⚛️ quantum physics

The Entropy Flow of a Laser Beam

The paper demonstrates that an ideal laser beam with phase diffusion possesses an extensive entropy flow defined by the remarkably simple relation $\dot{S} = k_B \sqrt{\dot{N}\ell}$ , where $\dot{N}$ is the photon number flow and $\ell$ is the spectral width.

Original authors: Howard M. Wiseman

Published 2026-02-12

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CC BY 4.0

Original authors: Howard M. Wiseman

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

The "Wandering Spotlight" Problem: Explaining the Entropy of a Laser

Imagine you are at a concert. There are two types of light hitting the crowd.

First, there is the ambient glow of the room—a messy, chaotic mix of light bouncing off walls, people, and dust. This is "thermal light." It’s like a crowd of people all talking at once in a noisy cafeteria; there is no pattern, just a soup of information and disorder.

Then, there is the laser beam. It is a tight, intense, organized spear of light. For a long time, physicists treated a laser like a perfect, unchanging arrow. They assumed it was "pure"—meaning it had zero disorder (zero entropy).

But this paper argues that even the most perfect laser has a "secret" amount of messiness.

1. The "Wandering Spotlight" (The Problem)

Think of a laser not as a frozen arrow, but as a spotlight being held by a person who is standing on a boat in a slightly choppy ocean.

The light is incredibly bright and stays in one direction, but because of the tiny, microscopic "waves" of quantum physics, the phase (the internal timing or "rhythm") of the light is constantly wobbling. It’s not moving left or right, but its internal clock is drifting.

Because that rhythm is constantly drifting, the laser isn't actually "pure." It is a "mixture" of many slightly different rhythms. In physics terms, this means it has entropy—a measure of disorder or "lost information."

2. The Discovery (The Math)

The author, Howard Wiseman, set out to calculate exactly how much "disorder" flows out of a laser beam every second.

He found a surprisingly beautiful and simple rule. He discovered that the amount of disorder (entropy flow) depends on two things:

How many photons are flowing (How bright the light is).
The linewidth (How much the "rhythm" is wobbling).

The formula he found is essentially: The disorder is the square root of (Brightness $\times$ Wobble).

3. The "Branching Paths" Analogy (The Intuition)

Why does this formula make sense? Imagine you are watching a single particle of light traveling through the beam.

Because of that tiny "wobble," every once in a while, the light's rhythm shifts just enough that it becomes "distinguishable" from what it was a moment ago. It’s like a traveler walking down a path that constantly splits into two new, slightly different paths.

The more the light "splits" into these different possible rhythms, the more "information" is lost, and the higher the entropy. The paper calculates exactly how often these "splits" happen.

4. Laser vs. Heat (The Comparison)

The most striking part of the paper is comparing the laser to a "thermal beam" (like the light coming off a hot radiator).

Thermal Light is like a massive, disorganized riot. It carries a huge amount of entropy because it is fundamentally chaotic.
Laser Light is like a highly disciplined marching band. Even though the band members might be slightly out of step (the wobble), they are still incredibly organized compared to the riot.

The paper shows that for the same amount of power, a laser is vastly more orderly than thermal light. In fact, the difference is so massive that it’s like comparing a single, perfectly timed heartbeat to the roar of a hurricane.

Why does this matter?

If we want to build ultra-precise tools—like the atomic clocks that run our GPS systems—we need to understand every tiny bit of "wobble" and "disorder" in our light sources. This paper provides the mathematical "speedometer" to measure exactly how much order is leaking out of our most precise light beams.

Technical Summary: The Entropy Flow of a Laser Beam

Author: Howard M. Wiseman
Subject Area: Quantum Optics / Statistical Mechanics

1. Problem Statement

In standard quantum optics textbooks, a laser beam is frequently modeled as a pure coherent state $|\alpha\rangle$ . However, this is an idealization that ignores the fundamental quantum noise inherent in any laser. In reality, even a laser without technical noise undergoes phase diffusion due to quantum fluctuations. This phase wandering means the laser is actually in a mixed state, not a pure one.

The author identifies a significant gap in the literature: while the coherence and spectral properties of lasers are well-studied, the entropy flow ( $\dot{S}$ )—the rate at which entropy is carried by the beam—has never been formally calculated for such a state.

2. Methodology

To solve this, the author employs several advanced mathematical and physical frameworks:

Model Definition: The laser is modeled as a constant-amplitude beam with a wandering phase $\phi(t)$ . The phase follows a stochastic process (a Wiener process) representing pure phase diffusion, characterized by a linewidth $\ell$ .
Rényi 2-Entropy: Because the von Neumann entropy ( $S_1$ ) is mathematically difficult to compute for this continuous-time process, the author utilizes the Rényi 2-entropy ( $S_2$ ). In the asymptotic limit (long time $T$ ), the difference between $S_1$ and $S_2$ is merely a constant factor, making $S_2$ a valid proxy for determining the entropy flow.
Feynman–Kac Technique: The calculation of the trace of the squared density matrix ( $\text{Tr}[\rho^2]$ ) is transformed into a problem of evaluating an expectation value over a Wiener process. This is solved by mapping the problem onto a differential equation (the Feynman–Kac formula), which is shown to be mathematically equivalent to a 1D Schrödinger equation with a $\sin^2$ potential.
Harmonic Approximation: In the high-coherence limit (where the number of photons per coherence time $C$ is extremely large), the $\sin^2$ potential is approximated as a harmonic oscillator potential to find the lowest eigenvalue ( $\lambda_0$ ), which dictates the decay rate of the density matrix and thus the entropy.

3. Key Results

The paper derives a remarkably simple and elegant expression for the entropy flow of an ideal laser beam:

$\dot{S}_2 = k_B \sqrt{\dot{N}\ell}$

Where:

$k_B$ is the Boltzmann constant.
$\dot{N}$ is the mean photon flow (photons per second).
$\ell$ is the spectral linewidth (FWHM).

Intuitive Interpretation: The author provides a physical explanation: the entropy increases when the phase wanders enough to make the coherent state "distinguishable" from its previous state. This occurs at a rate determined by the geometric mean of the photon flow and the linewidth.

4. Comparative Analysis

The author compares the laser's entropy flow to that of a unidirectional thermal beam of the same power $P$ .

Thermal Beam Entropy: Scales with the square root of power ( $\dot{S}_\Theta \propto \sqrt{P}$ ).
Laser Beam Entropy: Scales with $\sqrt{P/\hbar} \cdot \sqrt{\ell/\omega_0}$ .

The comparison reveals that for a given power, a laser beam has significantly lower entropy than a thermal beam. The difference is proportional to $Q^{-1/2}$ , where $Q$ is the beam's quality factor ( $\omega_0/\ell$ ). For high-end lasers where $Q \approx 10^{10}$ , the laser's entropy flow is orders of magnitude smaller than thermal light, quantifying its "ordered" nature.

5. Significance and Contributions

Theoretical Completion: It fills a long-standing gap in quantum optics by providing the first formal calculation of entropy flow for a phase-diffusing laser.
Fundamental Insight: It provides a rigorous way to distinguish the "information content" or "order" of laser light versus thermal light using statistical mechanics.
Applications: The result has implications for the study of optical clocks and the fundamental question of irreversibility in high-precision quantum measurements. It confirms that the "purity" of a laser is limited by its quantum-induced phase noise.