Tutorial
Vol. 37, No. 4 / April 2020 / Journal of the Optical Society of America B 1153
Quantum electrodynamics in modern optics and
photonics: tutorial
David L. Andrews,
1,
* David S. Bradshaw,
1
Kayn A. Forbes,
1
AND A. Salam
2,3,4
1
School of Chemistry, University of East Anglia, Norwich Research Park, Norwich NR4 7TJ, UK
2
Department of Chemistry, Wake Forest University, Winston-Salem, North Carolina 27109-7486, USA
3
Physikalische Institut, Albert-Ludwigs-Universität-Freiburg, Hermann-Herder-Strasse 3, D-79104, Freiburg, Ger many
4
Freiburg Institute for Advanced Studies (FRIAS), Albertstrasse 19, D-79104, Freiburg, Germany
*Corresponding author: d.l.andrews@uea.ac.uk
Received 15 November 2019; revised 19 February 2020; accepted 20 February 2020; posted 21 February 2020 (Doc. ID 383446);
published 19 March 2020
One of the key frameworks for developing the theory of light–matter interactions in modern optics and photonics is
quantum electrodynamics (QED). Contrasting with semiclassical theory, which depicts electromagnetic radiation
as a classical wave, QED representations of quantized light fully embrace the concept of the photon. This tutorial
review is a broad guide to cutting-edge applications of QED, providing an outline of its underlying foundation and
an examination of its role in photon science. Alongside the full quantum methods, it is shown how significant dis-
tinctions can be drawn when compared to semiclassical approaches. Clear advantages in outcome arise in the pre-
dictive capacity and physical insights afforded by QED methods, which favors its adoption over other formulations
of radiation–matter interaction. © 2020 Optical Society of America
https://doi.org/10.1364/JOSAB.383446
1. INTRODUCTION
Formulations of theory represent the foundation for describing
and interpreting all forms of optical interaction with matter. As
such, they not only represent the basis for technical and quan-
titative descriptions; they also offer frameworks for conceiving
and understanding the nature of such interactions, and their
underlying mechanisms. In a wide-ranging field of applications
in optics and photonics—including the science and technology
of optical materials, spectroscopic analysis, optical sensors, laser
frequency conversion, nanoscience, photophysics, photochem-
istry, and photobiology—one may observe that two essentially
different kinds of theory are commonly applied.
The most prevalent is semiclassical theory (SCT), a framework
in which light is treated classically, usually as a sinusoidal wave,
and matter is treated by methods based on quantum mechanics
[1]. Such a formulation formally entails classical electrodynam-
ics based on wave optics, as it has been taught for well over a
century. Sustained by its remarkably broad applicability, SCT
has found wide acceptance in the successive generations of
textbooks from which practitioners usually learn the basics, with
the tacit assumption that it is easier to grasp than full quantum
theories. Indeed, much of the theory in physical optics is almost
entirely classical, in the sense that material parameters such as
bulk optical susceptibilities are treated phenomenologically,
in terms of scalar or tensor parameters whose mathematical
constructs are not always of direct concern. However, it would
have to be acknowledged that the correct form and quantita-
tive behavior of those parameters are ultimately only derivable
from quantum mechanical representations. Of course, even
the band structures of bulk materials can be interpreted in no
other way—but these primarily concern materials, not light.
Despite its shortcomings, an SCT formulation of optics can still
be considered serviceable.
The other mainstream representation of theory for optical
interactions is quantum electrodynamics (QED) [2–10]. Here,
both matter and radiation are treated with the full rigor of quan-
tum mechanics, and as such they naturally express processes and
interactions in terms of light quanta, i.e., photons [11]. In the
sphere of optics, this cast of theory is commonly applied in a
formulation that treats space and time nonrelativistically, since
none of the salient charges move at anything approaching the
speed of light. Just as with SCT, this framework too is based on
Maxwell’s equations for the electromagnetic fields, but here,
those fields are promoted from regular variables to acquire the
status of quantum operators, which act on radiation field state
vectors. In consequence, quantum principles are consistently
applied to the entirety of each and every system, whether or not
charges or photons are present. In particular, for applications
in optics, this version of theory intrinsically subsumes both
quantum optics and quantum mechanics.
It is significant that, while the notion of photons is invariably
deployed at some stage in describing even the simplest optical
processes, such as the electronic transitions that result from
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1154 Vol. 37, No. 4 / April 2020 / Journal of the Optical Society of America B
Tutorial
absorption—even in SCT—the photon is strictly a concept
that is at odds with the assumptions of that theory. The Planck
relation between energy and frequency is clearly not a construct
that can be elicited from any classical wave representation.
Nonetheless, conventional SCT treatments of absorption and
emission afford neat, tractable introductions to the applica-
tions of time-dependent perturbation theory [12–14], while
the development of higher-order responses leads gently into
nonlinear optics [15–17]. As such, this branch of theory has
become an almost incidental assumption, frequently sustained
in the descriptions of experimental phenomena, even at the
research level.
One of the most important reasons for the sustained popu-
larity of SCT is that, when it is allowed the logically inconsistent
deployment of the Planck relation in judicious places, it often
provides results that are in correct agreement with experiment,
and with QED, to the level of experimental error. Yet SCT is
flawed at a fundamental level—not just because the world of
light and matter is, in fact, comprehensively quantum mechani-
cal, but for several other reasons. Where failures of SCT arise,
they are found not only in some obvious departure of agreement
with experiment in relatively exotic or high-precision physics,
where QED has scored some of its most celebrated successes
(calculations of the fine-structure constant, relativistic effects in
heavy atoms, and fine-structure level splitting afford significant
examples [18–20]). Despite notable attempts by Jaynes and
co-workers [21,22], it proves impossible to elicit a correct repre-
sentation of spontaneous emission [5,23], the generic term for
all familiar forms of fluorescence, phosphorescence, and lumi-
nescence. This is the most striking and ultimately unacceptable
manifestation of a fundamental flaw in SCT.
In SCT, any electronic excited state secured as a solution of
the time-independent Schrödinger equation is necessarily a
stable, stationary state: there is no system operator to act as a
perturbation and provide for state decay, unless, of course, light
of the appropriate wavelength is present, as in the special case
of stimulated emission. Thus, isolated excited atoms might
incorrectly be expected to have an infinite excited state lifetime,
precisely because they are stationary eigenstates of the atomic
Hamiltonian. Under SCT, the electric field of radiation e is zero
when no light is present, whereas in QED, a multipolar coupling
operator such as −µ · e representing an electric dipole (E1) µ
engaging with an electric field vector e is never zero, because
µ and e are both operators, one acting on the states of the matter
and the other on the radiation field. The same applies when this
dipole coupling is more formally expressed (as we do later) in
terms of the electric displacement field. Moreover, the system
Hamiltonian in QED comprises not only terms for the matter
and multipolar (or other) coupling to radiation, but also a radi-
ation Hamiltonian that is always present [5,8]. Consistent with
the finite ground state of every quantum mechanical harmonic
oscillator, the physical corollary is that vacuum fluctuations
perturb every material excited state, providing a mechanism for
decay transitions to occur.
Terms such as “vacuum fluctuations” or “vacuum field”
might, incidentally, be considered unfortunate, being suggestive
of exotic phenomena that can be identified only in regions of
space devoid of matter, whereas they are, in fact, universally
present [24]. However, they represent only one form of physical
interpretation of the relatively simple mathematics; it is not
entirely necessary to deploy such a viewpoint to attain the cor-
rect QED form of spontaneous emission. Loose terminology
is always a problem: as a counterargument, it could be said that
several terms more widely used in the semiclassical literature
can also be misleading, such as the common description of an
absorber as an “oscillator” or a “dipole,” without identifying the
key distinction between static and transition forms of oscillation
and dipole.
There are two widely held misconceptions concerning QED.
One is a common perception that the subject is substantially
more difficult and complicated to apply than SCT. In fact, it is
easy to demonstrate that, in describing common optical proc-
esses, QED is no more difficult than SCT: the latter is scarcely
any simpler even when the complexities and assumptions of
classical wave theory are hidden away, as is often the case in text-
book treatments. A second objection is that (despite its unique
success with spontaneous emission) the application of QED
is required only for high-precision calculations [25]. Both the
premise and conclusion of such logic are, of course, spurious;
the success of QED in calculating the fine-structure constant
and other such quantities, with unmatched precision, serves to
underscore its complete scientific validity.
An unwarranted and unnecessary gulf thus seems to have
become established, principally for historical reasons, between
the fully quantized formalisms of quantum optics, and the
largely semiclassical treatments of atomic and molecular inter-
actions with light. It is legitimate to pose the question of why one
should choose to deploy kinds of theory that do not correctly
reflect the quantum nature of light, for the problems and faults
of SCT presented above are not its only deficiencies. At a time
when the whole sphere of technology is being transformed by
engineered photonics (a field commonly called “quantum tech-
nologies”), and science has moved into what has already been
dubbed the “century of the photon,” the attraction of framing
theory in a formulation that duly represents the quantum nature
of electromagnetic radiation is increasingly evident. The photon
concept is clearly a requisite for understanding and correctly
representing operations in quantum technology. While SCT
often provides extremely similar results to QED for systems with
large quantum numbers, it is important to bear in mind that any
such notion of large numbers makes sense only with regard to
a specified interaction volume, a few implications of which are
discussed in Appendix A.
A comparison of QED and SCT is widely available in both
textbooks, such as Ref. [1], and the literature, including the
in-depth review article by Milonni [26]. Without heavy repeti-
tion of the earlier work, this tutorial review aims to objectively
illustrate not only applications—primarily in the sphere of
condensed phase optics and photonics—but also insights into
mechanisms afforded by a photon-based perspective. Several
recent examples are drawn from the fields of optical manipu-
lation and structured light. While the nature of the subject is
of course mathematical, and demands the language of math-
ematics, our intention is to keep equations to a minimum, in
order to focus on physical and interpretive aspects. The height-
ened predictive power of QED is also exhibited in areas where
there are evident shortcomings in theory cast in a classical or
semiclassical guise.
Tutorial
Vol. 37, No. 4 / April 2020 / Journal of the Optical Society of America B 1155
2. BASIC FORMULATIONS
To address the mathematical detail, it will be helpful to begin
with a summary of key differences in the foundational equations
of QED and SCT. The most substantial difference between
these theories is encapsulated in their distinct forms of system
Hamiltonian, exemplified in the following analysis. Here, we
shall use the term “molecule” in a generic sense, to signify any
material entity that is electrically neutral, and which has an
identifiable electronic integrity. As such, the description in
principle applies to free atoms, molecules, chromophore groups
within molecules, and even larger systems with extended quan-
tum behavior such as “quantum dots.” With care, application
can also be made to guest species in a host lattice—notably,
rare-earth dopants in crystal media with significantly different
absorption features.
First, we consider the optical interactions of any such
single “molecule.” The SCT and counterpart QED system
Hamiltonians are cast as
H
SCT
sys
= H
mol
+ H
int
; (1)
H
QED
sys
= H
mol
+ H
rad
+ H
int
. (2)
Here, in both equations, H
mol
represents the molecular
Hamiltonian, while in the second expression, H
rad
repre-
sents a Hamiltonian for the radiation field. In both cases, the full
Hamiltonian is expressible in the form H = H
0
+ H
int
, where
the interaction Hamiltonian term provides for perturbations
that allow transitions within a basis set of states, which are the
eigenstates of H
0
. Nonetheless, a difference of meaning in H
0
in each of these interpretations signifies that the corresponding
eigenstates take a significantly different form in the two theories.
In the case of SCT, the basis states are simply eigenstates of
H
mol
, expressible in the form |moli = |E
ξ
m
i for a molecule ξ
in its mth electronic state: additional labels may be inserted
within the ket, as necessary, to specify other internal degrees
of freedom. Formally, these quantum states are formulated
in a Hilbert space. However, with H
rad
included in the treat-
ment from the outset in QED theory [5,8], the unperturbed
Hamiltonian H
0
comprises the sum of H
mol
and H
rad
. This
means that the basis states employed in this construct are
separable molecule-field product states, given in general by
|moli|radi = |mol; radi = |E
ξ
m
; n(k, η)i, i.e., products of
Hilbert states for the material and Fock states for the radiation.
Here, number state representations are used to specify the
electromagnetic field, n(k, η), with n designating the number
of photons of wave vector k and polarization index η: the latter
two variables together designate a plane-wave photon mode.
While the radiation field may be described in other ways, for
example as a coherent state, the number state approach usually
enables a more direct connection to the light quanta: it also has
the advantage of being the simplest basis, in terms of which any
other can in principle be expressed. For example, variations in k
vector are accommodated in the quantum description of struc-
tured light (a point we shall return to in Section 5). Notably,
in the QED formulation, H
int
is an operator in the space of
both matter and radiation states, and therefore it is always part
of the system Hamiltonian, whereas it only features in SCT if
electromagnetic radiation is present.
By extension to an assembly of molecules, individually distin-
guished by a label ξ , the QED multipolar Hamiltonian takes a
particularly simple form, when deploying the standard, Power–
Zienau–Woolley (PZW) formulation [9,27–29],
H
QED
sys
=
X
ξ
H
mol
+ H
rad
+
X
ξ
H
int
. (3)
This equation deserves a number of comments before we go
further. It is striking that, while QED was first formulated to
tackle the interactions of fundamental charges and photons, the
subsequent development of molecular QED wrought this beau-
tiful simplicity to the interactions of larger, electrically neutral
species—interactions both with light, and among themselves.
Notably, there is no direct interaction between “molecules” in
the exact multipolar form of system Hamiltonian, as is evident
from the lack of any double-sum in the structure of Eq. (3).
In this formalism, intermolecular interactions operate only
through mediation of the electromagnetic field, which in the
quantum formulation has to mean photons (whether real or
virtual, with the latter being unobserved).
To be clear, the lack of any intermolecular terms in Eq. (3)
relating to direct Coulombic interactions formally signifies
that such static or “longitudinal” fields cancel out exactly, in
the detailed multipolar form [3]. All forms of electrodynamic
coupling between molecules are necessarily mediated by the
exchange of photons [5,8,9]. (The differences that arise in
the “minimal-coupling” formulation are well documented in
these three references and elsewhere: they are not relevant to
the comparisons with SCT, and they lie beyond the scope of the
present review.) This does not mean, however, that the effects
of static fields cannot be accommodated in the theory. In fact,
while externally applied static electric and magnetic fields can
be treated as additional zero-frequency perturbations—see,
for example, Refs. [30,31]—their effect can also be correctly
identified as the result of virtual photon coupling with static
multipoles of the corresponding electric or magnetic kind.
Adopting this form of coupling enables proper account of the
influence of local molecular dipoles and surface potentials, as for
example in a recent study of the fluorescence energy transfer at
membrane surfaces [32].
The requirements for using a molecular QED formulation
are principally that the component particles are slow-moving,
and electronically distinct. These constraints preclude only
direct application (without due modifications being imple-
mented) to particles moving at relativistic speeds, or those with
wave function overlap leading to exchange integral energies.
Somewhat misleadingly, the term “dilute gas” is sometimes
deployed to signify insignificant overlap between the wave func-
tions of component particles. In practice, the notion of distinct
electronic integrity suffices, enabling perfectly sound applica-
tions to be made to real gases, and to most liquids and solutions.
By interpreting the “molecular” Hamiltonian appropriately, it is
entirely possible to take into account both electronic and nuclear
motions, most commonly through a Born–Oppenheimer sepa-
ration of intramolecular vibrations. Hence, for example, many
1156 Vol. 37, No. 4 / April 2020 / Journal of the Optical Society of America B
Tutorial
studies of the molecular Raman effect have been pursued using
QED methods; see, for example, Refs. [5,33–38].
With limitations, molecular solids are also valid objects of
such theory: here, the constraint on direct applicability simply
means nonconductivity, and an exclusion of delocalized phonon
or polariton excitations. By a logical extension, the formulation
of molecular QED can be further applied to regular dielectric
solids, for which the elementary components may be regarded
as comprising unit cells [39,40]. Indeed, the foundations of
“classical” nonlinear optical susceptibility are based on an
exactly similar premise of dependence on atom density in most
classic texts; see, for example, Refs. [17,41,42]. Nonetheless,
there is a caveat here, for it is evident that departures from the
simplicity of a conventional molecular QED formulation
must start to be apparent whenever coupling occurs between
the intraparticle eigenstates of H
mol
and delocalized states of
the condensed phase bulk. Such coupling can engender ther-
malization and dissipation processes, some most obviously
manifest as damping [43]—a topic we shall shortly return to in
another connection.
Returning again to Eq. (3), we now focus on the middle term.
In the second quantized representation, the Hamiltonian for the
radiation field may be written as
H
rad
=
X
k,η
a
†(η)
(k)a
(η)
(k) +
1
2
~c k, (4)
and described as a collection of harmonic mode oscillators
with circular frequency ω = c k. The noncommuting oper-
ators a
(η)
(k) and a
†(η)
(k) are lowering (annihilation) and
raising (creation) operators, respectively. In the case of the
number states, which are exact eigenstates of the radiation
Hamiltonian, these operators serve to decrease or increase
by one the number of photons of mode (k, η) in the radia-
tion field, via a
(η)
(k)|n(k, η)i = n
1/2
|(n − 1)(k, η)i and
a
†
(η)
(k)|n(k, η)i = (n + 1)
1/2
|(n + 1)(k, η)i. The ordered
product a
†(η)
(k)a
(η)
(k) is called the number operator, n(k, η)
signifying an integer number of photons. For some types of laser
beam, to correctly represent a number that is subject to fluc-
tuation and aspects of photon statistics, it is expedient to employ
other forms of radiation state containing a suitable superposi-
tion, typically, the previously mentioned coherent states. From
Eq. (4), it is apparent that the entire energy of the radiation
field is identical to that of the populated subset of an infinite
set of quantum harmonic oscillators, H
rad
= (n + 1/2)~ω,
n = 0, 1, 2, . .., where the second term denotes the zero point
energy of the vacuum, meaning n = 0, i.e., an absence of
photons.
Before proceeding further, it is interesting to observe an
immediate conclusion that can be drawn from a compari-
son between the forms of Eqs. (1) and (2). The radiation
Hamiltonian, H
rad
, of Eq. (4) does not commute with H
int
,
whose leading terms have a linear dependence on the photon
creation and annihilation operators. (As we shall see, each indi-
vidual engagement of the electric or magnetic field operators
necessitates the involvement of one or other of these boson
operators.) Consequently, stationary states of H
SCT
cannot be
stationary states of the complete H
QED
. This, in a nutshell, is
why SCT ultimately fails to correctly account for spontaneous
emission, among much else. In fact, since the photon creation
operator acts on the radiation vacuum state to give a nonzero
result—as follows from the equations above with n = 0—it is
a simple matter, using methods to be detailed in the next sec-
tion, to derive a correct expression for the rate of spontaneous
emission,
h0i =
ω
3
3πε
0
~c
3
|µ
m0
|
2
. (5)
The above result, a textbook example [5], represents the rate
for an isotropic source undergoing an E1 allowed transition,
from the mth excited state down to the ground state, subject
to energy conservation E
m0
≈ ~ω, where ω = c k is the cir-
cular frequency. The result, Eq. (5), can be easily extended to
incorporate the influence of a dielectric medium [39,40].
In the next section, we outline the formal components for
a QED development of theory, providing the formal basis for
both simple and substantially more intricate forms of optical
interaction. This establishes the rigorous basis for such a devel-
opment; it will then be unnecessary to rehearse the entire basis in
each and every implementation.
3. DEVELOPING QED THEORY FOR
APPLICATIONS TO SPECIFIC OPTICAL
PROCESSES
It is not unreasonable to assert that every formulation of theory
in the realm of optics has, at its roots, one or more of Maxwell’s
equations. As observed earlier, this is just as true for SCT as
for QED; the key difference is that, in the latter, the fields
appearing in those equations are promoted to operator sta-
tus. Since Maxwell’s work represents an accepted common
ground, we shall begin the following development at a higher
level: interested readers may find the underlying develop-
ment, beginning with Maxwell’s equations, in the standard
textbooks [1,5,8,44–47].
A. Field Expansions
To elicit the mechanistic form of optical interactions, it is
necessary to represent the fundamental nature of the engage-
ment between light and matter. Once again, there is a widely
accepted representation (though, in this case, an understood
acknowledgement that it is inexact): the E1 approximation.
Physically, this may be argued on the basis that, at least for
dielectric materials, the electric field of any optical radiation
will be the dominant influence, acting upon local charge distri-
butions to produce shifts in the equilibrium centers of positive
and negative charge. Although this E1 approximation does not
have to be applied either in QED or in SCT, light–molecule
interactions mediated by an E1 are much more efficient than
those for an electric quadrupole (E2), any other higher-order
electric multipole, or even a magnetic dipole (M1)—although
both M1 and E2 forms of coupling become important in studies
on chirality [48–51]. Therefore, for ease of explanation, we
will restrict ourselves to the E1 approximation. In this case, the
interaction (PZW) Hamiltonian is expressible as [5,8]
H
int
= −ε
−1
0
µ
i
(ξ)d
⊥
i
, (6)
Tutorial
Vol. 37, No. 4 / April 2020 / Journal of the Optical Society of America B 1157
where µ
i
(ξ) is the electric-dipole moment operator. The Latin
subscript denotes a Cartesian component with an implied
summation convention for repeating indices. Moreover, d
⊥
i
is a
component of the transverse electric displacement operator for a
plane wave, expressible (in its Fourier mode expansion form) as
d
⊥
i
(
r
)
= i
X
k,η
~c kε
0
2V
1
2
×
n
e
(
η
)
i
(
k
)
a
(
η
)
(
k
)
e
i k·r
− ¯e
(
η
)
i
(
k
)
a
†
(
η
)
(
k
)
e
−i k·r
o
.
(7)
Here, e
(η)
i
(k) is the electric polarization unit vector with the
complex conjugate variable denoted by an overbar, r is an
arbitrary position in space, and V is an arbitrary quantization
volume. Here, the linear dependence on photon creation and
annihilation operators, alluded to earlier with respect to H
int
, is
clearly obvious. In passing, it is worth noting that the magnetic
induction field of the radiation field, signifying the counterpart
field in electromagnetic radiation, has an exactly analogous
expansion, also linear in a
(η)
(k) and a
†(η)
(k) [5,8,10]. In conse-
quence, every interaction of light has to involve the annihilation
or creation of photons—or both. (Where these interactions
involve virtual photons, the creation and annihilation always
occur as paired events [49]).
B. Perturbation Theory
A frequently used analytical method for tackling time evo-
lution in all but the simplest complex quantum systems is
time-dependent perturbation theory. For SCT, this appears to
be a natural consequence of partitioning the Hamiltonian into
unperturbed, H
0
, and perturbed parts, according to whether or
not light (and hence, in this interpretation, H
int
) is present. In
QED, the distinction is equally effective, but its basis is perhaps
more subtle, since none of the operators can be identified with
zero: as should be the case in any quantum theory, the operators
are not to be defined in terms of a specific state of the system.
Nonetheless, in the realm of application of both formulations,
the coupling between electromagnetic radiation and matter,
represented by the interaction Hamiltonian, H
int
, is associated
with energies that are small (due to weak fields) relative to the
Coulomb binding interactions within the molecule itself (which
arise from a much higher field strength).
Before proceeding with the detail, it is worth observing that
in high-field optics, for both SCT and QED, perturbation
theory breaks down at high levels of optical intensity. The
threshold for departures from perturbation theory is com-
monly around 10
17
Wm
−2
; at such intensities, molecular bonds
typically begin to break and electrons detach. Under these cir-
cumstances, one common approach is Floquet theory, which
provides an exact solution to the time-dependent Schrödinger
equation for a Hamiltonian that is periodic in time [52]. In an
SCT framework it has, for example, been applied to the study
of short laser pulses of high intensity interacting with atoms
[53], in particular multiphoton resonances, high-harmonic
generation, and above-threshold ionization processes. For
quantum fields, Floquet theory is less commonly employed,
but examples include application to high-order harmonic
emission and hyper-Raman spectra [54], and the dynamics of
strong field coupling [55]. For a two-level system interacting
with a single-mode quantum radiation field, the well-known
Jaynes–Cummings model illustrates the intricate convolutions
necessary to secure a result using SCT [56]. Much more recently,
there has been renewed interest in the dynamical solution to this
model, in the context of the breakdown of perturbation theory
and the rotating-wave approximation (RWA) at the onset of the
ultrastrong coupling regime [57].
Let us return to the more widely adopted perturbation
theory methods. By employing the standard techniques of
time-dependent perturbation theory, the influence of the
perturbation in causing the total system to evolve from some
initial state, |ii, to some final state, | f i, may readily be evalu-
ated. The probability amplitude for the transition | f i ← |ii is
expressible in terms of a transition operator, T, whose matrix
elements are usually written as M
f i
=
h
f |T|i
i
. The operator
T is itself expanded as a series in powers of the perturbation
operator [58,59],
T = T
(1)
+ T
(2)
+ T
(3)
+ T
(4)
+ .. . , (8)
with
T
(1)
= H
int
, (9)
T
(2)
= H
int
1
E
i
− H
0
+ iε
H
int
, (10)
and so forth, where E
i
is the energy of state |ii. Notice again
that H
0
and its eigenenergies differ according to the defini-
tion of the system in SCT and QED. The introduction of an
infinitesimal quantity ε ensures analyticity, understanding that
it is to be taken in the limit ε →
+
0. Commonly, line-shape
factors associated with damping and dissipative losses are not
included in formulations at this level, primarily because they are
associated with material components and local fields beyond the
compass of the system under study, and commonly associated
with temporally stochastic or heterogeneous forms of interac-
tion. As such, these may, for example, include coupled motion
in structural vibrations, or the electronic influence of neigh-
boring matter: only in the case of atomic gases at low pressure
can it be assumed that radiative decay properly accounts for any
experimentally determined linewidth.
It is incorrect to assume that the arbitrary infinitesimal can
simply be replaced by any all-encompassing phenomenologi-
cal damping factor: such factors can be assimilated into the
theory, but they are not in general amenable to tractable ana-
lytic expression. Any inclusion of a generic phenomenological
representation of damping must therefore be essentially a prag-
matism, for either SCT or QED. Its introduction can certainly
obscure the rigor at the core of the QED formulation, since
breaking the symmetry of time-reversal leads unavoidably to
a non-Hermitian Hamiltonian [60,61]. The implications are
indeed more easily understood from the QED perspective, since
to secure real expectation values for the radiation fields—and
molecular state energies—the corresponding quantum opera-
tors in each respect have to be Hermitian. The eigenfunctions
of such operators cannot accommodate beam dissipation or
