Let ℭ be the number of photons in a coherence time. The standard (Schawlow-Townes) result is ℭ = O ( µ 2 ), with µ the laser cavity photon number. Here we establish the Heisenberg limit: ℭ = O ( µ 4 ).
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.
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.
Due to their high coherence, Lasers are a ubiquitous tool in science. The standard quantum limit for the phase coherence time was first introduced by [A. Schawlow and C. Townes, Phys. Rev. 112, 1940 (1958)], who showed that the minimum possible laser linewidth is determined by the linewidth of the laser cavity divided by twice the number of photons in the cavity. Later, Wiseman showed theoretically that by using Susskind-Glogower (SG) operators to couple the gain medium to the laser cavity it is possible to eliminate pump noise, but not loss noise. This decreases the minimum laser linewidth, though only by a factor of two. In this article, we show that by engineering the coupling between the laser cavity and the output port it is possible to eliminate most of the loss noise as well and construct a laser that has a vastly narrower linewidth, narrower than the standard quantum limit by a factor equal to the number of photons in the laser cavity. We establish a roadmap for building such a device in the laboratory by using Josephson junctions and linear circuit elements to build coupling circuits that behave like SG operators for a range of cavity photon occupancies and using them to couple the laser cavity to both the gain medium and the output port. This device could be an ultra-coherent, cryogenic light source for microwave quantum information experiments. Further, our laser provides highly squeezed light and could be modified to provide designer quantum light which is an important resource for CV/linear optical quantum computing, readout of quantum states in superconducting quantum computers, quantum metrology, and quantum communication. Finally, our proposal relies on the tools and elements of superconducting quantum information, and thus is a clear example of how quantum engineering techniques can inspire us to re-imagine the limits of conventional quantum systems such as the laser.
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.
Quantum-enhanced measurements use highly non-classical quantum states in order to enhance the precision of the measurement of classical quantities, like the length of an optical cavity. The major goal is to beat the standard quantum limit (SQL), i.e. a precision of order 1/N, where N is the number of quantum resources (e.g. the number of photons or atoms used), and to achieve a scaling 1/N, known as the Heisenberg limit. Doing so would have tremendous impact in many areas, but so far only three experiments have demonstrated a slight improvement over the SQL. The required quantum states are generally difficult to produce, and very prone to decoherence. Here we show that decoherence itself may be used as an extremely sensitive probe of system properties. This should allow for a new measurement principle with the potential to achieve the Heisenberg limit without the need to produce highly entangled states.