  Sympathetic cooling of trapped ions through collisions with neutral buffer
gases is critical to a variety of modern scientific fields, including
fundamental chemistry, mass spectrometry, nuclear and particle physics, and
atomic and molecular physics. Despite its widespread use over four decades,
there remain open questions regarding its fundamental limitations. To probe
these limits, here we examine the steady-state evolution of up to ten barium
ions immersed in a gas of three-million laser-cooled calcium atoms. We observe
and explain the emergence of nonequilibrium behavior as evidenced by
bifurcations in the ion steady-state temperature, parameterized by ion number.
We show that this behavior leads to limitations in creating and maintaining
translationally cold samples of trapped ions using neutral-gas sympathetic
cooling. These results may provide a route to studying nonequilibrium
thermodynamics.
