The dominant role of quantum fluctuations in determining the steady-state and transient response of a laser is demonstrated when there is a small number of particles in the system. In this regime, quantum fluctuations are found to suppress the lasing threshold and create a non-Poisson probability distribution for n discrete excited electronic states and s discrete photons. The correlation between n and s damps the averaged dynamic response of laser emission. Random walk calculations verify the master equation predictions and are used to connect to systems containing larger numbers of particles.
The dominant role of quantum fluctuations in determining the steady-state and transient response of a laser is demonstrated when there is a small number of particles in the system. In this regime, quantum fluctuations are found to suppress the lasing threshold and create a non-Poisson probability distribution for n discrete excited electronic states and s discrete photons. The correlation between n and s damps the averaged dynamic response of laser emission. Random walk calculations verify the master equation predictions and are used to connect to systems containing larger numbers of particles.
The random nature of spontaneous emission leads to unavoidable fluctuations in a laser's output. This is often included through random Langevin forces in laser rate equations, but this approach falls short for nanolasers. In this paper, we show that the laser quantum noise can be quantitatively computed for a very broad class of lasers by starting from simple and intuitive rate equations and merely assuming that the number of photons and excited electrons only takes discrete values. While the approach has seen previous success, we here derive it rigorously from an open quantum system master equation, whereas it was previously introduced only on phenomenological grounds. We further show that in the many-photon limit, the model simplifies to Langevin equations. We perform an extensive comparison of different approaches for computing quantum noise in lasers, identifying the best approach for different system sizes, ranging from nanolasers to macroscopic lasers, and different levels of excitation, i.e., cavity photon number. In particular, we show that below the laser threshold, stochastic fluctuations in the numerical solution to the Langevin equations can drive populations to unphysical negative values, requiring the introduction of population bounds, which in turn skew the noise statistics, leading to inaccuracies. The Laser Markov Chain model, on the other hand, is accurate for all pump values and laser sizes when collective emitter effects are excluded.
A quantitative analysis is made of quantum fluctuations in the number of photons in the resonator of a single-mode laser considered in a quasi-linear approximation. An expression is obtained for a general four-level system, which is exact above the lasing threshold, but can also be used to calculate the fluctuations in the case of three- and two-level lasers. Above the threshold when certain relationships between the spontaneous decay constants are satisfied, the photon statistics are of the sub-Poisson type in four- and three-level systems. However, for other relationships between the decay constants, pumping rate, and average number of photons, fluctuations in the resonator obey both the Poisson and super-Poisson statistics with variances much greater than the average number of photons. Both Poisson and super-Poisson statistics with large values of the variance are found for a two-level system. The results of calculations indicate that, in general, there is no coherent Glauber state of the field in a laser resonator when the threshold is significantly exceeded.
Using population rate equations, we have calculated the intrinsic intensity fluctuations expected in the output of 3-level and 4-level CW laser oscillators. Such fluctuations stem from the quantum shot noise implicit in a restriction of the atomic and photon population-operator eigenvalues to discrete integral values (in the manner of Shimoda, Takahasi, and Townes). The intrinsic fluctuations below threshold are typical of a Gaussian distribution ofe-mfield amplitudes (Bose-Einstein distribution of photons) and above threshold of an amplitude-stabilizede-mfield (Poisson distribution of photons). The noise properties of 3-level and 4-level lasers are qualitatively similar, although for a given output power the intensity fluctuations of 3-level oscillators are greater than (or equal to) those in 4-level oscillators, principally as a consequence of the increased spontaneous emission associated with the larger upper-state occupation necessary to achieve a net gain in 3-level systems.
Laser amplification kicks off with just a few photons in the cavity, a regime where standard additive Langevin noise cannot capture the real fluctuations. To understand how this amplification begins, we map out a single-mode photon-carrier model that links macroscopic lasers directly into nanolasers. Stationary rate equations reveal that stimulated emission takes over across a tight pump window. In nanolasers, however, spontaneous processes coexist across a much broader range. Because steady-state averages miss the individual events that lead to the macroscopic photon number, we frame the stochastic dynamics around the actual discrete transitions, deriving a state-dependent diffusion approximation. The destruction of an excited carrier concurrent with the generation of a photon creates a negative photon-carrier cross-correlation. This unavoidably leads to a correlated, multiplicative Gaussian-noise framework. We track the resulting discrete dynamics with an established, event-based Laser Markov Chain (LMC) which stochastically samples pump, relaxation, emission, and cavity-loss events keeping all physical quantities as non-negative integers. Comparing its predictions to those of our multiplicative model proves that the latter reproduces excellently well the statistical behavior of photons and carrier, as long as a continuous diffusion description holds. Once the emission drops into the few-photon regime, however, the discrete model becomes indispensable. On the other hand, at large photon counts relative fluctuations shrink to a level where the state-dependence of the diffusion matrix can be ignored, to leading order, thus returning to the classic Langevin description of the system's stochastic dynamics.
The steady-state intensity distribution for a laser operating somewhat above threshold is determined by solving a master equation. The master equation describes the interaction between a single radiation mode and N two-level atoms, it contains the effects of pumping and radiation loss from the cavity, and it is applicable to both homogeneous and Doppler broadening. Forms of the master equation applicable to both three- and four-level lasers are considered. Solutions are obtained under the assumption that the distribution of atoms and radiation intensity is a bivariate Gaussian. Fluctuations in the output of giant-pulse lasers are also considered.
We summarize the basic problems in laser coherence and the concepts and techniques needed to solve them. Special techniques needed to handle partition noise in semiconductor lasers will be touched on. We address the problem of how a self-sustained oscillator works. Why is the laser line so narrow? Why does a small amount of noise produce a broadening of the line as opposed to additive side bands? What is the role of zero point oscillations and spontaneous emission in determining the line width? In what sense can the transition to a lasing state be described as a phase transition? The concepts used to deal with the response of a laser system involve a quasi-Markoffian description. This in turn validates a quantum regression theorem which relates the spectrum of the noise to the transport response even in a system far from equilibrium. A significant simplification is made by introducing non-commuting (quantum) noise sources. The techniques required in the solution of laser noise problems involve a multi-time correspondence between quantum variables and c-number random variables that reduces a quantum stochastic problem to a classical stochastic problem. Adiabatic elimination is used to eliminate the fast variables to obtain a reduced problem for the fields alone or the fields and some total atomic variables (total inversion and polarization). Reduction to a rotating wave version of the van der Pol oscillator and numerical solution of the Fokker-Planck equation yields (a) the line shape (b) the spectrum of intensity fluctuations and (c) the distribution of photocounts observed over some time interval. Comparison of theoretical results with later experiments will be presented.