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.
Recent advances in laser theory have demonstrated that a quantum enhancement is possible for the production of coherence
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. In addition, inspired from these phase diffusion dynamics, we derive an upper bound on laser coherence
C≲1.1156μ4 -- 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.
It is shown that the Λ quantum beat system, into which a new atomic coherence is introduced, can produce bright two-mode sub-Poissonian light. The Mandel Q parameter is calculated using Scully-Lamb theory. On one hand, when quantum noises of the relative amplitude and relative phase drop to vaccum noise levels, the average mode exhibits sub-Poissonian photon distribution; when quantum noises of the average amplitude and average phase drop to vaccum noise levels, the relative mode exhibits sub-Poissonian photon statistics; the optimum Mandel Q parameter is -0.5. On the other hand, in the above two cases, each of the correlated spontaneous emission laser modes exhibits sub-Poissonian photon statistics with optimum Mandel Q parameter of -0.25.
A recent analysis of the influence of the pump statistics on the laser output fluctuations is applied to the experimentally feasible case of a laser being pumped by light coming from a separate pump field cavity that is driven by a broad-band squeezed vacuum plus a coherent driving field. In contrast to the 'squeezed-reservoir laser', where a reservoir of squeezed modes coming from all directions coupled to the dipole moment of the lasing transition and leads to atomic phase decay inhibition and phase locking of the laser field, here the squeezed light merely pumps the gain atoms, that is gives rise to inversion of the laser transition. In analogy to photodetection theory the photon statistics of the laser output is calculated as a function of the photon statistics of the absorbed squeezed pump light. For the case of a squeezed pump field chosen to display sub-Poissonian statistics, it is demonstrated that the Mandel Q-parameter of the laser cavity output can theoretically reach the optimum value of Q=- 1 / 2 , similar to the case of a deterministic pumping mechanism.
The author presents a tutorial review of the quantum theory of the micromaser which allows for arbitrary (sub- as well as super-Poissonian) fluctuations of the pumping beam. In conventional reservoir theory the rate of change of the cavity field is a sum of the changes due to separate interaction with the individual reservoirs, i.e. the interactions are uncorrelated. In their approach, which is based on discrete mapping rather than a maser equation, corrections to reservoir theory arise due to correlations between these interactions. The magnitude of these terms is characterized by the quantity p/N ex . Here p, the parameter describing pump beam fluctuations, is the negative of the Mandel Q parameter of the pump beam so that p=1 corresponds to regular pumping, p=0 to Poissonian one and p<0 to super-Poissonian pump beam fluctuations. N ex is the number of atoms passing through the cavity during the lifetime of the intercavity field. The conventional reservoir limit (standard laser theory) is recovered if p=0 and/or N ex is large. In all other cases the interactions with the gain and loss reservoirs are correlated. The author presents analytical results to demonstrate the effect of pump regularity on steady-state problems (photon statistics, mean number of photons and photon number variance) as well as on transient phenomena (correlation function and spectrum). In particular, it is shown that the approach to equilibrium can have non-Markovian character.