Forced synchronization of spaser by an external optical wave

1. Introduction

Joule (ohmic) losses in metallic nano-inclusions embedded in a host metamaterial medium are the main obstacle to realizing a practical “perfect” lens with subwavelength resolution as well as to other applications of metamaterials [1]. Losses can be compensated for, however, in amplifying media. The authors of [2,3] proposed making a stackable lens with alternating metamaterial and gain layers This can be accomplished by directly introducing gain inclusions such as molecules or quantum dots (QDs) into the matrix [4–6]. The latter scheme has recently been realized experimentally [7–10] demonstrating that it is possible to compensate for losses.

The combination of a gain medium and metallic nanoparticles (NPs) results in the emergence of a spaser (Surface Plasmon Amplification by Stimulated Emission of Radiation) first suggested by Bergman and Stockman [11] and realized experimentally in [12]. Spasers, as well as nanoplasmonics overall, is a new fast growing field with tremendous promise. Schematically, the spaser is a system of inversely excited two-level QDs surrounding metal nanoparticles [11,13]. Its principles of operation are analogous to those for a laser with the role of photons played by surface plasmons (SPs) localized at a NP that serves as the resonator [11,14,15]. In other words, in a spaser, near-fields of the NP are generated and amplified. The amplification of the SPs occurs due to non-radiative energy transfer from QDs. This process takes place due to the dipole-dipole (or any other near-field [16]) interaction between the QD and the plasmon NP. The efficiency of such a mechanism depends on the probability of the non-radiative excitation of the surface plasmon, which is ${\left(kr\right)}^{-3}$ 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 [18–21].

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 [18–21], 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 analytically and numerically 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.

2. Equations of motion of the spaser

The simplest model of the spaser consists of a two-level QD of size $r_{TLS}$which is positioned at a distance r from a metallic NP of size $r_{NP}$ [11]. The whole system is immersed into a solid dielectric or semiconductor matrix with dielectric permittivity ${\epsilon }_M$. The dipole moment of a typical QD of size $r_{TLS}\approx$ 10 nm is ${\mu }_{TLS}^{class}~$20 Debye [22]. For a NP with Ag-core and SiO₂-shell with radius $r_{NP}\approx r\approx 10\text{nm}$, the classical dipole moment near the frequency of the plasmon resonance, ω, can be estimated to be ${\mu }_{NP}^{class}~200\text{Debye}$. The energy of the NP-QD quasistatic dipole-dipole interaction is estimated to be $V=\hslash {\Omega }_R~{\mu }_{NP}^{class}{\mu }_{TLS}^{class}/r^3,$ which gives ${\Omega }_R\approx 5\cdot {10}^{12}{c}^{-1}~{10}^{-2}\omega .$ Thus, the Rabi frequency (the coupling constant) is two orders of magnitude smaller than the generation frequency. This allows for the use of the slowly varying envelope approximation in the investigation of the dynamics of the spaser.

The transition processes of the NP and two-level QD interacting in a spaser can be described by the model Hamiltonian

In the absence of the external field, for the NP near field we have ${\left(8\pi \right)}^{-1}{\displaystyle \int {\mathrm{Re}\epsilon |}_{{\omega }_{SP}}{E}_1{E}_1^{\ast }}dV=0$ [26]. The near field is not equal to zero at the resonance frequency only. However, in this case, the field vanishes at the infinity and ${\displaystyle \int {\mathrm{Re}\epsilon |}_{{\omega }_{SP}}{E}_1{E}_1^{\ast }}dV={\displaystyle \int {\mathrm{Re}\epsilon |}_{{\omega }_{SP}}\nabla {\phi }^{\ast }\cdot \nabla \phi }dV=-{\displaystyle \int {\phi }^{\ast }\nabla \cdot \left({\mathrm{Re}\epsilon |}_{{\omega }_{SP}}\nabla \phi \right)}dV$. The latter integral is equal to zero because ${\nabla }^2\phi =0$both inside and outside the NP. Furthermore, $\partial \epsilon /\partial \omega =0$ outside of the particle. This leads to the following result

Assuming that the frequencies of the QD transition and the frequency of the dipole SP are close, ${\omega }_{SP}\approx {\omega }_{TLS}$, we will look for the time dependencies of $\widehat{\tilde{a}}(t)$ and $\widehat{\tilde{\sigma }}(t)$ in the form $\widehat{\tilde{a}}(t)=\widehat{a}(t)\exp \left(-i{\omega }_{a}t\right)$ and $\widehat{\tilde{\sigma }}(t)=\widehat{\sigma }(t)\exp \left(-i{\omega }_{a}t\right)$, where $\widehat{a}(t)$ and $\widehat{\sigma }(t)$ are the slowly changing amplitudes and ${\omega }_a$ is the autonomous frequency of the spaser which we seek. In the rotating wave approximation [24] we can neglect rapidly oscillating terms, $\propto \exp \left(\pm 2i{\omega }_{a}t\right)$, and obtain the interaction operator $\widehat{V}$ in the form of the Jaynes–Cummings Hamiltonian [27]

The commutation relations for operators $\widehat{a}(t)$ and $\widehat{\sigma }(t)$ are standard: $\left[\widehat{a},{\widehat{a}}^{†}\right]=\widehat{1}$ and $\left[{\widehat{\sigma }}^{†},\widehat{\sigma }\right]=\text{}\widehat{D}$, where the operator $\widehat{D}$ describes the inversion of the occupancies of the ground and excited states of the QD,$\widehat{D}(t)={\widehat{n}}_e(t)-{\widehat{n}}_g(t)$, ${\widehat{n}}_e=|e〉〈e|$ and ${\widehat{n}}_g=|g〉〈g|$ are the occupancy operators for the ground and excited states, ${\widehat{n}}_g+{\widehat{n}}_e=\widehat{1}$. Using the Hamiltonian, Eqs. (1)-(3), and the commutation relations for operators $\widehat{a}(t)$ and $\widehat{\sigma }(t)$we obtain the Heisenberg equations of motion for the operators $\widehat{a}(t)$, $\widehat{\sigma }(t)$, and $\widehat{D}(t)$ [28,29]

For $D_0<D_{th}$ the system of Eqs. (8)-(10) has a trivial stationary solution $a=\sigma =0,$ $D=D_0$only, while for $D_0>D_{th}$ there are two stationary solutions. The trivial unstable solution corresponds to the absence of SPs, while the stable one corresponds to laser generation:

 

Fig. 1 Stationary amplitudes a and σ are shown by dash-double dotted and dash-dotted lines, respectively. The stable solution for D is shown by the solid line. The unstable solution appearing for $D>D_{th}$ is shown by the dashed line. For $D_0={D^{\prime }}_0$ (also shown in Fig. 2) the stable and unstable solutions of D are marked by red dots.

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3. The spaser in the external electromagnetic field

Let us consider the dynamics of the NP and QD in the field of the external optical wave, $E(t)=E\cos {\omega }_{f}t$. Assuming that the external field is classical and taking into account the dipole interaction only, we can write the Hamiltonian in the form

Below we assume for simplicity that ${\omega }_{TLS}={\omega }_{SP}$. Hence ${\Delta }_{SP}={\omega }_f-{\omega }_{SP}={\Delta }_{TLS}$$={\omega }_f-{\omega }_{TLS}=\Delta$. Note that without the QD, the polarization of the NP corresponding to the stationary solution of the system of Eqs. (13)-(15), $\dot{\widehat{a}}=\dot{\widehat{\sigma }}=0$, has the form

The quantities defined by Eqs. (13)-(15) differ from those in Eqs. (9)-(11). Without the external field the frequency ${\omega }_a$ in the substitutions $\widehat{\tilde{a}}(t)=\widehat{a}(t)\exp \left(-i{\omega }_{a}t\right)$ and $\widehat{\tilde{\sigma }}(t)=\widehat{\sigma }(t)\exp \left(-i{\omega }_{a}t\right)$ is found from the solution of stationary Eqs. (8)-(10). When the external field is present, we use the representations $\widehat{\tilde{a}}(t)=\widehat{a}(t)\exp \left(-i{\omega }_{f}t\right)$ and $\widehat{\tilde{\sigma }}(t)\equiv \widehat{\sigma }(t)\exp \left(-i{\omega }_{f}t\right)$. We look for a stationary solution of the system of Eqs. (13)-(15). The stationary values of D satisfy a cubic equation and are shown in Fig. 2 as a function of the amplitude of the external field E. For $\Delta =0$ (Fig. 2а) in the absence of the external field $E=0$ and $D_0\ge D_{th}$ two points, stable and unstable, correspond to the points marked by dots in Fig. 1.

 

Fig. 2 The stationary values of D as a function of the amplitude of the external field for (a) zero ($\Delta =0$) and (b) non-zero ($\Delta ={10}^{11}{s}^{-1}$) detuning. For both graphs ${\tau }_a={10}^{-14}s$, ${\tau }_{\sigma }={10}^{-11}s$, ${\tau }_D=0.5\cdot {10}^{-14}s$, ${\Omega }_R={10}^{13}{s}^{-1}$, $D_0={D^{\prime }}_0=0.55$.

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4. Spaser synchronization

Only the lower branch of the synchronization region corresponds to a stable solution. The stationary solutions at zero external field and with no detuning (${\delta }_{TLS}={\delta }_{SP}=\Delta =0$) are shown in Fig. 1. When detuning is present (we still assume that ${\delta }_{SP}={\delta }_{TLS}=0$ but $\Delta \ne 0$), the stable stationary solution exists for $E>E_{Synch}\left(\Delta \right)$ only (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. To obtain an analytical estimate of the Arnold tongue boundary, we find the solution of Eqs. (13)-(15) in the first approximation with respect to the field. For the case ${\delta }_{SP}=0$, using the substitution $a=\left|a\right|\exp \left(i\phi \right)$ and $\sigma =\left|\sigma \right|\exp \left(i\psi \right)$in Eq. (15) and equating its imaginary parts, we obtain

 

Fig. 3 The dependence of the plasmon dipole moment on the amplitude of the external field and the frequency detuning Δ. The speckle structure at low values of E corresponds to the chaotic behavior of the dipole moment.

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Since $E(t)=E\cos {\omega }_{f}t$, $\phi$ represents the phase shift between the external field and spaser oscillations. For $\left|a\right|$, $\left|\sigma \right|$, and $\cos \left(\psi -\phi \right)$ we can use their values obtained from Eq. (11) in the zeroth order with respect to the field approximation. As the result, we have

The phase dynamics can be viewed as the motion of a particle sliding along the potential profile $U\left(\phi \right)$ (Fig. 4 ) in a viscous liquid. For a small field and/or large detuning, $\left|\xi /\Delta \right|<1$, the phase difference of the system and the external field increases monotonically. For $\left|\xi /\Delta \right|>1$, the “particle” should be trapped in one of the minima of the potential function. In this regime the oscillation of the system is synchronized and the phase difference is time independent. In the approximation considered, the width of the synchronization region is proportional to the field amplitude. Within the Arnold tongue, if the energy of the interaction of the NP with the external field is smaller than the QD-NP interaction, the amplitude of the auto-oscillations depends weakly upon the amplitude of the external field, which in this case plays the role of a synchronizer.

 

Fig. 4 The potential $U(\phi )$ for $\left|\xi /\Delta \right|<1$ (dashed blue line) and $\left|\xi /\Delta \right|>1$ (solid red line).

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When $\Delta \to 0$ the threshold field value $E_{synch}$ is determined by the equation ${\left({\mu }_{NP}{E}_{Synch}/\hslash \right)}^2=\left(D_0-D_{th}\right){\Delta }^2\left({\tau }_a/4{\tau }_D\right).$ For large values of the detuning, $E_{synch}$ becomes independent of $\Delta$ (see Fig. 3). Taking into account that ${\tau }_a\sim {10}^{-14}{s}^{-1}$ and ${\tau }_{\sigma }\sim {10}^{-11}{s}^{-1}$ for plasmonic NPs, one can estimate the asymptotic value of the Arnold tongue boundary, $E_{Synch}^{*}$. Our numerical calculations show that $E_{Synch}(\Delta )$ tends to a plateau for $\Delta \sim 2\cdot {10}^{11}{s}^{-1}$, which gives $E_{Synch}^{*}\sim 3\cdot {10}^{3}V/m$. This agrees with calculations of [3,10] in which wave propagation in the system of spasers was considered. In [3,10] the amplitude of the incident wave was a few orders of magnitude larger than $E_{Synch}^{*}$. Such a field is comparable with the near field inside the spaser. In this case, spaser becomes synchronized with the external field for any value of detuning; it ceases to be an autonomous system and responds linearly to the external field as can be seen in Fig. 3. For such strong fields, losses cannot be compensated exactly. Moreover, our numerical calculations show that for a very strong field, the population inversion of the QD decreases, the excitation of SPs by QDs is inhibited, and the spaser becomes just a lossy NP. Thus, when detuning is substantial, spasers cannot be used as active inclusions for the perfect lens because both loss and amplification destroy the perfect image, so that the exact compensation of loss is required.

For applications it is important to know not only a response of a single spaser to an external electromagnetic wave but also the behavior of a system of interacting spasers randomly or regularly distributed in the dielectric matrix. The interaction between spasers may lead to a cooperative behavior of spasers. This problem is a subject of our future research.