Measuring an unknown phase with quantum-limited precision using nonlinear beamsplitters

High precision phase measurement is currently a central goal of quantum interferometry. In general, the precision is described by the phase estimation uncertainty $\Delta\theta$, which is characterized by two scaling behaviors, shot-noise limited with $\Delta\theta\sim 1/\sqrt{N}$ and Heisenberg limited with $\Delta\theta\sim 1/N$ (N the total particle number). According to Bayesian analysis, Heisenberg limited preciosion for $\theta=0$ can be achieved in a Mach-Zehnder interferometer with $(|N-1,N+1\rangle+|N+1,N-1\rangle)/\sqrt{2}$ as input state based and a single measurement or $|N,N\rangle$ input based on multiple measurements. As $\theta$ deviates from zero, both schemes degrade rapidly to worse than shot-noise-limited precision. In contrast, a Quantum Fourier Transform (QFT) based interferometer can measure an arbitrary $\theta$ at Heisenberg limited precision, but requires a quantum computer. To extend the range of precisely measurable $\theta$ without a quantum computer, we propose using nonlinear beam-spitters. We find that this can achieve nearly Heisenberg-limited precision over a wide range of $\theta$. This scheme can be implemented in a bimodal Bose-Einstein condensate (BEC) system with tunable scattering length. Numerical calculations show: i) at $\theta=0$, $\Delta\theta\sim 1/N$; and ii) as $\theta$ moves towards $\pm \pi/2$, the precision crosses over smoothly to $\Delta\theta\sim 1/\sqrt{N}$, providing a wide range over which the precision is nearly Heisenberg limited.