Molecular Quantum Electrodynamics: Developments of Principle and Progress in Applications

Abstract

Molecular quantum electrodynamics is a powerful and effective tool for the representation and elucidation of optical interactions with matter. Its history spans nearly a century of significant advances in its detailed theory and applications, and in its wider appreciation. To fully appreciate the development of the subject into its modern form invites a perspective on progressive technical progress in the theory, noting a growth in applications that closely mirrors advances in optical experimentation. The challenges and deficiencies of alternative approaches to theory are also taken into consideration.

1. Introduction

The origins of quantum electrodynamics lie in the realm of fundamental physics, and, in particular, the emergence of a new theory for the electrodynamic interactions of elementary particles. Coinage of the term ‘quantum electrodynamics’, or ‘QED’, is commonly attributed to Paul Dirac, whose work in the 1920s provided the first comprehensive, quantum-based theory to describe the interactions of light and matter, set in the framework of a relativistic field theory [1]. At much the same time, Gilbert Lewis introduced the term ‘photon’ to describe a quantum of light [2]—though the basic concept had originated in a much earlier paper by Albert Einstein [3]. Following another seminal contribution by Walter Heitler, in his 1936 treatise on the quantum theory of radiation [4], QED theory was powerfully developed during the 1940s, most notably in a series of major advances by Richard Feynman [5], Julian Schwinger [6], Shinichiro Tomonaga [7], and Freeman Dyson [8]. In that era, the community of physicists engaged in fundamental electrodynamics had their attention primarily focused on the study of elementary particles and simple atomic systems, where precision measurements could be matched to tractable—if at the time quite complicated—calculations.

The emergence of QED, thus, heralded the growth of a branch of theoretical physics whose sphere of direct relevance most apparently lay in elementary particle and atomic physics. This was, accordingly, the sphere of application with which QED quite quickly became associated. In this connection, even today, it is frequently accoladed as the most successful and stringently tested theory in modern physics. Its continued validation is exemplified by the extraordinary precision of its calculation of the fine structure constant—the most recent result agreeing with experiment to nine-figure accuracy [9]. The dawn of the laser era in 1960 also led to the framework of radiation quantization, paving the way for the rapid emergence of quantum optics as a separate discipline primarily concerned with the quantum properties of light itself [10,11,12]. Since then, the quantum theory of light is a subject that has experienced phenomenal growth, comprehensively reviewed in the classic monograph by Leonard Mandel and Emil Wolf [13].

During the couple of decades that followed the invention of the laser, rapid progress in the associated optical technology was accompanied by no less equally remarkable advances in microfabrication. Together, these advances paved the way for the birth of nanotechnology—the latter a term notably introduced as early as 1974 [14]. These parallel developments led to new spheres of application for quantum electrodynamics. One of the first was cavity QED, which, emerging from research work in the 1980s, with accelerating progress in the 1990s, in due course led to a Nobel Prize award to two of its keenest practitioners, Serge Haroche [15] and David Wineland [16]. This sphere of application exploits quantum features of emission in confined spaces, resulting from boundary conditions on the local optical modes. Subsequent achievements in the engineering of linear nanoscale optical channels led to further developments in the 21st century, and the emergence of waveguide QED [17,18]. Here, quantum features of absorption and emission are tailored for engagement with modes whose propagation is guided by waveguide boundaries. The quantization of modes for macroscopic electrodynamics is now itself a subject that is experiencing rapid progress [19,20,21].

Today, the term ‘photonics’ (first recorded, despite many other claims, in a 1952 paper by Eugen Sänger [22]) embraces many such developments, including nanoscale particle and surface engineering that generate quantum features in absorption, emission, and light scattering characteristics. Such topics lie beyond the scope of the present review, but it is worth recording how their growth has led to an increasingly broad appreciation of QED, and of the extent of its applicability in optical physics and engineering. Amongst other successes, QED methods have led to the prediction of numerous optical effects that have subsequently been identified by experiment—including optical binding, cooperative two-photon absorption, chiroptical harmonic scattering, six-wave second harmonic generation, and the chirality of twisted beams. Details of these and numerous other processes are given in the following.

2. Progress in the Development of Molecular QED

In its original application to elementary particle physics, QED theory was cast in a form specifically designed to accommodate relativistic scale motion. Most molecules, however—except under rare, low-pressure experimental conditions—are found in the condensed phase, and their translational motions can be treated as essentially classical. Even the internal electronic motions of molecules generally fall well short of the relativistic regime (the only significant exceptions arise for electrons in high atomic-number atoms). Therefore, a Lorentz non-covariant formulation of the dynamics is entirely appropriate, and a four-vector cast of theory can be eschewed. Thus, in molecular applications, a non-relativistic formulation of QED is well admissible, and its application has proved highly effective. A corollary is that spin effects, such as the distinction between electronic excited states of different spin multiplicity, or transitions between them, are not systematically built into the theory. Where necessary in specific applications, it is expedient for spin to be directly accommodated in the wavefunctions. Importantly, however, relativistic retardation and causality remain intrinsic for the electromagnetic fields—a feature that has proved highly significant in connection with intermolecular interactions.

The main foundations for molecular QED were set forth in a series of powerfully effective and impactful papers. In the first of these, a pioneering 1959 study by Edwin Power and Sigurd Zienau [23] led the way by establishing the rigorous basis for a multipolar representation of the interaction Hamiltonian. To this end, it was necessary to tackle the gauge latency of a ‘minimal coupling’ formulation, in which electromagnetic fields at a spatial position $\left(r\right)$ are cast in terms of an incompletely defined vector potential $A\left(r\right)$ and scalar potential $\varphi \left(r\right)$. A range of possible choices exists for imposing gauge conditions, all necessarily leading to identical results for experimentally observable quantities. Crucially, the relationship between different gauge formulations was rigorously established by exhibiting their connection through canonical transformation. This is equivalent to adding a total time-derivative to the system Lagrangian—representing a degree of freedom that makes no difference to the equations of motion in classical physics.

Adoption of the Coulomb gauge, in particular, imposes the condition $\mathrm{div} A\left(r\right)=0$, signifying a fully transverse character in the vector potential field. Since the electric and magnetic induction fields are related to $A\left(r\right)$ by $E\left(r\right)=-\partial /\partial tA\left(r\right)$ and $B\left(r\right)=\mathrm{curl} A\left(r\right)$. These two fields are also both transverse in the sense that their divergences both vanish. In the common case of plane waves (see Section 4), a corollary observation is that the ‘transverse’ terminology correctly indicates that the photon‘s electric and magnetic fields are orthogonally disposed with respect to the wave-vector. Notably, however, this connection in meanings is undermined in the case of structured beams, as is discussed in Section 8.6. Physically, the disappearance of longitudinal field components signifies that effects that might otherwise be treated as static interactions re-emerge through the mediating influence of transverse photons.

Thus, for example, in application to the electrostatic interaction between two neutral particles—in the absence of radiation—theory is formulated in terms of mediation by the exchange of virtual photons. These are photons that are physically unobservable and exist only through exchanges between material particles. This is indeed formally equivalent to the basis on which, in the late 1940s, Hendrik Casimir and Dirk Polder reformulated the theory for the pairwise long-range forces between neutral particles [24,25]. Since virtual photons are unobserved, quantum theory requires an unrestricted sum over radiation modes, i.e., photon wave-vectors that need not be collinear with the displacement vector connecting their positions of creation and annihilation. Hence, an interaction that is longitudinal in the sense of inter-particle direction emerges from a formulation in which the mediating photons are intrinsically transverse. The emerging long-range shift in the range dependence of the potential energy for the interaction of charge-neutral molecules exhibits the manifestation of relativistic retardation. The change from inverse sixth to inverse seventh power was fully validated by precise measurement many decades later [26].

Further seeds of molecular QED were planted in an influential 1964 treatise by Power [27] and in many of his subsequent collaborations. This lead was consolidated in a review article by Guy Woolley [28], which then served to further establish the foundations of molecular QED, focused upon the multipolar representation. In consequence, the precise QED formulation of a multipolar framework led to its distinctive form of quantum operator becoming known as the Power–Zienau–Woolley (PZW) Hamiltonian [29]

$$\begin{array}{c}{H}_{\mathrm{PZW}}={{\displaystyle \sum _{\alpha }\frac{\left|p_{\alpha }\right|}{2m_{\alpha }}}}^2-{\displaystyle \int P·E^{\perp }}d\tau -{\displaystyle \int M·B d\tau }+{\displaystyle \int {\displaystyle \int O:BB}} d\tau d{\tau }^{\prime }\\ +{\displaystyle \frac{1}{2{\epsilon }_0}}{\displaystyle \int P·P^{\perp }}d\tau +{\displaystyle \frac{{\epsilon }_0}{2}}{\displaystyle \int \left({\left|E^{\perp }\right|}^2+c^2{\left|B\right|}^2\right)}d\tau .\end{array}$$

(1)

In Equation (1), α counts the charges, ε₀ is the vacuum permittivity; dτ represents a three-dimensional volume element encompassing the system, within which the field elements are evaluated: $p_{\alpha }$ is the linear momentum of each charge of mass $m_{\alpha }$, P is the vector electric polarisation field, M is its magnetic counterpart, and O is the corresponding tensor diamagnetisation; $E^{\perp }$ and B are the transverse electric and magnetic fields, their explicit position dependence suppressed for conciseness. For vectors and second rank tensors, respectively, the dots and colons indicate inner products resulting in scalars; BB represents an outer product of the magnetic field vectors—thus representing a second rank tensor.

Consolidation of the formalism was continued in the early 1970s by several other researchers who became key players in the field, David Craig and Thiru Thirunamachandran joining Power at the forefront [30]. A connected series of papers by Power and Thirunamachandran [31,32,33,34,35] using a Heisenberg representation is especially notable for its establishment of field causality.

With key aspects of the subject firmly established [36,37], the first real molecular applications of QED began to emerge. The earliest of these represented the birth of applications that have subsequently grown into several recognizable categories. Broadly, these are: optically linear processes, i.e., absorption, emission, and scattering; chiroptical interactions; and multiphoton processes. Several of these initial applications were summarily addressed and, to a considerable extent, developed in a seminal 1984 monograph by Craig and Thirunamachandran [38], itself quite quickly recognized as a landmark development. Subsequently, these fields of application were supplemented by several others, most notably addressing optical forces, energy transfer and pooling, and nonlinear optics; Section 8 briefly reviews each of these utilizations.

In recent times, Woolley has pursued the most extensive analysis and defence of the molecular QED formulation [39,40], tackling the long-standing issue of infinities associated with self-energies. Observing that the precision of this method extends as far as the point-charge model allows, the broader implications of gauge freedom for nonrelativistic QED have also been pursued by Adam Stokes and Ahsan Nazir [41]. For non-relativistic, energy-conserving processes, the results are exact and precise. Regarding the six terms in Equation (1), while the middle four are individually arbitrary, their sum is invariant. In practice, the first term (representing the summed kinetic energy of charges in the system) and the longitudinal part of the fifth term (the potential energy of charge interactions) are subsumed into an unperturbed molecular Hamiltonian term, H_mol. (Here, and in the following, the subscript designation ‘mol’ is intended to be generic, and may be applied to any particle with electronic integrity, such as quantum dots, where associated parameters such as transition dipoles still apply [42,43]). The second, third, and fourth terms in Equation (1) represent a light-matter interaction H_int, and the sixth is an unperturbed radiation term H_rad.

It is worth re-emphasizing that the PZW Hamiltonian (1), in contrast to its minimal coupling equivalent, contains no terms representing inter-particle coupling—yet its form is exact, not the result of any approximation. As noted above, all forms of coupling between electronically distinct particles are mediated by virtual photons. Each particle exerts an electrodynamic influence in this means. Giuseppe Compagno, Roberto Passante and Francesco Persico memorably described the concept as a virtual photon cloud [44]. These virtual photons play a crucial role in the formal theory of numerous phenomena to be discussed in what follows; indeed, they are pivotal in numerous areas of nanoscale photonics [45,46].

It is also worth noting some other fundamental advances that owe their origin to developments in molecular QED theory, including a proof by canonical transformation of how ground-state dipole moments may be eliminated in the formulation of response tensors in quantum optics [47]. A powerful method for the tensor rotational averaging of observables for fluid samples is another example; so too is the adoption of irreducible Cartesian tensors for the elucidation of symmetry selection rules. Such methods have since found wide application, and not only to QED calculations. Those methods are nonetheless only peripherally relevant to the present review.

Following studies by Iwo and Zofia Białynicki-Birula [48] and a monograph by Akbar Salam with especially substantial applications of the virtual photon mechanisms [49], the publication in 2022 of a definitive textbook treatise by Woolley represents the culmination of developing foundational theory for molecular QED [50], supplementing a pragmatic tutorial on its applications in modern optics and photonics [51]. For applications to chemistry and materials science, a connection to density functional theory has also been established in Ref. [52].

3. QED Versus Semiclassical Theory

At this juncture, it is worth establishing context through quite a brief diversion into the parallel track and record of success for this most prominent alternative, semiclassical theory. In the pre-laser era, the prevalent representation of optical interactions with matter was one almost universally expressed and taught in terms of classical electrodynamics. In the rapidly evolving framework of theory for optical spectroscopy, for example, the classical wave depiction—already well enough established in solid-state optics—afforded an entirely appropriate expedient.

Condensed-phase spectroscopy with light sources of moderate intensity generally concerns samples of at least microscale dimensions, which as a whole experience a high photon flux. Bohr’s correspondence principle might have been taken to suggest that a classical wave could reasonably represent the incident light. Nonetheless, the basis for applying this supposition to spectroscopic transitions is, actually, fallacious. At any instant, any molecule undergoing transition can respond only to the time-relevant number of photons propagating within the effective span of its electron density. For conventional, incoherent light sources, quite simple calculations of the time-average statistics show that, at any moment, this number is never more than marginally above zero—i.e., under such conditions, only a relatively small fraction of molecules experience the instantaneous transit of a photon.

Despite its application strictly beyond its legitimate compass, the adoption of a classical wave representation quite quickly, nonetheless, proved remarkably effective in comparably simple consideration of absorption and scattering processes. An essentially hybrid branch of theory, thus, gained prevalence, later to be known as semiclassical theory, in which quantum principles were fully applied to matter, but not to light itself. Certainly, to properly address any form of spectroscopy, the material must be treated quantum mechanically, and there must still be a tacit assumption that quantum transitions in the matter conform to the Planck relation between optical frequency and photon energy. In standard textbook treatments, the feature that individual photons are necessarily involved in each transition is still to be taken into account. Semiclassical theory generally incorporates this feature through a ‘rotating wave approximation’ (RWA) contrivance that dispenses with spurious, rapidly oscillating terms. By such means, the theory for most of linear and nonlinear optics can be approached semiclassically, producing meaningful results and straightforward interpretation. An instructive example illustrating both QED and semiclassical approaches is afforded by Power and Bill Meath in their 1980s study on two-photon absorption [53].

From the 1970s onwards, the availability of laser methods began to transform the cutting edge of molecular spectroscopy. Even though the associated light-matter interactions are necessarily associated with quantum transitions, and these occur within intrinsically nanoscale molecular dimensions, the entrenched status and success of semiclassical theory has persisted. Indeed, it was not immediately recognized that QED could be profitably applied to molecular systems.

At that juncture, fifty years ago, the prospect of tackling the interactions of complex molecules by quantum methods still proved somewhat daunting—not only because of the prevailing intractability of calculating the electronic structures and wavefunctions, but also the additional features of complex rotational and vibrational states that need to be taken into account. This was, surely, equally a problem for detailed calculations based on semiclassical theory, where it long remained a rarity for any general analysis to be applied, through explicit wavefunction calculation, to any specific molecule. However, recognition of this complication appears to have compounded a view that quantizing the radiation field could only lead to additional, highly unwelcome levels of complexity.

Semiclassical theory certainly takes the centre ground in most of today’s textbooks and reference monographs in chemistry, physics, and their interdisciplinary subject areas. While in certain quarters, disparaging views of molecular QED have prevailed, the illogicality of treating one component of the light-matter system by classical means, and the other quantum mechanically, has not often been confronted [54]. And yet, more significantly, it has been shown that semiclassical theory is manifestly not gauge-invariant [55,56]; it is also known that it entirely fails in known cases—such as spontaneous emission, highlighted at the end of this Section.

In semiclassical theory, the system Hamiltonian is an operator that acts only upon molecular wavefunctions, generically expressible as follows:

$$H=H_{\mathrm{mol}}+H_{\mathrm{int}},$$

(2)

where H_mol is the conventional Schrödinger operator for the molecule, and H_int is a perturbation operator representing the molecular interaction with electromagnetic radiation. A key distinction of Equation (2) from the QED Hamiltonian (1) is the absence of a quantum radiation term. At simplest, for example, for a single molecule in the electric dipole (E1) approximation—the leading term in a multipole expansion—one might write

$$H_{\mathrm{int}}=-\mu ·E,$$

(3)

where μ is the dipole operator and E is the classical, oscillating electric field (the explicit time-dependence suppressed). Within the framework of time-dependent perturbation theory, this formulation serves to deliver an acceptable result for the rate of photon absorption.

However, semiclassical theory spectacularly fails to account for spontaneous emission—i.e., the mechanism responsible for all conventional fluorescence and phosphorescence processes. The reason is natural: with no radiation initially present, E = 0, and no perturbation arises: according to semiclassical theory, even excited molecules in the dark can never undergo a radiative decay. Even today, with a few notable exceptions such as the excellent and objective treatment in Ref. [57] major texts seldom draw attention to this stark failure of the semiclassical approach. Semiclassical theory also cannot account for the proven long-range (Casimir–Polder) form of all intermolecular forces.

4. Distinctive Features of Molecular QED

The excursion into semiclassical theory in Section 3 provides us with a basis to explain the core advantages of a QED methodology, before considering the more detailed technical developments and molecular applications of the latter.

As noted in Section 1, the primary virtue of QED is that it affords an internally consistent (i.e., thoroughly quantum-based) treatment of every system to which it is applied—naturally accommodating quantum optics and relativistic retardation. It is, indeed, the only framework in which the photon concept has a sound mathematical basis. Once electromagnetic radiation is treated in this way, an optical energy operator, H_rad is necessarily included in the system Hamiltonian:

$$H=H_{\mathrm{mol}}+H_{\mathrm{int}}+H_{\mathrm{rad}} ,$$

(4)

where H_rad is a volume integral of a Hamiltonian density

$$H_{\mathrm{rad}}={\displaystyle \frac{{\epsilon }_0}{2}}{\displaystyle \int \left(e^2\left(r\right)+c^2{b}^2\left(r\right)\right)} d\tau .$$

(5)

Here, $e\left(r\right)$ and $b\left(r\right)$ are operators for the electric and magnetic induction fields at vector position r, the use of lower-case symbols serving to identify local microscopic quantities. For simplicity of notation, a formulation cast in terms of the electric field is used here; technically, the electric displacement field $d\left(r\right)$ which assimilates local polarization field contributions is more exact. For media where local fields need to be explicitly taken into account, the E1 interaction Hamiltonian ought then to be cast in terms of the electric displacement field $d\left(r\right)$ rather than $e\left(r\right)$, as used in Craig and Thirunamachandran’s monograph [38]. The corresponding field expansions may nonetheless each be written in a form that differs only by a factor of the vacuum permittivity (though this does not signify that corresponding mode summands are individually the same). A more explicit representation of local field effects introduces considerable complication—see, for example, the comprehensive treatments by Gediminas Juzeliūnas [58,59]. Here, just for consistency with the broader optics literature, the discussion is adhered to the $e\left(r\right)$ formulation.

As befits the oscillatory nature of the electric and magnetic fields in optics, the quantum form of H_rad is congruent with that of a simple harmonic oscillator. Specifically, this operator is biquadratic in conjugate pairs of dynamical quantities—in this case, the non-commuting electric and magnetic field operators. In consequence, each optical field can be cast as a linear combination of non-commuting raising and lowering operators, representing creation and annihilation of a photon in each optical mode designated by wave-vector k, corresponding frequency ${\omega }_k=c\left|k\right|$, where c denotes the speed of light, and η is the polarization. Writing these operators as $a_{k,\eta }^{†}$ and $a_{k,\eta }$ respectively, H_rad can then be rewritten as

$$H_{\mathrm{rad}}={\displaystyle \sum _{k,\eta }\left(a_{k,\eta }^{†}{a}_{k,\eta }+\frac{1}{2}\right)}\hslash {\omega }_k ,$$

(6)

where ℏ is the reduced Planck constant.

The electromagnetic fields are also cast in operator form, in terms of Fourier sums over radiation modes. Any complete set suffices; commonly, a plane-wave basis is assumed, giving the following expressions:

$$e\left(r\right)=i{{\displaystyle \sum _{k,\eta }\left(\frac{\hslash {\omega }_k}{2{\epsilon }_{0}V}\right)}}^{1/2}(e_k^{\left(\eta \right)}{a}_{k,\eta }{e}^{ik.r}-{\overline{e}}_k^{\left(\eta \right)}{a}_{k,\eta }^{†}{e}^{-ik.r}) ;$$

(7)

$$b\left(r\right)=i{{\displaystyle \sum _{k,\eta }\left(\frac{\hslash {\omega }_k}{2c^2{\epsilon }_{0}V}\right)}}^{1/2}(b_k^{\left(\eta \right)}{a}_{k,\eta }{e}^{ik.r}-{\overline{b}}_k^{\left(\eta \right)}{a}_{k,\eta }^{†}{e}^{-ik.r}) ,$$

(8)

where V is the quantization volume. In this plane-wave expansion, the polarization label η designates an electric vector polarization $e_k^{\left(\eta \right)}$ (plane, circular, or elliptical) that is specifically orthogonal to the wave-vector; $b_k^{\left(\eta \right)}$ is its magnetic counterpart. The polarization sum in each of the field operators (7) and (8) can be taken over any mutually orthogonal polarization states, as, for example, may be represented by diametrically opposite points on the surface of a Poincaré sphere. It is essential to clarify that the term ‘plane wave’ here signifies a wave-vector and polarization vector that are independent of position. While this is an assumption commonly applicable to most samples subjected to well-collimated laser beams, it is an aspect whose extent of validity the discussion returns to in Section 4.

The structure of these field expansions leads to a highly significant corollary: each and every linear engagement of matter with an optical field can only release or capture a single photon. For example, only a quadratic dependence on the electric field can account for two-photon absorption processes, such as were first observed when lasers became available. Moreover, only terms associated with two annihilation operators then contribute. There are no counterparts to the terms that must be discarded through RWA arguments in semiclassical theory; it emerges that over measurable timescales, those are non-physical. The finding that optical nonlinearities such as multiphoton absorption or scattering are observed only with high-intensity laser light does, incidentally, undermine any notion that the Correspondence Principle supports treating light classically, even in linear optics. At levels of intensity corresponding to the arrival of photons singly, classical optics suggests a low but finite rate for the impossibility of two-photon absorption.

An especially significant validation of QED is its capacity to readily account for spontaneous as well as stimulated emission—the former just regarded as occurring in a system for which the initial photon number is zero. Since the electric field is now treated as an operator, rather than a dynamical variable, it cannot itself vanish—which means that the interaction operator sustains a capacity to mediate molecular transitions. Actually, it emerges that spontaneous emission owes its mechanistic origin to ‘vacuum fluctuations’—i.e., field oscillations associated with a residual zero-point energy [60].

Returning to the structure of the system Hamiltonian (4) for the widest range of applications, it is expedient for the molecular and interaction terms to both be summed over all electronically distinct units such as molecules or optical centres within the region of interaction, i.e.,

$$H=H_{\mathrm{rad}}+{\displaystyle \sum _{\xi }\left(H_{\mathrm{mol}}^{\xi }+H_{\mathrm{int}}^{\xi }\right)} ,$$

(9)

where ξ represents a label for each independent material unit. Explicitly involving the summation enables a correct distinction to be recognized between processes that are coherent—in the sense of phase-matched—and others that are incoherent. This is crucial as it determines how the observable is derived from the matrix element—see in Section 5 below. One may again, as in Section 2, note that in this PZW formulation, electronic coupling between electronically distinct components can only occur through the interaction of the molecular sub-systems with the quantized radiation field; there is no inter-particle term in the Hamiltonian (9) [30,55].

The exact summation of interaction terms, generally represented in Equation (9) as H_int, invites multipole development for its applications. The result is expressible as follows:

$$H_{\mathrm{int}}=-\mu ·e-Q:\nabla e\dots -m·b\dots +O:bb+\dots .$$

(10)

The first two contributions in Equation (10), before the first ellipsis (…), are the leading terms of an electric multipole series. When symmetry rules allow, the first of these contributions generally provides the predominant, i.e., electric dipole (E1) contribution. Its leading correction is afforded by the electric quadrupole (E2) term that couples the quadrupole operator Q with the gradient of the electric field. This term specifically represents an inner product expressible in the summation convention as ${\mathrm{Q}}_{ij}:{\nabla }_j{\mathrm{e}}_i$. The third term in Equation (10) signifies a magnetic dipole (M1) interaction coupling the corresponding operator with the magnetic field, consistent with the inclusion of E2—as just before, when symmetry allows. This term is itself the leading term of a magnetic multipole series, whose higher orders are indicated by the second ellipsis. The last term in Equation (10), involving the diamagnetization O, again the first of a series, is seldom significant—an instructive analysis by Kayn Forbes has identified how and why [61].

5. Implementation of the Quantum Matrix Element

In the sphere of molecular QED, as well as other formulations, the usual framework within which observables are calculated is time-dependent perturbation theory. For each phenomenon linking initial |I> and |F> final system states, $|F\rangle \leftarrow |I\rangle$, its application leads to an associated quantum ‘matrix element’ M_FI—essentially a quantum amplitude, whose significance is to be explained below in this Section. A full discussion of perturbation theory is beyond the remit of this review, but a few specifically QED-salient features are worth identifying. In this connection, the derivation and implementation of the quantum matrix element invite particular scrutiny.

Often, the derivations make use of graphical methods. Indeed, the progress in molecular QED has been significantly assisted by the widespread utilization of time-ordered diagrams—generally, a modified type of the kind associated with Feynman. The construction of a complete set of topologically distinct time-ordered graphs facilitates the full evaluation of M_FI. In an application involving n photon events, the form of the matrix element determined by nth order perturbation theory entails progression through (n − 1) virtual intermediate states, each expressed by summation over a complete basis set of molecular states.

During the 1990s, faced with the rapidly proliferating number of diagrams required for still relatively simple processes, a new alternative method was devised [62] and its general application was formally introduced around the turn of the millennium [63]. State-sequence diagrams represent a representational basis that essentially stands in a reciprocal relationship to time-ordered diagrams. Whereas in time-ordered diagrams, vertices represent distinct photon events, and line segments represent states, the converse applies for state-sequence diagrams: vertices signify system states, and line segments represent photon events. The key virtue of this form of representation is that all the distinct time-ordered graphs are subsumed into a single diagram.

The advantages of the state-sequence method become especially apparent in dealing with two-centre of multi-centre interactions such as optical binding and energy transfer—as shown in Section 8. Since their innovation, state-sequence diagrams have been applied to a variety of processes ranging from nonlinear optics [64,65,66] to energy harvesting [67,68].

Figure 1 illustrates the two kinds of representation for the simple case of Rayleigh scattering, which converts an incident photon of wave-vector k and polarization η to an emergent photon (k′,η′). The process is associated with the second-order term in the perturbation theory expansion (identified by the enclosed term in the formula at the top of Figure 1). The molecule begins in its ground state 0 and returns to it via an intermediate virtual state r. The system as a whole progresses from a system initial state $|I\rangle$ of energy E_I through a virtual state $|R\rangle$ of energy E_R, to a final state $|F\rangle$ having the same energy as E_I. The first and third order terms do not contribute to Rayleigh scattering.

For processes in which a system transitions from an initial to an observably different final state, the matrix element M_FI is generally deployed in a Fermi rule determination of the process rate:

$$\mathrm{\Gamma }=\frac{2\pi }{\hslash }{\left|M_{FI}\right|}^2{\rho }_F,$$

(11)

where ${\rho }_F$ is a density of final states appropriate for the specific interaction.. Thus, the modulus |M_FI| here features quadratically in the result. Later, in Section 8.3, the discussion returns to the converse, observably time-independent case, where the states nominally designated ‘initial’ and ‘final’ are identical. Here, as, for example, in the determination of intermolecular forces and potential energies, the same form of diagrammatic methodology can be deployed. Then, M_FI itself denotes the energy result. Such an application leads to particle position-dependent energies, generating potential energy surfaces whose gradients elicit forces, torques, and other kinds of mechanical effects.

For the majority of optical processes, the initial and final states of the system differ. Either the material system, individual modes of the radiation field, or both, undergo gains or losses of energy during the course of interaction. Since optical modes are characterized by both wave-vector and polarization state, this category therefore extends to formally elastic processes. For the consideration of systems comprising a relatively large number of independent material units, such as molecules or optical centres, an essential sub-classification is then necessary: an explicit distinction needs to be made between processes that are coherent, and others that are incoherent. This is crucial, as it determines how the observable is derived from the matrix element.

6. Coherence

Coherence, in the present sense, signifies a process in which each unit generates a phase-identical matrix element. An example is forward-emission second harmonic generation (SHG), in which the phase associated with each pairwise annihilation of pump photons is precisely compensated by the phase in the harmonic photon emission. For such optically parametric processes (those in which neither the material or the radiation field experiences an overall change in either energy or linear momentum), the matrix element values for differently situated molecules have a common phase. As such, the M_FI results are directly additive, and there is fully constructive interference between all the associated signals. This leads to a quadratic dependence of the rate on the number of particles—a significant determinant of efficiency for all such harmonic generation processes in condensed phase materials.

Incoherent processes, however, are those in which the radiation field gains or loses energy (such as emission or absorption) and/or linear momentum (such as non-forward Rayleigh scattering). Most spectroscopic studies are based on interactions of this kind. For these processes, individual molecules participating in different spatial locations are associated with different optical phases, such that there is an overall destructive interference between their associated quantum amplitudes. This leads to quite a simple additivity in the associated |M_FI|² values, and in consequence, the observed signal has a linear dependence on the number of participant molecules.

It is worth recording this crucial difference between coherent and incoherent signals. An unexpectedly common error is an accidental conflation of the axial frames of reference for material particles and the optical field. Optical frequency doubling affords a particularly straightforward example. Wave-vector matched SHG may occur with high efficiency as a result of its intrinsic coherence, whereas a harmonic emitted in any other direction (hyper-Rayleigh scattering) generates only quite a weak signal, due to fundamental incoherence. Nonetheless, in isotropic systems, the result of implementing an ensemble average is that SHG proves to be forbidden, whereas the weak hyper-Rayleigh effect is not. Both processes, to emphasize, are associated with the same matrix element construct, and hence the scope for potential confusion. The detailed arguments, whose origins can be traced back to an early QED treatise by Dietrich Marcuse [37], were first fully detailed in the mid-1990s [69].

Before continuing, it is also worth noting that, across the realm of optics, coherence is also a term with a multitude of other, context-specific meanings. It is therefore worth observing that it is only in a quantum-based description of light that features of an optical input such as squeezing, photon statistics, or the influence of number-phase uncertainty, are also intrinsic elements of the theory. Such is indeed the case with QED. A recent example highlights the role of quantum coherence in condensed-phase energy transfer, associated with the interference of virtual photon pathways, also leading to a form of damping, which is considered in Section 7 just below [70].

7. Resonance and Dissipative Systems

The classical theory of optics and electrodynamics generally introduces damping terms to represent dissipative losses arising at resonance, and the observed line-broadening under near-resonance conditions. For low-pressure atomic gases, where radiative emission is the key path to deactivation, it is indeed possible to analytically account for such effects. But the multiplicity of loss and line-broadening mechanisms that arise in molecules and condensed-phase matter rule out any analytically exact formulation. A wide variety of distinct factors contributes to spectral damping, including intramolecular energy redistribution within a manifold of vibrational states.

In a Green’s function analysis of temporal development, it is common to introduce an infinitesimal addendum, ε, to potentially resonant terms to avoid unnecessary singularities. Despite rigorous theory requiring that such artificial constructs are applicable only in the limit of ε tending to zero, some of the early semiclassical literature adopted a pragmatic, though fundamentally unsatisfactory, work-around of replacing ε with a finite value, to produce a physically realistic resonance line-shape and linewidth. Treating such issues phenomenologically generally entails recasting the energies of individual electronic states |r〉 as complex-valued quantities, $E_r\mp i\hslash {\Omega }_r$; in the case of losses, this denotes their association with an effective decay lifetime $2\pi /{\Omega }_r$, manifest in a Lorentzian optical response. Reference [71] provided one of the most straightforward algorithms for determining the appropriate sign, which can differ amongst the individual contributions to a response tensor (i.e., terms associated with different time-orderings).

Against this backdrop, it is striking that resonance damping is absent from the canonical Craig, Power and Thirunamachandran literature and its progeny: the fundamental rigour of their approach to theory precluded any such notion of entertaining phenomenological terms. Throughout such work, optical dispersion, in the sense of wavelength dependence, is therefore accommodated in response tensors only as it features in energy denominator terms—all of which are real and vanish on exact resonance, generating unphysical resonance singularities. A highly original and incisive analysis by Nick Blake, tackling the problem in terms of intermolecular interactions, regrettably attracted little attention [72]. A later QED development by Gregory Scholes and the author [73], in the context of energy transfer accommodating multipole contributions, was positively acknowledged, yet was subsequently extensively deployed in semiclassical applications.

The origin of the problem lies in the nature of the system Hamiltonian. As an energy operator, i.e., the quantum operator for a conserved observable, the Hamiltonian has to be Hermitian. The addition of loss or gain terms, actually, undermine a fundamental requirement of closed-system dynamics. It can be noted that, under certain conditions, it is possible for systems exhibiting optical gain or loss to be represented by an explicitly non-Hermitian Hamiltonian, and yet to still be associated with real observables [74,75,76,77]. In such a formulation, the electric field loosely plays the role of wavefunction, but not in any conventional sense; this is not a formulation in which light itself is truly quantized. Further discussion from a QED perspective is given in a recent review [78].

Given the necessarily non-Hermitian nature of any implicitly non-conservative system Hamiltonian, it is impossible to reconcile any internally consistent representation of damping with both the Hermitian character that befits measurable electromagnetic fields and the demands of time-reversal invariance. This is an issue that sparked lively debate in the end of previous century and early years of this century [79,80]. Attempts to reconcile an ensuing cast of theory with full temporal symmetry ultimately prove unsatisfactory, unless assumptions are made—such as the discounting of neighbour interactions—that rarely apply to molecules [81,82]. However, a later solution emerged which, for fundamental consistency, takes advantage of the explicit involvement of the sum in Equation (9). This approach does indeed facilitates properly addressing the electrodynamics of non-conservative systems [78].

With this objective, the system Hamiltonian is partitioned into components directly engaged with the process of interest, and surroundings whose electromagnetic influence engages in the form of ‘bath’ dissipation. In this ‘open QED’ formulation, the material particles can therefore be considered as sub-groups, distinguishing a set $\left\{S\right\}$ comprising the material part of the system of interest from the rest, which are physically separate. For example, $\left\{S\right\}$ might represent all the material within given physical boundaries, or the illuminated part of a continuous system, with all other particles beyond: in other applications, $\left\{S\right\}$ might represent optically significant solute molecules, or guests in a host lattice, counting the other particles as solvent, or host lattice. In broader applications, the term ‘bath’ may even include internal vibrations, where material is modelled only in terms of its electronic structure. Thus, one writes the following:

$$H=H_{\mathrm{rad}}+{\displaystyle \sum _{\xi \in \left\{S\right\}}\left(H_{\mathrm{mol}}^{\xi }+H_{\mathrm{int}}^{\xi }\right)}+{\displaystyle \sum _{\xi \notin \left\{S\right\}}\left(H_{\mathrm{mol}}^{\xi }+H_{\mathrm{int}}^{\xi }\right)} .$$

(12)

While the system as a whole is indeed subject to the corresponding time-dependent Schrödinger equation, its explicit compartmentalization into stimulus, material subject, and bath terms—the latter signifying the terms embraced by the second summation—signifies that the latter is (often) regarded as a source of net loss. Understandably, this no longer represents an isolated system within which energy can only be exchanged with a radiation field through the $H_{\mathrm{int}}^{\xi }$ operations of its constituent particles. Particles of matter within the system $\left\{S\right\}$ can lose (or gain) energy from the ‘external’ set of particles through their mutual engagement with the radiation field. In consequence the time-dependence of the basis functions for perturbation theory develops the intrinsically energy-conserving form $\exp \left(-i{E}_{\xi }t/\hslash \right)$ into $\exp \left(-i{\tilde{E}}_{\xi }t/\hslash \right)$, where ${\tilde{E}}_{\xi }$ signifies a ‘complex energy’ with an imaginary component that has a negative sign for losses (damping the resonance response), or a positive one for gain. To fully accommodate quantization of the bath Hamiltonian, a polariton first proposed by Jasper Knoester and Shaul Mukamel [83] was established after further development by Juzeliūnas [84], accommodating an arbitrary number of polariton resonances. The role of a quantized bath in polaritonic chemistry has been extensively reviewed recently.

8. Applications

Let us now briefly illustrate developments in the main categories of application for molecular QED. The simple enough cases of single-photon absorption, emission, and scattering need little further rehearsal: they were comprehensively treated in the launchpad monograph by Craig and Thirunamachandran [38]. However, much more recent study [85] has established the existence of distinctive effects in absorption and fluorescence due to the electrical influence of neighbouring molecules. Often associated with modified selection rules, such effects offer intriguing possibilities for all-optical switching.

Before addressing the most prominent of these, subsequent fields of application, to note, however, is that the original textbook treatment of two-photon absorption required modification in the ensuing development of QED theory. Craig and Thirunamachandran’s analysis [38] of the two-photon effect produced a result proportional to a product of two mean intensity factors; a beam irradiance $\overline{I}$ and a radiant energy density per unit frequency J. Allowing for a frequency spread in the input was intended to provide for its association with the density of states in the Fermi rule. However, such an approach fails to elicit a more straightforwardly anticipated quadratic dependence on a single measure of beam intensity. Moreover, the corresponding rate of two-photon absorption at the intersection of two beams is also undermined by dependence on an arbitrary labelling of ‘beam 1’ and ‘beam 2’, since the result entails a product of $\overline{I}$ for one beam with J for the other. Subsequent work on two, three, and four-photon absorption resolved the problem by a physically more realistic connection of the requisite density of states to the linewidth of the molecular excited state [86].

Over the years, five especially prominent areas of application have emerged for molecular QED. These are: chiroptical interactions, nonlinear optical processes, optical forces, energy transfer, bimolecular excitation phenomena, and the interactions of structured light.

8.1. Chiroptical Interactions

It is convenient to start with chiroptical interactions, which afforded the first prominent field of application. The term ‘chiroptical’ denotes optical processes that exhibit differences in response according to the handedness of chiral matter—often termed ‘optically active’ in the jargon of chemistry. The primary example is optical rotation, a long-established process wherein the passage of plane polarized light, through a medium (commonly, a liquid or solution) that comprises chiral molecules, results in a clockwise or anticlockwise rotation of the polarization plane, with the sense of rotation determined by the molecular handedness.

The first QED studies of optical rotation and birefringence were performed by Sir Peter Atkins and Laurence Barron [87]. Theory shows that optical rotation is quantitatively related to the optical rotatory strength of the responsible molecule. The latter is a molecular parameter arising from the trace of an E1M1 scattering tensor, whose existence conditions require transitions to be both E1 and M1 allowed. Since these have opposite spatial parity, such transitions are forbidden unless parity is inapplicable, as, for example, when the molecule is non-centrosymmetric. The two-state QED theory developed by Power and Thirunamachandran [88] led to a result in exact agreement with the much earlier classical result [89].

Subsequently, attention focused more on chiral effects that more specifically exhibited a differential response to a circularly polarized beam, according to the relative handedness of the material and the light. Circular polarizations have the maximum absolute value for the projection of spin onto the forward-propagating linear momentum, known as helicity. First, the then-recently discovered process of circular differential Rayleigh scattering—and its Raman counterpart [90] later known as Raman optical activity [91]—were subjected to QED analysis [38,92]. This effect admits interferences of E1 with both M1 and E2, electric quadrupole transitions. A similar QED construct of theory was also applied to circular dichroism—a simpler effect based on single-photon absorption of circular polarizations [93]. An extension to two-photon circular dichroism followed [94]—the latter affordable one of several notable examples where the predictions of both QED and semiclassical theory [95] significantly predated eventual experiments [96].

Molecular QED was also first to identify the capacity for helicity-specific features to arise in the spectroscopy of achiral (i.e., non-chiral) molecules, through the influence of a locally chiral electromagnetic field. Beyond the relatively trivial case of an achiral group chemically bonded to a chiral centre, there are two types of such conferred chirality. In the case of achiral molecules exhibiting circular dichroism, for example, chirality may be conferred either by the proximity of other molecules that are intrinsically chiral (through virtual photon coupling), or by the throughput of circularly polarized light. Both effects known, respectively, as induced circular dichroism [97] and laser-induced circular dichroism [98], were first identified by applying the methods of molecular QED.

Since these earliest studies, the same methods have been applied to numerous other kinds of chiroptical effect, many concerning circular differential effects in optical harmonic scattering [99] forty years prior to its experimental observation [100], others entertaining handedness-specific optical forces [101,102]. QED methods have also proved effective in eliciting definitive answers to questions over the possibility for enantioselective interactions of optical vortices (to which the discussion returns in Section 8.6) and distinguishing far-field and near-field optical chirality [103]—recently leading to a generalized polarization matrix approach to fully characterize the electromagnetic fields in the vicinity of a chiral emitter [104].

8.2. Nonlinear Optical Processes

The formalism of QED is readily applicable to most forms of optical nonlinearity—beyond those necessarily associated with the temporal development of an ensemble, such as optical bleaching and saturation. The simplest process, coherent SHG, is known to require a non-centrosymmetric medium for its observation. In such media the corresponding nonlinear optical susceptibility is a third rank tensor, generally cast in terms of a pure electric-dipole response (E1³). In consequence, the local breaking of symmetry at the surface of a centrosymmetric solid does allow SHG to occur, leading to its use in surface spectroscopy and microcavity harmonic generation [105,106].

It was long assumed, and in some quarters forcibly argued, that the forbidden nature of SHG in any isotropic system extended only as far as the E1 approximation. However, by a QED analysis, this proved to be untrue [107,108,109]. In such media, the forbidden nature of coherent SHG extends to all orders of multipole, both electric and magnetic: only incoherent signals can arise through E1²E2 coupling. As shown earlier, QED methods had already been applied to incoherent (non-forward) forms of harmonic scattering, including hyper-Rayleigh and hyper-Raman scattering [110,111] (the former occasionally referred to as second harmonic light scattering, SHLS) [112]. In the electric dipole approximation, these relatively weak effects are allowed in isotropic media only if the molecular scatterers have a finite hyperpolarizability—requiring that they are themselves non-centrosymmetric.

The issue of coherent SHG from centrosymmetric media arose again, in response to conjectures that optically induced static fields, by undermining local symmetry, could thereby permit static field-induced coherent SHG to occur. However, it again, required a QED analysis [113] to show that, for the requisite levels of pump intensity, such an effect is necessarily weaker than an alternative mechanism, a six-wave mixing process, i.e., 4hν → 2hν′, where hυ is the photon energy with h the Planck constant and υ the photon frequency, as illustrated by the state-sequence diagram in Figure 2. Since this entails a sixth-rank, i.e., even rank, tensor, this is indeed a mechanism that can operate in the bulk of centrosymmetric media, as later proven by experimental determination [114] to verify the QED prediction [113,115]. Much more recently, the case of SHG based on optical vortices has received new attention; see Section 8.6.

Amongst many other applications, QED methods first elicited the dependence on beam configuration for five-and six-wave mixing [116,117], also correctly accounting for significant quantum effects in down-conversion, not readily apparent from any classical treatment [66,118].

8.3. Optical Forces

When relatively large molecules, or larger particles of sub-wavelength dimensions, experience the passage of sufficiently intense light, they may experience optically induced forces arising from several distinct mechanisms. The direct transfers of linear momentum that arise from photon absorption (radiation pressure force) or non-forward scattering (known as a scattering force or radiation force) are readily calculable from the associated rate equations.

With intense, off-resonant laser light, scattering predominantly in the forward direction gives rise to other kinds of force due to optically induced shifts in the molecular energy levels. These shifts are intensity-dependent, and as laser beams have a non-uniform transverse spatial profile, the position-dependence of the effect produces a non-uniform potential energy surface. Gradient forces (also known as dipole forces), thus, arise as particles are drawn to the regions of lower energy, often the high-intensity core of the beam. In all such applications, there are notable differences from the theory applicable to gas-phase atoms. In particular, non-spherical shapes allow the possibility of orientation-dependent torques [119,120].

One especially notable success of QED is its application in the prediction, and later interpretation, of ‘optical binding’—an effect that results from the laser-induced interaction between particles separately trapped in a laser beam [121]. The effect is typically manifest in a damped oscillatory dependence in inter-particle distance. The pioneering 1980 prediction by Thirunamachandran [122] was duly cited in the first experimental measurements, a whole decade later [123]. The paper [122] also paved the way for later identifying a means of cancelling the short-range interaction between atoms in Bose-Einstein condensates [124]. Subsequently, the full mapping of laser-conferred potential energy landscapes was analysed with QED theory [125] and applied in experiments on silver nanoparticles [126], establishing a general mechanism for the role of optical binding in nanoparticle self-assembly [127]. The possibility of a similar multi-beam effect on larger particles was also identified using QED [128] and applied in numerous experiments, such as the optical binding of nanowires [129].

The QED method of tackling optical binding represents the electrodynamic interactions of the participant particles in terms of virtual photons traversing the intervening space. This accords with the absence of a static, longitudinal term in the PZW Hamiltonian (1); it also ensures conformance with causality. Although there are other mathematically equivalent means of representing such necessarily retarded (i.e., not instantaneous) interactions, the virtual photon approach facilitates direct construction of the relevant matrix element by the use of time-order or state-sequence diagrams, as illustrated in Figure 3.

8.4. Energy Transfer

As noted in Section 2 above, the first application of QED methods to interparticle interactions can be traced back to the pioneering studies of Casimir and Polder [24], on the retarded force interactions for which their names later became eponymous. At the same time, the physically simpler effect of electronic energy transfer—less prominent other than in association with atomic collisions in the gas phase—was also receiving attention. The first definitive consideration of the intermolecular process as it occurs in condensed-phase materials was delivered by Theodor Förster, in a quantum mechanical treatment that identified a mechanism also to become associated with his name [130]. His treatment neglected the effects of retardation—though in both cases, short-range, quasi-static effects commonly dominate. The Förster theory leads to a characteristic inverse sixth-power dependence on the separation R between the energy donor and the energy acceptor. However, the establishment of the Förster theory as a key mechanism for radiationless energy transfer led to it being understood as mechanistically unconnected to radiative transfer. It took a quantum electrodynamical approach, almost forty years later, to fully reveal their mechanistic link. Progress in the initial and more recent development of theory has recently been surveyed by Garth Jones and David Bradshaw [131].

The transfer of electronic energy is most simply expressed by a chemical equation

A* + B → A + B*,

which correctly exhibits the essential process in terms of progression from a well-defined initial state of the system, where A is electronically excited and B is not, to an experimentally recognisable end state where the converse applies. The initial state is often prepared by previous excitation of species A, often by optical means. The end state is frequently observed through subsequent fluorescent emission from B. Internal vibrational losses in each species commonly lead to B’s fluorescence occurring at a longer wavelength than the input used to excite A, thus allowing unequivocal evidence of the excitation transfer. Provided A and B are electronically distinct, then from a QED perspective, such a transfer can only occur through virtual photon coupling, the calculation requiring summation over a complete set of radiation modes.

Conversely, for any well-separated donor A and acceptor B, radiative coupling is generally assumed to occur through the release of a real photon by A and its capture by B. Radiative coupling is familiar from its R⁻², inverse square law distance dependence. Nonetheless, the intervening photon cannot be observed without undermining the identification of the sought system‘s final state. A fully consistent quantum mechanical viewpoint always requires summation over all pathways between observed initial and final states, again accommodating a complete set of radiation modes. The two processes, radiationless and radiative, are, thus, both described by a single mechanism, first conclusively shown in the study by the author and Brad Sherborne, which became known as the unified theory of resonance energy transfer, RET [132,133]. This study established that the R⁻⁶ and R⁻² dependences on A–B separation R are just the asymptotes of a more general function A; see Figure 4. Significantly, an entirely new feature was identified for the first time, through this analysis: the presence of an additional R⁻⁴ term that is equally significant at separation distances around k⁻¹ = λ/2π, where λ is the wavelength corresponding to the transfer energy.

Several other new insights emerged from Refs. [112,134] and subsequent studies [135,136,137,138]. First, the range-variation in the distance-dependence of RET exhibits the progressively dominant effect of retardation, as an ostensibly instantaneous transfer process succumbs to the causal effects of a necessarily discrete time interval over which energy transfer occurs. Secondly, the reason the short-range effect appears to conform to the static coupling envisaged by Förster is that the time-of-flight of the intervening virtual photons then admits a nearly unlimited range of frequencies into their state summation, consistent with the time-energy uncertainty principle. Equally, as B recedes from A, the lengthening flight-time diminishes the spread of frequencies until the coupling photon is now identified as real. The effects of retardation also exert an influence on the effective range-dependence of selection rules for the transfer of angular momentum in association with RET [137]. Due to the prevalence of quantum uncertainty effects in the near-zone, the communication of angular momentum does not, in general, map unambiguously between a donor and energy acceptor.

The effects on resonance energy transfer of coupling beyond the electric dipole approximation have been considered elsewhere, thereby allowing for the identification of distinctive effects in chiral media [73,139,140,141]. Establishing the QED mechanism for resonance energy transfer also paved the way for considering in detail the general effects, on the retarded propagation functions, of the local fields produced by surrounding media. The result was a series of papers based on the polariton (‘dressed photon’) model by Juzeliūnas et al., casting results in terms of a complex refractive index. This formulation directly delivers results that accommodate both Lorentz local fields and dissipative loss effects. The visual complexity of the associated results for high-order (multi-centre and/or multiphoton) processes is such that the ‘raw’ vacuum fields are often still used even for dielectric media applications; quite simple algorithms have been found that readily enable such results to correctly incorporate media corrections.

Distinctive effects specifically due to the electrical influence of neighbouring molecules can be modelled more directly, using the virtual photon coupling methodology. Again—just as arises for absorption and emission—such modifications to the energy transfer are often associated with modified selection rules, leading to a range of ‘third-body’ effects [142,143,144,145,146]. More generally, the influence of applied static fields can be accommodated by two means: either as a non-retarded zero-frequency field [146], or by incorporating the source of such a field into the picture [147].

Further developments of theory have also led to the identification of a proliferation of related processes, each of which hinges upon intermolecular energy transfer, including laser-assisted, optically controlled and directed energy transfer [148,149,150,151]. A recent QED approach has also tackled issues arising for atomic Dicke states, when one of an interacting pair is electronically excited, eliciting the significance of degeneracies in their resonance interactions [152]. Establishing the fundamental formalism of QED has also paved the way for the development of macroscopic quantum electrodynamics, increasingly focused on nanocavity and plasmon coupling effects on resonance energy transfer [153,154,155,156,157,158,159,160]. Other studies have focused on larger, multi-centre processes such as are involved in energy pooling and optical energy harvesting [67,161,162,163].

8.5. Bimolecular Photophysics

It emerges that virtual photon coupling can also play a role in multiphoton absorption processes. Actually, there is a range of nonlinear optical phenomena that necessarily involve the concerted interaction of two or more molecules, coupled by virtual photons [62]. One such example is two-photon energy transfer (two-photon absorption followed by energy transfer), yet again an effect for which the theory was devised [164] before the first experimental observations [165]. Another such case of QED predictions is cooperative two-photon absorption:

A + B + 2hν → A* + B*.

Here too, theory [166] was developed prior to the first experimental observations [167], assisting the subsequent mechanistic interpretation [168]. A further example, a counterpart to the cooperative two-photon case, is mean-frequency absorption. Here, the excitation of two neighbouring molecules arises through their concerted absorption of two photons mismatched from resonance by equal and opposite amounts:

2A + hν₁ + hν₂ → 2A*.

Determination of the process matrix element, including all mechanisms consistent with the leading non-vanishing order of perturbation theory (fourth-order), here entails the construction and interpretation of a total of 48 topologically distinct time-ordered diagrams in the general case [169,170]. Again, there is an apparent advantage in using the state-sequence diagrammatic formulation, enabling the full set of 48 matrix element contributions to be cast as a single diagram.

8.6. Structured Light Interactions

As noted in Section 4 above, the standard QED description of freely propagating light is commonly cast in terms of a plane-wave description, the wave-vector defined by the coordinate-free normal to the plane containing the electric and magnetic field vectors. The quantization volume whose photon population defines the intensity is considered arbitrary; no account is taken of the transverse intensity variation in any real laser beam. The general validity of this particular assumption mattered little, in either QED or semiclassical formulations, for applications to the interactions of molecules, or nanoscale particles, across whose dimensions the beam structure could be taken as constant.

However, the development of ‘twisted beam’ optical vortices conveying orbital angular momentum [171,172,173] and other kinds of structured beam [174], necessitated a reappraisal of this assumption. Laser beams propagating as Laguerre–Gaussian modes, for example, especially those with high topological charge, have potentially highly significant spatial variations in both intensity and phase across the beam. The emergence of vector vortex beams, with transverse variation in polarization, potentially has added further complications [175,176]. Initially, there was a degree of scepticism over whether the character of such beams could be conveyed by individual photons, as opposed to a beam collective property. Such doubts were laid to rest by studies unequivocally showing that individual photons did indeed convey structured beam properties [177,178]. QED methods were the first to identify a mechanism for the direct generation of an optical vortex from a suitably structured chromophore array [179].

Essentially, for laser light propagating in any such form of structure the modal basis comprises five degrees of freedom: two are accommodated by the polarization state, (as is the case for plane waves); the others are a wave-vector and two indices designating transverse structure. In the case of Laguerre-Gaussian beams, the transverse indices are nominally azimuthal (representing topological charge) and radial. By detailed formulation of the requisite mode expansions, the first QED treatment of optical vortex interactions tackled second harmonic generation [180], verifying the conservation of orbital angular momentum discovered in Ref. [181].

Rapid technical advances paved the way for structured light to develop into a major field of optics. The vast majority of studies reported in the literature focus on the efficient and controllable production and characterisation of such optical beams, especially vortex beams conveying orbital angular momentum [182,183,184,185,186], alongside increasing interest in their quantum properties and their utilisation for cryptography and optical communication [187,188,189]. Considerations of the symmetry and selection rule aspects of twisted beam interactions with matter have mostly focused on atoms and simple diatomic molecules [190,191,192,193,194,195,196]. Until a recent pioneering study [197] much less attention had been paid to quantum effects in the quantum interactions of vortex and other forms of structured light with intrinsically more complicated material structures—exactly the scope of molecular QED.

In an early consideration of molecular interactions, several research groups sought a resolution to the question of whether the chiral nature of an optical vortex could differentially interact with chiral molecules of opposite handedness. The first study deployed QED analysis to consider the case of a paraxial vortex beam interacting through electric and magnetic dipole transitions [198]. The chirality of vortex beams under such conditions was found to be incapable of delivering enantioselectivity of the kind observed with circular polarizations—a prediction later confirmed by experiments [199,200]. Subsequent QED studies additionally entertaining electric quadrupole interactions nonetheless elicited new and viable mechanisms for enantioselective forms of coupling with vortex beams. These mechanisms operate through the coupling of quadrupoles to field gradients [201], providing access to the topological charge that encodes beam chirality [202,203,204]. Subsequent QED studies further demonstrated a role for magnetic dipole interactions in non-paraxial regimes where longitudinal fields allow the chirality of vortex beams to be more directly engaged [205,206]. Again, as many cases discussed in this review, experiments have identified strong effects, powerfully validating these predictions [207,208,209,210].

9. Summary

Since its inception, the tools of molecular QED have been applied to a remarkably wide, and still growing, range of applications. Its advantages over other familiar approaches are several: (i) fundamental consistency, treating every part of the system as subject to quantum principles; (ii) universal applicability to wide-ranging optical and electrodynamic interactions; (iii) a conceptual framework that provides new and natural insights into mechanism; (iv) as detailed in various cases considered in this paper, the prediction of numerous processes and effects well in advance of experimental observations, each discovered through QED analysis.

In the sphere of atomic and molecular spectroscopy the unmatched precision of QED methods is also highly recognised [211], and the quantitative influence of external fields on material properties is now within purview [212]. At the high-energy end of spectroscopy, the signal-enhancing role of electromagnetic fields is firmly established [213], and it is of high interest to see whether links can be established with the corresponding visible/ultraviolet range effects [148]. And, beyond the optical topics discussed here, the adoption of QED principles in the realm of chemistry and nanomaterials now encompasses the role of polaritons, plasmons and cavity effects that significantly modify chemical reactions [158,214,215].

It is fitting to mark the turn of a century since the birth of quantum electrodynamics and, in so doing. this review has touched on only a fraction of its most prominent molecular and optical applications. The development and utilisation of QED methods continues to flourish: today, it is an area of theory that represents a firmly established, core component of modern optical physics.