Improved scheme for modeling a spaser made of identical gain elements

Tharindu Warnakula; Mark I. Stockman; Malin Premaratne

doi:10.1364/JOSAB.35.001397

Journal of the Optical Society of America B
Vol. 35,
Issue 6,
pp. 1397-1407
(2018)
•https://doi.org/10.1364/JOSAB.35.001397

Improved scheme for modeling a spaser made of identical gain elements

Tharindu Warnakula, Mark I. Stockman, and Malin Premaratne

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Tharindu Warnakula, Mark I. Stockman, and Malin Premaratne, "Improved scheme for modeling a spaser made of identical gain elements," J. Opt. Soc. Am. B 35, 1397-1407 (2018)

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Abstract

Simulation of the most general configuration of a spaser where each gain element is assumed to have distinct behavior is almost intractable with current computing technologies. Hence, here we consider a representative, yet simple enough system with all possible interactions allowed, in which all gain elements have identical features including two energy levels, to gain deeper understanding of the operational characteristics. We reason in an intuitively sound, yet mathematically rigorous, way how to exploit the symmetry of the most efficient method currently available to reduce the size of the problem by order $O (M)$ , where $M$ is the size of the truncated plasmon number state. Finally, we explore some physical results as predicted by our method.

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Equations (26)

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			Excitations
	Left Operator	Right Operator	$E x_{x}$	$E x_{y}$
1	${\hat{σ}}_{10}^{n}$	—	$↑$	—
2	—	${\hat{σ}}_{01}^{n}$	—	$↑$
3	${\hat{σ}}_{10}^{n}$	${\hat{σ}}_{01}^{n}$	$↑$	$↑$
4	${\hat{σ}}_{11}^{n}$	—	—	—
5	—	${\hat{σ}}_{11}^{n}$	—	—
6	$\hat{a}$	—	$↓$	—
7	${\hat{a}}^{†}$	—	$↑$	—
8	—	$\hat{a}$	—	$↑$
9	—	${\hat{a}}^{†}$	—	$↓$
10	$\hat{a} {\hat{a}}^{†}$	—	—	—
11	—	$\hat{a} {\hat{a}}^{†}$	—	—
12	${\hat{a}}^{†}$	$\hat{a}$	$↑$	$↑$
13	$\hat{a}$	${\hat{a}}^{†}$	$↓$	$↓$
14	${\hat{σ}}_{10}^{n} \hat{a}$	—	—	—
15	—	${\hat{σ}}_{01}^{n} {\hat{a}}^{†}$	—	—

Formalism	Dimensions	Actual Dimensions
Full Density Matrix	$O (4^{N} \times M^{2})$	$≊ 10^{64}$
Richter et al. [14]	$O (N^{3} \times M^{2})$	$≊ 10^{10}$
This paper	$O (N^{3} \times M)$	$≊ 10^{8}$

			Excitations
	Left Operator	Right Operator	$E x_{x}$	$E x_{y}$
1	${\hat{σ}}_{10}^{n}$	—	$↑$	—
2	—	${\hat{σ}}_{01}^{n}$	—	$↑$
3	${\hat{σ}}_{10}^{n}$	${\hat{σ}}_{01}^{n}$	$↑$	$↑$
4	${\hat{σ}}_{11}^{n}$	—	—	—
5	—	${\hat{σ}}_{11}^{n}$	—	—
6	$\hat{a}$	—	$↓$	—
7	${\hat{a}}^{†}$	—	$↑$	—
8	—	$\hat{a}$	—	$↑$
9	—	${\hat{a}}^{†}$	—	$↓$
10	$\hat{a} {\hat{a}}^{†}$	—	—	—
11	—	$\hat{a} {\hat{a}}^{†}$	—	—
12	${\hat{a}}^{†}$	$\hat{a}$	$↑$	$↑$
13	$\hat{a}$	${\hat{a}}^{†}$	$↓$	$↓$
14	${\hat{σ}}_{10}^{n} \hat{a}$	—	—	—
15	—	${\hat{σ}}_{01}^{n} {\hat{a}}^{†}$	—	—

Formalism	Dimensions	Actual Dimensions
Full Density Matrix	$O (4^{N} \times M^{2})$	$≊ 10^{64}$
Richter et al. [14]	$O (N^{3} \times M^{2})$	$≊ 10^{10}$
This paper	$O (N^{3} \times M)$	$≊ 10^{8}$

Tharindu Warnakula	https://orcid.org/0000-0002-6275-4671
Mark I. Stockman	https://orcid.org/0000-0002-6996-0806
Malin Premaratne	https://orcid.org/0000-0002-2419-4431

Abstract

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Figures (8)

Tables (2)

Equations (26)

Journal of the Optical Society of America B