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On-chip control of the coherence matrix of four-mode partially coherent light: rank, entropy, and modal Stokes parameters

This paper demonstrates the on-chip manipulation and tomographic reconstruction of four-mode partially coherent light using a hexagonal mesh of Mach-Zehnder interferometers to control coherence rank, entropy, and matrix structure, thereby validating the scalability of integrated photonics for large-scale optical information processing.

Original authors: Amin Hashemi, Abbas Shiri, Bahaa E. A. Saleh, Andrea Blanco-Redondo, Ayman F. Abouraddy

Published 2026-02-02

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CC BY 4.0

Original authors: Amin Hashemi, Abbas Shiri, Bahaa E. A. Saleh, Andrea Blanco-Redondo, Ayman F. Abouraddy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine light not just as a beam, but as a choir of four singers. In a perfectly "coherent" choir, every singer hits the exact same note at the exact same time, moving in perfect unison. This is like a laser pointer. But in the real world, most light is "partially coherent"—like a choir where the singers are slightly out of sync, or where some are singing loudly while others are whispering, and their timing drifts a bit.

This paper is about learning how to conduct this chaotic choir using a tiny, high-tech conductor's baton built directly onto a computer chip.

Here is the breakdown of what the researchers did, using simple analogies:

1. The Problem: The "Chaos" of Light

Usually, to study or use this "messy" light, scientists had to use bulky, table-sized equipment (like mirrors and lenses on a long optical table). It was unstable and hard to scale up. The researchers wanted to shrink this down to a microchip, similar to how we shrank computers from room-sized machines to the phone in your pocket.

2. The Tool: The "Smart Chip"

The team built a chip containing a hexagonal grid of 72 tiny switches called Mach-Zehnder Interferometers (MZIs).

Analogy: Think of these as tiny, programmable traffic circles for light. You can tell a specific beam of light to go left, right, split, or merge with another beam. By chaining these together, you can perform complex math on the light itself.

3. The Mission: Taming the Four Singers

The researchers started with four "singers" (four modes of light) that were completely out of sync with each other (maximally incoherent). They wanted to prove they could control three specific things about this group:

A. Controlling the "Rank" (How many singers are actually singing?)

The Concept: The "rank" is simply the number of independent voices in the choir.
- Rank 1: Only one singer is active; the other three are silent. This is a perfectly coherent laser beam.
- Rank 4: All four singers are active and independent. This is the most "messy" or random light.
The Action: The chip acts like a volume knob for each singer. By turning the volume of three singers down to zero, they could instantly turn a messy Rank-4 field into a clean Rank-1 field. They could also stop at Rank 2 or Rank 3. They proved they could switch between these states on the chip.

B. Tuning the "Entropy" (How random is the mix?)

The Concept: Entropy is a measure of randomness. If you have two singers, you can have them share the spotlight equally (high randomness) or have one dominate (low randomness).
The Action: The researchers showed they could adjust the "volume" of the singers to create specific levels of randomness.
- The Cool Discovery: They found that two different groups of singers could have the exact same amount of randomness (entropy) but still sound different. You couldn't just use a simple switch (a "unitary" transformation) to turn one group into the other; you needed a more complex adjustment (a "non-unitary" change) to reshape the mix. They successfully created these unique, "iso-entropy" states on the chip.

C. Shaping the "Structure" (How the singers relate to each other)

The Concept: Even if the singers are out of sync, they might still have a hidden relationship. The "coherence matrix" is a map of these relationships.
The Action: Using the chip's traffic circles, they performed a "dance" (a unitary transformation) that rearranged the relationships between the singers. They could take a group that had no connection and make them "talk" to each other, creating interference patterns (like ripples in a pond meeting). They could also control the phase of this relationship (whether the singers were slightly ahead or behind each other).

4. The Proof: Taking a "Snapshot"

To prove they actually did what they said, they had to "take a picture" of the light's state.

The Method: They used a mathematical tool called Kronecker-Pauli matrices (think of these as a specialized set of filters).
The Process: They ran the light through different filter settings and measured the intensity (brightness) at the end. By combining these measurements, they could reconstruct the entire "coherence matrix"—essentially a 4x4 map showing exactly how the four modes of light were behaving.
The Result: The map they built matched their theoretical predictions with very high accuracy (about 95-99% fidelity).

Summary

In short, this paper demonstrates that for the first time, scientists can take a messy, four-part light signal and, using a tiny microchip:

Select how many parts are active.
Adjust how random the signal is.
Rearrange how the parts relate to one another.
Verify the result by taking a precise mathematical snapshot.

This proves that the "coherence advantage" (using messy light to carry more information or resist noise) can be moved from big, unstable lab tables to stable, scalable computer chips, paving the way for future optical technologies.

Technical Summary: On-chip control of the coherence matrix of four-mode partially coherent light

Problem and Motivation
While partially coherent light has long been recognized for its unique spatiotemporal dynamics, recent findings have identified a specific "coherence advantage" in optical information processing where partially coherent light outperforms coherent light in tasks such as multi-mode communications and scattering-immune information carrying. Historically, experimental investigations of these phenomena have been restricted to bulk optics. However, realizing the full potential of this advantage requires integrated photonic platforms capable of providing the stability and scalability necessary for cascaded operations on large arrays of optical modes. While on-chip processing of quantum states and classical coherent light is well-established, the manipulation of partially coherent light—specifically the control of its coherence matrix—remains an unmet challenge in integrated photonics. Previous work has demonstrated two-mode control, but scaling this to higher dimensions (four modes and beyond) is critical for practical applications.

Methodology
The authors demonstrate the scalable on-chip manipulation of four-mode partially coherent optical fields using a hexagonal mesh of 72 programmable Mach-Zehnder interferometers (MZIs) on an integrated photonic chip (iPronics Smartlight Processor). The experimental workflow involves five distinct stages:

Source Generation: A generic four-mode mutually incoherent spatial field is generated off-chip using a short-coherence-length laser diode. By introducing fiber loops with lengths exceeding the laser's coherence length (0, $L$ , $2L$ , and $3L$ ) into four separate single-mode fiber paths, the authors ensure that the fields entering the chip are mutually incoherent, represented by a maximally incoherent $4 \times 4$ coherence matrix ( $G = \frac{1}{4}\hat{I}_4$ ).
Coherence Rank Control: The coherence rank (the number of non-zero eigenvalues of the coherence matrix) is controlled via non-unitary linear operations. By selectively attenuating specific modes using MZIs, the authors transition the field from a rank-4 state (maximally incoherent) down to rank-3, rank-2, and finally rank-1 (fully coherent) states.
Entropy Tuning: The field entropy is tuned without altering the rank by precisely adjusting the relative weights of the eigenvalues using MZIs. This allows for the synthesis of iso-entropy fields that cannot be inter-converted via unitary transformations alone.
Unitary Transformations: The structure of the coherence matrix is molded using $4 \times 4$ unitary transformations constructed from sequences of $2 \times 2$ unitaries acting on pairs of modes. These transformations introduce correlations between modes, creating off-diagonal elements (both real and imaginary) in the coherence matrix.
Tomographic Reconstruction: The full $4 \times 4$ Hermitian coherence matrix is reconstructed by measuring modal Stokes parameters. This is achieved by expanding the coherence matrix in terms of $4 \times 4$ Kronecker-Pauli matrices (outer products of standard $2 \times 2$ Pauli matrices). The required measurements are performed by applying specific unitary sequences to the modes and recording the resulting modal weights.

Key Results

Rank and Entropy Control: The authors successfully generated and verified coherence matrices of ranks 1 through 4. The reconstructed matrices showed high fidelity ( $F \approx 0.95$ for ranks 1–3 and $F \approx 0.99$ for rank 4) with theoretical expectations. The slight reduction in fidelity for lower ranks was attributed to leakage in the MZIs used for extinction.
Entropy Tuning: The system successfully tuned the entropy of rank-2 and rank-3 fields across their theoretical limits. The authors demonstrated the generation of iso-entropy rank-3 fields with different eigenvalue distributions, confirming that fields with equal entropy are not necessarily unitarily inter-convertible, thus requiring non-unitary operations for conversion.
Unitary Molding: By applying specific $4 \times 4$ unitaries ( $\hat{U}_B$ and $\hat{U}_C$ ) to diagonal coherence matrices, the authors successfully introduced significant off-diagonal elements, effectively creating correlations between previously uncorrelated modes. The reconstructed matrices maintained high fidelity ( $F \approx 0.95$ ) and demonstrated control over both the magnitude and phase of inter-modal correlations.
Scalability: The experiments confirmed that $2 \times 2$ on-chip building blocks can be concatenated to synthesize and reconstruct high-dimensional ( $4 \times 4$ ) coherence matrices.

Significance and Claims
The paper claims to provide the first experimental demonstration of scalable on-chip coupling, unitary/non-unitary manipulation, and reconstruction of partial coherence in the spatial domain for four-mode fields. The results confirm that integrated photonic platforms are capable of performing all critical tasks required for the reliable processing of multimode partially coherent light.

The authors assert that this work represents a decisive step toward large-scale on-chip manipulation of massively moded partially coherent light. They posit that this capability enables new fundamental studies of partial coherence (such as entropy concentration and swapping) and facilitates the utilization of these fields in optical information processing applications, including communications, cryptography, spectroscopy, sensing, and computation. The work establishes that the "coherence advantage" can be harnessed within the stable, compact environment of integrated photonics, moving beyond the limitations of bulk optics.