Measuring microwave quantum states: tomogram and moments

TL;DR

This paper discusses methods to measure and analyze microwave quantum states using tomograms and moments, providing relations, formalism, and evolution equations relevant for experiments involving harmonic oscillators.

Contribution

It introduces a formalism linking tomograms and moments, and derives linear differential equations for their time evolution in microwave quantum states.

Findings

Relations between tomogram and moments are established.

Time evolution equations for moments are derived.

Applications to harmonic and damped oscillators are demonstrated.

Abstract

Two measurable characteristics of microwave one-mode photon states are discussed: a rotated quadrature distribution (tomogram) and normally/antinormally ordered moments of photon creation and annihilation operators. Extraction of these characteristics from amplified microwave signal is presented. Relations between the tomogram and the moments are found and can be used as a cross check of experiments. Formalism of the ordered moments is developed. The state purity and generalized uncertainty relations are considered in terms of moments. Unitary and non-unitary time evolution of moments is obtained in the form of linear differential equations in contrast to partial differential equations for quasidistributions. Time evolution is specified for the cases of a harmonic oscillator and a damped harmonic oscillator, which describe noiseless and decoherence processes, respectively.