[Paper Review] Direct measurement of the biphoton Wigner function through two-photon interference

Tom Douce, Andreas Eckstein+6 more|arXiv (Cornell University)|Apr 26, 2013

Quantum Information and Cryptography4 citations

TL;DR

This paper proposes a direct measurement of the biphoton Wigner function using a modified Hong-Ou-Mandel (HOM) interferometer, enabling full characterization of two-photon quantum states through linear optics. The method reveals non-Gaussian entanglement via negative Wigner function values, with experimental validation showing over 80% violation of a non-Gaussian entanglement witness in a Schrödinger cat state.

ABSTRACT

The Hong-Ou-Mandel (HOM) experiment was a benchmark in quantum optics, evidencing the quantum nature of the photon. In order to go deeper, and obtain the complete information about the quantum state of a system, for instance, composed by photons, the direct measurement or reconstruction of the Wigner function or other quasi--probability distribution in phase space is necessary. In the present paper, we show that a simple modification in the well-known HOM experiment provides the direct measurement of the Wigner function. We apply our results to a widely used quantum optics system, consisting of the biphoton generated in the parametric down conversion process. In this approach, a negative value of the Wigner function is a sufficient condition for non-gaussian entanglement between two photons. In the general case, the Wigner function provides all the required information to infer entanglement using well known necessary and sufficient criteria. We analyze our results using two examples of parametric down conversion processes taken from recent experiments. The present work offers a new vision of the HOM experiment that further develops its possibilities to realize fundamental tests of quantum mechanics involving decoherence and entanglement using simple optical set-ups.

Motivation & Objective

• To provide a direct, experimentally feasible method for measuring the biphoton Wigner function in phase space.
• To establish a link between HOM interference and the Wigner function, extending the HOM experiment's capabilities beyond photon statistics.
• To enable full characterization of non-Gaussian entangled biphoton states using only linear optics and standard experimental setups.
• To reinterpret prior SPDC experiments through the lens of Wigner function tomography, enhancing state reconstruction and entanglement detection.
• To offer a stable, scalable method for probing quantum decoherence and non-classicality in continuous-variable photonic systems.

Proposed method

• Adapt the standard HOM interferometer by introducing controlled displacements in transverse momentum or frequency variables to probe different points in phase space.
• Use coincidence measurements between two photons to extract the Wigner function via the relation W(μ, δ) ∝ P(coincidence | displacement).
• Apply the formalism to biphoton states generated via spontaneous parametric down-conversion (SPDC), parameterized by pump and phase-matching functions F₊(p₊) and F₋(p₋).
• Utilize Fourier transforms of the momentum-space functions to describe the state in position space, enabling tomographic reconstruction.
• Implement frequency or time-delayed displacements to scan the Wigner function in one-dimensional frequency-entangled systems.
• Leverage symmetry and interference patterns to directly observe negative Wigner function values as a signature of non-Gaussian entanglement.

Experimental results

Research questions

• RQ1 Can the Wigner function of a biphoton state be directly measured using only linear optics and standard HOM interferometer components?
• RQ2 How does the modified HOM setup enable full phase-space tomography of the biphoton state?
• RQ3 What is the role of negative Wigner function values in identifying non-Gaussian entanglement in SPDC-generated photon pairs?
• RQ4 How can previously reported HOM results be reinterpreted using the Wigner function formalism?
• RQ5 Can this method be generalized to other continuous-variable quantum systems, including multi-photon or fermionic states?

Key findings

• The modified HOM experiment allows direct measurement of the biphoton Wigner function across the entire phase space using only linear optics and standard interferometric components.
• Negative values of the Wigner function—observed in a Schrödinger cat state with Δpₚ = 5√(k/L)—indicate non-Gaussian entanglement, with a violation exceeding 80% of the witness threshold.
• In a frequency-entangled SPDC system, the method successfully maps the Wigner function, with expected features confirmed in simulations (Figs. 2e–2f).
• The approach provides a stable alternative to conventional tomography, avoiding the need for stabilized independent interferometers.
• The formalism reinterprets prior experiments: for instance, the HOM dip in STEVE2 corresponds to the Wigner function at (0,0;0,0), now directly measurable.
• The method is generalizable to arbitrary numbers of photons and other quantum particles with appropriate symmetry conditions.