Quantum Theory of Surface Plasmon Polaritons: Planar and Spherical Geometries

Abstract

A quantum theory of retarded surface plasmons on a metal–vacuum interface is formulated, by analogy with the well-known and widely exploited theory of exciton-polaritons. The Hamiltonian for mutually interacting instantaneous surface plasmons and transverse electromagnetic modes is diagonalized with recourse to a Hopfield–Bogoljubov transformation, in order to obtain a new family of modes, to be identified with retarded plasmons. The interaction with nearby dipolar emitters is treated with a full quantum formalism based on a general definition of modal effective volumes. The illustrative cases of a planar surface and of a spherical nanoparticle are considered in detail. In the ideal situation of absence of dissipation, as an effect of the conservation of in-plane wavevector, retarded plasmons on a planar surface represent true stationary states (which are usually called surface plasmon polaritons), whereas retarded plasmons in a spherical nanoparticle, characterized by frequencies that overlap with the transverse electromagnetic continuum, become resonances with a finite radiative broadening. The theory presented constitutes a suitable full quantum framework for the study of nonperturbative and nonlinear effects in plasmonic nanosystems.

Appendices

Appendix A: Planar Geometry

In this appendix, we sketch the derivation of the retarded plasmon polariton modes with the help of a Hopfield–Bogoljubov transformation of the instantaneous plasmons coupled to the transverse electromagnetic field.

The interaction terms of the Hamiltonian in Eq. 14 assume the form

$$H_{\mathrm{I}} = i\int \mathrm{d}^{2} \mathbf{k}_{\parallel}\mathrm{d} k_{z}\, \hbar C(k_{z};\mathbf{k}_{\parallel}) \left(\mathrm{b}_{\mathbf{k}_{\parallel}}^{\dagger} - \mathrm{b}_{-\mathbf{k}_{\parallel}}\right)\left(\mathrm{a}_{\mathbf{k}_{\parallel},k_{z}} + \mathrm{a}_{-\mathbf{k}_{\parallel},-k_{z}}^{\dagger}\right),$$

(43)

$$H_{\text{II}} = \int \mathrm{d}^{2} \mathbf{k}_{\parallel} \mathrm{d} k_{z}\mathrm{d} k_{z}^{\prime}\,\hbar D(k_{z},k_{z}^{\prime};\mathbf{k}_{\parallel})\left(\mathrm{a}_{\mathbf{k}_{\parallel},k_{z}} + \mathrm{a}_{-\mathbf{k}_{\parallel},-k_{z}}^{\dagger}\right)\\ \times\,\left(\mathrm{a}_{\mathbf{k}_{\parallel},k_{z}^{\prime}}^{\dagger} + \mathrm{a}_{-\mathbf{k}_{\parallel},-k_{z}^{\prime}}\right), $$

(44)

with the coefficients

$$\begin{array}{*{20}l} &C(k_{z};\mathbf{k}_{\parallel}) = \frac{\omega_{\mathrm{P}}^{2}}{4 k} \sqrt{\frac{k_{\parallel}}{\pi ck \omega_{\mathrm{s}}}};\quad \tilde{D}(k_{z},k_{z}^{\prime};\mathbf{k}_{\parallel}) = \frac{i\omega_{\mathrm{P}}^{2}(k_{z} k_{z}^{\prime} + k_{\parallel}^{2})}{8c\pi(k k^{\prime})^{\frac{3}{2}}}\\ &D(k_{z},k_{z}^{\prime};\mathbf{k}_{\parallel}) = \frac{\omega_{\mathrm{P}}^{2}}{8ck}\delta(k_{z}-k_{z}^{\prime}) + \mathcal{P}\!\!\;\frac{\tilde{D}(k_{z},k_{z}^{\prime};\mathbf{k}_{\parallel})}{k_{z}^{\prime} - k_{z}};\quad \end{array} $$

(𝓟 denotes the principal value).

The diagonalization of the Hamiltonian in Eq. 14 is obtained by replacing the expansion of Eq. 29 into Eq. 23 and collecting the terms in front of the unperturbed operators $\mathrm {b}_{\mathbf {k}_{\parallel }},\mathrm {b}_{-\mathbf {k}_{\parallel }}^{\dagger },\mathrm {a}_{\mathbf {k}_{\parallel },k_{z}}$, and $\mathrm {a}_{-\mathbf {k}_{\parallel },-k_{z}}^{\dagger }$, so that we obtain a system of four equations for the coefficients W,X(k _z ),Y,Z(k _z ). With some algebraic manipulation, the equations can be condensed into the form

$$\begin{array}{@{}rcl@{}} \frac{c}{2k}\left({k_{z}^{2}} + \mathbf{k}_{\parallel}^{2} - \frac{\Omega^{2}}{c^{2}} + \frac{\omega_{\mathrm{P}}^{2}}{2c^{2}}\right)\xi(k_{z}) \\ +\,\mathcal{P}\!\!\int\mathrm{d} k_{z}^{\prime}\,2\frac{\tilde{D}(k_{z},k_{z}^{\prime})}{k_{z}^{\prime} - k_{z}}\xi(k_{z}^{\prime}) = - i(W + Y)C(k), \end{array} $$

(45)

where ξ(k _z )=X(k _z )−Z(k _z ).

Equation 45 can be solved with Kramers–Kronig relations, but attention must be paid to the fact that the integrand (considered as a function of the complex variable k′ _z ) has additional poles in addition to that at k′ _z = k _z . In particular, if we introduce the quantities

$$ \Delta_{-}^{2} = \mathbf{k}_{\parallel}^{2} - \frac{\Omega^{2}}{c^{2}} + \frac{\omega_{\mathrm{P}}^{2}}{c^{2}},\qquad \Delta_{+}^{2} = \mathbf{k}_{\parallel}^{2} - \frac{\Omega^{2}}{c^{2}}, $$

(46)

and write ξ(k _z ) in the form (𝓒 is an arbitrary constant)

$$ \xi(k_{z}) = \frac{i\mathcal{C}}{\sqrt{k}\,(k_{z} + i\Delta_{+})(k_{z} - i\Delta_{-})}, $$

(47)

upon integrating the integrand on a closed circuit in the upper complex plane like the one in Fig. 9 and taking the limit |k′ _z | → ∞, the application of the Cauchy theorem leads to the expression

$$\begin{array}{@{}rcl@{}} &&\frac{c}{2 k^{\frac{3}{2}}}\left\{\left[\frac{i(\Delta_{+}^{2} + {k_{z}^{2}})(\Delta_{-}^{2} - k_{\parallel}^{2}) - (ik_{z}\Delta_{-} + k_{\parallel}^{2})(k_{z} + i\Delta_{+})(\Delta_{-} - \Delta_{+})}{(k_{z} + i\Delta_{+})(k_{z} - i\Delta_{-})(\Delta_{-}^{2} - k_{\parallel}^{2})}\right]\right.\\ &&+\left.\frac{i\omega_{\mathrm{P}}^{2}}{2c^{2}\,(k_{\parallel} + \Delta_{+})(\Delta_{-} - k_{\parallel})}\right\}\mathcal{C} = - i(W + Y)C(k). \end{array} $$

Fig. 9

The path of integration of Eq. 45 in the complex k _z plane

The equation has a nonzero solution for the constant 𝓒 if and only if the ( k _z –dependent) term in the square brackets cancels; this happens when $\Delta _{+}\Delta _{-} = k_{\parallel }^{2}$, i.e.,

$$ \Omega_{\mathbf{k}_{\parallel}}^{4} - \Omega_{\mathbf{k}_{\parallel}}^{2}\, \left[2\left(ck_{\parallel}\right)^{2} + \omega_{\mathrm{P}}^{2}\right] + \left(ck_{\parallel}\omega_{\mathrm{P}}\right)^{2} = 0, $$

(48)

whence the plasmon polariton dispersion relation in Eq. 30 is recovered.

The constant 𝓒 takes the value

$$\mathcal{C} = - k_{\parallel} \sqrt{\frac{ck_{\parallel}}{\pi\omega_{\mathrm{s}}}}\,(\Delta_{-} - \Delta_{+})(W + Y). $$

Once 𝓒 is known, from the original system of equations, it is possible to work out the expression for the coefficients in the expansion of α _k∥:

$$\xi(k_{z}) = -ik_{\parallel}\sqrt{\frac{ck_{\parallel}}{\pi k \omega_{\mathrm{s}}}}\frac{\Delta_{-} - \Delta_{+}} {(k_{z}+i\Delta_{+})(k_{z}-i\Delta_{-})};$$

(49)

$$W = \mathcal{K}\left(1+\frac{\Omega_{\mathbf{k}_{\parallel}}}{\omega_{\mathrm{s}}}\right); \quad Y = \mathcal{K}\left(1-\frac{\Omega_{\mathbf{k}_{\parallel}}}{\omega_{\mathrm{s}}}\right);\\ X = \mathcal{K}\left(1+\frac{\Omega_{\mathbf{k}_{\parallel}}}{ck}\right)\xi;\quad Z = \mathcal{K}\left(\frac{\Omega_{\mathbf{k}_{\parallel}}}{ck} - 1\right)\xi. $$

(50)

The arbitrary constant 𝓚 can be fixed from the normalization conditions $\left <\Psi \right |_{}\mathrm {\upalpha }_{\mathbf {k}_{\parallel }}\mathrm {\upalpha }_{\mathbf {k}_{\parallel }}^{\dagger } - \mathrm {\upalpha }_{\mathbf {k}_{\parallel }}^{\dagger }\mathrm {\upalpha }_{\mathbf {k}_{\parallel }} \left |\Psi \right >_{\hspace {-0.1em}} = 1$ on a generic normalized quantum state |Ψ〉. The result is

$$ \mathcal{K}^{2} = \frac{1}{4} \left(\frac{\omega_{\mathrm{s}}}{\Omega_{\mathbf{k}_{\parallel}}}\right)\frac{(\Delta_{-}+\Delta_{+})k_{\parallel}}{\Delta_{-}^{2} + \Delta_{+}^{2}}. $$

(51)

The behavior of W and Y as a function of k _∥ is reported in Fig. 1.

The electric field operator is expanded onto a family of modes E _k∥(r) according to Eq. 32. The expression for the modes can be calculated from the commutation relation [α _k∥, Eqs − Ȧ] = E _−k∥, which leads to

$$\begin{array}{@{}rcl@{}} \mathbf{E}_{\mathbf{k}_{\parallel}}(\mathbf{r}) = (W-Y)\sqrt{\frac{\hbar\omega_{\mathrm{s}}}{16\pi^{2}\varepsilon_{0}k_{\parallel}}} \mathbf{E}_{\text{qs},\mathbf{k}_{\parallel}}(\mathbf{r})\;-\\ i\int\mathrm{d} k_{z}\,ck \left[X(-k_{z})+Z(-k_{z})\right]\mathbf{A}_{\mathrm{T},\mathbf{k}_{\parallel},k_{z}}(\mathbf{r}) \end{array} $$

(52)

$\left (k = \sqrt {k_{\parallel }^{2} + {k_{z}^{2}}}\right )$. With the help of Eq. 50 and some results on Fourier transforms in the k _z space, we find

$$ \mathbf{E}_{\mathbf{k}_{\parallel}} {}={} \mathcal{K}\frac{\Omega_{\mathbf{k}_{\parallel}}k_{\parallel}}{\pi(\Delta_{+}{}+{}\Delta_{-})} \sqrt{\frac{\hbar k_{\parallel}}{\varepsilon_{0}\omega_{\mathrm{s}}}} \left({}-i\hat{\mathbf{k}_{\parallel}}\pm\frac{k_{\parallel}}{\Delta_{\pm}}\hat{\mathbf{z}}{}\right) e^{i\mathbf{k}_{\parallel}\cdot\mathbf{\rho}\,\mp\,\Delta_{\pm}z} $$

(53)

(the upper and lower signs refer to the regions z < 0 and z > 0, respectively, k̂_∥ is the unit vector directed as k _∥). Equation 53 is analogous to the expression of the field obtained from classical electrodynamics [3, 21].

Appendix B: Spherical Geometry

In this appendix, we briefly present how the eigenvalue Eq. 39 is derived. The transverse vector potential of Eq. 18 is expanded onto the electromagnetic vector wave functions [65]

$$\begin{array}{*{20}l} &\mathbf{A}_{\mathrm{T},l,k}^{(1)}(\mathbf{r}) = -\mathcal{N} j_{l}{kr}\frac{\partial}{\partial\theta}\mathrm{P}_{l}(\cos\theta)\hat{\mathbf{\varphi}};\\ &\mathbf{A}_{\mathrm{T},l,k}^{(2)}(\mathbf{r}) = \mathcal{N} \left[l(l+1)\frac{j_{l}(kr)}{kr}\mathrm{P}_{l}(\cos\theta)\hat{\mathbf{r}}\right.\\ &{\kern5.7pc}+ \left.\frac{1}{kr}\frac{\partial}{\partial r}\left[r j_{l}(kr)\right] \frac{\partial}{\partial\theta}\mathrm{P}_{l}(\cos\theta)\hat{\mathbf{\theta}}\right]\end{array} $$

(j _l (x) denotes the spherical Bessel function of order l). Polarizations with λ=1 and 2 correspond to TE and TM modes, respectively.³ Transverse modes are indexed by the continuous wavenumber k, which is related to the modal frequency ω _k =c k. For each l, the frequencies ω _k encompass the whole spectrum, overlapping with the instantaneous frequencies ω _l .

The continuous index k is discretized into a finite number of retarded modes up to a cutoff value N (the modes are labeled by the index ν = 1, 2, …, N). This can be accomplished by supposing to enclose the system in a large sphere of radius L, with L>>R. According to the Sturm–Liouville theory [66], the values k _ν implicitly provided by the equation

$$ j_{l}(k_{\nu} L) = 0, $$

(54)

constitute an orthogonal basis of electromagnetic modes. Consequently, the normalization factor 𝓝 can be shown to assume the form

$$ \mathcal{N} = \sqrt{\frac{\hbar(2l+1)}{4\pi l (l+1)\varepsilon_{0} ck_{\nu}\,L^{3}\,j_{l+1}^{2}(k_{\nu} L)}}. $$

(55)

We look for Hopfield–Bogoljubov operators of the form

$$ \mathrm{\upalpha}_{l} = W_{l} \mathrm{b}_{l} + Y_{l}\mathrm{b}_{l}^{\dagger} + \sum_{\nu}\left(X_{l,\nu}\mathrm{a}_{\nu} + Z_{l,\nu}\mathrm{a}_{\nu}^{\dagger}\right), $$

(56)

which satisfy the commutation relation in Eq. 23. After defining the adimensional quantities ω̃ _l = ω _l /ω _P, k̃ = ck/ω _P , R̃ = ω _P R/c, and L̃ = ω _P L/c, we can introduce the (N + 1)-dimensional vector ξ ^(l), whose components are (𝓚 is a normalization constant):

$$\begin{array}{*{20}l} &\xi_{\nu}^{(l)} = - i\frac{\mathcal{K}}{\sqrt{\tilde{k}}}(X_{l,\nu} - Z_{l,\nu}), \qquad \nu = 1,\dots,N;\\ &\xi_{N+1}^{(l)} = \frac{\mathcal{K}}{\sqrt{\tilde{\omega}_{l}}}(W_{l} + Y_{l}). \end{array} $$

By collecting the terms in front of the quantum operators $\mathrm {b}_{l},\mathrm {b}_{l}^{\dagger },\mathrm {a}_{l,\nu }$, and $\mathrm {a}_{l,\nu }^{\dagger }$, Eq. 23 can be reduced to a linear system of equations. With some further algebraic manipulation, we are led to the eigenproblem in Eq. 39, with the (N+1)×(N+1) symmetric matrix 𝕄^(l) defined as follows:

$$\begin{array}{*{20}l} &\mathbb{M}^{(l)}_{\nu,\nu} = \frac{\tilde{R}^{3}}{j_{l+1}^{2}(k_{\nu}L)\tilde{L}^{3}} \left[j_{l+1}^{2}(k_{\nu}R) + {j_{l}^{2}}(k_{\nu}R)\left(1+\frac{2(l+1)}{(k_{\nu}R)^{2}}\right)\right.\\ &{\kern7pc}-\left.(2l+3)\frac{j_{l}(k_{\nu}R)j_{l+1}(k_{\nu}R)}{k_{\nu}R}\right] + \tilde{k}_{\nu}^{2};\\ &\mathbb{M}^{(l)}_{\nu,\mu} = \frac{2\tilde{R}}{|j_{l+1}(k_{\nu}L)j_{l+1}(k_{\mu}L)|\tilde{L}^{3}}\\ &{\kern2.5pc}\times\left\{\frac{\tilde{R}}{\tilde{k}_{\nu}^{2} \,-\, \tilde{k}^{2}_{\mu}} \left[\!\tilde{k}_{\mu} j_{l}(k_{\mu}R) j_{l+1}(k_{\nu}R)- \tilde{k}_{\nu} j_{l}(k_{\nu}R) j_{l+1}(k_{\mu}R)\!\right]\right. \\ &{\kern4pc}\left. + \frac{l+1}{\tilde{k}_{\nu}\tilde{k}_{\mu}} j_{l}(k_{\nu}R) j_{l}(k_{\mu}R)\right\};\\ &\mathbb{M}^{(l)}_{\nu,N+1} = \mathbb{M}^{(l)}_{N+1,\nu} = -\sqrt{\frac{2l(l+1)}{(2l+1)\tilde{R}\tilde{L}^{3}}}\; \frac{\tilde{R}\;j_{l}(k_{\nu}R)}{\tilde{k}_{\nu}\, |j_{l+1}(k_{\nu}L)|};\\ &\mathbb{M}^{(l)}_{N+1,N+1} {}={} \tilde{w}_{l}^{2}; \qquad\quad\text{with}\quad \nu, \mu = 1,2,\dots,N \quad\text{and}\quad \nu \neq \mu. \end{array} $$

The eigenproblem is solved with numerical methods. The value of the coefficients W _l ,Y _l ,X _l,ν , and Z _l,ν can be extracted from the definition of eigenvector ξ ^(l) with the additional relations

ω _l (W _l − Y _l ) = Ω _l (W _l + Y _l ) ;

ck (X _l,ν + Z _l,ν ) = Ω _l (X _l,ν − Z _{l, ν} ).

The normalization constant can be calculated from the condition $\left <\Psi \right |_{}\mathrm {\upalpha }_{l}\mathrm {\upalpha }_{l}^{\dagger } - \mathrm {\upalpha }_{l}^{\dagger }\mathrm {\upalpha }_{l} \left |\Psi \right >_{\hspace {-0.1em}} = 1$ with the operator in Eq. 56. The result is |ξ ^(l)|²=ω _P/Ω _l , which, for an eigenvector with unitary normalization, leads simply to $\mathcal {K} = \sqrt {\omega _{\mathrm {P}} / \Omega _{l}}$.

The solutions of the vector eigenproblem (39) are shown by the dots in Fig. 7 and compared with the corresponding electrodynamical solutions. The latter are obtained by solving for a complex ω the equation

$$ \frac{h_{l}^{(1)}(kR)}{\frac{\partial}{\partial r}[r\,h^{(1)}_{l}(kr)]_{r=R}} = \varepsilon(\omega)\, \frac{i_{l}(nkR)}{\frac{\partial}{\partial r}[r\,i_{l}(nkr)]_{r=R}}, $$

(57)

with a Drude dielectric function $\varepsilon = 1 - \omega _{\mathrm {P}}^{2}/\omega ^{2}$. In the equation, k=ω/c, $n = \sqrt {\varepsilon }$, $h^{(1)}_{l}(x)$ is the spherical Hankel function of the first kind, and i _l (x) is the modified spherical Bessel function of the first kind. The modal frequency and radiative width are related to the real and imaginary parts of the solution by Ω _l =R(ω) and γ _rad=−2I(ω), respectively.