A nonlinear phase shift is introduced to a Mach-Zehnder interferometer (MZI), and we present a scheme for enhancing the phase sensitivity. In our scheme, one input port of a standard MZI is injected with a coherent state and the other input port is injected with one mode of a two-mode squeezed-vacuum state. The final interference output of the MZI is detected with the method of active correlation output readout. Based on the optimal splitting ratio of beam splitters, the phase sensitivity can beat the standard quantum limit and approach the quantum Cram{e}r-Rao bound. The effects of photon loss on phase sensitivity are discussed. Our scheme can also provide some estimates for units of $chi^{(3)}$, due to the relation between the nonlinear phase shift and the susceptibility $chi^{(3)}$ of the Kerr medium.
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