The laser has evolved from table size apparatuses to truly nano sized devices, in much the same way that computer chips have been continuously minimized. The few-emitter nanolaser represents an extreme in terms of size. The emitters can be either atoms or quantum dots, and they are coupled to a high-Q optical cavity. This system has previously been studied for the one and two emitters case, see e.g. [1, 2]. The emitters and cavity are modelled as two-level systems and a harmonic oscillator, respectively. The coupled system is modelled using the Jaynes-Cummings Hamiltonian and the two-level systems are pumped incoherently by a rate P. Solutions are found using the corresponding master equation. However, with cavity populations exceeding 100 and several emitters, the dimension of the Hilbert space of the system becomes too large to handle efficiently on a conventional computer. E.g. for four emitters and 100 photon states the density matrix has more than 2.5 × 106 elements. We have been able to simplify the problem significantly by adiabatically eliminating the photon-assisted polarizations and the correlations between emitters and cavity [3]. This results in a set of rate equations for the population of the cavity, na, and the occupation of the emitters. Fig. 1a) shows na for up to 4 emitters coupled to a cavity, as a function of the pumping rate P. The figure also shows results from a full master equation solution, and the correspondence is very good for large values of P. In Fig. 1b) the second order correlation functions g(2)(0) are shown, as obtained using the full model. The second order correlation functions become 1 for sufficiently large values of P signifying the onset of lasing. However, for larger values of P the laser quenches and the emission becomes chaotic (g(2)(0) = 2). This quenching effect is well-known in two-level models [3], but might not be realistic for semiconductor quantum dots. A proposed application for these nanolasers is in on-chip optical interconnects. In this application the modulation bandwidth is important. We have calculated the modulation bandwidth using a small-signal analysis of the rate equations, which has the advantage compared the full model of yielding a semianalytical result. The results are shown in Fig. 1c). The nanolaser has a high bandwidth, which is mostly limited by the cavity lifetime (/ 1/), but shows a prominent drop during the lasing regime. The modulation bandwidth again becomes limited by as the output becomes chaotic.
The laser has evolved from table size apparatuses to truly nano sized devices, in much the same way that computer chips have been continuously minimized. The few-emitter nanolaser represents an extreme in terms of size. The emitters can be either atoms or quantum dots, and they are coupled to a high-Q optical cavity. This system has previously been studied for the one and two emitters case, see e.g. [1, 2]. The emitters and cavity are modelled as two-level systems and a harmonic oscillator, respectively. The coupled system is modelled using the Jaynes-Cummings Hamiltonian and the two-level systems are pumped incoherently by a rate P. Solutions are found using the corresponding master equation. However, with cavity populations exceeding 100 and several emitters, the dimension of the Hilbert space of the system becomes too large to handle efficiently on a conventional computer. E.g. for four emitters and 100 photon states the density matrix has more than 2.5 × 106 elements. We have been able to simplify the problem significantly by adiabatically eliminating the photon-assisted polarizations and the correlations between emitters and cavity [3]. This results in a set of rate equations for the population of the cavity, na, and the occupation of the emitters. Fig. 1a) shows na for up to 4 emitters coupled to a cavity, as a function of the pumping rate P. The figure also shows results from a full master equation solution, and the correspondence is very good for large values of P. In Fig. 1b) the second order correlation functions g(2)(0) are shown, as obtained using the full model. The second order correlation functions become 1 for sufficiently large values of P signifying the onset of lasing. However, for larger values of P the laser quenches and the emission becomes chaotic (g(2)(0) = 2). This quenching effect is well-known in two-level models [3], but might not be realistic for semiconductor quantum dots. A proposed application for these nanolasers is in on-chip optical interconnects. In this application the modulation bandwidth is important. We have calculated the modulation bandwidth using a small-signal analysis of the rate equations, which has the advantage compared the full model of yielding a semianalytical result. The results are shown in Fig. 1c). The nanolaser has a high bandwidth, which is mostly limited by the cavity lifetime (/ 1/), but shows a prominent drop during the lasing regime. The modulation bandwidth again becomes limited by as the output becomes chaotic.
We investigate the dynamical properties of nanolasers comprising a few two-level emitters coupled to an optical cavity. A set of rate equations is derived, and is shown to agree very well with a solution of the full master equation model. Using a linearized version of these rate equations, we can analytically express the response of the nanolaser to a modulation of the pumping rate. These results are compared with the modulation response obtained directly from the master equation using a novel method. We show that contrary to conventional semiconductor lasers, the nanolaser is typically over-damped and displays a dip in the modulation bandwidth as the two-level systems become inverted. Both these features can be traced back to the use of discrete emitters that are incoherently pumped.
The development and physical understanding of high-beta nanolasers operating in regime of cavity-quantum-electrodynamics (cQED) is a highly interdisciplinary field of research, involving important aspects of nanotechnology, quantum optics, and semiconductor physics. Of particular interest is the quantum limit of operation, in which a few or even a single emitter act as gain material. The regime of strong light-matter coupling is typically associated with weak excitation. With current realizations of cQED systems, strong coupling may persevere even at elevated excitation levels sufficient to cross the threshold to lasing. In the presence of stimulated emission, the vacuum-Rabi doublet in the emission spectrum is modified and the established criterion for strong coupling no longer applies. Based on an analytic approach, we provide a generalized criterion for strong coupling and the corresponding emission spectrum that includes the influence of higher Jaynes-Cummings states. The applicability is demonstrated in a theory-experiment comparison of a state-of-the-art few-emitter quantum-dot (QD)–micropillar laser as a particular realization of the driven dissipative Jaynes-Cummings model [1]. Furthermore, we address the question if and for which parameters true single-emitter lasing can be achieved. By using a master-equation approach for up to 8 QDs coupled to the mode, we provide evidence for the coexistence of strong coupling and lasing in our system in the presence of background emitter contributions by identifying signatures in the mean-photon number, the photon-autocorrelation function, and the emission linewidth. [1] C. Gies et al., accepted for publication in PRA, arxiv:1606.05591
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.
Traditionally, the active material in a laser is modelled as independent emitters, but in recent years it has become increasingly clear that radiative coupling between emitters can significantly change the characteristics of small lasers. Collective effects in free space such as superradiance have been studied extensively [1,2], but the effects of inter-emitter correlation in micro- and nano-cavities need further examination to be put on firm theoretical ground. Several studies of collective effects in nano-cavities have been made [3-6], but the theoretical models employed are intricate, and numerical methods are needed both to generate the dynamic equations and to solve them. We propose a model where the complexity is strongly reduced, allowing analytical solutions [7]. We consider a collection of identical two-level emitters interacting with a single cavity mode. We start from Maxwell-Bloch equations, but instead of making the typical adiabatic elimination of the polarization, we allow the polarization decay rate to be of the same magnitude or smaller than other decay rates. Hence, the traditional laser rate equations for the photon number and the population inversion must be supplemented by equations for the emitter-field correlation and the emitter-emitter correlation. This gives us four generalized laser rate equations, which we solve analytically in steady state. Comparing with the steady state results obtained from the traditional laser rate equations we see that inclusion of collective effects leads to a reduction of the photon number for small pump rates, similarly to what is found in [4]. From the generalized laser rate equations, we derive a measure of the strength of collective effects in terms of laser parameters: This describes the difference between results with and without inter-emitter correlations, and it goes smoothly to zero as we approach parameter values where the traditional laser rate equations become valid. To gain insight into the photon statistics of the laser, we construct dynamic equations for higher order correlations of operators. We derive an analytical expression for the zero-delay photon auto-correlation function, and for low pump rates we find that the interaction of emitters results in super-thermal values of the auto-correlation. This feature is observed in experiments and numerical models [4-5], and with our analytical expressions, we are able to pinpoint the parameter combinations for which the collective effects have the largest impact. Considering the same model in terms of the Fourier components of the operators, we find results for the photon number that agree well with the previous approach, while allowing computation of the linewidth. Thus, we can examine how emitter-emitter correlation affects the line broadening of the laser.
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 Q 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 P 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.
We analytically calculate the optical emission spectrum of nanolasers and nano-LEDs based on a model of many incoherently pumped two-level emitters in a cavity. At low pump rates, we find two peaks in the spectrum for large coupling strengths and numbers of emitters. We interpret the double-peaked spectrum as a signature of collective Rabi splitting, and discuss the difference between the splitting of the spectrum and the existence of two eigenmodes. We show that an LED will never exhibit a split spectrum, even though it can have distinct eigenmodes. For systems where the splitting is possible, we show that the two peaks merge into a single one when the pump rate is increased. Finally, we compute the linewidth of the systems, and discuss the influence of inter-emitter correlations on the lineshape.