Interacting quantum plasmons in metal-dielectric structures

Over the past decade, quantum plasmonics maier-np13 underwent a rapid development fueled by a host of recently discovered phenomena such as strong exciton-plasmon coupling effects bellessa-prl04 ; sugawara-prl06 ; wurtz-nl07 ; fofang-nl08 ; hakala-prl09 ; gomez-nl10 , plasmon-assisted hot carrier generation park-nl11 ; melosh-nl11 ; halas-science11 , plasmonic laser (spaser) bergman-prl03 ; stockman-natphot08 ; noginov-nature09 ; zhang-nature09 ; gwo-science12 , plasmon tunneling baumberg-nature12 ; aizpurua-nl12 ; dionne-nl13 ; tan-science14 and more, along with a growing number of applications. Surface plasmons are collective electron excitations living at the metal-dielectric interfaces which can interact strongly with light and localized electron excitations such as excitons in molecules or semiconductors stockman-review . Although classical description of many experiments in terms of local field enhancement has largely been successful, a growing number of topics and applications require a more rigorous quantum approach hohenester-prb08 ; nordlander-nl09 ; stockman-jo10 ; waks-pra10 ; garcia-prl13 ; garcia-optica17 ; aizpurua-nanophot20 .

In nanoscale systems, the local fields can change strongly over the length scale well below the diffraction limit, and so the plasmon interactions with the electromagnetic (EM) field and excitons depend sensitively on the system parameters such as the geometry of a metal-dielectric structure and the exciton position relative to it. While the coupling parameters, characterizing these interactions, have been suggested in several forms by using analogy with the cavity modes khitrova-nphys06 ; andreani-prb12 ; shahbazyan-nl19 ; mortensen-rpp20 , these parameters have yet to emerge within a consistent quantization approach for interacting plasmons. Yet another longstanding challenge for quantum plasmonics is to account properly for strong optical dispersion and losses in metals that give rise to non-Markovian effects nori-prb09 ; tejedor-prb10 ; thanopulos-prb17 ; moradi-sr18 ; molmer-acsph19 .

Within the canonical quantization scheme, localized plasmon modes with discrete frequency spectrum $\omega_{m}$ are described by the Hamiltonian

where $\hat{a}^{\dagger}_{m}$ and $\hat{a}_{m}$ are, respectively, the plasmon creation and annihilation operators obeying the canonical commutation relations $[\hat{a}_{m},\hat{a}^{\dagger}_{n}]=\delta_{mn}$. Plasmon interactions with quantum emitters (QEs), modeled hereafter by two-level systems, are usually described by the Jaynes-Cummings interaction Hamiltonian

where $\hat{\sigma}^{\dagger}_{i}$ and $\hat{\sigma}_{i}$ are, respectively, the raising and lowering operators for the $i$th QE (related to the Pauli matrices for Fermionic systems), while $g_{im}$ is the QE-plasmon coupling, which, within the canonical approach, is an ad hoc parameter. Although widely employed, the canonical scheme has significant limitations when used in metal-dielectric structures characterized by a complex dielectric function $\varepsilon(\omega,\bm{r})=\varepsilon^{\prime}(\omega,\bm{r})+i\varepsilon^{\prime\prime}(\omega,\bm{r})$ since it ignores the medium optical dispersion and, hence, is unsuitable for describing non-Markovian effects in plasmonics.

On the other hand, the medium optical dispersion is accurately accounted for in the macroscopic electrodynamics approach based upon the fluctuation-dissipation (FD) theorem welsch-pra98 ; welsch-p00 ; philbin-njp10 . In this framework, the EM fields are quantized in terms of reservoir noise operators $\hat{\bm{f}}(\omega,\bm{r})$ driven by the Hamiltonian

and obeying the commutation relations

where $\bm{I}$ is the unit tensor. Interactions with QEs are described by the Hamiltonian term $\hat{H}_{\rm int}=-\sum_{i}\hat{\bm{p}}_{i}\cdot\hat{\bm{E}}(\bm{r}_{i})$, where $\hat{\bm{p}}_{i}$ and $\hat{\bm{E}}(\bm{r}_{i})$ are, respectively, the QE dipole moment and electric field operators. The latter is given by

where $\hat{\bm{P}}_{N}(\omega,\bm{r})=(i/2\pi)\sqrt{\hbar\varepsilon^{\prime\prime}(\omega,\bm{r})}\hat{\bm{f}}(\omega,\bm{r})$ is the noise polarization vector operator and $\bm{D}(\omega;\bm{r},\bm{r}^{\prime})$ is the EM dyadic Green function defined as

The macroscopic FD approach has been extensively used to model spontaneous emission, strong coupling effects and non-Markovian dynamics in metal-dielectric structures welsch-pra00 ; dzsotjan-prb10 ; hughes-prb12 ; andreani-prb12 ; hughes-prb13 ; zubairy-prb14 ; garcia-prl14 ; rousseaux-prb18 ; sivan-prb19 ; nori-prb09 ; tejedor-prb10 ; thanopulos-prb17 ; moradi-sr18 ; molmer-acsph19 . Its major drawback in relation to plasmonics is that, even though surface plasmons reside primarily at the metal-dielectric interfaces, the eigenstates of $\hat{H}_{N}$ extend over the entire system reservoir, i.e., the Hilbert space, spanned by the operators $\hat{\bm{f}}(\omega,\bm{r})$, is excessively large. Furthermore, the plasmons only appear as resonances in the classical EM Green function $\bm{D}$, so that, in practical terms, this approach is limited to systems that allow $\bm{D}$ to be evaluated analytically or numerically.

In principle, the Hamiltonians (1) and (2), along with the canonical commutations relations and QE-plasmon coupling, should be obtained within the macroscopic FD framework starting with a suitable mode expansion for the EM Green function that would define the basis set for normal mode expansion of the electric field operator (5) varguet-ol16 ; dzsotjan-pra16 ; hughes-prl19 . However, for systems of general shape, a straightforward expansion of $\bm{D}$ over a discrete set of EM modes that include radiation, such as quasinormal modes hughes-njp14 ; hughes-acsphot14 ; lalanne-pra14 ; hughes-pra15 ; muljarov-prb16 ; lalanne-prb18 ; lalanne-lpr18 , gives rise to dissipation-induced coupling between the modes which compicates the commutations relations hughes-prl19 . On the other hand, on the length scale well below the diffraction limit, the plasmon interactions with QEs take place primarily via the near field coupling rather than photon exchange, while the rate of nonradiative losses, determined by $\varepsilon^{\prime\prime}(\omega,\bm{r})$, by far exceeds the plasmon radiative decay rate. This implies that in nanoscale systems, the plasmons should be treated as primarily electronic excitations interacting with the EM field and QEs, and so the quantization of plasmons should be carried on its own, i.e., separate from the radiation field. Furthermore, since the electron motion in metals is unretarded, the plasmon modes in such systems can be described within quasistatic approximation stockman-review , which allows, as we show below, one to define the orthogonal basis set that is free of dissipation-induced coupling.

In this paper, we develop a consistent quantum approach to localized plasmons interacting with quantum emitters and the electromagnetic field which accurately accounts for medium optical dispersion and losses. First, starting within the macroscopic FD framework, we employ the near-field plasmon Green function shahbazyan-prl16 ; shahbazyan-prb18 do define the normal mode expansion for the plasmon field operators free of dissipation-induced coupling. Using this basis set, we explicitly obtain the plasmon Hamiltonian (1), along with the equal-time canonical commutations relations, and show that the canonical quantization scheme is valid only in the Markov approximation, i.e., if the dielectric function $\varepsilon(\omega,\bm{r})$ is replaced by its value at the plasmon frequency (i.e., $\omega=\omega_{m}$). In this way, we also obtain the microscopic coupling parameters that define the plasmon interactions with the EM field and QEs in terms of the plasmon local fields, system geometry and QE positions relative to the plasmonic structure. Second, moving beyond the Markov approximation, we present a new approach that bridges the macroscopic and canonical quantization schemes while accounting accurately for the medium optical dispersion and losses. In this approach, quantum plasmons are described in terms of a discrete set of bosonic modes with linear dispersion which reside at the metal-dielelectric interfaces, and whose operators span a reduced Hilbert space obtained by projecting the full reservoir states upon localized plasmon modes. The interactions of such projected reservoir modes with the EM field and QEs are mediated by classical plasmons, so that the classical plasmon enhancement effects are encoded in the Hamiltonian coupling parameters.

In this section, starting within the macroscopic quantization scheme welsch-pra98 ; welsch-p00 ; philbin-njp10 , we obtain the canonical Hamiltonian (1) and the equal-time commutation relations for localized surface plasmon in metal-dielectric structures of arbitrary shape.

Consider now the plasmon coupling to QEs and the EM field. In contrast to cavity modes, the plasmons are localized at the scale well below the diffraction limit and, therefore, interact with the EM field $\bm{\mathcal{E}}(t)$ in a way similar to point dipoles. The interaction Hamiltonian has the form $H_{\rm pl-em}=-\sum_{m}\hat{\bm{p}}_{m}\cdot\bm{\mathcal{E}}(t)$, where $\hat{\bm{p}}_{m}=\!\int\!dV\hat{\bm{P}}_{m}(\bm{r})$ is the plasmon dipole moment and $\hat{\bm{P}}_{m}(\bm{r})$ is the polarization vector operators. To determine $\hat{\bm{P}}_{m}(\bm{r})$, we present the Gauss law (7) as $\bm{\nabla}\!\cdot\!\bm{E}_{m}(\bm{r})+4\pi\bm{\nabla}\!\cdot\!\bm{P}_{m}(\bm{r})=0$, where $\bm{P}_{m}(\bm{r})=\chi^{\prime}(\omega_{m},\bm{r})\bm{E}_{m}(\bm{r})$ is the mode polarization vector and $\chi(\omega,\bm{r})$ is the system susceptibility. In the Markov approximation, converting this relation to operator form as $\hat{\bm{P}}_{m}(\bm{r})=\chi^{\prime}(\omega_{m},\bm{r})\hat{\bm{E}}_{m}(\bm{r})$ and using Eq. (15), for the EM field of the form $\bm{\mathcal{E}}(t)=\bm{\mathcal{E}}e^{-i\omega_{L}t}+\bm{\mathcal{E}}^{*}e^{i\omega_{L}t}$ that is uniform on the system scale, we obtain in the rotating wave approximation (RWA)

where $\bm{\mu}_{m}\equiv\bm{\mu}_{m}(\omega_{m})$, and we introduced the frequency-dependent transition matrix element,

to be used in the following section. The scaling factor $\sqrt{\hbar\omega_{m}/U_{m}}$ in Eq. (25) converts the plasmon energy $U_{m}$ to $\hbar\omega_{m}$ in order to match the energy of the EM field. With matrix element (25), the plasmon radiative decay rate is given by the standard expression

where $W_{m}^{rad}=p_{m}^{2}\omega_{m}^{4}/3c^{3}$ is the power radiated by dipole $\bm{p}_{m}$ scully-book .

We now turn to the interactions between plasmons and QEs situated at positions $\bm{r}_{i}$ with excitation frequency $\omega_{e}$ and dipole moments $\hat{\bm{p}}_{i}=\bm{\mu}_{i}(\hat{\sigma}^{\dagger}_{i}+\hat{\sigma}_{i})$, where $\bm{\mu}_{i}=\mu_{e}\bm{n}_{i}$ is the transition matrix element ($\bm{n}_{i}$ is the dipole orientation). Employing the mode expansion (15) in the interaction Hamiltonian $\hat{H}_{\rm pl-qe}=-\sum_{i}\hat{\bm{p}}_{i}\cdot\hat{\bm{E}}_{\rm pl}(\bm{r}_{i})$, we obtain, in RWA, the coupling Hamiltonian (2) with $g_{im}$ given by

Using Eq. (12), the QE-plasmon coupling $g_{im}$ can be recast in the form similar to cavities

where ${\cal V}^{(i)}_{m}$ is the projected plasmon mode volume shahbazyan-acsphot17 ; shahbazyan-prb18 , which characterizes the plasmon field confinement at a point $\bm{r}_{i}$ in the direction $\bm{n}_{i}$. Since the Gauss equation (7) is scale-invariant stockman-review , the coupling parameters (25) and (27) are independent of the overall field normalization. By rescaling the fields as $\tilde{\bm{E}}_{m}(\bm{r})=\sqrt{\hbar\omega_{m}/4U_{m}}\bm{E}_{m}(\bm{r})$, these parameters are brought to a more familiar form

To illustrate the sensitivity of QE-plasmon coupling to the system parameters, we present the results of numerical calculations of the coupling parameter $g_{em}$, given by Eq. (28), for a single QE situated at a distance $d$ from the tip of an Au nanorod in water (see Fig. 1). The nanorod was modeled by a prolate spheroid with semi-major and semi-minor axes $a$ and $b$, respectively, at aspect ratio $a/b=3.0$, the standard spherical harmonics were used for calculating the longitudinal plasmon fields, the QE dipole orientation was chosen along the nanorod symmetry axis, and the Au experimental dielectric function was used in all calculations. The calculated QE-plasmon coupling is normalized by full plasmon decay rate $\gamma_{m}$, which includes the nonradiative decay rate (13) and radiative decay rate (26) taken at plasmon frequency $\omega=\omega_{m}$. The coupling parameter behaves as $g_{em}\propto 1/\sqrt{{\cal V}_{m}}$, where, for small nanostructures, the plasmon mode volume ${\cal V}_{m}$ scales as the metal volume shahbazyan-prb18 ; shahbazyan-nl19 , resulting in the enhanced coupling for smaller nanorods. The coupling sharply increases as QE approaches hot spot at the nanorod tip characterized by very large plasmon field confinement (small mode volume).

Finally, the full canonical Hamiltonian for plasmons interacting with the EM field and QEs has the form $H=H_{\rm pl}+H_{\rm pl-qe}+H_{\rm pl-em}+H_{\rm qe}+H_{\rm qe-em}$, where we included the standard QE Hamiltonian $H_{\rm qe}=\hbar\omega_{e}\!\sum_{i}\hat{\sigma}_{i}^{\dagger}\hat{\sigma}_{i}$ and the coupling term $H_{\rm qe-em}=-\sum_{i}(\bm{\mu}_{i}\!\cdot\!\bm{\mathcal{E}}e^{-i\omega_{L}t}\hat{\sigma}_{i}^{\dagger}+{\rm H.c.})$ describing the QEs’ interactions with the EM field. We stress that for plasmons, the canonical quantization scheme is valid only in the Markov approximation that ignores the dielectric function dispersion. However, in metal-dielectric structures, the role of materials optical dispersion can be very significant, and so a quantum description that includes such effects is necessary, as we discuss in the following section.

In this section, we present yet another plasmon quantization approach that bridges the macroscopic and canonical schemes while combining the advantages of both. In this ”intermediate” scheme, the Hilbert space is restricted to plasmonic degrees of freedom, in contrast to macroscopic approach, while the medium optical dispersion is still accurately accounted for, in contrast to the canonical scheme.

In summary, we developed a quantization approach for surface plasmons in metal-dielectric structures that accounts for optical dispersion of the medium complex dielectric function and, hence, is suitable for describing non-Markovian effects in quantum plasmonics. Starting within the macroscopic quantization framework based on the dissipation-fluctuation theorem, we derived, in the Markov approximation, the canonical quantization scheme for interacting surface plasmons and obtained the relevant coupling parameters in terms of local fields and system geometry. Beyond Markov approximation, we developed a quantum approach in terms of discrete set of bosonic modes with linear dispersion whose interactions with the electromagnetic field and quantum emitters are mediated by the classical plasmons, providing resonant coupling between the system components. We have shown that, within this approach, the classical plasmonic enhancement effects, such as resonant plasmon excitation by incident light and resonance energy transfer from a quantum emitter to plasmonic structure, are obtained in the first-order of perturbation theory. The quantum dynamics of bosonic modes is restricted to the reduced Hilbert space of reservoir states which overlaps with the electric fields of plasmon modes.