Forced Synchronization of Spaser by an External Optical Wave

We demonstrate that when the frequency of the external field differs from the lasing frequency of an autonomous spaser, the spaser exhibits stochastic oscillations at low field intensity. The plasmon oscillations lock to the frequency of the external field only when the field amplitude exceeds a threshold value. We find a region of values of the external field amplitude and the frequency detuning (the Arnold tongue) for which the spaser synchronizes with the external wave.

and the plasmon NP. The efficiency of such a mechanism depends on the probability of the nonradiative excitation of the surface plasmon, which is ( ) 3 kr − greater than the radiation of the photon [17], where r is the distance between the centers of the QD and the NP and k is the photon wavenumber. Stimulated radiation from the QD into the plasmon mode results in spasing.
The excitation of the plasmon mode is carried out via the excitation of the QD.
In theoretical studies of a metamaterial with compensated losses, it is assumed that electromagnetic wave propagation can be described by the Maxwell equations with real valued effective electric permittivity and magnetic permeability. In its turn, the use of effective permittivities and refraction coefficients assumes that the dipole moment of a NP oscillates with the frequency of the external field and the amplitude of this oscillation is determined by the external field. In the absence of the external field, there should be no oscillations of a dipole moment. The classical linear description treats the gain medium as a system with a negative imaginary part of the permittivity [18][19][20][21].
However, when spasers are used as gain inclusions, their dipole moments are excited not only by the external field but also by the radiation produced by QDs, which actually tends to compensate for loss. Unlike the classical linear description [18][19][20][21], the semi-classical analysis shows that the spaser in the presence of pumping is a self-oscillating system. The dipole moment of the spaser's NP oscillates autonomically even in the absence of the external field. The autonomic frequency of this self-oscillation is determined by the plasmon frequency, transition frequency of the gain inclusion and characteristic times of relaxation in NP and excitation of gain inclusions [15]. Therefore, in order to develop a correct description of metamaterials with spasers, it is necessary to study the interaction of the spaser with the external field in details.
In this paper, we study the operation of a spaser driven by an external optical wave. We demonstrate that the pumping drastically changes the spaser's behavior in the optical field in comparison with the behavior of a passive QD-NP pair. In particular, when the frequency of the external field is detuned from the autonomous frequency of the spaser, even infinitesimally weak field drives the spaser to stochastic oscillations. The spaser can be synchronized to the external field only when the field amplitude exceeds a threshold value.
The simplest model of the spaser consists of a two-level QD of size TLS r which is positioned at a distance r from a metallic NP of size NP r [11].
describe the non-interacting NP and QD, respectively [11,23,24], the operator ˆT states of the QD, TLS e e g = μ r is the QD dipole moment matrix element. Quantization of the plasmon field without taking into account losses is carried out in the standard way [25]. Thus we obtain [11,15] where ( ) is the normalization factor and is the q-th eigenmode determined by the geometry of the problem [11]. In the dipole approximation, in Eq. (4) we retain only one dipole mode For a spherical NP this dipole mode is uniform inside the particle and has the form of the field of the dipole with a unitary dipole moment outside of the NP: 1 . Since for the field of the dipole ε ω ∂ ∂ = outside of the particle, we obtain . Thus, the operator for the dipole moments of the NP is Assuming that the frequencies of the QD transition and the frequency of the dipole SP are ).
where Rabi frequency is The commutation relations for operators ˆ( ) a t and ˆ( ) 1( ) , where Hermitian conjugation [11,[29][30][31]. The difference between the occupancies of the upper and lower levels D(t) must be a real valued quantity because the respective operator is Hermitian.
The quantities σ(t) and a(t) are the complex amplitudes of the dipole oscillations of the QD and SP, respectively.
The system of equations (7)-(9) has two stationary solutions. The trivial unstable solution corresponds to the absence of SPs, while the stable one corresponds to laser generation:  Fig. 1). The stationary values of a, σ, and D shown in Fig. 1. Let us consider the dynamics of the NP and QD in the field of the external optical wave, Assuming that the external field is classical and taking into account the dipole interaction only, we can write the Hamiltonian in the form   (Fig. 2b). This result is confirmed by our numerical simulation shown in Fig. 3. Such behavior of a self-oscillating system is referred to as synchronization by an external periodic influence [32]. The region in which the synchronization takes place is known as the Arnold tongue.
The phase dynamics can be viewed as the motion of a particle sliding along the potential profile ( ) U ϕ (Fig. 4) in a viscous liquid. For a small field and/or large detuning, / 1 ξ ∆ < , the phase difference of the system and the external field increases monotonously. For / 1 ξ ∆ > , the "particle" should be trapped in one of the minima of the potential function. This corresponds to the regime of synchronization: the phase of the system oscillations does not depend upon time.
The Arnold tongue is the set of parameters ξ and ∆ for which synchronization occurs. In the approximation considered, the width of this region is proportional to the field amplitude. Within the Arnold tongue, for small fields of the external wave (the energy of the interaction of the NP with the external field is smaller than the QD-NP interaction), the amplitude of the autooscillations depends weakly upon the amplitude of the external field, which in this case plays the role of a synchronizer.  . This agrees with calculations of Refs. [3,10] in which the wave propagation at the system of spasers was considered. In Refs. [3,10] the amplitude of the incident wave was by a few orders of magnitude greater than * Synch E . Such a field is comparable with the near field inside the spaser. In this case, spaser synchronizes with the external field for any value of the detuning; it ceases to be an autonomous system and responses linearly to the external field as can be seen in Fig. 3. For such strong fields there is no compensation of losses and only amplification can occur.
Authors are indebted to Yu.E. Lozovik and C. Z. Ning for useful discussions. This work was supported by RFBR Grants Nos. 10-02-91750 and 11-02-92475 and PSC-CUNY grant.