Nanomechanical resonators provide an unparalleled mass sensitivity sufficient to detect single biomolecules, viruses and nanoparticles. In this work we propose a scheme for mass sensing based on the hybrid opto-electromechanical system, where a mechanical resonator is coupled to an optical cavity and a microwave cavity simultaneously. When the two cavities are driven by two pump fields with proper frequencies and powers, a weak probe field is used to scan across the optical cavity resonance frequency. The mass of a single baculovirus landing onto the surface of the mechanical resonator can be measured by tracking the resonance frequency shift in the probe transmission spectrum before and after the deposition. We also propose a nonlinear mass sensor based on the measurement of the four-wave mixing (FWM) spectrum, which can be used to weigh a single 20-nm-diameter gold nanoparticle with sub-femtogram resolution.
© 2014 Optical Society of America
Nanomechanical resonators are being actively investigated as sensitive mass sensors for applications such as biological and chemical sensing because of their minuscule masses (10−15 − 10−21kg), high resonance frequencies (kHz–GHz), and high quality factors (103 − 107) [1, 2]. The operation of a nanomechanical mass sensor employs tracking the resonance frequency shifts of a nanomechanical resonator due to mass changes caused by accreted particles. Mass sensitivity, an important parameter to relate frequency shifts with added masses, is proportional to the resonance frequency of the resonator and inversely proportional to its mass . This has led to ever better levels of mass resolution–from femtogram (10−15g)  through attogram (10−18g)  and zeptogram (10−21g)  to yoctogram (10−24g) . In the past decade, nanomechanical mass sensors based on cantilevers [8, 9], nanowires [10–12], suspended microchannel resonators [13, 14] and carbon nanotubes [15, 16] have been used to weigh single molecules, viruses, and nanoparticles. Recently, Liu et al. demonstrated sub-pg mass sensing with an microtoroid optomechanical oscillator (OMO) supporting whispering-gallery mode (WGM) [17, 18]. Their analysis shows that femtogram level resolution is within reach even with relatively large OMOs. Furthermore, Shao et al. experimentally detected single nanoparticles and lentiviruses based on microtoroid microcavity by monitoring WGM broadening in microcavities, which is immune to both noise from the probe laser and environmental disturbances . Li and Zhu theoretically proposed a scheme for nonlinear mass sensor with a toroidal microcavity optomechanical system [20, 21].
Cavity optomechanics, where the mechanical resonator is coupled to the electromagnetic cavity via radiation pressure force, is drawing considerable research interest in recent years [22–24]. The optical response of optomechanical systems is modified due to the mechanical interactions, leading to effects such as normal mode splitting  and optomechanically induced transparency [26–28]. Likewise, the optomechanical interaction allows for the readout of the mechanical motion with high sensitivity [29, 30], quantum ground state cooling of the mechanical resonators [31, 32], and quantum coherent coupling between light and mechanical resonator [33,34]. More recently, Addrews et al. have successfully coupled a micromechanical resonator to both a microwave cavity and an optical cavity , which consists of a hybrid opto-electromechanical system [36–38]. The vibration of the resonator formed by a silicon nitride (Si3N4) membrane with low dissipation changes the resonance frequencies of both the optical and microwave cavities. When the cavities are driven by two pump fields with suitable frequencies and powers, respectively, and a weak probe field scans across the optical cavity resonance frequency, the resonance frequency of of the membrane can be obtained from the probe transmission spectrum. Consequently, the accreted mass deposited on the membrane can be weighed easily according to the frequency shift. The extremely narrow linewidth of the membrane  makes it a high-resolution mass sensor. Similar to the mass sensing demonstrated by Liu et al. [17, 18], the mass sensor proposed here is also based on monitoring the frequency shift of the mechanical resonator. However, only one laser provides the optical power both for actuation and high resolution monitoring of the mechanical resonant frequency in Liu’s experiment. In the hybrid opto-electromechanical system we consider here, two pump fields are used to drive the two cavities and one weak probe field is applied to track the resonance frequency of the oscillating mechanical mode by detecting the probe transmission spectrum. Such a pump-probe technique has been widely used in optomechanical experiments [26–28, 35].
2. Model and theory
The hybrid opto-electromechanical system under consideration is consisted of an optical cavity and a microwave cavity which are coupled to a common mechanical resonator. The schematic of this system is shown in Fig. 1, where the mechanical resonator with resonance frequency ωm and damping rate γm is formed by a thin silicon nitride membrane that is free to vibrate . The optical cavity with resonance frequency ωo is driven by a strong pump beam Eo with frequency Ωo and a weak probe beam Ep with frequency Ωp simultaneously. The microwave cavity with resonance frequency ωe, denoted by equivalent inductance L and equivalent capacitance C, is only driven by a strong pump beam Ee with frequency Ωe. When the membrane vibrates due to radiation pressure force, it moves along the optical intensity standing wave and changes the resonance frequency of the optical cavity . At the same time, the vibrating membrane modulates the capacitance of the microwave cavity, and thus its resonance frequency. The interaction Hamiltonian between the mechanical mode and cavity modes is Hint = −h̄goa†a(c† + c) − h̄geb†b(c† + c), where operators a, b, and c are the annihilation operators of optical cavity, microwave cavity, and mechanical resonator, respectively. go (ge) is the single-photon coupling rate between the mechanical mode and the optical (microwave) cavity mode. In a rotating frame at the pump frequency Ωo and Ωe, the Hamiltonian of the hybrid system can be written as H = H0 + Hint + Hdrive, where35]. The quantum Langevin equations governing the system can be obtained by applying the Heisenberg equation and adding the corresponding damping and input noise terms for the cavity and mechanical modes, 40] Eqs. (2)–(4) by setting all the time derivatives to zero, which are given by Eqs. (2)–(4) perturbatively by rewriting each Heisenberg operator as the sum of its steady-state mean value and a small fluctuation with zero mean value, (2)–(4) and assuming that |as| ≫ 1 and |bs| ≫ 1, we can obtain the following linearized Heisenberg-Langevin equations: 41]. In the following, since the drives are classical coherent fields, we will identify all operators with their expectation values, and drop the quantum and thermal noise terms . Then the linearized Langevin equations can be written as: Eqs (13)–(15), we make the ansatz  〈δa〉 = a+e−iδt + a−eiδt, 〈δb〉 = b+e−iδt + b−eiδt, and 〈Q〉 = Q+e−iδt + Q−eiδt. Upon substituting the above ansatz into Eqs. (13)–(15), we can obtain the following solution 43] , where aout (t) is the output field operator. Considering the output field of the optical cavity, we have
We can see from Eq. (21) that the output field contains two input components (Ωo and Ωp) and one generated four-wave mixing (FWM) component at the frequency 2Ωo − Ωp. FWM is a third-order nonlinear process , where two pump photons at the frequency Ωo are converted, due to the optomechanical intercation between photons and phonons, into one idler and one probe photon at the frequencies Ωidler and Ωp, respectively, in accordance with energy conversation: Ωidler = 2Ωo − Ωp. This process has been experimentally observed in an ultrahigh-Q toroid microcavity  and theoretically investigated in a single-mode optomechanical system . Based on the current experimental conditions, FWM can be observed in the hybrid opto-electromechanical systems. The transmission of the probe beam, defined by the ratio of the output and input field amplitudes at the probe frequency , is then given by3]
3. Results and discussion
For illustration of our numerical results, we choose the experimentally realizable hybrid opto-electromechanical system. The parameters used in the simulation are [31, 35]: ωo = 2π × 282 THz, ωe = 2π × 7.1 GHz, κo = 2π × 1.65 MHz, κe = 2π × 1.6 MHz, κo,ext = 0.76κo, κe,ext = 0.11κe, go = 2π × 27 Hz, ge = 2π × 2.7 Hz, ωm = 2π × 5.6 MHz, γm = 2π × 4 Hz, and the effective mass of the resonator is approximately meff = 45 pg. We take baculovirus and gold nanoparticle (the density of Au is ρAu =19300 kg/m3) as deposition sample, respectively. It should be noted that the hybrid system has to operate under cryogenic temperature to keep the superconducting circuitry working.
Mass sensing is based on detecting the frequency shift of the mechanical resonator before and after the adsorption of the accreted mass. Firstly, we would present a scheme for measuring the resonance frequency of the mechanical resonator based on this hybrid opto-electromechanical system. Figure 2(a) plots the transmission spectrum of the probe field as a function of the probe-cavity detuning Δp with Δo = Δe = 0 and Po = Pe = 0.01 μW. It can be seen clearly from this figure that there is a broad transmission dip when the probe field is resonant with the optical cavity, which corresponds to the cavity absorption. Furthermore, two sharp sideband peaks, representing the resonant amplification and absorption of the mechanical mode, locate exactly at Δp = ±ωm. The spectral width of the sideband peaks is the mechanical damping rate γm/2π = 4 Hz, which can be seen clearly from the enlarged image in Fig. 2(b). This extremely narrow spectral linewidth is beneficial in resolving the frequency shifts due to accreted mass. Therefore, Fig. 2 provides us an convenient method to measure the resonance frequency of the mechanical resonator. The measurement process can be illustrated as follows. (1) We apply a strong optical pump field and a strong microwave pump field to the respective cavity, and fix the pump fields at the resonance of the cavity frequencies (Δo = Δe = 0); (2) we then apply another weak probe field to the optical cavity, and scan the probe frequency across the cavity frequency. By detecting the probe transmission spectrum, one can easily obtain the resonance frequency of the mechanical resonator. The physical mechanism for this phenomenon can be understood as a result of the radiation pressure force oscillating at the beat frequency δ between the optical pump and probe fields. If δ is close to the resonance frequency ωm, the mechanical resonator starts to oscillate coherently. The induced motion leads to a mechanical sideband on the pump field that can interfere with the probe field and hence modifies the probe transmission spectrum.
After the resonance frequency of the mechanical resonator is determined via the probe transmission spectrum, we can subsequently measure the frequency shift caused by an object landing on the resonator. The frequency shift depends on the mass and position of the deposited object, so trapping an object at a known position would allow its mass to be determined directly. We assume for simplicity that the relationship between the accreted mass and the frequency shift satisfies Eq. (24) . If a single baculovirus lands onto the surface of the mechanical resonator, the total mass of the resonator would be increased, resulting in reduction of the resonance frequency. Figure 3 plots the probe transmission spectrum as a function of the probe-cavity detuning Δp before (solid curve) and after (dashed curve) the adsorption of a single baculovirus in the vicinity of the mechanical resonance frequency. We can see that the resonance frequency shift Δω = −2π × 93 Hz can be resolved in the transmission spectrum due to the increased mass of the resonator. Based on the resonance frequency shift and using Eq. (24), we found the added mass to be 1.5 fg (1 fg = 10−15 g), about the mass of a single baculovirus . Therefore, the hybrid opto-electromechanical system studied here could be used to weigh the mass of a single virus. Though the principle of mass sensing we propose is simple and feasible, it remains challenging to realize such a mass sensor in experiments. For example, the baculovirus and the mechanical resonator have to be dealt with by some specific process before the adsorption of the baculovirus . In addition, mass responsivity ℛ is an important parameter to evaluate the performance of the mechanical resonator for mass sensing. The inset of Fig. 3 shows the direct linear relationship between the resonance frequency shifts and the number of the baculovirus landing on the mechanical resonator. Such a linear relationship has been verified by various experiments [6, 8]. The negative slope of the line gives the mass responsivity of the resonator. Both smaller mass and higher resonance frequency of the resonator are crucial in obtaining higher mass responsivity. Sun and Zheng et al. have recently designed and experimentally demonstrated femtogram L3-nanobeam optomechanical cavities by embedding a doubly clamped nanomechanical double-beam resonator with mass as small as 25 fg in a finely tuned two-dimensional (2D) photonic crystal (PC) slab, where optical transduction of the fundamental flexural mode around 1 GHz was demonstrated [47–49]. They also pointed out that such femtogram-mass, high-mechanical resonance frequency structures could be used as ultrasensitive sensors of mass, force and displacement.
Furthermore, the hybrid opto-electromechanical system can be employed as a nonlinear mass sensor based on the third-order nonlinear optical effect. Figure 4 plots the FWM spectrum as a function of the probe-cavity detuning with Δo = Δe = 0 before and after a binding event of a gold nanoparticle of 20 nanometers in diameter. Similarly, a narrow peak appears at Δp = ωm due to the quantum interference effect, which can be used to measure the resonance frequency of the mechanical resonator. After the adsorption of a single nanoparticle, there is a frequency shift Δω = −2π × 5 Hz and the red dashed peak locates at ωm + Δω. According to Eq. (24), we can obtain the mass of the single 20-nm-diameter nanoparticle: (1 ag =10−18 g), which is close to the mass calculated by the density and volume of the gold nanoparticle. Compared with the traditional mass spectrometry, the nonlinear mass sensor based on the four-wave mixing effect has some obvious advantages. First, the particles needn’t to be ionized like traditional mass spectrometer , and their masses can be measured from the FWM spectrum conveniently. Second, the simultaneous presence of the pump and probe beams generates a beat wave to drive the mechanical resonator. Therefore, resonators with both high and low resonance frequency are suited for this mass sensing scheme. Third, the use of nonlinear optical spectrum may offer better performance over the linear optical spectrum in the presence of detection noise .
Finally, it should be pointed out that we do not take into account the noise processes in our theoretical simulations. Actually, there are various noise sources which impose the ultimate mass sensitivity limits for the resonator, such as thermomechanical noise generated by the internal loss mechanisms in the resonator, adsorption-desorption noise from residual gas molecules [3, 52], and detection noise in the readout circuitry . In the hybrid opto-electromechanical system we study here, the thermal noise of the mechanical motion is the dominant noise source. If the experiment could be done at the dilution refrigerator temperature of 40 millikelvin, the thermal noise would be greatly eliminated. In addition, added vibrational noise can be reduced by using mechanical resonators with higher quality factors .
In conclusion, we have theoretically demonstrated that the hybrid opto-electromechanical system, consisted of an optical cavity and a microwave cavity coupled to a common mechanical resonator, can be employed as an ultrasensitive mass sensor. Due to the quantum interference between the mechanical mode and the beat of the two optical fields, the resonance frequency of the mechanical resonator can be determined from the probe transmission spectrum as well as the FWM spectrum. Therefore, the mass of the accreted particles such as single viruses and gold nanoparticles landing onto the resonator can be obtained according to the relationship between the added mass and the corresponding frequency shifts. The scheme proposed here could be achievable in current experiments.
The authors gratefully acknowledge support from National Natural Science Foundation of China (Grant Nos. 11304110 and 11274230), Jiangsu Natural Science Foundation (Grant No. BK20130413), and Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 13KJB140002).
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