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arXiv:2205.13473 · Laser & Photonics Review

Quantum Density Matrix Theory for a Laser Without Adiabatic Elimination of the Population Inversion: Transition to Lasing in the Class‐B Limit

Alejandro M. Yacomotti, Zakari Denis, Alberto Biella, Cristiano Ciuti

Published Oct 20, 2022• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1002/lpor.202200377

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Quantum Density Matrix Theory for a Laser Without Adiabatic Elimination of the Population Inversion: Transition to Lasing in the Class‐B Limit

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Comment on "Thermal, Quantum Antibunching and Lasing Thresholds from Single Emitters to Macroscopic Devices"

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In a recent Letter [Phys. Rev. Lett. 126, 063902 (2021)], M. A. Carroll et al. derived a model to analytically determine regimes of thermal, collective anti-bunching, and laser emission for emitters in a cavity. According to their model, nanolasers exhibit a distinct threshold at which the coherent laser field emerges from a bifurcation at a finite pump rate. The amplitude of this coherent field increases with a further increase in the pump rate. Such a behavior contrasts the usual view of the transition to lasing in single-mode high-$\beta$ nanolasers, according to which strong spontaneous emission into the lasing mode results in a smooth transformation from the thermal state to the coherent state as the pump rate increases. Here, we demonstrate that the authors have ignored important terms in the equations of their model, which caused the bifurcation and the ideally monochromatic field to emerge at a finite pump rate.

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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.
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 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.
Abstract Starting from a quantum description of multiple Λ-type three-level atoms driven with a coherent microwave field and incoherent optical pumping, we derive a microscopic model of lasing from which we move towards a consistent macroscopic picture. Our analysis applies across the range of system sizes from nanolasers to the thermodynamic limit of conventional lasing. We explore the necessary conditions to achieve lasing without inversion in certain regimes by calculating the non-equilibrium steady state solutions of the model at, and between, its microscopic and macroscopic limits. For the macroscopic picture, we use mean-field theory to present a thorough analysis of the lasing phase transition. In the microscopic case, we exploit the underlying permutation symmetry of the density matrix to calculate exact solutions for N three-level systems. This allows us to show that the steady state solutions approach the thermodynamic limit as N increases, restoring the sharp non-equilibrium phase transition in this limit. We demonstrate how the lasing phase transition and degree of population inversion can be adjusted by simply varying the phase of the coherent driving field. The high level of quantum control presented by this microscopic model and the framework outlined here have applications to further understanding and developing nanophotonic technology.
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.
In a recent Letter [Phys. Rev. Lett. 126, 063902 (2021)], M. A. Carroll et al. derived a model to analytically determine regimes of thermal, collective anti-bunching, and laser emission for emitters in a cavity. According to their model, nanolasers exhibit a distinct threshold at which the coherent laser field emerges from a bifurcation at a finite pump rate. The amplitude of this coherent field increases with a further increase in the pump rate. Such a behavior contrasts the usual view of the transition to lasing in single-mode high-β\betaβ nanolasers, according to which strong spontaneous emission into the lasing mode results in a smooth transformation from the thermal state to the coherent state as the pump rate increases. Here, we demonstrate that the authors have ignored important terms in the equations of their model, which caused the bifurcation and the ideally monochromatic field to emerge at a finite pump rate.