Dynamical Structure Factor from Weak Measurements

TL;DR

This paper introduces a novel method to measure the dynamical structure factor in quantum systems using weak measurements, providing a practical alternative to traditional excitation-response techniques, with numerical validation on the Bose-Hubbard model.

Contribution

It proposes and analytically demonstrates a new weak measurement approach to directly obtain the dynamical structure factor, expanding measurement capabilities in quantum many-body systems.

Findings

Weak measurements can directly recover the dynamical structure factor.

Numerical simulations confirm the method's effectiveness in the Bose-Hubbard model.

The approach is feasible with limited imaging resolution in experiments.

Abstract

Much of our knowledge of quantum systems is encapsulated in the expectation value of Hermitian operators, experimentally obtained by averaging projective measurements. However, dynamical properties are often described by products of operators evaluated at different times; such observables cannot be measured by individual projective measurements, which occur at a single time. For example, the dynamical structure factor describes the propagation of density excitations, such as phonons, and is derived from the spatial density operator evaluated at different times. Conventionally, this is measured by first exciting the system at a specific wavevector and frequency, then measuring the response. Here, we describe an alternative approach using a pair of time-separated weak measurements, and analytically show that their cross-correlation function directly recovers the dynamical structure factor. We provide numerical confirmation of this technique with a matrix product states simulation of the one-dimensional Bose-Hubbard model, weakly measured by phase contrast imaging. We explore the limits of the method and demonstrate its applicability to real experiments with limited imaging resolution.