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On-chip measurement of the modal Stokes-Gell-Mann parameters for partially coherent three-mode light

This paper presents the first on-chip measurement of the eight Stokes-Gell-Mann parameters for partially coherent three-mode light using a photonic integrated platform with a hexagonal Mach-Zehnder interferometer mesh, enabling the reconstruction of the $3\times3$ coherence matrix and facilitating the exploration of iso-entropy fields for advanced optical applications.

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

Published 2026-06-11

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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 singers. In many optical experiments, we usually listen to a duet (two singers). We have a well-established way to describe how well these two singers are in tune with each other using a set of three numbers called "Stokes parameters." It's like a simple scorecard that tells us if they are singing in perfect harmony, completely out of sync, or somewhere in between.

However, what happens when you have a trio? What if your light is made of three distinct modes (three singers) instead of two? The old scorecard doesn't work anymore. The math gets messy, and until now, no one had figured out how to measure the "tune" of this trio directly.

This paper is the first time scientists have successfully built a machine to measure that trio's harmony. Here is how they did it, explained simply:

1. The Problem: The "Three-Mode" Puzzle

For decades, physicists have known that if you have three modes of light, you need a more complex scorecard. Instead of the old three numbers, you need eight new numbers, called Stokes-Gell-Mann (SGM) parameters. These numbers are named after a famous physicist who used similar math to describe particles in high-energy physics.

The challenge was that measuring these eight numbers is incredibly difficult. It's like trying to listen to three singers simultaneously while they are constantly changing their pitch and volume, and you need to figure out exactly how they relate to one another without a microphone for each specific relationship.

2. The Solution: A "Light Switchboard" on a Chip

The researchers built a tiny, microscopic circuit (a photonic chip) that acts like a sophisticated switchboard for light.

The Setup: They took a laser and split it into three separate paths (the three "singers"). To make sure the experiment was fair, they made sure the light in these three paths started out completely unconnected (incoherent), like three strangers who have never met.
The Mesh: The chip contains a honeycomb-like grid of tiny mirrors and beam splitters (Mach-Zehnder interferometers). Think of these as traffic controllers. They can take the light from any two paths, mix them together, delay one slightly, and then send them out again.
The Process: By carefully adjusting these traffic controllers, the researchers could mix the three light paths in specific patterns. After mixing, they measured the brightness (intensity) of the light coming out. By doing this in seven different specific mixing patterns, they could mathematically reverse-engineer the eight hidden numbers (the SGM parameters) that describe the light's state.

3. The Experiment: Tuning the "Entropy"

The researchers didn't just measure one type of light; they wanted to see how the system worked for different levels of "disorder," which they call entropy.

Low Entropy (Order): Imagine the three singers are all singing the exact same note. This is highly ordered.
High Entropy (Chaos): Imagine the three singers are all singing different, random notes. This is highly disordered.
The Middle Ground: They created light that was somewhere in between.

They tested their chip by creating light with three different levels of disorder (0.5, 1.0, and 1.5 bits of entropy). For each level, they created many different variations of the light by mixing the paths in different ways.

4. The Result: Reconstructing the Picture

After measuring the eight numbers for each variation, they used a computer to reconstruct the full picture of the light's coherence matrix (the 3x3 scorecard).

The Proof: They compared their reconstructed picture to the picture they intended to create.
The Score: The match was incredibly high (about 95% to 98% accurate). This proved that their method works. They successfully measured the eight parameters and rebuilt the full description of the three-mode light.

Why This Matters (According to the Paper)

The paper claims this is a major breakthrough because:

First of its kind: It is the first time these specific eight parameters (SGM) have been measured directly for light, either on a chip or in free space.
Odd Numbers: Previous experiments could only handle pairs (2 modes) or groups of four (4 modes). This is the first time they successfully handled an odd number (3 modes).
Future Tools: This opens the door to using complex, multi-mode light for better optical communications (sending more data), sensing, and information processing.

In a nutshell: The researchers built a tiny, programmable light mixer on a chip. They used it to listen to a trio of light waves, figured out exactly how they were related to each other using a new set of eight numbers, and proved they could perfectly reconstruct the "voice" of the trio. This gives scientists a new tool to control and use complex light for advanced technology.

Technical Summary: On-chip Measurement of Modal Stokes-Gell-Mann Parameters for Partially Coherent Three-Mode Light

Problem Statement
The manipulation of partially coherent light in integrated photonic platforms is emerging as a frontier for optical information processing, offering advantages in dense communications, scattering-free transmission, and optical cryptography. These applications rely on "structured coherence," where optical fields spanned by $N$ modes are characterized by an $N \times N$ coherence matrix $G$ . While the reconstruction of $2 \times 2$ (two-mode) and $4 \times 4$ (four-mode) coherence matrices has been achieved on-chip via Stokes tomography, a significant gap remains for odd-dimensional matrices, specifically $3 \times 3$ .

Conventional Stokes parameters (SPs) are defined as expansion coefficients of a $2 \times 2$ coherence matrix in terms of Pauli matrices. They cannot be directly applied to three-mode fields, which require a $3 \times 3$ coherence matrix. Although it has been theoretically proposed that the eight Gell-Mann matrices (generators of the SU(3) group) could serve as a basis for these fields, yielding eight "Stokes-Gell-Mann" (SGM) parameters, direct experimental measurement of these parameters has not been realized. Furthermore, while the space of iso-entropy fields (fields sharing the same entropy value) is well-defined, the distinction between iso-entropy fields that can be unitarily inter-converted and those that cannot has not been experimentally explored for three-mode systems.

Methodology
The authors present an on-chip measurement scheme utilizing a photonic integrated circuit (iPronics Smartlight Processor) featuring a hexagonal mesh of Mach-Zehnder interferometers (MZIs). The experimental approach involves three primary stages:

Source Preparation and Entropy Tuning:
A laser diode source is split into three single-mode fibers (SMFs) representing modes $|1\rangle$ , $|2\rangle$ , and $|3\rangle$ . Fiber loops exceeding the laser's coherence length are inserted to ensure the modes are mutually incoherent initially. The field entropy $S$ is tuned by independently adjusting the power in each mode using specific MZIs configured as non-unitary attenuators. This allows the generation of diagonal coherence matrices $G_D = \text{diag}\{\lambda_1, \lambda_2, \lambda_3\}$ with eigenvalues summing to unity, spanning the entropy range from $S=0$ (coherent) to $S=\log_2 3$ (maximally incoherent).
Unitary Transformation:
To explore the structure of the coherence matrix, the diagonal matrix $G_D$ is transformed by an arbitrary $3 \times 3$ unitary operator $\hat{U}$ . This unitary is constructed on-chip using a sequence of $2 \times 2$ unitaries acting on pairs of modes ( $\hat{U}_{12}, \hat{U}_{23}, \hat{U}_{13}$ ) and phase shifters, following the decomposition shown in the Reck or Clements style (specifically $\hat{U} = \hat{U}_{13}\hat{U}_{23}\hat{U}_{12}$ in this work). This step generates non-diagonal coherence matrices $G = \hat{U}G_D\hat{U}^\dagger$ while preserving the field entropy.
SGM Parameter Measurement:
The eight SGM parameters ( $s_0$ through $s_8$ ) are extracted by measuring modal intensities after applying specific unitary transformations.
- Diagonal Parameters ( $s_0, s_3, s_8$ ): Obtained by directly measuring the modal weights $I_1, I_2, I_3$ without additional mixing.
- Off-Diagonal Parameters ( $s_1, s_2, s_4, s_5, s_6, s_7$ ): Obtained by implementing specific $2 \times 2$ unitaries (analogous to Pauli matrix rotations) on pairs of modes ( $|1\rangle$ - $|2\rangle$ , $|1\rangle$ - $|3\rangle$ , and $|2\rangle$ - $|3\rangle$ ) before detection. For example, $s_1$ and $s_2$ are derived from measurements on modes $|1\rangle$ and $|2\rangle$ after applying specific unitaries $\hat{U}_2$ and $\hat{U}_3$ .
  The measured parameters are then used to reconstruct the full $3 \times 3$ coherence matrix via the expansion $G = \frac{1}{2}\sum_{j=0}^8 s_j \hat{\Lambda}_j$ , where $\hat{\Lambda}_j$ are the Gell-Mann matrices.

Key Contributions

First Direct Measurement: This work reports the first direct measurement of modal Stokes-Gell-Mann (SGM) parameters for partially coherent three-mode light, both in free space and on-chip.
Odd-Dimensional Reconstruction: It provides the first on-chip reconstruction of an odd-dimensional ( $3 \times 3$ ) coherence matrix, extending previous capabilities limited to even dimensions ( $N=2, 4$ ).
Iso-Entropy Exploration: The authors experimentally map the space of iso-entropy fields. They demonstrate two distinct categories of iso-entropy states:
1. Fields that can be inter-converted into one another via unitary transformations (sharing the same eigenvalues).
2. Fields that share the same entropy value but cannot be inter-converted unitarily (possessing different eigenvalue sets).
High-Fidelity Reconstruction: The system successfully reconstructed 15 distinct coherence matrices across three entropy levels ( $S = 0.5, 1.0, 1.5$ bits). The reconstructed matrices showed high fidelity ( $F \geq 0.95$ , typically $\approx 0.98$ ) compared to theoretical predictions, with estimated entropy values within 5% of the target.

Results
The experiment successfully tuned the field entropy from coherent states ( $S=0$ ) to maximally incoherent states ( $S=\log_2 3 \approx 1.585$ bits). By applying specific unitary sequences, the authors synthesized various coherence matrices and verified their structure through SGM parameter measurement. The data confirmed that for a fixed entropy $S > 1$ bit, the iso-entropy curve forms a closed loop within the eigenvalue space, containing rank-3 fields. At $S=1$ bit, the curve touches the boundaries of the eigenvalue triangle (rank-2 fields), and for $S < 1$ bit, the iso-entropy curves become disjoint near the vertices. The fidelity of the reconstructed matrices consistently exceeded 0.95, validating the on-chip measurement protocol.

Significance
The paper claims that this work represents a critical step toward generalizing Stokes tomography to on-chip reconstruction of coherence matrices of any dimension. By successfully measuring SGM parameters for $N=3$ , the authors establish a methodology that can be scaled to factorize measurements for larger odd-dimensional coherence matrices. This capability is essential for exploiting structured coherence in advanced applications such as optical communications, sensing, and information processing. The authors modestly note that while the current implementation uses on-chip waveguide modes, it does not yet solve the problem of measuring SGM parameters for non-paraxial 3D partially polarized light (which requires transducing polarization modes to waveguide modes), a task they identify as a direction for future work.