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Analytical results for laser models producing a beam with sub-Poissonian photon statistics and coherence scaling as the Heisenberg limit

Lucas A. Ostrowski, Travis J. Baker, Dominic W. Berry, Howard M. Wiseman

TL;DR

This work analytically treats Heisenberg-limited laser models producing beams with sub-Poissonian statistics, focusing on two main families, the $p,\lambda$- and $p,q$-models, in a linearized regime. It shows that cavity dynamics resemble an Ornstein–Uhlenbeck process for the photon number while the phase evolves via a physically realizable ensemble undergoing pure phase diffusion, with notable phase squeezing of the cavity state. The authors derive closed-form expressions for first- and second-order Glauber coherence functions, Mandel-$Q$ parameters, and the intensity-noise spectrum, establishing a tight upper bound on laser coherence $\mathfrak{C} \lesssim 1.1156\mu^4$ and demonstrating how sub-Poissonian pumping enhances coherence. The results provide deep analytic insight into the win-win relationship between sub-Poissonian statistics and HL coherence, with implications for ultra-coherent light sources and quantum metrology, and point to feasible experimental routes in platforms such as circuit QED.

Abstract

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 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 $\mathfrak{C}$. In addition, inspired from these phase diffusion dynamics, we derive an upper bound on laser coherence $\mathfrak{C} \lesssim 1.1156 μ^4$ -- 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.

Analytical results for laser models producing a beam with sub-Poissonian photon statistics and coherence scaling as the Heisenberg limit

TL;DR

This work analytically treats Heisenberg-limited laser models producing beams with sub-Poissonian statistics, focusing on two main families, the - and -models, in a linearized regime. It shows that cavity dynamics resemble an Ornstein–Uhlenbeck process for the photon number while the phase evolves via a physically realizable ensemble undergoing pure phase diffusion, with notable phase squeezing of the cavity state. The authors derive closed-form expressions for first- and second-order Glauber coherence functions, Mandel- parameters, and the intensity-noise spectrum, establishing a tight upper bound on laser coherence and demonstrating how sub-Poissonian pumping enhances coherence. The results provide deep analytic insight into the win-win relationship between sub-Poissonian statistics and HL coherence, with implications for ultra-coherent light sources and quantum metrology, and point to feasible experimental routes in platforms such as circuit QED.

Abstract

Recent advances in laser theory have demonstrated that a quantum enhancement is possible for the production of coherence 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 . In addition, inspired from these phase diffusion dynamics, we derive an upper bound on laser coherence -- which is tighter by a factor of 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.

Paper Structure

This paper contains 23 sections, 1 theorem, 99 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Heisenberg-Limited, Sub-Poissonain Laser Models
  3. Coherence Properties of a Laser Beam
  4. The $p,\lambda$-Family of Laser Models
  5. The $p,q$-Family of Laser Models
  6. The Linearized Regime
  7. Number Dynamics and Beam Photon Statistics
  8. Incoherent Cavity Dynamics
  9. The Second-Order Glauber Coherence Function
  10. Spectrum of Intensity Correlations and Beam Sub-Poissonianity
  11. Phase Dynamics and Beam Coherence
  12. Physically Realizable Ensembles and Pure Phase Diffusion
  13. Matching Cavity Dynamics with Pure Phase Diffusion
  14. Phase Squeezing in the Laser Cavity State
  15. The First-Order Glauber Coherence Function and Laser Coherence Calculations
  16. ...and 8 more sections

Key Result

Theorem 1

For a device which satisfies conditions 1--4 stated above, the coherence is bounded from above by in the asymptotic limit $\mu\to\infty$, where $\mu$ is the mean number of excitations stored within the device, and $z_A\approx-2.338$ is the first zero of the Airy function.

Figures (6)

  • Figure 1: (a): Diagonal matrix coefficients of the operators $\hat{L}^{(p,\lambda)\dagger}\hat{L}^{(p,\lambda)}$ (red) and $\hat{G}^{(p,\lambda)\dagger}\hat{G}^{(p,\lambda)}$ (green) in the number basis of the laser cavity for $\mu=50$, $p = 4$, and $\lambda = 0.5$. Linearized versions of these matrix elements, as defined in Eqs. (\ref{['Lsq2']}) and (\ref{['gain_lin']}), are displayed as dashed black lines. (b): Diagonal elements of the cavity steady state for the $p,\lambda$-family (orange) and $p,q$-family (blue) of laser models in the number basis. The Gaussian approximation of these distributions, as given in Eq. (\ref{['ss_lin']}), is given by the dashed black curve. Panels (c) and (d) depict the same as that in (a) and (b), respectively, but for parameter value choices that more closely satisfy the linearized regime. Note the restricted range of $n$ plotted in these latter cases.
  • Figure 2: (a): Numerical and analytical calculations of the second-order correlation functions (\ref{['g2ps']}) for the $p,\lambda$-family and $p,q$-family of laser models. Here, we consider the maximally sub-Poissonian case (i.e., $\lambda=0.5$ and $q=-1$, respectively for the two families), and have set $\mu = 250$ and $p=4.15$. (b) Corresponding intensity noise spectra computed from the data in (a). Panels (c) and (d) are the same as that shown in (a) and (b), respectively, but using $p=50$, which is clearly in the linearized regime.
  • Figure 3: Relative distance, as defined in the LHS of Eq. (\ref{['to_verify']}) against $p$. (a) considers the Poissonian limit for the $p,\lambda$-family (\ref{['lambda_master']}) and $p,q$-family (\ref{['q_master']}), where respectively one sets $\lambda\to0$ and $q\to0$ (in this scenario, the two families are equivalent). (b) considers the the maximally sub-Poissonian case for the $p,\lambda$-family (\ref{['lambda_master']}), where $\lambda\to0.5$. (c) considers the the maximally sub-Poissonian case for the $p,q$-family (\ref{['q_master']}), where $q\to-1$. Crosses present numerical data, while the solid black lines are power-law fits to the data pertaining to $\mu = 500$ and $p\in[10,50]$ in panels (a) and (b), and $\mu = 500$ and $p\in[10,20]$ in panel (c). The inset in (c) considers a fixed value of $p=50$ and plots the fitted power law $R_0 = 5.90p^{-1.17}$ [black line in (c)] subtracted from the relative distance as a function of $\mu$. Black dots depict numerical data and the red line is the fitted power law $159\mu^{-1.48}$.
  • Figure 4: Top: Short-time behaviour of the First-order Glauber coherence function for the $p,\lambda$- and $p,q$-families of laser models for $p=50$ and $\mu = 250$. Crosses represent numerical data and black curves depict the analytical formula (\ref{['g1_general_2']}). The "Poissonian limit" refers to the case in which the two families are described by the same master equation (respectively, with $\lambda\to0$ and $q\to0$). Bottom: Same as the top plot, but considering a much larger timescale, on the same order of magnitude as the coherence time of the laser cavities.
  • Figure 5: Top: Coherence of the $p,\lambda$- and $p,q$-families of laser models as a function of the parameter $p$, with $\mu = 250$. Numerical data is given by the coloured markers, while the analytically-derived formulas of Eqs. (\ref{['coh_lam_lin']}) and (\ref{['coh_q_lin']}) are represented by the black curves. Light-grey curves depict heuristically derived formulas for the coherence from Ref. Ostrowski2022a [see Eqs. (58a) and (58b) in that work]. The "Poissonian limit" refers to the case in which the two families are described by the same master equation (respectively, with $\lambda\to0$ and $q\to0$). Bottom: Relative error $\epsilon$ (red dots) between the numerical and analytical data given in the top plot. For a given value of $p$, the data points are essentially the same, regardless of the particular family considered. Black line shows a power law fitted to data pertaining to parameter values of $p\geq20$.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1: Upper bound on $\mathfrak{C}$ for an ideal laser beam
  • proof