# Quantum Electrodynamics of Dicke States:
Resonant One-Photon Exchange Energy and Entangled Decay Rate

U. D. Jentschura  Department of Physics and LAMOR, Missouri University of
Science and Technology, Rolla, Missouri 65409, USA  Hungarian Academy
Institute for Nuclear Physics (ATOMKI), Debrecen, Hungary  C. M. Adhikari
Department of Chemistry, Physics and Materials Science, Fayetteville State
University, Fayetteville, North Carolina 28301, USA

###### Abstract

We calculate the one-photon exchange contribution to the interatomic
interaction potential between electrically neutral, identical atoms, one of
which is assumed to be in an excited state, by matching the scattering matrix
($S$ matrix) element with the effective Hamiltonian. This approach allows us
to use covariant perturbation theory, where the two possible time orderings of
emission and absorption are summarized in a single Feynman amplitude. Our
results encompass the full retardation correction to the one-photon exchange
van-der-Waals potential. We employ the temporal gauge for the virtual photon
propagator. Based on the Feynman prescription, we obtain the imaginary part of
the interaction energy, which leads to an interaction-induced correction to
the decay rate. Our results lead to precise formulas for the distance-
dependent enhancement and suppression of the decay rates of entangled
superradiant and subradiant Dicke state, as a function of the interatomic
distance. We apply the result to an example calculation involving two hydrogen
atoms, one of which is in the ground state, the other in an excited $P$ state.

##  I Introduction

Normally, one assumes that interatomic long-range interactions (van der Waals
interactions) are associated with fluctuating dipole moments of the two
interacting atoms, which, in turn, are due to quantum fluctuations of the
electron positions in both atoms. In the dipole approximation, the interaction
Hamiltonian is equal to the scalar product of the dipole operator and the
second-quantized electric-field operator. If both atoms are in an initial
ground state (which is generally spherically symmetric), then both of them
undergo a virtual dipole transition to an excited state upon emission (atom
$A$) and absorption (atom $B$) of a virtual photon. The process is complete
upon the exchange of a second virtual photon, with the order of emission (atom
$B$) and absorption (atom $A$) reversed. The final state has both atoms in the
ground state, without any photons present (no excitations of the photon
field). The energy of the final state is equal to that of the initial state,
but energy conservation does not hold for the virtual transitions.

Because each emission and absorption of a virtual photon involves a single
power of the interaction Hamiltonian, the exchange of two photons (two
emissions and two absorptions) is a process of fourth-order perturbation
theory. The corresponding interaction energy can be calculated in both time-
ordered perturbation theory (see Ref. [1] and Chap. 5 of Ref. [2]) and by
matching the scattering amplitude to the effective Hamiltonian (see Ref. [3]
and Chap. 12 of Ref. [2]). The above situation pertains to two interacting
ground-state atoms, and the resulting expressions contain the polarizabilities
of both involved atoms. The result (for both identical as well as non-
identical ground-state atoms) smoothly interpolates between the non-retarded
short-range $1/R^{6}$ so-called van der Waals limit and the fully retarded
$1/R^{7}$ long-range limit of the interaction. The process involves the
exchange of two virtual photons (see Ref. [1] and Chap. 5 of Ref. [2]).

However, if we consider two identical atoms, one of which is in an excited
state, there can be energetically degenerate states of the two-atom system
which are connected with the initial state of the process by the exchange of a
single, not two, virtual photons. In the quantum-field theoretical picture,
this situation corresponds to a nonvanishing transition matrix elements of the
interaction Hamiltonian already in the second-order perturbation theory of the
second-quantized Hamiltonian. For example, if one of the atoms (atom $A$) is a
hydrogen atom in a $1S$ state, and the other (atom $B$) is a hydrogen atom in
a $2P$ state, then there can be a one-photon exchange process which connects
the initial state to a state where atom $A$ is in the $2P$, and atom $B$ is in
the $1S$ state. The final state is energetically fully degenerate with the
initial state of the process. This implies that the eigenstates of the
combined Hamiltonian of the individual atoms, and of the interaction, are
coherent superpositions of product states of atoms $A$ and $B$ in the
$|(1S)_{A}(2P)_{B}\rangle$, $|(2P)_{A}(1S)_{B}\rangle$ states. (An
illustrative discussion can also be found near the beginning of Chap. 7 of
Ref. [1].)

In the nonretarded approximation, this process has been treated in Refs. [4,
5, 6, 7]. One observes that the treatment in these references relies on the
use of the van-der-Waals interaction Hamiltonian

$H_{\rm
vdW}=\left(\delta_{ij}-3\hat{R}_{i}\,\hat{R}_{j}\right)\frac{d_{Ai}\,d_{Bj}}{4\pi\epsilon_{0}R^{3}}\,$
(1)

which involves the product of the dipole operators of both atoms. Here,
$\vec{R}$ is the interatomic separation vector, $\hat{R}=\vec{R}/R$ is its
unit vector, $\delta_{ij}$ is the Kronecker symbol, and $\vec{d}_{A}$ and
$\vec{d}_{B}$ are the dipole operators for the two atoms. It leads to a
nonvanishing matrix element in the energetically degenerate system in the
first order of perturbation theory. One might ask how can this be understood,
when we just said that one-photon exchange is a second-order perturbative
process. The answer is that the van der Waals Hamiltonian is derived without
field quantization, i.e., by simply expanding the electrostatic
(instantaneous) Coulomb interactions of the constituent electrons and nuclei
in the two atoms [4, 5, 6, 7], in powers of the distances of the electrons and
nuclei. This expansion does not use field quantization. A single nonretarded
Coulomb interaction is proportional to $e^{2}$, where $e$ is the electron
charge, and is thus of second order in the quantum-field theoretical picture,
where each photon emission or absorption vertex is considered to add an order
of perturbation theory. In the second-quantized picture, one uses temporal
gauge [see Eq. (9.133) of Ref. [2]] and describes the same process in second-
order perturbation theory, using two second-quantized interaction Hamiltonians
which are each proportional to the scalar product of dipole operators and
second-quantized electric field. Because the dipole operators are proportional
to $e$, the resulting interaction also is proportional to $e^{2}$. The
timelike component of the photon propagator vanishes in temporal gauge, and it
is therefore ideally suited to treat the retarded form of the van der Waals
interaction.

Implicitly, the temporal gauge actually is used in the derivation outlined in
Chap. 7 of Ref. [1], where the interaction with the radiation field is
formulated exclusively in terms of the dipole coupling term with the electric
field. A decisive difference to the derivation outlined here is that, in our
covariant approach, the consideration of two different time orderings of the
photon emission and absorption by the two atoms involved in the interaction.
Hence, by using the technique of matching the effective Hamiltonian with the
scattering matrix element, we can unify both time orderings into one single
Feynman diagram. The most interesting consequence of the use of the Feynman
prescription is the emergence of an imaginary part of the one-photon exchange
interaction, which leads to a modification of the decay rate.

The modification of the decay rate is especially interesting for  two-atom
Dicke states [8],  otherwise known as Bell states [9], which constitute
entangled states of the two-atoms system. If we denote the ground state as
$|\psi_{g}\rangle$ and the excited state as $|\psi_{e}\rangle$, then the Dicke
states are

$\displaystyle|\Psi_{\pm}\rangle=$
$\displaystyle\;\frac{1}{\sqrt{2}}\,\left[|(\psi_{e})_{A}(\psi_{g})_{B}\rangle\pm|(\psi_{g})_{A}(\psi_{e})_{B}\rangle\right]$
(2) $\displaystyle=$
$\displaystyle\;\frac{1}{\sqrt{2}}\,\left[|\psi_{e}\,\psi_{g}\rangle\pm|\psi_{g}\,\psi_{e}\rangle\right]\,.$
(3)

Let us briefly discuss the entanglement. In the basis of states

$|\psi_{1}\rangle=|\psi_{g}\rangle\,,\quad|\psi_{2}\rangle=|\psi_{e}\rangle$
(4)

one can form the following two-particle states,

$|\Phi_{1}\rangle=|\psi_{g}\,\psi_{g}\rangle\,,\quad|\Phi_{2}\rangle=|\psi_{e}\,\psi_{g}\rangle\,,\quad|\Phi_{3}\rangle=|\psi_{g}\,\psi_{e}\rangle\,,\quad|\Phi_{4}\rangle=|\psi_{e}\,\psi_{e}\rangle\,.$
(5)

Dicke states have the form

$\displaystyle|\Psi_{\pm}\rangle=$
$\displaystyle\;\frac{1}{\sqrt{2}}\,|\Phi_{2}\rangle\pm\frac{1}{\sqrt{2}}\,|\Phi_{3}\rangle=\sum_{i=1}^{2}\sum_{j=1}^{2}c^{\pm}_{ij}\;|\psi_{i}\psi_{j}\rangle\,,\qquad
c^{\pm}_{12}=\frac{1}{\sqrt{2}}\,,\qquad c^{\pm}_{21}=\pm c^{\pm}_{12}\,,$ (6)

while all $c^{\pm}_{ij}$ other than $c^{\pm}_{12}$ and $c^{\pm}_{21}$ vanish.
It is crucial to observe that these $c_{ij}$ cannot be written in the form
$c_{ij}=a_{i}\,b_{j}$, and the Dicke states are thus entangled.

As a clarifying remark, we do not consider transient phenomena connected to
the  excitation process [10, 11, 11, 12, 13, 14, 15, 16, 17], and [see also
Eqs. (5.10), (5.11) and (5.16) of Ref. [16]] and consider the entangled Dicke
states as the basis of our discussions. This paper studies the properties, not
the preparation, of the entangled Dicke states.  Following p. 200 of Ref.
[18], we remark that the concept of an intermolecular interaction energy for
the situation in which the initial state of A or B corresponds to an excited
state holds as long as the excited state or states in question are
sufficiently long lived relative to the time taken for the photon to propagate
between the two sites. The preparation of Dicke states by carefully engineered
light pulses has been discussed in Eqs. (4.21) and Eqs. (7.1), (7.2) and (7.3)
of Ref. [19]. Further considerations on suitable preparation algorithms have
been reported in Refs. [20, 21, 22]. We note that the entangled denoted here
as $|\Psi_{+}\rangle$ is known as the Bell state $|\Psi^{+}\rangle$ in the
literature on quantum computation [19, 9].

From a historical perspective, it is interesting to remark that the
possibility of resonant energy transfer between excited and ground states of
atomic and molecular systems via the exchange of resonant virtual photons has
been recognized in the early days of quantum mechanics [23, 24, 25, 26], and
summarized in reviews [27, 28, 29]. Thus, considerable effort has been
invested into the calculation of the retardation corrections to the
interaction potential given in Eq. (1) (see Refs. [30, 31, 32, 33, 34, 35, 36,
37, 29]).

The  three advantages of the second-quantized picture are that (i) it becomes
possible to study the effect of retardation, i.e., the effect of the finite
speed of light is the propagation of the interaction from atom $A$ to atom
$B$, and (ii) it is possible to obtain precise formulas for the distance-
dependent modification of the decay rate of Dicke states.  Furthermore, (iii)
the field-theoretical formalism employed here allows us to consistently
identify the position of the poles of the propagator denominators, in view of
a consistent application of the Feynman prescription [2] which is implicit in
the matching procedure employed here. Our goal is to obtain the interaction
potential between atoms, on account of retardation, as a function of the
variable $\omega R/c$, where $\omega$ is an angular frequency of a photon, $R$
is the interatomic distance, and $c$ is the speed of light.

Our calculation pertains to the exchange of one, not two, photons. In the
contrasting case of the two-photon interaction, $\omega$ is the modulus of the
frequency of either of the two virtual photons; the frequencies add up to zero
in view of energy conservation in the initial and final states [3]. Thus, the
relevant diagrams for the one-photon exchange are not those given in Fig. 5.1
of Ref. [2], not those given in Fig. 7.5 of Ref. [1], not those given in Fig.
1 of Ref. [38], but those given in Fig. 1 of Ref. [33], and Fig. 2 of Ref.
[29] (and in Fig. 1 here). Natural units with $\hbar=c=\epsilon_{0}$ are used
in the following unless stated otherwise.

##  II $\maybebm{S}$–Matrix and Effective Hamiltonian

It is useful to recall the principle of matching the $\maybebm{S}$ matrix with
the effective Hamiltonian. We consider two identical atoms $A$ and $B$ in the
initial states $|g\rangle$ and $|e\rangle$ (ground and excited),

$\psi_{A}(\vec{r}_{A})=\psi_{g}(\vec{x}_{A}-\vec{R}_{A})\,,\qquad
E_{A}=E_{g}\,,\qquad\psi_{B}(\vec{r}_{B})=\psi_{e}(\vec{x}_{B}-\vec{R}_{B})\,,\qquad
E_{B}=E_{e}\,,$ (7)

which scatter into the final states

$\psi_{A}^{\prime}(\vec{r}_{A})=\psi_{e}(\vec{x}_{A}-\vec{R}_{A})\,,\qquad
E^{\prime}_{A}=E_{e}\,,\qquad\psi_{B}^{\prime}(\vec{r}_{B})=\psi_{g}(\vec{x}_{B}-\vec{R}_{B})\,,\qquad
E^{\prime}_{B}=E_{g}\,,$ (8)

under the action of a potential $U$. The electron coordinates are
$\vec{x}_{A}$ and $\vec{x}_{B}$, and the coordinates of the nuclei are
$\vec{R}_{A}$ and $\vec{R}_{B}$. The energies of the states $|g\rangle$ and
$|e\rangle$ are assumed to differ by $\omega_{0}$, where

$\omega_{0}=|E_{A}-E_{B}|=|E^{\prime}_{A}-E^{\prime}_{B}|=|E_{e}-E_{g}|\,.$
(9)

The final state of the process has the energy of the two atomic states
interchanged and therefore the same energy as the initial state. The
corresponding Feynman diagrams are given in Fig. 1.

Figure 1:  Diagrams are given for the exchange of a virtual photon of angular
frequency $\omega_{0}$ between two identical atoms. The two different time
orderings are written out explicitly for illustration, even if, in the
language of Feynman diagrams of quantum electrodynamics [2], the diagrams are
considered to be identical.  The ground state is denoted as $|g\rangle$, the
excited state as $|e\rangle$.

For the matching, we consider the action of the potential
$U(\vec{r}_{A},\vec{r}_{B},\vec{R})$, where $\vec{R}=R_{A}-\vec{R}_{B}$ is the
interatomic distance. The corresponding (first-order) $S$-matrix element reads
as follows,

$\displaystyle S_{fi}=$
$\displaystyle\;-{\mathrm{i}}\,\int{\mathrm{d}}^{3}r_{A}\int{\mathrm{d}}^{3}r_{B}\;\psi_{A}^{\prime*}(\vec{r}_{A})\,\psi_{B}^{\prime*}(\vec{r}_{B})\,U(\vec{r}_{A},\vec{r}_{B},\vec{R})\,\psi_{A}(\vec{r}_{A})\,\psi_{B}(\vec{r}_{B})\,$
$\displaystyle\;\times\int{\mathrm{d}}t\,{\mathrm{e}}^{-{\mathrm{i}}\,(E_{A}+E_{B}-E^{\prime}_{A}-E^{\prime}_{B})\,t}$
$\displaystyle=$
$\displaystyle\;-{\mathrm{i}}\,T\,\int{\mathrm{d}}^{3}r_{A}\int{\mathrm{d}}^{3}r_{B}\;\psi_{A}^{\prime*}(\vec{r}_{A})\,\psi_{B}^{\prime*}(\vec{r}_{B})\,U(\vec{r}_{A},\vec{r}_{B},\vec{R})\,\psi_{A}(\vec{r}_{A})\,\psi_{B}(\vec{r}_{B})\,.$
(10)

We have assumed energy conservation
($E_{A}+E_{B}=E^{\prime}_{A}+E^{\prime}_{B}$) and denoted the (long) time
interval over which the transition from initial to final state occurs, as
$\int{\mathrm{d}}t=T$. In our calculations, we shall obtain the (manifestly)
non-forward scattering amplitude (the final states differ from the initial
states) in the functional form

$\displaystyle S_{fi}=$
$\displaystyle\;-{\mathrm{i}}\,\int{\mathrm{d}}^{3}r_{A}\int{\mathrm{d}}^{3}r_{B}\;{\psi^{\prime}_{A}}^{*}(\vec{r}_{A})\,{\psi^{\prime}_{B}}^{*}(\vec{r}_{B})\,S(\vec{r}_{A},\vec{r}_{B},\vec{R})\,\psi_{A}(\vec{r}_{A})\,\psi_{B}(\vec{r}_{B})$
$\displaystyle=$
$\displaystyle\;-{\mathrm{i}}\,\left<{\psi^{\prime}_{A}}\,{\psi^{\prime}_{B}}\left|S\right|\psi_{A}\,\psi_{B}\right>\,.$
(11)

The matching relation is thus

$\langle
S(\vec{r}_{A},\vec{r}_{B},\vec{R})\rangle=T\,U(\vec{r}_{A},\vec{r}_{B},\vec{R})\,,\qquad\vec{R}=\vec{R}_{A}-\vec{R}_{B}\,.$
(12)

The use of the notation $\langle S(\vec{r}_{A},\vec{r}_{B},\vec{R})\rangle$
indicates that we obtain the matching in the integrand of the scattering
matrix element. The final integration occurs over the wave functions of the
initial and final states of the two-atom system.

##  III Calculation of the Effective Hamiltonian

In time-dependent quantum electrodynamic (QED) perturbation theory, the
interaction is formulated in the interaction picture [39, 40, 2]. The second-
quantized operators in the interaction Hamiltonian have a time dependence
which is generated by the action of the free Hamiltonian (see Chap. 3 of Ref.
[2]). The designation of an interaction Hamiltonian being formulated in the
interaction picture is not redundant (see Chap. 3 of Ref. [2]). The
interaction Hamiltonian is

$V(t)=-\vec{E}(\vec{R}_{A},t)\cdot\vec{d}_{A}(t)-\vec{E}(\vec{R}_{B},t)\cdot\vec{d}_{B}(t)\,.$
(13)

Here, $\vec{d}_{k}=e\,\vec{r}_{k}=e\,(\vec{x}_{k}-\vec{R}_{k})$ with $k=A,B$
is the dipole operator for atom $k$ (for atoms with more than one electron,
one has to sum over all the electrons). The $\vec{R}_{A}$ and $\vec{R}_{B}$
are the positions of the atomic nuclei, while the $\vec{x}_{A}$ and
$\vec{x}_{B}$ denote the electron coordinates. The second-order contribution
to the $S$-matrix is

$\displaystyle\langle\phi^{\prime},0|S^{(2)}|\phi,0\rangle=$
$\displaystyle\;\frac{(-{\mathrm{i}})^{2}}{2!}\int{\mathrm{d}}t_{1}\int{\mathrm{d}}t_{2}\;\left<\phi^{\prime},0\left|{{\bf{T}}}[V(t_{1})V(t_{2}){\color[rgb]{0,0,0}]}\right|\phi,0\right>\,.$
(14)

Here, ${{\bf{T}}}$ denotes the ordering of all operators, pertaining to both
atomic dipoles and electric fields. We denote by $|0\rangle$ the “vacuum” of
the electromagnetic field. In the vacuum, there are no photons in the
radiation field. Let us also denote the initial state as
$|\phi\rangle=|\psi_{A}=\psi_{g},\,\psi_{B}=\psi_{e}\rangle$ and the final
two-atom state as
$|\phi^{\prime}\rangle=|\psi^{\prime}_{A}=\psi_{e},\,\psi^{\prime}_{B}=\psi^{\prime}_{g}\rangle$,
with the conditions given in Eqs. (7) and (8). Then,

$\displaystyle\left<\phi^{\prime},0\left|{\bf{T}}[V(t_{1})V(t_{2}){\color[rgb]{0,0,0}]}\right|\phi,0\right>=$
$\displaystyle\;\left<\phi^{\prime},0\left|{\bf{T}}\left[\left(-\vec{E}(\vec{R}_{A},t_{1})\cdot\vec{d}_{A}(t_{1})-\vec{E}(\vec{R}_{B},t_{1})\cdot\vec{d}_{B}(t_{1})\right)\,\right.\right.\right.$
$\displaystyle\;\left.\left.\left.\times\left(-\vec{E}(\vec{R}_{A},t_{2})\cdot\vec{d}_{A}(t_{2})-\vec{E}(\vec{R}_{B},t_{2})\cdot\vec{d}_{B}(t_{2})\right)\right]\right|\phi,0\right>$
$\displaystyle\sim$
$\displaystyle\;\left<\phi^{\prime},0\left|{\bf{T}}\left[\left(\vec{E}(\vec{R}_{A},t_{1})\cdot\vec{d}_{A}(t_{1})\right)\,\left(\vec{E}(\vec{R}_{B},t_{2})\cdot\vec{d}_{B}(t_{2})\right)\right]\right|\phi,0\right>$
$\displaystyle\;+\left<\phi^{\prime},0\left|{\bf{T}}\left[\left(\vec{E}(\vec{R}_{B},t_{1})\cdot\vec{d}_{B}(t_{1})\right)\,\left(\vec{E}(\vec{R}_{A},t_{2})\cdot\vec{d}_{A}(t_{2})\right)\right]\right|\phi,0\right>\,,$
(15)

where by $\sim$ we denote the omission of operators which pertain to the one-
loop self energy of the two atoms, and keep only the terms relevant for the
interaction energy. The time ordering of the electric field operators and
atomic dipole operators leads to

$\displaystyle S^{(2)}=$
$\displaystyle\;-\frac{1}{2}\int{\mathrm{d}}t_{1}\int{\mathrm{d}}t_{2}\,\left[\left<0\left|{\mathcal{T}}\left[E^{i}(\vec{R}_{A},t_{1})\,E^{j}(\vec{R}_{B},t_{2})\right]\right|0\right>\,\left<\psi_{e}\,\psi_{g}\left|{\mathrm{T}}\;d_{A}^{i}(t_{1})\,d_{B}^{j}(t_{2})\right|\psi_{g}\,\psi_{e}\right>\right.$
$\displaystyle\;-\frac{1}{2}\int{\mathrm{d}}t_{1}\int{\mathrm{d}}t_{2}\,\left[\left<0\left|{\mathcal{T}}\left[E^{i}(\vec{R}_{B},t_{1})\,E^{j}(\vec{R}_{A},t_{2})\right]\right|0\right>\,\left<\psi_{e}\,\psi_{g}\left|{\mathrm{T}}\;d_{B}^{i}(t_{1})\,d_{A}^{j}(t_{2})\right|\psi_{g}\,\psi_{e}\right>\right.\,.$
(16)

Here, ${\mathrm{T}}$ is the time ordering operator for the atomic dipole
moments, while the ${\mathcal{T}}$ operator time-orders the electric field
operators. Also, $i,j=1,2,3$ denote the Cartesian components. We use a
relativistic notation where the superscript denotes the Cartesian component.

We now recall a few known results from Ref. [3] and Chap. 12 of [2]. The time-
ordered product of electric-field operators can be evaluated as follows,

$\left<0\left|{\mathcal{T}}\left[E^{i}(\vec{R}_{A},t_{1})\,E^{j}(\vec{R}_{B},t_{2})\right]\right|0\right>={\mathrm{i}}\,\int\frac{{\mathrm{d}}\omega}{2\pi}\,\omega^{2}\,D_{F}^{ij}(\omega,\vec{r})\,{\mathrm{e}}^{-{\mathrm{i}}\omega(t_{1}-t_{2})}\,.$
(17)

Here,

$D_{F}^{ij}(\omega,\vec{R})=-\left(\delta^{ij}+\frac{\nabla^{i}\,\nabla^{j}}{\omega^{2}}\right)\,\frac{{\mathrm{e}}^{{\mathrm{i}}|\omega|R}}{4\pi
R}\,,\qquad|\omega|=\sqrt{\omega^{2}+{\mathrm{i}}\epsilon}\,,$ (18)

is the photon propagator in the mixed frequency-position representation, in
the temporal gauge [in the conventions outlined in Eq. (9.133) of Ref. [2]].
In the temporal gauge, the timelike component of the photon propagator
vanishes, and one has $D_{00}(\omega,\vec{R})=0$. We use the photon propagator
in the convention
${\mathrm{i}}D_{F}^{\mu\nu}(x-x^{\prime})=\left<0\left|{\mathcal{T}}A^{\mu}(x)\,A^{\nu}(x^{\prime})\right|0\right>$,
with $\mu,\nu=0,1,2,3$, where $A^{\mu}(x)$ is the quantized four-vector
potential propagator. The result for the spatial components of the photon
propagator  is found according to Refs. [41, 3],

$D_{F}^{ij}(\omega,\vec{R})=-\left[\alpha^{ij}+\beta^{ij}\,\left(\frac{{\mathrm{i}}}{|\omega|R}-\frac{1}{\omega^{2}\,R^{2}}\right)\right]\,\frac{{\mathrm{e}}^{{\mathrm{i}}|\omega|R}}{4\pi
R}\,,\quad\alpha^{ij}=\delta^{ij}-{\hat{R}}^{i}\,{\hat{R}}^{j}\,,\quad\beta^{ij}=\delta^{ij}-3{\hat{R}}^{i}\,{\hat{R}}^{j}\,,$
(19)

where $|\omega|=\sqrt{\omega^{2}+{\mathrm{i}}\epsilon}$, and the branch cut of
the square root is taken along the positive real axis [42, 43]. Now, let us
proceed to evaluate the time-ordered product of dipole operators,

$X^{ij}(t_{1}-t_{2})=\left<\psi_{e}\,\psi_{g}\left|{\mathrm{T}}\;d_{A}^{i}(t_{1})\,d_{A}^{j}(t_{2})\right|\psi_{g}\,\psi_{e}\right>=\left<\psi_{e}\,\psi_{g}\left|{\mathrm{T}}\;d_{A}^{i}(t_{1}-t_{2})\,d_{A}^{j}(0)\right|\psi_{g}\,\psi_{e}\right>\,.$
(20)

Introducing the Fourier transform $X^{ij}(\omega)$, we can write
$X^{ij}(t_{1}-t_{2})=\int\frac{{\mathrm{d}}\omega}{2\pi}\,{\mathrm{e}}^{-{\mathrm{i}}\,\omega\,(t_{1}-t_{2})}\,X^{ij}(\omega)$.
The Fourier transform of the time-ordered product of dipole operators can be
evaluated as follows,

$\displaystyle X^{ij}(\omega)=$
$\displaystyle\;\int_{-\infty}^{\infty}{\mathrm{d}}t\,{\mathrm{e}}^{{\mathrm{i}}\omega
t}\,X^{ij}(t)=\int_{-\infty}^{\infty}{\mathrm{d}}t\,{\mathrm{e}}^{{\mathrm{i}}\omega
t}\,\left<\psi_{e}\,\psi_{g}\left|{\mathrm{T}}\;d_{A}^{i}(t)\,d_{B}^{j}(0)\right|\psi_{g}\,\psi_{e}\right>$
$\displaystyle=$
$\displaystyle\;\int_{0}^{\infty}{\mathrm{d}}t\,{\mathrm{e}}^{{\mathrm{i}}\omega
t}\,\left<\psi_{e}\,\psi_{g}\left|d_{A}^{i}(t)\,d_{B}^{j}(0)\right|\psi_{g}\,\psi_{e}\right>+\int_{-\infty}^{0}{\mathrm{d}}t\,{\mathrm{e}}^{{\mathrm{i}}\omega
t}\,\left<\psi_{e}\,\psi_{g}\left|d_{B}^{j}(0)\,d_{A}^{i}(t)\right|\psi_{g}\,\psi_{e}\right>$
$\displaystyle=$
$\displaystyle\;\left(\int_{0}^{\infty}+\int_{-\infty}^{0}\right){\mathrm{d}}t\,{\mathrm{e}}^{{\mathrm{i}}\omega
t}\,\left<\psi_{e}\left|d_{A}^{i}(t)\right|\psi_{g}\right>\,\left<\psi_{g}\left|d_{B}^{j}(0)\right|\psi_{e}\right>\,.$
(21)

We have used the fact that the atoms are identical and undergo transitions
$|g\rangle\to|e\rangle$ and $|e\rangle\to|g\rangle$, respectively. We can thus
add the two integration domains and conclude that

$\displaystyle X^{ij}(\omega)=$
$\displaystyle\;\int_{-\infty}^{\infty}{\mathrm{d}}t\,{\mathrm{e}}^{{\mathrm{i}}\omega
t}\,\left<\psi_{e}\left|d_{A}^{i}(t)\right|\psi_{g}\right>\,\left<\psi_{g}\left|d_{B}^{j}(0)\right|\psi_{e}\right>$
$\displaystyle=$
$\displaystyle\;\int_{-\infty}^{\infty}{\mathrm{d}}t\,{\mathrm{e}}^{{\mathrm{i}}\omega
t}\,{\mathrm{e}}^{{\mathrm{i}}(E_{2}-E_{1})\,t}\,\left<\psi_{e}\left|d_{A}^{i}(0)\right|\psi_{g}\right>\,\left<\psi_{g}\left|d_{B}^{j}(0)\right|\psi_{e}\right>$
$\displaystyle=$
$\displaystyle\;2\pi\delta(E_{2}-E_{1}+\omega)\,\left<\psi_{e}\left|d_{A}^{i}(0)\right|\psi_{g}\right>\,\left<\psi_{g}\left|d_{B}^{j}(0)\right|\psi_{e}\right>\,.$
(22)

The two terms in Eq. (III) yield equivalent contributions, and we obtain, with
the help of the results obtained previously for the time-ordered products of
the electric-field and atomic dipole operators,

$\displaystyle S^{(2)}=$
$\displaystyle\;\frac{1}{2}\int{\mathrm{d}}t_{1}\int{\mathrm{d}}t_{2}\,\left<0\left|{\mathcal{T}}\left[E^{i}(\vec{R}_{A},t_{1})\,E^{j}(\vec{R}_{B},t_{2})\right]\right|0\right>\,\left<\psi_{e}\,\psi_{g}\left|{\mathrm{T}}\;d_{A}^{i}(t_{1})\,d_{B}^{j}(t_{2})\right|\psi_{g}\,\psi_{e}\right>$
$\displaystyle=$
$\displaystyle\;\int{\mathrm{d}}t_{1}\int{\mathrm{d}}t_{2}\,\left[-{\mathrm{i}}\,\int\frac{{\mathrm{d}}\omega}{2\pi}\,\omega^{2}\,D_{F}^{ij}(\omega,\vec{R})\,{\mathrm{e}}^{-{\mathrm{i}}\omega(t_{1}-t_{2})}\right]{\mathrm{e}}^{-{\mathrm{i}}\,(E_{1}-E_{2})\,(t_{1}-t_{2})}\,$
$\displaystyle\;\times\left<\psi_{e}\left|d_{A}^{i}\right|\psi_{g}\right>\,\left<\psi_{g}\left|d_{B}^{j}\right|\psi_{e}\right>$
$\displaystyle=$
$\displaystyle\;-{\mathrm{i}}\int{\mathrm{d}}(t_{1}-t_{2})\int{\mathrm{d}}t_{2}\int\frac{{\mathrm{d}}\omega}{2\pi}\omega^{2}\,D_{F}^{ij}(\omega,\vec{R}){\mathrm{e}}^{-{\mathrm{i}}(\omega+E_{1}-E_{2})\,(t_{1}-t_{2})}\left<\psi_{e}\left|d_{A}^{i}\right|\psi_{g}\right>\left<\psi_{g}\left|d_{B}^{j}\right|\psi_{e}\right>$
$\displaystyle=$
$\displaystyle\;-{\mathrm{i}}\,T\,(E_{2}-E_{1})^{2}\,D_{F}^{ij}(|E_{2}-E_{1}|,\vec{R})\,\left<\psi_{e}\,\psi_{g}\left|d_{A}^{i}\,d_{B}^{j}\right|\psi_{g}\,\psi_{e}\right>\,,$
(23)

where $d_{A}^{i}\equiv d_{A}^{i}(0)$ and $d_{Bj}\equiv d_{Bj}(0)$. Based on
the matching relation (12) and remembering Eq. (9), we read off the
interaction potential,

$\displaystyle U(\vec{r}_{A},\vec{r}_{B},\vec{R})=$
$\displaystyle\;(E_{2}-E_{1})^{2}\,D^{ij}(|E_{2}-E_{1}|,\vec{R})\,d_{A}^{i}\,d_{B}^{j}$
$\displaystyle=$
$\displaystyle\;-\omega_{0}^{2}\,\left[\alpha^{ij}+\beta^{ij}\,\left(\frac{{\mathrm{i}}}{\omega_{0}\,R}-\frac{1}{\omega_{0}^{2}\,R^{2}}\right)\right]\,\frac{{\mathrm{e}}^{{\mathrm{i}}\omega_{0}\,R}}{4\pi
R}\,d_{A}^{i}\,d_{B}^{j}$ $\displaystyle=$
$\displaystyle\;\left[\left(\delta^{ij}-3\frac{R^{i}\,R^{j}}{R^{2}}\right)\,\left(1-{\mathrm{i}}\omega_{0}R\right)-\left(\delta^{ij}-\frac{R^{i}\,R^{j}}{R^{2}}\right)\omega_{0}^{2}\,R^{2}\right]\,\frac{{\mathrm{e}}^{{\mathrm{i}}\omega_{0}\,R}}{4\pi
R^{3}}\,d_{A}^{i}\,d_{B}^{j}\,.$ (24)

This result confirms results previously obtained in Eqs. (13), (14), (40),
(41) and (42) of Ref. [31], Eqs. (12), (13), (14) of Ref. [33], Eq. (2.23) of
Ref. [34], Eqs. (4) and (5) of Ref. [35], Eq. (2.1) of Ref. [36], Eqs. (3.5),
(3.6) and (3.7) of Ref. [32], Eq. (14) of Ref. [37], and Eq. (5) of Ref. [29].
One important advantage of the method of derivation employed here is that the
imaginary part follows directly from the Feynman prescription for the
propagator denominators. In other approaches, additional considerations are
required to fix the location of the poles of the propagators in the complex
plane (for an illustrative discussion on this point, see Ref. [29]). This
potential can be separated into a real and an imaginary part,

$\displaystyle{\rm Re}[U(\vec{r}_{A},\vec{r}_{B},\vec{R})]=$
$\displaystyle\;\left[\beta^{ij}\,\left(\cos(\omega_{0}\,R)+\omega_{0}R\,\sin(\omega_{0}\,R)\right)-\alpha^{ij}\,\omega_{0}^{2}\,R^{2}\,\cos(\omega_{0}\,R)\right]\,\frac{d_{A}^{i}\,d_{B}^{j}}{4\pi
R^{3}}\,,$ (25a) $\displaystyle{\rm Im}[U(\vec{r}_{A},\vec{r}_{B},\vec{R})]=$
$\displaystyle\;\left[\beta^{ij}\,\left(\sin(\omega_{0}\,R)-\omega_{0}R\,\cos(\omega_{0}\,R)\right)-\alpha^{ij}\,\omega_{0}^{2}\,R^{2}\,\sin(\omega_{0}\,R)\right]\,\frac{d_{A}^{i}\,d_{B}^{j}}{4\pi
R^{3}}\,.$ (25b)

The real part had previously been given in Eq. (5) of Ref. [30] and in Eq.
(7.2.27) on page 149 in Chapter 7 of Ref. [1]. We have used the relations
given in Eq. (19). Using $\hat{R}=\vec{R}/R$, one has the static limit
($\omega_{0}R\to 0$),

$U(\vec{r}_{A},\vec{r}_{B},\vec{R})\to\left(\delta_{ij}-3\hat{R}_{i}\,\hat{R}_{j}\right)\frac{d_{Ai}\,d_{Bj}}{4\pi
R^{3}}\,\qquad\omega_{0}\to 0\,,$ (26)

which verifies the well-known expression for the non-retarded van-der-Waals
interaction given in Eq. (1). SI units can be restored by multiplication with
an additional overall factor $1/\epsilon_{0}$, and replacing the factor
$\omega_{0}\,R$ by the (dimensionless) factor $\omega_{0}\,R/c$.

Figure 2:  When the resonant virtual exchange photon becomes on-shell (a real
photon), a decay process is being described.  By the Cutkosky rules [44], this
is denoted by a vertical dashed line which cuts the diagram. Only one of the
diagrams in Fig. 1 contributes to the imaginary part. The internal state of
the diagram, which is cut open, has both atoms in the ground state and the
photon becoming real. Hence, it is a valid final state for the decay process
(see also Fig. 1 of Ref. [32], Sec. 2.1 of Ref. [35] and Ref. [34]). However,
the sign of the imaginary part of the exchange interaction potential given in
Eq. (25b) is oscillating. For a decay rate to be described by the imaginary
part of an energy shift, we would not expect such an oscillation; hence, a
careful interpretation is required.

##  IV Interpretation of the Imaginary Part

The imaginary part of the interatomic interaction potential given in Eq. (25b)
is oscillating in sign. It describes the one-photon resonant emission from the
decaying excited state which forms part of the entangled Dicke states given in
Eq. (2). in full analogy, the imaginary part of the one-loop self-energy of an
excited bound state in atom is naturally interpreted in terms of the decay
width of that same excited reference state [45, 46, 47]. The corresponding
Feynman diagram is given in Fig. 2. The oscillating character of the imaginary
part could be seen as a problem because the imaginary part of the resonance
energy of a decaying state is required to be negative [48, 49]; a positive
imaginary part would correspond to an anti-resonance and a negative decay
rate.  The solution is presented in the following. We can anticipate that the
negative decay rate which could otherwise naively result from the interaction
energy is compensated by the natural decay rate, to give the subradiant and
superradiant states their (nonnegative) decay rates.

One generally writes a resonance energy in terms of ${\rm
Re}\,E-{\mathrm{i}}\Gamma/2$, where $\Gamma$ is the width. Hence, it is useful
to define a decay rate operator $\widehat{\Gamma}(\xi)$, where
$\xi=\omega_{0}\,R$ is the dimensionless argument appearing in Eq. (25b). This
operator is related to the imaginary part of the exchange potential as
follows,

$\displaystyle\widehat{\Gamma}(\xi)=$ $\displaystyle\;-2\;{\rm
Im}[U(\vec{r}_{A},\vec{r}_{B},\vec{R})]\,,\qquad\xi=\omega_{0}\,R\,.$ (27)

We write the result given in Eq. (25b) somewhat differently as

$\displaystyle\widehat{\Gamma}(\xi)=$
$\displaystyle\;\frac{3}{2}\frac{\xi\,\cos(\xi)+[\xi^{2}-1]\,\sin(\xi)}{\xi^{3}}\,\frac{4\alpha}{3}\,\omega_{0}^{3}\,\left(\delta^{ij}-\frac{R^{i}\,R^{j}}{R^{2}}\right)r_{A}^{i}\,r_{B}^{j}$
$\displaystyle\;+3\frac{\sin(\xi)-\xi\,\cos(\xi)}{\xi^{3}}\frac{4\alpha}{3}\,\omega_{0}^{3}\,\frac{R^{i}\,R^{j}}{R^{2}}\,r_{A}^{i}\,r_{B}^{j}$
(28)

This expression can be conveniently written in terms of a transverse decay
operator $\widehat{\Gamma}_{\perp}(\xi)$, which describes transition whose
polarization axis is perpendicular to the interatomic distance vector
$\vec{R}$, and a longitudinal decay operator
$\widehat{\Gamma}_{\parallel}(\xi)$, which describes transition whose
polarization axis is parallel to the interatomic distance vector $\vec{R}$.
The result is

$\widehat{\Gamma}(\xi)=\widehat{\Gamma}_{\perp}(\xi)+\widehat{\Gamma}_{\parallel}(\xi)\,,\qquad\widehat{\Gamma}_{\perp}(\xi)=f_{\perp}(\xi)\,\widehat{\Gamma}_{\perp}(0)\,,\qquad\widehat{\Gamma}_{\parallel}(\xi)=f_{\parallel}(\xi)\,\widehat{\Gamma}_{\parallel}(0)\,.$
(29)

The weight functions $f_{\perp}(\xi)$ and $f_{\parallel}(\xi)$ and the
transverse and longitudinal decay operator at zero distance,
$\widehat{\Gamma}_{\perp}(0)$ and $\widehat{\Gamma}_{\parallel}(0)$, are given
as follows,

$\displaystyle\widehat{\Gamma}_{\perp}(0)=$
$\displaystyle\;\frac{4\alpha}{3}\,\omega_{0}^{3}\,\left(\delta^{ij}-\frac{R^{i}\,R^{j}}{R^{2}}\right)r_{A}^{i}\,r_{B}^{j}\,,$
(30) $\displaystyle\widehat{\Gamma}_{\parallel}(0)=$
$\displaystyle\;\frac{4\alpha}{3}\,\omega_{0}^{3}\,\frac{R^{i}\,R^{j}}{R^{2}}\,r_{A}^{i}\,r_{B}^{j}\,,$
(31) $\displaystyle\widehat{\Gamma}(0)=$
$\displaystyle\;\widehat{\Gamma}_{\perp}(0)+\widehat{\Gamma}_{\parallel}(0)=\frac{4\alpha}{3}\,\omega_{0}^{3}\,\delta^{ij}\,r_{A}^{i}\,r_{B}^{j}\,,$
(32) $\displaystyle f_{\perp}(\xi)=$
$\displaystyle\;3\frac{\xi\cos(\xi)+[\xi^{2}-1]\,\sin(\xi)}{2\xi^{3}}=1+\mathcal{O}(\xi^{2})\,,$
(33) $\displaystyle f_{\parallel}(\xi)=$
$\displaystyle\;3\,\frac{\sin(\xi)-\xi\,\cos(\xi)}{\xi^{3}}=1+\mathcal{O}(\xi^{2})\,,$
(34)

Here, $\widehat{\Gamma}(0)$ is the total decay operator at zero distance. It
fulfills the relation
$\langle\psi_{e}\,\psi_{g}|\widehat{\Gamma}(0)|\psi_{g}\,\psi_{e}\rangle=\Gamma(0)$,
where $\Gamma(0)$ (no hat symbol over the $\Gamma$) is the natural decay width
of the excited state in vacuum, which is obtained without consideration of the
atom-atom-interaction and its consequential modification of the decay rate.
One compares it with Eq. (3.37) of Ref. [2].

Let us now investigate the two Dicke states,

$\displaystyle|\Psi_{\pm}\rangle=$
$\displaystyle\;\frac{1}{\sqrt{2}}\,\left[|\psi_{e}\,\psi_{g}\rangle\pm|\psi_{g}\,\psi_{e}\rangle\right]\,.$
(35)

The expectation value of the decay rate operator can be written as follows,

$\displaystyle\langle\Psi_{\pm}|\widehat{\Gamma}(\xi)|\Psi_{\pm}\rangle=$
$\displaystyle\;\pm\langle\psi_{g}\,\psi_{e}|\widehat{\Gamma}(\xi)|\psi_{e}\,\psi_{g}\rangle$
(36) $\displaystyle=$
$\displaystyle\;\pm\left[f_{\perp}(\xi)\,\Gamma^{eg}_{\perp}(0)+f_{\parallel}(\xi)\,\Gamma^{eg}_{\parallel}(0)\right]\,.$
(37)

Here,

$\displaystyle\Gamma^{eg}_{\perp}(0)=$
$\displaystyle\;\langle\psi_{g}\,\psi_{e}|\widehat{\Gamma}_{\perp}(0)|\psi_{e}\,\psi_{g}\rangle=\frac{4\alpha}{3}\,\omega_{0}^{3}\,\left(\delta^{ij}-\frac{R^{i}\,R^{j}}{R^{2}}\right)\langle
g|r_{A}^{i}|e\rangle\,\langle e|r_{B}^{j}|g\rangle$ (38) $\displaystyle=$
$\displaystyle\;\frac{4\alpha}{3}\,\omega_{0}^{3}\,\left(\delta^{ij}-\frac{R^{i}\,R^{j}}{R^{2}}\right)\langle
g|r_{A}^{i}|e\rangle\,\langle e|r_{A}^{j}|g\rangle\,,$ (39)
$\displaystyle\Gamma^{eg}_{\parallel}(0)=$
$\displaystyle\;\langle\psi_{g}\,\psi_{e}|\widehat{\Gamma}_{\parallel}(0)|\psi_{e}\,\psi_{g}\rangle=\frac{4\alpha}{3}\,\omega_{0}^{3}\,\frac{R^{i}\,R^{j}}{R^{2}}\,\langle
g|r_{A}^{i}|e\rangle\,\langle e|r_{B}^{j}|g\rangle$ (40) $\displaystyle=$
$\displaystyle\;\frac{4\alpha}{3}\,\omega_{0}^{3}\,\frac{R^{i}\,R^{j}}{R^{2}}\,\langle
g|r_{A}^{i}|e\rangle\,\langle e|r_{A}^{j}|g\rangle\,,$ (41)
$\displaystyle\Gamma(0)=$
$\displaystyle\;\Gamma^{eg}_{\perp}(0)+\Gamma^{eg}_{\parallel}(0)=\frac{4\alpha}{3}\,\omega_{0}^{3}\,|\langle
g|\vec{r}_{A}|e\rangle|^{2}\,.$ (42)

We have repeatedly used the fact that the two atoms are identical, to replace
$r_{B}^{j}\to r_{A}^{j}$. Note that $\Gamma^{eg}_{\perp}(0)$ and
$\Gamma^{eg}_{\parallel}(0)$ can depend on the magnetic projections, but the
natural width $\Gamma(0)$ is independent of magnetic quantum numbers [2].

In the space spanned by the $|\psi_{g}\psi_{e}\rangle$ and
$|\psi_{e}\psi_{g}\rangle$, one finally obtains the following effective
Hamiltonian for the two-atoms system, which takes into account the one-photon
exchange and, with it, the imaginary part of the exchange energy,

$\mathbbm{H}=\left[E_{g}+E_{e}-\tfrac{{\mathrm{i}}}{2}\Gamma(0)\right]\,\left(|\psi_{g}\psi_{e}\rangle\,\langle\psi_{g}\psi_{e}|+|\psi_{e}\psi_{g}\rangle\,\langle\psi_{e}\psi_{g}|\right)\\\
+\left\\{|\psi_{g}\,\psi_{e}\rangle\,\left\\{{\rm
Re}\,U(\xi)-\frac{{\mathrm{i}}}{2}\left[f_{\perp}(\xi)\,\Gamma_{\perp}(0)+f_{\parallel}(\xi)\,\Gamma_{\parallel}(0)\right]\right\\}\langle\psi_{e}\,\psi_{e}|+\mbox{h.c.}\right\\}$
(43)

Here, $E_{e}={\rm Re}E_{e}$ is the real part of the energy of the excited
state, and we have supplemented the term $-{\mathrm{i}}\Gamma(0)/2$ in the
resonance energy of the unperturbed states. It means that in the basis of
states $|\psi_{g}\psi_{e}\rangle$, $|\psi_{e}\psi_{g}\rangle$, the Hamiltonian
matrix is

$\mathbbm{H}=\left(\begin{array}[]{cc}E_{0}&\delta E\\\ \delta E&E_{0}\\\
\end{array}\right)\,,\qquad
E_{0}=E_{g}+E_{e}-\frac{{\mathrm{i}}}{2}\,\Gamma(0)\,,\qquad\delta
E=E_{\gamma}-\frac{{\mathrm{i}}}{2}\,\Gamma_{\gamma}\,,$ (44)

Here,

$E_{\gamma}=\langle\psi_{g}\,\psi_{e}|{\rm
Re}\,U(\xi)|\psi_{g}\,\psi_{e}\rangle\,,\qquad\Gamma_{\gamma}=\langle\psi_{g}\,\psi_{e}|\widehat{\Gamma}(\xi)|\psi_{g}\,\psi_{e}\rangle\,,$
(45)

describe the real and imaginary parts of the one-photon exchange energy. We
write the vector representation of the Dicke states $|\Psi_{-}\rangle$ and
$|\Psi_{+}\rangle$ and $E_{-}=E_{-}(\xi)$, and $E_{+}=E_{+}(\xi)$, as follows,

$|\Psi_{\pm}\rangle=\frac{1}{\sqrt{2}}\,\left(\begin{array}[]{c}1\\\ \pm
1\end{array}\right)\,,\qquad E_{\pm}=E_{0}\pm\delta E\,.$ (46)

The resonance energies can be written as follows,

$\displaystyle E_{-}(R)=$ $\displaystyle\;{\rm
Re}\,E_{0}+E_{\gamma}-\frac{{\mathrm{i}}}{2}\left\\{\Gamma(0)-\left[f_{\perp}(\xi)\,\Gamma^{eg}_{\perp}(0)+f_{\parallel}(\xi)\,\Gamma^{eg}_{\parallel}(0)\right]\right\\}\,,$
(47) $\displaystyle E_{+}(R)=$ $\displaystyle\;{\rm
Re}\,E_{0}-E_{\gamma}-\frac{{\mathrm{i}}}{2}\left\\{\Gamma(0)+\left[f_{\perp}(\xi)\,\Gamma^{eg}_{\perp}(0)+f_{\parallel}(\xi)\,\Gamma^{eg}_{\parallel}(0)\right]\right\\}\,.$
(48)

The effective decay rates of the Dicke states are

$\displaystyle\Gamma_{-}(R)=$
$\displaystyle\;\Gamma(0)-\left[f_{\perp}(\xi)\,\Gamma^{eg}_{\perp}(0)+f_{\parallel}(\xi)\,\Gamma^{eg}_{\parallel}(0)\right]=\mathcal{O}(\xi^{2})\,,$
(49) $\displaystyle\Gamma_{+}(R)=$
$\displaystyle\;\Gamma(0)+\left[f_{\perp}(\xi)\,\Gamma^{eg}_{\perp}(0)+f_{\parallel}(\xi)\,\Gamma^{eg}_{\parallel}(0)\right]=2\Gamma(0)-\mathcal{O}(\xi^{2})\,,$
(50)

Hence, $|\Psi_{-}\rangle$ is the entangled subradiant state, while
$|\Psi_{+}\rangle$ is the superradiant state. In the short-range limit, the
subradiant state $|\Psi_{-}\rangle$ becomes metastable in view of the
entanglement, while the superradiant state $|\Psi_{+}\rangle$ acquires twice
the natural decay width.

##  V Verification of the Result

Let us include some cryptic remarks regarding a verification of the results
given in Eqs. (49) and (50) based on an alternative method. To this end, one
considers Eq. (7.2.4) of Ref. [1], but crucially takes into account that the
entangled photon emissions from the atoms $A$ and $B$ happen at different
positions. Thus, one considers the matrix element $M$,

$\displaystyle M=$
$\displaystyle\;\tfrac{1}{2}\left|\left<\psi_{g}\,\psi_{g}\left|\hat{\epsilon}\cdot\vec{r}_{A}\;{\mathrm{e}}^{{\mathrm{i}}\vec{k}\cdot\vec{R}_{A}}+\hat{\epsilon}\cdot\vec{r}_{B}\;{\mathrm{e}}^{{\mathrm{i}}\vec{k}\cdot\vec{R}_{B}}\right|\psi_{e}\,\psi_{g}\pm\psi_{g}\,\psi_{e}\right>\right|^{2}$
$\displaystyle=$
$\displaystyle\;\left|\left<\psi_{g}\left|\hat{\epsilon}\cdot\vec{r}_{A}\right|\psi_{e}\right>\right|^{2}\pm\left|\left<\psi_{g}\left|\hat{\epsilon}\cdot\vec{r}_{A}\right|\psi_{e}\right>\right|^{2}\,\cos(\vec{k}\cdot\vec{R})\,.$
(51)

where $\vec{R}=\vec{R}_{A}-\vec{R}_{B}$. In the last step, we have used the
indistinguishability of the two identical atoms. Furthermore, on resonance,
one has $|\vec{k}|=\omega_{0}$. The sum over the two photon polarizations
results in

$\sum_{\lambda}\epsilon_{\lambda}^{i}\,\epsilon_{\lambda}^{j}=\delta^{ij}-\frac{k^{i}\,k^{j}}{\vec{k}^{2}}\,,$
(52)

After some algebra, one shows the result

$\displaystyle\int{\mathrm{d}}\Omega\frac{3}{8\pi}\left(\delta^{ij}-\frac{k^{i}\,k^{j}}{\vec{k}^{2}}\right)\,\cos(\omega_{0}\hat{k}\cdot\vec{R})=\left(\delta_{ij}-\frac{R^{i}\,R^{j}}{\vec{R}^{2}}\right)\,\,f_{\perp}(\omega_{0}R)+\frac{R^{i}\,R^{j}}{\vec{R}^{2}}f_{\parallel}(\omega_{0}R)\,,$
(53)

where the previously obtained weight functions $f_{\perp}(\omega_{0}R)$ and
$f_{\parallel}(\omega_{0}R)$ [see Eqs. (33) and (34)] naturally appear. Based
on these identities and the formalism outlined in Chaps. 3 and 4 of Ref. [2],
one can verify the results given in Eqs. (49) and (50). The first term on the
right-hand side of Eq. (53) gives the term $\Gamma(0)$ in Eqs. (49) and (50),
while the  second term on the right-hand side of Eq. (53) gives the second
term on the right-hand side of Eqs. (49) and (50).

Conversely, the introduction of the Feynman prescription into the formalism
outlined in Ref. [1], which was initially designed to obtain the real part of
the resonant one-photon exchange interaction only, leads to the imaginary part
obtained here. Notes on this point are collected in the Appendix.

##  VI Example of Superradiant and Subradiant States

We would like to conclude this work with a concrete example, namely, the
hydrogen atom, where the $1S$ state is the ground state and the excited state
could be one of three $2P$ states, which we take in a Cartesian basis.
Furthermore, we assume that the interatomic distance is along the $z$ axis,

$\vec{R}=R\,\hat{\mathrm{e}}_{z}\,.$ (54)

The wave functions are well known,

$\displaystyle\psi_{g}(\vec{r})=$
$\displaystyle\;\psi_{1S}(\vec{r})=\frac{1}{\sqrt{\pi}\,a_{0}^{3/2}}\,\exp\left(-\frac{r}{a_{0}}\right)\,,\qquad
r=|\vec{r}|\,,$ (55a) $\displaystyle\psi_{e,x}(\vec{r})=$
$\displaystyle\;\psi_{2P,x}(\vec{r})=\frac{r\,\sin\theta\,\cos\varphi}{4\sqrt{2\pi}\,a_{0}^{5/2}}\,\exp\left(-\frac{r}{2a_{0}}\right)\,,$
(55b) $\displaystyle\psi_{e,y}(\vec{r})=$
$\displaystyle\;\psi_{2P,y}(\vec{r})=\frac{r\,\sin\theta\,\sin\varphi}{4\sqrt{2\pi}\,a_{0}^{5/2}}\,\exp\left(-\frac{r}{2a_{0}}\right)\,,$
(55c) $\displaystyle\psi_{e,z}(\vec{r})=$
$\displaystyle\;\psi_{2P,z}(\vec{r})=\frac{r\,\cos\theta}{4\sqrt{2\pi}\,a_{0}^{5/2}}\,\exp\left(-\frac{r}{2a_{0}}\right)\,.$
(55d)

The nonvanishing dipole transition matrix elements are as follows,

$\langle\psi_{g}|x|\psi_{e,x}\rangle=\langle\psi_{g}|y|\psi_{e,y}\rangle=\langle\psi_{g}|z|\psi_{e,z}\rangle=\frac{2^{7}}{3^{5}}\,\sqrt{2}\,a_{0}\,,$
(56)

while all other combinations vanish (e.g.,
$\langle\psi_{g}|x|\psi_{e,y}\rangle=0$). The resonance frequency is

$\omega_{0}=\frac{3}{8}\,\alpha^{2}m$ (57)

and the natural decay width is (see Ref. [50])

$\Gamma(0)=\frac{2^{8}}{3^{8}}\alpha^{5}m\,.$ (58)

The $x$ and $y$ polarized $P$ states cannot decay via a $z$-polarized
transition, while the $z$ polarized $P$ states decays exclusively via a
$z$-polarized transition. We have the results

$\Gamma_{\perp}^{0x}(0)=\Gamma_{\perp}^{0y}(0)=\Gamma_{\parallel}^{0z}(0)=\Gamma(0)\,,\qquad\Gamma_{\parallel}^{0x}(0)=\Gamma_{\parallel}^{0y}(0)=0\,,\qquad\Gamma_{\perp}^{0z}(0)=0\,.$
(59)

One can thus define the distance-dependent transverse and longitudinal decay
rates, $\Gamma_{\perp}(\xi)$ and $\Gamma_{\parallel}(\xi)$,

$\displaystyle\Gamma_{\perp}(\xi)=$
$\displaystyle\;\Gamma_{\perp}^{0x}(\xi)=\Gamma_{\perp}^{0y}(\xi)=f_{\perp}(\xi)\,\Gamma(0)\,,$
(60) $\displaystyle\Gamma_{\parallel}(\xi)=$
$\displaystyle\;\Gamma^{0z}_{\parallel}(\xi)=f_{\parallel}(\xi)\,\Gamma(0)\,,\qquad\xi=\omega_{0}\,R\,.$
(61)

At short range, the decay-rate admixtures expand as follows,

$\displaystyle\frac{\Gamma_{\perp}(R)}{\Gamma(0)}=$
$\displaystyle\;1-\frac{(\omega_{0}R)^{2}}{5}+\mathcal{O}((\omega_{0}R)^{4})\,,$
(62) $\displaystyle\frac{\Gamma_{\parallel}(R)}{\Gamma(0)}=$
$\displaystyle\;1-\frac{(\omega_{0}R)^{2}}{10}+\mathcal{O}((\omega_{0}R)^{4})\,,$
(63)

Figure 3:  The plot shows the function $\chi^{\rm T}(\xi)=1\pm
f_{\perp}(\xi)$, which is the ratio of the effective decay rate of the
superradiant and subradiant entangled Dicke states as a function of the
interatomic distance. The red curve corresponds to the positive sign
(superradiant), while the blue curve corresponds to the negative sign
(subradiant state). The decay rate of the transverse superradiant state, in
the short-range limit, assumes a value equal to twice the natural decay rate.

From the $x$ or $y$ polarized $P$ states, we have the following subradiant
entangled Dicke states,

$|\Psi^{x}_{-}\rangle=\frac{1}{\sqrt{2}}\,(|\psi_{g}\,\psi_{e,x}\rangle-|\psi_{e,x}\,\psi_{g}\rangle)\,,\qquad|\Psi^{y}_{-}\rangle=\frac{1}{\sqrt{2}}\,(|\psi_{g}\,\psi_{e,y}\rangle-|\psi_{e,y}\,\psi_{g}\rangle)\,.$
(64)

They have the following decay rates,

$\displaystyle\Gamma^{\perp}_{-}(R)=$
$\displaystyle\;\Gamma(0)-\Gamma_{\perp}(R)=\Gamma(0)\,[1-f_{\perp}(\xi)]$
(65) $\displaystyle=$
$\displaystyle\;\Gamma(0)\times\left\\{\begin{array}[]{cc}\left[\frac{(\omega_{0}R)^{2}}{5}+\mathcal{O}((\omega_{0}R)^{4})\right]&\omega_{0}\,R\to
0\\\\[9.09888pt]
\left[1-\frac{3\sin(\omega_{0}R)}{2\omega_{0}R}+\mathcal{O}\left(\frac{1}{(\omega_{0}R)^{2}}\right)\right]&\omega_{0}\,R\to\infty\\\
\end{array}\right.\,.$ (68)

In the short-range limit, the decay is suppressed, while in the long-range
limit, the natural decay width is approached, albeit with a long-range,
sinusoidal modification. For the $z$ polarized (longitudinal), subradiant
state,

$|\Psi^{\parallel}_{-}\rangle=\frac{1}{\sqrt{2}}\,(|\psi_{g}\,\psi_{e,z}\rangle-|\psi_{e,z}\,\psi_{g}\rangle)\,,$
(69)

one finds for the decay rate

$\displaystyle\Gamma^{\parallel}_{-}(R)=$
$\displaystyle\;\Gamma(0)-\Gamma_{\parallel}(R)=\Gamma(0)\,[1-f_{\parallel}(\xi)]$
(70) $\displaystyle=$
$\displaystyle\;\Gamma(0)\times\left\\{\begin{array}[]{cc}\left[\frac{(\omega_{0}R)^{2}}{10}+\mathcal{O}((\omega_{0}R)^{4})\right]&\omega_{0}\,R\to
0\\\\[9.09888pt]
\left[1+\frac{3\cos(\omega_{0}R)}{(\omega_{0}R)^{2}}+\mathcal{O}\left(\frac{1}{(\omega_{0}R)^{4}}\right)\right]&\omega_{0}\,R\to\infty\\\
\end{array}\right.\,.$ (73)

Figure 4:  The plot shows the function $\chi^{\rm P}(\xi)=1\pm
f_{\parallel}(\xi)$, where the red curve corresponds to the positive sign,
while the blue curve corresponds to the negative sign. The positive sign
corresponds to the longitudinal superradiant state, whose decay rate, in the
short-range limit, assumes a value equal to twice the natural decay rate.

From the $x$ or $y$ polarized $P$ states, we calculate the following
superradiant entangled Dicke states,

$|\Psi^{x}_{+}\rangle=\frac{1}{\sqrt{2}}\,(|\psi_{g}\,\psi_{e,x}\rangle+|\psi_{e,x}\,\psi_{g}\rangle)\,,\qquad|\Psi^{y}_{+}\rangle=\frac{1}{\sqrt{2}}\,(|\psi_{g}\,\psi_{e,y}\rangle+|\psi_{e,y}\,\psi_{g}\rangle)\,.$
(74)

One finds the decay rates,

$\displaystyle\Gamma^{\perp}_{+}(R)=$
$\displaystyle\;\Gamma(0)+\Gamma_{\perp}(R)=\Gamma(0)\,[1+f_{\perp}(\xi)]$
(75) $\displaystyle=$
$\displaystyle\;\Gamma(0)\times\left\\{\begin{array}[]{cc}\left[2-\frac{(\omega_{0}R)^{2}}{5}+\mathcal{O}(R^{4})\right]&\omega_{0}\,R\to
0\\\\[9.09888pt]
\left[1+\frac{3\sin(\omega_{0}R)}{2\omega_{0}R}+\mathcal{O}\left(\frac{1}{(\omega_{0}R)^{2}}\right)\right]&\omega_{0}\,R\to\infty\\\
\end{array}\right.\,,$ (78)

In the short-range limit, the decay rate is twice the natural width, while, in
the long-range limit, the natural width is approached, with a long-range
sinosoidal modification. One also finds the following superradiant
longitudinal entangled Dicke state,

$|\Psi^{\parallel}_{+}\rangle=\frac{1}{\sqrt{2}}\,(|\psi_{g}\,\psi_{e,z}\rangle+|\psi_{e,z}\,\psi_{g}\rangle)\,.$
(79)

The distance-dependent decay rate is

$\displaystyle\Gamma^{\parallel}_{+}(R)=$
$\displaystyle\;\Gamma(0)+\Gamma_{\parallel}(R)\Gamma(0)\,\chi_{+}^{\parallel}(\omega_{0}R)$
(80) $\displaystyle=$
$\displaystyle\;\Gamma(0)\times\left\\{\begin{array}[]{cc}\left[2-\frac{(\omega_{0}R)^{2}}{10}+\mathcal{O}((\omega_{0}R)^{4})\right]&\omega_{0}\,R\to
0\\\\[9.09888pt]
\left[1-\frac{3\cos(\omega_{0}R)}{(\omega_{0}R)^{2}}+\mathcal{O}\left(\frac{1}{(\omega_{0}R)^{3}}\right)\right]&\omega_{0}\,R\to\infty\\\
\end{array}\right.\,.$ (83)

The results are illustrated in Figs. 3 and 4.

##  VII Conclusions

We have considered retardation corrections to the one-photon exchange between
identical atoms. In Sec. II, we have considered the matching of the $S$ matrix
element and the effective Hamiltonian. Specifically, our calculation has been
based on the non-forward scattering matrix element induced by an interaction
potential of the functional form $U(\vec{r}_{A},\vec{r}_{B},\vec{R})$ which
depends on the relative coordinates $\vec{r}_{A}$ and $\vec{r}_{B}$ of the two
atoms, and the interatomic distance $\vec{R}$. For the matching to be
successful, we need to assume the initial and final states of the process to
have identical energy. This is the case if, e.g., the initial state is a
combination of one of the atoms in the ground state, and the other atom in an
excited $P$ state. The final state has the energies of the two states
reversed, with the possibility of different degeneracy indices for the initial
and final states. Because the energies of the quantum states of the individual
states have been exchanged in the initial and final states of the identical
atoms, the total energy of the final state is equal to that of the initial
state, and the matching of the non-forward $S$ matrix element to the effective
Hamiltonian can proceed.

This program is realized in Sec. III, where the calculation is realized in the
temporal gauge for the virtual photon. In this gauge, the timelike component
of the photon propagator vanishes, which implies that it is the ideal gauge
for the calculation of the retardation corrections to the van der Waals
potential given in Eq. (1). The final result is given in Eq. (III). The
retarded potential has both a real and an imaginary part. The imaginary part
is suppressed for the short-range interaction, as an expansion of Eq. (25b)
for $R\ll a_{0}/\alpha$ immediately shows. The interpretation of the imaginary
part of the resonant one-photon exchange energy is discussed in Sec. IV. In
the quantum-field theoretical picture, the imaginary part is connected with
the virtual resonant exchange photon becoming on shell, and thus, being
emitted as a real photon by the entangled two-atom system. It is found that a
completely consistent picture is obtained when one calculates the Hamiltonian
matrix including the unperturbed resonance energies of the ground and excited
states, as well as the one-photon resonant exchange energy and its imaginary
part. The modification of the decay rate of the entangled Dicke state is
consistently obtained, and the result allows us to obtain consistent formulas
for the distance-dependent decay rates of the superradiant and subradiant
Dicke states.

A possibility for an independent verification of the results based on Fermi’s
Golden Rule is sketched in Sec. V. An example calculation involving hydrogen
$1S$ and $2P$ states is given in Sec. VI, culminating in the results presented
in Fig. 3 and 4. The modification of the decay rates has a long-range, $1/R$
tail. Our calculations show that the imaginary part of the retarded one-photon
exchange as obtained by the Feynman prescription finds a completely consistent
physical interpretation in terms of the distance-dependent modification of the
decay rate of entangled superradiant and subradiant Dicke states of the
identical two-atom system.

##  VIII Appendix: Comparison with the Literature

In Chap. 7 of Ref. [1], the authors carry out a related calculation using
time-ordered perturbation theory. Specifically, in Eq. (7.2.21) of Ref. [1],
the authors obtain an expression for the retarded van der Waals interaction
which, in our notation, reads as follows,

$\displaystyle U^{\prime}(\vec{r}_{A},\vec{r}_{B},\vec{R})]=$
$\displaystyle\;\lim_{\gamma\to
0^{+}}\frac{1}{2\pi^{2}}\int_{0}^{\infty}{\mathrm{d}}p\,p^{4}\,\frac{\tau_{ij}(p\,R)}{\omega_{0}^{2}-p^{2}}\,{\mathrm{e}}^{-\gamma
p}\,,$ (84) $\displaystyle\tau_{ij}(p\,R)=$
$\displaystyle\;\alpha_{ij}\frac{\sin(p\,R)}{p\,R}+\beta_{ij}\left(\frac{\cos(p\,R)}{p^{2}\,R^{2}}-\frac{\sin(p\,R)}{p^{3}R^{3}}\right)\,.$
(85)

For clarity, we observe that $\omega_{0}$ is denoted as $k$ in Ref. [1], and
that $\epsilon_{0}$ is explicitly written out in Ref. [1]. In Chap. 7 of Ref.
[1], the authors evaluate the integral over $p$ as a principal-value integral,
using a convergent factor, and obtain the real part of our result,

$U^{\prime}(\vec{r}_{A},\vec{r}_{B},\vec{R})]={\rm
Re}[U(\vec{r}_{A},\vec{r}_{B},\vec{R})]\,.$ (86)

The Feynman prescription of quantum electrodynamics is implemented by the
substitution

$\frac{\tau_{ij}(p\,R)}{\omega_{0}^{2}-p^{2}}\to\frac{\tau_{ij}(p\,R)}{\omega_{0}^{2}-p^{2}+{\mathrm{i}}\epsilon}=\frac{\tau_{ij}(p\,R)}{(\omega_{0}+{\mathrm{i}}\epsilon-p)\,(\omega_{0}+{\mathrm{i}}\epsilon+p)}$
(87)

in Eq. (84). As a function of $p$, the integrand then obtains poles in the
complex $p$ plane at $p=\pm\sqrt{\omega_{0}^{2}+{\mathrm{i}}\epsilon}$, i.e.,
at $p=\omega_{0}+{\mathrm{i}}\epsilon$ and at
$p=-\omega_{0}-{\mathrm{i}}\epsilon$, where $\epsilon$ denotes an
infinitesimal positive imaginary part. In order to consider the contribution
from the pole, one symmetrizes the integral on the domain $-\infty<p<\infty$,
and inserts the infinitesimal imaginary part, leading to

$U^{\prime\prime}(\vec{r}_{A},\vec{r}_{B},\vec{R})]=\lim_{\epsilon\to
0^{+}}\lim_{\gamma\to
0^{+}}\frac{1}{4\pi^{2}}\int_{-\infty}^{\infty}{\mathrm{d}}p\,p^{4}\,\frac{\tau_{ij}(p\,R)}{\omega_{0}^{2}-p^{2}+{\mathrm{i}}\epsilon}\,{\mathrm{e}}^{-\gamma|p|}\,.$
(88)

The real part of $U^{\prime\prime}(\vec{r}_{A},\vec{r}_{B},\vec{R})$ can be
evaluated by principal value, confirming the result given in Eq. (86). The
imaginary part is obtained by considering the pole at
$p=\omega_{0}+{\mathrm{i}}\epsilon$ which is encircled in the mathematically
positive sense,

${\mathrm{i}}\,{\rm
Im}U^{\prime\prime}(\vec{r}_{A},\vec{r}_{B},\vec{R})]=2\pi{\mathrm{i}}\,\mathop{\rm
Res}_{p=\omega_{0}}\frac{1}{4\pi^{2}}p^{4}\,\frac{\tau_{ij}(p\,R)}{\omega_{0}^{2}-p^{2}}=-\frac{{\mathrm{i}}}{4\pi}\,\omega_{0}^{3}\,\tau_{ij}(\omega_{0}\,R)={\mathrm{i}}\,{\rm
Im}U(\vec{r}_{A},\vec{r}_{B},\vec{R})]\,,$ (89)

where the latter equality is obvious by inspection of Eqs. (85), (88) and
(25b). We conclude that, if the infinitesimal imaginary part of the photon is
supplemented in the treatment outlined in Chap. 7 of Ref. [1], then  the
result for the interaction potential $U$ given in Eq. (III) can be obtained,

$U^{\prime\prime}(\vec{r}_{A},\vec{r}_{B},\vec{R})=U(\vec{r}_{A},\vec{r}_{B},\vec{R})\,.$
(90)

We should also briefly discuss certain approximations underlying our treatment
and that discussed in Chap. 7 of Ref. [1]. First, we note the Born–Oppenheimer
approximation, which allows us to consider the motion of the electrons and
nuclei separately and corresponds to the limit of infinitely heavy atomic
nuclei (vanishing electron-nucleon mass ratio). In the same light, we neglect
the recoil energy upon photon emission, and we assume that the wave functions
of the two atoms do not overlap. Higher-order multipoles as well as multi-
photon transitions are also neglected. The same approximations underlie the
treatments given in Refs. [51, 52] for systems consisting of unlike atoms with
close excitation energies.

###### Acknowledgements.

The authors acknowledge support by the National Science Foundation (grant
PHY–2110294).

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