Photon statistics analysis of h-BN quantum emitters with pulsed and continuous-wave excitation

This study characterizes the photon statistics of hexagonal boron nitride quantum emitters under pulsed and continuous-wave excitation across various temperatures, demonstrating that the Mandel Q parameter remains largely temperature-independent, aligns with theoretical two-level emitter models when accounting for collection efficiency, and effectively supports high-speed random number generation.

The Gist

Imagine you have a tiny, magical light bulb embedded in a sheet of hexagonal boron nitride (h-BN). This isn't a normal light bulb that glows continuously; it's a Quantum Light Bulb. Its superpower is that it can be coaxed into releasing exactly one particle of light (a photon) at a time, on demand.

Scientists love these because they are the building blocks for future technologies like ultra-secure communication and super-fast quantum computers. But to use them, you need to be sure they are actually behaving like "one-at-a-time" bulbs and not accidentally spitting out two photons or none at all.

This paper is about a team of scientists checking the "personality" of these light bulbs to see how perfectly they follow the "one-at-a-time" rule. They used a special scorecard called the Mandel Q parameter to grade them.

Here is the breakdown of their adventure, explained simply:

1. The Scorecard: What is "Mandel Q"?

Think of the Mandel Q parameter as a traffic cop for light.

• Q = 0: This is a standard laser. It's like a steady stream of rain. The drops fall regularly, but sometimes you get two drops at once, and sometimes you get a gap. It's predictable but not "perfectly single."
• Q > 0: This is a thermal light source (like a light bulb or the sun). It's chaotic, like a crowd of people rushing through a door. You get huge bursts of light followed by silence.
• Q = -1: This is the Holy Grail. It means the light source is a perfect single-photon emitter. It releases exactly one photon, then waits, then releases exactly one more. No more, no less.

The scientists wanted to see how close their h-BN emitters could get to that perfect -1 score.

2. The Two Ways to Turn on the Light

The team tested the light bulbs in two different ways:

• Pulsed Excitation (The Flashlight Method): They hit the bulb with a quick, sharp flash of laser light, like a camera strobe. They waited for the bulb to blink, then hit it again.
  • Result: They got a score of -0.002. This is very close to perfect! It means the bulb is almost always releasing just one photon per flash.
• Continuous Wave (The Stream Method): They turned the laser on and left it on, creating a steady stream of light to keep the bulb glowing.
  • Result: They got a score of -0.0025. Even with a constant stream, the bulb managed to keep its "one-at-a-time" discipline.

3. The Temperature Test: Does Cold Make it Better?

Usually, when you cool down electronics, they work better. The scientists put their light bulbs in a cryogenic freezer (down to -266°C or 7 Kelvin) to see if the cold made the "one-at-a-time" behavior more perfect.

• The Surprise: The temperature didn't change much. The score stayed roughly the same whether it was hot or freezing cold.
• The Analogy: Imagine a dancer. You might think they would dance better in a quiet, cold room than a hot, crowded one. But this dancer (the light bulb) was already so good that the temperature didn't really matter. They were consistent in both environments.

4. The "Dead Time" Problem

One tricky part of this experiment is the detector (the camera taking the photos). After the camera sees a photon, it needs a tiny moment to "reset" before it can see the next one. This is called dead time.

• If photons arrive too fast, the camera misses the second one, making it look like the light source is more perfect than it actually is.
• The scientists were careful to adjust their timing so the photons arrived slowly enough that the camera could catch every single one, ensuring their score was honest.

5. The Simulation: The "Virtual Twin"

To prove their measurements were real and not just a fluke, they built a virtual twin of the light bulb on a computer. They used complex math (the Extended Jaynes-Cummings Model) to simulate how a single atom should behave when excited.

• When they compared the real bulb's score to the virtual twin's score, they matched perfectly. This confirmed that their "Quantum Light Bulb" is indeed behaving exactly as quantum physics predicts.

6. The Real-World Application: Making Random Numbers

Why does this matter? The scientists showed a practical use: Random Number Generation.

• The Problem: Computers are terrible at making truly random numbers. They usually follow a pattern.
• The Solution: Quantum physics is naturally random. If you have a perfect single-photon source, you can use it to generate true randomness.
• The Experiment: They tried two methods to generate random bits (0s and 1s).
  • Method 1: Just count every photon that hits a detector. (Result: Failed some randomness tests).
  • Method 2: They used their Mandel Q score to pick the perfect time window where the bulb was most likely to send exactly one photon. (Result: Passed all randomness tests!)

The Takeaway: The better the Mandel Q score (the closer to -1), the faster and more reliable you can generate random numbers. It's like having a better dice roller; you get a true random result faster.

Summary

This paper is a report card for a new type of quantum light bulb. The scientists proved that:

1. These bulbs are excellent at releasing single photons (getting a near-perfect score on the Mandel Q test).
2. They work well whether you flash them or shine a steady light on them.
3. They work just as well in a freezer as they do at room temperature.
4. This "perfection" is crucial for making secure, random numbers for the next generation of quantum technology.

In short: They found a tiny, reliable, one-photon-at-a-time light switch that works great, and they proved it with math, cold temperatures, and a bit of digital magic.

Technical Summary

Here is a detailed technical summary of the paper "Photon statistics analysis of h-BN quantum emitters with pulsed and continuous-wave excitation."

1. Problem Statement

Hexagonal boron nitride (h-BN) is a promising 2D material for quantum applications due to its stable single-photon emitters (SPEs) at room temperature and high Debye-Waller factors. While the quantum nature of these emitters is often confirmed using the second-order correlation function, $g^{(2)}(0)$, this metric alone does not fully characterize the photon statistics required for specific quantum applications like random number generation (RNG) or quantum computing.

The Mandel Q parameter, which quantifies the deviation of photon statistics from a Poissonian distribution (where $Q=0$), is a more direct measure of sub-Poissonian statistics ($Q < 0$). However, previous studies have largely focused on pulsed excitation, and the behavior of Mandel Q under continuous-wave (CW) excitation remains under-explored. Furthermore, the influence of experimental factors—such as photon collection efficiency, detector dead time, temperature, and pump power—on the measured Mandel Q value needs rigorous characterization to validate h-BN emitters for practical on-demand single-photon sources.

2. Methodology

The authors employed a comprehensive experimental and theoretical approach to analyze h-BN quantum emitters:

• Sample Preparation: Exfoliated h-BN flakes were placed on Si/SiO$_2$ substrates and annealed. Emitters were identified via confocal microscopy, specifically targeting those with a narrow zero-phonon line (ZPL) around 585–588 nm.
• Experimental Setup:
  • Excitation: Both pulsed (532 nm, 100 ns pulse-to-pulse) and continuous-wave (CW) 532 nm lasers were used.
  • Detection: A Hanbury Brown-Twiss (HBT) interferometer setup with two single-photon avalanche diodes (SPADs) was used. Time-tagging electronics recorded photon arrival times.
  • Conditions: Measurements were conducted at room temperature and in a cryogenic microscope (7 K to 250 K).
  • Data Processing: Time-gating (removing photons arriving >10 ns after the pulse) was used to minimize background noise. For CW excitation, specific time-bin sizes were optimized to calculate Mandel Q.
• Theoretical Modeling: An extended Jaynes-Cummings Model (eJCM) was developed. This model treats the emitter as a two-level system interacting with multiple field modes (discretized Fourier transform of a temporal pulse). It incorporates:
  • Spontaneous emission dynamics.
  • Initial thermal noise.
  • Excitation efficiency ($p_E$).
  • Experimental losses modeled via an imbalanced beam splitter (simulating the ~0.5% collection efficiency).

3. Key Contributions

• Comprehensive Mandel Q Characterization: The study provides one of the few detailed analyses of Mandel Q for h-BN emitters under both pulsed and CW excitation regimes.
• CW Excitation Analysis: The authors demonstrate a method to measure sub-Poissonian statistics under CW excitation by optimizing time-bin sizes and accounting for detector dead time, a regime previously less studied.
• Temperature Independence: The paper investigates the temperature dependence of photon statistics from 7 K to 250 K, concluding that Mandel Q is largely independent of temperature within experimental error.
• Simulation-Experiment Validation: A custom simulation (using Boeing's Quantum Design Studio) based on the eJCM successfully reproduces experimental Mandel Q values when accounting for collection efficiency, validating the two-level emitter model.
• Application to Random Number Generation (RNG): The authors link the Mandel Q parameter directly to the performance of quantum RNG, demonstrating how $Q$ values influence bit generation rates and randomness quality.

4. Key Results

• Pulsed Excitation:
  • Achieved a Mandel Q of -0.002.
  • Comparison with control sources confirmed the sub-Poissonian nature: Laser light yielded $Q \approx 0$ (within error), and thermal light yielded $Q > 0$.
  • Theoretical simulations matched experimental results when the low collection efficiency ($\eta \approx 0.005$) was included, explaining why $Q$ did not reach the ideal value of -1.
• Temperature Dependence:
  • Measurements at 7 K, 50 K, 100 K, 150 K, 200 K, and 250 K showed that Mandel Q fluctuates weakly around zero (minimum -0.0002).
  • The lack of strong temperature dependence suggests the emitter's photon statistics are robust across a wide thermal range.
• CW Excitation:
  • Under CW pumping, the minimum Mandel Q ($Q_{min}$) was found to be -0.0025.
  • Pump Power Effect: $Q_{min}$ showed a non-monotonic dependence on pump power. This was attributed to "blinking" (ON/OFF state switching). When analyzing only the "ON" states, $Q_{min}$ was inversely correlated with pump power (higher power $\rightarrow$ lower/negative $Q$).
  • Time Binning: An optimal time bin size ($T_{Qmin}$) was identified to maximize the sub-Poissonian character.
• Random Number Generation (RNG):
  • Two protocols were tested: (1) Direct channel assignment (Channel 0 = 0, Channel 1 = 1) and (2) Time-binned single-photon selection.
  • NIST Testing: Protocol 1 failed 4 out of 16 NIST randomness tests. Protocol 2 (using time bins optimized for $Q_{min}$) passed all 16 tests.
  • Generation Rate: A clear correlation was found: as Mandel Q becomes more negative (closer to -1), the rate of generating valid random bits increases. This is because a lower $Q$ implies a higher probability of exactly one photon per interval, reducing the need to discard multi-photon or zero-photon events.

5. Significance

This work establishes the Mandel Q parameter as a critical metric for evaluating the utility of solid-state quantum emitters beyond simple antibunching ($g^{(2)}(0)$).

• Practical Utility: It demonstrates that while h-BN emitters are excellent single-photon sources, their efficiency in applications like RNG is heavily dependent on minimizing noise and optimizing excitation conditions to drive $Q$ closer to -1.
• Design Guidance: The study provides a roadmap for optimizing quantum devices, showing that pump power can be used as a "knob" to tune photon statistics in CW regimes.
• Validation of Models: The agreement between the simplified eJCM simulation and experimental data confirms that h-BN emitters can be effectively modeled as two-level systems for statistical analysis, provided experimental losses are accounted for.
• Application Impact: By linking Mandel Q directly to RNG performance, the paper offers a quantitative framework for selecting and tuning quantum emitters for high-speed, high-fidelity quantum information processing tasks.