Quantum field theory and inverse problems: Imaging with Entangled Photons

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.

💡 Research Summary

The paper develops a rigorous mathematical framework for an inverse scattering problem that arises in quantum field theory (QFT) when a pair of entangled photons interacts with a medium consisting of two‑level atoms. The authors adopt a scalar model of the electromagnetic field, which leads to a system of three coupled, non‑local partial differential equations (PDEs) for the two‑photon amplitude ψ₂(t,x₁,x₂), the mixed atom‑photon amplitude ψ₁(t,x₁,x₂), and the double‑excitation amplitude a(t,x₁,x₂). The equations contain fractional Laplacians (−Δ)^{1/2} that encode the non‑local propagation of photons, and they involve the atom density ρ(x) as a spatially varying coefficient. The coupling constants g (atom‑photon interaction strength) and Ω (atomic transition frequency) are assumed positive.

The system is rewritten in matrix form A u = 0 with a four‑component state vector u and an initial condition u|_{t=0}=f, where f is a symmetric, compactly supported function representing the initial two‑photon state. The authors prove well‑posedness of this direct problem by constructing self‑adjoint operators L_k (the square roots of the Laplacians) on Sobolev spaces H^k(ℝ^{2n}) and a bounded perturbation B_k that encodes the interaction with the atoms. Using the Hille–Yosida theorem and the bounded perturbation theorem, they show that the generator P_k = −i(L_k + B_k) generates a strongly continuous semigroup, guaranteeing a unique, smooth solution u_f(t) for any smooth initial data f.

The inverse problem is formulated via a measurement operator Λ. In the experimental configuration, one photon of the pair is detected by a spatially resolved point detector in region W₁, while the other photon, after interacting with the medium, is recorded by an integrating detector that averages over region W₂ (characterized by a smooth cutoff χ). The measurement at time t and point x₁∈W₁ is defined as  Λ f(t,x₁) = ∫_{ℝ^n} χ(x₂) |u_f⁰(t,x₁,x₂)|² dx₂, where u_f⁰ is the first component of the solution vector. The goal is to recover the unknown atom density ρ(x) from knowledge of Λ for all admissible source functions f.

A crucial geometric condition (Condition 4.5) is imposed on the source region S, the detector regions W₁, W₂, and the support Σ = supp ρ. Roughly, every point of Σ must lie on a line segment joining a source point z₂∈S to some detector point in W₂, ensuring that the second photon’s ray probes the entire medium. Simultaneously, for each such direction there must exist a reference ray of equal length from a point x₁∈W₁ to the source point z₁ that avoids Σ, allowing the contribution of ρ to be isolated by comparing the two measurements. Under these assumptions the authors prove Theorem 1.1: the map Λ uniquely determines ρ.

The proof relies on microlocal analysis. Although the operator A is not a standard pseudodifferential operator because its symbol contains non‑smooth terms, the authors show that when the source f is supported away from the symbol’s singularities, A can be treated as a pseudodifferential operator locally. They then apply geometric optics constructions to trace singularities of the solution, obtaining line integrals of ρ along the photon paths. By varying the source and detector geometry, a full set of such integrals is generated, and the classical inversion of the X‑ray transform yields ρ uniquely. A key insight is that the initial state f must be non‑factorizable (i.e., genuinely entangled); if f were separable, the two photons would propagate independently and Λ would contain no information about ρ.

Appendix A provides detailed semigroup arguments establishing existence, uniqueness, and smoothness of the direct problem. Appendix B treats a variant where the system is driven by a time‑dependent source term rather than initial data; in this case the geometric constraints can be relaxed, and a similar uniqueness result is proved under weaker hypotheses.

Overall, the paper bridges quantum field theoretic scattering with inverse problem theory, demonstrating that entangled photon measurements can, in principle, recover microscopic material properties that are inaccessible to classical scattering. This establishes a solid theoretical foundation for quantum imaging techniques such as ghost imaging and imaging with undetected photons, suggesting potential improvements in resolution and signal‑to‑noise ratio beyond classical limits.