High Q mg-scale monolithic pendulum for quantum-limited gravity measurements

We present the development of a high Q monolithic silica pendulum weighing just 7 mg. The measured Q value for the pendulum mode at 2.2 Hz was $2 \times 10^6$, the highest value to date for a mg-scale oscillator. By employing this suspension system, the optomechanical displacement sensor for gravity measurements we recently reported in Phys. Rev. Lett. 122,071101 (2019) can be improved to realize quantum-noise-limited sensing at several hundred Hz. In combination with the optical spring effect, the amount of intrinsic dissipation measured in the pendulum mode is enough to satisfy requirements for measurement-based quantum control of a massive pendulum confined in an optical potential. This paves the way not only for testing dark matter via quantum-limited force sensors but also Newtonian interaction in quantum regimes, namely, between two mg-scale oscillators in quantum states.

Introduction.-The development of quantum-limited massive sensors is a key component for the direct measurement and investigation of macroscopic quantum mechanics, the quantum nature of Newtonian interaction [1,2], as well as direct detection of dark matter by looking at fifth forces [3]. In recent years, cavity optomechanics has pioneered the development of low-loss mechanical oscillators in a variety of different architectures [4][5][6], opening the door to measurement-based control of mechanical oscillators in the quantum regime [7][8][9]. On the other hand, recent proposals to investigate gravitational interactions at the mg scale [10] have motivated the top-down approach relying on techniques utilizing macroscopic suspended pendulums, getting inspiration from gravitational-wave detectors [11]. However, the development of a mechanical oscillator with the possibility of quantum-limited sensitivity, while at the same time being massive enough to measure gravitational interaction has yet to be realized.
To achieve quantum-limited displacement sensing and measurement-based quantum control, the oscillator must satisfy two basic requirements. The first demands for the frequency of oscillation to exceed the thermal decoherence rate, which induces heating from the mechanical bath into the system i.e., ω m >nγ m . Heren is the average phonon number of the oscillating mode, ω m /2π is its resonant frequency, and γ m is the oscillating mode's dissipation. This translates into the commonly named f Q condition: which establishes a lower bound on the quality factor Q m = ω m /γ m of the mode, necessary to undergo at least one coherent mechanical oscillation before one phonon from the thermal bath enters the mode. Here k B is the Boltzmann constant,h is the reduced Planck constant, and T is the temperature of the thermal bath. The second requirement is closely related to the min-imum readout noise required to resolve the zero-point motion of the oscillator x zpf = h/2mω m in a measurement time scale faster than the thermal decoherence rate.
In optomechanical systems using massive pendulums, we can set the standard quantum noise limit for a free mass S SQL = 2h/mω 2 (hereinafter free mass SQL) [12] as the reference readout noise level. This is because the sum of the main readout noises, like shot noise and mirror thermal noise, can be designed to be close to the free mass SQL at several hundred Hz [13][14][15]. This translates into our noise requirement in terms of dissipation: In these expressions ω is the Fourier frequency. This second requirement is critical for the oscillator to be implemented in any type of measurement-based quantum experiment like feedback cooling. In optomechanical experiments implementing pendulums, an optical spring can be used to trap and shift the pendulum mode to higher frequencies. This effect does not add excess thermal fluctuating forces on the pendulum, since the effective temperature of each photon in the optical spring (e.g. ≈ 15, 000 K for 1064 nm wavelength laser) greatly exceeds the temperature of the background thermal bath (300 K). Effectively, this means the optical field is almost at its ground state and acts as a zero-temperature heat bath with zero entropy. Thus, when the second condition is satisfied at some frequency, the first condition can also be satisfied by changing the pendulum's frequency around that frequency band. In terms of optomechanical paremeters, this means the quantum cooperativity is close to 1 in that range of frequencies.
In this Letter, we present the development of a monolithic 7 mg silica pendulum with an intrinsic pendulum quality factor of Q m = 2 × 10 6 at 2.2 Hz, capable of satisfying both requirements between 400-1800 Hz. Implementing an optical spring in this frequency range is arXiv:1912.12567v1 [quant-ph] 29 Dec 2019 within the capability of previously reported experiments [16][17][18], and therefore paves the way to the study of a mg-scale oscillator's motion in the quantum regime, and test of the intersection between gravitational and quantum regimes.
Pendulum as system.-Under the fluctuationdissipation theorem, interaction with the environment produces a fluctuating force on a mechanical oscillator dependant on its dissipation [19]. A pendulum system by itself can allow for the pendulum's mode Q factor to exceed by many orders of magnitude that one which would be imposed by the pure material dissipation. Therefore, massive oscillators have traditionally been isolated via suspension pendulums to achieve minimal dissipation. This effect is termed dissipation dilution because the energy loss is being diluted by the intrinsically lossless gravitational potential, where most of the energy is stored. The ratio of the spring constants k g /k el is termed the enhancement factor, and for a pendulum of a single wire [20], where l is the length of the wire, r is its radius, m is the mass, E is the Young's modulus of the material, and Q mat is its intrinsic quality factor of the material. It is thus evident that in order to achieve maximum dilution the choice of material, as well as minimizing (maximizing) the radius (length) of the wire have to be taken into consideration. Dissipation in the system can originate principally through energy loss from internal or external channels, and the total loss will be given by a sum of all the losses. Internal losses take into account material losses, surface losses, and thermoelastic losses. On the other hand, external losses can come from residual gas losses, clamping losses, and bonding losses. In general, the study of different loss mechanisms is critical to achieve minimum dissipation in the pendulum [21]. Regarding the dissipation's frequency dependence, the pendulum is known to follow the structural damping model [20,22] in frequencies where high-order modes are sufficiently sparse. Energy loss generates from internal material losses, and the dissipation is not constant (as opposed to viscous damping, where the mode is assumed to be damped by external friction) but depends on frequency: and where the quality factor of the pendulum mode is related to the constant loss angle by φ m = 1/Q m . This is advantageous, since the displacement noise spectral density of a structurally damped pendulum falls faster than a viscously damped oscillator (x th ∝ 1/ω 2.5 vs 1/ω 2 ), lowering the noise floor of the suspension thermal noise at higher frequencies.
Fabrication.-We fabricate a 1 µm fiber diameter with a length of 5 cm starting from a 125 µm diameter fused silica fiber. This is done by pulling the fiber while heating it with a hydrogen torch (HORIBA, OPGU-7100).
The fiber is pulled about 30 cm by programable motorized stages (SIGMAKOKI, SHOT-GS, OSMS26-300), and its taper region follows the model in [23]. Improvements since [16] are the addition of a mass flow controller (HORIBA, SEC-E40MK3) to reduce surface imperfections and an increased fiber fabrication length of 1 cm to 5 cm. The former has improved the intrinsic material quality factor by an order of magnitude, while the latter directly affects gravitational dilution. The material quality factor, measured via a ring-down measurement of the pendulum's yaw mode, is estimated to be Q yaw = 1.3 × 10 4 . This value is close to the limiting Q due to the fiber's surface losses ≈ 2 × 10 4 as estimated in [24]. An example of this improvement is shown later in figure 2 (b), where we show measurements for other pendulums fabricated with this method. The repeatability of the fiber-pulling rig has been confirmed by SEM measurements.
Once the ultra-thin and long fiber is fabricated, we proceed to mount it on a bench implementing a CO 2 laser (Coherent, Diamond C-30A) for welding the mass, fiber, and the silica block support at the top. The laser spot is focused to a 30 µm beam spot, which allows localizing the welding point as shown in the left picture of figure  1. The monolithic aspect of this approach is critical in reducing loss mechanisms since, in contrast with kg-scale systems [21,25,26], previous reports utilizing tabletop mg-to g-scale test masses have until now been unable to achieve comparable levels of dissipation [27][28][29]. The test mass is a 7 mg silica disk of 3 mm in diameter and 0.5 mm in width to emulate a suspended mirror in a cavity optomechanics experiment.
Results and discussion.- Figure 2 shows several averaged ring-down measurements after excitation of the pendulum. The position of the pendulum is measured by detecting the intensity modulation due to the shadow cast onto a Si photodiode (HAMAMATSU S1223-01) from a laser (Coherent, Mephisto 500) intersecting the test mass' path. A bandpass filter is applied to the data around the resonance of interest, and the envelope of the time trace is then extracted and fitted to an exponential function to calculate the Q value. The results of multiple measurements are then aggregated into time bins, where the average and statistical error are shown in figure 2. Further, to neglect residual gas damping, the experiment was performed at low pressure. In our case, the experiment was performed at pressures lower than 10 −5 pascals, which would limit the Q m at around 10 9 × ( ωm/2π 2.2 [Hz] ). The black line in the figure is the pendulum utilized in our previous reported experiment and is shown for comparison. That pendulum was a 1 cm long and 1 µm in diameter fused silica fiber bonded to a silica mirror by epoxy glue, and clamped at the top by a pair of stainless steel plates. The system had performed with a quality factor of 1 × 10 5 and had a resonant frequency of 4.4 Hz.
This time, we report a 40 fold decrease in terms of dissipation, since we measure a quality factor of Q m = 2×10 6 at a resonance frequency of 2.2 Hz. Our pendulum surpasses the requirement of maintaining at least one coherent oscillation before thermal decoherence, since Q eff ω eff /2π = 9.2 × 10 12 , under the same modified effective frequency of 280 Hz as in our last report [16]. To calculate Q eff and ω eff , we work with the assumption that the effective oscillating mode is the pendulum mode as modified by the optical spring once the suspended mirror is confined in the optical trap [16,[30][31][32]. Because the optical spring is effectively loss-less, it allows the oscillator to undergo further dilution given by the enhancement factor k eff /k g = (ω eff /ω m ) 2 , where the effective rigidity k eff = k opt + k g + k el , and k opt , and k el are the optical and material rigidity respectively. In the enhancement factor, we have ignored the material rigidity k el because k opt >> k g >> k el . Thus, the achievable quality factor scales as Q eff = Q m × (ω eff /ω m ) 2 . We note here that when analyzing the pendulum mode's spectrum we observed fluctuations of its resonance frequency on the order of a few µHz, resulting in phase decoherence. However, although the origin of these fluctuations is at the moment unknown, because in our frequencies of interest the optical rigidity is much larger than the bare pendulum's rigidity (k opt /k g ≈ 10 4 ), these fluctuations are negligible in the effective frequency even to first order since k eff = k opt + k g → ω eff = ω opt 1 + (ω m /ω opt ) 2 .
The biggest gain in terms of dissipation was achieved when the clamping parts were removed and instead welded. This agrees with the assumption that the pendulum mode has most of its bending and energy loss at the top of the fiber, not near the mass [33]. Although prior attempts of welding only the disk and the fiber (green data in fig 2) showed some ammount of gain, most of the 6 fold decrease in dissipation from the black data to the green data can be explained by the increase in length of the pendulum from 1 cm to 5 cm. In fact, although our pendulum is still two orders of magnitude away from reaching the ideal quality factor given by equation 3 (we believe this may be due to welding losses [34]), the current state is enough to fulfill both requirements and further improvement would be masked by other dissipation mechanisms. Instead, future attempts at improving thermal noise will be better focused on the mirror's thermal noise and thermoelastic noise. For our system, when considering a model including nonlinear thermoelastic losses [24] it is possible to tune the fiber radius to effectivly cancel out thermoelastic losses at our frequency of interest [35]. In terms of dissipation, the expected dissipation of the pendulum following the structural damping model (Eq. 4) satisfies the second requirement (Eq. 2) at frequencies above 400 Hz. Figure 3 shows the noise budget considering higher order modes and mirror thermal noise (substrate plus coating thermal noise [36]). The spectrum is calculated using the analitic model in [22], derived by solving the elastic beam equation with boundary conditions correspoding to a rigid mass of finite size. Due to the 40 fold decrease in dissipation, this pendulum's suspension thermal noise is estimated to be roughly 6 times lower than that of our previous report [16]. Also, the improvement in the material quality factor suggests this fabrication method can be advantageous for mg-scale torsion pendulum experiments [37]. In figure 3 we do not include the optical rigidity, and see that our pendulum's thermal noise goes below the SQL between 400 -1800 Hz, meaning quantum fluctuations dominate the noise spectrum. Because the ratio between the suspension thermal noise and quantum noise does not change by adding an optical spring, quantum-limited sensing can be achieved from 400 Hz up to around 1 kHz on an optically trapped pendulum's resonance. We note that to achieve this level of mirror thermal noise, state of the art coatings like crystalline coatings [38] should be implemented.
Conclusion.-We report the fabrication of a completely monolithic mg-scale pendulum meeting the requirements for performing quantum control experiments. To the best of our knowledge, this is the lowest dissipation ever achieved at this mass scale, which combined with the optical spring effect can open the door for experimentation in the intersection between quantum theory and gravity.