Nonlinear-linear duality for multipath quantum interference

TL;DR

This paper introduces a generalized duality in quantum optics linking nonlinear multipath interference setups with linear optical systems, enabling new insights into quantum photonic device development beyond low-gain regimes.

Contribution

It proposes and proves a duality that replaces PDCs with hypothetical wavelength-shifting beamsplitters, extending the understanding of quantum interference in complex optical systems.

Findings

Generalized duality links nonlinear and linear quantum optical setups.

Cascaded PDCs can be modeled as optical cavities using the Redheffer star product.

Normalization coefficients include contributions from looping photons inside cavities.

Abstract

In quantum optics, the postselection amplitude of a nondegenerate parametric down-conversion (PDC) process is linked to a beamsplitter (BS) via partial time reversal, up to a normalization coefficient which is related to the parametric gain [Proc. Natl. Acad. Sci. USA 117, 33107 (2020)]. A special example where the gain is low is reminiscent of Klyshko's advanced-wave picture in quantum imaging. Here, we propose and prove a generalized duality for multiple spatial paths connecting a quantum nonlinear interference setup consisting of nondegenerate PDCs and linear optical systems to a linear one, where the PDCs are directly replaced by hypothetical wavelength-shifting BSs. This replacement preserves the geometry of the original setup, and cascaded PDCs become optical cavities whose calculation involves the Redheffer star product. Additional terms in the normalization coefficient are related to the contribution of looping photons inside the cavities. Then, we discuss the case of coherent state input and postselection for $Q$-function calculation. This theorem will be helpful in the development of quantum photonic devices beyond the low-gain limit.