  Spatial correlation functions provide a glimpse into the quantum correlations
within a quantum system. Ions in a linear trap collectively form a nonuniform,
discretized background on which a scalar field of phonons propagates. Trapped
ions have the experimental advantage of each having their own "built-in"
motional detector: electronic states that can be coupled, via an external
laser, to the ion's vibrational motion. The post-interaction electronic state
can be read out with high efficiency, giving a stochastic measurement record
whose classical correlations reflect the quantum correlations of the ions'
collective vibrational state. Here we calculate this general result, then we
discuss the long detection-time limit and specialize to Gaussian states, and
finally we compare the results for thermal versus squeezed states.
