# Cavity optomechanical spectroscopy constraints chameleon dark energy scenarios

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## Abstract

The chameleon scalar field is a matter-coupled dark energy candidate with the screening mechanisms. In the present paper,we propose a quantum cavity optomechanical scheme to detect the possible signature of chameleons via the optical pump-probe spectroscopy. Compare to the previous experiment the sensitivity can be improved by the using of electrostatic shield and a pump-probe scheme to read the weak frequency splitting. We expect that this work will be a useful addition to the current literature on proposals to detect effects of dark energy.

## 1 Introduction

The nature of dark energy is a central mystery in cosmology, one possibility is that it consists of the scalar fields which may drive the acceleration of the expansion of the universe directly. This new light degrees of freedom will couple to matter fields and leads to long-range fifth forces [1]. But this new forces have not yet been detected on earth or in the solar system [2, 3]. One way to alleviate this tension between theory and observation is through the introduction of screening mechanisms. The archetype of this screening mechanism is the chameleon model [4, 5, 6] which proposes that the coupling to matter depends on the local environmental matter density. In dense regions, such as in the laboratory, the coupling is very small, and the resulting force mediated by the chameleon is short ranged, shielding the chameleon interaction from detection. In regions of low density, such as in space, the coupling can be much stronger, and the resulting force mediated by the chameleon is long ranged. In a laboratory vacuum, the extremely low density ensures that sufficiently small objects are not screened from the scalar field and thus the force arising from the dark energy scalar could become significantly stronger. There have been a number of ways for probing the chameleon screening mechanism in the laboratory proposed or implemented in high vacuum, which have made efforts to derive new limits on the chameleon parameters. These include torsion-balance experiments [7, 8], gravity resonance spectroscopy [9, 10]. Recent searches using microscopic test masses such as atom, neutron [11, 12, 13, 14, 15, 16] and the levitated microspheres [17] often provide the strongest constraints.

On the other hand, optomechanical coupling between the electromagnetic degrees of freedom and the mechanical motion of mesoscopic objects are promising approaches for studying the transition of a macroscopic degree of freedom from the classical to the quantum regime [18]. These systems can also be of considerable technological for improved measurements of displacements [19], forces [20] and masses [21]. Recently, the optical pump-probe technique has become a popular topic, which affords an effective way to investigate the light-matter interaction. Most recently, this optical pump-probe scheme has also been realized experimentally in cavity optomechanical systems [22, 23, 24]. Several phenomena have been demonstrated in different kinds of optomechanical systems based on the optical pump-probe technology such as optomechanically induced transparency [25], the large change in light velocity [26], optically-tunable delay [27], and light storage [28]. The mechanism underlying these effects can be explained as the four-wave mixing (FWM) process. The optical pump-probe technology uses a strong pump laser to stimulate the system to generate coherent optical effect while used the weak laser for probing. This method is of great interest for applications in nonlinear optical measurement within the cavity optomechanical system.

## 2 Theory framework

We consider the two microspheres are placed inside a vacuum cavity consisting of two fixed mirrors. As shown in Fig. 1, the approach is based on optically trapping a microsphere with low natural mechanical frequency in the anti-node of an optical standing wave. The dielectric microsphere is attracted to the anti-node of the field. The resulting gradient in the optical field provides a sufficiently deep optical potential well which allows the particle to be confined in a number of possible trapping sites, with precise localization due to the optical standing wave [29, 30, 31, 32]. We use the two spheres with the same radius \(R_{i}\), density \(\rho _{i}\), and mass \(M_{i}\)(\(i=1,2\)). A shield membrane is placed between the sphere A and sphere B to minimize the electrostatic and Casimir background forces by preventing direct ac coupling between the masses. We use a silicon die which is etched into a frame bearing a \(1~\upmu \hbox {m}\) thick membrane of silicon nitride across an area of \(\sim 1~\hbox {mm}^{2}\). The entire shield wafer die, including the membrane, is coated with gold on both sides, Because of the geometry and the tensile stress in the membrane [33], the membrane is expected to be sufficiently stiff. Previous experimental works have been successful at testing gravity at ultrashort distances by use of the Faraday shield [34, 35], and similar techniques may work for the setup proposed here. Thus the experiment’s force noise is presently induced by the electromagnetic interaction between the shield and the microspheres as we will discuss at last. The shield separates the left side of the cavity from the right side. In our model, the left cavity modes are strongly driven by the pump and probe pulses. Since the shielding membrane is apparently fully reflecting, we don’t pump the right cavity which only provides the optical trapping of the sphere B.

*g*is the optomechanical coupling rate. Therefore the Hamiltonian of the system, in a frame rotating with the input laser frequency, is then given by

A chameleon scalar field [4] is characterized by an effective potential density \(V_{eff}(\phi )=V(\phi )+A(\phi )\rho _{i}.\)The self-interaction \( V(\phi )=\Lambda ^{4+n}\phi ^{-n}\) is characterized by strength of the self interaction \(\Lambda \). The chameleons can drive the cosmic acceleration observed today if \(\Lambda \) is close to the cosmological-constant scale, \( \Lambda \simeq 2.4~\hbox {meV}\). The coupling function to matter \(A(\phi )=e^{\phi /M} \) is characterized by an energy scale *M*, which is expected to be below the Planck mass. The chameleon profile due to an arbitrary static distribution of matter can be obtained by solving the non-linear Poisson equation \(\nabla ^{2}\phi =\partial V_{eff}/\partial \phi \).

## 3 Forecasts and constraints

To illustrate the experimental feasibility of the proposal, we will choose a set of experimental parameters.

(1) Dielectric object. We assume microspheres fabricated from fused silica with a radius \(R_{i}=3~\upmu \hbox {m}\), density \(\rho _{i}=2.3~\hbox {g}/\hbox {cm}^{3}\), and a dielectric constant \(\epsilon =2\). (2) Cavity. We assume a low-finesse cylinder cavity of length \(L=10~\hbox {cm}\), the radius of mirrors \(R_{vac}=5~\hbox {cm}\), and finesse \(F_{c}=10\) leading to a cavity decay rate \(\kappa =c\pi /2F_{c}L=5\times 10^{8}~\hbox {Hz}\). Then we obtain the standard optomechanical coupling rate \(g\approx 0.25~\hbox {Hz}\) with waist of the trapping field \(W=120~\upmu \hbox {m}\) [37]. (3) Lasers. The cavity is impinged by a pump pulse of power \( P_{p}=\Omega _{p}^{2}\hbar \omega _{c}/2\kappa =3\times 10^{-5}~\hbox {W}\) with wavelength \(\lambda =1064~\hbox {nm}\), and we scan the probe frequency across the pump frequency in the spectrum. We assume the probe pulse of power \( P_{pr}=10^{-7}~\hbox {W}\) and the pulse frequency \(\sim \omega _{i}(10^{5}~\hbox {Hz})\).

*P*the gas pressure.

*M*and \(\Lambda \). For both the test and source objects are unscreened. The CIC strength can be expressed as

*M*. Without the presence of chameleons \((M\rightarrow \infty )\), we find a enhanced peak located at the fundamental frequency \(\omega _{i}\) of the two oscillators as shown in the first picture. As the chameleons appear with the strength \(M=10^{-3.6}M_{Pl}\), the probe transmission spectrum will present an asymmetric splitting depicted by the second picture. Here \(M_{Pl}=1/\sqrt{8\pi G}\) is the reduced Planck mass. The two peaks displaying in Fig. 2 for given energy scales \(M=10^{-3.8}M_{Pl}\) and \( M=10^{-4}M_{Pl}\) represent the chameleon interactions between the two spheres. The splitting becomes larger with increasing

*M*. Thus the dark energy signature can be observed distinctly in the pump-probe spectrum. The splitting distance

*D*has a simple relationship with the CIC strength \(\Psi \) via \(\Psi =D/2\), which provides a straight way to measure the CIC strength on the spectrum. According to the Eq. (30), the minimal detectable field-matter coupling strength \(M_{\min }\) is mainly determined by the full width at half maximum (FWHM) of the resonance peak on the probe spectrum. Considering \(D_{\min }=FWHM\approx 10^{-7.8}~\hbox {Hz}\) in Fig. 2, one can obtain \( \Psi _{\min }\approx 8\times 10^{-9}~\hbox {Hz}\), hence \(M_{\min }=10^{-3.5}M_{Pl}\).

*M*, but be changed by the strength of the self interaction \(\Lambda \). Substituting the minimal detectable CIC strength \(\Psi _{\min }=D_{\min }/2\approx 8\times 10^{-9}~\hbox {Hz}\) into Eq. (31), we find that the dark energy scale can be measured down to a precision of the order of \(\Lambda \sim 10^{-1}~\hbox {meV}\).

## 4 Background noise

### 4.1 Background forces

One can place a stiff metallized shield between the drive and test masses to minimize the effects of electromagnetic forces by preventing direct ac coupling between the masses. This method lead to the conclusion that the experiment’s force resolution is presently limited by an environmental effect, most likely an electrostatic interaction between the shield and the test mass. Basically, there is a relatively large background force present at all times between the microspheres and shield membrane. Although electrically neutral microspheres are used, they still contain \(10^{14}\) charges and interact primarily as electric dipoles.

### 4.2 Thermal noise

*T*is the resonator temperature, \( \Delta f\) is the measurement bandwidth,

*Q*is the quality factor of the microsphere which can be given by \(Q=\omega _{i}/2\pi \gamma _{i}\approx 10^{12}\). \(E_{c}=m\omega _{i}^{2}\langle x_{c}^{2}\rangle \) represents the maximum drive energy of the microsphere, where we defined \(\langle x_{c}\rangle \) as the maximum rms level still consistent with producing a predominantly linear response. For a Gaussian field distribution, the nonlinear coefficients are given by \(\xi =-2/W^{2}\) [45], Considering the beam waist radius \(W=120~\upmu \hbox {m}\), for small displacements \( \left| x_{c}\right| \ll \left| \xi \right| ^{-1/2}=8\times 10^{-5}~\hbox {m}\), the nonlinearity is negligible. In our considerations, \(x_{c}\) is taken to be 3 orders of magnitude smaller, we choose \(x_{c}=8\times 10^{-8}~\hbox {m}\), thus the frequency stability will reach \(\delta \omega =1.5\times 10^{-8}~\hbox {Hz} \) with the measurement bandwidth \(\Delta f=10^{-5}~\hbox {Hz}\) at the room temperature (\(T=300~\hbox {K}\)). Here, \(\Delta f\approx 1/2\pi t_{m}\) and is dependent upon the measurement averaging time \(t_{m}\), thus we get \(t_{m}\approx 1.6\times 10^{4}~\hbox {s}\).

## 5 Conclusion

We discuss dynamics of the chameleonic coupled quantum cavity optomechanical oscillators, then suggest feasible methods to detect the chameleon dark energy with high sensitivity based the pump-probe technology.Using two levitated microspheres in an ultrahigh-vacuum chamber, we reduced the screening mechanism. We find that the normal mode splitting can be observed due to the chameleonic induced coupling strength. The splitting distance has a simple linear relationship with the CIC strength, which strongly reveals the new light degrees of freedom coupled to matter fields. We use a stiff metallized shield to provide the attenuation of electrostatic and Casimir forces between two spheres, then discuss the background forces between the shield membrane and the levitated sensor. The splitting distance is independent of the interactions between them, leading an effective way to avoid strong background force noise. Moreover, several other methods are also proposed to minimize the noises.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 11274230 and 11574206), the Basic Research Program of the Committee of Science and Technology of Shanghai (No. 14JC1491700).

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