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High-sensitivity force sensing using a phonon laser in an active levitated optomechanical system

Yutong He, Zijian Feng, Yuwei Jing, Wei Xiong, Xinlin Chen, Tengfang Kuang, Guangzong Xiao, Zhongqi Tan, and Hui Luo

Open Access Open Access

Abstract

Force detection with high sensitivity is of paramount importance in many fields of study, from gravitational wave detection to investigations of surface forces. Here, we propose and demonstrate a force-sensing method based on gain-enhanced nonlinearity in a nonlinear phonon laser. Experimental and simulation results show that the input force leads to the frequency shift of phonon laser, due to nonlinearity. In addition, we further investigate the influences of the pumping power, numerical aperture, and microsphere’s refractive index on the performance of this force-sensing system, regarding the sensitivity and the linear response range. Our work paves a new way towards the realization of precise metrology based on the nonlinearity of phonon laser.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Force detection has always taken an important role in precision measurement since all the changes of motion were concerned with the force due to Newton’s laws. Thus, it facilitates a variety of applications including magnetic resonance force microscopy [1], space experiments [2], tests of gravitational physics at short range [3,4], investigations of surface forces including the Casimir effect [5], as well as inertial sensing [6,7]. In particular, levitated optomechanics is showing the potential for precise force measurements [8,9]. Levitated optomechanical sensing technology uses an optical trap in a vacuum to achieve levitation and confinement of the optical-trapped mechanical oscillator at the micro- and nanoscale [10–12]. Theoretically, the nano- and microspheres are entirely isolated from external thermal noise and mechanical vibration. Thus, it exhibits low thermal noise in translational motion, which, in principle, is limited mainly by the background gas pressure, photon recoil [13] and shot noise [14]. Then, systems with optically levitated and trapped dielectric nano- and microspheres are generally believed to be a promising platform for precision sensing.

In recent years, various breakthroughs have been achieved in the field of precision measurement with this technology. The sensitivity of force sensing has been improved from 10−18 N/√Hz to 10−21 N/√Hz. Besides, the resolution and continuous working time have been improved to the 10−22 N and kilo-second, respectively [15–17]. We note that displacement measurement of the trapped microspheres is the key to the works mentioned above. The main methods of displacement measurement in levitated optomechanical systems include quadrant photodetector [18], balanced photodetector [19] and fiber optical detection [20,21]. All of the methods mentioned above are based on the measurement of spatial characteristic of scattered light, and their accuracies are limited by the resolving power of the light intensity, shot noise, and so on.

Phonon lasers, which exploit coherent amplifications of phonons, have gradually become one of the emerging frontiers in the last decades, and have promising applications in quantum sensing and precise measurement [22–30]. Very recently, we demonstrated a multi-color phonon laser by taking advantage of the gain-enhanced nonlinearity of an active levitated optomechanical (LOM) system [31]. In that work, strong nonlinear optical force emerges, benefiting from dissipative coupling between the intracavity-trapped microsphere and the active cavity. A remarkable result is that a nonlinear phonon laser with multiple harmonics was first realized. We stress that this unique nonlinear system also has the potential for precise metrology.

In this paper, we propose a strategy for direct force sensing by using the active LOM system. In such nonlinear system, the movement of the trapped microsphere would lead to its frequency shifting, due to nonlinearity-induced frequency shift. Thus, the input force can be easily detected by measuring the real-time frequency of the phonon laser. This new method has a promising potential in breaking the limits of light intensity resolution in traditional levitated optomechanical systems.

2. Experimental setup and principles

The experimental setup is depicted in Fig. 1 which includes a dual-beam optical tweezer (green), an active LOM system (red) and an electric field force generator (blue). In the experiment, a charged silica microsphere (diameter, D = 2 μm; mass, 11.1 ng; charge, ∼103e) is optically trapped in a vacuum with the dual-beam optical tweezers. In addition, the free-space laser path of the active LOM system is vertical to the dual-beam optical tweezer in the z-direction as illustrated in Fig. 1(a). The active LOM system contains the Yb3+-doped gain fiber as the gain medium, pumped by a semiconductor laser at 976 nm. The clockwise (CW) and counterclockwise (CCW) beams in the ring cavity are expanded by two collimators C1 and C2 with opposite directions into the free-space laser path respectively. Two lenses L1 and L2 (NA = 0.25) focus the CW and CCW beams onto the trapped microsphere. Collimators C1 and C2 are then used to couple the transmitted light through the trapping region back into the fiber loop. In the free-space laser path, the dissipative coupling between the microsphere’s position and the intracavity laser is formed. In addition, the needle as an electrode is directly below the microsphere in the x direction as shown in Fig. 1(b). It is made of polished stainless steel with a tip radius of 200 μm. The distance d0 between the trap center and the needle is measured to be about 3 mm. Meanwhile, a power amplifier is installed to amplify the electronic field, with an amplification factor of 92.8.

 figure: Fig. 1.

Fig. 1. Schematic of the experimental set-up on (a) the yoz plane and (b) the xoy plane. WDM, wavelength division multiplexer; C1 and C2, collimators; L1-L4, lenses; PBS, polarizing beam splitter; λ/2, half-wave plate.

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The Langevin equation of microsphere is quite crucial to discuss the relationship between the fundamental frequency of nonlinear phonon lasers and the electrostatic forces. The microsphere carries multiple charges, which causes it subjected to the additional electric field force Fe. Then, the Langevin equation of the microsphere in this system can be described as

$$m\frac{{{d^2}x}}{{d{t^2}}} + \gamma \frac{{dx}}{{dt}} + m\omega _0^2x = {F_{Brown}} + {F_{IC}} + {F_e}$$
where the γ is the gas damping, related to the radius of the microsphere and the viscosity of the gas. ω0 is the natural frequency determined by the optical force on the microsphere in a passive cavity. ${F_{Brown}} = \sqrt {2{k_B}T\gamma } W(t)$ is the Brownian motion force originated by a random process that satisfies the fluctuation-dissipation theorem. ${F_{IC}} = Q(x)[P(x) - {\textstyle{{dP} \over {dx}}}{\tau _c}(x){\textstyle{{dx} \over {dt}}}]$ is the optical force exerted by the laser in the ring cavity, where Q(x) is the trapping efficiency, P(x) is the intracavity power, and τc(x) is the intracavity photon lifetime. In addition, the intracavity power P(x) can be described as [32]
$$P(x) = {P_0}[\frac{{\mu {P_p}}}{{{\delta _i} + {\delta _s}(x)}} - 1]$$
where P0 is the saturation power constant, Pp is the pumping power, and δi is the scatting loss caused by the microsphere. Fe arises from the Coulomb interaction of the charged microsphere with the external electric field E(t). Approximating the needle tip as a sphere of radius rt, then according to Gauss’s Law, the equation could be listed as $\oint {Ed{s_t}} = \oint {{\textstyle{{dv} \over {dr}}}4\pi {r_t}dr} = 4\pi {r_t}V = {\textstyle{{{q_t}} \over {{\varepsilon _0}}}}$, where V is potential, qt is the charge at the needle tip, ε0 is the vacuum permittivity. Thus, Fe can be expressed as [15]
$${F_\textrm{e}} = \frac{{{q_t}{q_p}}}{{4\pi {\varepsilon _0}{d_0}^2}} = \frac{{{q_p}{r_t}V}}{{{d_0}^2}}$$
where qp is the charge of the microsphere, and d0 is the distance between the microsphere and the needle tip. In addition, the displacement x of the microsphere is negligible with respect to the distance d0. Thus, d0 is approximately independent of x.

To sum up, the Langevin equation of the microsphere can be converted as

$$m\frac{{{d^2}x}}{{d{t^2}}} + [\gamma + {\gamma _{opt}}(t)]\frac{{dx}}{{dt}} + m\Omega _0^2x = \sqrt {2{k_B}T\gamma } W(t) + {F_e}$$
where ${\gamma _{opt}}(x) = Q(x){\textstyle{{dP} \over {dx}}}{\tau _c}(x)$ is the optical damping rate, related to the microsphere’s displacement; Ω0 is the tuned oscillating frequency, expressed as
$$\Omega _0^2 = \omega _0^2 - \frac{1}{m}\frac{{dQ(x)P(x)}}{{dx}}. $$

As shown in Eq. (5), the oscillating frequency is determined by the nonlinear feedback forces Q(x)P(x) from the active LOM system. We note that the displacement of the trapped microsphere can regulate the nonlinear feedback forces. Thus, the displacement signal can be transformed to the frequency shift of the phonon laser, i.e., nonlinearity-induced frequency shift. Then, the force-sensing resolution of this method is decided by the oscillating linewidth of the phonon laser, instead of spatial distribution of scattered laser. This new feature has promising potential in future applications, since researchers have mastered the technology for precise frequency measurement, i.e., time measurement [33].

We define two key parameters to quantify the performance of the sensor, including the sensitivity S and the linear response range ξ. The sensitivity S can be written as

$$S = \frac{{\varDelta {\Omega _0}}}{{\varDelta {F_\textrm{e}}}}$$
where ΔΩ0 is the frequency shift, and ΔFe is the variation of electrostatic force. In addition, the determination coefficient R2 reflects how well the linear regression curve fits the observed values, and it is defined as
$${R^2} = \frac{{SSR}}{{SST}} = \frac{{\sum\limits_{i = 1}^n {{{({{\hat{y}}_i} - \bar{y})}^2}} }}{{\sum\limits_{i = 1}^n {{{({y_i} - \bar{y})}^2}} }}$$
where SSR is the regression sum of squares, SST is the total sum of squares, yi is the i-th data to be fitted, $\bar{y}$ is the mean value of y, and ${\hat{y}_i}$ is the i-th fitted data. Then, the linear response range in experiments and simulations could be defined as a region with a determination coefficient greater than 0.95. Notably, R2 is closer to 1, indicating that the fitted region is more linear.

3. Results and discussions

Driving a nonlinear phonon laser is the foundation of the experiment, as the force sensing is based on its nonlinearity. In our system, the active feedback plays a key role in compensating losses and achieving the nonlinear phonon laser. By regulating the active cavity position, the dissipative coupling strength is enhanced. Then, when the central frequency of the intracavity power spectrum is consistent with the trapped microsphere’s oscillating frequency, the nonlinear phonon laser emerges, also see Ref. [31].

To investigate the nonlinearity-induced frequency shift, we apply the DC signal to the electrode with three different voltages 300 mV, 0 mV and -300 mV. The resultant power spectral densities are shown in Fig. 2(a), where three oscillating peaks with evident distinction are observed. We find that the fundamental frequency Ω0 of the phonon laser shifts towards the positive axis for a positive DC voltage induced force + Fe, whereas a negative DC voltage induced force -Fe leads to a negative shift. That is a symbolic phenomenon resulting from the nonlinearity of our unique active LOM system.

 figure: Fig. 2.

Fig. 2. Nonlinearity-induced frequency shift. (a) Normalized power spectral density with three typical electrostatic forces. (b) Frequency shift versus the electrostatic force Fe for experiment (dotted) and simulation (dashed).

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We further discuss the shift of the fundamental frequency with different electrostatic DC forces. As depicted in Fig. 2(b), we scan the electrostatic forces from -1.3 pN to 1.3 pN to investigate the linear response range. The fundamental frequencies of phonon laser with different electrostatic forces are shown in Fig. 2(b), where three symbolic results are revealed. First, when the electrostatic forces are within a certain range (blue-shaded), the fundamental frequencies of the phonon laser are linear with the electrostatic forces. The linear response range is around 1.8 pN, and the sensitivity is 833 Hz/pN. The performance can be improved by further optimizing experimental parameters. Second, the direction of the electrostatic force has a distinct effect on the fundamental frequency. The spectrum has a blue shift when applying a positive force, while it has a red shift when applying a negative force. That means we can recognize the force’s direction. The dashed line is the simulation result calculated by solving Eq. (4). We can find that the simulation results are consistent with the experiments, indicating that our model can well simulate the phonon lasing dynamic.

To further explore higher sensitivity and larger linear response range, we numerically discussed the influences of the pumping power Pp, the refractive index np of the microsphere and the numerical aperture NA of the lens on the fundamental frequency shift as below.

Characteristic of the fundamental frequency shift versus pumping power Pp is shown in Fig. 3. Figure 3(a) illustrates the frequency shift as a function of electrostatic force for three typical pumping powers. Result shows that the pumping power is a crucial factor to optimize the sensitivity S and the linear response range ξ. We extract the sensitivity S for different pumping power according to Eq. (6), as shown in Fig. 3(b). When the pumping power increases, the sensitivity S firstly increases and then decreases with a corner pumping power of 34 mW. Below the corner pumping power, the intracavity tweezers provide stronger nonlinearity with the pumping power’s increasing, and thus a greater shift of phonon laser’s frequency emerges. When the pumping power exceeds 34 mW, the optical force gradually becomes dominant. As a result, the sensitivity decreases. The linear response range as a function of the pumping power is calculated according to Eq. (7), as presented in Fig. 3(c). It shows that the linear response range rises with the pumping power’s increasing. As a result, the higher sensitivity could be achieved with a moderate linear response range in practice.

 figure: Fig. 3.

Fig. 3. Characteristics of the fundamental frequency shift versus pumping power. (a) Frequency shift versus the electrostatic force for three typical pumping powers. (b) Sensitivity S and (c) linear response range ξ versus pumping power.

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NA is also a decisive factor for the nonlinearity of our system, as illustrated in Fig. 4(a). The sensitivity and linear response range of fundamental mode frequency show distinctive tendency for different NA. When the NA increases, the sensitivity decreases gradually, as shown in Fig. 4(b). We attribute it to the enhancement of intracavity optical force. Since the dissipative coupling strength is enhanced for a larger NA, the intracavity power will increase and motion of the trapped microsphere is further restricted within a weak nonlinear region. The linear response range versus NA is shown in Fig. 4(c). The linear response range initially rises until the NA reaches 0.45. However, with the further increase of NA, the linear response range starts to decrease, as the consequence of the apparent decrease in intracavity power resulting from excessive scattering loss. Taking both sensitivity and linear response range into account, setting NA around 0.45 is optimal with a maximum linear response range at 2.45 pN and a relatively high sensitivity.

 figure: Fig. 4.

Fig. 4. Characteristics of the fundamental frequency shift versus NA. (a) Frequency shift versus the electrostatic force for three typical NAs. (b) Sensitivity S and linear response range ξ versus NA.

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The influence of the refractive index np of microsphere on the characteristics of fundamental frequency shift, is shown in Fig. 5(a). The refractive index mainly affects intricate momentum change acted on the particle. As a result, the influence of the refractive index on the fundamental mode shift is quite complex. Figure 5(b) shows that the sensitivity increases gradually with the refractive index of the microsphere increasing. In addition, the linear response range versus the refractive index of the particle is shown in Fig. 5(c). The linear response range firstly decreases with the growth of the refractive index. However, it begins increasing, when the refractive index increases beyond 2.

 figure: Fig. 5.

Fig. 5. Characteristics of the fundamental frequency shift versus refractive index. (a) Frequency shift versus the electrostatic force for three typical refractive indexes. (b) Sensitivity S and (c) linear response range ξ versus microsphere refractive index.

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4. Conclusions

In this work, we propose and demonstrate a force-sensing method based on gain-enhanced nonlinearity in a nonlinear phonon laser. Experimental results show that the external force would lead to frequency shift of phonon laser, thanks to nonlinearity-induced frequency shift. A theoretical model is built to analyze the relationship between external force and frequency shift of phonon laser. Then, we fully discuss the influences of the pumping power, numerical aperture, and microsphere’s refractive index on the sensitivity and the linear response range of this force-sensing system.

In the experiment, the sensitivity is 833 Hz/pN and the linear response range is 1.8 pN. Very recently, ultra-stable levitated oscillator with oscillating linewidth of 80 nHz is realized [34]. With that parameter, the force sensing resolution of our method can research ∼10−22 N by a nanogram microsphere. We stress that our method has potential to realize higher resolution with optimal parameter. The pumping power, numerical aperture, and microsphere’s refractive index show distinctive effect to the sensitivity and the linear response range. A higher sensitivity can be obtained for a higher NA and a larger refractive index. The optimal pumping power is around 34 mW. Meanwhile, a larger pumping power is beneficial to broaden the linear response range.

The unique feature of this work is that the displacement signal can be read from the frequency shift of the phonon laser. Since the frequency measurement accuracy is extremely high, this method provides a basis for achieving higher sensing accuracy. Besides of force sensing, our method is also available for other regions of precise metrology. Our work opens new valuable directions for the application of nonlinear phonon laser and paves the way for precise metrology with flexible platforms.

Funding

National Natural Science Foundation of China (61975237); Scientific Research Project of the National University of Defense Technology (ZK20-14); Natural Science Foundation of Hunan Province (2021JJ40679).

Disclosures

The authors declare no competing financial 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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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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Figures (5)
Fig. 1.
Fig. 1. Schematic of the experimental set-up on (a) the yoz plane and (b) the xoy plane. WDM, wavelength division multiplexer; C1 and C2, collimators; L1-L4, lenses; PBS, polarizing beam splitter; λ/2, half-wave plate.

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Fig. 2.
Fig. 2. Nonlinearity-induced frequency shift. (a) Normalized power spectral density with three typical electrostatic forces. (b) Frequency shift versus the electrostatic force Fe for experiment (dotted) and simulation (dashed).

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Fig. 3.
Fig. 3. Characteristics of the fundamental frequency shift versus pumping power. (a) Frequency shift versus the electrostatic force for three typical pumping powers. (b) Sensitivity S and (c) linear response range ξ versus pumping power.

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Fig. 4.
Fig. 4. Characteristics of the fundamental frequency shift versus NA. (a) Frequency shift versus the electrostatic force for three typical NAs. (b) Sensitivity S and linear response range ξ versus NA.

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Fig. 5.
Fig. 5. Characteristics of the fundamental frequency shift versus refractive index. (a) Frequency shift versus the electrostatic force for three typical refractive indexes. (b) Sensitivity S and (c) linear response range ξ versus microsphere refractive index.

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Equations (7)

Equations on this page are rendered with MathJax. Learn more.

m d 2 x d t 2 + γ d x d t + m ω 0 2 x = F B r o w n + F I C + F e
P ( x ) = P 0 [ μ P p δ i + δ s ( x ) − 1 ]
F e = q t q p 4 π ε 0 d 0 2 = q p r t V d 0 2
m d 2 x d t 2 + [ γ + γ o p t ( t ) ] d x d t + m Ω 0 2 x = 2 k B T γ W ( t ) + F e
Ω 0 2 = ω 0 2 − 1 m d Q ( x ) P ( x ) d x .
S = Δ Ω 0 Δ F e
R 2 = S S R S S T = ∑ i = 1 n ( y ^ i − y ¯ ) 2 ∑ i = 1 n ( y i − y ¯ ) 2
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