Noninvasive Measurement of Dynamic Correlation Functions

The measurement of dynamic correlation functions of quantum systems is complicated by measurement backaction. To facilitate such measurements we introduce a protocol based on weak ancilla--system couplings. Different choices of the coupling operator give access to the real and imaginary parts of the dynamic correlation function. This protocol reduces disturbances due to the early time measurements to a minimum, and we quantify the deviation of the measured correlation functions from the theoretical, unitarily-evolved ones. Implementations of the protocol in trapped ions and other experimental platforms are discussed. For spin-$1/2$ models and single-site observables we prove that measurement backaction can be avoided altogether, allowing for the use of ancilla-free protocols.