Three-level Spaser System: a Semi-Classical Analysis

Scaling down electronic and optical systems to harness their optimum efficiency has been a goal of today’s scientific and industrial research. Among the research fields, working in this direction, nanoplasmonics plays an important role due to its unique possibilities and broad applications [1, 2, 3]. In nanoplasmonics, the light is confined within a sub-wavelength scale, which results in a strong enhancement of the optical field. Some of the notable application of nanoplasmonics are in the areas of near-field optics [4, 5], bio-sensing [6, 7, 8], surface plasmon-based photo-detectors [9, 10], spaser[11, 12, 13] and many others. Spaser (surface plasmon amplification by stimulated emission of radiation), which has experienced a vast development over the last decade, plays a special role in this list. It was introduced in the early 2000’s [14] by David. J. Bergman and Mark. I . Stockman [15, 16], and throughout the time has paved its way up as a miniature source of spectrally tunable stimulated emission [17, 18]. Apart from the mainstream research, spaser has found numerous applications in different areas such as opto-electronic systems[19, 20, 21, 22, 23, 24, 25], sensing in biological and chemical agents[26, 27, 28] and also as a biological probe[29, 30] in disease therapeutics and diagnostics.

The whole idea of spaser is based on the existence of localized surface plasmons, which are characterized by a high concentration of optical energy within a nanoscale range[31]. Such strong concentration of optical field combined with stimulated emission [14] process allow to design a nanoscale laser - spaser. Different variations of such nanoscopic lasers were proposed theoretically and realized experimentally. The first of the type[32] was introduced in 2008 and was based on an array of plasmonic resonators. A year later, a spaser, based on Localized Surface Plasmons Resonance[33, 34](LSPR), was demonstrated experimentally[35], which contained a gold sphere embedded inside a dye. The same design has been also been studied for a cancer diagnosis and treatment by Ganzala et al[29] in 2017

The design of spaser based on a gain nanorod placed near plasmonic metal with the dimensions in micrometers has been considered in Ref. [36, 37]. Recently, the topological spasers of type I and II were introduced[38, 39, 40], where the spaser dynamics is protected by nontrivial topological properties of either a plasmonic system[38] or a gain medium[39, 40].

Spaser can be understood as a nanoplasmonic counterpart of a normal laser[41]. It consists of two major components: a metal resonator and a gain medium. Usually, the metal is silver due to its low non-radiative losses, however, gold or aluminum can also be used for the spaser operating at high frequency. The gain medium is usually dye molecules, with the gap close to the SP frequency. Apart from dyes, materials[42, 43, 44] with non-linear effects that exhibit topological resonance, have also been studied as a suitable gain. Additionally, the system may also be placed in an appropriate dielectric environment, which can be used to adjust the SP frequency of a nanosphere.

The two basic possibilities of how a gain medium can be introduced into a spaser system are shown in Fig. 1: i) gain encapsulating the solid nanosphere ii) gain placed inside a metal nanoshell. Below, we consider only the first case when the gain is placed outside the solid silver metal nanosphere as shown in Fig. 1a. Additionally, this system is placed in water which will further help to adjust the LSP frequency($\omega_{\mathrm{sp}}$).

Theoretical modeling[18] of such nanolaser[35] is mostly in agreement with the experimental results, apart from the behavior of the spaser at a large value of electric pumping. In the present paper, we are using the same theoretical model[18] of spaser with some modifications, which can address the unexplained behavior. Namely, we consider a three-level model for the gain while in Ref. [18] only two levels were introduced for the gain medium.

We consider a gain medium consisting of three energy levels with the corresponding populations $n_{0}$, $n_{1}$ and $n_{2}$. This approach is similar to the previously studied 2-level [18] system, but with an additional energy level added to account for the finite relaxation rate of electron population from the highest level to the second excited level. Many other noticeable works have been done previously to deal with the effects of multi-level[45, 46] on the spasing process. However, this paper describes an elegant way to account for coherent processes in gain, which makes our findings in close appropriation with the experimental ones as discussed below. The general schematics of transition within the system is shown in Fig. 2.

The gain system is pumped by an external light, which excites the system from the ground state $|0>$ to the second excited state $|2>$ with the transition (gain) rate $g$. The excited states of the system are also characterized by relaxation rates $\gamma_{21}$ and $\gamma_{10}$, which represent the transitions $|2>\rightarrow|1>$ and $|1>\rightarrow|0>$ respectively. The gain medium is coupled to the plasmonic system through the field-dipole interaction and it is at the almost resonant condition with the frequency $\omega_{12}$ is close to the surface plasmon frequency, $\omega_{\mathrm{sp}}$, see Fig. 2.

For more than two mediums present in the system, we use the standard Laplace equation approach as adopted in a popular spaser research[29] to calculate the field of the Localized Surface Plasmons. In the spherical system, we consider the electric potential of the dipole mode to be of the form

$\displaystyle\phi_{i}(\mathbf{r})=\left(\frac{a_{i}}{r^{2}}+b_{i}r\right)Y_{10}(\mathbf{r}),$

(1)

where $i$ labels the medium ($i=1,2,3$), $a_{i}$ and $b_{i}$ are coefficients corresponding to medium $i$, and $Y_{10}(\mathbf{r})$ is a spherical harmonics $(l=1$ and $m=0)$. The Maxwell’s continuity equations across the interfaces of the layers are given by the following expressions

	$\displaystyle\phi_{i}(\mathbf{r_{i}})$	$\displaystyle=\phi_{i+1}(\mathbf{r_{i}}),$		(2)
	$\displaystyle\epsilon_{i}\frac{\partial{}}{\partial{r}}\phi_{i}(\mathbf{r_{i}})$	$\displaystyle=\epsilon_{i+1}\frac{\partial{}}{\partial{r}}\phi_{i+1}(\mathbf{r_{i}}),$		(3)

where $\epsilon_{i}$ is the permittivity of medium $i$. For our system, which consists of 3 layers, silver sphere, dye, and water, we solve Eqs. (2) and (3) to obtain permittivity of silver ($\epsilon_{s}$) as a function of permittivities of dye ($\epsilon_{d}$) and water ($\epsilon_{w}$), i.e.,

$\displaystyle\epsilon_{s}\approx\epsilon_{s}(\epsilon_{d},\epsilon_{w}).$

(4)

The frequency of LSPs is then obtained by equating $\epsilon_{s}$ to the experimental value $\epsilon_{\mathrm{sil}}(\omega)$[47]

$\displaystyle\epsilon_{s}=\real[\epsilon_{\mathrm{sil}}(\omega_{\mathrm{sp}})]$

(5)

The plasmon dipole mode creates a highly localized dipole-field as shown in the cross-section diagram in Fig. 3. The corresponding operator of electric field[14, 18] can be expressed in terms of creation and annihilation operators, $\hat{a}$ and $\hat{a}^{*}$, of SP,

$\displaystyle\mathbf{E}(\mathbf{r},t)$

$\displaystyle=-A_{\mathrm{sp}}(\nabla\phi(\mathbf{r})\hat{a}e^{-i\omega t}+\nabla\phi^{*}(\mathbf{r})\hat{a}^{*}e^{i\omega t}),$

(6)

where

$\displaystyle A_{\mathrm{sp}}=\sqrt{\frac{4\pi\hbar}{s_{1}\derivative{\epsilon_{\mathrm{sil}}(\omega)}{\omega}}}\Big{|}_{\omega=\omega_{\mathrm{sp}}}$

(7)

Here, the geometrical parameter $s_{1}$ is given by the following expression

$\displaystyle s_{1}=\frac{\displaystyle{\int_{\mathcal{V}_{metal}}|\nabla\phi_{i}(\mathbf{r})|^{2}d^{3}\mathbf{r}}}{\displaystyle{\int_{\mathrm{All~{}Space}}|\nabla\phi_{i}(\mathbf{r})|^{2}d^{3}\mathbf{r}}}.$

(8)

The Hamiltonian of a spaser can be expressed in terms of individual Hamiltonians of surface plasmons and the gain medium, and the dipole type interaction Hamiltonian of the SP and the gain:

$$\mathcal{\hat{H}}_{total}=\hbar\omega_{\mathrm{sp}}\hat{a}^{*}\hat{a}+\mathcal{\hat{H}}_{gain}+\int_{V}\mathbf{E(r,t)}\mathbf{\hat{d}}~{}d\mathbf{{}^{3}r}.$$

(9)

Here $V$ is the total volume of the gain medium and $\mathbf{\hat{d}}$ is the transition dipole moment operator of the gain medium.

In this article, we adopt a quasi-classical approach [18, 2] to study the properties of SPs, where the operators $\hat{a}$ and $\hat{a}^{*}$ are treated as classical variables represented in the form of a time dependent variable $\hat{a}=a_{0}e^{-i\omega t}$ with $a_{0}$ being the slow varying amplitude. Then, the number of SPs in a given mode then can be written as $N_{n}=\absolutevalue{a_{0}}^{2}$.

Below we assume that the corresponding dipole matrix elements are nonzero only between the levels $|0>$ and $|1>$ of the gain system, i.e., only the transitions between these levels can generate SPs. These interactions can be also characterized by the Rabi frequency[48], which is given by the following expression

$\displaystyle\Omega_{10}(\mathbf{r},t)=\frac{\mathbf{E}(\mathbf{r},t)\mathbf{d_{01}}}{\hbar},$

(10)

where $\mathbf{d_{01}}=<0|\mathbf{\hat{d}}|1>$.

We describe the gain system within the density matrix approach with the corresponding equation of motion

$$i\hbar\dot{\mathcal{\hat{\rho}}}(\mathbf{r,t})=[\mathcal{\hat{\rho}}(\mathbf{r,t}),\mathcal{\hat{H}}],$$

(11)

where $\mathcal{\hat{\rho}}$ is the density matrix of three level gain system. Using Rotating Wave Approximation(RWA), we can express $\mathcal{\hat{\rho}}(\mathbf{r,t})$ as slow-varying diagonal terms and fast varying non-diagonal terms with frequency $\omega\approx\omega_{\mathrm{sp}}$.

$\displaystyle\mathcal{\hat{\rho}}(\mathbf{r})$

$\displaystyle=\left(\begin{array}[]{ccc}{\rho}_{22}(\mathbf{r},t)&0&0\\ 0&{\rho}_{11}(\mathbf{r},t)&{\rho}_{10}(\mathbf{r},t)e^{i\omega t}\\ 0&{\rho}_{01}(\mathbf{r},t)e^{-i\omega t}&{\rho}_{00}(\mathbf{r},t)\end{array}\right).$

(15)

It is convenient to introduce the following notations: $n_{0}=\rho_{00}$, $n_{1}=\rho_{11}$ and $n_{2}=\rho_{22}$. Then equations for the elements of the density matrix, which can be derived from Eq. (11), take the following form

	$\displaystyle\dot{{\rho}}_{10}(\mathbf{r})=[-i(\omega-\omega_{10})-\Gamma_{10}]{\rho}_{10}(\mathbf{r})+$
	$\displaystyle\hskip 85.35826ptin_{10}(\mathbf{r}){\Omega}_{10}^{*}(\mathbf{r})a_{0}^{*},$		(16)
	$\displaystyle\dot{n}_{2}=gn_{0}-\gamma_{21}n_{2},$		(17)
	$\displaystyle\dot{n}_{1}=-\gamma_{10}n_{1}+\gamma_{21}n_{2}-2\int_{V}d\mathbf{{}^{3}r}\imaginary\left(\rho_{10}(\mathbf{r})a_{0}\Omega_{10}(\mathbf{r})\right),$		(18)
	$\displaystyle\dot{n}_{0}=-gn_{0}+\gamma_{10}n_{1}+2\int_{V}d\mathbf{{}^{3}r}\imaginary\left(\rho_{10}(\mathbf{r})a_{0}\Omega_{10}(\mathbf{r})\right),$		(19)

where $\omega_{10}$ is the transition frequency between $|1>$ and $|0>$ levels of the gain medium and $g$ is the rate of excitation from the $|0>$ level to the $|2>$ level by an external pulse. In the above equations, we also introduced the relaxation rates: polarization relaxation rate $\Gamma_{10}$ and the spontaneous relaxation rates $\gamma_{10}$ and $\gamma_{21}$ between the corresponding states as indicated by the indices.

The equation of motion for SPs is obtained from Hamiltonian (9) and is given by the following expression

	$\displaystyle\dot{a}$	$\displaystyle=-a_{0}\gamma_{\mathrm{sp}}(\omega)+i(\omega-\omega_{\mathrm{sp}})a_{0}+$
		$\displaystyle\hskip 85.35826pti\int_{V}d\mathbf{{}^{3}r}\left(\rho_{10}^{*}(\mathbf{r})\Omega_{10}^{*}(\mathbf{r})\right),$		(20)

where the plasmon relaxation rate $\gamma_{\mathrm{sp}}(\omega)$ is introduced, $\gamma_{\mathrm{sp}}(\omega)=\frac{\Im{\epsilon_{\mathrm{sil}}(\omega)}}{\frac{\real\epsilon_{\mathrm{sil}}(\omega)}{\partial\omega}}$. Since the transitions between the levels $|0>$ and $|1>$ are due to coupling to the SPs, the corresponding relaxation rate, $\gamma_{10}$, can be expressed as

$$\gamma_{10}={|\Omega_{10}|}^{2}\frac{2(\gamma_{\mathrm{sp}}+\Gamma_{10})}{(\omega_{\mathrm{sp}}-\omega_{10})^{2}+(\gamma_{\mathrm{sp}}+\Gamma_{10})^{2}}.$$

(21)

Equations (16)-(20) determine the dynamics of three level spaser. First, we analyze the stationary solution of these equations, i.e., the continuous wave regime of a spaser. In this regime, the time derivatives in the left hand sides of Eqs. (16)-(20) are zero. It is convenient to introduce the population inversions through the following expressions

	$\displaystyle{n}_{10}={n}_{1}-{n}_{0},$		(22)
	$\displaystyle{n}_{21}={n}_{2}-{n}_{1}.$		(23)

Then taking into account that ${n}_{0}+{n}_{1}+{n}_{2}=1$, we can express populations of different levels in terms of $n_{10}$ as follows

	$\displaystyle n_{0}$	$\displaystyle=\frac{\gamma_{21}n_{10}-\gamma_{21}}{2\gamma_{21}+g},$		(24)
	$\displaystyle n_{1}$	$\displaystyle=-\frac{-\gamma_{21}-gn_{10}-\gamma_{21}n_{10}}{2\gamma_{21}+g},$		(25)
	$\displaystyle n_{2}$	$\displaystyle=-\frac{gn_{10}-g}{2\gamma_{21}+g}.$		(26)

Substituting Eqs. (24)-(26) into the system of equations (16)-(20) we obtain the following solution of the stationary equations

$\displaystyle\rho_{10}(\mathbf{r})=-\frac{\left(a_{0}^{*}n_{10}\Omega_{10}^{*}(\mathbf{r})\right)}{i\Gamma_{10}-\omega_{s}+\omega_{10}},$

(27)

$$n_{10}=\frac{(\omega_{s}-\omega_{\mathrm{sp}})(\omega_{10}-\omega_{s})+\gamma_{\mathrm{sp}}\Gamma_{10}}{V\rho\Omega_{10}^{2}},$$

(28)

	$\displaystyle N_{n}=|a_{0}|^{2}=$	$\displaystyle\frac{\Gamma_{10}^{2}+(\omega_{s}-\omega_{10})^{2}}{2n_{10}\Gamma_{10}}\times$
		$\displaystyle\frac{\gamma_{21}\left(g-\gamma_{10}-n_{10}\left(\gamma_{10}+g\right)\right)-\gamma_{10}gn_{10}}{\left(2\gamma_{21}+g\right)\Omega_{10}^{2}}.$		(29)

In the above equations we assumed that the Rabi frequency is constant within the gain medium and the corresponding integrals in Eqs. (18) and (19) can be replaced by $V$. Here the spasing frequency $\omega_{s}$ is given by the following expression

$\displaystyle\omega_{s}=\frac{\omega_{10}\gamma_{\mathrm{sp}}+\Gamma_{10}\omega_{\mathrm{sp}}}{\Gamma_{10}+\gamma_{\mathrm{sp}}},$

(30)

where $\omega_{\mathrm{sp}}<\omega_{s}<\omega_{10}$, and in the absence of detuning, i.e., if $\omega_{\mathrm{sp}}=\omega_{10}$, it is equal to $\omega_{\mathrm{sp}}$.

From the above expressions we can identify the effect of relaxation rate $\gamma_{21}$ on the main spaser characteristics such as population inversion $n_{10}$, spasing frequency $\omega_{s}$, threshold, and the number of generated plasmons $N_{n}$. From Eqs. (28), (30) one can see that both the population inversion and the spasing frequency do not depend on $\gamma_{21}$.

The spasing threshold $g_{th}$ can be found from Eq. (29), where the threshold is determined from the condition $N_{n}=0$,

$$g_{th}=\gamma_{10}\frac{1+n_{10}}{1-n_{10}-\frac{\gamma_{10}}{\gamma_{21}}n_{10}}.$$

(31)

Thus the spaser threshold depends on the relaxation rate, $\gamma_{21}$, but since $\gamma_{10}\ll\gamma_{21}$ this dependence is weak. Taking into account that $\gamma_{10}/\gamma_{21}\ll 1$, we can find the correction to the threshold $g_{th}^{(0)}$ determined by the two-level spaser model

$$g_{th}\approx g_{th}^{(0)}\left(1+\frac{\gamma_{10}}{\gamma_{21}}\frac{n_{10}}{1-n_{10}}\right),$$

(32)

where $g_{th}^{(0)}=\frac{1+n_{10}}{1-n_{10}}\gamma_{10}$.

Finite relaxation rate $\gamma_{21}$ also affects the number of generated plasmons, see Eq. (29). When $\gamma_{21}$ is large enough, i.e., $\gamma_{21}\gg g$, we can consider $\frac{1}{\gamma_{21}}$ as a small parameter and find expansion of Eq. (29) in the powers of $\frac{1}{\gamma_{21}}$ as follows

	$\displaystyle N_{n}=\frac{\left(\Gamma_{10}^{2}+(\omega_{s}-\omega_{10})^{2}\right)\left(g-\gamma_{10}-n_{10}\left(\gamma_{10}+g\right)\right)}{4n_{10}\gamma_{10}\Gamma_{10}\Omega_{10}^{2}}-$
	$\displaystyle\frac{g\left(1-n_{10}\right)\left(g-\gamma_{10}\right)\left(\Gamma_{10}{}^{2}+(\omega_{s}-\omega_{10})^{2}\right)}{8n_{10}\gamma_{10}^{2}\Gamma_{10}\Omega_{10}^{2}}\left(\frac{1}{\gamma_{21}}\right)-$
	$\displaystyle\frac{g^{2}\left(1-n_{10}\right)\left(\Gamma_{10}^{2}+(\omega_{s}-\omega_{10})^{2}\right)}{16n_{10}\gamma_{10}^{2}\Gamma_{10}\Omega_{10}^{2}}\left(\frac{1}{\gamma_{21}}\right)^{2}.$		(33)

The first term in this expansion is the result for the two-level spaser system[18], for which $\gamma_{21}\to\infty$. In this case, the number of generated SPs is proportional to the gain, $g$. The second and the third terms give the correction due to finite relaxation rate, $\gamma_{21}$. These terms introduce quadratic dependence on $g$.

Spaser is a unique system where the coupling of the plasmonic system and the gain medium results in the coherent generation of the localized plasmons at the nanoscale. The theories of spaser that are based on the two-level system of gain have their limitation in explaining the spasing behavior at the high pumping rates. In the present paper, we considered, within a semi-classical approach, a three-level gain system to identify the effects of relaxation between the non-spasing levels on the spaser dynamics. Our results show that the number of generated surface plasmons strongly depends on the relaxation rate $\gamma_{21}$. At large values of $\gamma_{21}$, i.e., fast relaxation, the three-level system converges to the regular two-level spaser system[18] with linear dependence of the number of generated plasmons on the pumping rate. However, at smaller values of $\gamma_{21}$, the dependence of the number of plasmons on the pumping rate becomes parabolic, which is more pronounced at large pump intensity. Such behavior is consistent with experimental results[29, 45]. Our spaser model can be beneficial in improving the efficacy of existing models and comparison with experimental results.

Major funding was provided by Grant No. DE-FG02-01ER15213 from the Chemical Sciences, Biosciences and Geosciences Division, Office of Basic Energy Sciences, Office of Science, US Department of Energy. Numerical simulations were performed using support by Grant No. DE-SC0007043 from the Materials Sciences and Engineering Division of the Office of the Basic Energy Sciences, Office of Science, US Department of Energy.