Experimental realization of phonon shift operation in a trapped ion system

We report an experimental realization of phonon shift operation in a trapped ion system. The shift operation is a pure addition or subtraction without the modification of a state amplitude by $\sqrt{n}$, which transfer the state from $|n>$ to $|n+1>$ or from $|n> to |n-1>$ for any n. We implement the pure addition by applying $\pi$ pulse of a blue-sideband transition $|\downarrow,n> -> |\uparrow,n+1>$ followed by a $\pi$ pulse of resonant carrier transition of spin $|\uparrow,n+1> -> |\downarrow, n+1>$. For subtraction we exchange the order. The essence is in an adiabatic blue-sideband $\pi$ operation with high-fidelity and speed regardless of phonon number n (0 to 10) through the same operation in Ref. [1]. Although the pure shift operations are different from creation $\hat{a}^{\dagger}$ and annihilation $\hat{a}$, it produces a non-classical state of phonon [2]. We observe a negative probability in the Wigner function of a phonon state after the pure shift operation on various phonon states. The scheme can be used for the projective measurement of phonon number states and the test of boson sampling problem [3].\\[4pt] [1] Junhua, et al., PRA 89, 013608 (2014).\\[0pt] [2] Daniel K, et al., PRL 110, 210504 (2013).\\[0pt] [3] Shen, et al., arxiv:1310.4860