# Thermal-difference states of light: quantum states of heralded photons

## TL;DR

This paper introduces thermal-difference states (TDS), a new family of non-Gaussian quantum states, and analyzes how heralded photons generated via parametric down-conversion exhibit nonclassical features influenced by brightness and loss.

## Contribution

The paper defines TDS as a new family of non-Gaussian states and links them to heralded photons, providing a framework to analyze nonclassicality under loss and brightness variations.

## Key findings

- Heralded photon states are a subset of TDS with nonclassical properties.
- Nonclassicality measures depend on brightness and transmittance.
- TDS serve as benchmarks for analyzing single-mode bosonic states.

## Abstract

We introduce the thermal-difference states (TDS), a three-parameter family of single-mode non-Gaussian bosonic states whose density operator is a weighted difference of two thermal states. We show that the states of "heralded photons" generated via parametric down-conversion (PDC) are precisely those among the TDS that are nonclassical, meaning they have a negative $P$-function. The three parameters correspond in that context to the initial brightness of PDC and the transmittances, characterizing the linear loss in the signal and the idler channels. At low initial brightness and unit transmittances, the heralded photon state is known to be a single-photon state. We explore the influence of brightness and linear loss on the heralded state of the signal mode. In particular, we analyze the influence of the initial brightness and the loss on the state nonclassicality by computing several measures of nonclassicality, such as the negative volume of the Wigner function, the sum of quantum Fisher information for two quadratures, and the ordering sensitivity, introduced recently by us [Phys. Rev. Lett. 122, 080402 (2019)]. We argue finally that the TDS provide new benchmark states for the analysis of a variety of properties of single-mode bosonic states.