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arXiv:2512.01230

The Entropy Flow of a Laser Beam

Howard M. Wiseman

quant-phSubmitted Dec 1, 2025• Announced Dec 2, 2025
v1 · Dec 1, 2025v2 · Feb 11, 2026v3 · Jul 14, 2026
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Abstract

A laser beam is often modelled by a pure coherent state. In fact its state is mixed, even if it has coherent-state photon-number statistics (Poissonian), because the phase must vary. We consider such an ideal laser beam, with phase diffusion rate $\ell$, equal to its (Lorentzian) spectral width. We show that the beam entropy is extensive, with an entropy flow of $\dot{S} = \sqrt{\dot{N}\ell}$, where $\dot{N}$ is the number flow. We give an intuitive explanation for this remarkably simple result, and compare it to a unidirectional thermal beam.

Journal reference
Optics Letters 51, 3977-3979 (2026)
DOI
10.1364/OL.599036
Comments
3 pages

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A laser beam is often modelled by a pure coherent state. In fact its state is mixed, even if it has coherent-state photon-number statistics (Poissonian), because the phase must vary. We consider such an ideal laser beam, with phase diffusion rate ℓ\ellℓ, equal to its (Lorentzian) spectral width. We show that the beam entropy is extensive, with an entropy flow of S˙=N˙ℓ\dot{S} = \sqrt{\dot{N}\ell}S˙=N˙ℓ​, where N˙\dot{N}N˙ is the number flow. We give an intuitive explanation for this remarkably simple result, and compare it to a unidirectional thermal beam.
We examine the phase relationships between different packets of a beam of light emitted from a cavity through a partially transmitting mirror. We find that the beam has a pronounced phase coherence even when the intracavity field has a completely indeterminate phase such as, for example, when it is in a pure photon number state. In this case, the phase coherence can be directly attributed to entanglement caused by the partially transmitting mirror and, even though the external light has phase coherence, it has no identifiable phase. Consequently there is no phase diffusion but there is diffusion of the phase difference between a packet and the intracavity field. Remarkably, we find that the phase coherence of the beam is almost independent of the intracavity state, provided this has a reasonably narrow number-state distribution about a large mean value.
The stochastic dynamics of the quantum state of a laser conditioned on continuous intensity and phase-sensitive measurements on its output are examined. We first develop the essential model for a Poissonian laser, and then generalize it for regularly pumped lasers. We show that the rate of phase diffusion (which gives the laser linewidth) is not affected by the regularity of the pump. In both cases, heterodyne detection causes the phase variance in the conditioned laser state to become very small, yet remain significantly above (at least three times) that of a coherent state. The phase diffusion is manifest by a random walk undertaken by the mean phase, with the stochasticity arising from the local-oscillator shot noise. In contrast, intensity measurements have no effect on a Poissonian laser. For a perfectly regularly pumped laser, coarse-grained intensity measurements (for which an approximate theory is developed here) collapse the state to one with an arbitrarily small photon-number variance-to-mean ratio.
The author presents a study of the harmonic wave properties (intensity and coherence) in connection with the statistics of the fundamental: coherent (ideal laser), chaotic and non-ideal laser cases are considered for the fundamental. Two approximations of different order are considered and their validity range is studied. The author shows that a coherent wave generates a coherent harmonic only it its photon number is great, while a chaotic one always creates a harmonic with different statistics. He shows also that a fundamental with Lorentzian spectral width generates a harmonic with a coherence time four times smaller.
Recent advances in laser theory have demonstrated that a quantum enhancement is possible for the production of coherence C\mathfrak{C}C by a continuous-wave laser device. Curiously, natural families of laser models that achieve Heisenberg-limited scaling for coherence produce the most coherence when the beam exhibits sub-Poissonian photon statistics. In this work, we provide an analytical treatment of those novel families of laser models by specializing to a parameter regime that permits a linearization. We characterize the dynamics of each laser system, and find that some of the intuitions from standard laser theory may be applied here. Specifically, the intracavity number dynamics are well-described as an Ornstein-Uhlenbeck process, while the intracavity phase dynamics are well-described in terms of a physically realizable ensemble of pure states, which evolve according to pure phase diffusion. Unlike a standard laser, however, we find that the pure states comprising the ensemble in the Heisenberg-limited lasers are substantially phase squeezed. From our dynamical analysis, we deduce various quantities of the beam for each laser family, including the first- and second-order Glauber coherence functions, intensity noise spectrum, Mandel-Q parameter and coherence C\mathfrak{C}C. In addition, inspired from these phase diffusion dynamics, we derive an upper bound on laser coherence C≲1.1156μ4\mathfrak{C} \lesssim 1.1156 μ^4C≲1.1156μ4 -- which is tighter by a factor of 3/83/83/8 when compared to that derived in [Baker et al., Nat. Phys. 17 179 (2021)] -- by making one of the assumptions of that paper slightly stronger.
The laser is treated as a thermodynamic machine where production of radiation is the result of entropy creating spontaneous flows of energy and polarization through the system. One of the laws of thermodynamics governing flows of extensive properties is the symmetry principle of Curie and Prigogine. It determines how much flows are coupled. Going back to the first principles of statistical mechanics and using aforementioned symmetry requirements, a set of alternative Bloch equations is derived. The impact of the new formulation on laser physics is developed. It leads to a set of new conclusions for spontaneous pulsating regimes, for the spectral line shape of coherent radiation, for saturation effects, etc.
Abstract Using the operator correspondence of the real and fictious modes in the thermo entangled state representation, we solve the quantum master equation describing the diffusion channel and obtain the Kraus operator-sum representation of its analytical solution. we find that the pure coherent states evolve into the new mixed thermal superposed states in the diffusion channel. Also, we investigate the statistical properties of the initial coherent states and their entropy evolutions in the diffusion channel, and find that the entropy evolutions are only related to the decay time and without the amplitudes of the initial coherent states.