Compactifying linear optical unitaries using multiport beamsplitters

TL;DR

This work addresses realizing any $N$-dimensional unitary $U_N$ with a scalable, low-loss optical architecture. It combines a Lie-group-based finite product of identical fixed MBS blocks with the Bell-Walmsley-Clements (BWC) framework to synthesize $U_N$, yielding $N+2$ phase masks, $N-1$ fixed MBSs, and $N-1$ BSs. Key contributions include achieving $\mathcal{O}(N)$ fixed components versus $\mathcal{O}(N^2)$ fixed BSs in the Clements scheme, providing explicit $N=3$ reconfigurable circuits using four fixed tritters, and demonstrating a USD 3D unitary realized by this scheme; a topological proof guarantees universality for $N \ge 4$. The results enable compact, loss-tolerant photonic interferometers with potential impact on scalable LOQC and high-dimensional state discrimination.

Abstract

We show that any $N$-dimensional unitary matrix can be realized using a finite sequence of concatenated identical fixed multiport beamsplitters (MBSs) and phase shifters (PSs). Our construction is based on a Lie group theorem applied to existing decompositions. Using the Bell-Walmsley-Clements framework, we prove that any $N$-dimensional unitary requires $N+2$ phase masks, $N-1$ fixed MBSs, and $N-1$ BSs. Our scheme requires only $\mathcal{O}(N)$ fixed, identical components (MBSs and BSs) compared to the $\mathcal{O}(N^2)$ fixed BSs required by conventional schemes (e.g., Clements), all while keeping the same number of PSs. Experimentally, these MBS can be realized as a monolithic element via femtosecond laser writing, offering superior performance through reduced insertion losses. As an application, we present a reconfigurable linear optical circuit that implements a three-dimensional unitary emerging in the unambiguous discrimination of two nonorthogonal qubit states.

Figures (6)

• Figure 1: Panels (a)-(c) depict variants of two-port Mach-Zehnder interferometers (MZIs): (a) the antisymmetric MZI (aMZI) used in the Clements scheme, (b) a modified aMZI, and (c) the symmetric MZI. Panel (d) shows an $N$-dimensional controllable multiport BS (MBS), composed of PSs (thick vertical lines) and $N-1$ BSs (thick horizontal lines), with beam-splitting ratios as indicated. When a single photon enters the first input port, it exits with equal probability across all output ports.
• Figure 2: (a) The elementary 2-dimensional block illustrating our theorem in the two-dimensional case. (b) The shaded region nullifies the off-diagonal element of a given two-dimensional unitary, while the remaining PS accounts for the diagonal matrix. (c) Elementary $3 \times 3$ tritter block composed of two PSs and two BSs, each with reflectivity $\eta=1/2$. (d) Proposed scheme for realizing an arbitrary $U_3$ using four fixed tritter blocks (shown in rotated ellipses). The shaded rectangular region implements any desired $U_3$ [see Eq. (\ref{['td1b']})], following the Clements decomposition scheme. (e) Bell-Walmsley-Clements scheme that uses two fixed tritters and two BSs to realize any $U_3$.
• Figure 3: We assume that each BS or tritter can be fabricated as a single integrated block. Under this assumption, the realization of any $U_3$ requires: (a) in the original Clements scheme, either 6 identical blocks or 5 blocks with optimized placement; (b) in the modified Clements scheme, 4 non-identical blocks with varied configurations; (c) in the proposed scheme shown in Figure \ref{['fig-tri']}(d), only 4 identical fixed tritter blocks. In both (a) and (b), tritters are shown in rotated ellipses.
• Figure 4: Panels (a)-(c) illustrate the Bell-Walmsley-Clements (BWC) decomposition for dimensions $N=4, 5$, and $6$ respectively. Each shaded region represents an MBS corresponding to that dimension. This architecture clearly demonstrates the $\mathcal{O}(N)$ linear scaling of fixed identical MBS blocks required for universal unitary realization, needing $N+2$ phase masks in total.
• Figure 5: Illustration of the topological proof of the theorem. We show that $G=W$, and $\bar{W}$ is a null set.