A numerically exact solution to the many emitter--cavity problem as an open many body system is presented. The solution gives access to the full, nonperturbative density matrix and thus the full quantum statistics and quantum correlations. The numerical effort scales with the third power in the number of emitters. Notably the solution requires none of the common approximations such as good/bad cavity limit. As a first application the recently discussed concept of coherent surface plasmon amplification---spaser---is addressed: A spaser consists of a plasmonic nanostructure that is driven by a set of quantum emitters. In the context of laser theory it is a laser in the (very) bad cavity limit with an extremely high light matter interaction strength. The method allows us to answer the question of spasing with a fully quantized theory.
A numerically exact solution to the many emitter--cavity problem as an open many body system is presented. The solution gives access to the full, nonperturbative density matrix and thus the full quantum statistics and quantum correlations. The numerical effort scales with the third power in the number of emitters. Notably the solution requires none of the common approximations such as good/bad cavity limit. As a first application the recently discussed concept of coherent surface plasmon amplification---spaser---is addressed: A spaser consists of a plasmonic nanostructure that is driven by a set of quantum emitters. In the context of laser theory it is a laser in the (very) bad cavity limit with an extremely high light matter interaction strength. The method allows us to answer the question of spasing with a fully quantized theory.
We present a quantum theory of a spaser-based nanolaser, under the bad-cavity approximation. We find first- and second-order correlation functions g(1)(τ) and g(2)(τ) below and above the generation threshold, and obtain the average number of plasmons in the cavity. The latter is shown to be of the order of unity near the generation threshold, where the spectral line narrows considerably. In this case the coherence is preserved in a state of active atoms in contradiction to the good-cavity lasers, where the coherence is preserved in a state of photons. The damped oscillations in g(2)(τ) above the generation threshold indicate the unusual character of amplitude fluctuations of polarization and population, which become interconnected in this case. Obtained results allow to understand the fundamental principles of operation of nanolasers.
We present a theoretical discussion on the feasibility of spasing for a specific spaser setup. For this purpose we develop a numerically exact and feasible solution to the open system Tavis-Cummings model in the Born- Markov Lindblad formalism. The complexity of the solution scales with the third power in the number of two level systems- a considerable advance compared to the exponential scaling of the brute force solution. The question of spasing is answered negatively in agreement with the literature.
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.
With the aid of supercomputer models we numerically investigate the exact solutions for a small number of atoms (typically 1–10) interacting with a single quantized mode of an electromagnetic cavity that is driven by an external coherent field. Effects of spontaneous emission and a nonzero cavity decay rate are included. We investigate the strong coupling regime where the atom-field coupling constant is much greater than the dissipation rates to the external environment. In this regime, the usual system size expansions of quantum statistical physics are not applicable and a full numerical solution is required. We model the recently reported experimental observations of Rabi oscillations in the second order coherence function1 g(2)(τ) and compare and contrast the results with an effective single atom-cavity model with an appropriately scaled coupling constant. In addition, the steady state field distribution for the few-atom-cavity system is explored. A multipeaked phase distribution is found that generalizes the bimodal phase distribution found in the single-atom-cavity case.2 Spectra for the few-atom-cavity system are investigated in the weak and strong driving field limits.
We calculate the quantum state of the plasmon field excited by an ensemble of molecular emitters, which are driven by exchange of electrons with metallic nanoparticle electrodes. Assuming identical emitters that are coupled collectively to the plasmon mode but are otherwise subject to independent relaxation channels, we show that symmetry constraints on the total system density matrix imply a drastic reduction in the numerical complexity. For Nm three-level molecules, we may thus represent the density matrix by a number of terms scaling as (Nm+8)!/(8!Nm!) instead of 9Nm, and this allows exact simulations of up to Nm=10 molecules. Our simulations demonstrate that many emitters compensate strong plasmon damping and lead to the population of high plasmon number states as well as a narrowed emission, which indicates a transition from spontaneous emission to amplified spontaneous emission. For large Nm, our exact results are reproduced by an approximate approach based on the plasmon reduced density matrix. With this approach, we have extended the simulations to more than 100 molecules and shown that the plasmon number state population follows a Poisson-like distribution, which indicates the formation of a plasmon coherent state. This clearly indicates a transition from amplified spontaneous emission to lasing. An alternative approach based on nonlinear rate equations for the molecular state populations and the mean plasmon number also reproduce the main lasing characteristics of the system.
Open optical or plasmonic resonators are placed on and connected through surfaces or via waveguides, forming complex lightguiding nanostructures, e.g. for integrated photonic quantum devices. We derive general boundary conditions for quasinormal modes that account for the structure's specific geometry. We then present a general quantization scheme for multiple, interacting quasinormal-mode cavities coupled to quantum emitters and to a non-bosonic bath of propagating photons on waveguides or a surface. We derive a system-bath Hamiltonian with rigorously defined coupling elements that can be computed using Maxwell solvers, including light-matter coupling between the electromagnetic field and quantum emitters. We define system-bath correlation functions for an effective, bath-mediated, and time-delayed interaction between the quasinormal modes and quantum emitters, which is a main ingredient commonly used to simulate open quantum system dynamics.