Summary form only given. The interaction strength of light and matter can be greatly influenced by altering the environment of the emitters and the electric field. In a photonic cavity the emitter-field coupling strength may become sufficiently large that the emitters and the field can exchange an excitation multiple times before it decays, giving rise to Rabi oscillations. In a passive cavity with one (several) two-level emitter(s), this manifests itself as a (collective) Rabi splitting of the transmissions spectrum [1-3], and it has recently been shown that incoherently pumped single emitters in cavities may exhibit Rabi splitting of the emission spectrum at large coupling strengths [4,5]. We present a fully analytical model describing a large collection of incoherently pumped two-level emitters in a single-mode cavity (e.g. quantum dots in a nano-cavity) and show that collective Rabi splitting of the emission spectrum can occur in these systems as well [6]. We derive a condition in terms of system parameters for the splitting to occur, and show that this is different from the condition typically associated with strong emitter-field coupling. Moreover, we show that the collective Rabi splitting decreases with increasing pump rate P, giving a transition from the strong-coupling regime to the weak-coupling regime, and possibly lasing at large P (Fig. 1(a)).
Summary form only given. The interaction strength of light and matter can be greatly influenced by altering the environment of the emitters and the electric field. In a photonic cavity the emitter-field coupling strength may become sufficiently large that the emitters and the field can exchange an excitation multiple times before it decays, giving rise to Rabi oscillations. In a passive cavity with one (several) two-level emitter(s), this manifests itself as a (collective) Rabi splitting of the transmissions spectrum [1-3], and it has recently been shown that incoherently pumped single emitters in cavities may exhibit Rabi splitting of the emission spectrum at large coupling strengths [4,5]. We present a fully analytical model describing a large collection of incoherently pumped two-level emitters in a single-mode cavity (e.g. quantum dots in a nano-cavity) and show that collective Rabi splitting of the emission spectrum can occur in these systems as well [6]. We derive a condition in terms of system parameters for the splitting to occur, and show that this is different from the condition typically associated with strong emitter-field coupling. Moreover, we show that the collective Rabi splitting decreases with increasing pump rate P, giving a transition from the strong-coupling regime to the weak-coupling regime, and possibly lasing at large P (Fig. 1(a)).
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.
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.
We present a systematic study of the properties of systems composed of N two-level quantum emitters coupled to a single cavity mode, for light-matter interaction strengths ranging from the weak to the ultrastrong and deep-strong coupling regimes. Beginning with an analysis of the energy spectrum as a function of the light-matter coupling strength, we examine systems with varying numbers of emitters, from a pair to large collections, approaching the thermodynamic limit $(N\ensuremath{\rightarrow}\ensuremath{\infty})$. Additionally, we explore the emission properties of these systems under incoherent excitation of the emitters, employing a general theoretical framework for open cavity-QED systems, which is valid across all light-matter interaction regimes and preserves gauge invariance within truncated Hilbert spaces. Furthermore, we study the influence of the emitter-environment interaction on the spectral properties of the system. Specifically, when each emitter interacts independently with its own reservoir, we observe the emergence of an emission peak at the cavity's resonant frequency for even values of N. Our analysis also clarifies the evolution of the system as the number of emitters increases, ultimately converging toward an equivalent system composed of two interacting single-mode bosonic fields.
The regime of strong light-matter coupling is typically associated with weak excitation. With current realizations of cavity-QED 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. We provide a generalized criterion for strong coupling and the corresponding emission spectrum, which includes the influence of higher Jaynes-Cummings states. The applicability is demonstrated in a theory-experiment comparison of a few-emitter quantum-dot--micropillar laser as a particular realization of the driven dissipative Jaynes-Cummings model. Furthermore, we address the question if and for which parameters true single-emitter lasing can be achieved and provide evidence for the coexistence of strong coupling and lasing in our system in the presence of background emitter contributions.
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
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.