\section{Noise Isolation Mechanism} \label{sec:response}

To study the noise isolation mechanism of the friction dampers, we need to measure the cavity noise in a controlled fashion. 
We achieve this by placing the cavity apparatus on a vibrating platform, which is driven along the cavity axis at a known voltage and frequency using a piezo motor (Piezosystem Jena GMBH Translation System Tritor 101 CAP).
The noise isolation performance is deduced from the gain ratio $\xi=x/a$ between the cavity and the platform vibration amplitudes, $x$ and $a$ respectively, across different driving frequencies $f$. 
We measure $x$ with the PDH technique described in Section \ref{sec:apparatus}, but using a cavity with a lower finesse to observe a larger range of the movement (up to $\sim$5~nm). 
On the other hand, $a$ is determined from the movement of the fixed mirror (mirror attached to the cavity mount in Figure \ref{fig:attocube cavity}) with respect to the optical table. 
The mirror movement is measured via a Michelson interferometer formed with another mirror fixed to the optical table. 
We placed the friction dampers with mass around 1~g either to the top corner site (one damper case) or to all three sites (3 dampers case) and compare the gain ratios with the case without any friction dampers. 
As the cavity and platform vibration amplitudes span around 5 orders of magnitude, we drive the piezo motor with different amplitudes and reconstruct the gain ratio correspondingly. 

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{figures/fig_isolation/fig_isolation.pdf}
    \caption{Frequency-dependent gain ratio $\xi$ between the cavity and the platform vibration amplitudes for three different damper configurations: no damper (top black line), one damper (middle red line), and three dampers (bottom blue line). The shaded regions represent an estimate of the standard deviation over repeated measurements.}
    \label{fig:isolation}
\end{figure}

Figure~\ref{fig:isolation} shows the vibration gain ratio $\xi$ of the cavity system with different driving frequencies. 
Without any damper, there is a peak of $\xi\approx100$ at 320 Hz, which we attribute to the `tuning-fork' mechanical resonance of the cavity structure. 
The dampers restrict this `tuning-fork' motion and reduce the gain ratio on this resonance by a factor of $\sim$100. 
Furthermore, the dampers also provide further noise isolation for frequencies lower than $\sim$400~Hz. 
The three-damper configuration also provides additional noise isolation at $\sim$600~Hz, compared with the one-damper configuration, as they restrict more modes due to the inclusion of the damper at the bottom corner.
For higher frequencies, the dampers no longer provide any noise isolation, which may be due to the structure resonating at higher harmonics.

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{figures/fig_stickslip/fig_stickslip.pdf}
    \caption{Stick to slip transition. 
    By driving the the cavity structure near its mechanical resonance (330 Hz) with increasing platform vibration, we observe a jump in the vibration amplitudes of the attocube-side mirror (top) and in their relative phases (bottom). The solid lines and the shaded regions represent the medians and the ranges of values observed over seven repeated measurements, with increasing damper masses (from left to right). 
    The measurement of the vibration amplitudes is limited to the linear region of the interferometric intensity, around $\lambda/4\pi \approx 60~\mathrm{nm}$.}
    \label{fig:stickslip}
\end{figure}

Having demonstrated the noise isolation obtained from adding the friction dampers, we show that the damper friction force acts on the cavity structure in the stick regime. 
We place a friction damper with masses ranging from 0.03~g to 0.17~g on the top corner site, and monitor the vibration amplitudes of both mirrors, representing the cantilever and platform vibrations, via two Michelson interferometers.
Figure~\ref{fig:stickslip} shows that, near the `tuning-fork' mechanical resonance, the cantilever vibration amplitude increases with higher platform vibration, until around 15 to 20~nm where it abruptly jumps to significantly higher values. 
We identify this as the stick-to-slip transition of the friction regime. As the noise isolation mechanism described in previous sections operates at much lower vibration amplitudes ($\lesssim 1~\mathrm{nm}$), the dampers stick on their contact points and reduce motional modes along the dampers, similar to a caged system, when operating at small deviations from the setpoint. 
Furthermore, the slipping mechanism of the friction dampers allow large movements necessary for the alignment of the mirrors. 

\begin{figure}
    \centering
    \includegraphics{figures/fig_damper_ratio/fig_damper_ratio.eps}
    \caption{The effect of putting friction dampers of various masses on the DC gain of the translation stage (blue square) and the total vibration noise (red circle). The damping effect ratio quantifies the ratio of the respective measurements results with the case without any friction dampers.
    }
    \label{fig:damper_ratio}
\end{figure}

Figure~\ref{fig:stickslip} also indicates that the effect of damping increases with larger damper mass, which we suspect to be due to a combination of following two factors. 
First, dampers with higher mass have higher cross-section areas, increasing its stiffness.
In the stick regime, the movement between the two contact points can be effectively reduced with higher damper stiffness.
Second, the effective areas of contact between the damper and the mounts increase with larger normal force from heavier damper mass~\cite{archard1957elastic}. 

It is difficult to model the interplay between these factors with a relatively complex structure that includes the flexural translation stages in attocube and the mirror mounted to it with the cantilever.
Instead, we examine experimentally the effects of using different damper masses, placed on the top corner site, on two relevant parameters -- DC gain of the translation stage and total vibration noise (see Figure~\ref{fig:damper_ratio}). 
To measure the DC gain of the translation stage, we drive the translation stage in the direction of the cavity axis in the low frequency regime ($\sim$10~Hz), well below any mechanical resonances. 
Next, we measure the corresponding cavity vibration amplitude via the PDH technique. 
We observe a reduction of the DC gain after placing the friction dampers, which is expected as the damper restricts some movement between the cavity mounts in the stick friction regime.
In addition, we also quantify the total vibration noise correspondingly with the technique described in the previous section.
We observe a significant reduction in the total noise, even just by placing one 0.05~g friction damper on one corner site. This damping effect seems to saturate at around 1~g of damper mass, with an enhanced noise reduction of around $70\%$, or equivalently around $90\%$ reduction of the noise power when we use heavier dampers. 
