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

Spatial correlation functions for the collective degrees of freedom of many trapped ions