## Abstract

We model spaser as an *n*-level quantum system and study a spasing geometry comprising of a metal nanosphere resonantly coupled to a semiconductor quantum dot (QD). The localized surface plasmons are assumed to be generated at the nanosphere due to the energy relaxation of the optically excited electron-hole pairs inside the QD. We analyze the total system, which is formed by hybridizing spaser’s electronic and plasmonic subsystems, using the density matrix formalism, and then derive an analytic expression for the plasmon excitation rate. Here, the QD with three nondegenerate states interacts with a single plasmon mode of arbitrary degeneracy with respect to angular momentum projection. The derived expression is analyzed, in order to optimize the performance of a spaser operating at the triple-degenerate dipole mode by appropriately choosing the geometric parameters of the spaser. Our method is applicable to different resonator geometries and may prove useful in the design of QD-powered spasers.

© 2013 Optical Society of America

## 1. Introduction

The emerging era of nanoplasmonics is expected to improve the speed and efficiency of optical devices, by allowing miniaturization beyond the diffraction limit using surface plasmons in circuits [1–3]. Despite their advantages in miniaturization, surface plasmons which excite at metal-dielectric interfaces are highly dissipative and vanish in a few wavelengths when traveling [1, 4]. Therefore, the energy must be transferred from an external source to the surface plasmon wave to sustain its existence in nanoplasmonic circuits [1]. The spaser, which is the nanoplasmonic counterpart of a conventional laser is the prime device for generating these surface plasmon waves and amplifying them during propagation [5, 6].

Bergman and Stockman proposed the theory of the spaser [5] by claiming that a nanosystem with an excited active medium can transfer energy nonradiatively to a closely located plasmonic resonator and excite localized fields inside it. This energy transfer takes place due to the interaction between the resonator and active medium through the near field. It was stated that, electronic transitions in active medium are stimulated by the surface plasmons which already exist in the system resulting in a multiplication of the plasmon population, very much resembling the coherent feedback strategy used in conventional laser cavities [5, 7].

Since its theoretical formulation, many experimental efforts have been carried out to fabricate a spaser. Seidel *et al.*[ 8] proved the possibility of the stimulated emission of surface plasmon by amplifying surface plasmons at the interface between a flat continuous silver film and a liquid containing organic dye molecules. The first demonstration of a spaser was done by Noginov *et al.*[ 9] using a gold nanosphere with a 7 nm radius, surrounded by the active medium, which is a 15 nm thick silica shell containing dye molecules. When the dye is optically pumped, it releases energy to the gold nanosphere causing excitation of localized surface plasmons. Another realization of a spaser was reported by Flynn *et al.*[ 10] who sandwiched a gold-film plasmonic waveguide between optically pumped InGaAs quantum wells.

Several other structures have also been proposed for spasers, including a V-shaped metallic nanoparticle attached to quantum dots (QDs) [5], a two-dimensional array of split-ring shape plasmonic resonators supported by a substrate acting as the active medium [11], a bowtie-shaped metallic structure, in which QDs are placed in the bowtie gap and multiple quantum wells are located in the substrate [12], and a metal groove with QDs placed at its bottom [13]. Even though most of these spaser designs employ QDs in the active medium, it is possible to have optically pumped rare-earth ions, dyes or bulk electrical injection as the excitable gain element [1]. QDs are widely used in many lasing setups. As the charge carriers in a QD are confined in all three dimensions to a very small size, its density of states almost resembles a delta function, mimicking an atomistic behavior with a well defined spectral response. They also promise a better stability over temperature variations [14–17].

Although ample research has been done on spasers, it is seen that results depend on the model. Furthermore, most of these studies very much focus on the role of the active medium. In this paper, instead of only considering the states of the active medium, we analyze the electronic and plasmonic states of the whole spaser as a single quantum system. Degeneracy of plasmon modes is also considered. This is a more general approach to describe the spaser quantum mechanically compared to available literature. We rigorously describe our model using a simple spaser geometry comprising of a spherical metal nanoparticle and a QD. A major importance of the spaser is its ability to be effectively used in nanoplasmonic circuits, to generate surface plasmon waves. This paper is based on studying the capability of a spaser to excite these surface plasmons and finding all the parameters affecting the excitation rate. We also consider dissipations to the environment and pay attention on all the material properties influencing the spaser operation. The presenting work also provides clear design guidelines for a spherical spaser. These guidelines allow one to select its operating wavelength, possible values for the size parameters of the spaser components, and optimum QD placement.

It is important to compare our work with the related literature. Many theoretical studies have been performed on interaction between the gain elements and plasmonic systems with either localized or propagating surface plasmons. In order to determine spaser characteristics such as threshold spasing condition, spasing frequency and spaser linewidth, Stockman [5, 18, 19] analyzed the interaction between a metal nanoparticle and closely located chromophores which are modeled as two level systems (TLS). Introducing a ‘dipole nanolaser’, Protsenko *et al.*[ 20] derived the equations of motion for spasers. Khurgin *et al.* have also discussed the threshold spasing condition, linewidth and coherence of a spaser in [21, 22]. Considering a simple spaser formed by a two level QD attached to a metal nanoparticle, Andrianov *et al.* have examined some attributes of spasers such as dipole response to an optical wave [23], external forced synchronization [24] and occurrence of Rabi oscillations [25]. Among the other investigations on radiation emitted from nanoparticle-TLS ensembles, [26] which inspect the resultant photon statistics, and [27] which presents the spectral properties of these photons are conspicuous. The theory of spaser also suggests that propagating surface plasmons can be amplified by placing a gain medium in the vicinity [1]. This idea has been exploited to compensate the losses in metamaterials. A few of such utilizations are described in [28, 29] where plasmon-TLS interactions and [30, 31] in which plasmon-four level system interactions are analyzed.

Among the quantum mechanical models of the spaser, the work of Stockman [5, 18, 19] and Protsenko *et al.*[ 20] are noteworthy. However, they only analyze the quantum states of the active medium assuming it as a TLS. We improve this model by considering the whole spaser as a single *n*-level quantum system with a 3-level active medium. Stockman considers a third level [18] but assumes that its population is negligible and electron-hole pairs rapidly relax to the second level. In our model, there is a finite third level population and electron-holes pairs relax to the second level by interacting with the environment at a defined rate. Having a three level system, a designer has more control on choosing a suitable pump frequency such that it doesn’t overlap with any surface plasmon resonance frequency of the resonator. In addition, we consider dissipations through the generalized master equation in the form of interaction with the system’s bath. Here, all the major dissipation rates at each level are well represented. Since our intention is to find all the parameters affecting the plasmon excitation rate, it is required to consider all the forms of losses.

In our analysis, we first find the electric field and the eigenfrequencies of the localized surface plasmon modes in the resonator, then characteristics of the active medium, followed by the derivation of the spaser Hamiltonian, which is needed to analyze the spaser kinetics using the density matrix theory. Then we obtain an expression for the plasmon excitation rate of the spaser, and study how the geometric parameters affect the operation of a dipole spaser.

## 2. Spaser model

A spaser consists of a plasmonic resonator (or cavity), which supports surface plasmon modes, and an active medium, which amplifies the surface plasmons [5]. Figure 1 shows a schematic diagram of our spaser model where the resonator is a metal nanosphere surrounded by dielectric shell, and a QD is embedded in the shell as the active medium. Spasing occurs as a consequence of the nonradiative energy transfer from the QD to the nanosphere, exciting localized surface plasmon modes inside it.

The spherical core-shell structure is one of the most studied geometries for surface plasmon resonance [32–34], because it is one of the few geometries where Maxwell’s equations can be analytically solved [35, 36]. Suppose the inner radius of the metal nanosphere is *R*_{1} and it is smaller than the skin depth of the metal [37], which is usually around 25 nm for noble metals. The latter assumption is necessary to allow the localized surface plasmon modes to penetrate everywhere within the nanosphere and excite coherent electron cloud oscillations [18, 38]. Hence, the nanosphere radius is very smaller compared to the wavelength of incident light. Further, we assume that *R*_{1} is greater than *v _{F}/ω* (where

*v*is Fermi velocity and

_{F}*ω*is the surface plasmon frequency), which is about 1 nm for noble metals, to avoid the effects of Landau damping [38]. The outer radius of the shell,

*R*

_{2}must be chosen such that the shell thickness is large enough to the QD be entirely embedded within the dielectric (i.e.

*R*

_{2}−

*R*

_{1}> QD diameter).

When the nanosphere radius and the shell thickness are fixed, the resonator supports a series of plasmon modes (dipole, quadrupole, *etc.*) with unique energies [32, 33]. These plasmon modes may overlap with the QD in different degrees, and hence experience gain in different amounts. The plasmon mode receiving the highest gain survives and becomes the dominant spaser mode [9, 38]. The gain to the spasing mode is received as a result of the electronic transitions in the QD causing recombination of an electron-hole pair. To ensure continuous spasing, we facilitate population inversion of the two energy levels pertaining to these transitions using a suitable pump source. Here we opt for optical pumping for the sake of design simplicity. Even though QD is the target of the pumping field, it may also excite the resonator in the vicinity [39]. To minimize this extraneous effect, we intentionally select the optical pumping frequency with considerable detuning from the surface plasmon resonances of the resonator. Therefore, the excited field is much weaker compared with the spaser mode which overlaps with QD emission lines. However, we have not made any restrictive assumption here because one may completely avoid this situation by adopting a different pumping mechanism such as direct electric injection to QD as suggested in [40].

#### 2.1. Localized surface plasmon modes of the resonator

To find the electric field and eigenfrequencies of the localized surface plasmon modes, we assume that QD’s presence does not perturb the electric field in the system because it is very small compared to the nanosphere and therefore the resultant permittivity change of the shell is insignificant. We adopt a standard spherical coordinate system (*r,θ,ϕ*) where the origin coincides with the center of the spherical shell (see Fig. 1). The supported electromagnetic modes can be found by solving the vector Helmholtz equation for the spherical shell following the method of Debye potentials [35, 36]. The solution gives a series of localized surface plasmon modes denoted by the angular momentum number *l _{p}* and its projection

*m*,

_{p}*ω*

_{lp}is the angular frequency of the plasmon mode

*l*. The stationary electric field is given by the expression

_{p}*j*(

_{l}*kr*),

*y*(

_{l}*kr*),

*h*(

_{l}*kr*) for

*ν*= 1, 2, 3, ${P}_{l}^{m}$ are associated Legendre polynomials,

*k*is the wavenumber, and

*a*,

_{l}*d*,

_{l}*g*and

_{l}*f*are constants assuring the field’s continuity at boundaries.

_{l}Plasmon modes expressed by Eq. (1) possess a unique energy determined by *h̄ω*_{lp}, which can be found by solving the dispersion relation of the resonator. In order to obtain the dispersion relation, we first write a set of equations expressing the constants *a _{l}*,

*d*,

_{l}*g*and

_{l}*f*in Eq. (2) by equating the tangential components of the field at the metal-dielectric and dielectric-ambient interfaces. This results in a homogeneous system of linear equations in which we apply the condition for a nontrivial solution and obtain the following dispersion relation for the

_{l}*l*th mode surface plasmon resonance:

*s*

_{1}= −

*ψ*(

_{l}*k*

_{m}R_{1}),

*s*

_{2}=

*ψ*(

_{l}*k*

_{d}R_{1}),

*s*

_{3}= −

*χ*(

_{l}*k*

_{d}R_{1}), ${k}_{1}=\sqrt{{\epsilon}_{2}}{\xi}_{l}^{\prime}\left({k}_{0}{R}_{2}\right)$,

*k*

_{2}= −

*ψ′*(

_{l}*k*

_{d}R_{2}),

*k*

_{3}=

*χ′*(

_{l}*k*

_{d}R_{2}),

*m*

_{1}=

*ξ*(

_{l}*k*

_{0}

*R*

_{2}),

*m*

_{2}= −

*ψ*(

_{l}*k*

_{d}R_{2}) and

*m*

_{3}=

*χ*(

_{l}*k*

_{d}R_{2}). Here the Reccati-Bessel functions are

*ψ*(

_{l}*x*) =

*xj*(

_{l}*x*), ${\xi}_{l}\left(x\right)=x{h}_{l}^{\left(1\right)}\left(x\right)$ and

*χ*(

_{l}*x*) = −

*xy*(

_{l}*x*);

*k*,

_{m}*k*, and

_{d}*k*

_{0}are the wavenumbers of the electromagnetic field in the metal, dielectric, and ambient, respectively.

According to Eq. (3), the energy of the plasmon mode, *h̄ω _{l}* is a function of

*R*

_{1},

*R*

_{2},

*ε*

_{1},

*ε*

_{2}, and

*ε*

_{3}, which implies that only the size parameters of the resonator determine the energy of the spaser mode when the spaser materials are chosen. Furthermore, this nontrivial solution of the boundary conditions allows us to express the coefficients

*a*,

_{l}*d*,

_{l}*g*in terms of

_{l}*f*as

_{l}*f*as a normalization constant, we follow the procedure of secondary quantization for dispersive media [41, 42], and equate the total energy of the electric field of the plasmon mode to

_{l}*h̄ω*,

_{l}*V*implies the volume integral over three dimensional Euclidean space. We can note that owing to this equality,

*f*implicitly depends on the spaser shell parameters

_{l}*R*

_{1}and

*R*

_{2}.

#### 2.2. Active medium

Characteristics of the active medium determine the strength of its electron-hole pairs’ interaction with the plasmon modes in the resonator and the amount of amplification that each plasmon mode receives. The characteristics of the QD includes its wavefunction and the energy levels of the electron-hole pairs. Electronic transitions between these levels fuel spasing and we assume that, as in the case of lasers, those transitions representing recombination of electron-hole pairs would excite the plasmon modes in the resonator. Normally, an excitable QD like this can be effectively described using three energy levels to account for pumping and stimulated transitions. We denote the ground level of the QD by state vector |0* _{e}*〉 and two excited levels by |1

*〉 and |2*

_{e}*〉. To assign energies to these three levels, we have to take the quantum confinement effects into account. To do this, we approximate the QD by a sphere with a radius*

_{e}*R*, confined by an infinite potential barrier (i.e. potential

_{q}*V*(

*r*) = 0 for

*r*<

*R*and it is infinite otherwise). The resulting time-independent Schrodinger equation is separable in spherical coordinates and possesses a solution similar to the hydrogen atom and given by [43]

_{q}*n*,

_{q}*l*,

_{q}*m*are principal, azimuthal and magnetic quantum numbers describing the states,

_{q}*ξ*

_{nqlq}is the

*n*th root of the spherical Bessel function of the first kind (i.e

_{q}*j*(

*ξ*

_{nqlq}) = 0), and ${Y}_{{l}_{q}}^{{m}_{q}}$ are spherical harmonics.

The corresponding eigenvalue of the wave function *ψ*_{nq,lq,mq} gives the excitable energy levels of an electron-hole pair:
${\mathcal{E}}_{{n}_{q},{l}_{q}}={\mathcal{E}}_{g}+\frac{{\overline{h}}^{2}}{2\mu {R}_{q}^{2}}{\xi}_{{n}_{q},{l}_{q}}^{2}$, where
$\mu ={m}_{e}^{*}{m}_{h}^{*}/\left({m}_{e}^{*}+{m}_{h}^{*}\right)$ is the reduced mass of an electron-hole pair,
${m}_{e}^{*}$ and
${m}_{h}^{*}$ are effective masses of an electron and a hole, and *ℰ _{g}* is the bandgap of the QD material. These energy levels denoted by

*ℰ*

_{nq,lq}are 2

*l*+ 1 times degenerate. For a transition from an initial state |

_{q}*s*〉 with quantum numbers

_{i}*s*= (

_{i}*n*) to a final state |

_{i}, l_{i}, m_{i}*s*〉 with quantum numbers

_{f}*s*= (

_{f}*n*,

_{f}*l*,

_{f}*m*), the absorbed energy from the system will be Δ

_{f}*ℰ*

_{sf,si}=

*ℰ*

_{nf,lf}−

*ℰ*

_{ni,li}. Energy will be released in case this quantity is negative.

Having this knowledge, we map three QD energy levels. For |1* _{e}*〉 and |2

*〉, we select two lowest energy levels [i.e (*

_{e}*n*) = (1, 0), (1, 1)], assuming that probability of populating the higher energy levels are very small, and |0

_{q}, l_{q}*〉 is mapped to the ground level. This mapping also enables us to calculate the QD radius,*

_{e}*R*required for an efficient energy transfer to the resonator where the energy received by the spaser modem,

_{q}*h̄ω*

_{lp}matches the energy released by the QD, −Δ

*ℰ*

_{s0e,si}, giving the resonance QD radius:

## 3. Spaser kinetics

In the previous section we analyzed the characteristics of the isolated electronic and plasmonic subsystems. Here, we let them to interact with each other and create the functioning spaser system, as shown in Fig. 2. In the electronic subsystem, we have three states denoted by |0* _{e}*〉, |1

*〉, and |2*

_{e}*〉, as described in Section 2.2. We assume that active medium strongly interacts only with the plasmon mode*

_{e}*l*, which is the spaser mode. Here we note that this assumption is not valid for higher modes when frequency spacing between them become smaller than the QD emission linewidth. Let us define the state |0

_{p}*〉 with zero plasmons, and 2*

_{pl}*l*+ 1 states $|{1}_{pl}^{\left({l}_{p},{m}_{p}\right)}\u3009$ with one plasmon, as the plasmonic subsystem. Then we amalgamate these two subsystems to makeup the total system shown in Fig. 2(c) with

_{p}*n*= 2

*l*+ 4 product states defined as |1

_{p}*〉 ≡ |0*

_{s}*〉 |0*

_{e}*〉, |2*

_{pl}*〉 ≡ |1*

_{s}*〉 |0*

_{e}*〉, |3*

_{pl}*〉 ≡ |2*

_{s}*〉 |0*

_{e}*〉, |4*

_{pl}*〉 ≡ |0*

_{s}*〉 $|{1}_{pl}^{\left({l}_{p},-{l}_{p}\right)}\u3009$, |5*

_{e}*〉 ≡ |0*

_{s}*〉 $|{1}_{pl}^{\left({l}_{p},-{l}_{p}+1\right)}\u3009,\dots ,|{n}_{s}\u3009\equiv |{0}_{e}\u3009|{1}_{pl}^{\left({l}_{p},{l}_{p}\right)}\u3009$.*

_{e}The product states |1* _{s}*〉, |2

*〉 and |3*

_{s}*〉 are associated with zero plasmons and |4*

_{s}*〉, |5*

_{s}*〉,...,|*

_{s}*n*〉 possess one plasmon of the spaser mode. The |1

_{s}*〉 → |3*

_{s}*〉 transition, which is the excitation of ground electron-hole pairs to the highest energy level in our model, occurs due to the electron-hole pairs’ interactions with the pump light which we analyze classically. Transitions from the state |2*

_{s}*〉 to one of the states |4*

_{s}*〉, |5*

_{s}*〉,...,|*

_{s}*n*〉 is the driving force for the phenomena of spasing because they excite plasmon modes in the resonator. Some transitions may occur from the state |

_{s}*j*〉 to |

_{s}*i*〉 due to the interaction with the bath. They can be considered as dissipations. Having this model, we analyze the kinetics of the

_{s}*n*-level system by first constructing its Hamiltonian, and then deriving the density matrix equations to find the corresponding active state populations.

#### 3.1. Hamiltonian of the spaser

Hamiltonian *H* of the spaser contains the non interacting electronic and plasmonic Hamiltonians, *H _{e}* and

*H*, and Hamiltonian,

_{pl}*H*, of the interacting subsystems:

_{i}*H*=

_{e}*h̄ω*

_{1e}|1

*〉 〈1*

_{e}*| +*

_{e}*h̄ω*

_{2e}|2

*〉 〈2*

_{e}*|, and ${H}_{pl}={\sum}_{{m}_{p}=-{l}_{p}}^{{l}_{p}}\overline{h}{\omega}_{{l}_{p}}{b}_{{l}_{p},{m}_{p}}^{\u2020}{b}_{{l}_{p},{m}_{p}}$. Here ${b}_{{l}_{p},{m}_{p}}^{\u2020}$,*

_{e}*b*

_{lp,mp}are the creation and annihilation operators of the surface plasmons corresponding to the quantum numbers

*l*and

_{p}*m*.

_{p}The interacting Hamiltonian *H _{i}* can be decomposed to represent the interactions between the electron-hole pairs and pump light as

*H*and the interactions between electron-hole pairs and surface plasmons as

_{e,L}*H*:

_{e,pl}*V*

_{sf,si}is the matrix element for the transition |

*i*〉 → |

*f*〉,

*i, f*= {0

*, 1*

_{e}*, 2*

_{e}*}, ${g}_{{l}_{p}}=\sqrt{{\omega}_{{l}_{p}}/2{\epsilon}_{2}{V}_{n}}$, where*

_{e}*V*is the normalization volume,

_{n}*φ*(

*t*) and

*ω*are the envelope function and frequency of the pump light, and c.c. represents the complex conjugate [44–46]. Relaxation process from the state |2

_{L}*〉 to |1*

_{e}*〉 is taken into account through the relaxation constant*

_{e}*ξ*

_{23}in spaser kinetics. The matrix element ${V}_{{s}_{f},{s}_{i}}^{{l}_{p},{m}_{p}}$ corresponding to plasmon’s interaction with electron-hole pairs can be written as

*u*〉 and |

_{c}*u*〉 are the Bloch functions, |

_{v}*s*〉 and |

_{i}*s*〉 are the envelope wavefunctions of the initial and final electronic states characterized by the sets of quantum numbers

_{f}*s*and

_{i}*s*and

_{f}**r̃**is the displacement vector of the electron-hole pairs. Assuming the equality

**E**

_{lpmp}=

*E*

_{lpmp}

**e**

_{lpmp}, we may write

*P*[47, 48] as, $\u3008{u}_{c}\left|{\mathbf{e}}_{{l}_{p}{m}_{p}}.\tilde{\mathbf{r}}\right|{u}_{v}\u3009=\sqrt{2}P/{\mathcal{E}}_{g}$ and the second matrix element, $\u3008{s}_{f}\left|{E}_{{l}_{p}{m}_{p}}\right|{s}_{i}\u3009={\mathrm{\Upsilon}}_{{s}_{f},{s}_{i}}^{{l}_{p},{m}_{p}}$, can be expressed by

*V*,

_{QD}**r**=

**r**

_{0}+

**r′**is the electron position inside the quantum dot, and

**r**

_{0}is radius vector of the quantum dot’s center.

*E*

_{lpmp}(

**r**) can be derived from Eq. (2) and

*ψ*

_{sf}is taken from Eq. (5). In case QD is very small compared to the nanosphere, it is reasonable to assume that

*E*

_{lpmp}(

**r**) is approximately constant over the QD’s volume, therefore the integral in Eq. (13) can be simplified to ${\mathrm{\Upsilon}}_{{s}_{f},{s}_{i}}^{{l}_{p},{m}_{p}}={E}_{{l}_{p}{m}_{p}}\left({\mathbf{r}}_{0}\right){\delta}_{{s}_{f},{s}_{i}}$. Hence, from Eq. (12), the matrix element for the spaser mode’s interaction with electron-hole pairs can be given by

*V*

_{s2e,s0e}, electric field term

*E*

_{lpmp}in Eq. (11) should be replaced by the electric field caused by pump light.

#### 3.2. Plasmon excitation rate of the spaser

Having the Hamiltonian calculated, we analyze the *n* states system comprises of the product states |1* _{s}*〉, |2

*〉,...,|*

_{s}*n*〉, using the density matrix formalism. We define the populations of those states by

_{s}*ρ*

_{11},

*ρ*

_{22},...,

*ρ*and assume that the system has a short-term memory [49] and coupled to a bath which is the reservoir for system’s dissipations. Then the relaxation superoperator, which is added to the commuted Hamiltonian and density operator to incorporate dissipations, consists of a set of constants that define the relaxation kinetics of diagonal and off-diagonal elements of the reduced density matrix [44, 49]. Using the Markov and secular approximations [50], master equation for the system can be given by

_{nn}*γ*is the population relaxation rate of the state |

_{μμ}*μ*〉,

_{s}*γ*= (

_{μν}*γ*+

_{μμ}*γ*)

_{νν}*/*2 +

*γ̂*is the coherence relaxation rate between the states |

_{μν}*μ*〉 and |

_{s}*ν*〉,

_{s}*γ̂*is the pure dephasing rate, and

_{μν}*ξ*is the transition rate from state |

_{νκ}*κ*〉 to state |

_{s}*ν*〉 due to interaction with the bath. We assume that the lifetime of the ground state |1

_{s}*〉 is very large, by setting*

_{s}*γ*

_{11}= 0. The parameters

*γ*

_{44},

*γ*

_{55},...,

*γ*define the dissipation of degenerated states of the plasmon mode denoted by

_{nn}*l*quantum numbers. However, since the energy and dielectric properties are common for all the degenerated plasmon states, we assume that all these plasmon dissipation constants are equal and can be denoted by

_{p}, m_{p}*γ*.

_{pl}Populations *ρ*_{44}, *ρ*_{55},...,*ρ _{nn}* represent the excitation rates for the plasmon modes

*l*= (

_{p}, m_{p}*l*−

_{p},*l*), (

_{p}*l*, −

_{p}*l*+ 1)

_{p}*,*...,(

*l*) respectively. The sum of the plasmon populations of

_{p}, l_{p}*l*th mode,

_{p}*ℛ*

_{lp}gives the number of plasmons excited in the spaser at a given time. Hence, this quantity can be referred as the effective plasmon excitation rate of the spaser. We solve the system of partial differential equations given in the Eq. (15) for the continuous wave (CW) operation assuming that

*φ*(

*t*) = 1 and obtain an expression for the total plasmon excitation rate:

_{L}_{3}=

*ω*−

_{L}*ω*

_{3}, Δ

_{2}

*=*

_{j}*ω*

_{2}−

*ω*when

_{j}*ω*

_{2},

*ω*

_{3},

*ω*and

_{j}*ω*are the energies of the states |2

_{L}*〉, |3*

_{s}*〉, |*

_{s}*j*〉 and pump light respectively. As all the degenerate plasmon modes have the same energy,

_{s}*ω*=

_{j}*ω*

_{lp}∀

*j*and therefore Δ

_{2}

*= Δ*

_{j}_{2}

*. We can assume that detuning of the energy of the pump light with the energy of the state |3*

_{p}*〉, Δ*

_{s}

_{L}_{3}≪

*γ*

_{13}and hence, $\frac{{\gamma}_{13}}{{\gamma}_{13}^{2}{\mathrm{\Delta}}_{L3}^{2}}\simeq \frac{1}{{\gamma}_{13}}$. Further, we also assume that all the degenerate states of the plasmon modes decay equally and therefore

*γ*=

_{jj}*γ*and

_{pl}*γ*

_{2}

*=*

_{j}*γ*

_{2}

*∀*

_{p}*j*. With these simplifications, Eq. (16) can be further simplified to

*γ*of the spaser mode can also incorporate the resultant of both radiative and nonradiative decay rates. In addition, decay rates could potentially be different for other plasmon modes [52], which are not taken into account assuming that they weakly overlap with the QD emission spectrum. As discussed in Section 2.1, energy of the spaser mode

_{pl}*ω*

_{lp}depends on resonator’s size parameters (i.e.

*R*

_{1}and

*R*

_{2}) when the materials of the spaser components are chosen. Therefore, it can be observed from Eq. (17) that the total plasmon excitation rate of the spaser,

*ℛ*

_{lp}mainly depends on the matrix elements for the electron-hole pair-plasmon interaction, ${V}_{{s}_{1e},{s}_{0e}}^{{l}_{p},{m}_{p}}$ for each degenerate state and the detuning Δ

_{2}

*because we assume that matrix element for the pump light-QD interaction*

_{p}*V*

_{s2e,s0e}is constant under CW operation. In the case of exact resonance (i.e.

*R*is given by Eq. 6), Δ

_{q}_{2}

*= 0 and we achieve the highest plasmon excitation rate.*

_{p}For fixed materials and size parameters *R*_{1}, *R*_{2}, there is a unique spaser mode energy *h̄ω*_{lp} and *R _{q}* may be chosen according to

*ω*

_{lp}to achieve a higher plasmon excitation rate. Then the position of the QD plays a major role deciding the amount of amplification of the plasmons as the electric field of the spaser mode changes with the location according to Eq. (2). To analyze these factors, we investigate the spaser’s behavior with respect to spaser size parameters and the QD’s location in the following section taking a dipole spaser as a case study.

## 4. Case study: A dipole spaser

Let us consider a spaser whose dipole mode (*l _{p}* = 1) is amplified by the active medium. We construct this spaser using gold for the nanosphere, and coating silica (SiO

_{2}) over it, making up a dielectric shell. We embed a CdSe QD in this shell. Once the materials for spaser components are chosen, we are ready to investigate the dipole spaser’s operation according to its geometric parameters. Here, we use the analytical results obtained in Sections 2 and 3 on the resonator’s plasmonic properties and the characteristics of the resonator-QD interactions.

The knowledge of the frequency dependence of the permittivities are required to calculate the electric field of the spaser mode and the plasmon decay rate. For silica, we assume a non-dispersive permittivity, *ε*_{2} = 2.15 in contrast to gold in which we assume a frequency dependent permittivity. Furthermore, since our gold nanosphere is very small, we have to consider the size dependency modification of the permittivity as well [53, 54]. In order to incorporate this size effect, we use the model [54],
${\epsilon}_{1}\left(\omega ,{R}_{1}\right)={\epsilon}_{\infty}-\frac{{\omega}_{p}^{2}}{{\omega}^{2}+i\omega \mathrm{\Gamma}\left({R}_{1}\right)}$, where *ω _{p}* is the bulk plasma frequency of gold, Γ is the electron collision frequency with damping defined by Γ =

*γ*+

_{b}*v*2

_{F}/*R*

_{1},

*γ*is the bulk electron collision frequency of gold,

_{b}*v*is the Fermi velocity, and

_{F}*ε*

_{∞}is the contribution from the interband transitions obtained by fitting

*ε*

_{1}(

*ω*,

*R*

_{1})|

_{R1→∞}to the experimental data published by Johnson and Christy [55] for bulk material. Also we assume

*ω*= 1.36 × 10

_{p}^{16}s

^{−1},

*γ*= 3.33 × 10

_{b}^{13}s

^{−1},

*v*= 1.4 × 10

_{F}^{6}m/s, and

*ε*

_{∞}= 9.84 [54,56]. Spaser’s outer boundary is assumed to be in free space and hence

*ε*

_{3}= 1.

Using these permittivities, we solve the dispersion relation given in Eq. (3) for dipole mode and then plot energy of the spaser mode as a function of the nanosphere radius and shell thickness as shown in Fig. 3(a). The contour plot shows that there can be many (*R*_{1}, *R*_{2}) pairs which can result in the same spaser mode energy. For example, if the spaser mode energy is 2.385 eV (i.e. equivalent wavelength is approximately 520 nm), it traces a curve on the contour plot as marked. It is important to note that, although we refer energies in electron volts for convenience, they should be converted to SI units when substituted to equations.

To find the appropriate QD radius to match the operating point, we substitute the corresponding spaser mode energy in Eq. (6). For the previously set operating point, 2.385 eV, the substitution gives resonant QD radius to be 2.4 nm, when the QD material parameters are *E _{g}* = 1.74 eV,
${m}_{e}^{*}=0.13$, and
${m}_{h}^{*}=0.45$[57, 58]. Calculating the energy levels of the QD with this radius, and considering the fact that dipole mode is triply degenerate with

*m*= −1, 0 and 1, we redraw the total system diagram given in Fig. 2(c) for this dipole spaser to attain the system illustrated in Fig. 4. The corresponding electronic levels |0

_{p}*〉, |1*

_{e}*〉 and |2*

_{e}*〉 of the QD in this system posses energies 0, 2.385 and 3.059 eV respectively. The resulting total system has*

_{e}*n*= 6 states denoted by |1

*〉, |2*

_{s}*〉,...,|6*

_{s}*〉.*

_{s}Evaluating this 6 state system of the dipole spaser for the highest plasmon excitation rate gives the optimum size parameters. Just opting an arbitrary *R*_{1}, *R*_{2} pair on the preferred spaser mode energy curve may not result in the highest plasmon generation. To investigate the behavior of our dipole spaser, we follow the results derived in Section 3 by substituting *l _{p}* = 1. As we discussed there, the total plasmon excitation rate of the dipole spaser is represented by the sum of the populations of the states |4

*〉, |5*

_{s}*〉, and |6*

_{s}*〉,*

_{s}*ℛ*

_{1}=

*ρ*

_{44}+

*ρ*

_{55}+

*ρ*

_{66}. We use the expression for

*ℛ*

_{lp}given in Eq. (17) to study the plasmon excitation rate of the dipole spaser. This equation contains some relaxation constants,

*γ*

_{13},

*γ*

_{22},

*γ*

_{33},

*γ*

_{2}

*, and*

_{p}*ξ*

_{23}, which depend on the environment and used materials, but do not depend on spaser’s geometry. Therefore, we keep them constant and compare the plasmon excitation rate by investigating the normalized plasmon population in the relevant plots, as our objective is to study how the spaser’s geometrical parameters result in relatively high or low plasmon populations. While calculating the matrix elements, for the sake of simplicity, we assume that QD is located on the nanosphere’s dipole axis in

*θ*= 0 direction.

The only relaxation rate influenced by spaser’s geometry is the decay rate of surface plasmons, *γ _{pl}* is a function of the spaser mode energy given by
${\gamma}_{pl}\simeq \text{Im}\left[{\epsilon}_{1}\left(\omega \right)\right]/{\frac{\partial}{\partial \omega}\text{Re}\left[{\epsilon}_{1}\left(\omega \right)\right]|}_{\omega ={\omega}_{{l}_{p}}}$[19]. Since the nanosphere is very much smaller than the wavelength, here we consider only nonradiative decay assuming that its radiation loss is negligible [51]. By substituting the model for nanosphere’s permittivity in this expression, the plasmon decay rate can be simplified to
${\gamma}_{pl}\simeq \frac{\mathrm{\Gamma}}{2}\left[1+{\left(\mathrm{\Gamma}/{\omega}_{{l}_{p}}\right)}^{2}\right]$. Since both the quantities Γ and

*ω*

_{lp}depend on size parameters

*R*

_{1}and

*R*

_{2},

*γ*also depends on them for a given material. Continuing with these simplifications, we plot the plasmon excitation rate,

_{pl}*ℛ*

_{1}of the dipole spaser with respect to nanosphere radius and shell thickness, as shown in Fig. 3(b), when the QD’s location is fixed to the middle of the dielectric shell and

*R*is fixed to the resonant radius given by Eq. (6). It can be noted that plasmon excitation rate is higher for smaller nanosphere radii and shell thicknesses. It monotonically decreases when the total volume increases. This plot helps us to figure out the (

_{q}*R*

_{1},

*R*

_{2}) pair that gives the highest plasmon excitation rate for a preferred spaser mode energy marked on Fig. 3(a). Hence, a designer has the freedom to optimize the spaser geometry by tuning either nanosphere radius or dielectric shell thickness.

Let us continue with the previous example and draw the curve corresponds to the spaser mode energy 2.385 eV on Fig. 3(b). Based on this curve, we can observe that the plasmon excitation rate varies with geometric parameters, even for the same spaser mode energy. Hence, we can select parameters for the optimum design by picking the *R*_{1}, *R*_{2} pair which results in the highest plasmon excitation rate. In our example, parameters (*R*_{1}, *R*_{2}) = (6 nm, 11 nm) offer the highest spaser mode amplification for the 2.385 eV energy curve. Here we emphasize that, although we obtained this result for a single QD dipole spaser, a better spaser configuration may consists of many QDs to achieve a higher gain. If this dipole spaser consists of *N* QDs, then
$N\propto \left({R}_{2}^{3}-{R}_{1}^{3}\right)/{R}_{q}^{3}$, if the QDs are uniformly distributed inside the dielectric shell. In that case, having a bigger shell volume to accommodate more QDs may result in a higher plasmon excitation rates because all QDs contribute the total amplification. However, each QD does not contribute evenly in many QDs case as the location of the QD and its size play vital roles in deciding the plasmon excitation rate.

In order to examine how the location of the QD affects the plasmon excitation rate of the spaser, we fix *R*_{1} and *R*_{2}, and plot *ℛ*_{1} for the case of resonant QD radius (i.e. *R _{q}* is also fixed). Here we vary QD’s location

*r*

_{0}within the dielectric shell from the innermost to the outermost position with respect to the nanosphere’s center. Such plots for four different

*R*

_{1},

*R*

_{2}pairs are shown in Fig. 3(d). It can be observed from these plots that the plasmon excitation rate rapidly decreases when the QD is moved away from the nanosphere. This happens because the interactions between plasmon modes and electron-hole pairs in QD gets weaker towards the shell’s outer boundary as the matrix element for interactions, ${V}_{1e,0e}^{{l}_{p},{m}_{p}}$ decreases in the radial direction according to Eqs. (2), (13) and (14). The decreasing rate is higher for smaller nanosphere radii. Especially, when there are many QDs, their contribution to the spasing mode won’t be uniform. Hence, when designing a multiple QD spaser, we must set the optimum size parameters such that a large number of QDs are closely located to the nanosphere center.

Thus far, we have only considered the case where the QD radius is tuned according to Eq. (6) such that its emission energy exactly equals to the energy of the spaser mode. To investigate the influence of the QD radius on spasing, we vary *R _{q}* within an interval of 1–3 nm and plot the plasmon excitation rate, as shown in Fig. 3(c), for different shell thicknesses keeping the nanosphere radius fixed to 10 nm. According to this plot, highest plasmon excitation rate is observable in the case of exact resonance and it rapidly decreases when the

*R*deviates from the resonant QD radius. For example, if the the QD radius deviates much (i.e. about 0.5 nm) from the resonant value, the resultant plasmon excitation rate will tend to zero.

_{q}The plots in Figs. 3(a)–3(d), clearly explicate how the geometrical parameters of spaser design can be optimized to attain an elevated plasmon excitation rate. In addition, we need to discuss the threshold power required from optical pumping which is a condition of spasing. Since we use a normalized electric field in this analytical treatment, it is important to calculate the required threshold gain, denoted by *g _{th}*. Threshold gain can be found by applying the condition of population inversion:

*ρ*

_{22}≥

*ρ*

_{11}. It can be shown that it does not depend on spaser geometry, and is a function of the dielectric constants of the spaser materials and the spaser mode frequency given by [19]

*R*

_{1}and

*R*

_{2}and nanosphere’s permittivity is a function of

*R*

_{1}, threshold gain also becomes a function of these two geometrical parameters as we plot in Fig. 3(e). According to the graph, the gain does not vary much with the size parameters but the required threshold gain for spasing is little higher when the nanosphere is smaller. We mark the 2.385 eV curve, which we used in the previous example, on threshold pump power plot and it provides an idea of choosing the correct geometrical parameters. It should be noted that, the expression for threshold gain is derived assuming that stimulated emission dominates in the spaser. However, there can be cases where spontaneous emission dominates spaser kinetics when the number of plasmons is one. In such situations, the semiclassical concept of threshold is not applicable and plasmon population in the resonator increases linearly with the pump rate [59]. In addition, extending this analysis to find the required pumping intensity will be useful in practical implementations. The developed model can also be used to study the polarization of resultant plasmons, which has been discussed in [20], as it allows us to consider the case where QD dipole moment is neither parallel nor orthogonal to the line connecting the centers of nanosphere and QD. However, in this study, we only focus on the optimization of spaser geometry to enhance plasmon generation based on the introduced quantum mechanical model.

The case study which we performed throughout this section provides us design guidelines on choosing the optimum size parameters of spaser components and where to place QDs to achieve a maximum plasmon excitation rate. In the process of designing a spaser, one might choose its operating wavelength as the first step, then pick possible values for the size parameters *R*_{1}, *R*_{2} and *R _{q}* by examining the resulting in plasmon excitation rate. Then, it is necessary to figure out where to embed the QD. Placing the QD closer to the nanosphere will give a higher plasmon excitation rate. However, if multiple QDs are to be placed, then one need to consider about the precise placement because not all QDs can be placed close to the nanosphere’s boundary.

Although we studied a spaser realization with spherical structure, the derivations done in Section 3 is valid for a spaser with any geometry. It might not be possible to obtain the electric field and the energies of plasmon modes analytically for most of the other resonator geometries as we did in Section 2.1. But numerical solutions for the field can be admitted into the expressions derived in Section 3 to analyze the spaser kinetics and investigate the plasmon excitation rate which characterize its performance.

## 5. Conclusions

We have theoretically modeled spaser as an *n*-level quantum system formed by amalgamating spaser’s electronic and plasmonic subsystems. With this model, we studied a simple spaser geometry consists of a metal nanosphere resonantly coupled to a QD. The energy transfer between the QD and the nanosphere accompanied the relaxation of electron–hole pairs resonantly generated inside the QD by a continuous-wave laser. By employing the density matrix theory, we analytically found the excitation rate of surface plasmons where three nondegenerate electron–hole pair states are coupled to a single plasmon mode of arbitrary angular momentum. The obtained expression was then examined numerically for the special case of a spaser operating at the triple-degenerate dipole mode. It was shown that the plasmon excitation rate can be significantly enhanced by appropriately choosing the geometric parameters of the spaser.

## Acknowledgments

The work of C. Rupasinghe is supported by the Monash University Institute of Graduate Research. The work of I. D. Rukhlenko and M. Premaratne is supported by the Australian Research Council, through its Discovery Early Career Researcher Award DE120100055 and Discovery Grant DP110100713, respectively.

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