Kramers-Kronig detection in the quantum regime

TL;DR

This paper extends Kramers-Kronig detection from classical optics to the quantum regime, demonstrating its ability to measure quantum states' quadratures and proposing a spectral tomography protocol for single-photon states.

Contribution

It shows that Kramers-Kronig detection can be used as a coherent quantum measurement up to first order in the local oscillator amplitude, enabling quantum state reconstruction.

Findings

Kramers-Kronig detection acts as a Gaussian measurement in the quantum regime.

It can reconstruct quadratures of quantum states such as coherent, pure, and mixed states.

A spectral tomography protocol for single-photon states is proposed based on this detection method.

Abstract

We investigate the quantization of Kramers-Kronig detection technique initially developped for classical optical communications. It consists in mixing the unknown field with a strong monochromatic local oscillator on an unbalanced beamsplitter. A single output of the beamsplitter undergoes a direct detection of the optical intensity by means of a single photodiode. When the measured output verifies signal processing constraints, namely, the minimal phase and the single sideband constraints, Kramers-Kronig detection reconstructs the phase of the signal from the intensity measurements via a digitally computed Hilbert transform. The local oscillator being known, Kramers-Kronig detection allows for reconstructing the quadratures of the unknown field. We show that this result holds in the quantum regime up to first order in the local oscillator amplitude and thus that Kramers-Kronig detection acts as a coherent detection able to measure both quadratures, making it a Gaussian measurement similar to double homodyne detection. We also study in details the phase information measured by Kramers-Kronig detection for bosonic coherent states, monomode pure states and mixed states. Finally, we propose and investigate a spectral tomography protocol for single-photon states that is inspired by Kramers-Kronig detection and relies on a spectral engineering of the single-photon.