Nonlinear-linear duality for multipath quantum interference

TL;DR

The paper addresses how to relate nonlinear quantum interference arising from nondegenerate PDCs to linear optics through a generalized nonlinear-linear duality for multipath configurations, using partial time reversal (PTR) with beamsplitter replacements $R=\tanh r$ and $T=\operatorname{sech} r$. It proves that an arbitrary network of nondegenerate PDCs and linear components has a PTR counterpart obtained by replacing all PDCs with hypothetical beamsplitters and combining them via the Redheffer star product, with the postselection amplitude related by a normalization coefficient $C_U$. The normalization coefficient is given explicitly as $C_U=\beta\,T_1T_2\cdots T_N$ with $\beta=\prod_{n=1}^{N-1}[1+R_{n+1}(\mathcal{U}_{\star n})_{s1,i1}]^{-1}$, and additional terms account for intracavity looping photons. The authors provide a Gaussian-state proof, a Q-function relation, and explicit simple examples to illustrate the duality, offering a practical tool for designing quantum photonic devices beyond the low-gain limit.

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.

Figures (3)

• Figure 1: (a) The nonlinear-linear duality for a single device proposed by Cerf and Jabbour. The $m$ and $n$ symbols stand for the initial and postselected photon numbers respectively. Green and red arrows represent s and i lights. The reflected light from $s1$ to $i1$ at the hypothetical BS $\hat{\mathcal{B}}$ gains a phase $\pi$. (b) If a nonlinear system $\hat{U}$ satisfies the duality which links it to a linear setup $\hat{\mathcal{U}}$ which contains hypothetical BSs, a combination of it and a PDC $\hat{G}$ also does. A cavity forms in its partially time-reversed (PTR) setup. (c) The decomposition of a general nonlinear system with nondegenerate PDCs $\hat{G}_n$ and linear systems $\hat{L}_{sn},\hat{L}_{in}$ and its PTR setup.
• Figure 2: Examples of the duality. (a) Two cascaded PDCs with two phase plates (the only possible type of single-path linear lossless setups) inserted. (b) The PDC $\hat{G}_1$ acting on paths $s1$ and $i$, a $\pi$ phase shift on $s1$, $\hat{G}$ acting on $s2$ and $i$, and $\hat{G}_2$ acting on $s1$ and $i$.
• Figure 3: (a) Setup of the quantum teleportation experiment. The s light from crystal 1 passes through wave plates (WPs) for polarization preparation. Crystal 2 produces polarization-entangled photon pairs. The s photons from two crystals interfere at the BS. The i photon from crystal 2 obtains the prepared polarization if detectors $D_{i1}$, $D_{s1}$, and $D_{s2}$ click. (b) Setup in the four-photon AWP. The fiber rotates the polarization by $90^\circ$ if the two polarization analyzers in front of it have orthogonal orientations.