Relative phase and dynamical phase sensing in a Hamiltonian model of the optical SU(1,1) interferometer

TL;DR

This paper reformulates the SU(1,1) interferometer using a Hamiltonian approach, revealing optimal sensing points, quantum Fisher information scaling, and measurement strategies for phase estimation in optical quantum sensors.

Contribution

It introduces a Hamiltonian model of the SU(1,1) interferometer, identifying optimal operating points and measurement strategies for phase sensing, advancing the understanding of quantum optical sensors.

Findings

Optimal phase sensing at specific points ($\,\phi=\pi\,$ and $\theta=0$).

Quantum Fisher information exhibits Heisenberg and logarithmically modified Heisenberg scaling.

Measurement of total photon number is suboptimal; weighted shift operators are optimal at high nonlinearity.

Abstract

The SU(1,1) interferometer introduced by Yurke, McCall, Klauder is reformulated starting from the Hamiltonian of two identical optical downconversion processes with opposite pump phases. From the four optical modes, two are singled out up to a relative phase by the assumption of exact alignment of the interferometer (i.e., mode indistinguishability). The state of the two resulting modes is parametrized by the nonlinearity $g$, the relative phase $\phi$, and a dynamical phase $\theta$ resulting from the interaction time. The optimal operating point for sensing the relative phase (dynamical phase) is found to be $\phi = \pi$ ($\theta=0$) with quantum Fisher information exhibiting Heisenberg scaling $E^{2}$ (logarithmically modified Heisenberg scaling $\left({E\over \ln E}\right)^{2}$). Compared to the predictions of the circuit-based model, we find in that in the Hamiltonian model: 1. the optimal operating points occur for a non-vacuum state inside the interferometer, and 2. measurement of the total photon number operator does not provide an estimate of the relative or dynamical phase with precision that saturates the quantum Cramer-Rao bound, whereas an observable based on weighted shift operators becomes optimal as $g$ increases. The results indicate a first-principles approach for describing general optical quantum sensors containing multiple optical downconversion processes.