# Environment-modified three-body energy transfer

Madeline C. Waller and Robert Bennett  School of Physics & Astronomy,
University of Glasgow, Glasgow G12 8QQ, United Kingdom

###### Abstract

Resonant energy transfer from a donor to an acceptor is one of the most basic
interactions between atomic and molecular systems. In real-life situations,
the donor and acceptor are not isolated but in fact coupled to their
environment and to other atoms and molecules. The presence of a third body can
modify the rate of energy transfer between donor and acceptor in distinctive
and intricate ways, especially when the three-site system is itself
interacting with a larger macroscopic background such as a solvent. The rate
can be calculated perturbatively, which ordinarily requires the summation of
very large numbers of Feynman-like diagrams. Here we demonstrate a method
based on canonical perturbation theory that allows us to reduce the
computational effort required, and use this technique to derive a formula for
the rate of three-body resonance energy transfer in a background environment.
As a proof-of-principle, we apply this to the situation of a dimer positioned
near a dielectric interface, with a distant third molecule controlling the
rate, finding both enhancement or suppression of the rate depending on system
parameters.

##  I Introduction

The transport of energy between atoms and molecules is important in many
diverse areas of science. It is present as a fundamental process in the
transport of energy in plants, and has applications in, for example,
artificial light-harvesting [1] and the “spectroscopic ruler”, a technique
used to estimate intermolecular distances within macromolecules [2]. A closely
related process, Interatomic Coulombic Decay (ICD) [3], could also be relevant
in radiation biology [4]. The mechanism that governs the transfer of energy
from one atom/molecule (donor) to another (acceptor) depends chiefly on the
strength of the light-matter coupling and the interatomic/intermolecular
distance. When the distances are ultra-short, the resulting energy transfer is
governed by Dexter theory [5], where electronic orbitals overlap allowing the
electrons to migrate between molecules. For longer distances where the
electronic orbitals no longer overlap, energy transfer is instead mediated by
a photon, with electrons of the donor and acceptor remaining bound to the
nuclei. For near field interactions where light-matter coupling is weak
relative to the intramolecular (e.g. vibrational) coupling, this energy
transfer is governed by Förster theory [6], leading to the well-known $R^{-6}$
dependence on the separation distance $R$. For far field interactions,
relativistic properties of the mediating photon become significant, leading to
an $R^{-2}$ distance dependence [7]. When intramolecular couplings are
comparable to (or weaker than) intermolecular ones, completely different
approaches must be taken (see, e.g., [8, 9, 10]).

The weak coupling interactions are described by molecular quantum
electrodynamics (QED), which can be most easily understood using Feynman
diagrammatic techniques. This is a unified, fully quantum theory, which
produces the $R^{-6}$ and $R^{-2}$ dependence of the short- and long-range
interactions respectively as limiting cases [11, 12, 13].

Describing the interaction between a single two-level donor dipole and similar
acceptor is relatively straightforward, even in the presence of a background
environment [14, 15]. However, the complexity increases significantly with the
addition of other levels, additional atoms/molecules [16, 17, 18, 19, 20, 21]
and higher multipole moments [22]. Calculations which account for a third
interacting body have only been done for the simplest case of a vacuum
environment — here we will generalise these to arbitrary environments. In
order to describe energy transfer within an external environment (see figure
1), the molecular QED framework can be combined with macroscopic QED [23, 24],
allowing interactions in the presence of arbitrarily-shaped dispersive and
absorptive media to be described.

Figure 1: System of three atoms or molecules (donor, mediator and acceptor) in
the presence of an arbitrary external environment.

In this paper, a three-body system of a donor, acceptor and a polarisable
mediator is studied within macroscopic QED. In section II, we will use
canonical perturbation theory to eliminate some of the computational
complexity arising from the presence of the third body, and in section III,
use macroscopic QED to model the effects of the external environment. Similar
calculations have been carried out for three-body ICD in a vacuum by
considering the mediator as part of the environment of the two-body system
[21], but this makes it awkward to extend the calculations to complex
geometries. We obtain a general formula for the rate of three-body resonance
energy transfer in an arbitrary environment, and in section IV we apply this
to a situation of experimental interest, namely a dimer trapped near a surface
controlled by a distant mediating agent (e.g. [25]).

##  II Hamiltonians

We consider a system of a donor, acceptor and mediator, as seen in Fig. 1.
Energy from the donor is released and transferred to the acceptor, either
directly (Fig. 2a) or via the mediator (Fig. 2b).

Figure 2: System of three two-level atoms/molecules transmitting energy
through the electromagnetic field due to resonance energy transfer. The donor
begins in an excited energy state, the acceptor in the ground state and the
mediator in its lower state. (a) Direct interaction. Energy is emitted from
the donor, is transmitted through the field and absorbed by the acceptor which
excites. (b) Mediated interaction. Energy emitted from the donor is absorbed
by the mediator causing it to become temporarily excited. The mediator
releases this energy again and it is absorbed by the acceptor which then
becomes excited.

We model this via the following Hamiltonian:

$H=H_{0}+H^{\mathrm{A}}_{\mathrm{int}}+H^{\mathrm{D}}_{\mathrm{int}}+H^{\mathrm{M}}_{\mathrm{int}},$
(1)

where

$H_{0}=H_{\mathrm{rad}}+H^{\mathrm{A}}_{\mathrm{mol}}+H^{\mathrm{D}}_{\mathrm{mol}}+H^{\mathrm{M}}_{\mathrm{mol}},$
(2)

$H_{\mathrm{rad}}$ is the Hamiltonian of the radiation field,
$H^{\xi}_{\mathrm{mol}}$ is the Hamiltonian of the molecule $\xi$ for which we
assume that the eigenstates are known, and

$H^{\xi}_{\mathrm{int}}=-\bm{\hat{d}}_{\xi}\cdot\bm{\hat{E}}(\bm{r}_{\xi}),$
(3)

where $\hat{\bm{d}}_{\xi}$ is the transition dipole moment of molecule $\xi$,
and $\hat{\bm{E}}(\bm{r}_{\xi})$ is the electric field at the position,
$\bm{r}$, of the molecule $\xi$.

The initial and final states of the system are:

$\ket{i}=\ket{e_{\mathrm{D}},s_{\mathrm{M}},g_{\mathrm{A}};0},\qquad\ket{f}=\ket{g_{\mathrm{D}},s_{\mathrm{M}},e_{\mathrm{A}};0},$
(4)

where $g_{\mathrm{D}}(g_{\mathrm{A}})$ denotes the ground state of the donor
(acceptor), $e_{\mathrm{D}}(e_{\mathrm{A}})$ the excited state of the donor
(acceptor), $s_{\mathrm{M}}$ is an arbitrary state of the mediator and $0$ is
the ground state of the electromagnetic field. We need to take into account
the mediated interaction between the donor and acceptor, which involves all
three molecules. Figure 3a shows the resonant interaction, where the
excitations are transmitted through the field as a result of molecular
relaxation from excited to ground states. Likewise, we must also consider the
other time-orderings that lead to off-resonant contributions (where molecular
excitations are accompanied by emission of a photon), and the half-resonant
contributions (where one set of interacting molecules are excited while the
photon is transmitted and the other set are in their ground states). All of
these contributions have to be considered when calculating rates for the whole
process, which confusingly is also known as resonant energy transfer (RET).
Figure 3a shows us that this process would involve four emission/absorption
events, meaning that fourth order perturbation theory would be required. The
complexity of such a calculation means that it is useful to simplify the
Hamiltonian as much as possible before proceeding.

###  II.1 Reducing the order of perturbation theory

The fourth-order calculation consists of contributions from four one-photon
vertices. This can be simplified by ‘collapsing’ the two one-photon vertices
at the mediator into one two-photon vertex, therefore lowering the
perturbation theory required to third order (see Fig. 3) as applied at lower
orders in [26].

Figure 3: The 3-body resonant interaction diagrams (a) before and (b) after
the reduction of the order of perturbation theory.

We are aiming to create a new effective Hamiltonian that encodes the
information for the two one-photon interactions at the mediator into one term,
so this new coupling term will be second order in the electric dipole moment.
To this end, we consider just the interaction at the mediator and not at the
donor and acceptor, and perform a unitary transformation on the Hamiltonian,

$\displaystyle H^{\mathrm{M}}_{\mathrm{new}}=$ $\displaystyle
e^{iS}H^{\mathrm{M}}e^{-iS}=\sum_{n=0}^{\infty}\frac{1}{n!}\commutator{iS}{\commutator{iS}{...H}}$
$\displaystyle=$ $\displaystyle
H_{0}+H^{\mathrm{M}}_{\mathrm{int}}+\commutator{iS}{H_{0}}+\commutator{iS}{H^{\mathrm{M}}_{\mathrm{int}}}$
$\displaystyle+\frac{1}{2}\commutator{iS}{\commutator{iS}{H_{0}}}+...\quad,$
(5)

where $S$ is a generator that is assumed later to be first order in the
electric dipole moment and we have made use of the Baker–Campbell–Hausdorff
formula. We seek a Hamiltonian of second order in
$H^{\mathrm{M}}_{\mathrm{int}}$ (and thereby $\hat{\bm{d}}_{M}$), so we
eliminate the first order $H^{\mathrm{M}}_{\mathrm{int}}$ term by choosing
$\commutator{iS}{H_{0}}=-H^{\mathrm{M}}_{\mathrm{int}}$. This leaves, up to
second order in the electric dipole moment,

$H^{\mathrm{M}}_{\mathrm{new}}=H_{0}+\frac{1}{2}\commutator{iS}{H^{\mathrm{M}}_{\mathrm{int}}}$
(6)

where, from our chosen definition of the generator, for initial state
$\ket{M}$ and final state $\ket{N}$ we have

$\bra{N}iS\ket{M}=\frac{\bra{N}H^{\mathrm{M}}_{\mathrm{int}}\ket{M}}{E_{N}-E_{M}}.$
(7)

We now calculate the expectation value of the new seconder order interaction
term using the definition of generator $S$ given above;

$\displaystyle\frac{1}{2}$
$\displaystyle\bra{N}\commutator{iS}{H^{\mathrm{M}}_{\mathrm{int}}}\ket{M}$
$\displaystyle=-$
$\displaystyle\frac{1}{2}\sum_{\mathrm{I}}\bra{N}H^{\mathrm{M}}_{\mathrm{int}}\ket{I}\bra{I}H^{\mathrm{M}}_{\mathrm{int}}\ket{M}\left[\frac{1}{E_{\mathrm{I}}-E_{\mathrm{N}}}+\frac{1}{E_{\mathrm{I}}-E_{\mathrm{M}}}\right]$
$\displaystyle=-$
$\displaystyle\frac{1}{2}\sum_{r}\bra{N}H^{\mathrm{M}}_{\mathrm{int}}H^{\mathrm{M}}_{\mathrm{int}}\ket{M}$
$\displaystyle\times$ $\displaystyle\left[\frac{1}{E_{\mathrm{rs}}+\hbar
cp}+\frac{1}{E_{\mathrm{rs}}-\hbar cp}+\frac{1}{E_{\mathrm{rs}}+\hbar
cp^{\prime}}+\frac{1}{E_{\mathrm{rs}}-\hbar cp^{\prime}}\right],$ (8)

where $E_{rs}$ is the transition energy of the mediator going from the excited
$r$ state to its lower $s$ state.

Since in three-body RET, the mediator responds at the frequency of the donor
decay transition [16], we can replace $\hbar cp\mapsto\hbar
ck=E_{\mathrm{eg}}$ and $\hbar cp^{\prime}\mapsto-\hbar ck=-E_{\mathrm{eg}}$,
giving

$\frac{1}{2}\bra{N}\commutator{iS}{H^{\mathrm{M}}_{\mathrm{int}}}\ket{M}\\\
=-\sum_{r}\bra{N}H^{\mathrm{M}}_{\mathrm{int}}H^{\mathrm{M}}_{\mathrm{int}}\ket{M}\left[\frac{1}{E_{\mathrm{rs}}+E_{\mathrm{eg}}}+\frac{1}{E_{\mathrm{rs}}-E_{\mathrm{eg}}}\right].$
(9)

We can therefore define a new coupling term as given below:

$H_{2}=\frac{1}{2}\commutator{iS}{H^{\mathrm{M}}_{\mathrm{int}}}=-\sum_{p,p^{\prime}}\alpha^{\mathrm{M}}_{ij}(k)\hat{E}_{i}(\bm{r}_{\mathrm{M}},p)\\\
\hat{E}_{j}(\bm{r}_{\mathrm{M}},p^{\prime})$ (10)

where

$\alpha^{\mathrm{M}}_{ij}(k)=\sum_{r}|\bm{\hat{d}}^{\mathrm{M}}_{\mathrm{sr}}|^{2}\left[\frac{1}{E_{\mathrm{rs}}+\hbar
ck}+\frac{1}{E_{\mathrm{rs}}-\hbar ck}\right]$ (11)

is identified as the dynamic polarizability of the mediator. This means that
when considering 3-body RET, instead of fourth order perturbation theory being
required, now only third order is needed. This reduces the usual 24 time-
ordered diagrams required for this calculation to just six, as shown in figure
4. This process is, of course, equivalent to using a polarisability-based
Hamiltonian, as done in [27] for example.

Figure 4: The six time ordered diagrams for three-body RET once the two one-
photon vertices have been collapsed into one two-photon vertex.

We now go one step further and use the same techniques to create an effective
fourth order term that describes the entire interaction including all three
bodies. We begin with the new Hamiltonian,

$H_{\mathrm{new}}=H_{0}+H_{1}+H_{2},$ (12)

where $H_{2}$ is the defined as in equation (10) and, for convenience, the
acceptor and donor interaction terms have been condensed into a single term,
$H_{1}=H^{\mathrm{A}}_{\mathrm{int}}+H^{\mathrm{D}}_{\mathrm{int}}$, which is
linear in the electric dipole moment. Introducing the dimensionless constant
$\lambda$ which is proportional to the electric dipole moment, so that

$H_{\mathrm{new}}=H_{0}+\lambda H_{1}+\lambda^{2}H_{2},$ (13)

we perform a series of unitary transformations:

$H_{\mathrm{new}}^{(1)}=e^{i\lambda^{3}S_{3}}e^{i\lambda^{2}S_{2}}e^{i\lambda
S_{1}}H_{\mathrm{new}}e^{-i\lambda
S_{1}}e^{-i\lambda^{2}S_{2}}e^{-i\lambda^{3}S_{3}}$ (14)

with generators defined as follows:

$\displaystyle\commutator{iS_{1}}{H_{0}}$ $\displaystyle=-H_{1}$
$\displaystyle\commutator{iS_{2}}{H_{0}}$ $\displaystyle=-H_{2}$
$\displaystyle\commutator{iS_{3}}{H_{0}}$
$\displaystyle=-\commutator{iS_{1}}{H_{2}}-\commutator{iS_{1}}{\commutator{iS_{1}}{H_{1}}}.$
(15)

These are chosen such that any donor-mediator-acceptor interactions are
eliminated to order $\lambda^{3}$.

After dropping any second order terms that do not contain donor and acceptor
contributions, and any fourth order terms that do not contain mediator
contributions, we arrive at our new interaction Hamiltonian, the mediator-
dependent parts of which are fourth order in the electric dipole moment as
required;

$H_{\mathrm{int}}=\frac{1}{2}\left(\commutator{iS_{1}}{H_{1}}+\commutator{iS_{1}}{\commutator{iS_{1}}{H_{2}}}+\commutator{iS_{2}}{\commutator{iS_{1}}{H_{1}}}\right).$
(16)

This will form the basis of our perturbative treatment of mediated resonance
energy transfer.

###  II.2 Perturbation theory

We now model our three-body system using the Hamiltonian (16), and perform
perturbation theory to find the matrix element of the interaction that will
eventually lead to a RET rate via Fermi’s golden rule. Considering both the
direct and indirect interactions, we find the matrix element to be

$\displaystyle
M_{fi}=\sum_{p,p^{\prime},k}\bra{f}\Biggl{[}\frac{H_{\mathrm{A}}H_{\mathrm{D}}}{\hbar
cp-E_{\mathrm{eg}}}-\frac{H_{\mathrm{D}}H_{\mathrm{A}}}{\hbar
cp+E_{\mathrm{eg}}}$
$\displaystyle+\frac{H_{\mathrm{D}}H_{\mathrm{A}}H_{2}}{(\hbar
cp+E_{\mathrm{eg}})(\hbar
cp^{\prime}-E_{\mathrm{eg}})}+\frac{H_{2}H_{\mathrm{D}}H_{\mathrm{A}}}{(\hbar
cp-E_{\mathrm{eg}})(\hbar cp^{\prime}+E_{\mathrm{eg}})}$
$\displaystyle+\frac{H_{\mathrm{D}}H_{2}H_{\mathrm{A}}}{(\hbar
cp+E_{\mathrm{eg}})(\hbar
cp^{\prime}+E_{\mathrm{eg}})}+\frac{H_{\mathrm{A}}H_{2}H_{\mathrm{D}}}{(\hbar
cp-E_{\mathrm{eg}})(\hbar cp^{\prime}-E_{\mathrm{eg}})}\Biggr{]}\ket{i}$ (17)

where, using Eq. (3),

$\displaystyle H_{\mathrm{D}}$ $\displaystyle\equiv
H_{\mathrm{D}}^{\mathrm{int}}=-\bm{\hat{d}}_{\mathrm{A}}\cdot\bm{\hat{E}}(\bm{r}_{\mathrm{D}})$
$\displaystyle H_{\mathrm{A}}$ $\displaystyle\equiv
H_{\mathrm{D}}^{\mathrm{int}}=-\bm{\hat{d}}_{\mathrm{A}}\cdot\bm{\hat{E}}(\bm{r}_{\mathrm{A}})$
$\displaystyle H_{2}$
$\displaystyle=-\alpha^{\mathrm{M}}_{ij}(k)\hat{E}_{i}(\bm{r}_{\mathrm{M}},p)\hat{E}_{j}(\bm{r}_{\mathrm{M}},p^{\prime}).$
(18)

This matrix element contains the information from all of the different time
orderings, meaning we no longer have any explicit intermediate states. As a
result, the order of perturbation theory is further reduced to first order.
The first two terms in (17) are the direct (two-body) interaction terms, the
first corresponding to the resonant time ordering and the second the off-
resonant. The other terms describe different time orderings of the mediated
interaction. The third and fourth terms are the half-resonant contributions,
where the third term corresponds to (d) and (e) in Fig. 4 and the fourth term
to (a) and (b). The fifth term describes the completely off-resonant
interaction, shown in (c), and the sixth is the completely resonant
interaction, (f). We can then use this matrix element to calculate the rate of
interaction.

##  III Derivation of the Rate

So far, we have not considered the environment that the 3-body system is in.
To do this, we employ macroscopic QED [28, 24], which introduces macroscopic
objects into the quantum description. This means that the effects of an
environment near the system can be accounted for more readily than the
atomistic approach in [29], since the macroscopic media can be described by
their effective properties, such as overall permittivity and permeability.
This environment can include arbitrarily shaped, dispersing, and absorbing
material bodies.

In macroscopic QED, the electric field is expressed as;

$\bm{\hat{E}}(\bm{r})=\sum_{\lambda}\int_{0}^{\infty}d\omega\int
d^{3}\bm{r}^{\prime}\bm{G}_{\lambda}(\bm{r},\bm{r}^{\prime},\omega)\cdot\bm{\hat{f}}_{\lambda}(\bm{r}^{\prime},\omega)+H.c,$
(19)

where $\hat{\bm{f}}_{\lambda}(\bm{r},\omega)$ is an annihilation operator for
a polaritonic excitation at position $\bm{r}$ and with frequency $\omega$, and
its Hermitian conjugate is the corresponding creation operator. These
operators obey bosonic commutation relations

$\displaystyle\left[\bm{\hat{f}}_{\lambda}(\bm{r},\omega),\bm{\hat{f}}_{\lambda}(\bm{r}^{\prime},\omega^{\prime})\right]$
$\displaystyle=\left[\bm{\hat{f}}^{\dagger}_{\lambda}(\bm{r},\omega),\bm{\hat{f}}^{\dagger}_{\lambda}(\bm{r}^{\prime},\omega^{\prime})\right]=0$
(20)
$\displaystyle\left[\bm{\hat{f}}_{\lambda}(\bm{r},\omega),\bm{\hat{f}}^{\dagger}_{\lambda}(\bm{r}^{\prime},\omega^{\prime})\right]$
$\displaystyle=\bm{\delta}(\bm{r}-\bm{r}^{\prime})\delta(\omega-\omega^{\prime}).$
(21)

where
$\bm{\delta}(\bm{r}-\bm{r}^{\prime})=\mathrm{diag}(1,1,1)\delta(\bm{r}-\bm{r}^{\prime})$.
The matrix $\bm{G}_{\lambda}$ obeys the completeness relation

$\sum_{\lambda=e,m}\int
d^{3}\bm{s}\bm{G}_{\lambda}(\bm{r},\bm{s},\omega)\cdot\bm{G}_{\lambda}^{*T}(\bm{r}^{\prime},\bm{s},\omega)\\\
=\frac{\hbar\mu_{0}}{\pi}\omega^{2}\imaginary\bm{G}(\bm{r},\bm{r}^{\prime},\omega),$
(22)

where $\bm{G}$ satisfies

$\left[\nabla\times\frac{1}{\mu(\bm{r},\omega)}\nabla\times-\frac{\omega^{2}}{c^{2}}\times\varepsilon(\bm{r},\omega)\right]\bm{G}(\bm{r},\bm{r^{\prime}},\omega)=\delta(\bm{r}-\bm{r^{\prime}}).$
(23)

Known as the Green’s tensor or Green’s dyadic, $\bm{G}$ encodes all of the
information about the environment, including different geometries [30]. In
particular, $\bm{G}(\bm{r},\bm{r}^{\prime},\omega)$ describes an excitation
that is propagating out from point $\bm{r}^{\prime}$ and then being observed
at point $\bm{r}$.

By making use of the completeness relation (22), we can write each of the
terms in the matrix elements as frequency integrals of the form;

$\int^{\infty}_{0}d\omega\frac{\omega^{2}\imaginary\bm{G}(\bm{r},\bm{r}^{\prime},\omega)}{\omega_{\mathrm{D}}\pm\omega}.$
(24)

For example, the first term of the matrix element becomes:

$\displaystyle\sum_{k}$
$\displaystyle\bra{f}\frac{H_{\mathrm{A}}H_{\mathrm{D}}}{E_{\mathrm{eg}}-\hbar
ck}\ket{i}$
$\displaystyle=\frac{\mu_{0}}{\pi}d_{A_{i}}^{*}\int^{\infty}_{0}d\omega\frac{\omega^{2}\imaginary
G_{ij}(\bm{r}_{\mathrm{A}},\bm{r}_{\mathrm{D}},\omega)}{\omega-\omega_{\mathrm{D}}}d_{D_{j}},$
(25)

where we have defined

$\displaystyle\hbar cp$ $\displaystyle=\hbar\omega,$ $\displaystyle
E_{\mathrm{eg}}$ $\displaystyle=\hbar\omega_{\mathrm{D}}$ (26)
$\displaystyle\bm{d}$ $\displaystyle\equiv\bra{g}\bm{\hat{d}}\ket{e},$
$\displaystyle\bm{d^{*}}$ $\displaystyle\equiv\bra{e}\bm{\hat{d}}\ket{g}.$
(27)

Since the frequency integrals have real axis poles, we let the eigenenergies
of the atom take on a small imaginary part, $\epsilon$. This means that the
poles become $\pm(\omega_{\mathrm{D}}+i\epsilon)$, and we then evaluate the
integrals by closing the contour in the upper half-plane [31, 13]. We find
that;

$\lim_{\epsilon\rightarrow
0+}\int^{\infty}_{0}d\omega\frac{\omega^{2}\imaginary\bm{G}(\bm{r},\bm{r}^{\prime},\omega)}{\omega_{\mathrm{D}}+\omega+i\epsilon}\\\
=-\int^{\infty}_{0}d\xi\bm{G}(\bm{r},\bm{r}^{\prime},i\xi)\xi^{2}\frac{\omega_{\mathrm{D}}}{\omega_{\mathrm{D}}^{2}+\xi^{2}}$
(28)

and

$\lim_{\epsilon\rightarrow
0+}\int^{\infty}_{0}d\omega\frac{\omega^{2}\imaginary\bm{G}(\bm{r},\bm{r}^{\prime},\omega)}{\omega_{\mathrm{D}}-\omega+i\epsilon}\\\
=-\int^{\infty}_{0}d\xi\bm{G}(\bm{r},\bm{r}^{\prime},i\xi)\xi^{2}\frac{\omega_{\mathrm{D}}}{\omega_{\mathrm{D}}^{2}+\xi^{2}}-\pi\omega_{\mathrm{D}}^{2}\bm{G}(\bm{r},\bm{r}^{\prime},\omega_{\mathrm{D}}).$
(29)

Applying this method to each of the terms in the matrix element, and assuming
that all media involved are reciprocal, so that Lorentz reciprocity
$\bm{G}^{T}(\bm{r},\bm{r}^{\prime},\omega)=\bm{G}(\bm{r}^{\prime},\bm{r},\omega)$
can be used, we find that the surviving term for the direct contribution (see
Fig. 2a) to the matrix element is

$M_{fi}^{\mathrm{dir}}=\mu_{0}\omega_{\mathrm{D}}^{2}d_{A_{i}}^{*}{G}_{ij}(\bm{r}_{\mathrm{A}},\bm{r}_{\mathrm{D}},\omega_{\mathrm{D}})d_{D_{j}}$
(30)

and for the indirect contribution (see Fig. 2b)

$M_{fi}^{\mathrm{indir}}=-\mu_{0}^{2}\omega_{\mathrm{D}}^{4}d^{*}_{A_{i}}{G}_{ij}(\bm{r}_{\mathrm{A}},\bm{r}_{\mathrm{M}},\omega_{\mathrm{D}})\\\
\times\alpha^{\mathrm{M}}_{jk}(k){G}_{kl}(\bm{r}_{\mathrm{M}},\bm{r}_{\mathrm{D}},\omega_{\mathrm{D}})d_{D_{l}}.$
(31)

Summing these contributions to find the total matrix element
$M_{fi}=M_{fi}^{\mathrm{dir}}+M_{fi}^{\mathrm{indir}}$, we calculate the rate
from Fermi’s Golden rule as

$\displaystyle\Gamma$
$\displaystyle=\sum_{f}\frac{2\pi}{\hbar}|M_{fi}|^{2}\delta(E_{i}-E_{f})$
$\displaystyle=\frac{2\pi\mu_{0}^{2}\omega_{\mathrm{D}}^{4}}{\hbar}|\bm{d}_{\mathrm{A}}^{*}\cdot\big{[}\bm{G}(\bm{r}_{\mathrm{A}},\bm{r}_{\mathrm{D}},\omega_{\mathrm{D}})$
$\displaystyle+\mu_{0}$
$\displaystyle\omega_{\mathrm{D}}^{2}\bm{G}(\bm{r}_{\mathrm{A}},\bm{r}_{\mathrm{M}},\omega_{\mathrm{D}})\cdot\bm{\alpha}^{\mathrm{M}}(k)\cdot\bm{G}(\bm{r}_{\mathrm{M}},\bm{r}_{\mathrm{D}},\omega_{\mathrm{D}})\big{]}\cdot\bm{d}_{\mathrm{D}}|^{2}.$
(32)

which reduces to the three-body ICD formula found in [21] if the Green’s
tensor is replaced by its vacuum counterpart and the transition dipole moment
of the acceptor is re-expressed in terms of an ionisation cross section.
Equation (32) is the main result of our work, describing resonant energy
transfer mediated by a third polarisable body in the presence of an arbitrary
environment. The first term of (32) describes the direct interaction between
the donor and acceptor, where the field propagates from the donor at position
$\bm{r}_{\mathrm{D}}$ and is observed at the acceptor at
$\bm{r}_{\mathrm{A}}$, therefore corresponding to the resonant interaction.
The second term describes the mediated interaction, where the field propagates
from the donor to the mediator, and then from the mediator to the acceptor. We
can see therefore that while all diagrams in Fig. 4 contribute, the result is
what one would have obtained from the pole contributions to the resonant
diagram only (as was assumed without rigorous justification in [21]), with all
the remaining diagrams serving to cancel the non-pole parts of this.

##  IV Example application; external control of a molecular dimer

Formula (32) allows calculation of the rate of mediated energy transfer in an
arbitrary external environment. As a proof-of-concept, we demonstrate the use
of the formula for the simplest inhomogeneous environment, namely a semi-
infinite half-space. We will specialise to some asymptotic distance regimes in
order to be able to write down simple Green’s tensors in position space (i.e.
without using an angular spectrum representation), verifying these against
full results at the end. However we emphasise that the formula (32) is
applicable to any external environment, and could be used to calculate
interactions within far more complex systems using numerically calculated
Green’s tensors. These could include proteins and other biological systems
[32], in which the dipole moments are often randomly oriented, necessitating
calculating the rate averaged over all possible dipole alignments. The
procedure for this is well-known (see, for example [33]), in particular it
amounts to making the replacement for the outer product of a dipole moment
with itself.

$\bm{d}_{\mathrm{A/D}}^{*}\otimes\bm{d}_{\mathrm{A/D}}\to\frac{1}{3}|\bm{d}_{\mathrm{A/D}}|^{2}\mathbb{I},$
(33)

where $\otimes$ denotes the outer product and $\mathbb{I}$ is the $3\times 3$
identity matrix. Multiplying out Eq. (32), applying this rule and again taking
advantage of Lorentz reciprocity, we find;

$\displaystyle\Gamma^{\mathrm{iso}}$
$\displaystyle=\frac{2\pi\mu_{0}^{2}\omega_{\mathrm{D}}^{4}}{9\hbar}|\bm{d}_{\mathrm{A}}|^{2}|\bm{d}_{\mathrm{D}}|^{2}$
$\displaystyle\qquad\times\text{Tr}[\bm{F}(\bm{r}_{\mathrm{A}},\bm{r}_{\mathrm{M}},\bm{r}_{\mathrm{D}})\cdot\bm{F}^{*}(\bm{r}_{\mathrm{D}},\bm{r}_{\mathrm{M}},\bm{r}_{\mathrm{A}})],$
(34)

where

$\displaystyle\bm{F}(\bm{r}_{\mathrm{A}},$
$\displaystyle\bm{r}_{\mathrm{M}},\bm{r}_{\mathrm{D}})\equiv\bm{G}(\bm{r}_{\mathrm{A}},\bm{r}_{\mathrm{D}},\omega_{\mathrm{D}})$
$\displaystyle+\mu_{0}\alpha^{\mathrm{M}}\omega_{\mathrm{D}}^{2}\bm{G}(\bm{r}_{\mathrm{A}},\bm{r}_{\mathrm{M}},\omega_{\mathrm{D}})\cdot\bm{G}(\bm{r}_{\mathrm{M}},\bm{r}_{\mathrm{D}},\omega_{\mathrm{D}}).$
(35)

and we have let $\bm{\alpha}^{\mathrm{M}}\to{\alpha}^{\mathrm{M}}\mathbb{I}$
in order to effect an isotropic mediator.

For simplicity we will initially look at a colinear system, where all three
bodies are positioned along the $z$-axis, as shown in Fig. 5. Inspired by
schemes that aim to control enhancement of light-harvesting efficiency (e.g.
[25]), we choose to look at the case where the donor and acceptor are
positioned very close to each other and also to the half-space, but with a
distant mediator. This means that we can make further simplifications, namely
that the donor and acceptor are close enough to each other and the surface
that the environment-dependent direct interaction between them is in the non-
retarded limit, where the intermolecular distance between the donor and
acceptor is significantly less that the characteristic wavelength of the
transition. The other limit we impose is that the mediator is far enough away
from the donor, acceptor and surface that the opposite, retarded limit can be
used there.

We can therefore write an approximate rate
$\Gamma^{\mathrm{iso}}_{\mathrm{CL}}$ for our colinear system as;

$\displaystyle\Gamma^{\mathrm{iso}}_{\mathrm{CL}}$
$\displaystyle=\frac{2\pi\mu_{0}^{2}\omega_{\mathrm{D}}^{4}}{9\hbar}|\bm{d}_{\mathrm{A}}|^{2}|\bm{d}_{\mathrm{D}}|^{2}$
$\displaystyle\qquad\times\text{Tr}[\bm{F}_{\mathrm{CL}}(\bm{r}_{\mathrm{A}},\bm{r}_{\mathrm{M}},\bm{r}_{\mathrm{D}})\cdot\bm{F}_{\mathrm{CL}}^{*}(\bm{r}_{\mathrm{D}},\bm{r}_{\mathrm{M}},\bm{r}_{\mathrm{A}})],$
(36)

where

$\displaystyle\bm{F}_{\mathrm{CL}}$
$\displaystyle(\bm{r}_{\mathrm{A}},\bm{r}_{\mathrm{M}},\bm{r}_{\mathrm{D}})\equiv\bm{G}_{\mathrm{NR}}(\bm{r}_{\mathrm{A}},\bm{r}_{\mathrm{D}},\omega_{\mathrm{D}})$
$\displaystyle+\mu_{0}\alpha^{\mathrm{M}}\omega_{\mathrm{D}}^{2}\bm{G}_{\mathrm{R}}(\bm{r}_{\mathrm{A}},\bm{r}_{\mathrm{M}},\omega_{\mathrm{D}})\cdot\bm{G}_{\mathrm{R}}(\bm{r}_{\mathrm{M}},\bm{r}_{\mathrm{D}},\omega_{\mathrm{D}}).$
(37)

Figure 5: Colinear system made up of three bodies and a semi-infinite
dielectric half-space. The donor and acceptor are assumed close enough
together and to the surface to apply the non-retarded (NR) limit to their
direct interaction, and the mediator is assumed far enough away from both that
the retarded (R) limit can be applied.

In order to explicitly calculate the approximate colinear rate
$\Gamma^{\mathrm{iso}}_{\mathrm{CL}}$ defined in (36), we note that Green’s
tensors in inhomogenous environments can in general be split into the sum of a
translation-invariant bulk part $\bm{G}^{(0)}$ and a ‘scattering’ part
$\bm{G}^{(1)}$, so that each of the Green’s tensors shown in (37) are;

$\bm{G}_{\mathrm{NR/R}}(\bm{r},\bm{r}^{\prime},\omega)=\bm{G}_{\mathrm{NR/R}}^{(0)}(\bm{r},\bm{r}^{\prime},\omega)+\bm{G}_{\mathrm{NR/R}}^{(1)}(\bm{r},\bm{r}^{\prime},\omega)$
(38)

For our chosen system (dielectric material in the region $z<0$, vacuum
otherwise), the bulk part of the Green’s tensor is that of the vacuum which is
known analytically in closed form (see, e.g. [23] or [30]), but due to the
physical system we have chosen we will only quote its short and long distance
forms. In the non-retarded limit (with non-coincident position arguments
$\bm{r}\neq\bm{r}^{\prime}$), the vacuum Green’s tensor reads [23]

$\bm{G}_{\mathrm{NR}}^{(0)}(\bm{r},\bm{r}^{\prime},\omega)=-\frac{c^{2}e^{i\omega\rho/c}}{4\pi\omega^{2}\rho^{3}}(\mathbb{I}-3\bm{e}_{\rho}\otimes\bm{e}_{\rho}),$
(39)

where $\bm{\rho}=\bm{r}-\bm{r}^{\prime}$, $\rho=|\bm{\rho}|$ and
$\bm{e}_{\rho}=\bm{\rho}/\rho$. In the retarded limit it becomes

$\bm{G}_{\mathrm{R}}^{(0)}(\bm{r},\bm{r}^{\prime},\omega)=-\frac{c^{2}e^{i\omega\rho/c}}{4\pi\rho}(\mathbb{I}-\bm{e}_{\rho}\otimes\bm{e}_{\rho}).$
(40)

Moving onto the scattering contribution, we note that the full expression of
the scattering Green’s tensor for a general dielectric half-space can only be
written in terms of Fourier-transformed quantities (see, e.g. [23] or [30],
necessitating one or more frequency integrals before results can be obtained.
Fortunately, due to our choice of physical situation we can again use the
short and long distance special cases, which can be written as simple
expressions in position space. The non-retarded limit of the scattering
Green’s tensor on the positive $z$ axis for an environment containing a non-
magnetic (relative permeability of unity), semi-infinite half-space in the
region $z<0$ is [34];

$\bm{G}^{(1)}_{\mathrm{NR}}(\bm{r},\bm{r}^{\prime},\omega)=\frac{c^{2}}{4\pi\omega^{2}(z+z^{\prime})^{3}}r_{\mathrm{NR}}\begin{pmatrix}1&0&0\\\
0&1&0\\\ 0&0&2\end{pmatrix}$ (41)

and the retarded limit is;

$\bm{G}^{(1)}_{\mathrm{R}}\bm{r},\bm{r}^{\prime},\omega)=\frac{e^{i(z+z^{\prime})\omega/c}}{4\pi(z+z^{\prime})}r_{\mathrm{R}}\begin{pmatrix}1&0&0\\\
0&1&0\\\ 0&0&0\end{pmatrix},$ (42)

where

$\displaystyle r_{\mathrm{NR}}$
$\displaystyle=\frac{\varepsilon(\omega)-1}{\varepsilon(\omega)+1},$
$\displaystyle
r_{\mathrm{R}}=\frac{1-\sqrt{\varepsilon(\omega)}}{1-\sqrt{\varepsilon(\omega}},$
(43)

are the reflection coefficients of the half-space for the non-retarded and
retarded limits, $z$ and $z^{\prime}$ are the $z$ components of the distances
$\bm{r}$ and $\bm{r^{\prime}}$, and $\varepsilon(\omega)$ is the frequency-
dependent relative permittivity of the half-space. Equations (36)-(42)
together constitute an analytic formula for the donor-acceptor transfer rate
in the situation shown in Fig. (5). Substituting Eqs (37)-(42) into (36) and
simplifying, we find

$\displaystyle\Gamma^{\mathrm{iso}}_{\mathrm{CL}}=\frac{\mu_{0}^{2}c^{4}|\bm{d}_{\mathrm{A}}|^{2}|\bm{d}_{\mathrm{D}}|^{2}}{18\pi\hbar}$
$\displaystyle\times\Bigg{\\{}\left[\frac{r_{\text{NR}}}{\left(z_{\text{A}}+z_{\text{D}}\right)^{3}}+\frac{1}{\left(z_{\text{A}}-z_{\text{D}}\right)^{3}}\right]^{2}+\frac{|C|^{2}}{32\pi^{2}c^{4}}\Bigg{\\}},$
(44)

where

$\displaystyle C=\,$ $\displaystyle 4\pi
c^{2}\left(\frac{1}{\left(z_{\text{A}}-z_{\text{D}}\right){}^{3}}-\frac{r_{\text{NR}}}{\left(z_{\text{A}}+z_{\text{D}}\right){}^{3}}\right)$
$\displaystyle-\Bigg{\\{}\frac{\mu_{0}\omega_{\text{D}}^{4}\alpha_{\text{M}}e^{-{i\omega_{\text{D}}\left(z_{\text{A}}+z_{\text{D}}-2z_{\text{M}}\right)}/{c}}}{\left(z_{\text{A}}-z_{\text{M}}\right)\left(z_{\text{A}}+z_{\text{M}}\right)\left(z_{\text{M}}-z_{\text{D}}\right)\left(z_{\text{D}}+z_{\text{M}}\right)}$
$\displaystyle\times\left(r_{\text{R}}\left(z_{\text{A}}-z_{\text{M}}\right)e^{{2iz_{\text{A}}\omega_{\text{D}}}/{c}}-z_{\text{A}}-z_{\text{M}}\right)$
$\displaystyle\times\left(r_{\text{R}}\left(z_{\text{M}}-z_{\text{D}}\right)e^{{2i\omega_{\text{D}}z_{\text{D}}}/{c}}+z_{\text{D}}+z_{\text{M}}\right)\Bigg{\\}}.$
(45)

with $z_{\text{D}}<z_{\text{A}}<z_{\text{M}}$, as indicated in Fig. 5. In the
limit of vanishing mediator polarisibility $\alpha_{\text{M}}$, all terms
depending on the retarded reflection coefficient $r_{\mathrm{R}}$ vanish (as
is expected from the assumption that the mediator is at a retarded distance
from all other bodies), and one is left with

$\displaystyle\Gamma_{0}$
$\displaystyle\equiv\Gamma^{\mathrm{iso}}_{\mathrm{CL}}(\alpha_{\text{M}}\to
0)=\frac{c^{4}\mu_{0}^{2}|\bm{d}_{\mathrm{A}}|^{2}|\bm{d}_{\mathrm{D}}|^{2}}{36\pi\hbar}\Bigg{[}\frac{3r_{\text{NR}}^{2}}{\left(z_{\text{A}}+z_{\text{D}}\right){}^{6}}$
$\displaystyle+\frac{2r_{\text{NR}}}{\left(z_{\text{A}}-z_{\text{D}}\right){}^{3}\left(z_{\text{A}}+z_{\text{D}}\right){}^{3}}+\frac{3}{\left(z_{\text{A}}-z_{\text{D}}\right){}^{6}}\Bigg{]}.$
(46)

To the best of our knowledge, this formula for the two-body isotropically-
averaged rate near an dielectric interface does not appear anywhere in the
literature, the closest known result being the that for oriented (non-random)
dipoles near a perfect reflector reported in Eq. (20) of [15]. In Appendix A
we demonstrate that the result in [15] is exactly reproduced by the relevant
special case of Eq. (32). Finally, taking the limit of (46) where the surface
becomes transparent ($r_{\text{NR}}\to 0$), one finds;

$\displaystyle\Gamma(\alpha_{\text{M}}\to 0,r_{\text{NR}}\to 0)=$
$\displaystyle\frac{c^{4}\mu_{0}^{2}|\bm{d}_{\mathrm{A}}|^{2}|\bm{d}_{\mathrm{D}}|^{2}}{12\pi\hbar\left(z_{\text{A}}-z_{\text{D}}\right)^{6}},$
(47)

in agreement with the well-known result for two-body resonance energy transfer
in vacuum [6]

Figure 6: Plot of rate of energy transfer (IV) against mediator position for a
system in a vacuum and a system near a half-space (gray) modelled as a perfect
reflector (corresponding to $\varepsilon\to\infty$ so that
$r_{\mathrm{NR}}=1=-r_{\mathrm{R}}$). All rates are normalised to their
isotropic two-body rate
$\Gamma_{0}=\Gamma^{\mathrm{iso}}_{\mathrm{CL}}(\alpha^{\mathrm{M}}\to 0)$
[see Eq. (46)] in their respective environments, and the mediator position is
in units of the donor transition wavelength $\lambda_{\mathrm{D}}$. The
polarizability volume $\alpha^{\mathrm{M}}/4\pi\epsilon_{0}$ of the mediator
is chosen as $0.1\lambda_{\text{D}}^{3}$. The surface is positioned at $z=0$,
the donor is at $z/\lambda_{\mathrm{D}}=0.04$ and the acceptor is at
$z/\lambda_{\mathrm{D}}=0.08$ as indicated by the blue and red vertical lines
and dictated by the imposition of the non-retarded limit in that section of
the system. In order for the retarded approximation to hold in its section of
the system, the mediator should not be brought nearer than approximately a
wavelength away from the donor, acceptor or boundary — this is indicated by
the vertical line.

Returning to our three-body, environment modified rate formula (IV), based on
Ref. [25] we are particularly interested in how changing the position of the
mediator affects the rate of energy transfer between the donor and acceptor.
Figure 6 shows how the rate of energy transfer changes as the position of the
mediator is varied along the $z$-axis in the presence of a half-space, and
compares this with a vacuum environment, both using the approximate formula
(36) and for an exact numerical calculation using the full formula (34)
(carried out using the Fourier-transformed Green’s tensors found in, for
example, appendix B of [23]). It is clear from Fig. 6 that the approximations
we applied to write down Eq. (36) work where they are expected to (mediator
significantly more than one wavelength away from donor, acceptor and surface),
but fail outside of that. It is interesting to note from Fig. 6 that for this
particular situation the effect of the mediator is actually diminished by the
presence of the half-space. In other words, when the environment contains a
half-space, adding a controllable third body will have a less of effect on the
energy transfer rate between the donor and acceptor than if no half-space were
present.

This points towards a highly non-trivial dependence of the donor-acceptor
transfer rate when accompanied by a mediator and a nearby surface. To
investigate this (and to go beyond the colinear case) we use the full form of
the Green’s tensor for an environment containing a half-space (given in [23]),
of which the retarded (42) and non-retarded (41) forms are limits. A contour
plot showing the rate for different positions of the mediator in the $x$\- and
$z$-axes is shown in Fig. 7. The plot demonstrates the intricate dependence of
the mediator’s position on the rate of energy transfer between the donor and
acceptor even in the presence of a relatively simple environment, producing
both enhancement and suppression in different regions.

Figure 7: Rate of energy transfer for a donor and acceptor near a half-space
with reflection coefficient $r_{p}=1$. The donor is fixed at position
$\\{x_{\mathrm{D}},z_{\mathrm{D}}\\}/\lambda_{\text{D}}=\\{-1,1\\}$, the
acceptor at position
$\\{x_{\mathrm{A}},z_{\mathrm{A}}\\}/\lambda_{\text{D}}=\\{1,2\\}$, while the
mediator is free to move in the $x$-$z$ pane. The other parameters and
normalisation are the same as in Fig. 6. The grey regions around donor and
acceptor indicate where the rate enhancement goes off the colour scale.

##  V Conclusions

Here we have given a formula, (32), which can be used to find the rate of
mediated energy transfer in any external environment. This was derived using
extended canonical perturbation theory beyond second order. It would have been
possible to calculate this using standard perturbation theory and considering
all time orderings, but this would have been an extremely complex and unwieldy
process. We then applied this formula to a simple system, namely three bodies
near an external semi-infinite half-space, but it could be applied to any
environment for which the Green’s tensor is known either analytically or
numerically.

The arrangement of a dimer made up of a donor and acceptor trapped near a
surface controlled by an external agent is a situation of experimental
interest (for example in [25]), and we have shown how the presence and
position of a third molecule influences the rate of energy transfer.
Furthermore, long-range transfer in photosynthetic complexes may rely on the
type of mediated RET discussed here. The formula presented here is also a
minimal model of RET in a more complex environment and could be used as a
starting point for such investigations.

The work presented here could also indicate a potential way to observe
retardation in RET. Ordinarily the donor-acceptor rate at retarded distances
is extremely small compared to the corresponding (observable) rates at smaller
distances [35]. However, adding a distant mediator to a non-retarded, surface
enhanced reaction could be a way of observing the role of retardation in RET,
without the complication of such low rates.

It is interesting to note that the form of the rate equation found, (32), is
exactly as one would anticipate from intuition about dipole moments and the
Green’s tensor. As indicated in Casimir and Polder’s 1948 paper on interatomic
potentials [36], this could likewise point towards a simpler way to obtain
fully quantum formulae of this nature. This would be the start of a powerful
method to carry out more complex many-body calculations.

##  Appendix A Consistency check with mirror-modified two-body rates

In Eq. (20) of Ref. [15], a normal-mode QED-based formula is given for the
mirror-modified two-body rate $\Gamma_{\text{trans}}(QD)$ for oriented (non-
random) quantum dots modelled as dipoles. These are taken to be at a non-
retarded distance from each other but arbitrary distance from a perfectly
reflecting mirror. Similarly, the dipole moments in Ref. [15] are taken to be
aligned with each other and with the surface of the mirror. Therefore, the
(fully) non-retarded limit of Eq. (20) in Ref. [15] should agree with with the
perfect-reflector limit of the following special case of Eq. (32);

$\displaystyle\Gamma_{xx}$
$\displaystyle=\frac{2\pi\mu_{0}^{2}\omega_{\mathrm{D}}^{4}}{\hbar}|{d}^{\mathrm{A}*}_{x}{G}^{\text{NR}}_{xx}(\bm{r}_{\mathrm{A}},\bm{r}_{\mathrm{D}},\omega_{\mathrm{D}}){d}_{x}^{\mathrm{D}}|^{2},$
(48)

with ${G}^{\text{NR}}_{xx}$ being the $xx$ component of the sum of
$\bm{G}_{\mathrm{NR}}^{(0)}$ given by Eq. (39), and
$\bm{G}_{\mathrm{NR}}^{(1)}$ given by Eq. (41). Substituting these in and
carrying out the algebra, we find;

$\displaystyle\Gamma_{xx}=$
$\displaystyle\frac{|d_{x}^{\mathrm{A}}|^{2}|d_{x}^{\mathrm{D}}|^{2}}{8\pi\hbar\epsilon_{0}^{2}}\Bigg{[}\frac{1}{\left(z_{\text{A}}-z_{\text{D}}\right){}^{6}}$
$\displaystyle-\frac{2r_{\text{NR}}}{\left(z_{\text{A}}-z_{\text{D}}\right){}^{3}\left(z_{\text{A}}+z_{\text{D}}\right){}^{3}}+\frac{r^{2}_{\text{NR}}}{\left(z_{\text{A}}+z_{\text{D}}\right){}^{6}}\Bigg{]}.$
(49)

Translating the notation of Eq. (20) in Ref. [15] to that used here (namely
$R\to z_{\mathrm{A}}-z_{\mathrm{D}}$) gives

$\displaystyle\Gamma_{\text{trans}}(QD)=\frac{|d_{x}^{\mathrm{A}}|^{2}|d_{x}^{\mathrm{D}}|^{2}}{8\pi\hbar\epsilon_{0}^{2}}\Bigg{[}\frac{1}{\left(z_{\text{A}}-z_{\text{D}}\right){}^{6}}$
$\displaystyle-\frac{2\cos\left(2kz_{\text{D}}\right)}{\left(z_{\text{A}}-z_{\text{D}}\right){}^{3}\left(z_{\text{A}}+z_{\text{D}}\right){}^{3}}+\frac{1}{\left(z_{\text{A}}+z_{\text{D}}\right){}^{6}}\Bigg{]},$
(50)

the $kz_{\mathrm{D}}\to 0$ (non-retarded) limit of which exactly reproduces
the perfect reflector ($r_{\text{NR}}\to 1$) limit of Eq. (A) above.

###### Acknowledgements.

The authors acknowledge financial support from the UK Research and Innovation
Council through grants EPSRC/DTP 2020/21/EP/T517896/1 (M.C.W) and EP/V048449/1
(R.B.)

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