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Abstract
A detector with both broad operation range and high sensitivity is desirable in the measurement of weak periodic forces. Based on a nonlinear dynamical mechanism of locking the mechanical oscillation amplitude in optomechanical systems, we propose a force sensor that realizes the detection through the cavity field sidebands modified by an unknown external periodic force. Under the mechanical amplitude locking condition, the unknown external force happens to modify the locked oscillation amplitude linearly to its magnitude, thus achieving a linear scaling between the sideband changes read by the sensor and the magnitude of the force to be measured. This linear scaling range is found to be comparable to the applied pump drive amplitude, so the sensor can measure a wide range of force magnitude. Because the locked mechanical oscillation is rather robust against thermal perturbation, the sensor works well at room temperature. In addition to weak periodic forces, the same setup can as well detect static forces, though the detection ranges are much narrower.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In a scientific experiment, various forces such as pressure, pulling force, spring force, gravitational force, etc, are measured by a variety of methods. The measurements of many other physical quantities can be as well transformed to the measurement of forces. How to measure a force as precise as possible is an important task everywhere. Thanks to the realizable high quality, mechanical resonators are usually used to the sensing of force [1,2]. Combining an optical element, such as constructing an optomechanical system (OMS), the sensing precision can be improved significantly [2,3]. The famous LIGO is actually an OMS that realizes the detection of a gravitational-wave strain [4].
So far many different schemes or realistic experiments have studied the sensing of weak constant or periodic force. Here we focus on the weak periodic force sensing [5–22]. If the detected force frequency is resonant with the frequency of the used mechanical resonator, a sensitivity $15 aN/\sqrt {Hz}$ at room temperature can be achieved in an integrated hybrid system of a nanomechanical beam and a microdisk cavity [5]. Working at ultralow temperature of 1.2K, one has the sensitivity $12 zN/\sqrt {Hz}$ with a nanotube mechanical resonator [6]. Assisted with an ultracold atom cloud in a high-finesse optical cavity, the sensitivity $42\pm 13 yN/\sqrt {Hz}$ near the standard quantum limit (SQL) was reported in Ref. [7]. A sensitivity $11.2 aN/\sqrt {Hz}$, 1.5dB below the SQL is possible to the off-resonant force applied to a thin Si$_3$N$_4$ membrane at the temperature 10K [8]. Moreover, for the detected force frequency much less than the mechanical frequency, some theoretical schemes [23–26] suggest that the sensitivity beyond the SQL should also be possible at ultralow temperatures. Nevertheless most of the schemes demand the condition of ultralow temperature, thus bringing about a difficulty in their applications.
On the other hand, for any force sensor, its operation range should be ideally large enough. Within its operation range, a one-to-one correspondence between the directly measured quantity and the external force to be measured must be well established. A perfect linear scaling between these two quantities is very helpful to a convenient reading of the specific magnitude of the external force. In spite of their high sensitivity (even beyond SQL), the operation ranges of many above-mentioned methods are narrow, so they can be only applied to some special situations. The purpose of the current work is to provide a workable approach to improve both of these figures of merit, the force sensitivity and its operation range. This target can be achieved by an OMS driven by a two-tone laser field and operating in a nonlinear regime. Previously, OMS dynamics was well studied in the regimes of linearized dynamics, such as in mechanical ground state cooling [27–33] and optomechanical entanglement [34–41]. Beyond the more familiar nonlinear dynamical scenarios such as self-sustained oscillations [42,43], dynamical multistability [44,45] and optical pulse train generation [46,47], which are all based on a single driving field, our adopted mechanism by means of two simultaneous drive tones under a frequency condition realizes a kind of oscillation locking on the mechanical resonators [48,49], so that their stabilized mechanical oscillation become rather robust against thermal noise. Such proposed sensor can therefore operate at room temperature and accurately measure the cavity field sideband changes proportional to the measured force magnitude. Most of all, a desired linear scaling between the measured force and the quantity read in the measurement can be well established, and this linear scaling range is comparable to the applied amplitude of driving field.
The rest of the paper is organized as follows. In Sec. 2, we describe the setups and its dynamics to clarify the mechanism of the force sensor. The performance of the force sensor in different situations is discussed in Sec. 3. The corresponding working range and the effect of the existing thermal noise is discussed in Sec. 4, before the conclusion of the work in Sec. 5.
2. System dynamics and operation mechanism
A typical OMS with a movable mirror is illustrated in Fig. 1. Here we consider the OMS driven by a two-tone field. In terms of the cavity field mode $a$, which gives the cavity field photon number $|a|^2$, and the dimensionless displacement (momentum) $X_m$ ($P_m$) of the mechanical resonator (the movable mirror) with its frequency $\omega _m$ and damping rate $\gamma _m$, the dynamical equations for the system in Fig. 1 read
Fig. 1. Setup for force sensing. The optoemechanical system is driven by a two-tone laser field. When an external weak periodic force is applied to the mechanical resonator, the induced intensity change of cavity sidebands can be measured by a spectrum analysis. Different from the detection of the field quadratures [50,51], a linear relation between the intensity change and the force amplitude can be realized under the condition $|\omega _{L_1}-\omega _{L_2}|=\omega _m$ for the two drive tones. Under this condition the stabilized mechanical oscillations $X_m(t)$ are almost frozen, as shown with the sample amplitudes $E=5\times 10^5\kappa$ (red solid), $10^6\kappa$ (blue dashed), and $3\times 10^6\kappa$ (black dash-dotted). The system parameters are chosen as $g_m=10^{-5}\kappa, \omega _m=10\kappa, Q=\omega _m/\gamma _m=10^4$.
Under any sufficiently powerful driving laser, be it of single tone or multiple tones, the mechanical resonator will stabilize in an oscillation
with the amplitude $A_m$ and a pure displacement $d_m$ off the equilibrium position, due to a Hopf bifurcation after increasing the driving laser power over a threshold. The higher harmonic components of the mechanical motion are negligible here, because the contribution from the first sideband of $|a(t)|^2$ (with the frequency $\omega _m$) dominates as a linear resonance effect. There will be a special phenomenon under the condition for the two drive tones [48,49]—the mechanical oscillation amplitude $A_m$ will change slightly in spite of a variable drive power or drive amplitude $E$; see the example in Fig. 1 where the amplitudes $A_m$ due to different drive amplitudes $E$ appear to be close to one another. In this situation, the increased energy from a higher drive power will predominantly stay in the cavity field without affecting the mechanical oscillation significantly. This frozen mechanical motion will last over a considerable range of the drive amplitude $E$ until the amplitude $A_m$ will be locked to a higher value like a transition to another energy level. Within the difference up to $10^{-2}\kappa$, between $|\Delta _1-\Delta _2|$ and $\omega _m$, such amplitude locking can be well realized [49].The frozen or locked mechanical oscillation is rather robust against external perturbations. In Fig. 2 we compare the thermal noise induced modifications to the mechanical oscillation in both situations of single and double drive tones. It is found that, compared to the scenario of a single drive tone, such modification $\delta X_m(t)$ of the mechanical oscillation can be suppressed by hundreds of times if applying a two-tone field satisfying the condition in Eq. (3). This stability is very helpful to the operation of the proposed force sensor.
Fig. 2. The modification of the mechanical oscillation due to the thermal noise at $300$ K, which corresponds to $n_{th}=3.92\times 10^6$ for a mechanical resonator with the frequency $\omega _m=10$ MHz. In (a) the OMS is driven by a one-tone field with the detuning $\Delta =0$, and in (b) by a two-tone field with the detunings $\Delta _1=0$ and $\Delta _2=-\omega _m$. Here a present force has $\eta =100\kappa$, and the other parameters are $g_m=10^{-5}\kappa, \omega _m=10\kappa, Q=\omega _m/\gamma _m=10^4$, and $E=5\times 10^5\kappa$.
Another useful property is about the response of the mechanical oscillation to a coherent force, and we also illustrate it through a comparison with the single-tone scenario. Driven by a single-tone field, the cavity field has an analytical form [44,45]
where $\phi (t)=g_mA_m/\omega _m\sin (\omega _mt)$, andThis relation is verified by the numerical simulation based on the complete dynamics given in Eq. (1); see Fig. 3(a). As the result, each sideband in Eq. (5) will be modified as well, to have
where we have applied the first order Taylor expansion of the Bessel function $J_l(-g_mA_m/\omega _m)$ together with the fact $g_m\delta A_m/\omega _m\ll 1$.Fig. 3. The relations between the amplitude changing and the dimensionless force amplitude $\eta /\kappa$. (a) The OMS is driven by a one-tone laser field with detuning $\Delta =0$. (b) The OMS is driven by a two-tone laser field with the detuning $\Delta _1=0$ and $\Delta _2=-\omega _m$. The parameters are $g_m=10^{-5}\kappa, \omega _m=10\kappa, Q=10^4, E=5\times 10^5\kappa$.
In the case of a two-tone drive, there is no analytical form of the field sidebands. However, the mechanical oscillation in Eq. (2) can be read from a numerical calculation based on the system of nonlinear differential equations in Eq. (1). After plugging this numerically obtained mechanical oscillation into the first equation of Eq. (1), one will have a linear differential equation
for the cavity field. Then the cavity field under a two-tone drive will have an approximate formA particular feature under a two-tone drive satisfying the condition in Eq. (3) is that the modification of the mechanical motion happens to be in a linear relation
with the amplitude of an external force, as it is exemplified by the numerical calculation results in Fig. 3(b), which are based on the complete dynamics given in Eq. (1). This fact is related to the stability of the locked mechanical oscillations; the higher-order corrections to the $X_m(t)$ by the external force are highly suppressed to be negligible if the external force magnitude is not too large. The corresponding sideband amplitude change will thus be3. Force sensing performance
3.1 Detection of the forces with $\omega _f=\omega _m$
The intensity change of the cavity field sidebands can be read from the measured output field at a certain frequency. At the frequency $\omega _l$ defined in Eq. (11), the sideband intensity is
for $l\geq 1$. According to the input-output relation the corresponding output intensity is $n_{c,out}^{(l)}(\omega _l)=\kappa n_c^{(l)}(\omega _l)$. Here we choose the sideband $l\geqslant 2$ to distinguish the signal from the input field components. In a realistic experiment, however, the detection of output intensities is impaired by the existing shot noise due to a statistics property. Here we define the following signal-to-noise ratio (SNR)To demonstrate the performance of the force sensor, we obtain the relation between the SNR and the dimensionless force amplitude $\eta /\kappa$ in Fig. 4(a). Here we use three combinations of the drive detunings—(1) $\Delta _1=0,\Delta _2=-10\kappa$, (2) $\Delta _1=-5\kappa,\Delta _2=-15\kappa$, and (3) $\Delta _1=5\kappa,\Delta _2=15\kappa$, which all satisfy the condition in Eq. (3). Due to the dramatically increasing SNR in the range $\eta /\kappa =1\sim 100$, we adopt a logarithmic scale in the figure (these SNR will display an exact linear relation with $\eta$ after restoring to their ordinary axis scales). As expected, the SNR for the combinations increase linearly with the force amplitude. Certainly the linear relation also exists with other combinations of the drive frequencies, as long as the condition in Eq. (3) is satisfied. As shown in Fig. 4(a), the combination of $\Delta _1=0,\Delta _2=-10\kappa$ is the best among the three examples. The use of one resonant driving field together with another one laser field of blue detuned can definitely enhance the sideband amplitudes used for the measurement, thus having a better performance.
Fig. 4. (a) The tendency of the achieved SNR against the dimensionless force amplitude $\eta /\kappa$. Here we use the modification of the second sideband $\omega _2$ defined in Eq. (11) for an illustration, and the used drive amplitude is $E=5\times 10^5\kappa$. (b) The corresponding relation between SNR and the drive amplitude $E/\kappa$. For the detuning combinations $\Delta _1=0,\Delta _2=-10\kappa$ and $\Delta _1=5\kappa,\Delta _2=15\kappa$, increasing the drive amplitude will lower the SNR, while for the detuning combination $\Delta _1=-5\kappa,\Delta _2=-15\kappa$, it will increase the SNR instead. Here the measured force magnitude is fixed as $\eta =100\kappa$. The fixed parameters are the same as those in Fig. 1.
Besides the drive detuning, the mechanical quality factor affects the sideband amplitude change as well. If the mechanical damping rate $\gamma _m$ is reduced, the response of the mechanical resonator to an external force will be enhanced, leading to a higher sideband amplitude change and a higher sensitivity. However, the SNR and the damping rate are not simply in an inversely-proportional relation. For example, if the $\gamma _m$ is reduced from $10^{-3}\kappa$ to $10^{-4}\kappa$, the SNR will be only increased from $635$ to $895$, given the drive detuning combination $\Delta _1=0,\Delta _2=-10\kappa$ in the measurement of a force with $\eta /\kappa =1$.
A higher drive power would enhance the sideband amplitudes, seemingly from Eq. (11). However, increasing the drive power will also slightly deviate the mechanical amplitude $A_m$ from its locked average value, so that the Bessel function $J_l(-g_mA_m/\omega _m)$ will be modified to offset the increased sensibility by a higher $E$ (the varied $E$ can change the amplitude $A_m$ by more than one order, though it is still negligible compared to the overall tendency as in Fig. 1). Therefore, the relation between the sideband change and the drive amplitude $E$ will become more complicated, as demonstrated by the corresponding relations directly obtained by the numerical calculations in Fig. 4(b). For the detuning combinations $\Delta _1=0,\Delta _2=-10\kappa$ and $\Delta _1=5\kappa,\Delta _2=15\kappa$, their SNR become decreasing with the drive amplitude $E$ instead, while the SNR increases with $E$ for the detuning combination $\Delta _1=-5\kappa,\Delta _2=-15\kappa$.
The minimal detectable force amplitude $f_{min}$ that can be detected is determined by the SNR. Under the drive amplitude $E/\kappa =5\times 10^5$, the SNR for detecting the force of the dimensionless magnitude $\eta /\kappa =1$ is about $895$, requiring a laser power of about $32.4$ mW (given the damping rate $\kappa =1$ MHz and the wavelength $\lambda =1537$ nm of the driving laser field). The used system parameters ($Q=10^4$, $g_m/\kappa =10^{-5}$, $\omega _m/\kappa =10$ used in Fig. 4) are compatible with those achieved in the past experiments [27]. For example, the experimental setup in Ref. [28] has achieved the better parameters as $Q=3.4\times 10^5$, $g_m/\kappa =10^{-3}$, $\omega _m/\kappa =50$, while $\kappa /2\pi \sim 200$ kHz. Based on our parameters, the minimum magnitude $\eta _{min}=1.12\times 10^{3}$ Hz at the detection limit $\text {SNR}=1$ corresponds to a minimal force magnitude $f_{min}=\eta _{min}p_{zpf}=3.63\times 10^{-17}$ N (the mass of the mechanical resonator is given as $1$ ng).
3.2 Detection of the forces with $\omega _f\ll \omega _m$
In addition to sensing resonant external forces, the unknown forces with their frequencies off the mechanical frequency can be detected as well, though their associated sideband changes are generally less significant. One interesting case is that a force frequency is much less than the mechanical frequency, i.e. $\omega _f\ll \omega _m$, which has been studied in some other approaches [23–26]. After such a static force is applied, the stabilized mechanical motion can be approximated as
It is equivalent to adding an extra pure displacement from the equilibrium point of the mechanical resonator, thus leading to a sideband amplitude change
Without loss of generality, we adopt a detuning combination $\Delta _1=0,\Delta _2=-10\kappa$ as the example and work with the fourth sideband centered at the frequency $\omega _4$ defined in Eq. (11). The numerically simulated relations between the SNR and the force magnitude $\eta /\kappa$, as well as with the drive amplitude $E$, is displayed in Fig. 5. These results are consistent with the qualitative estimation in Eq. (18). The SNR will be about $2$ for the magnitude $\eta /\kappa =1$, if the system is driven under $E/\kappa =5\times 10^5$ and has a mechanical quality factor $Q=10^4$. The minimum force magnitude that can be measured is about $f_{min}=1.62\times 10^{-14}$ N. After the drive amplitude is increased to $E/\kappa =1.1\times 10^6$, the SNR will be about $4$ under the force $\eta /\kappa =1$, so that the minimum detectable force can be decreased to $f_{min}=8.12\times 10^{-15}$ N.
Fig. 5. The relations between the achieved SNR and the dimensionless force amplitude $\eta /\kappa$ and the drive amplitude $E/\kappa$. These relations are realized when the external force is close to be static. The used detuning combination is $\Delta _1=0,\Delta _2=-10\kappa$, and the other parameters are the same as those in Fig. 2.
From the above discussion, one sees that the difference of detecting a quasi-static force ($\omega _f\ll \omega _m$) from detecting a periodic force with its frequency $\omega _m$ lies in the different responses of the cavity field sidebands—the former is to a pure displacement while the latter is their modifications under a forced oscillation. Then the cavity field sideband changes will have two different scaling laws in the respective measurements, the one in Eq. (18) and the one in Eq. (13). The static force can be also detected through the second sideband [52], which gives a much higher sensitivity while the corresponding linear scaling range will be much narrower than the one in Fig. 5(a). The mechanism to detect a static force is through the cavity field transition due to a parallel displacement [52], $\Delta _{1(2)}\rightarrow \Delta _{1(2)}+\delta _F$, of the two driving field components, where $\delta _F$ is an extra detuning arsing from the modification of the optical cavity size by the static force. While the condition in Eq. (3) is still satisfied under such parallel displacement, the mechanical oscillation locking will be adjusted to another state so that sidebands of the stabilized cavity field can be considerably modified to see the effect of the applied static force. However, the range for a linear scaling between the measured force and the detect sideband change is generally narrower in this situation, as compared to the one realized in the same OMS by a periodic force. Static forces can be measured more accurately (to the level of $10^{-20}$ N [52] as compared to $10^{-17}$ N for the periodic force if measuring the second field sideband with the same illustrative OMS) but their detection ranges by the same system becomes much narrower.
3.3 Detection of the forces with $\omega _f=n\omega _m$
Another interesting case is that the force frequency is integer times of the mechanical frequency $\omega _m$. Without loss generality, we here use $\omega _f=2\omega _m$ as an example. In this situation one needs to consider the high-order components of the mechanical motion such as in the following exemplary one:
where $B_m\ll A_m$. This expression is without the presence of the external force. The corresponding cavity field is thus improved to the formIf an external force with the frequency $\omega _f=2\omega _m$ is applied to the mechanical resonator, the mechanical motion will be
modifying the second order oscillation amplitude $B_m$ to the amplitude $B_m-\frac {\eta }{3\omega _m}$. Since $g_m\eta /6\omega ^2_m\ll 1$, the variation of each factor $J_p(-g_mB_m/2\omega _m)$ due to the external force can be approximated as linear, thus giving a scaling of the sideband amplitude change at the frequency $\omega _{l+2p}$ with the force magnitude $\eta$.The above estimations can be verified by the numerical simulations based on the full nonlinear dynamics given in Eq. (1), which lead to the linear relations in Fig. 6. Their consistency with the above qualitative results is over range of $\eta /\kappa \sim 1000$. According to these simulations, the SNR is about $7$ under a force with $\eta /\kappa =10$, when the drive amplitude is $E/\kappa =5\times 10^5$ and the system has $Q=10^4$. The minimum detectable force amplitude is then $f_{min}=4.64\times 10^{-14}$ N approximately. If the drive intensity is increased to $E/\kappa =1.1\times 10^6$, the detection limit can be improved to $f_{min}=2.71\times 10^{-14}$ N.
Fig. 6. The relations between the achieved SNR and the dimensionless force amplitude $\eta /\kappa$ and the drive amplitude $E/\kappa$. These relations are realized when the periodic external force has the frequency $2\omega _m$. The used detuning combination is $\Delta _1=0,\Delta _2=-10\kappa$, and the other parameters are the same as those in Fig. 2.
4. Operation range and thermal noise effect
For a force sensor, the sensitivity and operation range are two important figures of merit. Many force sensing schemes have a trade-off between these two figures of merit; a very high sensitivity should be realized within a very narrow range of operation. In our scheme such limitation is overcome by a dynamical mechanism described in Fig. 1. As pointed out in Refs. [48,49], the mechanical amplitude can be locked to the same level over a wide range of drive amplitudes, i.e. between $E/\kappa =5\times 10^5$ and $E/\kappa =10^7$ for the example in Fig. 1. The external oscillatory force is found to modify the locked mechanical amplitude linearly, and this property offers a nice scaling for the measurement. Such a linear scaling will be destroyed only when the force magnitude $\eta /\kappa$ becomes comparable with the drive amplitude $E/\kappa$. We illustrate this fact in Fig. 7(a), where the ending point of a linear scaling is marked by a green cube. The scaling relations in Fig. 7(a) are consistent with Eq. (13). For a lower drive amplitude $E$ in the examples, the linear scaling is destroyed at a lower force magnitude $\eta /\kappa$. This could be interpreted with a stronger locking effect under a higher drive power; the mechanical oscillation is less perturbed by an oscillatory force if it is locked by a stronger two-tone driving field satisfying the condition in Eq. (3). However, the drive amplitude $E$ should be also limited within a range so that the locked mechanical oscillation does not jump to a higher amplitude, which is shown in [48,49]; for the illustrated system this limit is at $E\sim 10^7\kappa$.
Fig. 7. (a) The linear scalings between the sideband amplitude modification at $\omega _2$ and the dimensionless force amplitude $\eta /\kappa$, which are illustrated with three different drive amplitudes that realize the locking of the mechanical amplitude illustrated in Fig. 1. The green cube at each end of a line denotes the boundary of the linear scaling, beyond which the relation will be no longer linear. (b) The relations between the sideband amplitude change at $\omega _2$ and the dimensionless force amplitude $\eta /\kappa$, given some different environment temperatures. The applied detuning combination is $\Delta _1=0, \Delta _2=-10\kappa$, and the mechanical oscillation frequency is $\omega _m=10$ MHz, while the fixed parameters are the same as those in Fig. 1.
At room temperature the thermal noise effect should be taken into account. Many force sensors must work at low temperatures to reduce the noise effects. Due to the locking mechanism by two drive tones satisfying Eq. (3), the stabilized mechanical oscillation is rather robust against the thermal perturbation as seen from Fig. 2. The specific modification from a thermal environment to the sideband amplitude change can be found numerically, as we illustrate in Fig. 7(b). One finds that, even at room temperature, the existing thermal noise only slightly deviates the sideband modifications from those at $T=0$ K. The force sensor thus has an advantage of being able to work at room temperature.
5. Discussion and conclusion
Regarding the implementability of the proposal, one can design the sensor based on a simple OMS with its parameters close to those of various previously available setups [27]. An essential requirement is that the difference $|\Delta _1-\Delta _2|$ for two drive tones should be as close to the mechanical frequency $\omega _m$ as possible, which can be approached by adjusting the drive frequencies with acoustic-optic modulator (AOM) [53] or single-sideband modulator (SSB) [54]. The required amplitude locking can be achieved once the difference between $|\Delta _1-\Delta _2|$ and $\omega _m$ is within the order $10^{-2}\kappa$ ($\kappa$ is the cavity damping rate) [49]. During an operation under a higher drive power, the resonant cavity frequency $\omega _c$ was observed to shift from the original one due to the changed temperature by a thermo-optic effect [55]. It will however add a common detuning to the two drive tones without violating the condition $|\Delta _1-\Delta _2|=\omega _m$ for the accurate measurement performed after the system stabilizes to thermal equilibrium. Because the same dynamical evolution result of a nonlinear optomechanical process can be realized under the condition $g_mE=\text {constant}$ [56], the drive power proportional to $E^2$ can be significantly lowered to neglect the thermo-optic effect. For example, if the optomechanical coupling $g_m=10^{-5}\kappa$ adopted in our numerical simulations is increased to $g_m=10^{-3}\kappa$ of the experimental setup in Ref. [28], the driving laser power for implementing the measurement can be only in the order of $\mu$W. A considerable flexibility in manufacturing the setup exists with the advancement of the relevant technologies.
In addition to its own values, the measurement of weak periodic forces over a relatively large magnitude range is highly relevant to photo-acoustic technology [57,58]. Our proposed force sensing approach is different from all others in the applied mechanism. The mechanical oscillation locking mechanism realizes a good linear scaling between a measured quantity and an unknown external force to be detected. Under the condition of oscillation locking, the effect of thermal perturbation can be highly suppressed, so the force sensor can work at room temperature. Another important feature is that the stability of locked oscillation enables the mentioned linear scaling range for the measure force magnitude to be comparable with the applied pump drive amplitude. This property significantly widens the detection range by the same force sensor, making the setup more attractive in possible applications.
Funding
National Natural Science Foundation of China (11574093); Natural Science Foundation of Fujian Province (2020J01061); Agencia Nacional de Investigación y Desarrollo (Fondecyt 1221250).
Acknowledgment
The authors thank Dr. Yan Lei Zhang 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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