Analytic solution to degenerate biphoton states generated in arrays of nonlinear waveguides

TL;DR

This paper presents an analytic solution for the evolution of degenerate biphoton states in arrays of nonlinear waveguides, enabling efficient analysis and control of quantum light states in integrated photonics.

Contribution

It introduces a supermode approach providing explicit analytic expressions for biphoton state evolution in arbitrary waveguide arrays, reducing computational complexity.

Findings

Analytic solutions for biphoton states in waveguide arrays.

Application examples including symmetric injection and small arrays.

Method enables inverse problem solving for quantum state engineering.

Abstract

The arrays of nonlinear waveguides are a powerful integrated photonics platform for studying and manipulating quantum states of light. Also, they are a valuable resource for various quantum technologies. In this work, we employed a supermode approach to obtain an analytic solution to the evolution of degenerate biphoton states in arrays of nonlinear waveguides. The solution accounts for arrays of an arbitrary number of waveguides and coupling profiles. In addition, it provides an explicit analytic expression without the need of computationally-expensive steps. Actually, it only relies on the calculation of the eigenvalues and eigenvectors of the coupling matrix. In general, this needs to be performed numerically, but there are relevant instances in which analytic expressions are available. Thus, in certain cases, the procedure proposed here does not require the use of any numerical method. We analyze the general properties of the solution and show some application examples. Particularly, simple solutions that can be obtained by special symmetric injection profiles, the results for small arrays, and the properties of the propagation when only the center waveguide is pumped in a symmetric odd array. In addition, we present a proof-of-principle example on how to use the analytic solution to tackle inversion problems. That is, obtaining the initial conditions required to achieve a desired quantum state -- which is a valuable technological application. A relevant aspect of the method presented here is its scalability for large arrays due to the lack of resource-intensive steps -- both for the direct and inverse problems.