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Quantum field theory and inverse problems: Imaging with Entangled Photons

Matti Lassas, Medet Nursultanov, Lauri Oksanen, John C. Schotland

TL;DR

This work addresses recovering an atom density $\rho$ from quantum-field-theoretic scattering of entangled two-photon states, formulating the problem as a nonlocal PDE inverse problem. The authors prove a uniqueness result for $\rho$ using a structured measurement map $\Lambda$ that couples a spatial detector with an integrating detector, under a geometric condition that ensures all photon paths intersect the support of $\rho$. The analysis combines a well-posed direct problem for a four-component system with microlocal techniques to track singularities and extract line-integral data of $\rho$, culminating in a reduction to a partial X-ray transform. An Appendix extends the framework to time-dependent sources and shows the robustness of the uniqueness result under weaker assumptions. The findings advance quantum imaging by exploiting entanglement to overcome classical limitations in object reconstruction.

Abstract

We consider the quantum field theory for a scalar model of the electromagnetic field interacting with a system of two-level atoms. In this setting, we show that it is possible to uniquely determine the density of atoms from measurements of the source to solution map for a system of nonlocal partial differential equations, which describe the scattering of a two-photon state from the atoms. The required measurements involve correlating the outputs of a point detector with an integrating detector, thereby exploiting information about the entanglement of the photons.

Quantum field theory and inverse problems: Imaging with Entangled Photons

TL;DR

This work addresses recovering an atom density from quantum-field-theoretic scattering of entangled two-photon states, formulating the problem as a nonlocal PDE inverse problem. The authors prove a uniqueness result for using a structured measurement map that couples a spatial detector with an integrating detector, under a geometric condition that ensures all photon paths intersect the support of . The analysis combines a well-posed direct problem for a four-component system with microlocal techniques to track singularities and extract line-integral data of , culminating in a reduction to a partial X-ray transform. An Appendix extends the framework to time-dependent sources and shows the robustness of the uniqueness result under weaker assumptions. The findings advance quantum imaging by exploiting entanglement to overcome classical limitations in object reconstruction.

Abstract

We consider the quantum field theory for a scalar model of the electromagnetic field interacting with a system of two-level atoms. In this setting, we show that it is possible to uniquely determine the density of atoms from measurements of the source to solution map for a system of nonlocal partial differential equations, which describe the scattering of a two-photon state from the atoms. The required measurements involve correlating the outputs of a point detector with an integrating detector, thereby exploiting information about the entanglement of the photons.

Paper Structure

This paper contains 9 sections, 18 theorems, 219 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Inverse Problem
  4. Main Result
  5. Physical Considerations and Applications
  6. Direct problem
  7. Direct problem for auxiliary equation
  8. Microlocal analysis
  9. Inverse problem

Key Result

Theorem 1.1

Assume that Condition condition_intro holds. Then the measurement map $\Lambda$ determines the density $\rho$ uniquely.

Figures (2)

  • Figure 1: Schematic illustration of the measurement apparatus. One photon is detected by a spatially-resolved detector, while the second photon, after interacting with the medium, is recorded by an integrating detector. The correlation of the detector outputs defines the measurement $\Lambda f$.
  • Figure 2: Illustrating the geometric configuration of the sets $W_1$, $W_2$, $S$, and $\Sigma$, with rays from the source point $y$ to detection points $x_1 \in W_1$ and $x_2 \in W_2$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 25 more