1. Introduction

Recent research on distinctive structure of cavity optomechanics, based on its high-quality (high-Q) factor optical cavity and ultrasmall mass of mechanical object, has exhibited a great potential for studying all-optical mass detection [1–12]. For a high-Q optomechanical cavity, the powerful light intensity circulating inside the cavity triggers a strong radiation-pressure coupling between cavity mode and mechanical mode [13]. Simultaneously, optical mode splitting and deformation of mechanical mode mediated by radiation-pressure force give rise of unique optical phenomena, such as optomechanically induced transparency [14, 15],radiation-pressure-induced four-wave mixing [16] and nonlinear sideband spectra [17, 18]. Linear or nonlinear output spectra induced by these optical phenomena have an extremely narrow line width and a strong light intensity, which allows for the readout of mechanical deformation with high sensitivity [19]. Additionally, since the mass of micromechanical object is within ${10}^{-11}$-${10}^{-20}$kg [20], dynamical behaviors of cavity optomechanical system are sensitive to untrasmall mass-change of micromechanical object. A powerful combination between high-Q cavity and ultrasmall mass of mechanical object in cavity optomechanical system indicates a novel possibility to measure biological and chemical minuscule masses, including biomolecules, single cells, DNA molecules and nanoparticles.

On the other hand, nonlinear parametrically driven mass detector has been widely explored by tracking the resonant frequency shift of output nonlinear spectrum in micro or nanomechanical resonators [6–10].Comparing with linearly mass detection techniques, the mass detector operating in nonlinear parametrical region has versatile performances on enhancing both nonlinear interaction and output signal intensity without amplifying thermo mechanical noise. Thus, it is favorable for improving the signal-to-noise ratio and the detection sensitivity [6–10]. More interestingly, when nonlinear squeezed effect occurs in nonlinear mass measurement process, these intrinsic noises can be reduced to a low-noise level [7, 21], even to below the standard quantum limit [22–24]. However, all-optical mass detection using nonlinear sideband effect in cavity optomechanical system has yet been studied.

In this paper, we demonstrate that DOMS assisted by DPA is suggested to realize high-sensitive mass detection via nonlinear sum-sideband. Similar to nonlinear features of multiple-probe-field-driven optomechanical system [18], the generation of nonlinear sum-sideband results from parametrical frequency-conversion induced by nonlinear optomechanical coupling when DOMS is driven by a strong control field, a weak probe pulse and a signal field of DPA, as shown in Fig. 1(a). Such a mass detection scheme is dependent upon an observable correlation between the added mass of mechanical object and the conversion efficiency of output sum-sideband spectrum. Using experimentally achievable parameters, it shows that nonlinear gain of DPA plays a crucial role in improving both the generated sum-sideband efficiency and the sensitivity of nonlinear mass detection. In the presence of optimized excitation in DPA, our results also demonstrate that femtogram level resolution can be achieved in this mass detection of DOMS, when the method of mass detection relies on a direct relationship between maximum efficiency of sum-sideband and mass-change of membrane.

 

Fig. 1 (a) Schematic diagram of all-optical mass detection based on DOMS. A DPA is embedded in the optomechanical cavity, while a thin dielectric membrane is located at an antinode of the cavity field. When DOMS is driven by a strong control field (with frequency ω_d) and a relatively weak probe pulse (with frequency ω_p), the added mass deposited on the dielectric membrane can be measured by monitoring the conversion efficiency of sum-sideband. (b) Frequency spectrogram of nonlinear sum sideband in DOMS. The output fields at sum-sideband (with frequency $\pm {\mathrm{\Omega }}_{+}=\pm \left({\mathrm{\Delta }}_p+{\mathrm{\Delta }}_G\right)$ in a frame rotating at ω_d) are used to weigh the added mass deposited on the dielectric membrane.

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2. Theoretical model of mass detection in a dispersive optomechanical system

Our proposed dispersive optomechanical structure includes two fixed high-finesse mirrors separated from each other by distance L and a thin dielectric membrane with angular frequency ω_m, effective mass m_m and finite reflectivity R, as illustrated in Fig. 1(a). This special optomechanical device has been theoretically and experimentally studied in [25–29]. The ’dispersive’ optomechanical coupling features a linear relationship between position-dependent cavity frequency and square-displacement of the membrane, (i.e., $\omega \left(x\right)={\omega }_c+\frac{1}{2}\frac{{\partial }^2\omega }{\partial x^2}{x}^2$ with intrinsic cavity frequency ω_c), when the dielectric membrane is located at an antinode of intracavity field. We assume that the constant $G=\frac{1}{2}\frac{d^2\omega }{d{x}^2}{|}_{x=0}=\frac{8{\pi }^{2}c}{L{\lambda }_d^2}\sqrt{\frac{R}{1-R}}$ with the speed of light c in vacuum and the wavelength of control field λ_d describes the strength of dispersive optomechanical coupling [27, 28]. In this DOMS, the optomechanical cavity is driven by a strong control field (with amplitude ε_d and center frequency ω_d) and a weak probe pulse (with amplitude ε_p and center frequency ω_p). Simultaneously, the DPA with a second-order nonlinearity crystal is excited by a pump driving with the frequency $2{\omega }_G$, so that the signal light and the idler light in DPA have the same frequency ω_G. Subsequently, DOMS assisted by DPA is expected for creating a well-established optomechanical circumstance, where nonlinear sum-sideband process is excited by dispersive optomechanical coupling.

In a rotating frame at the frequency ω_d of control field, we begin our analysis by writing the interaction Hamiltonian of DOMS,

Here, the corresponding frequency detunings are defined as ${\mathrm{\Delta }}_c={\omega }_c-{\omega }_d$, ${\mathrm{\Delta }}_p={\omega }_p-{\omega }_d$ and ${\mathrm{\Delta }}_G=2{\omega }_G-2{\omega }_d$. $\widehat{x}$ and $\widehat{p}$ are the position and the momentum operators of membrane. $\widehat{a}$ (${\widehat{a}}^{†}$) denotes bosonic annihilation (creation) operator of the cavity mode. ${\epsilon }_{d,p}=\sqrt{P_{d,p}/\hslash {\omega }_{d,p}}$ represents the amplitudes of control field and probe pulse with $P_{d,p}$ being control field and probe pulse powers at the input of cavity. The total loss rate $\kappa ={\kappa }_0+{\kappa }_L+{\kappa }_R$ includes an intrinsic loss rate κ₀ and an external loss rate of left (right) mirror ${\kappa }_L={\eta }_L\kappa$(${\kappa }_R={\eta }_R\kappa$) with the coupling parameter ${\eta }_{L,R}$. G_a denotes nonlinear gain of DPA, which can be continuously adjusted by the pump driving. It should be noted that DPA inside optomechanical system can provide an additional functionality for reducing thermomechanical noise and photon shot noise by utilizing two-photon squeezing and sideband cooling of mechanical mode [22–30]. Besides, the total mass of membrane $m=m_m+{\delta }_m$ consists of the intrinsic membrane mass m_m and the added mass δ_m. As usual, this added mass attached to the membrane surface can modify the dynamics motion of membrane and optomechanical interaction between optical and mechanical modes, ultimately trigger a dramatical variation on the frequency shift or the conversion efficiency of output transmission spectrum [1–12].

When we take account of the cavity damping and the dissipation process in DOMS, the dynamics evolution of system can be governed by the Heisenberg-Langevin equations. By defining some shorthand for the Heisenberg operators, i.e., $\widehat{X}={\widehat{x}}^2$,$\widehat{P}={\widehat{p}}^2$ and $\widehat{Q}=\widehat{x}\widehat{p}+\widehat{p}\widehat{x}$, the Heisenberg-Langevin equations can be obtained as:

In order to solve the differential Eqs. (2)-(7), we apply the perturbation method when the control field ε_d is much stronger than the probe pulse ε_p and the driving of DPA. Then, all of the operators have perturbation form of $\mathcal{O}\left(t\right)={\mathcal{O}}_0+\delta \mathcal{O}\left(t\right)$ ($\mathcal{O}=a,x,p,X,P,Q$) that contains steady-state solution ${\mathcal{O}}_0$ and perturbation term $\delta \mathcal{O}\left(t\right)$. Here, we focus on the mean response of system, so above operators are reduced to their expectation values, i.e., $a\left(t\right)\equiv 〈\widehat{a}\left(t\right)〉$, $x\left(t\right)\equiv 〈\widehat{x}\left(t\right)〉$, $p\left(t\right)\equiv 〈\widehat{p}\left(t\right)〉$, $X\left(t\right)\equiv 〈{\widehat{x}}^2\left(t\right)〉$, $P\left(t\right)\equiv 〈{\widehat{p}}^2\left(t\right)〉$ and $Q\left(t\right)\equiv 〈\widehat{x}\left(t\right)\widehat{p}\left(t\right)+\widehat{p}\left(t\right)\widehat{x}\left(t\right)〉$. By substituting all of perturbation forms into Heisenberg-Langevin equations, the steady-state solutions of Eqs. (2)–(7) are obtained as $\left[a_0,x,p,X_0,P_0,Q_0]=[\frac{\sqrt{{\eta }_L\kappa }{\epsilon }_c}{\frac{\kappa }{2}+i{\overline{\Delta }}_c},0,0,\frac{P_0}{m^2{\omega }_m^2\left(1+2\alpha \right)},\left(1+2n_{th}\right)\frac{\hslash m{\omega }_m}{2},0\right]$ with the detuning of effective cavity resonance frequency ${\overline{\Delta }}_c={\mathrm{\Delta }}_c+G{X}_0$ and $\alpha =\frac{\hslash G{\left|a_0\right|}^2}{m{\omega }_m^2}$.

After excluding the optomechanical coupling process between linear terms x and p in Eqs. (2) and (3), the evolution of perturbation term in Eqs. (4)–(7) satisfies the following equations:

A further treatment for perturbation method is analyzed by setting the following ansatz:

Under the assumption that the output fields transmit through the left mirror of the cavity, the output spectrum can be obtained by employing the input-output relation $S_{out}=\sqrt{{\eta }_L\kappa }a-S_{in}$, as follows:

Next, we demonstrate how to achieve high-sensitive mass sensor in DOMS. As mentioned in introduction, the mass of micromechanical object is within ${10}^{-11}$-${10}^{-20}$kg [20], but it plays an important role in the dynamical evolution of both output transmission and sideband spectra [14, 16–18]. The minuscule mass of mechanical object renders the optomechanical system extremely sensitive to tiny perturbation, especially for the external mass that absorbs on the surface of mechanical object. Therefore, it is possible to measure the external mass by monitoring the conversion efficiency of generated upper sum-sideband in cavity optomechanical system. Generally, the detection noise introduced by the measurement system can become the dominant noise source. Nonlinear parametrically driven mass detection is proposed to reduce this noise and improve the signal-to-noise ratio, because nonlinear output signal in cavity optomechanical system can be enhanced without amplifying input noise [7–10]. Comparing with the linear harmonic mass sensing, Turner and coauthors [9, 10] experimentally investigated that the detection sensitivity of nonlinear parametrically driven mass sensor remains essentially unaffected when the detection noise increases tenfold. And sum-sideband effect induced by nonlinear frequency-conversion is an ubiquitous phenomenon in multiple-probe-field-driven optomechanical system [18]. These are two reasons that we choose the sum-sideband as the detection signal of nonlinear mass detection.

To evaluate the identification ability for the mass-change of membrane in mass detection process, the sensitivity of mass detection using in [2] is defined as

Above sensitivity for mass detection is actually the slope of sum-sideband spectrum with respect to the mass change. When the measured mass is fixed, the higher the detection sensitivity is, the more obvious the efficiency variation of sum-sideband is. That is to say, an excellent mass detection device need to be simultaneously supported by strong nonlinear signal and highly sensitivity. As we all know, when the same external mass is absorbed on the different location of dielectric membrane in present mass detection device, their optical responses of dielectric membrane including the effective cavity resonance frequency are different. Then, we need to define a position-dependent responsivity function $R\left(y\right)$ to describe the relationship between the shift of effective cavity resonance frequency and mass-change of membrane [11, 12]. In analogy with the responsivity function in cantilevered beam resonator [12], the external mass landing at the middle of dielectric membrane induces the maximum responsivity, while the external mass landing at the fixed end of dielectric membrane causes the minimum responsivity $R\left(y\right)=0$. In present scheme of mass detection, we assume that all of the external masses are distributed evenly along the dielectric membrane, so that the position-dependent responsivity function becomes a constant and does’t have a direct influence on the amplitude of sum-sideband and the sensitivity of mass detection.

According to recent experimental and theoretical works [25, 27], the system parameters are chosen as: cavity length $L=67mm$, total loss rate $\kappa =0.2{\omega }_m$, wavelength of control field ${\lambda }_d=2\pi c/{\omega }_d=532nm$ and cavity mode detuning ${\mathrm{\Delta }}_c=2{\omega }_m$. Additionally, the membrane parameters are set as: angular frequency ${\omega }_m=2\pi \times 0.1MHz$, mass $m_m=100pg$, reflectivity $R=0.45$ and mechanical quality factor $Q={\omega }_m/{\mathrm{\Gamma }}_m=\pi \times {10}^4$.

 

Fig. 2 (a) The conversion efficiency η_s of upper sum-sideband and (b) the sensitivity S (in units of pg) of mass detection as a function of the probe-pulsed detuning Δ_p for different signal field detuning of DPA. Other parameters are$m_m=100pg$, ${\omega }_m=2\pi \times 0.1MHz$, $Q={\omega }_m/{\mathrm{\Gamma }}_m=\pi \times {10}^4$, $L=67mm$, $T=50K$, $\kappa =0.2{\omega }_m$, ${\mathrm{\Delta }}_c=2{\omega }_m$, ${\eta }_L=0.499$, ${\epsilon }_p=0.05{\epsilon }_d$, $P_d=230\mu W$, ${\delta }_m=0pg$ and $G_a=0.01\kappa$, respectively.

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3. Numerical results and discussions

Within above practical parameter set, first of all we analyze the optical properties of output sum-sideband in an optomechanical circumstance without the participation of added mass, i.e., ${\delta }_m=0pg$. In Fig. 2, we show that the conversion efficiency η_s of upper sum-sideband and the sensitivity S of mass detection versus the probe-pulsed detuning Δ_p for three different signal field detuning of DPA, i.e., ${\mathrm{\Delta }}_G=0.2{\omega }_m$, $0.3{\omega }_m$, $0.4{\omega }_m$, when the nonlinear gain of DPA is very small, i.e., $G_a=0.01\kappa$. From Fig. 2(a), it can find that sum-sideband spectrum experiences two emission peaks. One of emission peaks is located at ${\mathrm{\Delta }}_p=2.15{\omega }_m$, and immune to the change of signal field detuning of DPA. This interesting result is caused by the resonance condition between probe pulse and effective cavity resonance frequency ${\mathrm{\Delta }}_p\simeq {\overline{\Delta }}_c$, where the radiation-pressure induced by the beating between control field and probe pulse accurately matches with square-displacement motion of membrane. The other emission peak occurs in the matching condition of ${\mathrm{\Delta }}_p+{\mathrm{\Delta }}_G={\overline{\Delta }}_c$, which arises from the nonlinear sum-sideband process, as illustrated in frequency spectrogram of Fig. 1(b). In this case, the radiation-pressure induced by the beating between probe pulse and signal field of DPA matches with square-displacement motion of membrane [18]. Direct comparison of two emission peaks implies that, whether the frequency-shift or the conversion efficiency in left emission peak of η_s has a more obvious change than that in right emission peak. Associating the conversion efficiency η_s in Fig. 2(a) with the sensitivity S in Fig. 2(b), one can see that the locations of two emission peaks in Fig. 2(a) are entirely consistent with the local maximums of detection sensitivity in Fig. 2(b). Accordingly, based on this high-consistency between η_s and S, it may provide a possibility to realize high-sensitive mass detection by monitoring the local maximums of the conversion efficiency for nonlinear sum-sideband generation.

 

Fig. 3 Contour maps of (a) the conversion efficiency η_s of upper sum-sideband and (b) the sensitivity S (in units of pg) of mass detection as a function of the probe-pulsed detuning Δ_p and nonlinear gain G_a of DPA. Other parameters are the same as in Fig. 2 except for ${\mathrm{\Delta }}_G=0.2{\omega }_m$.

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For exploring the influence of nonlinear gain of DPA on nonlinear sum-sideband, in Fig. 3, we plot the conversion efficiency η_s of upper sum-sideband and the sensitivity S of mass detection varying with the probe-pulsed detuning Δ_p and nonlinear gain G_a, when the signal field detuning of DPA is fixed as ${\mathrm{\Delta }}_G=0.2{\omega }_m$. Here, we only investigate one of the emission peaks that corresponds to sum-sideband process in DOMS and is located in the region of ${\mathrm{\Delta }}_p\in \left[1.9{\omega }_m,2{\omega }_m\right]$, because it sufficiently supports the present scheme for all-optical mass detection. From Fig. 3, it is clear that the local maximum values, whether the maximum efficiency η₂ in Fig. 3(a) or the maximum sensitivity S in Fig. 3(b), approximately occur at the location of ${\mathrm{\Delta }}_p\simeq 1.95{\omega }_m$. This location of maximum efficiency results from the matching condition of ${\mathrm{\Delta }}_p+{\mathrm{\Delta }}_G={\overline{\Delta }}_c$ in nonlinear sum-sideband process, and closely depends on the correlation between the frequencies of probe pulse, signal field of DPA and effective cavity field. Moreover, as nonlinear gain of DPA increases from 0 to $1\kappa$, both the conversion efficiency η₂ in Fig. 3(a) and the sensitivity S in Fig. 3(b) are significantly enhanced. Physically, the signal field of DPA and the probe pulse simultaneously contribute to nonlinear sum-sideband process. The increase of nonlinear gain of DPA means that DOMS can provide more capacity for developing the frequency-conversion of nonlinear sum-sideband.

 

Fig. 4 Contour maps of (a) the conversion efficiency η_s of upper sum-sideband and (b) the sensitivity S (in units of pg) of mass detection as a function of the probe-pulsed detuning Δ_p and the added mass δ_m. Other parameters are the same as in Fig. 2 except for $G_a=1\kappa$ and ${\mathrm{\Delta }}_G=0.2{\omega }_m$.

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Now we examine how DOMS is used to realize mass detection by detecting the conversion efficiency of generated sum-sideband. The detailed measurement process can be illustrated as follows: (1) after the added mass is deposited on the dielectric membrane, DOMS is driven by the strong control field with fixed frequency detuning ${\mathrm{\Delta }}_c=2{\omega }_m$ and DPA pump driving with corresponding detuning ${\mathrm{\Delta }}_G=0.2{\omega }_m$; (2) then, we apply another probe pulse to scan across the DOMS, and detect the nonlinear transmission spectrum. By seeking the relationship between the efficiency of sum-sideband and the added mass, the amount of added mass can be identified. Based on this process, in Fig. 4 we plot the conversion efficiency η_s of upper sum-sideband and the sensitivity S of mass detection as a function of the probe-pulsed detuning Δ_p and the added mass δ_m by choosing an optimized excitation of DPA with nonlinear gain $G_a=1\kappa$. When we analyze the influence of added mass on sum-sideband spectrum, there are two main characteristics for the conversion efficiency η₂ in Fig. 4(a) and the sensitivity S in Fig. 4(b). Firstly, as the added mass δ_m increases, both the local maximum efficiency η_s in Fig. 4(a) and the maximum sensitivity S in Fig. 4(b) slightly shift toward red detuning of probe pulse. Nevertheless, the maximum efficiency η_s always keeps the same dynamical evolution with the maximum sensitivity S, which is favorable for high-sensitive mass detection. Secondly, with the added mass δ_m decreasing, both the maximum efficiency η₂ and the corresponding maximum sensitivity S exhibit a remarkable enhancement. In other words, the smaller the added mass is, the more sensitive the detection for the emission peak of sum-sideband spectrum is.

 

Fig. 5 (a) The maximum efficiency ${\eta }_s^{max}$ of upper sum-sideband and (b) the maximum sensitivity $S_{max}$ (in units of pg) of mass detection versus the added mass δ_m. Other parameters are the same as in Fig. 2 except for $G_a=1\kappa$ and ${\mathrm{\Delta }}_G=0.2{\omega }_m$.

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To make an intuitive picture that directly understands the functionality of this mass detection, we extract the maximum efficiency ${\eta }_s^{max}$ and its corresponding maximum sensitivity $S_{max}$ from the same sideband spectrum of Fig. 4. Then, the maximum efficiency ${\eta }_s^{max}$ of upper sum-sideband and the maximum sensitivity $S_{max}$ of mass detection versus the added mass δ_m are shown in Fig. 5(a) and Fig. 5(b), respectively. From Fig. 5, one can find that, as the added mass δ_m increases, both the maximum efficiency ${\eta }_s^{max}$ and the corresponding maximum sensitivity $S_{max}$ decrease synchronously, which agrees with the results of Fig. 4. As we know that the detection resolution is another important index in the present mass detection, expect for the detection sensitivity. Generally, the resolution of mass detection refers to the identification ability for the mass difference between two nearly added mass, only when the maximum efficiency of sum-sideband spectrum resulting from this two kind of added mass can be distinguished by heterodyne detection scheme [32]. Typically, the inset of Fig. 5(a) shows that, in the case of the added mass difference with $\mathrm{\Delta }{\delta }_m=10.001pg-10pg=1fg$, two maximum efficiencies ${\eta }_s^{max}$ that correspond to two nearly added mass ${\delta }_m=10.001pg$ and ${\delta }_m=10pg$ are clearly identified. Consequently, the resolution of mass detection based on present DOMS reaches fg level at least.

 

Fig. 6 (a) The conversion efficiency η_s of upper sum-sideband and (b) the sensitivity S (in units of pg) of mass detection varying with the probe-pulsed detuning Δ_p for different materials-induced loss ${\mathrm{\Gamma }}_m^m$. Other parameters are the same as in Fig. 2 except for ${\mathrm{\Gamma }}_m^d=20Hz$, $G_a=1\kappa$, ${\mathrm{\Delta }}_G=0.2{\omega }_m$ and ${\delta }_m=20pg$.

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Last but not least, we also consider the influence of materials-induced loss on the mass detection of DOMS. It is well known that mechanical dissipation of membrane mainly includes the damping of mechanical excitations ${\mathrm{\Gamma }}_m^d$ and the materials-induced losses ${\mathrm{\Gamma }}_m^m$. The former is caused by interactions with the surrounding thermal bath, while the latter is caused by the relaxation of extrinsic defect states in the surface of membrane when external masses are absorbed evenly on the surface of membrane. Various dissipation processes contribute independently to the overall mechanical losses, and hence add up incoherently, i.e., ${\mathrm{\Gamma }}_m={\mathrm{\Gamma }}_m^d+{\mathrm{\Gamma }}_m^m$ [13]. In Fig. 6, we plot that the conversion efficiency η_s of upper sum-sideband and the sensitivity S of mass detection varying with the probe-pulsed detuning Δ_p for different materials-induced loss. From Fig. 6(a) and Fig. 6(b), it is readily found that both the conversion efficiency and the sensitivity decrease with the materials-induced loss increasing. In view of the fact that different measurable materials can induce different materials-induced loss, developing different measurement standard is important for detecting the mass of different materials.

4. Conclusion

In conclusion, nonlinear sum-sideband process has been theoretically investigated in DOMS assisted by DPA, which enables this composite optomechanical system to be designed a high-sensitivity mass detector. Based on the distinctive structure of cavity optomechanics, a novel method for detecting the added mass is reported by seeking an observable correlation between the conversion efficiency of nonlinearsum-sideband and the added mass of mechanical object. With the help of DPA, we reveal that nonlinear gain of DPA not only enhances the conversion efficiency of sum-sideband generation, but also improves the sensitivity of mass detection beyond these previous works. More importantly, by applying a direct relationship of the maximum efficiency of sum-sideband and the mass-change of membrane, this mass detection of DOMS can achieve fg level resolution. Our proposed scheme offers a practical opportunity to design on-chip light manipulation and optical component in precision measurement.