Phase-space open-systems dynamics of second-order nonlinear interactions with pulsed quantum light

TL;DR

This paper develops an efficient phase-space framework for modeling second-order nonlinear interactions of broadband quantum pulses, enabling analysis of multimode quantum state transformations relevant for ultrafast quantum optics.

Contribution

It introduces the generalized Bloch-Messiah decomposition (GBMD) to simplify the description of open multimode quantum systems undergoing nonlinear interactions.

Findings

Derived a method to compute output Wigner functions via Gaussian convolution.

Analyzed the effects of entanglement breakage on the output quantum states.

Demonstrated the framework with examples involving THz and optical frequency regimes.

Abstract

The theoretical description of broadband, multimode quantum pulses undergoing a second-order $\chi^{(2)}$-nonlinear interaction can be quite intricate, due to the large dimensionality of the underlying phase space. However, in many cases only a few broadband (temporal) modes are relevant before and after the nonlinear interaction. Here we present an efficient framework to calculate the relation between the quantum states at the input and output of a nonlinear element in their respective relevant modes. Since the number of relevant input and output modes may differ, resulting in an open quantum system, we introduce the generalized Bloch-Messiah decomposition (GBMD), reducing the description to an equal number of input and output modes. The GBMD enables us to calculate the multimode Wigner function of the output state by convolving the rescaled Wigner function of the reduced input quantum pulse with a multivariate Gaussian phase-space function. We expand on this result by considering two examples input states: A Fock state in a single broadband mode and a two-mode squeezed vacuum, both in the THz-frequency regime, up-converted to a single output broadband mode of optical frequencies. We investigate the effect, the convolution and thermalization due to entanglement breakage have on the output Wigner function by calculating the von Neumann entropy of the output Wigner function. The methods presented here can be used to optimize the amplification or frequency conversion of broadband quantum states, opening an avenue to the generation and characterization of optical quantum states on ultrafast time scales.