# Time-dependent quantum correlations in phase space

## TL;DR

This paper introduces a regularized, smooth quasi-probability distribution in phase space to visualize and analyze time-dependent quantum correlations of light, applicable even for highly singular states, and demonstrates its use in characterizing optical parametric processes.

## Contribution

It develops a regularized, multi-time P functional for quantum correlations, enabling experimental access and analysis of complex quantum states in phase space.

## Key findings

- Successfully characterizes optical parametric processes with frequency mismatch
- Provides a measurement scheme for direct experimental access to the P functional
- Extends the applicability of phase-space methods to highly singular quantum states

## Abstract

General quasi-probabilities are introduced to visualize time-dependent quantum correlations of light in phase space. They are based on the generalization of the Glauber-Sudarshan P function to a time-dependent P functional [W. Vogel, Phys. Rev. Lett. 100, 013605 (2008)], which fully describes temporal correlations of radiation fields on the basis of continuous phase-space distributions. This approach is nontrivial as the P functional itself is highly singular for many quantum states and nonlinear processes. In general, it neither yields a well-behaved nor an experimentally accessible description of quantum stochastic processes. Our regularized version of this multi-time-dependent quasi-probability is a smooth function and applies to stronger divergences compared to the single-time and multi-mode scenario. The technique is used to characterize an optical parametric process with frequency mismatch. A measurement scheme, together with a sampling approach, is provided which yields direct experimental access to the regularized P functional from measured data.