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arXiv:2208.14082 · Physical Review A

Optimized laser models with Heisenberg-limited coherence and sub-Poissonian beam photon statistics

L. A. Ostrowski, Travis J. Baker, S. N. Saadatmand, Howard M. Wiseman

Published May 3, 2023• 6 citations
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Abstract

Recently it has been shown that it is possible for a laser to produce a stationary beam with a coherence (quantified as the mean photon number at spectral peak) which scales as the fourth power of the mean number of excitations stored within the laser, this being quadratically larger than the standard or Schawlow-Townes limit [Baker et al., Nat. Phys. 17, 179 (2021)]. Moreover, this was analytically proven to be the ultimate quantum limit (Heisenberg limit) scaling under defining conditions for CW lasers, plus a strong assumption about the properties of the output beam. In our related work [Ostrowski et al., Phys. Rev. Lett. 130, 183602 (2023)] we show that the latter can be replaced by a weaker assumption, which allows for highly sub-Poissonian output beams, without changing the upper bound scaling or its achievability. In this paper we provide details of the calculations given in our related paper and introduce three families of laser models which may be considered as generalizations of those presented in that work. Each of these families of laser models is parameterized by a real number, $p$, with $p=4$ corresponding to the original models. The parameter space of these laser families is numerically investigated in detail, where we explore the influence of these parameters on both the coherence and photon statistics of the laser beams. Two distinct regimes for the coherence may be identified based on the choice of $p$, where for $p>3$, each family of models exhibits Heisenberg-limited beam coherence, while for $p<3$, the Heisenberg limit is no longer attained. Moreover, in the former regime, we derive formulas for the beam coherence of each of these three laser families which agree with the numerics. We find that the optimal parameter is in fact $p\ensuremath{\approx}4.15$, not $p=4$.

DOI
10.1103/physreva.107.053702

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Recently it has been shown that it is possible for a laser to produce a stationary beam with a coherence (quantified as the mean photon number at spectral peak) which scales as the fourth power of the mean number of excitations stored within the laser, this being quadratically larger than the standard or Schawlow-Townes limit [Baker et al., Nat. Phys. 17, 179 (2021)]. Moreover, this was analytically proven to be the ultimate quantum limit (Heisenberg limit) scaling under defining conditions for CW lasers, plus a strong assumption about the properties of the output beam. In our related work [Ostrowski et al., Phys. Rev. Lett. 130, 183602 (2023)] we show that the latter can be replaced by a weaker assumption, which allows for highly sub-Poissonian output beams, without changing the upper bound scaling or its achievability. In this paper we provide details of the calculations given in our related paper and introduce three families of laser models which may be considered as generalizations of those presented in that work. Each of these families of laser models is parameterized by a real number, ppp, with p=4p=4p=4 corresponding to the original models. The parameter space of these laser families is numerically investigated in detail, where we explore the influence of these parameters on both the coherence and photon statistics of the laser beams. Two distinct regimes for the coherence may be identified based on the choice of ppp, where for p>3p>3p>3, each family of models exhibits Heisenberg-limited beam coherence, while for p<3p<3p<3, the Heisenberg limit is no longer attained. Moreover, in the former regime, we derive formulas for the beam coherence of each of these three laser families which agree with the numerics. We find that the optimal parameter is in fact p\ensuremath≈4.15p\ensuremath{\approx}4.15p\ensuremath≈4.15, not p=4p=4p=4.
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 Heisenberg limit to laser coherence C-the number of photons in the maximally populated mode of the laser beam-is the fourth power of the number of excitations inside the laser. We generalize the previous proof of this upper bound scaling by dropping the requirement that the beam photon statistics be Poissonian (i.e., Mandel's Q=0). We then show that the relation between C and sub-Poissonianity (Q<0) is win-win, not a tradeoff. For both regular (non-Markovian) pumping with semiunitary gain (which allows Q→-1), and random (Markovian) pumping with optimized gain, C is maximized when Q is minimized.
The ultimate limits to estimating a fluctuating phase imposed on an optical beam can be found using the recently derived continuous quantum Cram\'er-Rao bound. For Gaussian stationary statistics, and a phase spectrum scaling asymptotically as ${\ensuremath{\omega}}^{\ensuremath{-}p}$ with p>1p>1p>1, the minimum mean-square error in any (single-time) phase estimate scales as ${\mathcal{N}}^{\ensuremath{-}2(p\ensuremath{-}1)/(p+1)}$, where N\mathcal{N}N is the photon flux. This gives the usual Heisenberg limit for a constant phase (as the limit $p\ensuremath{\rightarrow}\ensuremath{\infty}$) and provides a stochastic Heisenberg limit for fluctuating phases. For p=2p=2p=2 (Brownian motion), this limit can be attained by phase tracking.
As discussed in the preceding paper [Wiseman and Vaccaro, preceding paper, Phys. Rev. A 65, 043605 (2002)], the stationary state of an optical or atom laser far above threshold is a mixture of coherent field states with random phase, or, equivalently, a Poissonian mixture of number states. We are interested in which, if either, of these descriptions of ${\ensuremath{\rho}}_{\mathrm{ss}}$ as a stationary ensemble of pure states, is more natural. In the preceding paper we concentrated upon the question of whether descriptions such as these are physically realizable (PR). In this paper we investigate another relevant aspect of these ensembles, their robustness. A robust ensemble is one for which the pure states that comprise it survive relatively unchanged for a long time under the system evolution. We determine numerically the most robust ensembles as a function of the parameters in the laser model: the self-energy $\ensuremath{\chi}$ of the bosons in the laser mode, and the excess phase noise $\ensuremath{\nu}.$ We find that these most robust ensembles are PR ensembles, or similar to PR ensembles, for all values of these parameters. In the ideal laser limit $(\ensuremath{\nu}=\ensuremath{\chi}=0),$ the most robust states are coherent states. As the phase noise or phase dispersion is increased through $\ensuremath{\nu}$ or the self-interaction of the bosons $\ensuremath{\chi},$ respectively, the most robust states become more and more amplitude squeezed. We find scaling laws for these states, and give analytical derivations for them. As the phase diffusion or dispersion becomes so large that the laser output is no longer quantum coherent, the most robust states become so squeezed that they cease to have a well-defined coherent amplitude. That is, the quantum coherence of the laser output is manifest in the most robust PR ensemble being an ensemble of states with a well-defined coherent amplitude. This lends support to our approach of regarding robust PR ensembles as the most natural description of the state of the laser mode. It also has interesting implications for atom lasers in particular, for which phase dispersion due to self-interactions is expected to be large.