Abstract Despite the enormous technological interest in micro and nanolasers, surprisingly, no class‐B quantum density‐matrix model is available to date, capable of accurately describing coherence and photon correlations within a unified theory. In class‐B lasers—applicable for most solid‐state lasers at room temperature—, the macroscopic polarization decay rate is larger than the cavity damping rate which, in turn, exceeds the upper level population decay rate. Here, a density‐matrix theoretical approach for generic class‐B lasers is carried out, and closed equations are provided for the photonic and atomic reduced density matrix in the Fock basis of photons. Such a relatively simple model can be numerically integrated in a straightforward way, and exhibits all the expected phenomena, from one‐atom photon antibunching, to the well‐known S‐shaped input–output laser emission and super‐Poissonian autocorrelation for many atoms (), and from few photons (large spontaneous emission factors, ) to the thermodynamic limit ( and ). Based on the analysis of , it is concluded that super‐Poissonian fluctuations are clearly related to relaxation oscillations in the photon number. A strong damping of relaxation oscillations with an atom number as small as is predicted. This model enables the study of few‐photon bifurcations and nonclassical photon correlations in class‐B laser devices, also leveraging quantum descriptions of coherently coupled nanolaser arrays.
A fully quantum-mechanical treatment is given of field and population fluctuations in a homogeneously broadened laser. The field and population reservoirs have been replaced by dissipation coefficients (or transition probabilities) and noncommuting noise sources. The atomic polarization variables are adiabatically eliminated. The mean motion of an arbitrary operator function $M(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, {N}_{1}, {N}_{2})$ of the field variables
b, ${b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and the population variables
N1 and
N2 is calculated. The equation of motion for the density matrix $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, {N}_{1}, {N}_{2}, t)$ is deduced, as well as the Fokker-Planck equation for the associated classical distribution function $P(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, {N}_{1}, {N}_{2}, t)$. The diffusion coefficients in the latter equation demonstrate the shot-noise origin of the fluctuations. The Fokker-Planck equation for
P is replaced by Langevin equations for $\ensuremath{\beta}$, ${\ensuremath{\beta}}^{*}$,
N1, and
N2 to facilitate adiabatic elimination of the population variables (when valid). The resulting Langevin equations for $\ensuremath{\beta}$, ${\ensuremath{\beta}}^{*}$ are converted to a Fokker-Planck equation for $P(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, t)$, to an operator equation for $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, t)$, and to an equation for ${\ensuremath{\rho}}_{\mathrm{mn}}$, the field density matrix in the photon-number representation. The transformation $\ensuremath{\beta}={I}^{\frac{1}{2}}{e}^{\ensuremath{-}i\ensuremath{\phi}}$ to an ($I, \ensuremath{\phi}$) representation leads (a) to the steady-state solution
P(I) valid at all operating levels, and (b) to the phase linewidth. The transformation to new variables
N=21(m+n), $k=m\ensuremath{-}n$, permits the equation for ${\ensuremath{\rho}}_{\mathrm{mn}}$ to be replaced by a Fokker-Planck equation for $\ensuremath{\rho}(N, k, t)$. The latter permits a verification of the previously obtained phase linewidth and a solution for the steady-state $\ensuremath{\rho}(N)$.
Considerable attention has been directed recently to the study of lasing without the need of population inversion. The optical coherence and quantum interference associated with the light amplification may lead to interesting statistical properties in inversionless lasers. Here we analyze a laser model that consists of an ensemble of closed multi-level (three or four levels) atoms driven by a single coherent field. Unlike conventional lasers where population inversion is maintained and kept at the threshold value for arbitrary laser output, the multi-level laser analyzed here can start lasing from population inversion and evolves from population inversion into non-inversion with the increasing atomic pumping. In other words, the system makes transition from light amplification by population inversion into light amplification by coherence. On the other hand, the system can be also made to start lasing without population inversion in any state basis. In both cases, initial inversion and non-inversion, which are controlled by the ratio of spontaneous decay rates, the laser well above threshold is always maintained by lasing without inversion (light amplification by coherence), and the laser intensity fluctuations may drop below the shot noise limit. Two factors contribute to the noise reduction: first, the disappearance of the population inversion leads to the depleted atomic population in the upper lasing state, which reduces the spontaneous emission noises; second, the fast coherent cycling of electrons between excited states leads to the highly regulated absorption and emission processes. We found that the combination of these two mechanisms can lead to the noise reduction in the laser output more than 50% below the shot-noise limit.
The Search and Discovery piece by Ashley Smart in the April 2017 issue of Physics Today (page 18) reported the excellent work on magnetic stopping and trapping of methyl radicals by the group of Takamasa Momose (University of British Columbia). I am writing to provide a different perspective on the field than what was conveyed by the author. The very first sentence of Smart’s article brings up a whole discussion: “By and large, physicists have succeeded in their quest to tame the atom.”The past few decades have seen remarkable advances in the ability to control and cool atoms in the gas phase. Those advances have been enabled by laser-cooling, which produces ultracold atomic gases. The atoms are then further cooled by evaporation to create quantum degenerate gases. However, those methods are far from ideal, even for atoms.Laser cooling is not a general method. Due to the requirements of a cycling transition and available lasers, it works on only a small set of elements in the periodic table. Evaporative cooling requires the right balance between elastic and inelastic collisions, again severely limiting generality. Furthermore, evaporative cooling leads to a large loss of atoms. For the alkalis, the optimum cases for laser cooling and evaporative cooling, the flux of ultracold atoms has been limited by optical density to around 109 atoms per second.It is natural to ask whether one can invent new methods that work on most elements and that break the current limit on flux. Such a development would open many new possibilities for fundamental science and for real-life applications. The challenges and limitations have motivated several groups to explore alternative approaches. One method is buffer-gas cooling, pioneered by John Doyle at Harvard University.11. J. M. Doyle et al., Phys. Rev. A 52, R2515 (1995). https://doi.org/10.1103/PhysRevA.52.R2515 A different approach was developed independently and in parallel by Frédéric Merkt and his group at ETH Zürich and by my group at the University of Texas at Austin: magnetic stopping and trapping of paramagnetic atoms in a supersonic beam.22. S. D. Hogan et al., Phys. Rev. Lett. 101, 143001 (2008). https://doi.org/10.1103/PhysRevLett.101.143001,33. E. Narevicius et al., Phys. Rev. Lett. 100, 093003 (2008). https://doi.org/10.1103/PhysRevLett.100.093003 That work was summarized in an invited review article I wrote for Science.44. M. G. Raizen, Science 324, 1403 (2009). https://doi.org/10.1126/science.1171506 Since then, many new developments have continued to advance the field.Now that the first sentence of the Search and Discovery piece is out of the way, I can discuss the rest—in particular, the motivation of studying quantum chemistry and the question of whether one actually needs to trap molecules. The goal of studying chemical reactions at ultralow temperatures is to observe unique quantum pathways that can dominate the reaction dynamics. Until recently it has been an elusive goal. Stopping and trapping molecules, either by electrostatic or magnetic forces, has produced a phase-space density that is too low for the study of chemical reactions. Molecules at much higher phase-space density can be produced by starting with Bose–Einstein condensates and “making” molecules with lasers. However, those experiments have so far been limited to alkali chemistry.Work by the Narevicius group at the Weizmann Institute of Science in Israel provided a major breakthrough in quantum chemistry without the need for stopping and trapping. Their approach relies on the merging of two supersonic beams by magnetically deflecting one of them.55. A. B. Henson et al., Science 338, 234 (2012). https://doi.org/10.1126/science.1229141 The copropagating beams have controllable collision energies down to temperatures of several millikelvin. The resulting chemical reactions, observed as a function of energy, reveal striking quantum resonances in the reaction dynamics. That work is continuing, and it demonstrates the power of merged beams, which do not require trapping and cooling. To study slower chemical reactions, trapping at high phase-space density is necessary. New ideas and work along those lines are in progress and will hopefully prove successful.ReferencesSection:ChooseTop of pageReferences <<CITING ARTICLES1. J. M. Doyle et al., Phys. Rev. A 52, R2515 (1995). https://doi.org/10.1103/PhysRevA.52.R2515, Google ScholarCrossref, ISI2. S. D. Hogan et al., Phys. Rev. Lett. 101, 143001 (2008). https://doi.org/10.1103/PhysRevLett.101.143001, Google ScholarCrossref3. E. Narevicius et al., Phys. Rev. Lett. 100, 093003 (2008). https://doi.org/10.1103/PhysRevLett.100.093003, Google ScholarCrossref, ISI4. M. G. Raizen, Science 324, 1403 (2009). https://doi.org/10.1126/science.1171506, Google ScholarCrossref5. A. B. Henson et al., Science 338, 234 (2012). https://doi.org/10.1126/science.1229141, Google ScholarCrossref, ISI© 2017 American Institute of Physics.