SPASER as a complex system: femtosecond dynamics traced by ab-initio simulations Juan Sebastian Totero Gongora1;∗,Andrey E. Miroshnichenko2, Yuri S. Kivshar2 and Andrea Fratalocchi1 1PRIMALIGHT, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia 2Nonlinear Physics Centre, Australian National University, Canberra, ACT 0200, Australia *[email protected], http://primalight.org Abstract| We study the temporal and spatial dynamics of the spaser emission by means of ab- initio simulations at the femtosecond scale. We reveal that the spaser's dynamics demonstrates different operating regimes which range from multipolar pulsed emission to coherent emission with rotational dynamics. These complex dynamics are explained by exploiting a novel quantum- mechanical approach which describes the spaser emission in terms of a dynamically-disordered magnetic system. The possibility of implementing the spaser as a nanoscale source of coherent light is a topic of great interest [1, 2, 3]. While recent theoretical and experimental studies have provided useful insights on the lasing conditions, many fundamental questions regarding the spaser's action are still unanswered. In analogy with laser physics, the "spasing" action is generally interpreted as a non-equilibrium phase transition from spontaneous to stimulated emission [4]. However, the details of the spaser's transition are still debated. Different spaser configurations, in fact, have been reported to exhibit distinctive emission regimes, ranging from pulsed to purely coherent emission [5, 6]. The origin of these distinct operating regimes in the lasing stage could be a signature of the different temporal dynamics leading to the spaser's action. At the same time, the mutual interaction of the spatial modes of the system could play a key role in the way the system achieves a coherent spaser's phase. In this context, the use of ab-initio numerical simulations could provide new insights on the dynamics underlying the spaser emission, due to the possibility of describing its temporal and spatial dynamics with femtosecond and nanometer accuracy. In order to address these fundamental questions, we investigate the spaser's action by means of Finite-Differences in Time- Domain (FDTD) simulations of a single core-shell spaser. We model the interaction between the resonant medium, the metallic structure and the electromagnetic radiation by means of a quantum- mechanical set of dispersive Maxwell-Bloch equations, which are solved using our own-made FDTD solver NANOCPP [7, 8]. In our simulations, the material parameters describing the amplifying medium are taken from the literature and correspond to Rhodamine 800 doped silica [9, 10]. The pumping rate, which is assumed to be constant, is controlled by varying the density of polarizable atoms in the excited state Na. The emission dynamics can be further controlled by setting the characteristic relaxation times of the excited and ground state τ1 and τ0. In order to characterize the different regimes of the spaser, we conducted an extensive campaign of massively parallel simulations for different values of τ0/τ1 and Na whose results are shown in Fig. 1. Interestingly, once the system meets the "spasing" conditions, it is characterized by different regimes of operation (Fig. 1-a). For low values of τ0/τ1, the spaser exhibits a pulsed multipolar emission at lower pumping rates (Fig. 1-b), while at higher pumping rates the system reaches a fully CW rotational emission (Fig. 1-d). By increasing the ratio τ0/τ1 an additional intermediate state appears, and it is characterized by temporal incoherence and by the lack of any definite spatial distribution (Fig. 1-c). The origin of such diverse regimes can be identified by developing a thermodynamic model for the spatial modes of the system. In our model the emitted field is expressed as a superposition of angular plane waves, with each mode identified by an angle α and a complex-valued amplitude aα = σα ∗exp(iθα). By employing the Coupled Mode Theory formalism, the dynamics of the system are related to the evolution and mutual interaction of the amplitudes σα and phases θα of the single modes. In our model, these quantities play the role of interacting spin variables in an equivalent magnetic-like thermodynamic system, whose thermodynamic temperature is proportional to the pumping rate Na. Consequently, the different operating regimes of the spaser can be interpreted as thermodynamic phases of the overall system, with the multipolar emission corresponding to (b) 5 ×10 4 87.5 0.2 (a) 0 0 y [µm] E [a.u.] -0.2 -5 0 20 40 60 80 -0.2 0 0.2 t [ps] x [µm] 62.5 Spontaneous Emission (c) 4 ×10 5 0.2 2 1 0 /τ 37.5 0 0 y [µm] τ E [a.u.] E -2 -0.2 12.5 Metastable Emission -4 0 20 40 60 80 -0.2 0 0.2 8.75 t [ps] x [µm] 6.25 (d) 1.5 ×10 6 3.75 0.2 1.25 Multipolar Emission 0.50 0 0 0.25 Rotational y [µm] 0.10 E [a.u.] 0.6 0.7 0.8 0.9 1 2 6543 -0.2 -1.5 N [1027 part/m3] 0 20 40 60 80 -0.2 0 0.2 a t [ps] x [µm] Figure 1: (a) Phase diagram of spacer's emission from FDTD simulations. (b-d) Electric field evolution (left) and electromagnetic energy distribution (right). By varying the pumping rate Na and the ratio between the excitation and emission characteristic times τ0/τ1, the system exhibits different regimes of operation, including (b) pulsed emission, (c) metastable incoherent emission, and (d) purely CW emission. These different regimes correspond to different spatial phases for the emitted field: multipolar (b, right), non- stationary superposition of modes (c, right) and purely rotational emission (d,right). an ordered ferromagnetic phase and the rotational emission corresponding to a high-temperature paramagnetic phase. The existence of an intermediate metastable phase is due to the different time scales of evolution of the amplitudes and phases, which are directly related to the characteristic decay times τ0 and τ1 of the gain medium. As a result, the complex dynamics underlying the spaser's emission are explained as the competition of two interacting systems evolving at different time scales. 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