Mechanical parametric feedback-cooling for pendulum-based gravity experiments

Assume a pendulum with resonance frequency ω₀, with a point-like mass suspended by a massless rod of length l subject to viscous damping and the suspension point executing a harmonic vertical motion $z(t)={z}_{0}\sin {\rm{\Omega }}t$ (figure 1). The equation of motion is given by [19]

$$\begin{eqnarray}&&\ddot{\phi }(t)+\gamma \dot{\phi }(t)+\displaystyle \frac{1}{l}(g+\ddot{z}(t))\phi (t)=0,\end{eqnarray} \tag{ 1 }$$

where ϕ ≪ 1 is the deflection angle of the pendulum, γ is the natural damping rate, defined as the reciprocal of the 1/e energy damping time and g is the gravitational acceleration. $\ddot{z}(t)={z}_{0}{{\rm{\Omega }}}^{2}\sin {\rm{\Omega }}t$ is the vertical acceleration of the suspension point with amplitude z₀, which acts as a parametric drive. By performing the substitution $\phi =\xi {e}^{-\tfrac{\gamma }{2}t}$, equation (1) is transformed to [20]:

$$\begin{eqnarray}&&\ddot{\xi }+\left({\omega }_{0}^{2}-{\left(\displaystyle \frac{\gamma }{2}\right)}^{2}+\displaystyle \frac{\ddot{z}(t)}{l}\right)\xi =0,\end{eqnarray} \tag{ 2 }$$

where ${\omega }_{0}=\sqrt{g/l}$. This is a form of the Mathieu equation [20]. In general, the Mathieu equation has the two independent solutions

$$\begin{eqnarray}&&{\xi }_{1}(t)={e}^{\tfrac{\mu }{2}t}{p}_{1}(t),\ {\xi }_{2}(t)={e}^{-\tfrac{\mu }{2}t}{p}_{2}(t),\end{eqnarray} \tag{ 3 }$$

with periodic functions p₁, p₂ and a real or purely imaginary characteristic exponent μ [21]. In the case of a real μ, one of the solutions diverges exponentially. This phenomenon is called parametric resonance and it occurs around excitation frequencies Ω ≈ 2ω₀/n with integer n [21]. Figure 2 shows the real part of the characteristic exponent versus drive frequency and amplitude. For a given small parametric amplitude z₀ ≪ l, the effect is strongest for Ω = 2ω₀. In this case, the functions p₁, p₂ can be approximated by $\cos ({\rm{\Omega }}t/2)$ and $\sin ({\rm{\Omega }}t/2)$, respectively and the exponent μ is approximately μ ≈ 2ω₀ z₀/l [21]. This means an initial excitation that is in phase with $\cos ({\rm{\Omega }}t/2)$ will be exponentially excited while an excitation in phase with $\sin ({\rm{\Omega }}t/2)$ will be damped. The novelty in our approach is the focus on the damped component. After observing the phase of the existing pendulum mode excitation, we control the phase of the parametric drive such that the pendulum oscillation is in phase with the damped component. This situation is depicted in figure 1. Since the phase of the pendulum oscillation changes due to external influences such as seismics, the phase of the parametric drive needs to be adjusted repeatedly during operation to maintain damping. This can be realized using a phase lock loop. To account for friction, one can simply reverse the substitution performed in equation (2) to see that the natural damping rate γ adds to the parametric damping rate μ to form a total damping rate δ = γ + μ.

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Parametric damping is not only able to compensate transient excitations and seismic noise, but also allows for reducing the root mean square amplitude of the motion below the value of a thermalized state and therefore makes cooling of a single mode possible [9, 13]. In view of spectral force measurements, the effect of parametric damping has to be described in the frequency domain. The coupling to a thermal bath can be understood as a random thermal force F_th acting on the oscillator. It has an autocorrelation given by the fluctuation-dissipation-theorem [23, 24] as $\langle {F}_{\mathrm{th}}(t){F}_{\mathrm{th}}(t^{\prime} )\rangle =2m\gamma {k}_{B}T\delta (t-t^{\prime} )$, where m is the oscillator's effective mass, γ is it's natural damping rate, k_B is Boltzmann's constant and T is the temperature. The single-sided power spectral density of the displacement x of a purely thermally excited oscillator is then S_xx (ω) = ∣χ(ω)∣² S_FF,th(ω), where χ is the transfer function of the oscillator and S_FF,th = 4m γ k_B T is the single-sided power spectral density of the thermal force, which is the Fourier-transform of the autocorrelation function $\langle {F}_{\mathrm{th}}(t){F}_{\mathrm{th}}(t^{\prime} )\rangle$. For a harmonic oscillator with the equation of motion $\ddot{x}+\gamma \dot{x}+{\omega }_{0}^{2}x=\tfrac{{F}_{\mathrm{th}}}{m}$ this evaluates to

$$\begin{eqnarray}&&{S}_{{xx},\mathrm{th}}(\omega )=\displaystyle \frac{4{k}_{B}T}{m}\displaystyle \frac{\gamma }{{\left({\omega }_{0}^{2}-{\omega }^{2}\right)}^{2}+{\gamma }^{2}{\omega }^{2}}.\end{eqnarray} \tag{ 4 }$$

Parametric damping by driving at the frequency Ω = 2ω₀ changes the oscillator's displacement spectral density in a way that it modifies the damping rate (γ → γ + μ), yielding

$$\begin{eqnarray}&&{S}_{{xx},\mathrm{th}}(\omega )=\displaystyle \frac{4{k}_{B}T}{m}\displaystyle \frac{\gamma }{{\left({\omega }_{0}^{2}-{\omega }^{2}\right)}^{2}+{\left(\gamma +\mu \right)}^{2}{\omega }^{2}}.\end{eqnarray} \tag{ 5 }$$

The damping rate in the numerator is not modified since it represents the fluctuating thermal force exciting the oscillator, which is not influenced by parametric damping. This aspect is key for the cooling capability of parametric damping as we explain below. Figure 3 displays the impact of parametric damping on the thermal noise power spectral density of the motion of a thermalized pendulum mode. Parametric damping (solid) reduces the on-resonance thermal noise spectral density without increasing the thermal noise above resonance. The latter is relevant in precision experiments on the gravitational field like interferometric gravitational wave detectors and can be reduced by combining parametric damping with pendula of lower natural damping. In contrast, passive damping always raises the natural damping rate (dashed) [25] and comes with an increase in off-resonance thermal noise, the only exception being when the dissipative element used for passive damping is connected to a thermal bath with lower temperature [26].

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Often the residual motion of a mechanical system that is actively damped by linear feedback is not due to thermal noise, but due to noise originating from the feedback-loop sensor or actuator. Sensor noise comes with the sensor signal (error signal), is amplified as well by the (frequency-dependent) complex-valued feedback loop gain, and thus transferred to the mechanical degree of freedom to be damped even if the actual sensor signal is zero. Sensor and actuator noise are issues in linear feedback systems, especially in those for suspension control in gravitational wave detectors [27]. Parametric feedback damping has fundamental advantages. First, the sensor signal and the noise accompanying it are not linearly translated to the actuator output. Instead, it is used to generate the phase locked actuator signal at twice the resonance frequency. Second, the coupling of the actuator movement is proportional to the pendulum's deflection equation (1), which reduces the relevance of actuator noise when the pendulum is at rest. The drawback of parametric feedback damping is a disturbance at precisely twice the frequency of the mechanical oscillator. Fortunately, this disturbance is generally of low magnitude since it is off-resonant and orthogonal to the horizontal, critical degree of freedom. Furthermore, it is rather harmless for the spectrum of the oscillator's motion because it has an arbitrary narrow linewidth, which is limited by the bandwidth of the phase lock loop used to generate the actuator signal. Even if some non-linear effects lead to up-conversion of the oscillating actuation signal into the measurement band, the resulting features will have a similarly narrow linewidth. Since these features will be correlated with the known applied actuation, the effect can even be mitigated in post-processing.

Once a parametrically damped oscillator has reached a stationary state described by the power spectrum in equation (5) we can associate a reduced temperature T_r based on the mean square position fluctuation $\left\langle {x}^{2}\right\rangle$ obtained by integrating its power spectral density. Using the equipartition theorem, ${k}_{B}{T}_{r}=m{\omega }_{0}^{2}\left\langle {x}^{2}\right\rangle$, one finds [9]

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{r}}{T}=\displaystyle \frac{\gamma }{\gamma +\mu }.\end{eqnarray} \tag{ 6 }$$

In other words, parametric damping allows the cooling of an oscillator below the temperature of its thermal environment. To achieve significant temperature reduction relative to the environment, the natural damping rate γ must be small compared to the parametric damping rate μ. In practice however, the minimal reachable excitation of a parametrically damped oscillator will not just be limited by the parametric damping rate but by the measurement of the phase of oscillation, which is required for effective parametric feedback, and additional sources of excitation such as seismic noise.

The usual criterion for an oscillator for being in its quantum ground state is a mean phonon occupancy 〈n〉 below unity. Since 〈n〉ℏ ω₀ = k_B T_r we find the requirement for the parametrically induced damping rate to reach the quantum mechanical regime near the ground state

$$\begin{eqnarray}&&\displaystyle \frac{\mu +\gamma }{\gamma }\gt \displaystyle \frac{{k}_{B}T}{{\hslash }{\omega }_{0}}.\end{eqnarray} \tag{ 7 }$$

With that we find a model system that would fulfill this condition. We could imagine a pendulum with a resonance frequency of ω₀ = 2π · 1 Hz placed in a dilution refrigerator at a temperature of 2 mK [28]. In that case the parametric damping rate needs to surpass the natural damping rate by a factor of about 4e7. If the parametric damping rate corresponds to a Q-factor Q_par = ω₀/μ as low as 100, which is certainly within reach, following the experimental part of this paper, the natural Q-factor Q_nat = ω₀/γ needs to have a value around 4e9. Q-factors in this regime have already been measured on violin modes of pendulum suspensions in gravitational wave detectors [29] and pendulum mode Q-factors around 10⁸ have been demonstrated in fused silica pendulum suspensions [30, 31]. With an optimized choice of materials and suspension geometry, Q-factors in the regime of 10¹⁰ might be realizable for pendulum modes as well [32].

The main layout of the optical setup is shown in figure 4(a). The polished front surface of the pendulum mass reflected a coherent light field of 1550 nm wavelength. The reflected field was analyzed in two ways. First, a small part of the light returning from the test mass was branched off onto a quadrant segmented photodiode (QPD). This allowed the measurement of the beam deflection, which was caused by tilting of the test mass surface in the pendulum mode. The photocurrents of the upper half of the QPD were subtracted from the ones of the lower half. This delivered a signal that was roughly proportional to the pendulum's deflection. The phase of the pendulum mode was continuously reconstructed from the oscillating signal by recording segments of 5 seconds by the QPD in figure 4 and fitting a sine function to each segment. This phase in combination with the previously measured natural pendulum frequency were then used to adjust the parametric actuation feedback phase to achieve damping. The actuation signal continued with this phase until the next segment was recorded and analysed. Before reaching the actuator, the actuation signal was low-pass filtered to prevent sudden jumps in the signal when the phase was updated.

The amplitude was reconstructed using a Michelson interferometer configuration. With this, the phase modulation of the reflected field, caused by the back-and-forth motion of the test mass, was converted to an amplitude modulation. The intensity arriving at the output port was

$$\begin{eqnarray}&&I({x}_{m})=\bar{I}\left(1+V\sin \left({\phi }_{0}+\displaystyle \frac{4\pi {x}_{m}}{\lambda }\right)\right),\end{eqnarray} \tag{ 8 }$$

where $\bar{I}$ is the mean intensity, V < 1 is the interference visibility, ϕ₀ is a phase offset and x_m is the position of the test mass along the beam axis. This nonlinear and ambiguous relationship made it inadequate for phase and frequency reconstruction but gave the ability to measure the oscillation amplitude with more precision than the QPD. Figure 5 shows an example time trace of the two signals recorded by the quadrant photodiode and the interferometer. We utilized an algorithm that counted the number of interference fringes crossed per oscillation to reconstruct the amplitude. Since the pendulum always covered multiple interference fringes during each swing, this method provided sufficiently precise results in our proof-of-principle experiment.

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To measure the achievable parametric damping rate, we parametrically excited the pendulum with the maximal amplitude of our actuator of z₀ = .15 mm. For this, the feedback phase was reversed. Subsequently, the ringdown of the pendulum amplitude was recorded with the parametric damping feedback enabled. Multiple measurements with varying feedback phases were taken until a value optimized for a maximal damping rate was found. For this optimal feedback phase, the measurement was repeated three times. Four measurements with only the natural damping after excitation served as the reference. Example measurements with and without parametric damping are shown in figure 6. Upon reaching an amplitude of approximately 10 μm, the QPD was no longer capable of determining the oscillation phase with sufficient precision and the parametric damping was switched off automatically.

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For each measurement, the overall damping rate was determined by fitting a linear function to the curve of the logarithm of the amplitude as seen in figure 6. The average natural Q-factor Q_nat = ω₀/γ of the pendulum was measured as 2740 ± 40. With parametric damping enabled, the Q-factor dropped to Q_tot = 477 ± 6. This implies that the combined parametric and natural damping rates μ + γ exceeded the natural damping rate γ by a factor of 5.7. If the pendulum had no natural damping at all, the Q-factor would have been ${Q}_{\mathrm{par}}^{\exp }({z}_{0})\approx 580$ enforced by the parametric damping rate alone as the Q-factors add with inverse law, 1/Q_tot = 1/Q_nat + 1/Q_par.