Coupling plasmonics with quantum systems

Over the last decade, the subject of metamaterials has become one of the most rapidly evolving areas of research in modern optics and electrodynamics. Metamaterials are artificially engineered media with exotic electromagnetic properties, and offer a wide range of potential applications ranging from super-resolution optical microscopy to science-fiction invisibility and cloaking devices.¹ However, there are a number of fundamental and technological challenges, one of which is to minimize or (ideally) compensate for dissipative losses, which are inevitable in the optical domain and severely limit metamaterial performance.^2,3

Among the most promising solutions is an engineered hybrid metamaterial system, in which a plasmonic nanoresonator (the building block of an optical metamaterial) is coupled to an active element (for example, a quantum dot or dye molecule) that provides energy transfer to the plasmonic resonator, see Figure 1(b). Such a system is known as a nanolaser or spaser.^4,5 One formalism describing the behavior of the hybrid metamaterial combines classical and quantum approaches, and was first developed and implemented in 2005.⁶ The physical effects resulting from the interaction of classic and quantum elements are not limited to loss compensation. A number of experimental observations suggest that the Purcell effect plays a crucial role in controlling the luminescence of quantum dots located in the vicinity of plasmonic nanoresonators.^7,8 The nonlinear response of carbon nanotubes exhibits multifold enhancement due to coupling with plasmonic nanoresonators.⁹ Moreover, in the GHz frequency domain, a system comprised of a coupled superconducting quantum interference device (SQUID) and a strip resonator¹⁰ exhibits the same behavior as its optical counterparts, and can be described by the same quantum-classical formalism. This opens up a new perspective for testing the physics of hybrid metamaterials at GHz frequencies, where technological requirements for sample fabrication are significantly relaxed.

Figure 1. Examples of elementary building blocks of a quantum magnetic metamaterial based on quantum dots (QD), one of the possible active quantum elements. (a) QD-dimer, where an anti-symmetric mode is formed through direct interaction of QDs through electric dipole fields. (b) QD-nanoparticle cluster, where a resonant plasmonic nanoparticle mediates interaction between the QDs. (c) Core-shell QD, where the core and shell are comprised of different materials.

Our model consists of two components: the classical describes the resonator through the multipole approach extended beyond Protsenko et al;⁶ and the quantum models the quantum element (QE, see Figure 1). The latter is based on a density matrix approach, which includes relaxation processes due to coupling with a thermostat and the external fields.

As has been shown recently, rather complicated plasmonic dynamics in a metamaterial nanoresonator can be satisfactorily described in the context of a multipole approach using a set of coupled harmonic oscillator equations.¹¹ In the case of a simple nano-resonator (a metal strip), a single harmonic oscillator equation is sufficient, yielding an electric-dipole approximation. The variable of the harmonic oscillator equation describes electron dynamics, and is proportional to the field generated by the electron. The generated field appears external to the QE, affecting its dynamics. The QE in turn produces a field, which is proportional to the non-diagonal elements of the density matrix and affects the electron dynamics closing the feedback loop.

Assuming closely-spaced resonance frequencies for the harmonic oscillator and QE, our master set of coupled equations describing the dynamics of the hybrid system has the following form (important for describing experimental situations, to be discussed shortly):

Here, ρ₂₂, ρ₁₁, ρ₁₂, and ρ₁₂* are the diagonal and non-diagonal matrix density elements of the QE, respectively. τ₂ and τ₁ are the constants accounting for phase and energy relaxation processes due to interaction with a thermostat. is the transition frequency between levels 2 and 1. μ_QE is the dipole moment of a quantum element. W is the pump rate. N=ρ₂₂−ρ₁₁and is the displacement of electrons in the nano-resonator. γ and ω₀ are the loss coefficient and resonance frequency of the plasmon resonator, respectively. A is the amplitude of the external electric field. This set of equations describes six experimental situations, discussed next.

In a nanolaser (spaser), , and Equation System 1 gives transition and stationary nanolaser dynamics. Adding stochastic Langevin terms for the first and last equations, one can calculate laser bandwidth in analog with the well-known Schawlow-Towns approach.^12,13 For luminescent enhancement, as with a spaser, , but is assumed to be small (saturation is absent) and describes zero field fluctuations, causing spontaneous emission. The Purcell effect appears as a multiplicator for the field A and takes into account density of states detuning for this field. We emphasize that the Purcell effect affects neither relaxation time τ₂ nor τ₁, which appears from the interaction with the thermostat (until the density of states of the thermostat is unchanged by the nanoresonator).

Nonlinearity of the active element appears due to the saturation effect, and essentially requires neither positive N₀ nor a nanoresonator. Enhancement of the saturation is caused by the external field, which transfers energy to the active element through the nanoresonator in addition to direct pumping. Taking into account the field enhancement effect near the plasmonic nanoresonator, the model adequately describes the increased strength of the nonlinear response experimentally observed by Nikolaenko et al.⁹

An enhanced magnetic dipolar response results from marginal modification of Equation System (1), allowing one to model enhancement of a high-order multipole response in the hybrid metamaterial. Specifically, complex nanoresonators (such as double-wire or split-ring resonators) support an anti-symmetric mode of excitation, which is responsible for the magnetic dipolar response.¹¹ It can be adequately described by two (instead of one) coupled harmonic oscillator equations. In the case of sufficiently strong pumping, where , the energy transferred from the appropriately-positioned QE will support excitation of the anti-symmetric mode.

Quantum magnetic metamaterials (see Figure 1) result from the combination of active QE (such as quantum dots) with specially-designed plasmonic nanoresonators. They provide magnetization at optical frequencies based not only on the plasmonic modes plasmonic modes, but also on coherently-coupled QE modes. Those structures could be used as building blocks for lossless metamaterials with a strong magnetic response at optical frequencies.¹³ The linear and nonlinear response of SQUIDs coupled with a microwave resonator¹⁰ also falls within the range of phenomena described. This allows us to model optical effects by investigating the behavior of microwave systems.

In summary, Equation System (1) provides a universal platform for describing the dynamics of various physical systems, consisting of classical and quantum elements in different frequency domains. In the future, our proposed model will be further investigated by comparing its predictions with experimental data and the results of rigorous numerical simulations. We note that analytical or semi-analytical modeling does not substitute numerical simulations, but rather serve as complimentary tools that allows us to quickly test various operation regimes, predict new physical phenomena, and gain valuable physical insight.