Optimal and near-optimal probe states for quantum metrology of number conserving two-mode bosonic Hamiltonians

TL;DR

This paper identifies optimal and near-optimal quantum probe states for estimating parameters in two-mode bosonic Hamiltonians, revealing connections to NOON states and SU(2) coherent states, and proposing variational states for high-precision measurements.

Contribution

It introduces new classes of variational superposition probe states that nearly saturate quantum Cramér-Rao bounds for various parameters in two-mode bosonic systems.

Findings

Optimal states for dephasing and interaction estimation relate to NOON states.

Superpositions of antipodal SU(2) coherent states optimize tunneling amplitude estimation.

Ground states near a quantum phase transition can serve as high-performance probes.

Abstract

We derive families of optimal and near-optimal probe states for quantum estimation of the coupling constants of a general two-mode number-conserving bosonic Hamiltonian describing one-body and two-body dynamics. We find that the optimal states for estimating the dephasing of the modes, the self-interaction strength, and the contact interaction strength are related to the NOON states, whereas the optimal states for estimation of the intermode single particle tunneling amplitude are superpositions of antipodal SU(2) coherent states. For estimation of the amplitude of pair tunneling and the amplitude of density-dependent single particle tunneling processes, respectively, we introduce classes of variational superposition probe states that provide near perfect saturation of the corresponding quantum Cram\'{e}r-Rao bounds. We show that the ground state of the pair tunneling term in the Hamiltonian has a high fidelity with the optimal states for estimation of a single particle tunneling amplitude, suggesting that high-performance probes for tunneling amplitude estimation may be produced by tuning the two-mode system through a quantum phase transition.