Tomography in Loop Quantum Cosmology

TL;DR

This paper develops a tomographic representation for the quantum cosmological model within Loop Quantum Cosmology, enabling the characterization of quantum states through measurable probability distributions.

Contribution

It introduces a novel tomographic approach to analyze quantum states in Loop Quantum Cosmology, linking Wigner functions to symplectic tomograms for the first time in this context.

Findings

Successfully derived symplectic tomograms from Wigner functions.

Interpreted quantum properties using quadrature operators and uncertainty principles.

Calculated expectation values of the volume operator consistent with existing literature.

Abstract

We analyze the tomographic representation for the Friedmann-Robertson-Walker (FRW) model within the Loop Quantum Cosmology framework. We focus on the Wigner quasi-probability distributions associated with Gaussian and Schr\"odinger cat states, and then, by applying a Radon integral transform for those Wigner functions, we are able to obtain the symplectic tomograms which define measurable probability distributions that fully characterize the quantum model of our interest. By appropriately introducing the quantum dispersion for a rotated and squeezed quadrature operator in terms of the position and momentum, we efficiently interpret the properties of such tomograms, being consequent with Heisenberg's uncertainty principle. We also obtain, by means of the dual tomographic symbols, the expectation value for the volume operator, which coincides with the values reported in the literature. We expect that our findings result interesting as the introduced tomographic representation may be further benefited from the well-developed measure techniques in the areas of Quantum optics and Quantum information theory.