Quantum metrology for nonlinear phase shifts with entangled coherent states

We investigate the phase enhancement of quantum states subject to nonlinear phase shifts. The optimal phase estimation of even entangled coherent states (ECSs) is shown to be better than that of NOON states with the same average particle number <n> and nonlinearity exponent k. We investigate the creation of an approximate entangled coherent state (AECS) from a photon-subtracted squeezed vacuum with current optical technology methods and show that a pure AECS is even better than an even ECS for large <n>. Finally, we examine the simple, but physically relevant, cases of loss in the nonlinear interferometer for a fixed average photon number <n>.