Emergence of second-order coherence in superfluorescence

This paper experimentally demonstrates that second-order quantum coherence emerges during the decay of a superfluorescent burst from a chirally coupled ensemble of cesium atoms, revealing striking similarities between the collective dynamics of cascaded and symmetrically coupled superradiant systems.

The Gist

Imagine a crowded room full of 900 people (the atoms), each holding a tiny flashlight (the quantum state). Usually, if you tell everyone to turn on their flashlights at once, they just flicker on randomly. The room gets a bit brighter, but the light is messy and uncoordinated. This is how normal atoms behave.

But in this experiment, the scientists did something special. They put these "people" in a very narrow hallway (an optical nanofiber) where the rules of the hallway force them to talk to each other in a specific order. If Person #1 speaks, Person #2 hears them. If Person #2 speaks, Person #3 hears them. But Person #3 cannot speak back to Person #1. This is called a cascaded system—a one-way street for information.

Here is the story of what happened when they turned on the lights:

1. The Setup: A One-Way Street

The scientists trapped cesium atoms near a super-thin glass fiber. Because of the physics of light in this tiny space, the atoms could only "talk" to the atoms in front of them, not behind them. It's like a line of dominoes: if the first one falls, it knocks over the second, which knocks over the third, and so on. But the third domino can't push the first one back.

They then used a laser pulse to "wind up" all the atoms, putting them in a state where they were ready to release a burst of energy. They prepared them in two different ways:

• The "Perfectly Wound" State: Everyone is ready to fire, but they are all slightly out of sync with each other.
• The "Slightly Off" State: Everyone is ready, but the laser left them with a little bit of a shared rhythm.

2. The Big Surprise: The "Silent" Start

When they let the atoms go, they expected a chaotic mess of light.

• In the "Perfectly Wound" state: At the very first split-second, the light was indeed chaotic. It was like a crowd of people shouting randomly. The scientists measured this chaos using a tool called the second-order coherence function (let's call it the "Chaos Meter"). A reading of 2 meant "total chaos" (random, independent flashes).
• The Magic Shift: As the light burst continued, something amazing happened. The Chaos Meter dropped from 2 down to 1.
  • What does this mean? It means the atoms suddenly stopped shouting randomly and started singing in perfect harmony. Even though they were in a one-way line where they couldn't talk back and forth, the act of releasing light together forced them to synchronize. They went from a chaotic crowd to a choir.

3. The "Off-Rhythm" State

When they started with the "Slightly Off" state (where the atoms already had a shared rhythm from the laser), the Chaos Meter started at 1 and stayed there. The light was already coordinated, so it stayed coordinated. It was like a choir that was already singing in tune; they just kept singing in tune.

4. The "Shot-to-Shot" Mystery

The scientists also noticed something weird about the timing of the bursts.
Imagine you have a group of 900 people trying to clap in unison. Even if you tell them to clap "now," sometimes the group waits 5 seconds, sometimes 7 seconds, before they all clap together.

• In the experiment, the scientists fired the laser millions of times.
• They found that the exact moment the "big burst" of light started varied slightly every single time.
• By looking at the correlation between early photons and late photons, they proved that this delay wasn't a mistake in their machine. It was a fundamental quantum quirk. The burst is triggered by a tiny, random "nudge" from the vacuum of space (vacuum fluctuations), which is why the start time is slightly different every time.

5. The Big Picture: Why This Matters

For a long time, physicists thought that for atoms to synchronize and create a super-bright burst of light (called superfluorescence or superradiance), they needed to be able to talk to everyone at once (like a giant circle where everyone hears everyone).

This paper proves that you don't need a circle. You can have a one-way line (a cascade), and the atoms will still learn to dance together.

• The Analogy: It's like a line of people passing a secret note. Even though they can only pass it forward, the act of passing the note eventually makes everyone in the line move in perfect sync.

Summary

The scientists took 900 atoms, forced them into a one-way line, and watched them go from a chaotic mess of random light to a perfectly synchronized beam of light. They proved that even without a "feedback loop" (talking back), nature finds a way to make things work together. It's a beautiful demonstration of how individual particles can spontaneously organize into a collective masterpiece, even in a system designed to be one-way.

Technical Summary

1. Problem Statement

The paper addresses the fundamental dynamics of collective light emission in quantum systems, specifically focusing on superfluorescence (the spontaneous build-up of coherence in an initially incoherent, inverted ensemble) and superradiance. While the phenomenon of superradiant bursts has been studied extensively in symmetrically coupled systems (the Dicke model), there is a lack of experimental understanding regarding the photon statistics and second-order quantum coherence ($g^{(2)}$) in cascaded quantum systems.

In a cascaded system, emitters interact unidirectionally (atom $i$ influences atom $j$ only if $j > i$), unlike the symmetric coupling in the standard Dicke model where all atoms couple equally to a common mode. The authors aim to determine:

• How the second-order coherence function $g^{(2)}(t_1, t_2)$ evolves during the decay of a superradiant burst in a cascaded system.
• Whether the spontaneous synchronization of dipoles (characteristic of superfluorescence) occurs in the absence of feedback loops inherent to symmetric coupling.
• How the initial state (specifically the average dipole moment) influences the coherence dynamics.
• The nature of shot-to-shot fluctuations in the burst delay.

2. Methodology

Experimental Setup

• System: The experiment utilizes an optical nanofiber (ONF) with a diameter smaller than the guided light wavelength. Approximately 900 cesium atoms are optically trapped in the evanescent field of the fiber.
• Chiral Coupling: Due to spin-momentum locking, the atoms are coupled chirally to the guided mode. The polarization of the atomic transition ($|g\rangle \to |e\rangle$) overlaps significantly ($\beta \approx 0.01$) with the forward-propagating mode but negligibly with the backward mode. This creates a cascaded quantum system where emission from an atom propagates forward to influence subsequent atoms.
• State Preparation:
  • Atoms are cooled to the motional ground state.
  • A resonant Rabi pulse (duration 4 ns) excites the ensemble. By varying the pulse area $A$, the authors prepare the ensemble in different initial states ranging from partial inversion to maximal inversion (where the average dipole moment vanishes).
  • Maximal inversion corresponds to the product state $|\psi_0\rangle \approx \bigotimes |e_k\rangle$, where atoms are initially independent.
• Detection:
  • A Hanbury-Brown and Twiss (HBT) interferometer setup is used to measure the two-time second-order coherence function, $g^{(2)}(t_1, t_2)$.
  • The burst light is split, with one path delayed by ~100 ns, allowing the measurement of correlations between photons emitted at different times ($t_1$ and $t_2$).
  • Photon coincidence rates are recorded over $10^6$ excitation cycles to build statistical averages.

Theoretical Modeling

• Truncated Wigner Approximation (TWA): The authors employ a stochastic simulation based on the TWA for spins. This method maps the quantum dynamics of $N$ emitters to a set of $2N$ coupled nonlinear stochastic differential equations. It is capable of handling large ensembles ($N \approx 900$) and accurately predicts both the optical power and $g^{(2)}$ dynamics, including absorption effects and inhomogeneous coupling.
• Symmetric Dicke Model: For comparison, the authors numerically solve the standard symmetric Dicke model (assuming perfect coupling, $\beta=1$) to contrast the dynamics of cascaded vs. symmetric systems.

3. Key Contributions and Results

A. Emergence of Second-Order Coherence

• Initial State ($t=0$):
  • When the ensemble is prepared at maximal inversion ($A \approx 1.1\pi$), the average dipole moment is zero. The initial emission exhibits bunching with $g^{(2)}(0,0) = 2$, characteristic of independent, incoherent emitters (thermal statistics).
  • When the pulse area deviates from maximal inversion ($A \neq 1.1\pi$), a non-zero average dipole moment exists. The atoms are phase-locked to the driving laser, resulting in coherent emission with $g^{(2)}(0,0) \approx 1$ immediately.
• Dynamics during the Burst:
  • Maximal Inversion Case: As the burst evolves, the authors observe a clear transition where $g^{(2)}(t,t)$ decreases from 2 to 1. This signifies the spontaneous build-up of second-order coherence. The initially independent atoms synchronize via the shared forward-propagating mode, transitioning from a product state to a collective state.
  • Non-Maximal Inversion Case: If the initial dipole moment is non-zero, $g^{(2)}(t,t)$ remains constant at $\approx 1$ throughout the burst, as the coherence is imprinted by the initial excitation rather than emerging spontaneously.
• Threshold Behavior: TWA simulations reveal that this coherence build-up only occurs when the atom number $N$ exceeds a threshold ($N_{thr} \approx 100$). Below this threshold, the system decays independently, and $g^{(2)}$ remains constant at 2.

B. Comparison with Symmetric Dicke Model

• Despite the fundamentally different Hamiltonian (unidirectional cascaded coupling vs. symmetric all-to-all coupling), the cascaded system exhibits striking similarities to the Dicke model.
• Both systems show the emergence of coherence from an incoherent start and a reduction of $g^{(2)}$ from 2 to 1 during the burst.
• However, the TWA model (cascaded) provides a more accurate fit to the experimental time dynamics than the symmetric Dicke model, which requires an effective atom number scaling ($N_{Dicke} \approx \beta N$) to match qualitative features.

C. Shot-to-Shot Fluctuations

• By analyzing $g^{(2)}(t_1, t_2)$ for $t_1 \neq t_2$, the authors detect anti-correlations ($g^{(2)} < 1$) for large time delays.
• Interpretation: This anti-correlation is evidence of fluctuations in the delay time of the burst onset from shot to shot. Since the burst is triggered by vacuum fluctuations (in the maximal inversion case), the exact start time varies. When averaging many shots, the burst appears broadened, but in a single realization, the probability of detecting two photons separated by a time longer than the burst duration is low. This confirms the stochastic nature of the superfluorescence trigger.

4. Significance

1. Validation of Collective Dynamics in Cascaded Systems: The work demonstrates that the hallmark features of superfluorescence—specifically the spontaneous emergence of coherence and the transition from thermal ($g^{(2)}=2$) to coherent ($g^{(2)}=1$) statistics—are robust even in the absence of feedback loops, provided the system is unidirectionally coupled.
2. Benchmarking Quantum Simulations: The experimental data serves as a rigorous benchmark for advanced many-body simulation techniques like the Truncated Wigner Approximation, validating their ability to model complex, large-scale quantum optical systems.
3. Fundamental Quantum Optics: The study clarifies the role of the initial dipole moment in determining whether coherence is imprinted (laser-driven) or emergent (superfluorescence).
4. Future Applications: Understanding the photon statistics and coherence properties of cascaded systems is crucial for developing quantum networks, quantum memories, and potentially superradiant lasers where the emission is driven by collective atomic dynamics rather than cavity feedback.

In summary, the paper provides the first experimental observation of the spontaneous emergence of second-order coherence in a cascaded quantum system, bridging the gap between theoretical predictions for symmetric and chiral quantum optics and confirming that collective behavior is a universal feature of dense quantum emitters.