\section{Apparatus and Characterisation of Noise} \label{sec:apparatus}
The system of interest is a symmetric Fabry-Perot optical cavity with an adjustable mirror alignment enabled by a flexural stage piezo scanner (attocube ANSxyz100).
This piezo scanner, which we refer to as attocube henceforth, can move in all three spatial directions, hence allowing us to adjust the mirror alignments with great flexibility. We incorporate it into the cavity by mounting a mirror onto a cantilever, which is subsequently anchored to the piezo scanner via screws. 
The second mirror is secured to a larger mount, which has a base to house the piezo scanner. 
This compact arrangement has a cross-section of 24.5 mm by 24 mm, and spans a length of 55 mm. 
This structure serves several purposes such as (a) bringing the concentric cavity towards its concentric point, (b) combating misalignment of the mirrors that occur from mechanical resonances and thermal drifts, (c) allowing sufficient optical access in atomic experiments.


\begin{figure}
    \includegraphics[width=\columnwidth]{figures/attocube_cavity_new.eps}
    \caption{3D drawing (SolidWorks) of the assembled cavity (left) and exploded view (right). One mirror is connected to the piezo scanner (refered to as attocube) while the other is mounted to a rigid base. The cavity alignment can be adjusted by using attocube to move one mirror relative to the other in all three dimensions. The overall system is compact, having a cross-section of 24.5 mm by 24 mm, and spanning 55 mm.}
    \label{fig:attocube cavity}
\end{figure}

\subsection{Measuring the noise}
The mechanical noise in a cavity is complicated to model since it depends on the mechanical structure of the cavity and its coupling to the environment. 
We take the heuristic approach that the net effect of the noise is equivalent to an overall change in the longitudinal alignment (along the optical axis) of the cavity mirrors. 
Whenever there is a perturbation away from the resonant cavity length, the laser is effectively detuned from the cavity, lowering the transmission of the cavity mode.
By considering the effects of length changes about the resonance condition, the detuning is given by
 \begin{equation}
    \frac{\delta\omega}{\kappa}=-\frac{2\mathcal{F}}{\lambda/2}\delta L, \label{eqn: frequency detuning from length fluctuations}
\end{equation}
where $\delta \omega$ and $\delta L$ are the deviations from the resonant values of angular frequency (s$^{-1}$) and cavity length respectively, $\mathcal{F}$ is the finesse of the cavity, $\kappa$ is the half-width-at-half-max (HWHM) of the cavity transmission in units of angular frequency, and $\lambda$ is the wavelength of the cavity mode. 
Eqn \eqref{eqn: frequency detuning from length fluctuations} demonstrates that the sensitivity of the cavity to mechanical noise is proportional to the finesse.
Assuming a rms noise $\sim1$ nm results in $\delta \omega/\kappa\sim10^{-3}\mathcal{F}$. 
The noise is easily noticeable at values of finesse as low as in the hundreds.

We design a test cavity with a similar length as the concentric cavity in \cite{nguyen2018operating}, except the test cavity has a planar geometry and its finesse is in the thousands. 
The higher finesse is useful in increasing the sensitivity of the cavity to the mechanical noise according to Eqn \eqref{eqn: frequency detuning from length fluctuations} so that the noise present and any subsequent improvement of the passive stability are more readily observed. 
As a consequence, the noise in the cavity typically exceeds one HWHM, leading to an obvious reduction in the cavity transmission. 

The method we use to measure the noise of the cavity is the Pound-Drever-Hall (PDH) technique \cite{eric2001pdh}. 
A frequency-stabilised laser (780 nm) is phase-modulated with a homemade electro-optic modulator (EOM) at a frequency of 20.1 MHz. 
We use a control loop with only a low integral gain to lock the laser to the cavity via the PDH error signal. 
This weak lock keeps the laser resonant with the cavity in the long term, but does not feedback the laser on short timescales. 
As a result, any detuning between the cavity and the laser that arises due to the mechanical noise remains uncorrected, allowing us to measure the noise by monitoring the error signal. 

Typically, the control loop is operated in the linear region of the error signal, where it is one-to-one (injective), hence one value of detuning corresponds to only one value of the error signal. 
Consequently, we can simply invert the error signal to find the detuning.
Monitoring the time trace of the error signal essentially informs us of the noise in the cavity. 
In our case, the noise of the cavity causes the detuning to exceed one $\kappa$, which causes the error signal to go beyond the linear region. 
As a result, the error signal is no longer injective and we cannot infer the detuning solely from the error signal. 
On the other hand, the transmission of the cavity can be taken as two-to-one in the sense that each value of transmission gives a unique value of detuning, up to a sign ambiguity. 
We resolve the ambiguity by corroborating with the sign of the error signal, which is correct even beyond the linear region. 


\subsection{Sources of noise}
The noise of our system mainly comes from the inherent mechanical vibration of attocube, which we have measured using the PDH method mentioned above to be $\sim$2~nm. The noise spectrum of this mechanical vibration displays several resonances in the 100-1000 Hz band (black line in Figure \ref{fig:isolation}). As we show in subsequent sections, the noise from this band contributes significantly to the fluctuation in cavity length. Once they are suppressed, the passive stability of the cavity is greatly improved. We have also observed environmental noise coupling into the cavity through the structure housing the set-up. The contribution from the environmental noise is typically an order of magnitude lower.

    
    
