Sp(2N, R) interferometry in multi-mode Gaussian bosonic systems for optimal metrology and quantum control

Abstract

Multi-mode interferometers for bosons in Gaussian states are important systems for quantum metrology with precision beyond the standard quantum limit and for bosonic quantum computing. However, there is a lack of theoretical foundation for generic $N$-mode Gaussian interferometry. In this work, we study quantum metrology and quantum control in multi-mode bosonic systems with quadratic Hamiltonians, exploiting the fundamental Sp$(2N,R)$ symmetry of such interferometers. We show that the optimal quantum control to maximize sensitivity requires aligning squeezing and displacement in the same direction. We propose Sp$(2N,R)$ echo, a multi-mode generalization of the SU$(1,1)$ interferometry, to achieve the sensitivity of phase estimation set by the quantum Fisher information. In addition, we introduce a geometrical means for reversing many-body dynamics with Sp$(2N,R)$ dynamical symmetry, such as dynamics of the bosonic Kitaev chain. Our schemes are readily realizable in optical, atomic, and mechanical platforms.