1. Introduction

A long-standing issue of how to find the mass of an object as precisely as possible affects many aspects of modern society. When the mass of a macroscopic body is measured, people resort to a comparison with those of standard objects. However, this routine method becomes invalid when the masses of more microscopic objects, such as nanoparticles, cells, etc, need to be known. An effective way to overcome the barrier is through a basic law of simple harmonic oscillation—the mechanical frequency $\omega _m=\sqrt {k/m}$, where $k$ is the elastic modulus and $m$ the mass of the oscillator. When a number of nanoparticles, for example, are attached to a thin membrane as the mechanical oscillator, their mass can be found by the frequency shift $\delta \omega _m$ of the membrane through the relation $\delta m/2m=-\delta \omega _m/\omega _m$ [1]. This approach has been widely applied with low-dimensional nanostructures such as carbon nanotubes, nanowires, and others [1–20], to realize highly sensitive measurements of mass to the levels of zg [17], sub-zg [1], and even yg [14].

Previously, almost all those measurements were realized by a direct determination of the changed mechanical resonant frequency by a mass to be measured, applying the response of a mechanical resonator to external drive. The power of the sensitivity lies in the relative mechanical frequency shift $\delta \omega _m/\omega _m$ that can be achieved in measurement. One way to increase detection precision is to reduce the mass $m$ of the nanomechanical resonator, thus increasing its corresponding mechanical frequency $\omega _m$. More recently, the resonant mechanical frequency was increased to the order of GHz [7,14], which can achieve the mass sensing to the level of yg ($10^{-24}$g) [14], corresponding to the ratio $\delta \omega _m/\omega _m\sim 10^{-6}$. All these measurements should be implemented in ultra-low temperature environment, e.g., $T=4.3$ K in Ref. [14], since the coupling of mechanical resonator to an thermal environment induces the random motion of the mechanical oscillator and thus impairs the detection precision for mechanical frequency. Such requirement obviously limits the practicality of the mass sensing technology.

In reality, there exists an inevitable loss of mechanical energy due to the friction quantified by the damping rate $\gamma _m$. A mechanical frequency shift $\delta \omega _m$ should be larger than this damping rate so that it can be measured precisely. It is possible to surpass the limit by increasing the detection time, but the detection efficiency will be inevitably lowered. The efforts to go beyond the limit of mechanical damping rate reflect a recent focus in the relevant researches. Among the various candidates, the optomechanical system (OMS) was regarded as a promising one to realize ultra-high resolution sensing of nanoparticles or single-molecules; see, e.g. [21–26]. So far OMS have been widely studied in many other interesting aspects, such as optomechanically induced transparency (OMIT) [27–29], optomechanically induced amplification (OMIA) [30–32], high-order sideband nonlinearity [33–36], quantum criticality [37–40], and were also proposed to measure the mass of nanoparticles beyond the mentioned limit (see, e.g. [13,41–44]). The sensitivity in determining $\delta \omega _m$ can reach $0.01\gamma _m$ by the application of OMIT [41], high-order sidebands [13,42], or quantum criticality [45]. However, ultra-low temperatures are also required in those schemes, because the increase of environment temperature lowers the sensitivity very quickly.

Here we put forward a previously unexplored approach of ultra-high resolution mass sensing, which works with an OMS driven by two equally strong fields or a two-tone field with equally strong components. One of the frequency component of the driving field is resonant with the resonant cavity frequency, i.e. its detuning $\Delta _1=\omega _c-\omega _1=0$, but the other one component is blue detuned as $\Delta _2=\omega _c-\omega _2<0$, where $\omega _c$, $\omega _1$, $\omega _2$ are the resonant frequency of the cavity and the frequencies of the two drive components, respectively. The mechanism of the concerned mass sensing is a nonlinear effect that arises when the difference in the frequencies of driving components is exactly equal to the mechanical frequency $\omega _m$ [46,47]. Based on the mechanism, very tiny frequency shift of the mechanical resonator, e.g. on the level of $7$ order narrower than the mechanical damping rate, can be discriminated through the changed high-order sidebands of the output cavity field, thus offering an indirect way of measuring the mechanical frequency shift $\delta \omega _m$ together with the corresponding mass $\delta m$ that causes the mechanical frequency shift. On the other hand, the requirements for the setup are less stringent, such as with a relatively low resonant mechanical frequency $\omega _m=10$ MHz and a moderate mechanical quality factor $Q=\omega _m/\gamma _m=10^3$. The mass sensing can be also performed at room temperature since the thermal noise does not seriously affect the output signals.

2. Setup and system dynamics

The physics of OMS varies with different types of driving field. If one red detuned drive at the point $\Delta =\omega _m$ is applied, it will be possible to achieve a ground state cooling of the mechanical resonator [48–52]. On the other hand, a considerable mechanical-field entanglement can be created if one applies a sufficiently strong driving field blue-detuned at $\Delta =-\omega _m$ [53–57]. Two fields, a strong one of red detuned together with another weak one of resonance, can bring about the phenomena of OMIT and OMIA [27–32]. More interestingly, when two equally strong driving fields have their frequencies well matched, a nonlinear phenomenon of the simultaneous amplitude and phase freezing of the mechanical oscillation will come into being [46,47]. Here, we use two optical fields or one field with two equal frequency components for the purpose of mass sensing. In the rotation frame with respective to the resonant cavity frequency $\omega _c$, the dynamical equations of the system are given in terms of the dimensionless cavity quadratures $X_c$, $P_c$ and the dimensionless mechanical displacement $X_m$ and momentum $P_m$ as

 

Fig. 1. Setup for mass sensing. A pump laser with two frequencies $\omega _1$ and $\omega _2$ is applied to the optical cavity with a thin membrane suspended inside. If one of the two equally strong frequency components is resonant with the intrinsic cavity frequency ($\omega _1=\omega _c$) and the other is detuned to the resonant blue detuning point ($\omega _2=\omega _c+\omega _m$), the mechanical motion of the membrane can be frozen into a series of orbits like energy levels. There exists a unique cavity field spectrum corresponding to each mechanical energy level (see a similar phenomenon in Ref. [46]); whenever the system maintains on one mechanical energy level, a varied drive amplitude $E_1=E_2=E$ only proportionally changes the amplitudes of the sidebands together, without altering the field spectrum pattern. After a tiny mass $\delta m$ is added onto the membrane, its intrinsic oscillation frequency $\omega _m$ will be shifted by $\delta \omega _m$, so that the difference of the driving frequencies $|\omega _1-\omega _2|=\omega _m$ will no longer match the mechanical frequency shifted by $\delta \omega _m$. Under a number of such mechanical frequency shifts, e.g. $\delta \omega _{m,i}$ for $i=1,2,3$ as in the above, the stabilized cavity field intensity or photon number $\mathcal {E}_c(t)$) will be changed according to the magnitude of $\delta \omega _{m,i}$ (an exaggerated view is shown here), while the orbit of the mechanical membrane is deformed. The amplitudes of the higher order sidebands ($l=2,3$) can be modified more significantly with the same mechanical frequency shift.

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The nonlinearity of the system is enhanced by the drive amplitude $E_1=E_2=E$, so it is impossible to adopt the perturbation approach to tackle the system dynamics. Here we apply the numerical simulations to the evolutions of $X_c(t), P_c(t), X_m(t)$, and $P_m(t)$ according to Eq. (1). When the mechanical membrane stabilizes in an oscillation, the numerically simulated cavity field is found to have the form

A side issue is about the high-order corrections

3. Mechanism for mass sensing

The mechanism for mass sensing is sketched in the lower part of Fig. 1. Under the condition of drive frequency match, i.e., $|\Delta _1-\Delta _2|=\omega _m$ or $|\omega _1-\omega _2|=\omega _m$, the stabilized motion of the suspended membrane will be fixed into a series of orbits when the drive field intensity is sufficiently strong [46]. However, an added mass $\delta m$ slightly modifies the oscillation frequency of the mechanical membrane, to violate such frequency match by the corresponding shift $\delta \omega _m$ proportional to $\delta m$. This change due to a very small $\delta \omega _m$ may only slightly deform the mechanical orbit, but will bring about observable effect to the sidebands of the cavity field, especially to the higher order ones. In Fig. 2 we detail more about the evolution of the system to the fixed orbits like energy levels $n\geq 1$, when the amplitudes $E_1=E_2=E$ of the driving fields are higher than $1.9\times 10^5\kappa$ (given the system parameters used for the figure) and their frequencies satisfy the condition $|\Delta _1-\Delta _2|=\omega _m$—the evolutions with the sample drive amplitudes are shown in Fig. 2(a). On such a energy level, the mechanical motion $X_m(t)$ is almost fixed to the same as if it were frozen on the level (see the appendix at the end of the article). Plugging such approximately identical $X_m(t)$ into the first two equations of Eq. (1), one will have the cavity field quadratures $X_c$ and $P_c$ that grow proportionally to $E_1=E_2=E$, and the stabilized dimensionless cavity field energy or cavity photon number

 

Fig. 2. Frozen mechanical motion and its disappearance due to unmatched drive frequencies. (a) The examples of evolving to the first energy level under the frequency match condition, which displays the evolutions of the mechanical energy due to $E=1.3\times 10^6 \kappa$ (red), $E=1.8\times 10^6 \kappa$ (indigo), and $E=2.3\times 10^6 \kappa$ (black). The mechanical energy due to a drive amplitude $E$ within a certain range only evolves to the first level $n=1$. Under the same frequency match condition, the evolved cavity energy or cavity photon number will have the same pattern. Here we show such evolved pattern on the first level $n=1$. (b) The energy levels will be destroyed if the frequency match condition is violated by a mechanical frequency deviation $\delta \omega _m$. Here the black curve no longer evolves to the first energy level, while the red and indigo one still approach to the level. The mechanical frequency shift in the example is $\delta \omega _m=0.05\kappa$. The corresponding cavity fields that have stabilized will be modified due to violating the frequency match condition. The red and indigo curves also have obvious change, though their corresponding mechanical motions still evolve toward the first energy level. The used parameters for the simulations are $g_m=10^{-5}\kappa, \omega _m=10\kappa$, and $\gamma _m=10^{-3}\kappa$, and the logarithmic scale is used here to show the overall pictures of evolution.

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In the current study we consider one resonant driving field $\Delta _1=0$ and one blue-detuned driving field $\Delta _2=-\omega _m$. Another combination of one resonant field plus one red-detuned field in Ref. [46] can be also applied for mass sensing. However, the mechanical energy level $n=1$ formed by the driving fields of resonant and blue-detuned is more sensitive to the drive frequency match condition. In Fig. 2 the frozen mechanical orbits on the first energy level have been partially lost when there is a mechanical frequency deviation on the level $\delta \omega _m\sim 0.01\kappa$; the orbits due to stronger driving fields escape from the level $n=1$ as seen from Fig. 2(b). The locking of mechanical oscillation on the energy level will be completely lost even by a smaller $\delta \omega _m$, and it gives rise to detectable modification on the sideband amplitudes of the cavity field. If the drive frequencies consist of one resonant and one red-detuned as in Ref. [46], the mechanical oscillation will be less deviated from the locked state and cavity field sideband will accordingly have less modification, given the same amount of $\delta \omega _m$. It is the reason for us to apply one resonant and one blue-detuned field for the detection of mechanical frequency shift.

For more details of how the modified cavity field sidebands are relevant to a mechanical frequency shift, we provide the numerical simulations of the relevant spectrum in Fig. 3, where the second and third-order sidebands, together with the evolved limit cycles of the mechanical resonator, are displayed for various relative intensities of the driving fields. All displayed driving field combinations make the system stabilize in dynamical oscillation, having the sidebands with their peaks around the frequencies $l\omega _m$ ($l$ are the integers). We start from a single resonant driving field in Fig. 3(a). Under a sufficiently strong field that is resonant with the intrinsic cavity frequency $\omega _c=cN\pi /L$ without optomechanical effect, where the integer $N$ counts the nodes of the electromagnetic standing wave over the distance $L$ in cavity, the radiation pressure will be relatively high to cause a larger pure displacement $\langle d\rangle$ of the mechanical membrane (its time-average position). The effective cavity frequency $cN\pi /(L+\langle d\rangle )$, therefore, becomes obviously different from $\omega _c$. This effect exhibits as shifting the peaks of the sidebands from the locations $l\omega _m$; see Fig. 3(a), where the peaks of the sidebands stay away from the locations $l\omega _m$. When there is a small change $\delta \omega _m$ of the mechanical frequency, the sidebands of the cavity field will be deviated form those of the original $\omega _m$ (compare the different sets of sidebands in the figure), though the corresponding change to the limit cycles of the mechanical membrane is not perceivable by the used scale. However, the direct discrimination of the two different sets of sidebands, which are respectively shifted from $l\omega _m$ and $l(\omega _m+\delta \omega _m)$, is very hard, especially when the mechanical frequency shift $\delta \omega _m$ is narrower than the damping rate $\gamma _m$. The similar sideband shifts also manifest in Fig. 3(e), where the system is under a single blue-detuned driving field instead.

 

Fig. 3. Illustration of the mechanism for mass sensing. From (a) to (c), the blue detuned field is increased till it is equal to the resonant one. From (c) to (e), the resonant field is lowered till there will be only the blue-detuned field acting on the optical cavity. In (a) and (e), the sidebands are shifted to different central positions for two mechanical frequencies that are differed by $\delta \omega _m=10^{-3}\kappa$. With the enhancement of the blue detuned field, the cavity field sidebands manifest the different variations in their amplitudes for such two different mechanical frequencies. The illustrated limit cycles are mostly indistinguishable unless with a magnified scale, but the difference in both second-order and third-order sideband become obvious, when the amplitudes of the two driving fields become closer. We choose $E_1=3\times 10^5\kappa$ in (a)-(c) and $E_2=3\times 10^5 \kappa$ in (d)-(e). The system parameters are the same as those in Fig. 2.

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There will exist another nonlinear effect if two driving fields (or two driving frequency components) under the frequency match condition $|\Delta _1-\Delta _2|=\omega _m$ act simultaneously on the OMS. An enhancement of the blue-detuned field, as seen from Fig. 3(a) to Fig. 3(c), will gradually lock the mechanical orbits to a fixed energy level. Meanwhile, the frequency shifts from $l\omega _m$ will disappear as the averaged pure displace $\langle d\rangle$ becomes lowered in the process. If the mechanical frequency is changed by $\delta \omega _m$ due to a small mass added to the membrane, the frequency match condition will be broken since the frequencies of the two driving fields are fixed. Given the system parameters used in Fig. 3, the mechanical orbits will no longer be frozen on the first energy level if the mechanical frequency shift is up to $\delta \omega _m\sim 0.1\kappa$, and then the system will be reduced to a quasi-linear one since the realistic optomechanical constant $g_m$ is rather small. Within a smaller range of $\delta \omega _m$, however, those frozen mechanical orbits will be only slightly deformed, but the tiny $\delta \omega _m$ will bring about modification to the cavity field spectrum. The shift $\delta \omega _m= 10^{-3} \kappa$ in Fig. 3 can cause obvious difference in the sideband amplitudes, though most of the limit cycles in Fig. 3(a)-Fig. 3(e) do not show perceivable difference by the used scale of coordinates.

The underlying mechanism is to have the mechanical oscillation $X_m(t)$ frozen under the drive frequency match condition $|\Delta _1-\Delta _2|=\omega _m$ and, at the same time, the cavity field will have a fixed pattern [for example, the one displayed in Fig. 2(a)], which has its spectrum amplitudes only grow proportionally with the increased $E_1=E_2=E$ not beyond the magnitude to reach another mechanical energy level. In principle any $\delta \omega _m$ induces a certain deviation of the cavity field from the fixed pattern under the mentioned drive frequency match condition. Such change, even if it could be very tiny, can be read out through the analysis of cavity field sidebands (especially the higher order ones). The mechanical frequency shift $\delta \omega _m$, as well as the corresponding tiny mass $\delta m$, can thus be measured with a high accuracy.

4. Mass sensing performance

Next, we will study the capability of the concerned system to detect the mechanical frequency shift $\delta \omega _m=\omega _m-\omega _m^0$ from the original $\omega _m^0$ of the mechanical membrane, which is proportional to the mass to be detected. Here we use the second-order sideband to illustrate the mass sensing approach. For the current example system, the cavity damping rate is set at $\kappa =1$ MHz, and the original mechanical frequency is $\omega ^0_m=10$ MHz, together with the mechanical damping rate at $\gamma _m=10$ kHz or $100$ Hz, corresponding to the quality factor $Q=10^3$ or $10^5$, respectively. These system parameters are well within the current experimental technology [48]. In the operation of the system we have the dimensionless drive amplitude $E/\kappa =3\times 10^5$, corresponding to a pump laser power of about $12$ mW (the wavelength of the laser is around $\lambda =1537$ nm).

We first give a couple of examples to show how the mechanical frequency shift within the range $\delta \omega _m=\omega _m-\omega _m^0\in [-0.0110\kappa, -0.0100\kappa ]$ (the minus sign arises from the added mass through the relation $\delta \omega _m=-\delta m/2m\times \omega _m^0$), which spans a length of $10^{-3}\kappa$, is related to a difference $\delta n^{(2)}_c$ in the second-order sideband intensity. Here the sideband difference

In addition to the changed sideband intensity due to a mechanical frequency shift, there is a tougher requirement on the exact determination of a shift $\delta \omega _m$. For example, the different shifts $\delta \omega _m \in [-0.0109010\kappa, -0.0109000\kappa ]$, i.e. the patch within the small circles pointing to the insets in Fig. 4(a), which spans over a distance $10^{-6}\kappa$ on the horizontal axis, must be well discriminated to realize a high-resolution performance. The changed output fields in the refined range are shown in the upper inset to Fig. 4(a)—a logarithmic scale is used there to show the trends for two different $Q$ together. However, the reading of the exact values of $n^{(l)}_{c,out}$, which are used to determine $\delta \omega _m$, are always limited by a shot noise from statistics. The shot noise has the magnitude $\sqrt {n^{(l)}_{c,out}}$. To distinguish between the output fluxes $n^{(l)}_{c,out}$ corresponding to two very close mechanical frequencies $\omega _m$, we define a relative signal-to-noise ratio (SNR) as

 

Fig. 4. Mass sensing performance in terms of the variation of the second-order sideband. (a) The sideband photon number difference due to the shift $\delta \omega _m$. Here $\omega _m^0=10\kappa$ and $E_1=E_2=E=3\times 10^5 \kappa$, and the difference on the right side of the indicated range is set to be zero. The insets show the details in a further refined range, the patch of $\delta \omega _m$ within the indicated small circles. The SNR at each $\delta \omega _m$ in this sample range is calculated by the sideband difference of the two close points $\omega _m^0+\delta \omega _m-5\times 10^{-9}\kappa$ and $\omega _m^0+\delta \omega _m+5\times 10^{-9}\kappa$ to a shifted mechanical frequency $\omega _m=\omega _m^0+\delta \omega _m$, and is larger than $100$ for $Q=10^5$ and about $2.6$ for $Q=10^3$. The corresponding output flux in the range is displayed by a logarithmic scale. (b) The SNR for different $\omega _m^0$ in an even refined range. Here SNR$=4.9$ for $\omega _m^0=10\kappa$, while the damping rate $\gamma _m=10^{-3}\kappa$ and the ratio $g_mE/\omega _m^0=0.5\kappa$ is fixed. (c) The sideband change and the achievable SNR in a sample range near $\omega _m^0$, where the SNR, which is $1.02$ for $Q=10^3$, is determined by the sideband difference from the points $\omega _m^0+\delta \omega _m-10^{-10}\kappa$ and $\omega _m^0+\delta \omega _m+10^{-10}\kappa$ (except for the point $\delta \omega _m=0$). The fixed parameters are the same as those in Fig. 2.

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In the small window $\delta \omega _m \in [-0.010901\kappa, -0.010900\kappa ]$ extracted from Fig. 4(a), each $\delta \omega _m$ from the original $\omega _m^0$ should be determined to the level of at least $10^{-7}\kappa$ so that different $\delta \omega _m$ in the range can be distinguished from one another. For the purpose we choose $\omega _{m,1}=\omega _m^0+\delta \omega _m-5\times 10^{-9}\kappa$ and $\omega _{m,2}=\omega _m^0+\delta \omega _m+5\times 10^{-9}\kappa$ in Eq. (9) to see how good each shift $\delta \omega _m$ inside the small window can be ascertained within the error $\pm 4.5\times 10^{-7}\delta \omega _m=\pm 5\times 10^{-9}\kappa$. For the system with $Q=10^5$ the SNR is larger than $100$, while it is about $2.6$ given $Q=10^3$, as the results in the lower inset of Fig. 4(a). The SNR can be improved further by increasing the original mechanical frequency $\omega ^0_m$, under the condition that the effective coupling rate $g_mE/\omega _m^0$ for the system keeps invariant. For example, increasing the mechanical frequency in this way from $\omega ^0_m=10\kappa$ to $\omega ^0_m=100\kappa$ will enhance the SNR from $4.9$ to $475$ as in Fig. 4(b). In many dynamical processes of OMS, including a dynamical cooling process [51] and the evolutions toward the frozen mechanical orbits under the matched frequencies of the driving fields [46], the effective coupling $g_mE/\omega _m^0$ determines the system behavior. If the rate $g_mE/\omega _m^0$ keeps the same, the dynamical evolutions according to the full nonlinear dynamics will be similar, except that the real-time cavity and mechanical energy only differ by a factor of $E^2$ [63]. A higher $E$ to compensate for a larger $\omega _m^0$ (while the ratio $g_mE/\omega _m^0$ keeps invariant) provides enhanced output fluxes $n^{(2)}_{c,out}$ to improve the discrimination of two very close mechanical frequency shifts.

The corresponding sideband differences exactly around the original mechanical frequency $\omega _m^0$ are shown in Fig. 4(c). Their tendencies are similar to those in Fig. 4(a). Within this sample range of $\delta \omega _m\sim 10^{-8}\kappa$, the SNR is found to be $1.02$ for ascertaining the tiny shifts $\delta \omega _m$ to $\pm 0.01\delta \omega _m$ by a system with $Q=10^3$. However, with a higher $Q$, the capacity for discriminating such tiny $\delta \omega _m$ can be significantly improved.

The limit performance (the SNR is only demanded to be $1$) of our example systems is summarized in Table 1. Around the original mechanical frequency $\omega _m^0$, this limit specifies a sideband change

Table 1. Mass sensing performance with different mechanical frequencies $\omega ^0_m$ and quality factors $Q$, if the $\delta \omega _m$ exactly around $\omega _m^0$ should be determined by a SNR higher than $1$. The drive amplitude is set to be $3\times 10^5 \kappa$, and the system parameters as those used in Fig. 2. Such performance can be improved by choosing a different drive amplitude.

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5. Choice of working point

The measurement of the small mass through the relation $\delta m/2m=-\delta \omega _m/\omega _m^0$ is achieved by the violation of the perfect frequency match condition $|\Delta _1-\Delta _2|=\omega _m^0$ with a small shift $\delta \omega _m$. A nonlinear mechanism due to the drive frequency match leads to the formation of a series of frozen orbits of the mechanical membrane [46], and which specific mechanical orbit can be reached is simply determined by the given drive amplitude $E_1=E_2=E$. The performance of the mass sensing is therefore highly relevant to the used $E$, as we will illustrate in what follows.

Without loss of generality we consider a possibility that the mechanical oscillator frequency is shifted from the original mechanical frequency $\omega ^0_m=10\kappa$ with a uncertainty to $\omega _{m,1}=9.99000000\kappa$ or $\omega _{m,2}=9.98999999\kappa$, given a setup of $Q=10^3$. The calculated SNR in Fig. 5 indicates how their difference in the order $10^{-8}\kappa$ can be distinguished by choosing different working points for the system. If the drive amplitude is below a threshold to form the first energy level, the dynamical evolutions of the system will be in a regime of critical slowing down [46], where the system takes extremely long time in approaching stability so that a stable spectrum of the cavity field is hard to come by. This regime is indicated in the inset of Fig. 5, and it excludes the lower $E$ for the implementation of the concerned mass sensing. Above the threshold around $E/\kappa =1.9\times 10^5$, the system will completely evolve to the first mechanical energy level to have a stable SNR until $E/\kappa =2.4\times 10^6$. Near the boundary of the instability and stability regions as in Fig. 5, a tiny mechanical frequency difference will result in larger difference in the sidebands; hence a high SNR exists near the threshold. This is related to a fact that the energy level is more fragile near the threshold and can be modified more rapidly with the shift from the original mechanical frequency $\omega ^0_m$. An evidence of the fact is that the stable energy level disappears as the deviation from the original mechanical frequency increases. For example, at $E=1.9\times 10^5\kappa$ near the threshold, the system is still on the first energy level when $\omega _m=9.99\kappa$, but it will enter the critical slowing down regime when the mechanical frequency is shifted further to $\omega _m=9.90\kappa$ (as compared to $\omega _m^0=10\kappa$). In a practical operation, therefore, one has to choose a drive amplitude that keeps a certain distance from the threshold. It should be noted that one of the examples in Fig. 4 shows the performance at point $P$ in Fig. 5, so there is a big room to improve the performance illustrated there.

 

Fig. 5. Capability of detecting a small shift $\delta \omega _m$ over the range of drive amplitude. Here we consider an example that the original mechanical frequency $\omega _m^0=10\kappa$ is shifted to $\omega _m=9.99000000\kappa$ or $9.98999999\kappa$ (two values with a difference $10^{-8}\kappa$) due to the uncertainty of added mass. The SNR for distinguishing between these two possibilities is high near the threshold for entering the first mechanical energy level, but will quickly drop with the increased drive amplitude $E$. If the drive amplitude $E>2.4\times 10^6 \kappa$, the distribution of SNR will be irregular due to the existence of the transition to the second energy level. Note that the horizontal axis is in the logarithmic scale. The corresponding distribution of the mechanical energy levels is in the inset, which shows how the stabilized average mechanical energy $<\mathcal {E}_m>$ under the matched drive frequencies distributes with the drive amplitude $E$. Here we consider a lower mechanical quality factor $Q=10^3$, and the fixed system parameters are the same as those in Fig. 2.

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Another flexibility in the range of lower drive amplitude $E$ is that the mass sensing can be implemented without a perfect frequency match condition, i.e., $|\Delta _1-\Delta _2|$ can have a slight distance from the exact mechanical frequency $\omega ^0_m$. As seen from Fig. 2(b), the mechanical orbits are still partially frozen for the lower $E$ (the exemplary evolutions under two lower $E$ reach an approximately fixed mechanical orbit in the figure). It is workable to apply an approximately frozen state as the reference to see the sideband changes due to the mechanical frequency further modified by the added mass.

The two minima of SNR in Fig. 5 correspond to where the second-order sideband of cavity field is most stable under small deviations from the drive frequency match condition. Other sidebands can be used there to have higher SNR. The SNR increases with $E$ across the second minimum, and it is partially due to a higher sensitivity of the energy level to the deviation of the drive frequencies match—one example is in Fig. 2(b) where the evolution due to a higher drive amplitude ($E=2.3\times 10^6\kappa$) escapes from the first energy level; here in Fig. 5 the shift $\delta \omega _m$ is much smaller to keep all evolutions toward that energy level. An irregular pattern of discontinuous SNR (the zigzag jumps between the SNR values) manifests when the drive amplitude $E$ reaches $E=2.4\times 10^6\kappa$. This phenomenon originates from the sudden transitions between the different mechanical energy levels when the drive amplitude $E$ is further enhanced [46]—the inset of Fig. 5 shows the sudden transitions between energy levels occur on the right part where the energy levels seemingly overlap with respect to the horizontal axis (by a magnified scale, however, one sees that only one such energy level can be reached given a specific $E$ [46]). The spectra on two different energy levels are different so that the calculated SNR according to the definition exhibits the irregular pattern. This regime should be also excluded for the performance of mass sensing.

6. Influence of noise perturbations

In addition to the shot noise arising from a statistical property in measurement, other types of noise, such as the fluctuation in the pump fields and the thermal noise due to the environment of non-zero temperature, exist to our concerned system. These noise driving terms $\xi _i(t)$ ($i=1,2$) and $\xi _m(t)$ in Eq. (1) can be simulated by the random functions generated by the computation platform MATLAB. One measures the output field for the purpose of mass sensing, so the influence of these noise drives on the output signals should be clarified. A particular advantage for our mass sensing method is that the concerned mechanical energy levels are rather robust against noise perturbations—the existing noise perturbations only affect the evolution of the system at the beginning stage; after the system stabilizes, those perturbations can hardly influence the system [46]. Around the mechanical energy level the concerned noise sources only modify the stabilized cavity field as the examples in Fig. 6(a).

 

Fig. 6. Influence of the driving field fluctuations and thermal noise. (a) The effect of the noise drives on the stabilized cavity field pattern, while the mechanical motion evolves to around the first energy level. The examples show the small modifications (viewed with a magnified scale) at $\delta \omega _m=-0.117505\kappa$. (b) The modification of the drive fluctuations to the sideband change $\delta n^{(2)}_c$. At each sample point of $\delta \omega _m=\omega _m-\omega _m^0$, we put a small variation $\chi =10^{-8}\kappa$ around that point so that the defined $\delta n^{(2)}_c$ is constant in the ideal situation without noise. (c) The similar modification due to the thermal noise at different temperatures. The results are obtained with the drive amplitude $E=3\times 10^5\kappa$, and the system parameters are the same as those in Fig. 2.

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In the determination of a small shift $\delta \omega _m$ from the original mechanical frequency $\omega _m^0$, we use the difference

The noise due to the fluctuation in the driving field mainly takes an effect in the regime where the SNR v.s. $E$ curve in Fig. 5 goes down in the form of a steep drop after entering the first energy level. In this regime the sidebands $n^{(l)}_c$ can become more sensitive to the driving field fluctuation. This regime also includes the working point $P$ at $E=3\times 10^5\kappa$ in Fig. 5 (note that the turning point at SNR$=0$ in Fig. 5 is due to taking the absolute value of sideband difference). The drive noise only adds an irregular deviation of the sideband difference $\delta n^{(2)}_c$ from the horizontal line without noise effect. The contribution from the noise with a larger magnitude $\sqrt {\langle \xi ^2_{1(2)}\rangle /\kappa }=100$ modifies the cavity photon number more seriously, as it is expected. The sideband change $\delta n^{(2)}_c$ shown in Fig. 6(b) indicates that, for the example system, the uncertainty of $\delta \omega _m$ can be determined to at least $1\times 10^{-8}\kappa$ even if there are driving field fluctuations. In practice the fluctuation of pump laser fluctuation can be well suppressed to very low level.

On the other hand, the thermal noise is an important factor that limits the detection sensitivity for many other schemes of mass sensing. The thermal noise induces a random motion of the mechanical membrane so that the shift $\delta \omega _m$ narrower than the mechanical damping rate is hard to measure directly. The thermal noise considerably modifies the amplitude of the second-order sideband, though the overall cavity field is not so much changed as in Fig. 6(a). However, even at the room temperature $T=300K$, the existing thermal noise modification in Fig. 6(c) will not qualitatively change the sideband difference $\delta n^{(2)}_c$ we use to discriminate $\delta \omega _m$. After considering the thermal noise effect at $T=300K$, one still has the SNR higher than $100$ (calculated with $\chi =10^{-8}\kappa$) at where the $\delta n^{(2)}_c$ is suppressed below the constant value of $\delta n^{(2)}_c$ obtained in the ideal situation without noise perturbation. This SNR can easily realize the discrimination of $\delta \omega _m$ within an error $\pm 8.5\times 10^{-7} \delta \omega _m=\pm 10^{-8}\kappa$, corresponding to a sensitivity $\chi /\gamma _m=10^{-5}$. Such a performance is achieved with moderate system parameters ($\omega _m^0=10$ MHz, $Q=10^4$, $g_m=10$ Hz) and at room temperature. The defined constant $\delta n^{(2)}_c$ without noise effect, the horizontal level line of $\delta n^{(2)}_c$ in Figs. 6(b) and 6(c), will be lowered to around $1\times 10^4$ when $\chi$ is reduced to $3\times 10^{-9}\kappa$, which can determine any $\delta \omega _m$ in the sample range to the error of $\pm 2.55\times 10^{-8}\delta \omega _m=\pm 3\times 10^{-9}\kappa$. Then the fluctuating thermal noise modification, as seen from the sample points that locate below the level line by $10^4$, would make the difference $\delta n^{(2)}_c$ hard to read. This $\chi$ specifies the highest precision available at room temperature for our example system.

7. Conclusion

We have presented the overall aspects of a novel approach to mass sensing. A setup illustrated in Fig. 1 allows an indirect measurement of the mechanical frequency shift $\delta \omega _m$, which could be too tiny to measure especially when it is less than the mechanical damping rate or mechanical line width $\gamma _m$. Our approach to measuring the shift $\delta \omega _m$ proportional to the mass $\delta m$ is based on a nonlinear dynamical mechanism of an OMS, which is driven by two driving fields or two frequency components of one driving field. The system will evolve to one of the fixed mechanical orbits under the condition that the difference of the drive frequencies matches the mechanical oscillation frequency. A particular cavity field pattern corresponding to each mechanical energy level will also come into being under the condition. The violation of the exact drive frequency match due to a tiny shift $\delta \omega _m$ will bring about detectable changes of the cavity field sidebands. The modification on the higher order sidebands, which are more sensitive to the drive frequency match condition, can be detected more easily.

Our illustrated examples working with the second-order sideband show that, given a mechanical membrane within a realistic range of intrinsic oscillation frequencies, one can easily realize the detection of its frequency shift $\delta \omega _m$ to the levels from $5$ to $7$ order narrower than its mechanical damping rate. Accordingly, a small mass $\delta m$ can thus be detected to the levels between $10^{-9}m$ and $10^{-11}m$, three to five orders better than those of the previous experiments and theoretical proposals, when it is attached to a membrane with the mass $m$. Such performance is achieved with moderate values of mechanical frequency $\omega ^0_m$ and quality factor $Q=\omega ^0_m/\gamma _m$, and can be improved further with a better fabrication of the setup. Because only the differences in output field influxes are required in the performance of mass sensing, a thermal noise does not impair their highly precise readings even at room temperature, in contrast to many other mass sensing schemes that should be implemented in ultra-low temperature environment. It is expected that the experimental exploration of the similar scenarios of mass sensing would be undertaken in the near future.

Appendix: Formation and deformation of mechanical energy levels

The formation of the mechanical energy levels is through a nonlinear mechanism that locks the oscillation amplitude and phase of a mechanical oscillator under the condition $|\Delta _1-\Delta _2|=\omega _m$ [46,47]. Due to the mechanism the mechanical motion in an OMS with the parameters in Fig. 2 will only evolve to one of the states corresponding to the energy levels $n\geq 1$ for a macroscopic object like mechanical membrane, as it is detailed in Fig. 7(a), once the drive amplitude $E_1=E_2=E$ is higher than $1.9\times 10^5\kappa$. Over the threshold the system can only evolve to one of the mechanical energy levels without settling down in anywhere else. This locking phenomenon belongs to the category of synchronization in nonlinear dynamical systems [64,65].

 

Fig. 7. Mechanical energy levels and their deformation due to mechanical frequency shift. (a) A group of sample evolutions to three different energy levels of the mechanical membrane. (b) A detailed view of stabilized mechanical energy $\mathcal {E}_m(t)$ on the first energy level, which change with their different pure displacements $d$ in Eq. (13). All of their corresponding $X_m(t)$, however, look the same as in the inset, because the pure displaces $d$ are much less than the frozen amplitude $A_1$ so that the difference in $X_m(t)$ is not perceivable by the used scale. The different amplitudes of the term $Ad\cos (\omega _m t+\phi )$ in Eq. (13) are magnified by $A$, so the stabilized $\mathcal {E}_m(t)$ better distinguishes the evolved results for different drive amplitudes $E$ that give rise to different $d$. (c) Examples of changing the curve of $E=1.5\times 10^6\kappa$ due to mechanical frequency shifts $\delta \omega _m$. The mechanical orbits have been off the energy level. The system parameters used here are the same as those in Fig. 2.

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Only the frequency component $\omega _m$ among the spectrum of the oscillating cavity field energy $\mathcal {E}_c(t)$, which is independent of any used observational frame, contributes to the mechanical motion significantly, because it provides the only driving force that is resonant with the mechanical frequency. The stabilized mechanical motion in all concerned situations (even if the mechanical energy levels are not formed), therefore, takes a rather simple form

(12)$$X_m(t)= A\cos(\omega_m t+\phi)+d.$$

Under the condition $|\Delta _1-\Delta _2|=\omega _m$, the mechanical motion $X_m(t)$ will evolve to the orbits with fixed $A_n$ for $n\geq 1$, while the pure displacement $d$ from the equilibrium point without radiation pressure is much smaller than the mechanical oscillation amplitude $A_n$. The corresponding dimensionless mechanical energy or phonon number

(13)$$\begin{aligned} \mathcal{E}_m(t)&=\left(X^2_m(t)+P^2_m(t)\right)/2\\ &= A^2/2+d^2/2+Ad\cos(\omega_m t+\phi)\\ &+A(\dot{d}/\omega_m)\sin(\omega_m t+\phi)+\dot{d}^2/(2\omega_m^2) \end{aligned}$$

under the condition $|\Delta _1-\Delta _2|=\omega _m$ will only evolve to one of the fixed stable states in Fig. 7(a). Their time-average

(14)$$\langle\mathcal{E}_m(t)\rangle=A_n^2/2+\langle d\rangle^2/2\approx A_n^2/2$$

are those of the illustrated energy levels. The stabilized mechanical energy oscillations for different drive amplitudes $E$ on the first energy level are detailed in Fig. 7(b). In the small inset of Fig. 7(b), all evolved mechanical displacements $X_m(t)$ to stability, which are due to these different $E$, appear as the same oscillation since the varied pure displacements $d$ with $E$ are negligible as compared to the fixed amplitude $A_1$. This phenomenon of complete locking the amplitudes and phases of mechanical oscillation exists under the conditions of sufficiently high $E$ and $|\Delta _1-\Delta _2|=\omega _m$. In this situation all evolved mechanical orbits to a fixed $A_1$ become approximately identical though drive amplitude $E$ can be different, as if they were frozen. Plugging such frozen $X_m(t)$ into the first two equations of Eq. (1), one will have a fixed cavity field pattern having its spectrum amplitudes to grow proportionally with the drive amplitude $E_1=E_2=E$.

Once the condition $|\Delta _1-\Delta _2|=\omega _m$ is violated by any amount of shift $\delta \omega _m$, the complete locking of the mechanical motion on the energy levels will be lost to some extent and the corresponding cavity field pattern will be deformed accordingly. Obvious deviations from an energy level are seen in Fig. 7(c), where the shifts $\delta \omega _m$ of the similar magnitudes to those in Fig. 2 have been relatively large. Given a much smaller $\delta \omega _m$, the deviation from the mechanical energy level could be imperceivable when viewed with a scale like the one for the limit cycles in Fig. 3, but the tiny difference can be detected from the modified spectrum of cavity field. This is the mechanism of our mass sensing technique.

Funding

National Natural Science Foundation of China (11574093); Natural Science Foundation of Fujian Province (2020J01061); Fondo Nacional de Desarrollo Científico y Tecnológico (1221250).

Acknowledgment

The authors thank Prof. Yun Feng Xiao and Prof. Chang Ling Zou for helpful discussions.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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