Non-Gaussian correlations in the steady-state of driven-dissipative clouds of two-level atoms

This paper reports experimental evidence that a laser-driven dense ensemble of $^{87}$Rb atoms emits light with non-Gaussian statistics, characterized by high-order correlations in the steady state despite the absence of first-order coherence.

The Gist

Imagine you are at a massive, crowded concert. Usually, when a crowd cheers, the sound is a chaotic, random roar. If you listen to two people in the crowd, their cheering is independent; if one stops, the other keeps going. In physics, this is called "Gaussian" or "chaotic" light. The relationship between how loud the crowd is (intensity) and how synchronized the cheering is (coherence) follows a strict, predictable rule called the Siegert relation. It's like a mathematical law of averages: if the crowd is random, the noise follows a specific pattern.

However, in this new study, scientists at the University of Paris-Saclay discovered something strange happening with a cloud of atoms.

The Setup: A Tiny, Crowded Dance Floor

The researchers took about 5,000 Rubidium atoms (think of them as tiny, glowing dancers) and squeezed them into a cigar-shaped cloud. They then hit this cloud with a powerful laser, forcing all the atoms to dance to the same beat.

In a normal, loose crowd (a "dilute" cloud), the atoms don't talk to each other. They just glow randomly. But in this tight, dense cloud, the atoms are so close that they start to influence one another, almost like a group of people in a small elevator who can't help but react to each other's movements.

The Surprise: Breaking the Rules

The scientists measured the light coming out of this cloud. They expected the light to follow the standard "Siegert rule" of random noise. Instead, they found the rule was broken.

Here is the analogy:
Imagine you are trying to predict the weather by looking at raindrops.

• The Rule (Siegert Relation): If the rain is random, the number of drops hitting your umbrella in two seconds should be perfectly predictable based on how hard it's raining.
• The Discovery: In this experiment, the "rain" (the light) was hitting the umbrella in a pattern that defied the prediction. The drops were arriving in a way that suggested they were secretly coordinating with each other, even though they weren't supposed to be.

The Big Mystery: Is Someone Conducting the Orchestra?

When the scientists saw this "broken rule," they had two guesses:

1. The Conductor Theory: Maybe the laser had secretly forced the atoms to march in perfect lockstep, creating a single, giant, coherent beam of light (like a laser pointer). If this were true, the "noise" would look different because there would be a strong, organized signal.
2. The Secret Society Theory: Maybe the atoms are still chaotic on the surface, but they are forming a complex, hidden "secret society" of correlations. They aren't marching in a line, but they are whispering secrets to their neighbors, creating a complex pattern that looks random but isn't.

The Investigation: Ruling Out the Conductor

The scientists played detective to figure out which theory was right.

• They checked the "volume" of the light. If the atoms were marching in lockstep (a coherent field), the brightness would have skyrocketed in a specific way (quadratic growth). Instead, the brightness grew linearly, just like a normal, messy crowd. The Conductor was fired.
• They checked the "rhythm" of the light. If there was a strong, organized signal, the rhythm would stay steady. Instead, the rhythm died out quickly, just like random noise.

The Conclusion: The Secret Society Wins

Since the light wasn't a single, organized beam, but it was breaking the rules of randomness, the scientists concluded that the atoms were in a non-Gaussian state.

What does this mean in plain English?
It means the atoms have developed a complex, high-level "group chat." Even though they aren't all singing the same note (no first-order coherence), they are coordinating their timing and intensity in a way that is mathematically impossible for a random crowd. They have created a "non-Gaussian" state of light—a kind of light that is neither a perfect laser nor random static, but something in between that holds complex quantum information.

Why Should We Care?

This is a big deal for the future of technology.

• New Materials for Light: Just as we discovered new materials like graphene, this discovery shows we can create new types of light that don't follow the old rules.
• Quantum Computing: These "secret societies" of atoms could be used to store and process information in ways that current computers can't.
• Understanding the Universe: It proves that even in a messy, chaotic system (driven by a laser and losing energy), nature can spontaneously organize itself into complex, stable patterns without a leader.

In short, the scientists found a way to make a crowd of atoms "talk" to each other in a secret language, creating a new kind of light that breaks the laws of physics we thought were unbreakable.

Technical Summary

1. Problem Statement

The paper addresses the challenge of characterizing the statistical properties of light emitted by dense, driven-dissipative ensembles of atoms. While collective effects like superradiance are known to modify emission rates and intensity, the nature of the photon statistics and atomic correlations in the steady state of such systems remains poorly understood.

Specifically, the authors investigate whether the light emitted by a strongly driven, dense cloud of two-level atoms obeys Gaussian statistics. In standard chaotic light (from statistically independent emitters), the second-order coherence function $g^{(2)}(\tau)$ is related to the first-order coherence $g^{(1)}(\tau)$ via the Siegert relation:
$$g^{(2)}_N(\tau) = 1 + |g^{(1)}_N(\tau)|^2$$
A violation of this relation implies either the emergence of a coherent mean field ($\langle \hat{E}^- \rangle \neq 0$) or the presence of non-Gaussian statistics arising from higher-order correlations within the atomic medium. The goal is to determine which mechanism is at play in a driven-dissipative many-body system.

2. Methodology

The researchers utilized a dense cloud of approximately 5,000 $^{87}$Rb atoms confined in an optical dipole trap.

• System Preparation:

  • Atoms were prepared in a two-level configuration ($|g\rangle = |5S_{1/2}, F=2, m_F=2\rangle$ to $|e\rangle = |5P_{3/2}, F=3, m_F=3\rangle$) using optical pumping and a magnetic field.
  • The cloud had a cigar shape with a radial size $\ell_{rad} \approx 0.6\lambda$ and an axial size $\ell_{ax} \approx 20\text{--}25\lambda$.
  • The system was driven by a resonant, $\sigma^+$-polarized laser propagating perpendicular to the long axis of the cloud. The Rabi frequency $\Omega$ was kept high ($\Omega > 2\Gamma$), ensuring saturation of the excited state population.

• Measurement Setup:

  • Hanbury-Brown and Twiss (HBT) Interferometry: Fluorescence was collected in two directions: along the cloud axis ($\hat{u}_z$) and perpendicular to it ($\hat{u}_\perp$).
  • Photon arrival times were recorded using avalanche photodiodes (APDs) and a time-tagging module to compute the second-order coherence function $g^{(2)}_N(\tau)$.
  • Heterodyne Detection: Used to measure the first-order coherence $g^{(1)}_N(\tau)$ and verify the absence of a coherent mean field.
  • Steady-State Analysis: Data was collected during a 400 ns pulse, focusing on the final 250 ns where the system reached a steady state.

• Calibration:

  • Experiments were first performed on a dilute cloud (expanded in free flight) where atoms are statistically independent. In this regime, the measured $g^{(2)}_N(\tau)$ perfectly matched the Siegert relation, validating the detection system's resolution and confirming the absence of systematic errors.

3. Key Contributions and Results

A. Violation of the Siegert Relation

In the dense regime, when measuring along the cloud axis ($\hat{u}_z$), the authors observed a clear violation of the Siegert relation:
$$g^{(2)}_N(\tau) < 1 + |g^{(1)}_N(\tau)|^2$$
Specifically, $g^{(2)}_N(\tau)$ dropped below 1 at certain time delays, a phenomenon impossible for Gaussian chaotic light. This violation was direction-dependent: it occurred along the axis (where superradiance is observed) but not in the perpendicular direction, where the Siegert relation held.

B. Exclusion of a Coherent Mean Field

A violation of the Siegert relation could theoretically arise if the system developed a non-zero average electric field ($\langle \hat{E}^- \rangle \neq 0$). The authors ruled this out through two independent tests:

1. Intensity Scaling: The steady-state intensity $\langle \hat{I} \rangle$ scaled linearly with the atom number $N$. If a coherent field existed, intensity would scale quadratically ($\propto N^2$) due to constructive interference. The linear scaling implies $\langle \hat{E}^- \rangle \approx 0$.
2. First-Order Coherence: Measurements of $g^{(1)}_N(\tau)$ showed that the coherence decays to zero at long times ($\tau \gg 1/\Gamma$), consistent with the behavior of a single atom and inconsistent with a macroscopic coherent field.

C. Evidence of Non-Gaussian Statistics

Since the mean field is zero, the violation of the Siegert relation must stem from non-Gaussian statistics. The authors quantified this by calculating the connected (cumulant) correlations $C(\tau)$:
$$g^{(2)}_N(\tau) = g^{(2)}_{\text{Gauss}}(\tau) + C(\tau)$$
They found that $C(\tau) \neq 0$ for large atom numbers, indicating the presence of high-order atomic correlations ($\langle \hat{\sigma}^+_i \hat{\sigma}^+_j \hat{\sigma}^-_k \hat{\sigma}^-_l \rangle_c \neq 0$). These correlations emerge spontaneously in the steady state due to the competition between the driving laser and collective dissipation, without external imposition.

D. Robustness

The non-Gaussian behavior persisted across a wide range of driving strengths ($\Omega = 2\Gamma$ to $10\Gamma$) and atom numbers ($N \approx 500$ to $5000$), demonstrating that these correlations are a stable feature of the driven-dissipative steady state.

4. Significance

• Fundamental Physics: The work provides the first experimental evidence that non-Gaussian correlations can stabilize in the steady state of a driven-dissipative many-body system in free space. This challenges the assumption that such systems inevitably relax to Gaussian or mean-field descriptions.
• Superradiance Connection: It links the phenomenon of superradiance (typically associated with intensity enhancement) to the emergence of complex quantum correlations that modify photon statistics.
• Quantum Resources: The ability to generate non-Gaussian states of light without a coherent mean field opens new avenues for quantum information processing. Non-Gaussian states are essential resources for quantum error correction and continuous-variable quantum computing.
• Future Directions: The authors propose measuring the Wigner function of the radiated field to check for negativity (a definitive signature of non-classicality) and extending these studies to the transient dynamics of superradiant bursts.

In summary, this paper demonstrates that a dense, driven atomic cloud acts as a source of non-Gaussian light characterized by high-order correlations, fundamentally altering the statistical properties of the emitted radiation beyond standard chaotic light models.