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arXiv:1301.5652 · Physical Review A

Quantum theory of a single-emitter nanolaser

Ларионов Николай Владимирович, Mikhail I. Kolobov

Published Jul 26, 2013• 16 citations
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Abstract

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.

DOI
10.1103/physreva.88.013843

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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.
We provide analytical results obtained in the framework of the quantum theory for a single-emitter nanolaser: an incoherently pumped single two-level system interacting with a single-cavity mode of finite finesse. In the good-cavity limit we analytically calculate the linewidth of such a laser, its amplitude fluctuation spectrum, and the intracavity Mandel QQQ parameter. Our analytical results are in very good agreement with numerical simulations of the master equation.
We consider the model of a single-emitter laser: an incoherently pumped single two-level system interacting with a single cavity mode of finite finesse. In the stationary regime using normally-ordered diagonal representation of the density matrix over the coherent states we derive equation for the phase-averaged Glauber-Sudarshan P function. In the strong-coupling regime between atom and cavity mode the asymptotic solution of this equation is obtained. This solution is a linear combination of two basis solutions. We analyze one of them which was ignored in previous work without any discussions.
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.
Using the master equation approach, we investigate the stationary regime of a single-emitter laser: an incoherently pumped single two-level system interacting with a single cavity mode of finite finesse. We analyze the closed equation for the phase-averaged Glauber-Sudarshan P function and obtain its approximate solution. This solution has some advantages over the previous results. In particular, it describes conversion from the classical-like probability distribution to the oscillating quasi-probability when transition from weak- to strong-coupling regime takes place.
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.