Analytical formalism for the interaction of two-level quantum systems with metal nanoresonators

Hybrid systems made of quantum emitters and plasmonic nanoresonators offer a unique platform to implement artificial atoms with completely novel optical responses that are not available otherwise. However, their theoretical analysis is difficult and since many degrees of freedom have to be explored, engineering their optical properties remains challenging. Here we propose a new formalism which removes most limitations encountered in previous analytical treatments and allows a flexible and efficient study of complex nanoresonators with arbitrary shapes in an almost fully analytically way. The formalism brings accurate closed-form expressions for the hybrid-system optical response, and provides an intuitive description based on the coupling between the quantum emitters and the resonance modes of the nanoresonator. The ability to quickly predict light scattering properties of hybrid systems paves the way to a deep exploration of their fascinating properties and may enable rapid optimization of quantum plasmonic metamaterials or quantum information devices. Table of


I. INTRODUCTION
Localized surface plasmon resonances of metallic nanoresonators have strong local field enhancements in deep-subwavelength mode volumes, and offer unique opportunities to tailor light-matter interaction at the nanoscale [1,2]. Such opportunities become all the more remarkable as the near field of nanoresonators is dressed by two-level quantum emitters, such as atoms, molecules or quantum dots. The resulting hybrid artificial system [3][4][5][6][7][8][9] indeed behaves very differently from its constituting elements and offers novel optical properties at nanometer scale. For instance, under excitation by an external driving field, various Fano-like spectral responses could be observed, such as super-absorption in ultrasmall metallic particles [9], 'cloaking' of the nanoresonator by the nearby quantum emitter [7,9], and unexpectedly broadband responses, much broader than the Purcell broadening as will be shown. Hybrid systems can be seen as artificial excitons [8], whose optical properties (the resonance frequency, line shape, far-field radiation pattern and near-field spatial and spectral distribution …) can be engineered with great flexibility, and could become the building blocks for future generations of quantum plasmonic metamaterials or quantum information devices.
In theoretical works, to avoid quenching, two-level quantum emitters are assumed to be located not in the immediate vicinity of the metal so that the wavefunctions of the emitters and of the electron gas do not spatially overlap. The nanoresonator is then described by a dispersive permittivity and in this limit, both the electromagnetic fields and the emitter degrees of freedom are treated as operators. The theoretical predictions of the optical properties generally rely on fully-vectorial Green-tensor calculations. This necessitates tedious computations, which hide simple interpretations and need to be repeated whenever key parameters, such as the driving laser field polarization, angle of incidence or frequency, the quantum emitter polarization or location, are tuned for the analysis. Computational loads become prohibitive for hybrid systems made of multiple quantum objects [10][11][12] or complex metallic nanoparticles [3]. Clearly, the design of artificial plasmonic excitons requires new formalisms that restore physical intuition and analyticity.
A first classical approach relies on the quasi-static approximation and models the nanoresonator as an electric dipole [4][5][6]13]. Full analyticity and intuition are immediately achieved, but the approach is drastically limited to asymptotic cases. Analyticity is also achieved using Mie theory for textbook cases of spherical nanoresonator shapes [14]. For arbitrarily complex shapes, it was shown that a strong physical intuition can be restored [8] by an aposteriori fit of all possible coupling parameters involved in the relaxation dynamics, but the approach intrinsically relies on ab-initio fully-vectorial computations of the system response.
Hereafter we combine the recently developed quasi-normal mode (QNM) formalism [15,16] of plasmonic nanoresonators with the density matrix formalism of two-level quantum systems [9,17,18], and propose a totally new formalism which removes most limitations encountered in previous analytical approaches and allows us to handle complex nanoresonator shapes in an almost fully analytically way. One just needs to calculate the fundamentals QNMs of the nanoresonator, but once the QNMs are known for a given nanoresonator shape, the hybrid-system response is obtained with closed form expressions for any driving laser frequency, quantum emitter locations, polarization … Computational loads are drastically reduced, analyticity favors intuition and reveals general properties. The formalism is derived and tested for the canonical, albeit important in practice, case of hybrids consisting of an arbitrary nanoresonator supporting one or several bright or dark modes dressed by a single two-level quantum emitter in the weak coupling regime, but the formalism can be extended to other configurations with multiple quantum objects supporting collective effects or with three-level quantum systems . Other physical regimes, such as nonlinear regime (with strong driving field) [4] or strong emitter-plasmon coupling regimes [5] may also be explored in future works. For our canonical geometry, we derive accurate and analytical expressions of the Fano line shape spectral responses and evidence that analyticity offers a great flexibility for harnessing the optical properties of artificial plasmonic excitons. For instance, we reveal a new property, namely the possibility to achieve spectral responses that are much broader than the spontaneous decay rate of the emitter; we deepen the general knowledge on the topic by proving that intriguing 'cloaking' properties reported for some specific geometries in earlier works [7,9] are in reality observed with arbitrary geometries, the only requirement being that the emitter decay rate is boosted by the presence of the nanoresonator; we also predict and explain that the spectral response of the near-field electromagnetic intensity of geometries supporting higher-order (often dark) QNMs strongly varies in space, a fascinating property for developing new spectroscopic techniques at the nanoscale [19].

II. QUASI-NORMAL-MODE AND DENSITY-MATRIX FORMALISMS
Both formalisms are already documented in the literature, and it is not our intention to present them again hereafter. Rather, we focus on how they may be coupled to provide a fully-analytical quantum treatment of hybrid plasmonic systems. As a reminder, the formalisms are first briefly presented and then the master equations leading to the coupling are derived in full for self-consistency. Technical details are appended in the methods.
The quantization of the electromagnetic radiation in dispersive and absorptive media has been initially developed by various authors [20,21]. These works were extended to inhomogeneous materials, see e.g. [22] by writing an Hamiltonian for the quantized field in the vicinity of a metallic nanoparticle, the emitter and the interaction. We adopt here the approach developed in [9,17] and consider the quantum emitter as a two level system (TLS) with an electric dipolar transition. Under the rotating wave approximation, and assuming a weak TLS-plasmon interaction, the dynamics of the expectation values is given by optical Bloch equations and the expectation value of the total field operator at the driving laser frequency ω L reads as where ( ) r E L , the coherent driving field, is treated as a classical quantity, that represents the total field in the absence of the atom.

r E TLS
is the expectation value of the field radiated by the TLS alone, which is treated as a point dipole source placed at r 0 in the vicinity of the plasmonic resonator. Note that an exp(iω L t) dependence is assumed throughout the manuscript.
A rigorous quantum treatment [9,23] shows that the self-action of the induced dipole reduces to a frequency shift (called Lamb shift in a vacuum electromagnetic environment [24]), so that the Rabi frequency depends only on Including the shift into the TLS resonance frequency, and neglecting the Rabi broadening, the expectation value for the induced dipole moment reads as [9] γ δ where eg μ is the electric dipole moment of the electric dipolar transition between the ground and excited states. Hereafter, we assume a linearly polarized atomic transition, and an ideal TLS whose damping is only driven by spontaneous radiative decay, but the formalism could be easily generalized to encompass saturation effects and non-radiative damping [9]. In Eq. (2), is the detuning between the laser driving frequency and the TLS resonance frequency ω TLS , and γ denotes the modified spontaneous decay rate of the excited state. The corresponding decay rate is derived from the Fermi's golden rule [25], Quasi-normal modes are the natural electromagnetic resonance modes of plasmonic resonators. They are eigensolutions of source-free Maxwell's equations with complex eigenfrequencies (due to the finite lifetime of the resonance). In the QNM formalism, the fields scattered by dissipative and dispersive resonant photonic or plasmonic nanoresonators are expressed as a superposition of modes. Even if the completeness of the modal expansion is not proven for dissipative plasmonic systems, it was shown that, close to resonance, accurate predictions can be obtained with expansions based on a few dominant QNMs only [15,26]. Hereafter we adopt the approach developed in Refs. [15,16], in which reciprocity arguments allow to normalize the QNMs and to define an orthogonality relationship. The force of the QNM formalism resides in the fact that, once the QNMs are identified and properly normalized, one may analytically derive the electromagnetic response for various excitations and considerably lower the computational burden.
To combine both formalisms, we use a scattered-field formulation [16]. The total where ( ) where ( ) which provides an analytical expression for the expectation value of the total field. The derivation relies on the sole approximation that plasmonic resonances are spectrally much broader than TLS resonances, an approximation largely justified even if the spontaneous decay of the TLS is boosted by a strong Purcell effect. In Eq. (5), is the frequency detuning normalized by the Purcell-broadened TLS linewidth is a complex number that is called the modal Fano coefficient hereafter. It plays a key role in the following analysis; every QNM possesses its own Fano spectral response, with a line shape characterized by The master equation (5), which is illustrated in Fig , and one exactly recovers the formula introduced by Ugo Fano himself [27]. The analytical formalism clearly evidences that the pure Fano's line shape results from the coupling of the laser and the TLS to the resonance mode of the resonator, and not from an intricate interference with the background fields.

III. IMPLEMENTATION, EFFECTIVENESS, NUMERICAL TESTS
In practice, to calculate Eq. (5), one first needs to compute the QNMs. This computation can be performed with the general method described in [16]. Hereafter we use an open-source code [28]. The calculation per QNM, which requires 4-5 iterations at complex frequencies starting from an initial guess, generally lasts less than 5 minutes and a low-speed workstation. Once the QNMs are known, then E L (r), the Rabi frequency and the Purcell factor are obtained analytically, as explained in Appendix A. The expectation value <d> of the induced dipole is then known and the total response is calculated from Eq. (5). Supplementary technical details are given in Appendices B and C.
To test the validity of the semi-analytical formalism, we consider a canonical arrangement, consisting of a small 5nm-diameter gold sphere embedded in a glass background (n = 1.5) and coupled to a TLS with a radial dipole moment. The responses of such systems have already been studied intensively [4][5][6]9] by resorting to either Mie theory or a dipolar approximation. To compare with data available from Ref. [9], the TLS is assumed to be placed 18 nm above the sphere, see Fig. 2(b), with a resonance wavelength λ TLS = 545 nm (Purcell factor ~ 2.2) and a natural linewidth γ 0 = 75 MHz in vacuum. The same TLS will be considered throughout the article. The gold sphere exhibits a dipolar plasmonic resonance around 545 nm, with a complex eigen-wavelength λ e = 548−22i nm.
We analyze the response of the hybrid system to a linearly polarized plane wave, whose electric-field is collinear to the TLS dipole moment. There, we assume that a single dipolar QNM contributes to the field expansion, and compute the value of the modal Fano coefficient, f =1+33i, from Eqs. (4) and (10). Afterwards, both the scattering (blue) and absorption (red) cross-sections, respectively denoted by σ sca and σ abs , are easily obtained (see Appendix C). Figure 2(a) compares the predictions of our QNM formalism with "exact" computational results obtained with Mie theory [9]. Quantitative agreement is reached for the nearly-symmetric line shape of σ sca and the asymmetric Fano line shape of σ abs with a large enhancement at zero detuning and a strong reduction at L δ = 16.5. The difference in lineshapes of both cross sections is due to the fact that However σ abs that depends on the field inside the nanoparticle has a typical Fano line shape.
The capability and accuracy of the QNM formalism is further evidenced in Fig. 2b, where the flux lines calculated at zero-detuning with the QNM formalism are shown to be almost superimposed with the ones provided by Mie theory. For the sake of compactness, we present an accuracy test just for a single particular canonical arrangement; however we have verified that the QNM formalism provides accurate predictions for many other arrangements, including all those studied in the following sections.
Equation (5) offers the possibility to analytically calculate the spectral response of the hybrid system for any driving laser frequency, shape or polarization or quantum emitter locations and polarizations. This removes most limitations encountered in previous approaches and allows us to handle complex resonator shapes in an almost fully analytically way. In return, the approach is approximate. For instance, it is likely not be able to account for quenching, which may require to calculate very high-order QNMs. Another drawback is that it is difficult to estimate the error made by retaining only a few QNMs. This would require a comparison with fully-vectorial data, but note that this would require only a few comparisons, since one may reasonably expect that the accuracy of the method does not change much by varying the geometrical parameters. This has been noticed for some particular structures, such as cylinders. Finally note that in experiments, the exact shape of the resonator, location and/or orientation of the emitter may be known only approximately and that highly accurate predictions are not critical to well understand the physics of the problem at hand (symmetries, quenching, number of modes …) and to interpret the experimental data. In fact, the present method may help. For example by introducing a statistical variation of geometry, index of refraction, etc… and by computing a distribution of modes, one may obtain a statistical distribution of extinction or absorption spectra, directly related to experimental uncertainties, in a rapid way.

IV. FANO LINESHAPES OF MONOMODE HYBRIDS
In this section, we consider hybrids that are dominantly driven by a single QNM . We will see that these monomode hybrids exhibit a large richness of optical responses that are easily tuned by slightly changing the hybrid geometry and are elegantly predicted by the analytical model with a single complex parameter, the Fano coefficient.
As is well known, the optical response of monomode hybrids in the general case presents a non-symmetric line shape with a local maximum (peak) and a local minimum (dip), and the overall shape largely varies by tuning some key internal parameters of the hybrid. This is due to the interference that takes place between a continuum (the broad plasmonic resonance) and the sharp TLS resonance.
As illustrated in Fig. 3, the great variety of response patterns is well described under monomode approximation with a single generic Fano function ( ) ~L δ F , in which the modal Fano coefficient f varies as the internal parameters are tuned. We emphasize that the different lineshapes in Fig. 3 are all obtained analytically with a single Fano function whose Fano coefficient is known analytically, and that we have checked that the Fano function accurately predicts the absorption spectra of each of the cases taken into consideration. With Fig. 3(a) , remains present in the spectrum.
The contrasted response of Fig. 3(a) admits a simple physical interpretation by considering that the TLS acts as an antenna with a scattering cross-section that is much larger than that of the nanoresonator, i.e., ( ) β >> TLS ω α . Surprisingly we find that the frequency interval between the peak and the dip is not bounded by the spontaneous decay rate γ of the excited level as one may intuitively expect; actually it may be more than ten times larger than the Purcell broadened linewidth. We also note that the frequency interval rapidly varies with the separation distance Δ, an important property that may be helpful for experimental characterization and that is analytically predicted with the modal formalism. A further important case in practice corresponds to hybrid systems operating with large Purcell factors. Generally, the position r 0 at which a TLS experiences large decay accelerations is always associated with a strong field enhancement under far-field laser excitation, provided that the TLS and the laser have the same polarization indeed. Thus assuming that

from Eq. (6), it is found that the implementation of large
Purcell factors requires Re(f) ≈ 1, as consistently observed in Fig. 3(b) (blue curve only) and 3c, as well as with other results (not shown) obtained for a spherical dimer [29]. Although the relation Re(f) ≈ 1 largely restricts the variety of possible response patterns, Im(f) is not imposed and, by tuning the TLS resonant frequencies within the plasmonic resonance bandwidth, the Purcell factor remains large and the spectral responses may still be significantly tuned, from a symmetric line shape with a dip at zero detuning for Im(f) = 0, to asymmetric lineshapes with dip frequencies that may be either red-or blue-shifted, see Fig. 3(c). Note that for large emission rates, the Fano function F determines both the scattering and absorption cross-sections. Thus deep dips of |F|, such as those of Fig. 3c, result in very weak extinction cross-sections, which make the nanoresonator hardly visible (or cloaked) for the far field.

V. MULTIMODE ARRANGEMENTS
So far, we have been considering the interaction of quantum emitters with metal nanoparticles supporting a single resonance. One of the great values of the present method is its capability to handle multi-resonance cases in a very intuitive manner, since the impact of each resonance is naturally and separately considered in the master equation. Otherwise, intuition would be considerably weakened and computational loads would be drastically enhanced.
From Eq. (5), the expectation value of each field component, X = E x , E y , E z , H x , H y , or H z , may be conveniently written as Thus, in sharp contrast to single-mode cases, hybrid systems made of multimode plasmonic resonator offer spectral line shape responses that considerably vary in the near-field zone of the nanoresonator. In particular, even if the far field response may remain driven by a predominant single radiative QNM, high-order modes such as electric quadrupoles that weakly radiate could efficiently couple to the TLS and offer a myriad of contrasted spectral responses in the near-field, a fascinating property that could be exploited to develop new imaging and spectroscopic techniques at the nanoscale [19]. The TLS acts as an auxiliary resonant structure that effectively transfers the far-field radiation to high-order dark modes of the resonator, via the resonant excitation of the lowest-order bright mode. Based on this mode conversion, scattered optical near-fields can be well manipulated by utilizing the interference between different resonator modes.
To exemplify our purpose, we consider a heterodimer composed of two gold nanorods of different sizes (radii R 1 = 20 nm and R 2 = 85 nm, lengths L 1 = 80 nm and L 2 = 145 nm, gap size 45 nm), of which the optical response in the spectral range 700~1200 nm is mainly driven by two QNMs with eigen-wavelengths λ e1 = 933−77i nm and λ e2 = 958−108i nm [15]. In the center of the dimer gap, a z-polarized TLS is placed, with a transition wavelength λ TLS = 990 nm, at which the Purcell factor reaches a maximum value of ≈155. Figure 4(a) shows the absolute value of Fano coefficient f ez (r) and f ex (r) of the z-and x-components of the electric field in the x-z mid-plane. The spatial distributions are highly non-uniform, and strongly differ from one component to the other, especially around the upper-arm. As a consequence, for different field component X, distinctive  Fig. 4(b) for sampling locations labeled by A, B, C, and D, in the near field region of the dimer. Lineshapes at A, C and D are similar to those obtained for the monomode case in Fig. 3, but the one at B is remarkably different because of its unusual Fano coefficient value (see the caption). In Fig. 4(c), we show the corresponding lineshapes of the local field expectation value |<E>|, for which |<E z >| ≈ |<E>| is dominant. Clearly, the local-field spectra drastically vary at the nanometer scale, due to the interference between several modes, with distinct and spatially-dependent modal Fano coefficients.

VI. CONCLUSION
In summary, we have derived an analytical formalism based on quasi-normal mode expansions for the study of the optical properties of plasmonic systems interacting with quantum objects (e.g., molecules, atoms, or quantum dots). In agreement with previous works [4][5][6][7][8][9], we show that the spectral response exhibits Fano resonances resulting from the interaction between the quantum objects and the plasmon resonance. Importantly and uniquely to our knowledge, the formalism allows the derivation of accurate closed-form expressions for the Fano factor, which can be directly calculated without fit even for complex geometries. The flexibility added by the analyticity is amazingly comfortable in practice, since it favors an in-depth understanding and allows a very efficient computation of the optical responses when some key physical parameters are modified. As a straightforward application, let us mention the possibility to rapidly evaluate ensemble-averaged responses over a distribution of TLS orientations, positions, and resonance frequencies. This could help the interpretation of experimental measurement on dilute samples, in which the artificial plasmonic excitons are neither identical nor identically oriented. With the present modal approach, analyticity can be preserved for more complicated systems, such as amplifying systems by modifying the optical Bloch equations to model the gain with a TLS driven out of thermodynamic equilibrium by a pump field [30]. The formalism can also be directly extended towards hybrids formed by a nanoresonator dressed by several TLS to study superadiance or spasing. Our recent efforts in that direction are very encouraging.
Generally speaking, we believe that theoretical formalisms, which explicitly rely on the natural resonances of the plasmonic nanoresonator, represent an enabling approach for the description of plasmon-exciton interactions in complex systems.
In order to compute the Rabi frequency where ( ) ( )

APPENDIX C. Calculation of σ abs , σ ext and σ sca
The response of the hybrid system can be quantified by the absorption cross-section σ abs , extinction cross-section σ ext and scattering cross-section σ sca = σ extσ abs . It should be first stated that, the entire hybrid system is a passive system, excited only by the driving laser beam, so that all the cross-sections are defined with respect to the incident power of the driving field. We calculate the absorption cross-section σ abs with the following formula Therefore when a single QNM is excited, the spectral lineshape of abs σ is determined by 2 β α + or by the square of the modulus of the Fano function of Eq. (7).
For a good accuracy, we calculate the extinction cross-section σ ext of the hybrid system as Equation (C2) does not correspond to the usual expression for extinction cross section [31], but allows us to distinguish the TLS and resonator contributions ( )   Fig. 1(b) in [9]: actually, the spectral width of the optical response is much larger than the Purcell broadening.