Coherent optical propagation and ultrahigh resolution mass sensor based on photonic molecules optomechanics

Optics Communications 382 (2017) 73–79

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Coherent optical propagation and ultrahigh resolution mass sensor based on photonic molecules optomechanics Hua-Jun Chen n, Chang-Zhao Chen, Yang Li, Xian-Wen Fang, Xu-Dong Tang School of Science, Anhui University of Science and Technology, Huainan, Anhui 232001, China

art ic l e i nf o

a b s t r a c t

Article history: Received 24 May 2016 Received in revised form 8 July 2016 Accepted 10 July 2016

We theoretically demonstrate the coherent optical propagation properties based on a photonic molecules optomechanics. With choosing a suitable detuning of the pump field from optomechanical cavity resonance, both the slow- and fast-light effect of the probe field appear in the system. The coupling strength of the two cavities play a key role, which affords a quantum channel and influences the width of the transparency window. Based on the photonic molecules optomechanical system, a high resolution mass sensor is also proposed. The mass of external nanoparticles deposited onto the cavity can be measured straightforward via tracking the mechanical resonance frequency shifts due to mass changes in the probe transmission spectrum. Compared with the single-cavity optomechanics mass sensors, the mass resolution is improved significantly due to the cavity–cavity coupling. The photonic molecules optomechanics provide a new platform for on-chip applications in quantum information processing and ultrahigh resolution sensor devices. & 2016 Elsevier B.V. All rights reserved.

Keywords: Cavity optomechanics Photonic molecules Coherent optical propagation Mass sensor

1. Introduction Cavity optomechanics systems [1], consisting of high-quality optical cavities coupled to mechanical resonators via radiation pressure force, have progressed enormously in recent years as they reveal and explore fundamental quantum physics, offer a promising route to making precision measurements in both the applied and fundamental science domains, and pave the way for potential applications of optomechanical devices. Studies of optomechanical systems have enabled the experiments to realize plenty of prominent phenomena, including strong coupling between an optical cavity and a mechanical resonator [2], cooling of mechanical modes to their quantum ground states [3–5], optomechanically induced transparency (OMIT) [6–9], coherent interconversion between optical cavity and mechanical modes [10,11], and the realization of squeezed light [12–14]. Taking full advantage of the above phenomena, kinds of applications in sensors including force [15], torque [16], and acceleration [17], especially in OMIT based optical storage [6–9] and ultrasensitive mass sensing [18] have been investigated remarkably. The underlying physics of OMIT [6–9] is formally similar to that of electromagnetically induced transparency (EIT) in three-level Λ -type atoms [19], which provide an effective approach of controlling electromagnetic fields and the optical characteristics of n

Corresponding author. E-mail address: [email protected] (H.-J. Chen).

http://dx.doi.org/10.1016/j.optcom.2016.07.027 0030-4018/& 2016 Elsevier B.V. All rights reserved.

matter. In optomechanical systems, mechanically mediated delay (slow-light) and advancement (fast-light) of signals based on OMIT have been observed both in optical [9] and microwave domains [11,20,21], which will offer new prospects for on-chip solid-state architectures capable of storing, filtering, or synchronizing optical light propagation. Theoretical investigations show that OMIT has also potential applications in precision measurement including phonon number [22], coupling-rate [23], and electrical charge [24]. The above phenomena still lie in a single optical mode coupled to a single mechanical mode. Recently, multimode optomechanical systems have become a tendency for further researching of optomechanics and their potential applications in quantum information processing. Three-mode coupled optomechanical systems are the typical multimode systems, which include two optical or microwave cavity modes coupled to a single mechanical mode [25] (or vice versa [26]), or a hybrid optomechanical system including a mechanical resonator coupled to superconducting microwave cavity and an optical cavity [27,28,30]. Based on the multimode optomechanical systems, the transfer of a quantum state [28], OMIT-like ground-state cooling [29], coherent optical wavelength conversion [25], optomechanical dark mode [31], and phonon-mediated electromagnetically induced absorption [32] have been investigated. In this work, we will demonstrate coherent propagation properties based on multimode cavity optomechanical systems. On the other hand, the properties of minuscule masses, high resonance frequencies, and high quality factors of optomechanical resonator enable them to realize ultra-sensitive mass sensing

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[18,33,34]. The nanomechanical resonator can be used for mass sensing because the deposited mass on the mechanical mode will result in a change of the resonance frequency in the resonator [35], and the basic principle of mass sensing is measuring the shift of the resonance frequency. There are two approaches for implementing the mass sensing techniques, i.e., electrical [36,37] and optical detections [18]. In the mass sensing scheme based on electrical measurement, the heat effect and energy loss in the electric circuitry will broaden the electrical response spectrum which finally affect the sensitivity of the frequency detection. For eliminating the weaknesses, all-optical mass sensors without the electrical environment are proposed [18]. Furthermore, an optical detection scheme for single nanoparticles and lentiviruses based on microtoroid microcavity by monitoring whispering-gallery mode (WGM) broadening in microcavities, which is immune to both noise from the probe laser and environmental disturbances, was also demonstrated experimentally [34] . To obtain high sensitivity of the mass sensing, a hybrid quantum dot-metal nanoparticle (QD-MNP) complex structure was considered implanting into a nanomechanical resonator [38]. Here, we present an ultrahigh resolution mass sensing scheme based on a photonic molecules optomechanical system with cavity–cavity coupling strength J, which can be implemented experimentally with a double-WGM cavity optomechanical model [39–41]. The photonic molecules optomechanical system is composed of two WGM cavities, in which one cavity is driven by a pump field and a probe field induced mechanical radial breathing modes (RBMs) [42] via the radiation pressure force, while another cavity is only driven with a pump field as shown in Fig. 1. The presence of a pump field and a probe field induces an anti-Stokes and Stokes field produced from the scattering of light from the pump field. The anti-Stokes field and Stokes field interfere with the intracavity probe field leading to phenomena of OMIT [6–9] and the signal amplification [20], respectively. In the present article, we demonstrate the mechanically mediated slow-light and fast-light of the probe field based on the photonic molecules optomechanical system under different detuning. In addition, when the weak probe field scans across the optical cavity resonance frequency, the resonance frequency of the mechanical mode can be obtained from the probe transmission spectrum. Consequently, the accreted mass deposited on one cavity can be weighed easily according to the frequency shift. The extremely narrow linewidth of the

mechanical mode makes it a high-resolution mass sensor.

2. Model and theory 2.1. The Hamiltonian of the system The system under consideration consisted of a WGM cavity a with frequency ωa coupled to a mechanical resonator b with frequency ωm and a WGM cavity c with frequency ωc as shown in Fig. 1. The light field coupled into the cavity a induces the RBM due to the radiation pressure force, which in turn modulates the frequency of cavity mode ωa. The single-photon coupling rate is g = g0x0 ( g0 = ωc /R ) for the WGM resonator, and the zero-point fluctuation of the mechanical oscillator's position is x0 =  /2Mωm with effective mass M and the mechanical resonator frequency ωm. Quantizing the cavity mode a and mechanical mode b with the creation (annihilation) operators for photons a+(a) and phonons b+(b), respectively, the interaction Hamiltonian between the cavity mode and the mechanical resonator [6–9] is Hint = ga+ a(b+ + b). The cavity a and the cavity c are coupled with the coupling strength J via exchanging energy and J depends on the distance between the two cavities, and the cavity–cavity coupling Hamiltonian [29,41] can be described as Hac = J (a+c + ac+). We apply one pump field with frequency ωp (the amplitude Ea) and one probe field ωs (the amplitude Es) to drive the WGM cavity a, and while only one pump field with frequency ωp (the amplitude Ec) to drive the WGM cavity c, which has the relation with the power Ea = Pa/ωp ( Ec = Pc /ωp , Es = Ps/ωs ). The laser field in the optical fiber couples to the cavities with the photon escape rate κex due to the external coupling, and then the transmitted field power can be detected by using a balanced homodyne detection scheme. In the adiabatic limit, only one cavity mode ωa is driven, and the cavity-free spectrum range c /2πR (c is the speed of light in the vacuum and R is the radius of WGM cavity) is much larger than the frequency of cavity vibration. Thus, the scattering of photons to the other cavity modes can be ignored. In the rotating frame at the pump field frequency ωp, the whole Hamiltonian of the system reads [7,25,29,41]

H = Δa a+a + Δc c +c + ωmb+b + J (a+c + ac+) − ga+ a(b+ + b) + i κ ae Ea(a+ − a) + i κ ae Es(a+e−iδt − aeiδt ) + i k ce Ec (c + − c ).

(1)

The first three items indicate the free Hamiltonian of the cavity a, cavity c, and the mechanical resonator b, respectively. Δa = ωa − ωp and Δc = ωc − ωp are the corresponding cavity-pump field detuning. The last three items are the probe laser field and the pump laser fields coupled to the two cavities. δ = ωs − ωp is the detuning between the probe laser field and the pump laser field. The decay rate of cavities mode κ = κc = κa = κex + κ0 with the intrinsic photon loss rate κ0 and κex describes the rate at which energy leaves the optical cavity into propagating fields [7]. In this article, we only consider the condition for simplicity: κex = κ0 = κae = κce . 2.2. The quantum Langevin equations and coherent optical spectrum

Fig. 1. Schematic diagram of the photonic molecules optomechanical system. The WGM cavity mode a is coupled to the mechanical mode b via radiation pressure force with a strong pump field and a weak probe field. The WGM cavity mode c is only driven by a pump field. The two cavities couple with each other via exchanging energy and the coupling strength J depends on the distance between the two cavities. The nanoparticles are deposited onto the surface of the cavity a.

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 [7,29,41]:

ȧ = − (iΔa + κ a)a + igaX − iJc +

κ ae (Ea + Ese−iδt ) +

2κ a ain,

(2)

H.-J. Chen et al. / Optics Communications 382 (2017) 73–79

c ̇ = − (iΔc + κ c )c − iJa +

κ ce Ec +

2κ c cin,

X¨ + ΓmẊ + ωm2 X = 2gωma†a + ξ ,

(3)

(4)

where the position operator X = b + b, and Γm is the decay rate of the resonator, ain and cin are the δ-correlated Langevin noise operator, which has zero mean and obeys the correlation † ′) † ′) ain(t )ain (t = cin(t )cin (t = δ (t − t′). The resonator function mode is affected by a Brownian stochastic force with zero mean value, and ξ(t ) has the correlation function

where aout(t) is the output field operator. The transmission spectrum of the probe field defined by the ratio of the output and input field amplitudes at the probe frequency, which shows [6–9]

t (ωs ) =

Es −

γm ωm

∫ d2ωπ ωe−iω(t − t′)[1 + coth( 2κωBT )], where kB and T are

Boltzmann constant and temperature of the reservoir of the coupled system, respectively. The probe field is much weaker than the pump field, following the standard methods of quantum optics, we rewrite each Heisenberg operator as the sum of its steady-state mean value and a small fluctuation with zero mean value: a = a¯ + δa , c = c¯ + δc , X = X¯ + δX . In this case, the steady-state values are governed by the pump power and the small fluctuations by the probe power. In the steady-state, disregarding the probe field, the time derivatives vanish, and the steady-state equation set consisting of

a¯ =

κ ae Ea(iΔc + κ c ) − iJ κ ce Ec , (iΔ¯a + κ a)(iΔc + κ c ) + J2

(5)

c¯ =

κ ce Ec (iΔa + κ a) − iJ κ ae Ea , (iΔ¯a + κ a)(iΔc + κ c ) + J2

(6)

2g a¯ 2 X¯ = , ωm

(7)

are related to the two intracavity photon numbers ( na = a¯ 2, and nc = c¯ 2) which are determined by

na =

κ aeEa2(Δc2 + κ c2) + κ ceEc2J2 − 2JΔc κ aeκ ce EaEc , (Δ¯a + κ a2)(Δc2 + κ c2) + 2J2 (κ aκ c − Δ¯a Δc ) + J 4

(8)

nc =

κ ceEc2(Δa2 + κ a2) + κ ceEa2J2 − 2JΔa κ aeκ ce EaEc , (Δ¯a + κ a2)(Δc2 + κ c2) + 2J2 (κ aκ c − Δ¯a Δc ) + J 4

(9)

2

where Δ¯a = Δa − 2g na /ωm . For the equation set of small fluctuation, we make the ansatz

δa = A− e−iδt + A+ eiδt ,

[43]:

δX = X −e

−iδt

δc = C −e−iδt + C+eiδt

and

iδt

+ X+e . Solving the equation set and working to the

lowest order in Es but to all orders in El, we can obtain

A− =

(iΔ2 − κ a) κ ae εp (iΔ1 + κ a)(iΔ2 − κ a) + g 2na2χ 2

,

(10)

χ = 2gωm/(ωm2 − iΓmδ − δ 2), Δ1 = Δ¯a − δ − gna χ − iJ 2/[i(Δc − δ ) + κc ] , and Δ2 = Δ¯a + δ − gna χ − iJ 2/[i(Δc + δ ) − κc ]. Using the standard input–output relation [44] aout (t ) = ain(t ) − κex a(t ), we obtain where

aout (t ) = (Ep − −

κ ex A− )e−i(Ω + ωp)t

κ ex A + e−i(Ω − ωp)t

= (Ep − −

κ ex a¯ )e−iωpt + (Es −

κ e a¯ )e−iωpt + (Es −

κ ex A + e−i(2ωp − ωs)t ,

κ ex A− )e−iωst (11)

κ ex A− Es

†

ξ†(t )ξ(t′) =

75

=1−

(iΔ2 − κ a)k ex (iΔ1 + κ a)(iΔ2 − κ a) + g 2na2χ 2

.

(12)

The transmission group delay can be expressed as [7]

τg =

d{arg[t (ωs )]} dϕ |ω = |ωs = ω . p dωs s = ω p dωs

(13)

2.3. The mass sensor Based on the photonic molecules optomechanical system, we further present an ultrahigh resolution mass sensing via the probe transmission spectrum. As it was demonstrated that when nanoparticles are deposited on the edge of WGM microcavities, the nanoparticles behave as a scatterer that will induce a counterpropagating optical mode circulating in the WGM cavity [45]. A portion of the scattered light is lost to the environment, creating an additional damping channel, whereas the rest couples back into the resonator and induces coupling between the counter-propagating WGMs, which will induce the mode splitting [46]. The scheme for detecting nanoparticles with mode splitting indeed provides an effective means for classifying biomolecules and also paves a way for practical single molecule microcavity sensing applications. However, the mass sensing mechanism in our scheme is different from the mode splitting, and we pay more attention to mass changes when nanoparticles are deposited on the surface of the WGM cavity that will induce the frequency shift. In an optomechanical oscillator the nanoparticles interact with the mechanical mode that is activated and interrogated by a high-Q optical mode. In our mass sensor scheme the evanescent optical field does not play any role and the optical mode is completely isolated from the nanoparticles [33]. In reality, the frequency shift depends on both the deposited nanoparticles and their position of adsorption on the surface of the WGM cavity. Here we have assumed that the deposited nanoparticles distribute uniformly onto the surface of the WGM cavity a and the nanoparticles are near the center of the cavity a. We therefore neglect the lights scattering from the nanoparticles deposited on the surface of the cavity. In addition, the statistical distribution of frequency shifts has been investigated by building the histogram of event probability versus frequency shift for small ensembles of sequential single molecule or single nanoparticle adsorption events [47]. The optomechanical resonators act as mass sensors due to their resonant-frequency sensitivity to the mass adsorbed by them. The optomechanical resonator can be described by a harmonic oscillator with an effective mass M, a spring constant k, and a fundamental resonance frequency [18]:

ωm =

k . M

(14)

Mass sensing monitors the shift δf of ωm induced by the adsorption onto the resonator. Typically the particle mass δm to be weighed is deposited on the WGM cavity, whose resonant frequency ωm will be shifted to ωm + δf . By detecting the frequency shift δf , the mass of deposited particle can be weighed. There are two steps in mass sensing. The first one is to determine the initial frequency of the resonator. The second one is to measure the new frequency of the resonator when additional mass is deposited on the surface of the WGM cavity, and then the frequency shift δf can be obtained. The frequency shift and the deposited particle mass obey the following equation as [48]

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Fig. 2. (a) The normalized magnitude of the probe transmission spectrum t 2 as a function of probe-cavity detuning Δs = ωs − ωa with four coupling strengths J with the pump power of cavity a Pa = 10 μW and the pump power of cavity c Pc = 2 μW . (b) The detail of the transmission window around Δs = 0 . (c) The phase of the probe transmission as a function of Δs for three coupling strengths J, and (d) the group delay as a function of the pump power Pa.

δm = R−1δf ,

(15)

where R = ( − 2M /ωm)−1 is the mass responsivity. Sub-pg mass sensing and measurement with an WGM cavity optomechanical oscillator have been demonstrated experimentally recently [33]. In their WGM cavity optomechanical system, a high quality factor optical mode simultaneously serves as an efficient actuator and a sensitive probe for precise monitoring of the mechanical eigen-frequencies of the cavity structure. When the optical field in the silica fiber taper couples to the WGM cavity, the large circulating optical power results in a strong coupling between the optical and mechanical modes through radiation pressure. Compared with the mass sensing scheme in Ref. [33], we present an all-optical mass sensing with the pump-probe technique based on a realistic two WGM cavities optomechanical system [39]. The pump-probe scheme will generate a beat wave to drive the mechanical resonator, which allows both the high and low frequency of mechanical resonator. The cavity–cavity coupling in the photonic molecules optomechanics will result in the extremely narrow linewidth of the mechanical resonator, and finally realize an ultrahigh resolution mass sensor.

3. Numerical results and discussions The realistic photonic molecules optomechanical system is considered and the parameters of cavity a used are [7] (g0, Γm, ωm, κa)/2π ¼(12 GHz/nm, 41 kHz, 51.8 MHz, 15 MHz) and (M, λ, Q, Pa)¼

(20 ng, 750 nm, 1500, 10 μW). For cavity c, we consider ωc = ωa − ωm , Pc = 2 μW , and κc = κa . J is the coupling strength between the two cavities which strongly depend on the distance between the two cavities [39,40], and the coupling strength we expect J/2π ∼ MHz . Fig. 1 presents the setup of the photonic molecules optomechanics and nanoparticles are deposited onto the surface of one WGM cavity a. OMIT has been observed in kinds of cavity optomechanical systems [6–9] induced by destructive interference between the anti-Stokes field and the probe field. However, in the photonic molecules optomechanical system, strong coupling between the two cavities is vital for an obvious transparency window in the probe transmission spectrum. Fig. 2(a) shows that the transparency window as a function of the probe-cavity detuning Δs = ωs − ωa with several coupling strengths J between the two cavities under the pump power of cavity a is Pa = 10 μW and the pump power of cavity c is Pc = 2 μW . In addition, we set the cavity a is driven on its red sideband, i.e., Δa = ωm , while the cavity c is driven on its blue sideband, i.e., Δc = − ωm , which is the so-called optimal condition that can suppress the heating process and reach the ground-state cooling [29]. The black curve in Fig. 2(a) indicates the OMIT in one cavity optomechanical system, while with increasing the coupling strength J, the transparency window is widening with a shift due to the detuning, and Fig. 2(b) gives the detail of the transparency window around Δs ≈ 0. Therefore, we indeed obtain OMIT in the photonic molecules optomechanical system, which indicates a tendency to reach OMIT-based slowlight effect.

H.-J. Chen et al. / Optics Communications 382 (2017) 73–79

77

Fig. 3. The normalized magnitude of the probe transmission spectrum t 2 as a function of Δs for four coupling strengths J with the pump power of cavity a: Pa = 10 μW and the pump power of cavity c: Pc = 2 μW . (b) The detail of amplification window around Δs = 0 . (c) The phase of the probe transmission as a function of Δs for three coupling strengths J¼ 0, 1.0κa , 1.5κa , and (d) the group delay as a function of the pump power Pa.

When the radiation pressure force acts on the mechanical resonator, the change of the displacement of the resonator from its equilibrium position will alter the resonance frequency of the cavity. Therefore, the effective refractive index seen by the propagating probe field changes and a phase shift is induced under the red sideband. We find that when the narrow transparency window appears in the photonic molecules optomechanics, there will be a very steep positive phase dispersion around Δs = 0 at red sideband Δp = ωm , which will result in a tunable group delay (i.e., the slow-light effect) of the transmitted probe beam as shown in Fig. 2(c). In Fig. 2(d) we plot the transmission group delay τg of the probe beam as a function of the pump power Pa for the coupling strengths J ¼0, 1.0κa, 1.5κa. This figure suggests that the slow light can easily be realized. For a fixed coupling strength, there is an optimum group delay. It is obvious that at a very low pump power the propagation velocity of the probe pulse can slow down quickly. Switching the cavity-pump field detuning Δa of cavity a from the red sideband ( Δa = ωm ) to the blue sideband ( Δa = − ωm ) and still setting the cavity c is driven on its blue sideband ( Δc = − ωm ), we display the transmission spectra of the probe field as a function of Δs for several coupling strengths J, which indicates the amplification of the probe field as shown in Fig. 3(a). These phenomena can be understood as follows. The radiation pressure force drives the vibration of the resonator near its resonant frequency under resonant cavity pump detuning, i.e., Δa = ± ωm . When the beat frequency δ between the probe and the pump field is close to the resonance frequency ωm of the resonator, the mechanical mode starts to oscillate coherently, which will induce Stokes and antiStokes scattering of light from the strong intracavity field. When the microwave cavity is driven on its blue sideband ( Δa = − ωm ), it is the Stokes field that interferes with the near-resonant signal

Fig. 4. The probe transmission spectrum around the frequency of resonator ωm with two different coupling strengths J¼ 0 (the blue curve) and J = 0.5κa (the green curve). The WGM cavity a is driven at resonance ( Δa = 0 ), the WGM cavity c is driven at the blue sideband ( Δa = 0 ), the pump power of cavity a is Pa = 5 μW , and the pump power of cavity c is Pc = 1 μW . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

field and thus modifies the probe spectrum. Therefore, the constructive interference between the Stokes field and the probe field amplifies the weak probe field. The coupling strength J plays an important role in the progress of amplification. However, the amplification of the probe field is always decreasing with increasing the coupling strengths J as detailed in Fig. 3(b). In addition, except the amplification progress around Δs = 0, there are also the phenomena of mode splitting in the probe transmission

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bandwidth (about 1.1 kHz) and the stronger intensity (about 270) with an obvious frequency-shift appears due to the cavity–cavity interaction between the two cavities as shown in Fig. 4. Therefore, Fig. 4 provides a straightforward method to measure the resonance frequency of the WGM resonator. The mass responsivity is an important parameter to evaluate the performance of a mechanical resonator for mass sensing, which is defined as R = ∂ωm/∂M = − ωm/(2M ). Obviously, the lower mass, higher vibration frequency and higher quality factor of the resonator may improve the sensitivity of the sensor effectively. Here, in our scheme, R = 1.036 × 1016 Hz/g . Eq. (15) suggests the

Fig. 5. The probe transmission spectrum before and after landing the nanoparticles on the surface of WGM cavity a, and the color curves show the mechanical frequency-shifts at (a) J¼ 0 and (b) J = 0.5κa . The inset shows the linear relationship between the frequency-shifts and the mass of the nanoparticles, and the natural number near every data point denotes the mass of nanoparticles. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

spectra with increasing the coupling strengths J, which has been demonstrated experimentally in two cavity modes system [49]. We further plot the phase of the transmitted probe field as a function of the probe-cavity detuning Δs for three coupling strengths J as shown in Fig. 3(c), where a steep negative phase dispersion appears in Δs = 0 which may result in fast-light effect. Fig. 3(d) presents the group delay τg as a function of the pump power. The group delay is negative, and then the fast-light effect can obtain when the WGM cavity a is driven on its blue sideband. Compared with the situation at the red sideband, we propose a scheme to efficiently switch from pulse delay to pulse advancement by adjusting the pump-cavity detuning. Fixing the probe field with frequency ωs = ωa , and then scan the pump frequency across the cavity resonance frequency ωa, one can efficiently switch from the probe pulse delay to advancement without appreciable absorption or amplification as the pump detuning Δa equals to ωm or −ωm . On the other hand, realizing the ultrahigh resolution mass sensing based on the photonic molecules optomechanical system is another potential application. To implement mass sensing, the first step is to determine the original frequency of the resonator. We here provide a scheme to determine the original frequency of the resonator from the probe transmission spectrum, where the cavity a is driven on the condition of resonance ( Δa = 0), the cavity c is driven at the blue sideband ( Δc = − ωm ), the pump power of cavity a is Pa = 5 μW and the pump power of cavity c is Pc = 1 μW . Fig. 4 shows the probe transmission spectra at the probe frequency in two cases of J¼ 0 and J = 0.5κa . When the coupling strength J ¼0, the optomechanical system reduces to a single WGM cavity optomechanical system. In such optomechanical system, the probe transmission spectrum shows a bandwidth about 41 kHz and an intensity is about 10, and the sharp peaks (see the blue curve) exactly locate at the resonator frequency ( Δs = − 51.8 MHz) induced by its vibration. While in the photonic molecules optomechanical system ( J = 0.5κa ), one can obtain the narrower

linear relationship between external accreted mass and frequencyshift of mechanical resonator. For example, we shall weigh the mass of nanoparticles (such as biomolecules, metal nanoparticles, etc.), and the order of magnitudes of nanoparticles are about femtogram ( 1 fg = 10−15 g ). We deposit a few nanoparticles onto the surface of the WGM cavity a and observe the frequency-shift of the mechanical resonator. We assume that nanoparticles are distributed uniformly on the resonator and the added mass of viruses does not affect the spring constant of the resonator. In Fig. 5(a), we first consider the condition of single WGM cavity (J ¼0), the black curve is the original probe transmission spectrum without landing any nanoparticles. With the increasing number of nanoparticles, the increasing mass of the mechanical resonator gives rise to the frequency-shifts in the probe transmission spectra. The frequencyshifts can be observed by optical detector. However, only the deposed mass is bigger than 10 fg, the frequency-shifts can be observed in the probe transmission spectra. However, if we take the coupling strength between the two cavities ( J = 0.5κa ) into consideration, even the frequency-shift induced by 2 fg nanoparticles can still be observed in the probe transmission spectra. The coupling strength J indeed enhance the resolution of the mass sensing. The inset of Fig. 5(b) indicates the direct linear relationship of the frequency-shifts and the mass of the nanoparticles deposited on the mechanical resonator. There are several advantages of the optical mass sensor based on the photonic molecules optomechanical system. The first one is the experimental feasibility. The photonic molecules systems based on double-WGM cavity have been implemented experimentally [39,40], additionally, on-chip single nanoparticle detection and sizing by mode splitting [45] and detection of single nanoparticles and lentiviruses using microcavity resonance broadening in an ultrahigh-Q microresonator have also been demonstrated experimentally [34]. These accomplished experiments indicate that the optical mass sensing based on the photonic molecules optomechanics is feasibility. The second advantage is its ultrahigh resolution. To enhance the mass sensitivity and resolution of optical mass sensor, a coupled metal nanoparticle-quantum dot mechanical resonator system is demonstrated due to the surface plasmon field in metal nanoparticle. However, putting together a number of ingredients (i.e., the quantum dot, the metal nanoparticle, and the nanomechanical resonator) is difficult to combine experimentally [38]. In the photonic molecules optomechanical system, the ultrahigh resolution of the optical mass sensor is controllable with manipulating the coupling between the two cavities which strongly depends on the distance of the two cavities [39,40]. Compared with the mass sensor based on the single WGM optomechanical systems [50], the resolution and sensitivity of the photonic molecules optomechanical sensor enhance significantly, and it can be easily implemented based on the current experimental conditions [7,39,40]. 4. Conclusion We have demonstrated the coherent optical propagation properties in a photonic molecules optomechanics consisted of a

H.-J. Chen et al. / Optics Communications 382 (2017) 73–79

WGM optomechanical cavity and a common WGM cavity coupled with exchanging energy depending on the distance between the two cavities under different driving conditions. Destructive interference between the probe beam and the anti-Stokes field (Stokes field) leads to a transparency window (amplification progress) in the probe transmission spectrum in conjunction with a steep positive phase dispersion (negative phase dispersion), giving rise to the corresponding slow light (fast light) effect. The coupling strength of the two cavities is a very significant factor for the coherent optical propagation in the photonic molecules optomechanics. In addition, the photonic molecules optomechanical system can be employed as an ultrahigh resolution mass sensor. The resonance frequency of the mechanical resonator can be determined from the probe transmission spectrum, and the coupling strength between the two cavities will enhance both the line width and the intensity, which will be beneficial to implementing mass sensing. Therefore, the mass of the accreted nanoparticles landing onto the resonator can be obtained according to the relationship between the added mass and the corresponding frequency shifts.

Acknowledgment The authors gratefully acknowledge support from the National Natural Science Foundation of China (No. 11404005, No. 51502005, No. 61272153, and No. 61572035).

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