How do we know that light travels slower in denser medium? What experiment can we do to verify it? How do we know that light travels slower in denser medium? What experiment can we do to verify it?
How did Fermat knew that light travels with different speed in different medium?
 A: The usual method of making precision measurements of the index of refraction of gasses is to put a tube on one leg of a two-leg interferometer (Michelson, Mach-Zender or similar), and rig it so that you can control the gas pressure inside the tube all the way down to vacuum. Then you observe the fringe shifts as you run the pressure up from (or down to) vacuum.
Now we have to make the connection between the observations (of fringe shifts as pressure change) to the requested basis (speed as a function of density).
For the density issue, we rely on the fact that dilute gasses are roughly ideal which means that pressure and density are in approximately linear relationship (a very good approximation until the pressure gets to be significant).
For the relationship between fringe shifts and speed we've a bit of a chain to follow.


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*Interferometers are sensitive to phase differences so fringe shifts mean the phase difference is changing, but the only thing that changes is the gas density in the tube so fringe shifts mean a change in the phase incurred on that leg.

*Phase differences can come from changes in the source phase, changes in frequency, changes in wavelength, or changes in distance. But the instrument uses the same source for each leg, the boundary conditions require that the frequency is the same inside the tube as outside, and the instrument has a fixed geometry, so we can rule out all these possibilities except changes in wavelength inside the tube.

*A general rule for waves holds that we can find the speed as $v = \lambda f$, so a change in the wavelength means a change in the speed.
The overall results is that this thing measures the way the speed of light changes as a function of the gas density.
