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 δθ, which is characterized by two scaling behaviors, shot-noise limited with δθ∼1/√N and Heisenberg limited with δθ∼1/N (N the total particle number). According to Bayesian analysis, Heisenberg limited preciosion for θ=0 can be achieved in a Mach-Zehnder interferometer with (|N-1,N+1>+|N+1,N-1>)/√2 as input state based and a single measurement or |N,N> input based on multiple measurements. As θ 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 θ at Heisenberg limited precision, but requires a quantum computer. To extend the range of precisely measurable θ 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 θ. This scheme can be implemented in a bimodal Bose-Einstein condensate (BEC) system with tunable scattering length. Numerical calculations show: i) at θ=0, δθ∼1/N; and ii) as θ moves towards ±π/2, the precision crosses over smoothly to δθ∼1/√N, providing a wide range over which the precision is nearly Heisenberg limited.