Detecting the effects of quantum gravity with exceptional points in optomechanical sensors

In this manuscript, working with a binary mechanical system, we examine the effect of quantum gravity on the exceptional points of the system. On the one side, we find that the exceedingly weak effect of quantum gravity can be sensed via pushing the system towards a second-order exceptional point, where the spectra of the non-Hermitian system exhibits non-analytic and even discontinuous behavior. On the other side, the gravity perturbation will affect the sensitivity of the system to deposition mass. In order to further enhance the sensitivity of the system to quantum gravity, we extend the system to the other one which has a higher-order (third-order) exceptional point. Our work provides a feasible way to use exceptional points as a new tool to explore the effect of quantum gravity.


I. INTRODUCTION
Cavity optomechanics [1], exploring the interaction between light and mechanical systems, has made a profound impact in recent years due to its wide variety of applications including optomechanical sensors. Optomechanical sensors have achieved ultrasensitive performance in gravitational wave detection [2,3], high-precision measurements [4], detection for mass [5], acceleration [6], displacement [7], and force [8][9][10]. For practical applications, the optomechanical system is unavoidably coupled with its surroundings, leading to a non-Hermitian optomechanical system. Several earlier studies have shown that non-Hermitian spectral degeneracies, also known as exceptional points (EPs) [11][12][13][14][15], governs the dynamics of parity-time (PT ) symmetric system subject to environment. In contrast to level degeneracy points in Hermitian systems, the EP is associated with level coalescence, in which the eigenenergies and the corresponding eigenvectors simultaneously coalesce [16,17]. Besides, the intriguing phenomena of EPs in unidirectional invisibility [18], topological chirality [19] and low-threshold lasers [20,21] have been predicted.
In recent years, sensitivity enhancement of the sensor operating at EPs has been explored both theoretically [12,22,23] and experimentally [14,[24][25][26] in a number of systems including nanoparticle detector [12,14], mass sensor [22], and gyroscope [23][24][25]27]. These studies have shown that if a second-order exceptional point (EP2) where the coalescence of two levels occurs is subjected to a perturbation of strength ǫ, the frequency splitting (the energy spacing of the two levels) is typically proportional to the square root of the perturbation strength ǫ. This is the so-called complex square-root topology. * yixx@nenu.edu.cn Moreover, the splitting is significantly enhanced for sufficiently small perturbation. This suggests that the use of EP can enhance the sensitivity of a quantum sensor.
In standard quantum mechanics, on the basis of Heisenberg uncertainty principle ∆x∆p ≥ 2 [28], the position x and the momentum p of an particle cannot be simultaneously measured to arbitrary precision, however, the uncertainty of x can reach zero in case ∆p approaches infinity. This is not the case when the quantum gravity is taken into account. It has been suggested that the uncertainty relation should be modified when gravitational effects have been taken into consideration [29]. Such generalized uncertainty principle (GUP) is found in various approaches to quantum gravity, such as the finite bandwidth approach to quantum gravity [30,31], string theory [32][33][34][35][36], the theory of double special relativity [37,38], relative locality [39] and black holes [40].
Here µ = β0 (Mpc) 2 = L 2 p 2 2 , β 0 is a dimensionless parameter, M p is Planck mass, M p c 2 is Planck energy and L p is Plank length. This inequality means that ∆x ≥ L p √ β 0 . So if β 0 = 1 [41], the minimal length is equal to the Planck length (L p ) beyond which the concepts of time and space will lose their meaning. The generalized uncertainty principle (GUP) has been extensively explored in various fields, including high energy physics, cosmology and black holes [42]. Due to experiments that can test minimal length scale directly require energies much higher than that currently available, most of the work has been devoted to find indirect evidences of quantum gravity in high energy particle collisions and astronomical observations [43,44].
In this manuscript, we theoretically propose the other sensing scheme to explore the effects of quantum gravity via GUP. We will consider a binary and a ternary mechanical system separately within an optomechani-cal configuration. Controlling the gain and loss of the mechanical oscillators and driving the two cavities with blue and red detuned lasers as well as manipulating the strength of the electromagnetic field (α in ), we can set the system into a self-sustained regime for the mechanical oscillations. In the absence of gravity perturbation, the system is controlled to be in the EP2 state. When the gravity perturbation comes, the supermodes of the system are shifted away from the EP2. The frequency splitting induced by gravitational effects can be read out in the mechanical spectrum. This result has been further extended to a third-order exceptional point (EP3) by taking a more complicated ternary mechanical system into account. Compared with the scheme utilizing EP2, optomechanical sensor based on EP3 performs better. The physics of this EP-based sensor is that the eigenvalues of non-Hermitian Hamiltonian may exhibit non-analytic and even discontinuous behavior, which in principle enables an unlimited spectral sensitivity.
The rest of this paper is organized as follows. In Sec. II, we introduce the physical model and derive a set of equation for the dynamics of our system. In Sec. III, we study the sensitivity of the system to gravity perturbation at EP2. In Sec. IV, we extend the study on the sensitivity of the system to the gravity perturbation to EP3. In Sec. V, we discuss experimental feasibility of the proposed sensing scheme and analyze the limitation of the proposed quantum gravity sensor. Finally, the conclusions are drawn in Sec. VI. In Appendix A, we present details of derivation for the effective Hamiltonian.

II. GENERAL FRAMEWORK
We start by briefly recalling the description of harmonic oscillator with mass m when the effect of gravity is taken into consideration. Afterwards we would apply this result to our model, in which the mechanical resonator is modeled as a harmonic oscillator.

A. Deformed harmonic oscillations under quantum gravity
From the aspect of communtation relation, the gravity would modify the relation leading to the generalized uncertainty principle (GUP) given in Ref. [35], Define [41] where x, p satisfying the (non deformed) canonical commutation relations [x, p] = i . It is easy to see that Eq.
(2) is written up to the first order in µ (the terms of order µ 2 and higher are neglected). For a harmonic oscillator with frequency ω and mass m, we assume that the Hamiltonian takes H = 1 2 mω 2 x 2 + p 2 2m [45]. In terms of p, the Hamiltonian of harmonic oscillator can be rewritten as Introducing canonical creation and annihilation operators, and rewriting the Hamiltonian (3) in terms of b, we can get The first term represents the free Hamiltonian of the harmonic oscillator. The second term comes from the gravitational effects.
FIG. 1: Sketch of the studied system. An open system consisted of two coupled mechanical resonators. Each resonator is coupled to an optical cavity. The two cavities are driven by red-and blue-detuned pump laser, respectively.

B. Modeling and dynamical equations
We consider two coupled identical resonators with optomechanically induced gain and loss. Each of the resonators is characterized by frequency ω j , damping rate γ m and coupling strength J. The schematic diagram is presented in Fig. 1. In this configuration, the theory states that only the mechanical commutation relation are modified, while the optical commutation relation remains unchanged, i.e., [a, a † ] = 1 [46]. The total Hamiltonian of the whole system can be written as ( = 1) where, In this expression, H f represents the sum of free Hamiltonian of the optomechanical system, a † j (b † j ) and a j (b j ) are the creation and annihilation operators of the jth cavity (mechanical resonator) (j = 1, 2). The frequencies of the cavities and mechanical resonators are ω a,j and ω j , respectively. H i describes the interaction Hamiltonian of the configuration. The first term represents the coupling of the cavities to the corresponding mechanical resonators with optomechanical coupling strength g. The second term describes the coupling between the two mechanical resonators with coupling strength J. H d indicates that the two cavities are driven by external fields with amplitude E and frequency ω p,j . H g describes the gravitational effects in mechanical resonators. The effective mass of the jth mechanical mode is m j . In the frame rotating at the input laser frequency ω p , the Hamiltonian of the system reads, where, ∆ j = ω p,j − ω a,j represents the detuning of the driving field with respect to the cavity. As we are interested in the classical limit, where photon and phonon numbers are assumed large in the model. Thus, we replace the quantum operators with their mean values, i.e., α j = a and β j = b . By introducing dissipation terms, the evolution of the system operators is obtained as follows [1], where κ and γ m are the intrinsic damping rates of the cavities and mechanical resonators, respectively. E = √ κα in j is the amplitude of the driving field, where α in j = pin ωp,j characterizes the input field driving the cavity. For the sake of simplicity, we assume the two cavities and mechanical resonators identical, this means ω j = ω m , m j = m, and µ j = µ. We apply the input lasers with the same power (p in ) to drive the two mechanical resonators, i.e., α in j = α in . Throughout the work, the parameters satisfy the following condition, γ m , g ≪ κ ≪ ω m , similar to those chosen in Ref. [47,48]. Under this hierarchy, the amplitude and phase of the mechanical resonators slowly evolving on the time scale of the cavity dynamics.
We will pay our attention to the steady state of the mechanical resonators. In this regime, β j (t) =β j + B j e −iθ e −iω l t [49,50], whereβ j is the center of the mechanical oscillations and amplitude B j can be regarded as a slowly evolving function of time. In the limit-cycle states, both mechanical resonators start oscillating with a locked frequency (ω l ). On this point, it can be seen from its Fourier spectrum, where the peak of the spectrum is much larger than the corresponding amplitude of other frequency components [51]. Throughout this paper, we set θ = 0. In parallel, we removed all terms in mechanical dynamics except for the constant one and the term oscillating at ω l . Using this analytic approximation, we solve the equation for α j assuming a fixed mechanical amplitude and then substitute the result into the equation for β j , resulting in the following set of equations of motion describing this effective mechanical system (see Appendix A): where, Θ j = µmω 2 j B 2 j (j = 1, 2). ω j ef f = ω j + Ω j and γ j ef f = γ m +Γ j (j = 1, 2) represent the effective frequency and damping of the jth mechanical oscillator (j = 1, 2), respectively. The modal field evolution in this configura- T is the state vector and t represents time. H ef f is the associated 4×4 non-Hermitian Hamiltonian (see more details in Appendix A): Here, Ω j (Γ j ) represents the optical spring effect (the optomechanical damping rate). These quantities are given as (see more details in Appendix A) and Firstly, we focus on the case without gravitational effect, the eigenvalues of the above effective Hamiltonian are given by where, Here, ω l = Replace the conventional vibrational modes, we now have new mechanical modes, which can be called as the mechanical supermodes. The effective frequencies and spectral linewidths of the system are defined as the real (ω) and imaginary (γ) parts of eigenvalues, respectively. The solid lines in Fig. 2 show the real and imaginary parts of the eigenvalues vs the driving strength α in before the perturbation introduced by gravity effects. At the specific point, both these pairs of effective frequencies and effective dampings of the system coalesce. It is evident that for a critical driving strength

III. SENSITIVITY AT THE SECOND-ORDER EXCEPTIONAL POINT
A. Sensitivity of a system at the second-order exceptional point to the gravity effect For the case with gravitational effects, we numerically solve the eigenvalues of this mechanical effective Hamiltonian and show the results in Fig. 2. The effective Hamiltonian has 4 eigenvalues forming two pairs, one pair is due to the apperarnace of β * j in the dynamics. As shown in Fig. 2, we see that the splitting of effective frequency (real part of the eigenvalue) and linewidth (imaginary part of the eigenvalue) increases as the mass of the mechanical resonators increases. This is attributed to the fact that gravity effect is enhanced by larger system mass. A typical detection strategy is to observe the associated mode response, usually the frequency splitting or the frequency shift, before and after the perturbation induced by gravitational effects taking place. In this paper, in order to quantify the frequency splitting caused by the gravity effect, we define the sensitivity as follows, The perturbation of gravitational effects can shift the EP2, and thereby the degeneracy of the effective frequencies are released and cause the supermodes to split. The frequency splitting caused by gravitational effects can be fitted using ∆ω ≈ ξ(µm) 1/2 .
Here, ξ is the fitting coefficient. Fig. 3 shows ∆ω as a function of the µm near the EP2. The blue dashed lines represent the fitting result according to Eq. (17) with ξ = 30.12ω 3/2 m , which is consistent with the mechanical frequency splitting in our model. Therefore, it can be inferred that the mechanical frequency splitting in response to the µm obeys the square root behavior. Due to the intrinsic properties of EP2, we can claim that the sensitivity is significantly enhanced by exploiting EP2 for sufficiently small perturbation strength, proving the efficiency of the EP2 sensor in detecting gravity effect.
B. Sensitivity of the system to deposition mass at the second-order exceptional point with gravity effect In order to gain insight into the influence of gravity effects on mass sensing, we assume that a mass δm has been deposited on the mechanical oscillator driven by the blue-detuned electromagnetic field, which would induce the frequency shift given in Eq. (14), i.e., replacing ω 2 ef f with ω 2 ef f +δω. For an ordinary mass sensor, the relation between the deposited mass δm and the frequency shift δω is given by [52] where ζ represents the mass responsivity of the mechanical resonator. We can define the gap as Figure 4 shows that for mechanical frequency shift δω = 10 −2 ω m , the larger the mass of the mechanical resonators, the larger the gap between effective frequencies before and after gravitational effects being considered. However, the gap between the effective dampings (the imaginary part of the eigenvaules) does not change significantly. Therefore, small mass of the mechanical resonators can decrease the disturbance caused by gravitational effects.

IV. SENSITIVITY OF A SYSTEM AT THE THIRD-ORDER EXCEPTIONAL POINT TO THE GRAVITY EFFECT
Inspired by these results, we now extend this scheme to the higher-order exceptional points (EPs). A possible configuration that supports a third-order exceptional point (EP3) would be a system consisting of two cavities and three coupled mechanical oscillators where the two cavities are symmetrically driven by red-and bluedetuned lasers, and the corresponding mechanical resonators are coupled together (see Fig. 5). Proceeding in a similar way, one can write the following the Hamiltonian of the system, with From Eq. (21), one can write the following nonlinear equations of motion, Here, Ξ j = 1 3 iµmω 2 j (j = 1, 2, 3), α j = a j (j = 1, 2), and β j = b j (j = 1, 2, 3). For the convenience of discussion, we assume ω j = ω m (j = 1, 2, 3). In Fig. 6, we show the overall properties of the steady state solutions of the mechanical resonators. It is easy to find that the amplitudes of the the mechanical resonators change very slowly over time [see Fig. 6 (a)]. Fig. 6 (b) shows the corresponding Fourier spectra. It is easy to see that all three mechanical resonators start oscillating with a same frequency, i.e., ω l = The inset of Fig. 6 (b) shows limit cycle oscillations at α in = 160 √ ω m and J = 2.2×10 −3 ω m . So in this case, the formal solution for β j (t) is still applicable. By the use of this formal solution, Eq. (22) can be further reduced to Here Θ j = µmω 2 j B 2 j (j = 1, 2, 3). The modal field evolution in this configuration obeys i dψ dt = H ef f ψ, where ψ = (β 1 , β 2 , β 3 , β * 1 , β * 2 , β * 3 ) T represents the modal state vector and t represents time. H ef f is the associated 6 × 6 non-Hermitian Hamiltonian, It is easy to find that the effective Hamiltonian has 6 eigenvalues forming two pairs, one pair is due to the apperarnace of β * j in the dynamics. This characteristic feature of the EP3 has been demonstrated in Fig. 7 (a) and (b), where we show the dependence of the eigenvalues on driving strength α in . Now to take this discussion further to show how the system reacts around the EP3. The real [ Fig. 7 (c)] and imaginary parts [ Fig. 7 (d)] of the eigenvalues are plotted as a function of µm. Moreover, it is easy to see that the power (p in ) required to reach the third-order exceptional point is lower than that required by EP2.
The difference between two effective frequencies (in this case, ω 2 and ω 3 ) is also plotted (Fig. 8) as a function of µm. The frequency splitting caused by µm can be fitted using ∆ω ≈ ς(µm) 1/3 .
Here ς is the fitting coefficient. The blue dashed line represents the fitting results according to Eq. (25) with ς = 2.874ω 4/3 m , which is consistent with the mechanical frequency splitting in our system, confirming thus that the mechanical frequency splitting in response to µm obeys the cube root behavior. This indicates that it is feasible to further enhance the sensitivity by means of third-order exceptional point (EP3).

V. EXPERIMENTAL FEASIBILITY AND ULTIMATE LIMITS OF THE SENSING SCHEME
There are many types of optomechanical systems. For concreteness, we choose one of them, where the mechanical degree of freedom is a dielectric membrane placed inside a Fabry-Perot cavity [53]. Here we use two coupled Si beams, which possess the mass of m = 5.3 × 10 −3 ng and thickness t = 80 nm [54]. Here we take the EP2based sensor as an example, as shown in Fig. 9 (a) and (b). In general, various basic physical noise processes will limit the sensitivity of the sensing scheme. For the nanomechanical resonators, the main noise source is the thermomechanical noise [54]. In order to obtain this basic limits imposed upon measurements by thermomechanical fluctuations, we need to consider the minimum detectable frequency shift (δω) that can be resolved in a practical noisy system. An estimate for δω can be obtained by [54] δω ≈ Here Q is the mechanical quality factor, K B is the Boltzmann constant, T is the effective temperature of the mechanical resonator, and E c = mω 2 m x 2 c , which describes the maximum drive energy. x c can be approximated as [54] x c ≈ 0.53t. (27) In order to obtain the ultimate sensitivity limits of the system to the effect of gravity, we assume that the frequency splitting caused by gravitational effects is exactly equal to the minimum measurable frequency shift (δω) determined by the thermomechanical fluctuations, i.e., ∆ω min = δω. We plot ∆ω min as a function of the bandwidth for thermomechanical fluctuations in Fig. 9 (c). The result shows that small bandwidth ∆f and high quality factor Q of the mechanical resonator are essential for the superresolution. Assuming that ∆f = 10 −10 Hz, we can obtain the quantum-noise-limited sensitiv- ity of the system to gravitational effects with Eq.(26), ∆ω min ∼ 10 −12 Hz for Q = 10 12 .

VI. CONCLUSION
In conclusion, we have presented a scheme for sensing the effect of quantum gravity. Starting with a system consisting of two coupled resonators with driving and dissipation, we show that the system eigenenergy is sensitive to the effect of quantum gravity when the system is in an second-order exceptional point. The response of the binary mechanical system to the gravity exhibits square root behaviour, and the sensitivity of the system at EPs increases significantly with the decrease of the perturbation. Moreover, we found that small mass of the mechanical resonator benefits the sensitivity of the system to deposition mass. In order to further enhance the sensitivity of the system to the effect of gravity, we extend the sensing scheme to a third-order exceptional point by taking a more complicated ternary mechanical system into account. The response of the ternary mechanical systems to perturbation exhibits cube root behaviour. The quantum-noise-limited sensitivity of the system to gravitational effects due to thermomechanical noise is also discussed. It is worthwhile to note that our scheme could, in principle, be extended to various photonic and phononic systems with optomechanically induced gain and loss. These findings may pave the ways for utilizing EPs as a novel tool to probe effect of quantum gravity.