Attempting to freeze matter under extreme pressure?
I tried finding some research and found lots about freezing matter or putting it under extreme pressure, but not trying both.
My thought experiment involved taking a small portion of ideal of matter (not sure what), artificially applying extreme pressure to it and than attempt to lower its temperature and if possible, apply even more pressure before trying to lower its temperature - taking it as low as possible under the highest pressure you could.
I assumed there's likely to be a conflict between pressure - thus increasing vibration/wave properties of the material - and how it would be possible to reduce those energetic wave properties.
Most experimental research in matter under extreme pressures is concerned with recreating conditions within the interiors of planets and stars (the latter falls under the field of high energy density physics). The temperatures involved therefore tend to be very high. However, there's no inherent conflict between high pressures and low temperatures, it's just that temperature tends to increase when you compress something. Compress an ideal gas, for example, and it will heat up. Let it sit in its compressed state for a while though, and it will cool back down despite remaining under high pressure.
This is true for solids and liquids too (putting any phase transitions aside), though they are much less compressible. The core of the Earth will eventually cool too, though it's currently kept at high temperature by the radioactive decay of heavy elements. Diamond anvil cells, however, can reach pressures exceeding those at the center of the earth in a laboratory setting, and some DACs can even be cooled to cryogenic temperatures. This figure on Wikipedia suggests cryo-DACs can be used to reach pressures up to 350 GPa at cryogenic temperatures. As an example, a quick search turns up a paper (arxiv version) that makes use of a DAC to study media at liquid nitrogen temperatures and pressures up to 10 GPa (~3% the pressure at the center of the Earth). Search around and I'm sure you can find others.
You might be interested in supercritical fluids, which are fluids at high enough pressures and low enough temperatures (but not high enough pressures or low enough temperatures to solidify) that they act as both a liquid and a gas: https://en.wikipedia.org/wiki/Supercritical_fluid
I was about to correct you with "superfluid" but I remembered that that's a completely different thing (low temp and low pressure), but also utterly fascinating.
For anyone who's never heard of it, here's part 1 of a series on the liquid Helium superfluid. It's pretty old, but still great stuff
You're thinking of gases. In condensed phases energy is just stored as a potential energy coming from the fact that molecules are too close to each other (going left of that minimum https://en.wikipedia.org/wiki/Lennard-Jones_potential). You don't have gases under such conditions, they condense