[Paper Review] From point processes to quantum optics and back

Rémi Bardenet, Alexandre Feller+7 more|arXiv (Cornell University)|Oct 11, 2022

Quantum Mechanics and Applications4 citations

TL;DR

This paper presents a cross-disciplinary synthesis linking point processes in probability theory to quantum optics, tracing the historical and theoretical foundations of permanental and determinantal point processes (DPPs) back to Odile Macchi's 1970s work. It demonstrates how quantum field theory—particularly through coherent states, creation/annihilation operators, and Wick’s theorem—naturally gives rise to DPPs and permanental processes, and shows how photodetection and electrodetection experiments realize these mathematical structures in physical systems, with key results linking anti-bunching to DPPs and bunching to permanental processes.

ABSTRACT

Some fifty years ago, in her seminal PhD thesis, Odile Macchi introduced permanental and determinantal point processes. Her initial motivation was to provide models for the set of detection times in fundamental bosonic or fermionic optical experiments, respectively. After two rather quiet decades, these point processes have quickly become standard examples of point processes with nontrivial, yet tractable, correlation structures. In particular, determinantal point processes have been since the 1990s a technical workhorse in random matrix theory and combinatorics, and a standard model for repulsive point patterns in machine learning and spatial statistics since the 2010s. Meanwhile, our ability to experimentally probe the correlations between detection events in bosonic and fermionic optics has progressed tremendously. In Part I of this survey, we provide a modern introduction to the concepts in Macchi's thesis and their physical motivation, under the combined eye of mathematicians, physicists, and signal processers. Our objective is to provide a shared basis of knowledge for later cross-disciplinary work on point processes in quantum optics, and reconnect with the physical roots of permanental and determinantal point processes.

Motivation & Objective

• To re-establish the physical origins of permanental and determinantal point processes (DPPs) in quantum optics, particularly through Macchi’s 1970s thesis on detection events in bosonic and fermionic systems.
• To unify perspectives from mathematics, physics, and signal processing by providing a modern, accessible introduction to the mathematical framework of point processes and quantum field theory.
• To demonstrate how quantum field theory—via coherent states, field operators, and Wick’s theorem—generates DPPs and permanental processes as correlation structures in photodetection and electrodetection.
• To explore open questions on the physical realizability of generalized point processes, such as α-DPPs and Pfaffian processes, and their connections to anyons and non-Gaussian states.
• To lay the foundation for Part II by connecting theoretical formalism to landmark experiments, including HBT effects and non-interacting trapped fermions.

Proposed method

• Uses the framework of quantum field theory, particularly Fock spaces and second quantization, to model systems of indistinguishable bosons and fermions.
• Applies creation and annihilation operators to derive the correlation functions of point processes from quantum states, especially coherent states and Gaussian states.
• Employs Wick’s theorem to show that Gaussian density matrices lead to determinantal or permanental correlation functions, depending on statistics.
• Models photodetection and electrodetection using first-order coherence functions and detector structure functions, linking them to second-order and higher-order coherence functions.
• Demonstrates that anti-bunching in single-photon sources corresponds to a DPP, while thermal light leads to a permanental process via the HBT effect.
• Uses the occupation number representation and mode decomposition to connect quantum field observables to point process correlation functions.

Experimental results

Research questions

• RQ1 How do determinantal and permanental point processes arise from the quantum statistics of bosons and fermions in photodetection experiments?
• RQ2 What is the role of coherent states and Gaussian states in generating point processes with tractable correlation structures?
• RQ3 Can generalized point processes such as α-DPPs be physically realized through quantum field theories with non-trivial statistics or interactions?
• RQ4 How do the symmetries and universality of low-energy quantum field theories relate to the universality observed in point processes like those from random matrix theory?
• RQ5 What physical systems underlie Pfaffian and other generalized point processes, and how do they extend the DPP framework?

Key findings

• The second-order coherence function in quantum optics directly determines the correlation functions of a point process, with anti-bunching (g(2)(0) < 1) corresponding to a DPP and bunching (g(2)(0) > 1) to a permanental process.
• Photodetection events in coherent (laser) light produce Poisson processes, while thermal light leads to permanental point processes due to photon bunching, as confirmed by the HBT experiment.
• Anti-bunching in single-photon sources is physically realized as a DPP, with the second-order correlation function ρ2(x1,x2) = 1 − |⟨ψ|a†(x1)a†(x2)a(x2)a(x1)|ψ⟩|² yielding a determinantal structure.
• Wick’s theorem for Gaussian states leads to correlation functions that are determinants (fermions) or permanents (bosons), providing a direct mathematical link between quantum field theory and point processes.
• The kernel of a DPP is not unique; multiple kernels can yield the same DPP, and the fermionic framework may help identify such invariances, as suggested by recent work of Olshanski (2020).
• α-DPPs for α ∈ [−1,1] can be constructed from fermionic systems via limit procedures, as shown by Cunden et al. (2019), suggesting a physical realization of generalized point processes.