🔬 optics

Phase synchronization dynamics of two mutually coupled InP lasers in a quantum entropy source

This paper investigates and quantifies how residual coupling between two mutually coupled InP lasers in a phase-diffusion quantum entropy source induces phase synchronization that reduces extractable entropy, utilizing both experimental data and a coupled stochastic rate equation model to derive an analytical framework for optimizing the device's performance.

Original authors: Berta Martínez-Pàmias, Miquel Rudé, Cristina Masoller

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Berta Martínez-Pàmias, Miquel Rudé, Cristina Masoller

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 you have two identical twins, Laser A and Laser B, living inside a tiny, high-tech city made of light called a Photonic Integrated Circuit (PIC). These twins are the heart of a "Quantum Random Number Generator" (QRNG), a super-secure device used to create uncrackable codes for the internet. Their job is to be perfectly chaotic. Every time they blink (which happens billions of times a second), they should flash with a completely random, unpredictable color phase, like two people flipping coins in separate rooms.

The Big Problem: The Twins Start Whispering
In a perfect world, these twins would never talk to each other. But in this tiny city, the walls are so close that a little bit of light from Laser A accidentally leaks into Laser B, and vice versa. It's like the twins are whispering secrets across a crowded room.

The paper investigates what happens when these "whispers" get too loud. The researchers found that this tiny leakage causes the twins to start syncing up their rhythms. Instead of being chaotic and random, they begin to march in lockstep. By the end of each blink (a pulse lasting about 5 nanoseconds), they stop being random and start flashing the exact same way.

The Main Finding: A Race Against Time
The authors discovered that this "phase synchronization" happens incredibly fast—on a timescale of about 0.14 nanoseconds.

Think of it like this: When the twins first wake up (the start of the pulse), they are totally wild and random. But as soon as they start whispering to each other, they quickly calm down and agree on a single rhythm. This is bad news for the QRNG because if the twins are synchronized, the "randomness" they produce disappears, leaving the system with less entropy (less true randomness) to generate secure keys.

What They Ruled Out
The paper explicitly argues against the idea that this weird behavior is caused by common glitches like "chirp" (a change in pitch), "jitter" (timing errors), or "relaxation oscillations" (wobbly starts). While those things exist, the authors show that the specific distortion they saw in the data—where the randomness fades away at the end of the pulse—is caused specifically by the coupling (the whispering) between the two lasers. They also ruled out that the heat from running two lasers at once was the main culprit; the temperature change was only about 2 °C, which wasn't enough to explain the effect.

How They Knew: A Mix of Real Data and Digital Twins
The researchers didn't just guess; they did two things:

Real Experiments: They built the device and measured the light pulses at speeds of 100 MHz and 1 GHz. They saw the randomness fade away in the real world.
Computer Simulations: They created a digital model of the lasers using "stochastic rate equations" (math that accounts for random noise). They fed the exact same electrical signals into the computer model as they did in the lab.

The results matched perfectly. The computer simulation showed the twins syncing up in the exact same way the real lasers did. This gives them high confidence that their model is correct.

The "Magic" Math Tool
One of the coolest parts of the paper is a new mathematical tool they built. Since you can't directly see the "phase" (the timing of the light wave) without breaking the system, they developed a way to look at the intensity (the brightness) of the light and work backward to figure out the phase.

They found that the phase doesn't just stay random or just stay locked. It follows a specific journey:

At the start: The phase difference is spread out like a uniform circle (totally random).
During the whisper: It turns into a "wrapped Cauchy distribution" (a specific shape where the randomness starts to collapse).
At the end: It settles into a "von Mises distribution" (a tight peak where the twins are locked in sync).

The Bottom Line
The paper suggests that to make these quantum random number generators work perfectly, we need to stop the lasers from whispering to each other. The authors didn't solve the problem of how to stop the whispering yet, but they provided the map to find it. They proved that the "leakage" between the lasers is the thief stealing the randomness, and they gave us the tools to measure exactly how much randomness is left at any given moment.

Key Numbers to Remember:

Pulse Speed: The lasers blink at 100 MHz (for detailed study) and 1 GHz (for normal operation).
Sync Speed: The lasers synchronize in about 0.14 nanoseconds.
Leakage: Only about 0.5% of the light leaks from one laser to the other (a coupling ratio of -23 dB), yet this tiny amount is enough to cause the problem.
Temperature: The heat difference between running one laser and two was only +2 °C, proving heat wasn't the main villain.

In short, the paper shows that even a tiny, almost invisible connection between two quantum light sources can ruin their randomness, and it gives us the math to catch them in the act.

Technical Summary: Phase Synchronization Dynamics of Two Mutually Coupled InP Lasers in a Quantum Entropy Source

Problem Statement
Quantum Random Number Generators (QRNGs) based on photonic integrated circuits (PICs) rely on the quantum phase diffusion of lasers to generate entropy. In the specific architecture studied here, two indium phosphide (InP) distributed feedback (DFB) lasers are gain-switched and their outputs interfere to produce random bits. Ideally, the optical phases of these lasers should be completely random, resulting in an interference intensity distribution that follows an arcsine law. However, experimental observations at 1 GHz modulation rates revealed deviations from this ideal behavior, specifically a narrowing of the probability density function (PDF) toward the end of the laser pulse. The authors hypothesize that this distortion arises from residual optical coupling between the two lasers caused by reflections within the PIC's multimode interferometer (MMI). This coupling can induce phase synchronization, where the lasers adjust their rhythms to maintain a constant phase difference, thereby reducing the extractable quantum entropy. While coupled laser dynamics are well-studied in steady-state regimes, the transient dynamics of phase synchronization during the short pulses used in QRNGs remain unexplored.

Methodology
The study employs a combined approach of experimental characterization and numerical simulation to quantify the impact of laser coupling on entropy generation.

Experimental Setup: The device is a monolithically integrated InP PIC containing two DFB lasers, an MMI, and photodetectors. The lasers are modulated at 100 MHz (to resolve transient effects more clearly than at the standard 1 GHz) with a 50% duty cycle. The system is tuned to zero detuning using integrated metallic heaters (IMHs). The interference signal is captured by a high-speed oscilloscope, and the PDF of the interference voltage is reconstructed from $2 \cdot 10^5$ pulses at various time steps within the pulse duration.
Physical Modeling: The system is modeled using stochastic rate equations for the optical fields and carrier densities of two coupled lasers. The model incorporates spontaneous emission noise and symmetric coupling terms. Crucially, the coupling delay ( $\approx 19$ ps) is treated as instantaneous relative to the relaxation oscillation period ( $\approx 110$ ps) and the modulation period.
Analytical Framework: To link the measurable interference intensity to the unmeasurable optical phase difference ( $\Delta\phi$ ), the authors derive an analytical model for the PDF of the interference intensity. They propose that the phase difference distribution evolves from a uniform distribution (ideal randomness) to a wrapped Cauchy distribution during the transient synchronization phase, and finally to a von Mises distribution in the steady state. The model uses the convolution of these two distributions to describe the full temporal evolution.
Parameter Estimation: Key parameters, including coupling strength ( $\eta$ ), spontaneous emission factor ( $\beta$ ), and carrier/photon lifetimes, are estimated by fitting the simulation results to experimental data. The fitting process minimizes the Jensen-Shannon divergence between measured and modeled PDFs.

Key Results

Transient Synchronization: Both experiments and simulations confirm that the interference intensity distribution evolves over time. At the beginning of the pulse ( $t \approx 0.3$ ns), the distribution resembles the ideal arcsine shape. However, within approximately 1 ns, the distribution narrows into a peak-like structure, indicating a reduction in phase randomness.
Phase Dynamics: The analysis of the fitted parameters ( $\gamma_{\Delta\phi}$ and $\kappa_{\Delta\phi}$ ) reveals that the phase difference distribution is initially dominated by the wrapped Cauchy component (characterizing the transient) and transitions to a von Mises distribution (characterizing the synchronized steady state).
Synchronization Time: The system synchronizes on a sub-nanosecond timescale. The extracted synchronization time constant is $\tau_{synch} \approx 0.14$ ns for both experimental and simulated data, confirming that the coupling effect is rapid enough to impact the entropy generation within the pulse duration.
Model Validation: The proposed analytical model, which accounts for the convolution of wrapped Cauchy and von Mises distributions, successfully reproduces the experimental PDFs of the interference signal. This allows for the indirect reconstruction of the optical phase difference distribution solely from intensity measurements.
Coupling Regime: The coupling strength is quantified at approximately -23 dB (0.5% power transfer), placing the system in the weak coupling regime. Despite being "weak," this coupling is sufficient to induce phase locking within the pulse duration.

Significance and Claims
The paper claims to bridge a gap in the understanding of QRNGs by identifying and quantifying laser coupling as a source of entropy reduction in phase-diffusion systems. The primary contribution is the development of a rigorous physical model that links the distortion of the interference intensity distribution to the transient phase synchronization dynamics of coupled lasers.

The authors assert that their analytical model provides a practical tool for certifying the quantum origin of randomness in such devices. By enabling the direct extraction of the quantum phase difference distribution from the measured interference signal, the work lays the groundwork for optimizing QRNG operating conditions to minimize synchronization effects and maximize entropy extraction. The study emphasizes that while the coupling is weak, its transient effects are significant enough to distort the probability distribution, necessitating careful modeling for the certification of quantum entropy sources.

More like this