Recent advances in laser theory have demonstrated that a quantum enhancement is possible for the production of coherence $\mathfrak{C}$ 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 considering 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 $\mathfrak{C}$. In addition, inspired from these phase diffusion dynamics, we derive an upper bound on laser coherence $\mathfrak{C}\lesssim 1.1156{\mu }^4$—which is tighter by a factor of $3/8$ 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.

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• Laser dynamics
• Open quantum systems & decoherence
• Optical coherence
• Photon statistics
• Quantum coherence & coherence measures
• Quantum optics
• Quantum states of light