Multiple photon Hamiltonian in linear quantum optical networks

TL;DR

This work addresses the problem of describing n-photon evolution through m-mode linear optical networks by constructing an explicit effective Hamiltonian for multiphoton states. It introduces a group-theoretic framework, using a photonic homomorphism $\varphi: U(m) \to U(M)$ and its differential $d\varphi$, to relate the single-photon generator $H_S$ to the multiphoton generator $H_U$ via $iH_U = d\varphi(iH_S)$, with $U = e^{iH_U}$ and $S = e^{iH_S}$. The paper provides explicit expressions for $H_U$, derives proportionality and equipartition rules for two-mode systems, and validates the approach with a two-photon, two-mode example that matches known beam-splitter behavior. The broad significance lies in delivering a compact Lie-algebraic toolkit for analyzing linear photonic networks, reducing the parameter burden and enabling straightforward integration with nonlinear components such as squeezing through Bogoliubov-type extensions.

Abstract

We give an alternative derivation for the explicit formula of the effective Hamiltonian describing the evolution of the quantum state of any number of photons entering a linear optics multiport. The description is based on the effective Hamiltonian of the optical system for a single photon and comes from relating the evolution in the Lie group that describes the unitary evolution matrices in the Hilbert space of the photon states to the evolution in the Lie algebra of the Hamiltonians for one and multiple photons. We give a few examples of how a group theory approach can shed light on some properties of devices with two input ports.

Theorems & Definitions (11)

• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 3.1
• proof
• Corollary 3.2
• Corollary 3.3