values chosen here,

γ_{tip}^{EP} / γ_{c} = 4

, see Fig. 2(c–d). Clearly,

γ_{tip}^{TP}

can be close to but not exactly coincides with

γ_{tip}^{EP}

, due to the fact that the TP depends on the detuning Δ_1,2 while the EP does not (for similar features, see also Ref. [58]).

Fig. 2 (a) Transmission rate of the coupled optical system as a function of Δ_P at three selected γ_tip/γ_c. (b) Transmission rate as a function of γ_tip when Δ_P is 0 and −11 MHz. Evolution of the real (c) and imaginary (d) parts of the eigenfrequencies of the supermodes as a function of the loss γ_tip. The parameters used here are γ₁ = γ₂ = γ_c = 6.43 MHz and J/γ_c = 2.

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Now we consider the role of COM coupling in LIT. For this aim, we express the dynamical variables as the sum of their steady-state values and small fluctuations to the first order, i.e.,

\begin{array}{l} x = x_{s} + δ x^{(1)} + \dots, \\ a_{i} = a_{i, s} + δ a_{i}^{(1)} + \dots (i = 1, 2), \end{array}

with which we can rewrite the equations of motion as

\begin{array}{l} \frac{d^{2}}{d t^{2}} (x_{s} + δ x^{(1)}) & = - Γ_{m} \frac{d}{d t} (x_{s} + δ x^{(1)}) - ω_{m}^{2} (x_{s} + δ x^{(1)}) + \frac{ℏ g}{m} (a_{1, s} + δ a_{1}^{(1)}) (a_{1, s} + δ a_{1}^{(1)}), \\ \frac{d}{d t} (a_{1, s} + δ a_{1}^{(1)}) & = [- i Δ_{L} - γ_{1} + i g (x_{s} + δ x^{(1)})] (a_{1, s} + δ a_{1}^{(1)}) + i J (a_{2, s} + δ a_{2}^{(1)}) \\ + ε_{L} + ε_{P} e^{- i ∊ t}, \\ \frac{d}{d t} (a_{2, s} + δ a_{2}^{(1)}) & = (- i Δ_{L} - γ_{2} - γ_{tip}) (a_{2, s} + δ a_{2}^{(1)}) + i J (a_{1, s} + δ a_{1}^{(1)}) . \end{array}

Here the higher-order terms such as

δ x^{(1)} δ a_{1}^{(1)}

will be neglected since they only contribute to the higher-order sidebands [30].

Then by using the ansatz:

(\begin{matrix} δ a_{i}^{(1)} \\ δ x^{(1)} \end{matrix}) = (\begin{matrix} δ a_{i +}^{(1)} \\ δ x_{+}^{(1)} \end{matrix}) e^{- i ∊ t} + (\begin{matrix} δ a_{i -}^{(1)} \\ δ x_{-}^{(1)} \end{matrix}) e^{i ∊ t} (i = 1, 2),

with the plus and minus sign denoting the upper and lower sidebands, we obtain the solutions for the fluctuation operators as

δ x_{+}^{(1)} = \frac{ℏ g ε_{P} a_{1, s}^{*} μ_{-}^{(1)} 𝒜_{1}^{(1)}}{𝒦^{(1)} 𝒜_{1}^{(1)} 𝒜_{2}^{(1)} + i ℏ g^{2} {| a_{1, s} |}^{2} (μ_{+}^{(1) *} 𝒜_{2}^{(1)} - μ_{-}^{(1)} 𝒜_{1}^{(1)})},

δ a_{+}^{(1)} = \frac{ε_{P} μ_{-}^{(1)} (𝒦^{(1)} 𝒜_{1}^{(1)} + i ℏ g^{2} {| a_{1, s} |}^{2} μ_{+}^{(1) *})}{𝒦^{(1)} 𝒜_{1}^{(1)} 𝒜_{2}^{(1)} + i ℏ g^{2} {| a_{1, s} |}^{2} (μ_{+}^{(1) *} 𝒜_{2}^{(1)} - μ_{-}^{(1)} 𝒜_{1}^{(1)})},

where

\begin{array}{l} 𝒦^{(1)} = m (- ∊^{2} - i ∊ Γ_{m} + ω_{m}^{2}), \\ 𝒜_{1}^{(1)} = μ_{+}^{(1) *} ν_{+}^{(1) *} + J^{2}, & 𝒜_{2}^{(1)} = μ_{-}^{(1)} ν_{-}^{(1)} + J^{2}, \\ μ_{\pm}^{(1)} = i Δ_{L} + γ_{2} + γ_{tip} \pm i ∊, & ν_{\pm}^{(1)} = i Δ_{L} + γ_{1} - i g x_{s} \pm i ∊ . \end{array}

With these results at hand, by using the standard input-output relation [64], we can obtain the transmission rate of the probe light

T_{p} = {| t_{P} |}^{2} = {| \frac{a_{1}^{out}}{a_{1}^{in}} |}^{2} = {| \frac{ε_{P} - 2 γ_{1} δ a_{1 +}^{(1)}}{ε_{P}} |}^{2} = {| 1 - \frac{2 γ_{1} δ a_{1 +}^{(1)}}{ε_{P}} |}^{2}

which describes the relation of the output field amplitude and the input field amplitude at the probe frequency.

Figure 3 shows the transmission rate T_P of the probe as a function of Δ_P and γ_tip. We see that (i) the strong-absorption regimes at Δ_p = ±11 MHz become transparent by increasing the tip loss, e.g., T_P is increased from zero to ∼ 0.35 or ∼ 0.6 for γ_tip/γ_c ∼ 3 or 8, see Fig. 3(b) which is same as the purely optical case. In contrast, (ii) for the resonant case (Δ_P = 0), the OMIT peak tends to be lowered, i.e., T_P decreases for more tip loss (see Fig. 3(a,c)), with its linewidth firstly decreased but then increased again (with the turning point γ_tip ∼ 1.5 γ_c, as numerically confirmed). More interestingly, (iii) for the intermediate regime (Δ_P = ±3 MHz), the feature as LIT in purely optical systems [58,59] in the resonant case can be clearly seen, i.e., T_P firstly drops down to zero but then increases again for more tip loss, with the turning point $γ_{tip}^{TP} / γ_{c} = 3$ , see Fig. 3(d). A more intuitive analysis of these phenomena in the various parametric regimes is shown in Fig. 3(e) and the TP and EP are illustrated in the figure. In Fig. 3(f), the transmission rate is plotted as a function of γ_tip and Δ_P which are both continuously varying to give a comprehensive view.

Fig. 3 (a)–(d) Transmission rate T_P of the probe light as a function of Δ_P. (e) Transmission rate of OMIT as a function of γ_tip at different Δ_P. (f) Transmission rate of OMIT as a function of γ_tip and Δ_P. The other parameters are ω_c = 193 THz, γ_c = 6.43 MHz, ω_m = 2π × 23.4 MHz, P_L = 1 mW, Δ_L = ω_m, g = ω_c/R, R = 34.5 μm, m = 5 × 10⁻¹¹ kg, J = 12.86 MHz and Γ_m = 0.24 MHz (see Ref. [11]).

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Based on the analyses above, we find that the LIT emerging at the resonance in the purely optical system now moves to the off-resonance regime of a specific detuning (Δ_P = ±3 MHz) in the COM system. We will give an analysis of this difference in the following section. By comparing the linearized equations of $δ a_{1 +}^{(1)}$ corresponding to the optical case and the COM case

(i Δ_{L} + γ_{1} - i ∊) δ a_{1 +}^{(1)} = i J δ a_{2 +}^{(1)} + ε_{P},

(i Δ_{L} + γ_{1} - i g x_{s} - i ∊) δ a_{1 +}^{(1)} = i g a_{1, s} δ x_{+}^{(1)} + i J δ a_{2 +}^{(1)} + ε_{P},

we can see that the differences are the COM terms:

- i g x_{s} δ a_{1 +}^{(1)}

,

i g a_{1, s} δ x_{+}^{(1)}

. Then we transform Eq. (17) into a form similar to Eq. (16) utilizing the linearized equation of

δ x_{+}^{(1)}

(i Δ^{'} + γ_{1}^{'} - i ∊) δ a_{1 +}^{(1)} = i J δ a_{2 +}^{(1)} + ε_{P},

where

Δ^{'} = Δ_{L} - g x_{s} - Re (C_{1}), γ_{1}^{'} = γ_{1} + Im (C_{1}),

and

C_{1} = \frac{𝒜_{1}^{(1)} ℏ g^{2} {| a_{1, s} |}^{2}}{𝒜_{1}^{(1)} 𝒦^{(1)} + i ℏ g^{2} {| a_{1, s} |}^{2} (- i Δ_{L} + γ_{2} + γ_{tip} - i ∊)} .

With chosen parameters above, we can numerically estimate a frequency shift of gx_s + Re(C₁) to be ∼ 3 MHz in the steady-state case. For comparison with purely optical system, we choose Δ_L = 0 and Δ_P = 0. This frequency shift caused by the COM interaction is matched well with the parametric regimes where we find the LIT at the detuning of Δ_P = ±3 MHz in the transmission spectrum of OMIT.

Now we turn to the slow-to-fast-light switch at the EP. Slowing or advancing of light can be associated with the OMIT process due to the abnormal dispersion [21]. This feature can be characterized by the group delay of the probe light

τ_{g} = \frac{d arg (T_{P})}{d Δ_{P}} .

Figure 4 shows the group delay as a function of γ_tip at different values of Δ_P. We find that at Δ_P = −3 MHz, the probe light experiences a fast-to-slow switch in the vicinity of the EP, a feature which is similar to the reverted OMIT reported previously in an active COM system [36]. This provides a new method to achieve coherent optical group-velocity switch by tuning the optical loss, which as far as we know, has not been demonstrated previously in purely optical systems. In view of the sensitive change of τ_g at the EP, this also could be used for e.g., EP-enhanced sensing of external particles entering into the mode volume of the resonator [61,67,68].

Fig. 4 (a) Group delay of the probe light as a function of γ_tip at different Δ_P. The pump power P_L is 1 mW. (b) Group delay of the probe light as a function of γ_tip and the pump power P_L at Δ_P = −3 MHz. The unit of group delay is μs.

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3. Second-order LIT in COM

In contrast to the linear systems [58, 59], the tip loss can also affect the higher-order process originating from intrinsic nonlinear COM interactions. In order to see this, we use the following second-order expansions of the operators [30]

x = x_{s} + δ x^{(1)} + δ x^{(2)} + \dots, a_{i} = a_{i, s} + δ a_{i}^{(1)} + δ a_{i}^{(2)} + \dots (i = 1, 2),

with

(\begin{matrix} δ a_{i}^{(2)} \\ δ x^{(2)} \end{matrix}) = (\begin{matrix} δ a_{i +}^{(2)} \\ δ x_{+}^{(2)} \end{matrix}) e^{- 2 i ∊ t} + (\begin{matrix} δ a_{i -}^{(2)} \\ δ x_{-}^{(2)} \end{matrix}) e^{2 i ∊ t} (i = 1, 2) .

Then by solving the Eq. (4), with the aid of Eqs. (11,22,23) and neglecting the higher-order terms more than second order, we get the second-order solution

δ a_{1 +}^{(2)} = \frac{- i ℏ g^{4} a_{1, s} {| a_{1, s} |}^{4} μ_{+}^{(1) *} μ_{+}^{(2) *} μ_{-}^{(2)} δ x_{+}^{(1) 2} + λ δ x_{+}^{(1)} δ a_{+}^{(1)}}{𝒜_{1}^{(1)} [𝒦^{(2)} 𝒜_{1}^{(2)} 𝒜_{2}^{(2)} + i ℏ g^{2} {| a_{1, s} |}^{2} (𝒜_{2}^{(2)} μ_{+}^{(2) *} - 𝒜_{1}^{(2)} μ_{-}^{(2)})]},

with

𝒦^{(2)} = m (- 4 ∊^{2} - 2 i ∊ Γ_{m} + ω_{m}^{2})

and

\begin{array}{l} 𝒜_{1}^{(2)} = μ_{+}^{(2) *} ν_{+}^{(2) *} + J^{2}, & 𝒜_{2}^{(2)} = μ_{-}^{(2)} ν_{-}^{(2)} + J^{2}, \\ μ_{\pm}^{(2)} = i Δ_{L} + γ_{2} + γ_{tip} \pm 2 i ∊, & ν_{\pm}^{(2)} = i Δ_{L} + γ_{1} - i g x_{s} \pm i ∊, \\ λ = i g μ_{-}^{(2)} 𝒦^{(2)} 𝒜_{1}^{(1)} 𝒜_{1}^{(2)} - ℏ g^{3} {| a_{1, s} |}^{2} & μ_{-}^{(2)} (𝒜_{1}^{(1)} μ_{+}^{(2) *} - μ_{+}^{(1) *} 𝒜_{1}^{(2)}) . \end{array}

As defined in Ref. [30], the efficiency of the second-order sideband process is

η = | - \frac{2 γ_{1}}{ε_{P}} δ a_{1 +}^{(2)} | .

Figure 5(a) shows the impact of the tip loss on the second-order sideband of OMIT. We find that, in contrast to the linear cases, the efficiency η increases by enhancing the loss γ_tip. A comparison between T_P and η is shown in Fig. 5(b–d). Figure 5(b) shows that T_P decreases by increasing γ_tip at the resonance while η is enhanced, a feature which was firstly revealed in Ref. [30]. However, for non-resonance cases, e.g., Δ_P = ±3 MHz or Δ_P = ±11 MHz, η increases by increasing γ_tip, which is evidently different from that for T_P, see Fig. 5(c–d). Clearly, these results on nonlinear OMIT process are beyond any linear EP picture [51]. We note that the presence of nonlinearity can lead to a shift of the EP position [69] or even the emergence of high-order EPs [51]. We also note that η_TP emerges at Δ_P = ±11 MHz, which is clearly different from the linear TP occurring at Δ_P = ±3 MHz. In fact, this frequency shift can also be identified by comparing the equations describing the linear process and its second-order sidebands, i.e.,

(i Δ^{'} + γ_{1}^{'} - i ∊) δ a_{1 +}^{(1)} = i J δ a_{2 +}^{(1)} + ε_{P},

(i Δ^{″} + γ_{1}^{″} - 2 i ∊) δ a_{1 +}^{(2)} = i J δ a_{2 +}^{(2)} + ℬ,

with

Δ^{″} = Δ_{L} - g x_{s} - Re (C_{2}), γ_{1}^{″} = γ_{1} + Im (C_{2}),

and

C_{2} = \frac{ℏ g^{2} {| a_{1, s} |}^{2} 𝒜_{1}^{(2)} 𝒜_{1}^{(1)}}{𝒜_{1}^{(2)} 𝒜_{1}^{(1)} 𝒦^{(2)} + i ℏ g^{2} {| a_{1, s} |}^{2} 𝒜_{1}^{(1)} μ_{+}^{(2) *}},

ℬ = \frac{i ℏ g^{2} a_{1, s} [- g^{2} {| a_{1, s} |}^{2} μ_{+}^{(2) *} μ_{+}^{(1) *} δ x_{+}^{(1) 2} - i g 𝒜_{1}^{(2)} a_{1, s}^{*} μ_{+}^{(1) *} δ x_{+}^{(1)} δ a_{1 +}^{(1)}]}{𝒜_{1}^{(2)} 𝒜_{1}^{(1)} 𝒦^{(2)} + i ℏ g^{2} 𝒜_{1}^{(1)} {| a_{1, s} |}^{2} μ_{+}^{(2) *}} + i g δ x_{+}^{(1)} δ a_{1 +}^{(1)} .

Here ℬ, in terms of

δ x_{+}^{(1)}

and

δ a_{1 +}^{(1)}

, can be taken as a constant. We see that in comparison with the linear process, the second-order sideband experiences a frequency shift |Re(C₂) − Re(C₁)| ∼ 10 MHz, i.e., agreeing with our numerical results in the order of magnitudes.

Fig. 5 (a) The efficiency of the second-order sideband process as a function of Δ_P. Subfigures (b), (c) and (d) show the comparisons between transmission rate of OMIT and efficiency of the second-order sideband as a function of γ_tip.

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4. Conclusions and discussions

We theoretically investigate the impact of loss on OMIT in a passive compound COM system by coupling an external nanotip to the optical resonator. Loss-induced transparency is found at the EP in the OMIT transmission spectrum, which is reminiscent of that as reported in Ref. [59], but here corresponding to the off-resonance case (i.e., with Δ_P = ±3 MHz). For the resonance case, however, increasing the tip loss leads to very minor changes for the OMIT peak. We also find that a slow-to-fast light switch can happen in the vicinity of the loss-induced EP. A detailed comparison between the linear OMIT process and its second-order sidebands, in the presence of a tunable tip loss, is also given, indicating that more exotic EP-assisted effects may happen in a nonlinear COM system.

We note that very recently, the emergence of EPs has also been experimentally demonstrated in COM or electromechanical devices with highly non-degenerate modes [62,63]. It can be of great interests to further extend our present study to such a generalized situation in future works. Also, in contrast to the previous study on reversed OMIT in active-passive-coupled resonators [36], our work here has significantly different new features: firstly, no active gain or complicated materials are needed here, instead, we find that increasing the loss can lead to the LIT-like effects at the vicinity of EP; secondly, the difference between the hybrid COM system and the purely optical LIT system is compared in details, for both the optical transmissions and the group delay; thirdly, the LIT-like effects in nonlinear OMIT process is also studied here. Hence our work sheds new light on manipulating or switching light with passive or lossy COM devices; for an example, in view of the very sensitive change of the optical velocity at EPs, a new way of EP sensing could be envisaged [61,67,68].

Funding

National Natural Science Foundation of China (NSFC) (11474087, 11774086); HuNU Program for Talented Youth.

References and links

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Loss-induced transparency in optomechanics

Abstract

1. Introduction

2. Model and results

3. Second-order LIT in COM

4. Conclusions and discussions

Funding

References and links

Cited By

Figures (5)

Equations (31)

Optics Express