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arXiv:2606.15101

Optimizing Wigner Negativity in Scattering Processes Using Energetic Cost Functions

Kian Hwee Lim, Kiarn T. Laverick, Sahil S. Jafar, Samyak P. Prasad, Maria Maffei, Alexia Auffèves

quant-phSubmitted Jun 13, 2026• Announced Jun 16, 2026• 0 citations• Revised Sep 2, 2026 (v3)
v1 · Jun 13, 2026v2 · Jun 18, 2026v3 · Sep 2, 2026
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Abstract

Wigner negativity is a key resource for quantum technologies but is difficult to optimize in multimode scattering systems. We study the scattering of coherent pulses by a two-level emitter coupled to a one-dimensional waveguide and introduce energetic cost functions that enable the optimization of Wigner negativity without reconstructing the full Wigner function. By decomposing the scattered energy into coherent, thermal, squeezing, and non-Gaussian contributions, we identify an energetic witness that strongly correlates with the achievable negativity across all driving regimes. This approach singles out optimal output temporal modes and uncovers operating points generating appreciable Wigner negativity with sub-photon input energies. We further identify a maximal energy-efficiency regime at spectral mode matching, where the emitter effectively implements a vacuum-selective $π$ phase shift, realizing a giant optical nonlinearity. These results establish energetic optimization as a practical route to engineering Wigner-negative photonic states in waveguide quantum electrodynamics and related bosonic scattering platforms.

Comments
5 Pages Main Text, 2 Page End Matter, 6 Pages Supplemental. 9 Figures (7 Main, 5 Supplementary). Comments are welcomed!

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Wigner negativity is a key resource for quantum technologies but is difficult to optimize in multimode scattering systems. We study the scattering of coherent pulses by a two-level emitter coupled to a one-dimensional waveguide and introduce energetic cost functions that enable the optimization of Wigner negativity without reconstructing the full Wigner function. By decomposing the scattered energy into coherent, thermal, squeezing, and non-Gaussian contributions, we identify an energetic witness that strongly correlates with the achievable negativity across all driving regimes. This approach singles out optimal output temporal modes and uncovers operating points generating appreciable Wigner negativity with sub-photon input energies. We further identify a maximal energy-efficiency regime at spectral mode matching, where the emitter effectively implements a vacuum-selective πππ phase shift, realizing a giant optical nonlinearity. These results establish energetic optimization as a practical route to engineering Wigner-negative photonic states in waveguide quantum electrodynamics and related bosonic scattering platforms.
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The theoretical description of broadband, multimode quantum pulses undergoing a second-order χ(2)χ^{(2)}χ(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.