Experimental measurement of correlation functions in trapped ions*

We measure the time-correlation functions of spins and phonons of a system that evolves under a Hamiltonian of Jaynes-Cumming model or Dirac equation with trapped 171Yb$+$ ions. The algorithm proposed in Ref. [1] requires only one ancilla qubit to obtain the time correlations of the observables in the system. In the experiment, conditional gates depending on the state of the ancilla, spin-dependent force and spin-independent force have been performed. We measure the spin-spin time-correlations in spin system and the spin-phonon time-correlation in bosonic field. According to the linear response theory, the time-correlations can be used to characterize relevant physical magnitude such as magnetic susceptibility. This scheme can be extended to a system including n-time spins and phonons correlations. [1] J. S. Pedernales, et al., Phys. Rev. Lett. 113, 020505 (2014).