"dampen the" is a correct and usable phrase in written English.
It is usually used in the context of dampening an emotion or enthusiasm. For example, "His words had a dampening effect on the enthusiasm of the crowd."
Exact(59)
It didn't dampen the mood.
That discrepancy did not dampen the enthusiasm.
That helped dampen the criticism a bit.
Clean it thoroughly and dampen the area.
That did not dampen the mood, however.
Trump did little to dampen the speculation.
Washington has ample reason to try to dampen the excitement.
Fortunately, that bath did little more than dampen the wood.
The accusations and concerns, however, did not dampen the mood.
"But that would do nothing to dampen the competition.
They flatten the space and dampen the imagination.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com