  Measuring heat flow through nanoscale systems poses formidable practical
difficulties as there is no `ampere meter' for heat. We propose to overcome
this problem by realizing heat transport through a chain of trapped ions. Laser
cooling the chain edges to different temperatures induces a current of local
vibrations (vibrons). We show how to efficiently control and measure this
current, including fluctuations, by coupling vibrons to internal ion states.
This demonstrates that ion crystals provide a suitable platform for studying
quantum transport, e.g., through thermal analogues of quantum wires and quantum
dots. Notably, ion crystals may give access to measurements of the elusive
large fluctuations of bosonic currents and the onset of Fourier's law. These
results are supported by numerical simulations for a realistic implementation
with specific ions and system parameters.
