# Detecting large extra dimensions with optomechanical levitated sensors

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## Abstract

Many experiments have been conducted to detect hypothetical large extra dimensions from sub-millimeter to solar system separations. However, direct evidence for such extra dimensions has not been found. Here we present a scheme to test the gravitational law in 4 + 2 dimensions at micron separations by optomechanical methods. We demonstrate the feasibility of the normal mode splitting in the optomechanical system under the gravitational interaction between two levitated resonators. The weak frequency splitting can be optically read by the optical pump-probe scheme. The sensitivity can be improved by suppressing the effect of the Casimir force coupling and the electrostatic interaction. Thus, we can detect the large extra dimensions at low noise levels based on the levitation optomechanics without the isoelectronic technique.

## 1 Introduction

Why is gravity so weak compared to the other fundamental forces in nature? This question can be recast in terms of the hierarchy problem, namely the seeming disparity (about 16 orders of magnitude) between the electroweak symmetry breaking scale and the Planck mass. This problem can be overcome by adding new dimensions in the large extra dimension (or the bulk) model, namely the ADD model, which was first developed by Arkani-Hamed, Dimopoulos, and Dvali [1, 2, 3]. They proposed that the extra dimensions could be as large as a millimeter and the measurements of gravity may observe the deviation from \(1/r^{2}\) Newtonian gravitation. Considering this possibility, they found that their model could describe the hierarchy between the Planck mass and the standard model electroweak scale in terms of the large size of the extra dimensions. Therefore testing general relativity and its Newtonian limit at short distances has become particularly important in recent theoretical developments [4, 5, 6, 7, 8, 9, 10, 11, 12, 13].

In the present work, by combining cavity optomechanics and the ADD model, we investigate the optomechanical system consisting of coupled quantum levitated oscillators and cavity modes. Then we propose a design for non-Newtonian gravity detection at short range with the levitated sensors. The results show that the sharp enhanced transparency peaks with ultra-narrow linewidth can be induced through the coupling between the optical cavity and the levitated resonator. We find that the two transparency peaks are apart near linearly with respect to the gravitational coupling strength. The feature reminds us of a practical application for precisely detecting large extra dimensions at micron separations. We also consider the Casimir effect as the main background force noise in the micro-scale optomechanical system. The constraints on the compactification distances are deduced as last, the super-resolution can be achieved by using the oscillators with high Q factor in vacuum. We expect that the proposed scheme could be applied to probe the large extra dimensions or set a new upper limit on the hypothetical long-range interactions which naturally arise in many extensions to the standard model [29, 30].

## 2 Theory framework

The micro- or nanoscale objects can be attracted to the anti-node of the field of an optical standing wave. The resulting gradient in the optical field provides a sufficiently deep optical potential well, which allows the object to be confined in a number of possible trapping sites with precise localization [25, 31, 32]. A schematic of our setup is sketched in Fig. 1a, where two dielectric microdisks are optically levitated between the two fixed mirrors by applying two trapping lasers, respectively. A Fabry–Pérot cavity consists of the left fixed mirror and the levitated microdisk 1 (the left one) which couples to the optical cavity field via radiation pressure. This cavity mode is driven by two light fields, one of which is the pump field with frequency \(\omega _{pu} \) and the other of which is the probe field with frequency \(\omega _{pr}\) as shown in Fig. 1a by the purple curve and green curve respectively. The mechanism can be explained as a quantum coupling induced normal mode splitting (NMS) in the four-wave mixing (FWM) process. Figure 1b is the energy level description of this process.

In the present paper, the trapped microdisks are treated as quantum-mechanical harmonic oscillators and their masses are \(m_{1}\) and \( m_{2}\), mechanical frequencies \(\omega _{1}\) and \(\omega _{2}\), and damping rates \(\gamma _{1}\) and \(\gamma _{2}\), respectively. The Hamiltonian can be regarded as \(H_{\omega }=\sum \nolimits _{{j=1,2}} \hbar \omega _{j}a_{j}^{+}a_{j}\), where \(a_{j}^{+}\) and \(a_{j}\) are the bosonic creation and annihilation operators for the two vibrational resonators. We use \( H_{c}=\hbar \omega _{c}c^{+}c\) to describe the Hamiltonian of the cavity mode; here \(\omega _{c}\) and \(c(c^{+})\) denote the oscillation frequency and the annihilation (creation) operator of the cavity. The radiation pressure of the cavity gives rise to the optomechanical coupling \(H_{g}=-\hbar g_{1}c^{+}c(a_{1}^{+}+a_{1})\), where \(g_{1}\) is the single-photon coupling rate between the microdisk 1 and the cavity mode. The parameter \(g_{1}\) has the typical value of \(\sim 2\pi \times 1.2\) Hz [32].

*n*is the number of extra dimensions. According to this model, the Planck scale is not fundamental but determined by the volume of the extra dimensions,

*c*is the vacuum speed of light,

*d*is the thickness of the disk, \( \varepsilon \) is the electrostatic value of the dielectric polarization of the substance, and \(\phi (\varepsilon )\) is the dielectric function. We use the silica disk with the dielectric constant \(\varepsilon =2\). According to Ref. [39], for \(1/\varepsilon =0.5\), numerical values of the dielectric functions \(\phi (\varepsilon )=0.35\). Then we can manipulate Eq. (7) with the same means and obtain the Casimir coupling rate,

## 3 Forecasts

We use silica microdisks with the same size and density \(\rho =2.3\) g/cm\( ^{3}\). The mechanical frequency \(\omega _{j}\) depends on the intracavity intensity; it can be modulated by the power of trapping beam [32]. If we use the trapping laser wavelength \(\lambda =1.5\) \(\upmu \hbox {m}\) and power \( P_{1}=P_{2}=5.06\) W, we can get the axial trapping frequency \(\omega _{1}/2\pi =\omega _{2}/2\pi =10\) kHz.

The quality factor is solely determined by the air molecule impacts. The random collisions with residual air molecules provide the damping rate \( \gamma _{j}=\omega _{j}Q_{j}^{-1}\), and thus the quality factor due to the gas dissipation can be defined as \(Q_{j}=\pi \omega _{j}\rho \nu d/32p\) for the microdisks [23, 24] where \(\nu =\sqrt{k_{B}T/m_{gas}}\) is the thermal velocity of the gas molecules, *p* is the air pressure, and \(m_{gas}\) is the mass of a gas molecule in the chamber. Let us consider levitated microdisks with the same radius \(r_{1}=r_{2}=75\) \(\upmu \hbox {m}\) and thickness \(d=1\) \(\upmu \hbox {m}\). We get \(Q_{1}=Q_{2}=10^{8}\) for the ultralow pressure \(p=10^{-7}\) mbar at room temperature (\(T=300\) K).

*n*. In the following, we take the pump-cavity detuning \(\Delta _{pu}=0\). Here we use the Heisenberg equation of motion to solve the Hamiltonian of the levitated microdisk-cavity system. By solving the Heisenberg equation, we can obtain the transmission of the probe beam, \( \left| t\right| ^{2}\), defined as the ratio of the output and input field amplitudes at the probe frequency [40] (see the supplementary materials).

Then we depict the transmission \(\left| t\right| ^{2}\) of the probe beam as a function of the probe-pump detuning \(\delta \) in Fig. 2. At first we assume \(\beta =\beta _{Casimir}=0\), then we get an enhanced peak which is located at \(\delta =\omega _{1}=10\) kHz; this just corresponds to the fundamental frequency of the levitated microdisk as shown by the black curve. Without the presence of extra dimensions \((n=0)\), we can only consider the Casimir coupling \(\beta _{Casimir}=2.5\times 10^{-4}\) Hz. Then we find that the resonance peak suffers a splitting in the spectrum characterized by the blue curve. Now, let us consider the large extra dimensions induced by the coupling between the two microdisks based on ADD theory. For 4 + 2 dimensions, \(n=2\), we have \(\beta =2\times 10^{-3}\) Hz. The result shows that the resonance frequency splitting can be amplified significantly in the spectrum as shown by the red curve. Here the transmitted spectrum of the probe laser can be effectively modulated by the number of extra dimensions. Without any interaction, one can obtain significant transmission of the probe laser at the resonant region. When gravity deviates from \(1/r^{2}\) in 4 + 2 extra dimensions and also the Casimir background force is taken into account, the enhanced peak splits and separates. Therefore we can adjust the resonance frequencies of the levitated microdisks to 10 kHz by modulating the trapping laser power; then the extra dimensions can be detected by the vibrational mode splitting of the microdisk resonators experimentally.

We define *L* as the separation between two peaks as shown in Fig. 2. The linear enhancement of *L* with \(\beta \) reminds us of the possibility to detect the gravity strength between the resonators by measuring the separation in the transmission spectrum. Their relationship can be expressed by \(L=2(\beta +\beta _{Casimir})\), which strongly reveals the deviation from gravitational inverse-square law, namely, a sign indicative of large extra dimensions. The resolution depends on the full width of half maximum (FWHM) of the peak, thus the minimal detectable coupling strength \(\beta _{\min }=L_{\min }/2=FWHM/2\). Considering the peak FWHM in Fig. 2 approximating \( 2\times 10^{-4}\)Hz, one can obtain \(\beta _{\min }\approx 0.1\) mHz.

*d*according to Eqs. (6) and (8). For \(d=1\) \(\upmu \hbox {m}\), we find that \(\beta _{Casimir}\approx \beta /8\); thus the Casimir coupling is 8 times smaller than the gravitational coupling in this case. We expect the contribution of the large extra dimensions will be able to show itself clearly on the probe spectrum with the low Casimir background noise.

In our scheme, the length of the cavity is 1 cm, while the separation between two microdisks is just 7 \(\upmu \hbox {m}\). If we trap two microdisks in the middle of the cavity, the distance between the microdisk and the mirror is much larger than the separation of two microdisks. Thus the Casimir and electrostatic force between the microdisk and the mirror can be safely neglected. In the Casimir force model we do not consider the frequency-dependent dielectric functions; there may be a few errors in the estimation of the Casimir force. But considering the gravity coupling in 4+2 dimensions is much larger than the Casimir induced coupling (about 8 times); those small errors will present no significant problems.

## 4 Electrostatic force background

If the two microdisks are electrically connected, electrons flow from the material with the smaller work function to the material with higher work function. This diffusion current builds up a double layer at the interface, resulting in an electrostatic potential (contact potential difference) \( \Delta \Phi \) given as \(\Delta \Phi =(\psi _{1}-\psi _{2})/e\) [41, 42], where \( \psi _{1}(\psi _{2})\) is the work function of the two samples. Since the two levitated microdisks are made of the same material (silica), they have the same work function and equal Fermi energies. Thus we have \(\psi _{1}=\psi _{2}\), and \(\Delta \Phi =0\). The surface potential difference is zero; thus it will not create electrostatic forces between the two levitated microdisks.

Optomechanical parameters of the levitated microdisks

Parameter | Units | Value |
---|---|---|

Separation distance \(r_0\) | \(\upmu \)m | 8 |

Microdisk radius \(r_{1}\) | \(\upmu \)m | 75 |

Microdisk thickness | \(\upmu \)m | 1 |

Microdisk frequency \(\omega _{1}\) | kHz | 10 |

Trapping wavelength \(\lambda \) | \(\upmu \)m | 1.5 |

Pump driving amplitude \(\Omega _{pu}\) | GHz | 10 |

Probe driving amplitude \(\Omega _{pr}\) | GHz | 1 |

Total cavity decay \(\kappa \) | GHz | 0.1 |

Air pressure | mbar | \(10^{-7} \sim 10^{-10}\) |

Room temperature | K | 300 |

Optomechanical coupling rate \(g_{1}\) | Hz | 7.5 |

Pump-cavity detuning \(\Delta _{c}\) | Hz | 0 |

Cavity length | cm | 1 |

Cavity finesse \(F_{c}\) | 1 | 470 |

The mechanism underlying these effects can be explained as FWM in a three-level system. The simultaneous presence of a pump field and a probe field generates a radiation pressure force at the beat frequency, which drives the motion of the oscillator near its resonance frequency. In Fig. 1b, we let \(\left| N\right\rangle \), \(\left| n_{1}\right\rangle \) and \(\left| n_{2}\right\rangle \) denote the number states of the cavity photon, left microdisk phonons, and right microdisk phonons, respectively. The \(\left| N,n_{1},n_{2}\right\rangle \leftrightarrow \left| N+1,n_{1},n_{2}\right\rangle \) transition changes the cavity field. The initial energy level of the cavity photon \(\left| N,n_{1},n_{2}\right\rangle \) is dressed by the mechanical modes \(\left| N,n_{1}+1,n_{2}\right\rangle \) via the radiation pressure. Thus the \( \left| N+1,n_{1},n_{2}\right\rangle \leftrightarrow \left| N,n_{1}+1,n_{2}\right\rangle \) transition is caused by the radiation pressure coupling. The mechanical modes of the right levitated microdisk adds a fourth level \(\left| N,n_{1},n_{2}+1\right\rangle \). In the system, the energy level \(\left| N,n_{1}+1,n_{2}\right\rangle \) can be mainly modified by the gravitational coupling between two resonators. The coupling breaks down the symmetry of the OMIT interference, the single OMIT transparency window is split into two transparency windows, which yields the quantum coupling induced NMS as shown in Fig. 2.

## 5 Constraints and limits

Figure 5 presents the constraints on \(R_{*}\) for the number of extra dimensions, \(n=1,2,3\). The sloping lines represent the calculated coupling rate \(\beta \) for different *n* and \(R_{*}\). We depict the horizontal dash black line to indicate the limits of the detectable coupling strength \(\beta _{\min }\). \(R_{*}\) and \(\beta \) are constrained to be larger than the values of the intersections in the picture. The solid lines in the figure represent the detectable parameter space, the dot lines represent the undetectable region. The arrow indicates that the sensitivity can be improved through the optomechanical oscillators with higher Q factors. Considering the minimum measurable gravitational coupling rate \( \beta _{\min }=0.1\) mHz, we get the precision for the force measurement as \( F_{\min }=300\) aN. The Q factor of the optically trapped particles is limited only by collisions with residual air molecules, thus the sensitivity can be improved by decreasing the air pressure. The lower pressure limit of the sputter-ion pumps is in the range of \(10^{-11}\) mbar. Lower pressures in the range of \(10^{-12}\) mbar can only be achieved when the sputter-ion pump works in a combination with other pump methods [49, 50]. In our considerations, a conservative value is taken, \(p=10^{-7}\) mbar, which is usually required for achieving ultrahigh-Q mechanical oscillators and the ultrasensitive measurements in the levitated optomechanical system [12, 19, 23, 24, 51]. If we choose a lower air pressure (\(p=10^{-11}\) mbar), this scheme will obtain the FWHM in the spectrum with \(FWHM=10\) nHz, corresponding to the force sensitivity of \(F_{\min }=0.2\) aN. Our scheme yields 2–3 orders of magnitude improvement for the force sensing in the microscale [5].

## 6 Conclusion

We study the dynamics of a driven optomechanical cavity coupled to a levitated resonator via the coupling induced by large extra dimensions, in which the splitting can be observed at the probe frequency. Our study reports a design for probing gravitational deviation in the range of 8 \(\upmu \) m with the pump-probe optical technology in cavity. Under the influence of non-Newtonian gravity, the transparency peak would show a distinct splitting in frequency space. We have shown that the coupling strength associated with such a hypothetical variation of gravity could be determined from a measurement of the splitting distance. The gravitational strength, characterized by the coupling strength \(\beta \), can also be determined by the splitting distance *L*. The Casimir coupling rate is about 8 times smaller than the gravitational strength in the system. The different surface potentials can be attenuated efficiently in our scheme, providing the precision measurement in the Casimir regime without the isoelectronic technique. We also hope that the precision can be significantly enhanced by experiments in ultrahigh vacuum.

## 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).

## Supplementary material

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