I placed back the magnetic translation stage and measure again the noise of the small sapphire sample. 

I took the measurement in two cases: with and without the small sapphire sample.

Acquisition time: 10 minutes, Rate 100ms, demodulation with the lockin internal oscillator frequency 420Hz.

Plot1 shows the 2D plot of the AC signal from the lockin after demodulation, divided by the DC.

Plot2 is a zooming of the first plot, the circles are centered on the mean of the signal over all the acquisition time and have a radius equal to the standard deviation.

It looks the system is noisier without the sample, this makes me think that vibrations of the sample don't make a lot of difference. I have the idea that most of the noise comes from the dust, and it depends on how we move the air during replacing of the sample or working inside the box.