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Physical Review A

Quantum theory of a thresholdless laser

Игорь Евгеньевич Проценко, P. Domokos, V. Lefèvre-Seguin, Jean Hare, J. M. Raimond, L. Davidovich

Published Feb 1, 1999• 71 citations
Publisher

Abstract

We develop a quantum theory of a single-mode thresholdless laser. We start from basic Heisenberg--Langevin equations of motion for the field and atomic operators, and obtain an approximate analytical solution to these operator equations. We compare the predictions of this model for the intensity and power spectrum of the field to the results of a Monte Carlo numerical simulation of the original Heisenberg-Langevin equations, and find them in excellent agreement. We also compare these predictions to those of a rate-equation model, which takes into account spontaneous emission. We show that our model gives more reliable results in the bad cavity limit at high intensities. Based upon these results, we propose a simple characterization of the thresholdless behavior. Finally, we apply our model to microsphere Nd-doped lasers at low temperatures, which are promising devices for a well-controlled thresholdless operation.

DOI
10.1103/physreva.59.1667

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We develop a quantum theory of a single-mode thresholdless laser. We start from basic Heisenberg--Langevin equations of motion for the field and atomic operators, and obtain an approximate analytical solution to these operator equations. We compare the predictions of this model for the intensity and power spectrum of the field to the results of a Monte Carlo numerical simulation of the original Heisenberg-Langevin equations, and find them in excellent agreement. We also compare these predictions to those of a rate-equation model, which takes into account spontaneous emission. We show that our model gives more reliable results in the bad cavity limit at high intensities. Based upon these results, we propose a simple characterization of the thresholdless behavior. Finally, we apply our model to microsphere Nd-doped lasers at low temperatures, which are promising devices for a well-controlled thresholdless operation.
Using two equivalent methods, the Heisenberg-Langevin and the master equation, we investigate the properties of a single-emitter nanolaser: an incoherently pumped single two-level system interacting with a single cavity mode of finite finesse. We are interested in a stationary generation regime when the mean number of photons inside the laser cavity is much larger than unity. In this case we linearize the Heisenberg-Langevin equations around the stationary classical solution and obtain analytical results for the linewidth, the quadrature fluctuation spectra, and the intracavity Mandel QQQ parameter. As the second method we use the master equation for the density operator of the laser field. Using the coherent-state representation of the density matrix we obtain the stationary solution for the Glauber PPP function of the laser mode and investigate the quantum statistics of the laser field and its threshold behavior. We compare the results of both methods with numerical simulations and find very good agreement. In the end we use our theoretical results for analyzing experiment with single-ion nanolasers.
A quantum theory of two-photon correlated-spontaneous-emission lasers (CEL's) is developed, starting from the exact atom-field interaction Hamiltonian for cascade three-level atoms interacting with a single-mode radiation field. We consider the situation where the active atoms are prepared initially in a coherent superposition of three atomic levels and derive a master equation for the field-density operator by using a quantum theory for coherently pumped lasers. The master equation is transformed into a Fokker-Planck equation for the antinormal-ordering Q function. The drift coefficients of the Fokker-Planck equation enable us to study the steady-state operation of the two-photon CEL's analytically. We have studied both resonant two-photon CEL for which there is no threshold, and off-resonant two-photon CEL for which there exists a threshold. In both cases the initial atomic coherences provide phase locking, and squeezing in the phase quadrature of the field is found. The off-resonant two-photon CEL can build up from a vacuum when its linear gain is larger than the cavity loss (even without population inversion). Maximum squeezing is found in the no-population-inversion region with the laser intensities far below saturation in both cases, which are more than 90% for the resonant two-photon CEL and nearly 50% for the off-resonant one. Approximate steady-state Q functions are obtained for the resonant two-photon CEL and, in certain circumstances, for the off-resonant one.
Nonlinear Heisenberg-Langevin equations are solved analytically by operator Fourier expansion for the laser in the light-emitting diode (LED) regime. Fluctuations of populations of lasing levels are taken into account as perturbations. The spectra of operator products are calculated as convolutions, preserving Bose commutations for the lasing field operators. It is found that fluctuations of population significantly affect spontaneous and stimulated emissions into the lasing mode, increase the radiation rate, the number of lasing photons, and broaden the spectrum of a bad cavity threshold-less and the superradiant lasers. The method can be applied to various resonant systems in quantum optics.
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.
We investigate spectral properties of the atomic fluorescence and the output field of the cavity-mode of a single-atom dressed-state laser in a photonic crystal. We pay particular attention to the behavior of the spectra in the presence of the frequency dependent reservoir and search for signatures of the thresholdless lasing. Although the thresholdless behavior has been predicted by analyzing the photon statistics of the cavity field, we find that the threshold behavior still exists in the spectrum of the cavity field. We find that the structure of cavity field spectrum depends strongly on the range of the pumping rate. For low pumping rates, the spectrum is not monochromatic, it is composed of a set of discrete lines revealing the discrete (quantum) structure of the combined dressed-atom plus the cavity field system. We find that for a certain value of the pumping rate, the multi-peak structure converts into a single very narrow line centered at the cavity field frequency. A physical explanation of the behavior of the spectra is provided in terms of dressed states of the system.
We have solved a master equation to obtain the diagonal elements of the density matrix for laser light. The master equation represents a generalization of a master equation derived earlier for a single radiation mode interacting with NNN stationary two-level atoms. The generalization takes into account the pumping scheme which characterizes a three-level laser. For positive temperatures and a lossless cavity, the equilibrium solution of the master equation gives the correct statistical-mechanical description of the atoms and radiation. For negative temperatures, the distribution function for the number of photons in the mode is either exponential or peaked, depending on whether the laser is operating below or above threshold. The calculated intensity fluctuations are in good agreement with semiclassical results for lasers operating slightly above threshold.