Perspectives of measuring gravitational effects of laser light and particle beams

We study possibilities of creation and detection of oscillating gravitational fields from lab-scale high energy, relativistic sources. The sources considered are high energy laser beams in an optical cavity and the ultra-relativistic proton bunches circulating in the beam of the Large Hadron Collider (LHC) at CERN. These sources allow for signal frequencies much higher and far narrower in bandwidth than what most celestial sources produce. In addition, by modulating the beams, one can adjust the source frequency over a very broad range, from Hz to GHz. The gravitational field of these sources and responses of a variety of detectors are analyzed. We optimize a mechanical oscillator such as a pendulum or torsion balance as detector and find parameter regimes such that -- combined with the planned high-luminosity upgrade of the LHC as a source -- a signal-to-noise ratio substantially larger than 1 should be achievable at least in principle, neglecting all sources of technical noise. This opens new perspectives of studying general relativistic effects and possibly quantum-gravitational effects with ultra-relativistic, well-controlled terrestrial sources.


I. INTRODUCTION
With the successful measurement of gravitational waves through the LIGO/Virgo collaboration, the measurement of gravitational signals from relativistic sources has gained a lot of interest as it is believed to lead to new insights about gravity, in particular, constraints on modifications of general relativity and potential effects of quantum gravity [1,[1][2][3][4][5]. However, such experiments are limited to detection since the experimenter has no access to the cosmic sources of the signal.
Starting already in the 1970s, proposals were formulated for constructing terrestrial relativistic sources and detectors of their gravitational signals. E.g. in [6][7][8] a cylindrical microwave resonator was proposed as source of a standing gravitational wave and a second concentric cylinder as detector based on photon creation in one of its modes. But it was clear that with the existing technology at the time it was not realistic to create a sufficiently strong source whose radiation could be detected. In recent years there has been renewed interest in the creation and detection of gravitational waves in the lab [9][10][11][12][13][14][15][16][17].
As technology has substantially progressed since some of the cited works have been published, both on the side of sources in the form of high-power lasers [35][36][37] and particle accelerators [38,39], and in the metrology of extremely weak forces [40][41][42][43][44][45], it is worthwhile to reassess the possibility to detect the gravitational effects of light and of ultra-relativistic particle beams. Indeed, progress in this direction would enable the test of general relativity (GR) in a new, ultra-relativistic regime (in the sense of special relativity), with an energymomentum tensor as the source term in Einstein's equations very different from the one that can be achieved with non-relativistic masses and purely Newtonian gravity, namely with a large off-diagonal component in Cartesian coordinates.
In this article we focus on the acceleration of non-relativistic sensor systems due to the gravitational field of light beams and ultra-relativistic particle beams such as the ones produced at the LHC. We add several new aspects that improve the outlook for experimental observation. Most importantly, we consider trapping of laser light in a cavity, through which the circulating power can be drastically enhanced. Secondly, we consider modulation of the gravitational sources with an adjustable frequency in order to match them to the optimal sensitivity of existing detectors. Several approaches are investigated to that end. The simplest one consists in having laser pulses oscillate to and fro in a cavity, such that the length of the cavity determines the oscillation frequency of the gravitational signal. We also examine the possibility of slowly (kHz frequency) modulating the power with which the cavity is pumped using a continuous wave (cw) or pulsed laser. With the pump power, the power circulating in the cavity is modulated, and thus, also the strength of the gravitational field. Thirdly, we extend the analysis to ultra-relativistic particle beams such as available at the LHC. And finally, we examine several possible sensors for their suitability for measuring the created gravitational fields.
Our work is also motivated by current developments towards measuring gravitational effects of sources in a quantum mechanical superposition as a possible experimental road to understanding quantum gravitational effects [40,42,46,47]. Creating quantum superpositions of sufficiently large masses is challenging, and it is therefore worthwhile to think about other sources that can be superposed quantum mechanically. We discuss perspectives in this direction for the gravitational sources studied in this paper in Sec.IV.

A. Laser pulses oscillating in a cavity
To create a strong, high frequency gravitational field, a source of high power and intensity is required. Modern femtosecond laser pulses can reach up to a Petawatt in pulse power.
One such laser pulse oscillating in a cavity, as illustrated in fig. 1, is a source of short bursts of high energy oscillating to and fro at high frequency. The perturbation to the metric and the resulting Riemann curvature tensor can be calculated within the theory of linearized gravity, as is done in appendix A and [28,29].
For a continuous-wave (cw)-laser with power P and circular polarization, the curvature component relevant to a non-relativistic sensor based on a mechanical resonator with axis perpendicular to the beam line of the laser is, for an observer in the x-z-plane (i.e. y = 0) and in the approximation of a vanishing opening angle, given by with G the gravitational constant, c the speed of light in vacuum, ρ 2 = x 2 + y 2 = x 2 , and Laser pulses were considered earlier in [18,21] in the approximation of an infinitely thin light pencil of length L. A further exploration in appendix A for the simplified case of box shaped pulses oscillating to and fro corroborates the result that close to the beam (ρ |z|, |z − D|, where z = 0 and z = D are the positions of the two mirrors) eq. (1) gives the correct result, limited, however, to a finite duration of the order of the length of the pulse (see Fig. 8 in [21]) but on the other hand with the cw-power P replaced by the power of the effective pulse in the cavity P cav p (see eq.(A23)). The curvature results in a tidal force between two infinitesimally separated points next to the beamline. However, there is no gravitational wave generated as this type of source is not quadrupolar in nature. Rather one can detect the gravitational near-field.
The average power at a given cross-section of the beam inside the cavity is P avg cav = 2τp τrt P cav p , where τ p is the length of the pulse, and τ rt = 2Lcav c is the round trip time in the cavity. As the power enters linearly into the gravitational potential, acceleration, and curvature, the considered gravitational effects will be proportional to the average power in the cavity.
A pump laser emitting very short pulses has a broad spectrum in the frequency domain.
Coupling these pulses into a cavity of high finesse F ≈ π(R 1 R 2 ) [48] leads to an electric field strength inside the cavitỹ is the field transfer function, with the intensity transmissivity T 1 of the mirror struck by the pump beam, and the intensity reflectivities of the two mirrors R 1/2 , where T 1 = 1 − R 1 .
An explicit calculation for the case of rectangular pulses can be found in appendix B, which is based on [49]. If the pulses are very short τ p τ rt and far apart 1/f rep τ L , where Figure 1: a) Laser pulse oscillating to and fro in a cavity. b) cw laser focused to a narrow waist inside a cavity. Its intensity is modulated to create a gravitational field oscillating at kHz frequency.
c) Ultrarelativistic particle bunches in an accelerator ring such as the LHC create a gravitational field very similar to that of laser pulses. In the vicinity of the waist of the laser beam or close to the beamline, a detector picks up resonant mechanical deformations due to the oscillating gravitational forces.
τ L ≈ 2F π τ rt is the 1/e energy decay time and f rep is the repetition rate of the pump laser, the pulse enters the cavity at an intensity T 1 I p , where the circulating power is enhanced by a factor 2F π , independent of the cavity length. Without any further modification the factors T 1 and 2F π at best cancel up to a factor of 4 (assuming R 1 , R 2 1), leaving little to be gained (see App.B, eq. (B8)). The ways one could imagine improving upon this all involve changes to the mirror that couples the pump laser pulses to the cavity: • An input coupler is a mirror which is significantly less reflective than what could be achieved with the best available mirrors. Combined with techniques such as impedance matching, it increases the power deposited into the cavity while also slightly reducing the cavity finesse [50]. For example, in [51] input couplers are employed to realize enhancement cavities with kilowatt-average-power femtosecond pulses, increasing the average power circulating in the cavity to 670 kW, 10 3 times the 420 W average power of the pump laser. Using larger laser spots on the mirrors of the cavity should allow for even stronger pump lasers to be used. With stronger pump lasers, such as the BAT laser in [52] with an average pump power of P pump = 300 kW, an average power within the cavity in the 100 MW range seems plausible.
• A switchable mirror would allow for the full pump beam power to enter the cavity, which means the average cavity power is expected to be the pump laser power enhanced by a factor 2F π . Depending on the cavity's length and the pump laser's repetition rate, the mirror has to be moved on a timescale of 10 −9 s to 10 −3 s, the slower end of which seems realistic. A mirror mounted on some mechanics might reduce the precision of its positioning and hence the cavity's finesse. Nonetheless, with a high finesse cavity ( 2F π ∼ 10 5 ) and high-average-power pump lasers (P pump ≈ 300 kW [52] ) an average power > 20 GW in the cavity would be achievable.
One limitation when scaling to higher powers is damage to the mirrors. In [53] the cw intensity threshold was determined to be at around 100 MW/cm 2 before thermal damage sets in. For sub-picosecond pulses the intensity threshold can be exceeded by at least an order of magnitude, as it is done in [51], as long as the average intensity on the mirrors does not exceed the thermal threshold. For 20 GW (100 MW) cavity power this needs a spot diameter on the mirrors of at least 16 cm (1.1 cm). For the input coupler the limitations are even stricter than for the end mirror as the power passes through the input coupler and creates more heating than when reflected at the surface of the reflecting mirror [51]. Large spot sizes require long cavities, as otherwise the mode in the cavity has a large opening angle and prevents positioning the sensor very close to the beam. For the cited spot-sizes of order 1-10 cm, a cavity length L cav 1 m suffices. In this work, the increase in power is accounted for by increasing the pulse duration by defining an effective pulse length For the BAT laser from [52], the repetition rate is f rep = 10 kHz and the pulse duration of the pump laser is T pump p = 100 fs. We further assume F = 10 5 and the signal to be at resonance with the sensor frequency τ rt = 2 2π ω 0 , see e.g. Table I. This is consistent with the image of creating a "train" of pulses (one could also imagine pulse stacking, i.e. increasing the pulse power instead). Laser pulses with far higher pulse powers exist. The National Ignition Facility achieves 5 · 10 12 W peak power [36] but is not as suitable for our purposes due to its low repetition rates. Peak powers of up to 10 · 10 15 W at repetition rates of up to 10 Hz exist [37] and others with peak powers on the order of 100 · 10 15 W are planned [54], but will need to achieve higher average intensities and repetition rates in order to lead to measurable gravitational effects.

B. Modulated cw-pumping
Instead of creating a periodic signal by having laser pulses oscillate in a cavity, one could also consider using a cw laser. To create a periodic signal, one can pump the cavity for part of the period and allow for the intensity inside the cavity to decay before switching the pump beam back on for the next period, thus creating a modulated signal with modulation period τ mod . Depending on τ L , the energy within the cavity as a function of time looks more like a periodic sequence of effective pulses that have the form of rectangles -in the case of τ L τ mod -or like a series of shark fins for τ L ∼ τ mod (see appendix B). We call P cav p the maximum power of the effective pulse in the cavity.
Using a cw pump laser, the coupling to the cavity is no longer detrimental as for ∆ω FWHM ∼ 1/τ L > ∆ω pump , where ∆ω pump is the line width of the pump beam, the pump beam couples almost fully to the cavity. The Newtonian gravitational potential for a thin light pencil in the form of a standing light wave in the cavity is (see [18] and appendix A) For long modulation periods τ mod τ L , the maximum power of the effective pulse in the cavity is P cav p = 2F π P pump , for approximately half the modulation period. Commercially available cw laser systems reach continuous powers of 500 kW in multi-mode operation and up to 100 kW in single-mode operation (see 1 , and e.g. [55]). Combining this with a high finesse cavity F ∼ 10 6 leads to an average circulating power in the cavity of The average power in the cavity can at most be a fraction < 1 − e −τ mod /(2τ L ) of the maximum power 2F π P pump . For slowly decaying cavities, where τ mod τ L , techniques such as 1 A single-mode has the advantage that one can focus it down to a spot size comparable to the wave length, i.e. one could get, at least in principle, much closer to the beam (oder 1 µm instead of ca. 100 µm. Thus, while loosing a factor 25 in power one gains a factor 100 in distance, i.e. there is an overall improvement by a factor 4 over the multi-mode case, if such small distances from the beam can indeed be realized Q-switching or switchable mirrors are necessary to adequately modulate the amplitude 2 .

C. The Large Hadron Collider (LHC)
Instead of laser light, one can also investigate ultra relativistic particle beams consisting of high-energy bunches, such as the one at LHC, as gravitational sources. A particle beam in the relativistic limit is, from a gravitational perspective, the same as a laser beam: for example, the rest mass of the protons m 938 MeV/c 2 makes a negligible contribution to their energy for achievable particle energies of about 6.5 TeV and both charge and spin are irrelevant [32][33][34]. To very good approximation, the energy-momentum relationship is then ω 0 with an effective pulse power P cav p = P LHC = 2P avg cav , where P LHC is the nominal average power of the LHC. The pulse power of the LHC beam is orders of magnitude smaller than that of extreme-power laser pulses, but the proton bunches are much longer (∼ 1 ns) than the laser pulses. This results in a higher average power of P avg cav ≈ 3.8·10 12 W, which is orders of magnitude larger than the average power of laser pulses 2 Shorter cavities lead to lower τ L at the same finesse and without decreasing the average power. The same considerations for the spot size and length of cavity as mentioned for the laser pulses apply also in the cw case for positioning the detector sufficiently close to the beam waist and neglecting higher order effects in the opening angle [28]. For a 1 m-long high-finesse cavity (F ∼ 10 6 ) the decay time τ L is in the low millisecond range, which is too slow for some of the proposed detector setups. This could be circumvented by implementing techniques such as Q-switching, with which the decay of energy within the cavity can be accelerated, and the aforementioned switchable mirrors. Also, the energy buildup can be modulated to a certain degree by pumping. For short modulation periods, τ mod ≈ τ L , the cavity is never fully pumped.
pulses in cavity 3 · 10 14 W 100 fs ·10 kHz 8·10 5 ω 0 † 2 · 10 10 W < 100 µm cw laser+cavity 2 · 10 11 W π ω 0 1 · 10 11 W < 100µm LHC 10 14 W 10 −9 s * 3.8 · 10 12 W 16 µm Table I: Comparison of relevant numbers of the LHC beam and the laser-based sources from II A and II B: P cav p pulse power, P avg power averaged over time, w B waist of beam. ω 0 is the desired signal frequency, assumed in Sec.III to be one of the resonance frequencies of the detector. † A switchable mirror is assumed for the pulses in the cavity. The pulses are assumed to be effectively stacked together to a larger circulating effective pulse, see eq.(3). * The effective pulse length T cav p for the LHC corresponds to a single proton bunch, but a much slower modulation of the beam on resonance with ω 0 can be envisaged.
oscillating in a cavity and about 40 times the average power that can be contained in a cavity pumped by the cw laser considered above (see table I). Therefore, from the perspective of the strength of the gravitational source, the LHC beam might be preferable. A potential drawback compared to the laser-based sources is the lack of flexibility in frequency. This can be compensated, however, by considering detectors with tunable resonance frequency.
Besides protons, it is also possible to use heavy nuclei, or partially ionized heavy atoms.
The latter have the advantage that the corresponding beams can be laser-cooled (see the discussion in Sec.IV B). Upgrades of the LHC to use heavy ions are currently considered [56], and also under development at Brookhaven National Lab [57].

III. DETECTORS
We consider three types of detectors, a mechanical rod, a detector based on superfluid helium-4 coupled parametrically to a superconducting microwave cavity, and a mechanical harmonic oscillator, motivated by the monolithic pendulum from [58,59] and the torsion balance from [40], with which recently very high levels of sensitivity for gravitational fields have been reached. The superfluid helium detector and the monolithic pendulum are optomechanical detectors close to the quantum limit. Quantum optomechanical detectors and different configurations have been studied in great detail over recent years, both theoreti-cally and experimentally [60][61][62]. They have been considered for high precision sensing [63] in particular, force sensing [64] and theoretical work has been performed to derive general limits for sensing of oscillating gravitational fields with such systems [65,66]. We take the mentioned types of detectors as starting points for examining the question what parameter values would need to be achieved such that they become suitable for measuring the gravitational forces considered in this paper.

A. Mechanical response of a rod
A spatially dependent gravitational acceleration compresses a 1D deformable resonator according to its Young modulus Y . The wave equation for the displacement field u(x, t), describing the relative position of an element of the rod from its equilibrium location x, is where the resonator is extended in the x direction, orthogonal to the beam, and m is its mass density. The length contraction due to modification of space-time is negligible in comparison to the elastic effect considered here, as it comes with an additional factor c 2 s /c 2 [68], where c s = Y / m is the speed of sound in the rod's material.
The displacement field can be expanded into the spatial eigenmodes of the free equation of motion complying with the boundary conditions, i.e. the tip of the resonator distant from the source was chosen to be fixed in place by the support (hence w n (L + ∆) has to vanish and ∂ x w n | x=∆ has to vanish at the other tip), where n ∈ N 0 , ∆ is the distance of the tip of the rod from the source, and L is the length of the resonator (see figure 1). The spatial eigenmodes are orthonormal with respect to the inner product The total displacement field is then given by u(x, t) = ∞ n=0 ξ n (t)w n (x). The differential equation for the temporal amplitude ξ n (t) resulting from the projection of (5) onto the nth spatial eigenmode is then given bÿ where ω n = c s n + 1 2 π L is the frequency of the mode and a linear dissipation term γ n ∂ t u(x, t) with rate γ n = m ωn Q was added to equation (5) in order to include dissipation from the elastic modes of the resonator.
In the case of resonant excitation, the amplitude of the steady state solution in the lowest eigenmode ξ 0 (t) = A(ω 0 ) sin(ω 0 t), reached after a transient time Q ω 0 is then given by where the integration of t over one mechanical period gives the Fourier component of the driving force corresponding to this mechanical mode. At this point we assumed the pulse to be centered around t = 0 and to be repeating at intervals of 2π ω 0 . With the periodic Newtonian potential from appendix A where P cav p is the pulse power and l| H| l is a sum of rectangular pulses of duration τ p . The integral over the oscillation period in eq. (7) returns With this, a resonant maximum amplitude of where P avg cav ≡ energy in the cavity oscillation period = is the power in the cavity averaged over one mechanical period, is reached in the steady state of prolonged driving. The logarithmic divergence of equation (11) for L ∆ is an artifact of idealizations of our model and will not be relevant in practice 3 .
Assuming for orientation numerical values of aluminum, c s = c Al s = 6420 m/s, ω 0 = 2π·10 9 Hz, ∆ = w B 4 , and Q = 10 6 for the rod, the laser cavity introduced in II A (P avg = 20 GW, w B ≈ 100 µm) would result in an amplitude of A ≈ 10 −34 m at the freely oscillating tip.
At resonance, the noise spectral density for a resonant-bar type detector is given by according to [67, p.440], where M eff = m A R (w 0 (x)) 2 dx is the effective mass of the mode with the rod cross-section A R , and A th is the amplitude resulting from the thermal noise after integration time τ int . At ω 0 = 2π · 1 GHz the thermal sensitivity limit for temperatures below T = 48 mK is already below the standard quantum limit (SQL) on noise spectral density for a resonant mass detector [69], At frequencies below the megahertz range, the thermal noise is the limiting factor. For Q = 10 6 , M eff = π 8 Al L 3 (assuming a constant aspect ratio) with the mass density of aluminum Al = 2.7 g/cm 3 and a frequency of ω 0 = 2π · 1 GHz the sensitivity is S SQL A ≈ 4 · 10 −17 m √ Hz , meaning that for 1 year of integration time at best an amplitude of 10 −20 m can be detected.
For the LHC, where the rate of bunches passing by is ν = 31.2 MHz, for the purposes of this estimation, we assume that the same amount of protons is split into 88925 bunches instead of 2808 such that we reach the frequency ω 0 = 2π · 1 GHz while keeping the same average power. With pulses filling half a period, the peak power is P cav p = 2P avg cav . Using the same c s = 6420 m/s and Q = 10 6 and the values of the LHC (P avg cav = 3.8 · 10 12 W, w B = 16 µm) one would expect the resonant amplitude to be A ≈ 9 · 10 −32 m, which is at least two orders of magnitude larger than that caused by the oscillating laser pulse from Sec.II A. For higher quality factors Q = 10 8 amplitudes of A ≈ 9 · 10 −30 m might be possible. At far lower frequencies, where the limit L ∆ becomes relevant in eq.(11), a lower speed of sound, for example c s = 100 m/s, is also beneficial. However, one quickly ends up with a meter long rod, outside the "close to the beam" limit, whilst still not within range of detection.
To probe the limit L ∆, we assume ω 0 = 2π · 1 kHz, Q = 10 6 , c s = 6420 m/s implying an extreme L ≈ 4 km. Then, the expected amplitude from the laser pulses in sec. II A is A ≈ 2 · 10 −25 m, for the cavity pumped with a modulated cw laser A ≈ 4 · 10 −25 m, while we expect an amplitude of A ≈ 8 · 10 −24 m for the LHC beam (which would have to be modulated to reach such low frequencies). Assuming a temperature of T = 5 mK the sensitivity is S th A ≈ 10 −17 m √ Hz , leaving the amplitudes still unmeasurable even for unreasonably long integration and rise times and an unreasonable rod length. For the ground mode, the system's description can be reduced to a one dimensional problem and treated as in section III A, but with two fixed ends instead of one. The spatial displacement amplitude is then given by w 0 = sin π L (x − ∆) . The position noise spectral density of the temporal displacement field ξ is given by eq. (13), when comparing to the result of Singh et al. [70] a factor of 2 has to be added to obtain the single sided density (ω 0 > 0). With the susceptibility on resonance χ = Q He iM eff ω 2 0 , this results in a thermal force noise spectral density (on resonance) of Which implies a lower bound to the detectable force over an integration time τ int , with 2σ uncertainty, ofF The Fourier component of the force corresponding to the considered lowest-frequency mode is given bȳ Note the similarity of the amplitude A(ω 0 ) to the case of the mechanical rod detector in equation (11) 5 . Here, both signal and noise are given as a force, for better comparability to [70].
To get a feeling for the orders of magnitude, we start off with the numbers from the actual experimental setup from Singh et al. [70]. We set τ int = 250 d, Q He = 6 · 10 10 , L = 4 cm, r = Choosing the LHC as a source, we assume ∆ = w B and set the average power to P cav p = P LHC = 3.8 · 10 12 W, τ p = 1 2 2π ω 0 , resulting inF eff ≈ 6.6 · 10 −24 N. Going further from the beamline (by less than L) to account for shielding and the Helium container only decreases the effective force slightly (for ∆ = 16 µm → ∆ = 3 cm, F eff decreases by a factor of 4) as there is limited contribution from the liquid Helium at the ends of the container to the ground mode.
Hence, at full amplitude and one week of integration time, the 4 cm prototype detector is lacking about 3.5 of magnitude in sensitivity. Under otherwise identical assumptions, the proposed first generation (0.5 m) detector will be about 2.5 orders of magnitude from being sensitive enough to detect the gravitational signal from the LHC. 5 In contrast to the spatial integral in (11), the one in (17) converges to ≈ 1.85 for L ∆. This is because of the different in boundary conditions, in particular, the logarithmic dependence stems from the overlap of the mode function with the steep end of the 1 x driving force, whereas the modes of the helium have to vanish at the end of the container. However, the missing logarithmic dependence is basically irrelevant on realistic length scales. Assuming once again a constant aspect ratio, i.e. M eff ∼ L 3 , we find a scaling ofF min ∼ L andF eff ∼ L 2 , implying that the force should be detectable if L is large enough. However, limitations apply as is discussed in section III A.

C. High-Q milligram-scale monolithic pendulum
In a recent publication, Matsumoto et al. [58] described the manufacturing of a pendulum and presented its properties. They found it to have a very high quality factor for a small scale system and even higher when combined with an optical spring. Different from the extended oscillators considered in the earlier subsections, the pendulum does not rely on the projection of the gravitational acceleration on an elastic mode but rather on the gravitational force on the pendulum mass relative to the support. A mechanical oscillator has to be of small scale to be close enough to the source for the gravitational acceleration to be significant, while the gravitational effects on the pivot point need to remain negligible.
For the l = 1 cm, m = 7 mg pendulum a mechanical Q-factor of Q m = 10 5 was measured in [58] at ω m = 2π · 4.4 Hz. Introducing an optical spring to shift the frequency, the effective Q-factor is expected to scale as for the damping model considered relevant for the pendulum (the effective frequency of the coupled system was renamed from ω 0 (Q 0 ) in the original work [58] to ω m (Q m ) for consistency). An additional feedback cooling is necessary to stabilize and cool the system to a temperature T fb , compensating the effect of heating through the optical spring. This reduces the Q-factor to Q fb , which has the benefit of allowing shorter driving times. At ω 0 = 2π · 280 Hz the authors of [58] demonstrated a sensitivity of 3 · 10 −14 m √ Hz with a Q-factor of Q fb = 250, with thermal motion the main source of noise. According to eq. (13) this corresponds to a temperature of a few Millikelvin.
In an update to this Cataño-Lopez et al. [59] described an improved version of this pendulum, with a measured mechanical Q-factor of Q m = 2 · 10 6 at a frequency of ω m = 2π · 2.2 Hz, which with the optical spring is tunable in the frequency range of 400 Hz < ω 0 2π < 1800 Hz. For a pulsed-beam source, the gravitational acceleration in radial direction for the duration of a pulse is given by where G is the Newton gravitational constant, P cav p is the pulse power, and ρ is the distance from the beam. For this setup ρ is limited by the radius of the pendulum mass (1.5 mm) and the beam width ( .5 mm), so ρ = 2 mm is a reasonable estimate which might be substantially increased, however, if a cryostat is needed.
The displacement resulting from prolonged (τ ∼ 2πQ fb ω 0 ) driving on resonance is given by whereā grav is the Fourier component of a grav (t) on resonance, and sin ω 0 τp 2 results from the overlap of the rectangular pulses with the sinusoidal oscillation calculated in eq.(10). We now consider how the pendulum would react to the different gravitational sources discussed above.

Cavity pumped with cw laser
For the pendulum from [58] at ω 0 = 2π · 280 Hz and the cw laser cavity from II B with a power in the cavity of P The SQL refers to amplitude-and-phase measurements of that position. In principle, due to the precisely known frequency, quantum non-demolition measurements allow continuous monitoring of the oscillation [71]. With a "single-transducer, back-action evading measurement", one can estimate a quadrature of the oscillator with an uncertainty that scales ∝ (βω 0 τ m ) −1/2 , where τ m is the relevant measurement time or inverse filter width, and β a numerical factor that can reach a value of order one (see eq. 3.21a,b in [71] and eqs. (32,33) in [72]). After upconverting the kHz signal to the GHz regime one can use modern microwave amplifiers with essentially no added noise [73][74][75][76]. Upconversion to the microwave frequency range was already discussed in the 1980s [72] and can be achieved by having the sensor modulate the resonance frequency of a microwave cavity.Additional sensitivity can be gained with a large number N of sensors arranged along the laser beam or particle beam.
Classical averaging their signal leads to a noise reduction of 1/ √ N in the standard deviation. When several sensors all couple to the same microwave cavity, one might even hope to achieve "coherent averaging", in which case the noise reduction scales as 1/N [77,78].
With N = 1 and a signal of 280 Hz, the sensitivity of the pendulum resulting from the standard quantum noise limit S SQL ≈ 7·10 −17 m √ Hz is 3 orders of magnitude lower than that given by the thermal noise. For 1 week of measurement, the thermal noise still exceeds the signal generated by the modulated cw laser (respectively train of laser pulses) by 8 (almost 9) orders of magnitude.

LHC beam
The minimum frequency of one bunch of ultra-relativistic protons going around the ring of the LHC is in the kHz range (see Sec.II C). Lower frequencies could be achieved by modulating the beam position with low frequency. The LHC as a source is expected to create almost 20 times larger amplitude than the considered cw-pumped cavity, due to the higher pulse power P cav p = P LHC where an "on-off" modulation of the LHC beam, similar to the cw cavity pumping scheme was assumed. After one week of measurement time one would be about 7 orders of magnitude off from measuring the signal with a single detector, 5 orders of magnitude starting at a temperature of T = 5 mK. Substantially more development will be needed to bridge this gap. Ideas in this direction are developed in the next section.

D. Optimizing the S/N
In this section we ask, what parameter values would be needed to achieve a signal-to-noise ratio comparable to 1 for a torsion balance or pendulum. We model both simply as damped harmonic oscillators, but keep in mind that their mechanial parameters and temperature can be substantially modified by using an optical spring and/or feedback cooling, and then compare to the existing setups described in [40,58,59]. We therefore continue to use T fb for the final temperature, Q fb for the final quality factor, and ω 0 as final resonance frequency ×2π, regardless of how they might be achieved.
1. Optimization of a mechanical oscillator as dectector and comparison to [40] According to eq.(5.60) in [69] the total position-noise power at frequency ω of a harmonic oscillator with (undamped) resonance frequency Ω measured with a transducer and amplifier that add back-action noise (referred back to the input) can be written as S xx,tot (T, ω, Ω, Q, m) = γ 0 γ 0 + γS xx,eq (T, ω, Ω, Q, m) +S xx,add (T, ω, Ω, Q, m) where γ 0 is the intrinsic oscillator damping without coupling to the transducer and γ ≡ γ(ω) the damping with the coupling. The equilibrium noise (comprising both quantum noise and thermal noise at temperature T ) reads with the quality factor Q ≡ Ω/(γ 0 +γ). To calculateS xx,add , one needs to know the force noise power of the detector and amplifier, butS xx,add is lower bounded byS xx,addMin = |Imχ xx |.
With this lowest possible value and the replacements Ω → ω 0 , Q → Q fb , T → T fb , one obtains for the total noise power to lowest order in γ/γ 0 (which slightly overestimates the contribution fromS xx,eq (T, ω, ω 0 , Q, m)) The maximum amplitude x grav of the harmonic oscillator is given by eq.(20) with sin(τ p ω 0 /2) = 1, but is reached only asymptotically as function of time, namely as x grav (t) = x grav (1 − exp(−ω 0 t/(2Q fb ))). We assume that the total time τ tot = 1 week for the experiment is split as τ tot = τ r + τ m into a time τ r needed for the amplitude of the harmonic oscillator to rise to a certain level, and a measurement time τ m used for reducing the noise.
The total signal-to-noise ratio on resonance is then given by where a distance ρ = 200 µm of the center of the detector mass from the beam axis was assumed. All quantities are in SI units. From this equation it is evident that the mass m should be as large as possible. At the same time, m cannot be made arbitrarily large, as otherwise the distance from the beam axis would have to be increased as well, which would lead to a decay of the signal ∝ 1/ρ for ρ ρ min , where ρ min is the minimum distance from the beam axis (which might contain a shielding of the particle beam in the case of the LHC, and which we assume to be ρ min = 100 µm for the LHC but might have to be substantially increased when using a cryostat). In principle, for a spherical detector mass, a scaling ∝ m 1/6 would still result, but it turns out that unrealistically large masses (larger than 1 kg) would be needed before this scaling gives an advantage over an alternative design with a cylindrical detector mass that allows to maintain ρ = 200 µm. If we allow that cylinder to become as long as L cyl = 0.5 m and determine the maximum mass as m = 0.9π Si (ρ − ρ min ) 2 L cyl (where 0.9 is a "fudge factor" that avoids that the detector mass touches the shielding), we find m = 33 mg.
With that value inserted in eq.(25), one can optimize S/N with respect to the parameters τ m , ω 0 , Q fb and T fb . With τ tot kept equal to 1 week, in the range 1 rad/s ≤ ω 0 ≤ 10 4 rad/s, 1 ≤ Q fb ≤ 10 8 , 1 nK ≤ T fb a maximum value S/N 0.6 is found for τ m = 3 · 10 5 s, ω 0 = 2π · 0.16 Hz, Q fb = 1.2 · 10 5 , and minimal T fb . The optimal value for ω 0 is at the lower end of the parameter range, but reasonably close to the one for the existing torsion balance in [40] (ω 0 = 2π × 3.59 mHz), where, however, the mechanical quality factor was Q = 4.9 and a mass of 92.1 mg was used. It remains to be seen if the parameters that result from the optimization can be reached. Problematic appears mostly whether the temperature of the cooled mode of about 1 nK can be reached, especially at low frequencies.
2. Assumption of Q-scaling and comparison to [58] The structural damping model used in [58] implies a quadratic scaling of the Q-factor with the resonance frequency (see eq. (18)). Including this scaling behavior and allowing the modification of the resonance frequency by means of an optical spring, leads to frequencies in the 100 Hz to 1 kHz range being preferred by the optimization. This ultra-high Q-factor is, however, not reachable in practice as the optical spring introduces heating, and so the mechanical oscillator has to be cooled to stabilize the system. In existing systems, feedback cooling [58,79], or a second optical spring tuned to the infrared [80] have been employed as cooling mechanisms. We assume an effective final temperature reached by feedback cooling as is expected in [58]. An initial temperature, before feedback cooling, of T bath = 5 mK is assumed and the parameter ranges are limited to 1 rad/s < ω m < 10000 rad/s, 1 rad/s < ω 0 < 10000 rad/s, 1 < Q m < 10 7 , and 1 < Q fb < 10 10 . We find S/N ≈ 0.077 for the optimal parameters of ω m = 2π · 0.16 Hz, ω 0 = 2π · 600 Hz, Q m = 10 7 , and Q fb = 1.6 · 10 8 .
Compared to the generic optimization as seen above this seems underwhelming but if the scaling of Q and temperature can be attained, the final temperature of T fb ≈ 23 nK would be more feasible than before.

Further possible improvements
A signal-to-noise ratio of 0.6 is still not good enough, but the planned upgrade of the LHC to the high-luminosity LHC [56] should increase S/N by a factor 10. Another factor 2.9 is expected to be gained by switching to tungsten (with mass density W = 19, 250 kg/m 3 ) as detector-mass material, all other optimized parameters remaining equal. Both factors combined lead to a S/N 16.
The maximum of S/N found in the optimization is rather flat, especially with respect to the feedback cooling quality factor, such that there is a wide range of values with similar signal-to-noise ratios that allow one to take into account other engineering constraints not considered here and without such extreme effective temperatures. Hence, with the highluminosity LHC and an optimized detector there is realistic hope that GR could be tested for the first time in this ultra-relativistic regime with a controlled terrestrial source and adapted optimized detector. Also without the upgrade of the LHC, further improvements from using a multitude of detectors (and possibly coherent averaging by coupling them all to the same read-out cavity [77,78]) or longer integration times can be envisaged that would bring S/N to order one. For the cases in which the main limiting factor is thermal noise, a temperature of 5 mK was assumed. Other parameters see text. * assuming the LHC beam can be modulated to produce a signal with appropriate frequency while maintaining the same average power.

IV. DISCUSSION
A. Perspectives for measuring the gravitation of light or particle beams We have theoretically investigated the fundamental limitations to measure the oscillating gravitational fields of lab-scale ultrarelativistic sources for three concrete examples: for laser beams, we have considered femtosecond-pulse lasers fed into a high finesse cavity, where they oscillate to and fro, and similarly, cw lasers used to pump a cavity periodically. For particle beams, we considered the LHC with its beam of proton bunches flying along the accelerator ring. All sources considered lead to oscillating curvature of space-time and acceleration of test particles with precisely controlled frequency up to the GHz range. In addition, we have given details on how modulations of these signals with much lower frequency, down to the kHz regime, can be achieved for all three example sources. In the latter regime, the LHC is the most promising ultrarelativistic source of gravity with a gravitational field strength 20 times stronger than the laser sources considered here.
We investigated three near-field detectors: A deformable rod offers force accumulation along its length thanks to its Young modulus. However, the spatial decrease of the studied gravitational effects limits the effects of force accumulation, resulting in immeasurably small amplitudes of the order of 8 · 10 −24 m even in the case of the LHC as a source. In the liquid helium chamber from Singh et al. [70], very high quality factors and low noise allow for sound wave buildup within the chamber. With the present experimental parameters [70], the gravitational force for the LHC is 3.5 orders of magnitude below the detectability limit of this detector with an averaging time of one week.
A pendulum from [59] and [58] or a recently demonstrated torsion balance [40] turned out to be the most promising detectors. In the present form of the monolithic pendulum [58], the fully built-up signal from the LHC is 5 orders of magnitude away from the sensitivity achievable within 1 week of averaging time (assuming a starting temperature of T = 5 mK and a final temperature of T fb = 12 nK after a shift of the resonance frequency via an optical spring and feedback cooling) with the benefit of a relatively small signal rise time.
Optimization of the signal-to-noise ratio of a mechanical oscillator as detector over its frequency, measurement time, quality factor and temperature in the parameter range provided in Sec.III D 1, leads to an expectation of a S/N of about 0.6 with the LHC as source within one week of signal rise time and averaging. By using a denser material such as tungsten for the detector mass and profiting from the planned high-luminosity upgrade of the LHC a S/N ratio 16 appears possible with one week of measurement time for a single detector.
Our considerations concerned fundamental limitations so far, so that a S/N ratio larger than 1 should be considered a necessary condition, but would still make for a very difficult experiment with additional noise and engineering issues to be overcome (see e.g. [42]).
Important additional noise sources that have not been considered in our work include, for example, seismic and thermal noise that may be reduced by moving to a higher frequency regime. Therefore, while very high source frequencies (GHz) turned out to be detrimental for the considered detectors, it may still be interesting to investigate an intermediate frequency range above the kHz regime. In their current design, the superfluid helium detector [70] and the pendulum detector [59] need a source oscillating with a frequency of the order of kHz and 400 Hz to kHz, respectively. The pendulum's operation at higher frequencies might be possible and relatively easy to achieve, given that the relevant noise terms in the kHz range stem from suspension eigenmodes, which are changeable by design. Also in [81] parametric cooling into the ground state for pendulum-style gravitational sensors was demonstrated, reducing problems from thermal noise and seismic noise in an even larger frequency range.
However, reaching the required low-temperatures in the nK regime in combination with the high quality factors will remain a huge challenge, even if the Q-scaling (18) and feedback cooling assumed in [58] is achieved.

B. Perspectives for quantum gravity experiments
The realization that the gravitational effect of light or high-energy particle beams might become measurable in the near future opens new experimental routes to quantum gravity, in the sense that it might become possible to study gravity of light or matter in a non-trivial quantum superposition. Concerning light, non-classical states of light, in particular in the form of squeezed light, have been studied and experimentally realized for a long time, and are now used for enhanced gravitational-wave-sensing in LIGO and Virgo [82,83]. While the current records of squeezing were obtained for smaller intensities than relevant for the gravitational sources we consider here [84], squeezing and entanglement shared by many modes was already achieved for photon numbers on the order 10 16 by using a coherent state in one of the modes [85]. This is substantially smaller than the ∼ 10 21 photons estimated in the cavity in the example of the cw laser leading to 100 GW circulating power considered above, but one might hope that technology progresses to achieve at least a small amount of squeezing also for the high-power sources relevant here.
As for the high-energy particle beams, transverse "coherent oscillations" of two colliding accelerator beams (including the ones at LHC) have already been studied [86][87][88][89][90] but these are of classical nature. Non-trivial quantum states of the beam are those that cannot be described by a positive semidefinite Glauber-Sudarshan P -function, a concept from quantum optics that is well established for harmonic oscillators and is hence applicable to smallamplitude transverse motion of the particle beam in the focusing regions where there is a linear restoring force. A stronger requirement would be a non-positive-semidefinite Wigner function, which can be applied to any system with a phase space. In order to reach such quantum states, it will be necessary to cool the particle beams. Efforts to do so are on the way or proposed for other motivations: cooling enhances the phase space density and hence the intensity of the beam in its center. In addition, new phases of matter in the form of classical crystalline beams attracted both theoretical and experimental interest at least since the 1980s [91][92][93][94][95][96]. Recently it was proposed to extend this work to create an "ultracold crystalline beam" and turning an ion beam into a quantum computer. For this, the beam should become an ion Coulomb crystal cooled to such low temperatures that the de Broglie wavelength becomes larger than the particles' thermal oscillation amplitudes [97].
Ideally, for our purposes, the center-of-mass motion of the beam should be cooled to the ground state of the (approximate) harmonic oscillator that restrains locally, at the detector position, the transverse motion, before interesting quantum superpositions can be achieved.
However, even superpositions in longitudinal direction would create an interesting experimental situation for which there is currently no theoretical prediction. Experimental progress in this direction would allow a different kind of search for quantum gravity effects compared to popular current attempts to detect deviations from canonical commutation relations between conjugate observables as predicted by various quantum-gravity candidates (see e.g. [98]). Different techniques for cooling particle beams are available (see e.g. [99,100] for electron cooling (absorption of entropy by a co-propagating electron beam of much lower energy and entropy), and a modern cousin of it, "coherent electron cooling" [101], under development at Brookhaven National Lab for ion energies up to 40 GeV/u for Au +79 ions [57,102]; and laser cooling, with which longitudinal temperatures on the order of mK have been reached for moderately relativistic ion beams [103,104]. Laser cooling is most efficient for longitudinal cooling, but transverse cooling can be achieved to some extent through sympathetic cooling [105]. Laser cooling is now proposed for an ultrarelativistic heavy-ion upgrade of the LHC [56]. Despite all these techniques, ground states of the transverse center-of-mass motion have never been reached in any ultra-relativistic particle beam as far as we know, nor was it perceived as an important goal. We hope that the perspective of winning the race to the first quantum gravity experiment will change this.
As Grishchuk put it [11]: "The laboratory experiment is bound to be expensive, but one should remember that a part of the cost is likely to be reimbursed from the Nobel prize money !". Following the calculation of the gravitational field of a box shaped laser pulse of length L emitted at z = 0 and absorbed at z = D from [18,21], we extend the calculation to an oscillation of a short pulse (L < D) between 0 and D. For a pulse propagating along the ±z-direction the energy momentum tensor is given by T 00 = T zz = ∓T 0z = ∓T z0 = u(z, t)δ(x)δ(y)A, where u(z, t) is the energy density of the electromagnetic field in 3D and A is the effective transverse area. This energy momentum tensor violates the continuity equation as the recoil of the mirrors is neglected. However, ultimately only positions very close to the beam will be considered where these contributions vanish [18]. The energy density is given by where delimit the profile of the pulse injected at t = 0 and reflected 2n times (2n + 1 times) traveling in positive (negative ) z-direction, and u p = Ep LA is the pulse energy density. From the wave equation in the Lorenz gauge the metric perturbation can be calculated using the Green's function Given the energy-momentum tensor, the metric perturbation can be decomposed into ρ = x 2 + y 2 , and u p A = P c . The box function χ n + imposes the additional boundaries of a n + < z < b n + , with a n + =z + Similarly, the box function χ n − adds the constraints of a n Following [21] the substitution ζ(z ) = z − z + ρ 2 + (z − z) 2 is used to further simplify the integration. The constraints turn into where r = ρ 2 + z 2 , and r D = ρ 2 + (z − D) 2 . P n 0 : 2nD < ct − r < 2nD + L causally connected to the reflection at z = 0, ct = 2nD n : 2nD + r + L < ct < (2n + 1)D + r D not causally connected to any reflection events, but causally connected to the pulse traveling from z = 0, ct = 2nD to z = D, ct = (2n + 1)D P n D : (2n + 1)D < ct − r D < (2n + 1)D + L causally connected to the reflection at z = D, ct = (2n + 1)D n : (2n + 1)D + r D + L < ct < (2n + 2)D + r not causally connected to any reflection events, but causally connected to the pulse traveling from z = D, ct = (2n + 1)D to z = 0, ct = (2n + 2)D.
The metric perturbation is then given by caused by the pulses starting from z = 0 and caused by the pulses returning from z = D, where t j := t − jD/c.
Following [21], the only independent non-vanishing elements of the Riemann curvature tensor are given by where i, j ∈ {x, y}.
In the limit ρ r, r D , the only independent components of the curvature in leading order are R 0i0j = − R 0izj = 4GP c 5 1 ρ 2 δ ij − 2 r i r j ρ 2 ≡ R ∀(z, t) ∈ P n 0 (A23) The geodesic equation for a test particle at position x µ is given in coordinate time t = 1 c x 0 by For a non-relativistic test particle this reduces tö with the linearized Christoffel symbol The acceleration a non-relativistic sensor experiences is therefore equivalent to that from a Newtonian potential for the duration of the pulse passing by (P 0 , P D ) with the potential vanishing at all other times.

Appendix B: Intensity in a Fabry-Pérot resonator
The considerations here follow those from [49] closely but are modified to reflect the setups used in this work.
For a Fabry-Pérot resonator consisting of two mirrors with field reflection coefficients √ R 1 , √ R 2 and field transmission coefficients √ T 1 , √ T 2 , the field in cavity (at the face of mirror 1) resulting from a pump beam striking mirror 1 can be written as in the time domain, where τ rt is the time for one round trip in the cavity. In the frequency domain this can be written as

Single monochromatic rectangular pulse
For a monochromatic pump field of frequency ω E and length τ p entering the cavity at t = 0 the pump field is given by For a very short pulse τ p τ rt , none of the addends will overlap and the intensity in the cavity is The pulse enters the cavity with an intensity reduced by T 1 and is reduced by a further factor R 1 R 2 for each subsequent round trip. The average power in the cavity relative to that of the pump laser is then given by For long pulses (τ p τ rt ) and a resonant cavity (ω E τ rt = 2πm) only addends from n min = max 0, t−τp τrt to n max = max 0, t τrt contribute for any given time, returning The intensity for this long-pulse resonant cavity case can be described as a "jagged shark fin" and is plotted in fig. 3

Series of monochromatic rectangular pulses
A series of periodic pulses separated by time τ rt can be written as a sum of pulses E Σ pump (t) = k E p pump (t − kτ rep ). As all of the operations on the field are linear the pump field eq. (B5) can be used to find For repetition times much longer than the lifetime of a pulse in the cavity τ rep τ L ∼ τrt(R 1 R 2 ) 1/4 1− √ R 1 R 2 , and the pulse length τ rep τ p , the addends barely overlap such that there is no interference between consecutive pulses. In this case the intensity in the cavity is just that