Reviews in Physics 1 (2016) 120–139

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Reviews in Physics

journal homepage: www.elsevier.com/locate/revip

Cavity quantum electrodynamics in application to plasmonics

and metamaterials

Pavel Ginzburg

a , b , ∗

a School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel b ITMO University, St. Petersburg 197101, Russia

a r t i c l e i n f o

Article history:

Available online 15 July 2016

Keywords:

Cavity quantum electrodynamics

Plasmonics

Metamaterials

Light-matter interactions

Purcell effect

a b s t r a c t

Frontier quantum engineering tasks require reliable control over light-matter interaction

dynamics, which could be obtained by introducing electromagnetic structuring. Initiated

by the Purcell’s discovery of spontaneous emission acceleration in a cavity, the concept

of electromagnetic modes’ design have gained a considerable amount of attention due to

development of photonic crystals, micro-resonators, plasmonic nanostructures and meta-

materials. Those approaches, however, offer qualitatively different strategies for tailoring

light-matter interactions and are based on either high quality factor modes shaping, near

field control, or both. Remarkably, rigorous quantum mechanical description might address

those processes in a different fashion. While traditional cavity quantum electrodynamics

tools are commonly based on mode decomposition approach, few challenges rise once

dispersive and lossy nanostructures, such as noble metals (plasmonic) antennas or meta-

materials, are involved. The primary objective of this review is to introduce key methods

and techniques while aiming to obtain comprehensive quantum mechanical description of

spontaneous, stimulated and higher order emission and interaction processes, tailored by

nanostructured material environment. The main challenge and the complexity here are set

by the level of rigorousity, up to which materials should be treated. While relatively big

nanostructured features (10 nm and larger) could be addressed by applying fluctuation–

dissipation theorem and corresponding Green functions’ analysis, smaller objects will re-

quire individual approach. Effects of material granularity, spatial dispersion, tunneling over

small gaps, material memory and others will be reviewed. Quantum phenomena, inspired

and tailored by nanostructured environment, plays a key role in development of quantum

information devices and related technologies. Rigorous analysis is required for both exam-

ination of experimental observations and prediction of new effects.

© 2016 Published by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license

( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

1. Introduction

The ability to control light phenomena is essential for achieving various functionalities in majority of opto-electronic

technologies, having inherent optical components. Continuous tendency in size reduction together with recent nano-

technological advances brought investigations of nanostructured devices into practical implementations. Noble metal-based

∗ Corresponding author at: School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel.

E-mail address: pginzburg@post.tau.ac.il

http://dx.doi.org/10.1016/j.revip.2016.07.001

2405-4283/© 2016 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

P. Ginzburg / Reviews in Physics 1 (2016) 120–139 121

structures, being the focus of this review, find use as auxiliary components in a large span of applications, among them

biomedicine [1] , photovoltaics [2] , flat optical devices [3] , sensing [4] and many others. The full potential of those elements

has not yet been fully explored, especially with regards to the context of quantum applications [5,6,7,8] . One of the most

prominent demands, to be achieved in order to enable those, is to confine and tailor propagation of electromagnetic energy

at nanoscale.

One of the very promising and already proven approaches for controlling electromagnetic fields on nanoscale is to employ

noble (plasmonic) metals, such as silver and gold

1 . The key optical property of those materials is their negative permittiv-

ity at the visible and infrared spectral ranges. Negative epsilon materials, being shaped into geometrical structures, have a

unique property of supporting localized plasmon resonances (LPR) with the resulting phenomenon of nano-scale electro-

magnetic energy confinement [9] . The ability of plasmonic structures to concentrate light beyond classical diffraction limit

[10–12] has the key advantage for tailoring light-matter interactions, opening roots for novel cavity quantum electrodynam-

ics (CQED) phenomena - plasmonic CQED (PCQED). The major fundamental difference between traditional CQED and PCQED

is in the way the interaction strength is controlled by an electromagnetic environment. The most prominent example, un-

derlining differences between those approaches, is the change of spontaneous emission rates of an emitter in respect to

a free space (Purcell effect [13] ). While free space vacuum fluctuations of electromagnetic modes are solely dependent on

fundamental physical constants, they can be modified by shaping surrounding material environment, in spite of the overall

conservation of the total density of states available for radiation [14] . In fact, electromagnetic structuring leads to spectral

and spatial redistribution of vacuum fluctuations. The relevant quantity here is the local density of states (LDOS) [9] – the

proportionality constant, modifying interaction Hamiltonians and, consequently, resulting emission rates. The Purcell factor

(the same quantity up to a constant) is proportional to the ratio of a quality factor of a confined mode to its volume (Q/V)

and can be influenced by manipulating both of the quantities. Traditional CQED affects the interaction dynamics by explor-

ing influences of quality factors, while PCQED acts via modal volumes for the most. This fundamental difference has to deal

with optical properties of material components and geometries. While dielectric cavities could have quality factors as high

as ∼10 10 [15,16] , their modal volumes are bounded from below by classical diffraction limit (roughly, a cubic wavelength in a

material). On the other hand, plasmonic structures could confine energy to small regions, while associated quality factors are

usually not exceeding hundreds [17] , with few exceptions available [18] . From the theoretical standpoint, the difference be-

tween CQED and PCQED is in the way electromagnetic field is quantized. Furthermore, standard definitions of Purcell factors

in PCQED could lead to ambiguous results and might be used only for preliminary estimations [19] . Main reasons, leading to

the contradiction with classical Purcell theory, are presence of additional nonradiative decays channels and open nature of

resonators, complicating rigorous definition of modal volumes. Those aspect were addressed in a series of recent works (e.g.

[20,21] ) and the revised formalism of Purcell factor was developed in [22] . Some of the above mentioned aspects along with

other probable impacts of absorptive and dispersive nature of plasmonic materials on light-matter interaction dynamics will

be discussed in details hereafter.

Large-scale nanostructures, having shaped plasmonic particles as unit cells 2 , enables achieving modified interactions for

macroscopic number of emitters. While, studies of modified light-matter interactions in CQED are typically restricted to

single-mode/single-emitter scenarios, collective phenomena, tailored by nanostructured media (metamaterials [23] ) gained

a considerable attention due to a span of promising frontier opto-electronic applications. For example, so-called hyperbolic

metamaterials (highly anisotropic artificially created crystals) hold a promise of broadband non-resonant enhancement of

spontaneous emission rates [24] with few experimental demonstrations already available (e.g. [25,26] ).

The main purpose of this review is to discuss few phenomena in the field of PCQED (including aspects of metamaterials)

by introducing major theoretical tools and relating them to existing experimental reports. Light-matter interaction dynamics

on nano-scale could be rather involved and, in contrary to well-defined isolated CQED-like problems, require conceptual

mesoscopic type of treatment. A generic reason for this occurrence is related to the fact that nano-scale systems could not

obey classical laws of electrodynamics (classical permittivity description breaks down), but yet too big to be treated with full

atomistic modeling [27] . One of the major questions, to be addressed before approaching a mesoscopic problem of this kind,

is whenever classical or quantum treatment of a phenomenon is required. In majority of cases, full quantum-mechanical

treatment of problems, lacking of analytical solutions, requires heavy and even unobtainable computational resources. In

order to bypass those limitations, capturing key mechanisms governing phenomena to the lowest level of sophistication is

needed. Few examples, demonstrating this general concept, will be revised and discussed. It is worth noting few recent

reviews on the topic of quantum plasmonics and metamaterials, giving a broad survey of theoretical and experimental

works, e.g. [8,28,29–31] . The focus of this review, apart from revising a set of major tools, is to introduce approaches for

mesoscopic models, which might require a unique treatment.

The review is organized as follows: first, general aspects of electromagnetic field quantization in the presents of material

bodies will be presented. Next, main geometries, subject to PCQED studies, will be introduced and few recent achievements

in the field will be reviewed. Mesoscopic models, enabling to bridge a gap between microscopic and macroscopic descrip-

tions of PCQED, will follow. Concepts of emulation experiments, aiming to address complex phenomena by means of analog

1 There is a venue for investigating and developing additional materials for plasmonics, such as aluminium [188] and transparent oxides [189] . 2 The concept of quantum metamaterial with semiconductor-based unit cells was introduced too (e.g. [190–192] ).

122 P. Ginzburg / Reviews in Physics 1 (2016) 120–139

simulation, will be introduced. Higher-order effects, tailored by plasmonic nanostructuring, will be briefly revised. Finally,

an extended outlook will conclude the manuscript.

2. Quantization of electromagnetic field in the presence of material bodies – main approaches

Quantization of electromagnetic field is a basis for an adequate description of light-matter interaction phenomena. Quan-

tum formulation is especially important for scenarios, where spontaneous emission or an operation on a few photon level

are involved. Nevertheless, the photon terminology is also useful for discussing interactions with macroscopic number of

light quanta (so-called c-number approximation, widely used in quantum optics and lasers theory [32] ). Hereafter, major

techniques for quantization of electromagnetic fields in a presence of material bodies will be revised.

2.1. Mode decomposition method

While the canonical formulation of field quantization is based on Lagrangian representation with imposing commutation

relations between conjugated coordinates [33] , exactly the same expressions could be obtained by utilizing Mode Decompo-

sition Method, e.g. [34] . Here the entire electromagnetic space is spanned by a set of Eigen functions (orthonormal modes)

and canonical creation/annihilation operators are attributed to each one of the modes. It is worth noting, that the modes

structure could be found by applying classical tools, such as standard Eigen solvers, based on diagonalization of construc-

tions, obtained from Maxwell’s equations. The validity of c-number approximation and semi-classical analysis of interaction

processes is the immediate consequence of this relation to the classical modal structure. The straightforward and simplest

example of the Mode Decomposition Method is a free space, where plane waves’ orthonormal basis spans the entire field.

In this case, the field operator is given by −→

E tot =

∑

k,λ

−→

E + k,λ

( −→

r , ω ) + c.c. , where −→

E + k,λ

( −→

r , t ) = i

√

h ω k 2 ε 0 V

ˆ a k,λe −i ω k t+ i � k � r has a clear

relation to the plane wave structure and the pre-factor regulates the normalization (vacuum energy in a mode if an external

excitation is absent). In the above expression c.c. stays for the complex conjugate, V is a space volume to be taken to an

infinity limit, and the rest are widely used notations [32] . If the volume contains material bodies with nontrivial geometries,

the modal structure could be obtained by solving the following differential equation:

�(� B m

( � r ) )

=

� ∇ ×

[� ∇ × �

B m

( � r )

ε r ( � r )

]=

(ω m

c

)2 � B m

( � r ) , (1)

where � B m

( � r ) is spatially dependent magnetic induction of m th mode, ε r ( � r ) relative position-dependent permittivity (could

be also a discontinuous function of space), ω m

eigenfrequency of the mode, and c the speed of light in vacuum. Eq. (1) ,

frequently referred as a Master Equation of photonic crystals [35] , defines a Hermitian eigenvalue problem only if ε r ( � r ) is

neither dispersive nor absorptive [36] . Under this strong assumption of hermiticity, the entire electromagnetic space could

be spanned with the help of eigenfunctions and canonical creation/annihilation modes’ operators could be defined. It is

worth underlining limitations of the Mode Decomposition Method, stemming from an inherent dispersion and absorption of

material components. Those properties are especially important in the case of Plasmonic and Metamaterial structures, made

of noble metals. If ε r ( � r ) in Eq. (1) is replaced by ε r ( � r , ω ) + i ε im

( � r , ω ) , the basic operator is neither Hermitian nor eigenvalue.

In fact, those two properties are tightly connected with each other via Kramers–Kronig relations, resulting from causality

principles. Another way to underline the same problem is to recall time-dependent behavior of a material response. In

the presence of strong dispersion, the polarizability depends on a certain history of electromagnetic interaction back in time

(convolution integral) and not just on the instantaneous field and its first derivative, as required for defining a Lagrangian. As

the result, the canonical Hamiltonian approach cannot be utilized for the field quantization, if materials’ dispersion plays an

important role in describing a phenomenon. As a remark, it is worth noting, that this method, being applied on a dispersive

case, will result in time-vanishing canonical commutation relations between field operators, violating basic postulates of

quantum-mechanical theory 3 . Furthermore, an attempt to define proper Eigen modes in lossy structures will result in an

inter-mode coupling, violating basic requirements of orthogonality. In the case of nonmagnetic medium, the operator in Eq.

(1) could be decomposed in to the sum of Hermitian ( �H ) and anti-Hermitian ( �A ) parts as [37] :

�(� B ( � r )

)= �H

(� B ( � r )

)+ �A

(� B ( � r )

),

�H

(� B ( � r )

)=

� ∇ ×

[ε r ( � r )

ε 2 r ( � r ) + ε 2 im

( � r ) � ∇ × �

B m

( � r )

],

�A

(� B ( � r )

)= −i � ∇ ×

[ε im

( � r )

ε 2 r ( � r ) + ε 2 ( � r ) � ∇ × �

B m

( � r )

]. (2)

im

3 There are still quite a few recent reports, introducing the damping term into the time evolution of field creation and annihilation operators. This

procedure will lead to vanishing in time canonical commutation relations, which is, of course, contradicts with the postulates of quantum-mechanical

theory. Introduction of Langevin noises is a one among possible solution, well understood in the field of Quantum Optics [32] . Nevertheless, majority of

experimental results could be explained with the help of the classical approach. It shouldn’t come as a surprise, as reported Purcell enhancements could

be explained classically via antenna impedance matching, as will be discussed hereafter (in fact, this is the c-number approximation, describing changes in

exponential lifetimes). However, pure quantum effects cannot be correctly addressed by adopting incomplete quantum description of relevant interactions.

P. Ginzburg / Reviews in Physics 1 (2016) 120–139 123

If dissipative losses are relatively small, the anti-Hermitian part could be treated as a perturbation. In this case Eigen

functions could be defined with the help of the main Hermitian operator, which spans the entire electromagnetic space.

However, the �A perturbation, solely inspired by the presence of losses, will introduce a coupling channel, which efficiency

is proportional to the spatial overlap between quasi-Eigen modes (labeled by subscripts a and b) and material losses -∫ ε im

( � r ) � E ∗a ( � r ) � E b ( � r ) d 3 � r . The dissipation induced coupling (optical analogue of Hanle effect) was proposed as a tool for efficient

polarization rotation in a metasurfaces-type configuration [37] . Set of Eqs. (2) underlines an additional possible artifact that

could be faced once applying Mode Decomposition Method on lossy structures.

2.2. Hopfield, Huttner and Barnett models

The major limitation of the Mode Decomposition Method results from ignoring material degrees of freedom, as permit-

tivities and permeabilities are considered as fully-classical parameters. Quantum theory of dissipation, however, requires

explicit introduction of a dissipation bath, having an infinite number of degrees of freedom [32] . Briefly, having an infinite

number of degrees of freedom ensures the relaxation and suppresses revivals, as the excitation has a vanishing probability

to hope back to the initial site. Dissipation in plasmonic and metamaterial systems could originate from several factors, such

as dissipation in materials components, coupling to dark modes (which subsequently decay to material heating), and other

solid-state related effects (for further partial discussion see, e.g. [38,39] ). All the above mentioned mechanisms could be

considered in a similar fashion, while physical implications (introduced via coupling coefficients, as will be described be-

low) will be completely different. Material degrees of freedom could be explicitly introduced in order to treat light-matter

interactions with electromagnetic structures and the Hamiltonian approach could be utilized for the field quantization. One

of the first widely-accepted models was developed by Hopfield [40] , who imitated a material by a collection of harmonic

oscillators, resonant at a certain frequency. The overall Hamiltonian of a system is given by:

ˆ H tot = h ω 0 a † ˆ a +

∑

k

h ω k q † k

q k + h

∑

k

g k (

ˆ q † k

a +

ˆ a † ˆ q k ), (3)

where ˆ a † ( a ) is creation (annihilation) operator of a cavity mode, { q †

k , q k } ∞

k =1 operators of harmonic oscillators, while the

last term in Eq. (3) stays for the coupling ( g k is a coupling strength). The overall Hamiltonian in Eq. (3) could be diagonal-

ized and true collective operators (polaritons – coupled light-matter excitations) could be obtained. Hopfield model enables

addressing material dispersion

4 , but does not account for absorption. This may be a reasonable assumption to make away

from a resonant transition (recall, that the imaginary part of the permittivity drops faster with the frequency than the real

part, if Lorentzian lineshape is adopted [41] ). The dissipation, nevertheless, could be included by introducing a continuous

number of degrees of freedom – each material dipole has its own dissipation bath. This model is called after Huttner and

Barnett [42] and the overall Hamiltonian could be still diagonalized. The explicit introduction of material degrees of free-

dom enables addressing dispersion and absorption of constitutive materials, but is rather complex procedure to be applied

on nontrivial geometries, as the diagonlization will not lead to analytic expressions. Additional discussions and limitations

of the above-mentioned models and their extensions could be found elsewhere, e.g. [43] .

2.3. PCQED and quantum theory of dissipation

Material losses and dispersion are of primary importance to plasmonic and metamaterial structures. While their impact

on classical aspect of interaction phenomena is well studied and understood [44] , those effects are still overlooked in PC-

QED. On the other hand, quantum theory of dissipation in CQED has a variety of well-developed tools, suggesting their

adaptation to the case of study here. For example, photon decay channels, either radiation leakage from a cavity or ohmic

losses, are generally described via interactions with thermal baths, having an infinite number of degrees of freedom. Those

variables could be either traced out by adopting the density-matrix approach or treated as stochastic noise operators [45] .

Equations of motion, corresponding to the latter approach, are derived from the quantum Heisenberg–Langevin evolution

and are preferable over the density-matrix formalism, if the time dependence of the operators is of particular interest [32] .

The Kossakowski–Lindblad equation, being extensively used for the nonunitary description of dissipation and decoherence

in systems, is however, restricted to the treatment of Markovian dynamics (in general – processes without a memory; here

– channels with exponential decays) and cannot be straightforwardly used for strong-coupling regimes [45] . Employment

of Jaynes–Cummings Hamiltonians [32] should be also carefully assessed, as those models, in principle, were proposed for

treatment of isolated cavities with well-defined modes (e.g. [43] for the discussion of the validity). Recently, Lindbald opera-

tors were introduced explicitly into time evolution of hybrid plasmonic sources and strong coupling regimes were addressed

[46] . In general, inclusion of several dissipation baths enables addressing competitive decay channels and investigating their

influence on quantum dynamics of photon emission (e.g. [38,47] ). A sufficient near-field feedback could also result in non-

exponential law of decay in various systems, e.g. [48] and requires proper account for material degrees of freedom. Examples

of nontrivial aspect of dissipation in plasmonic structures will be discussed hereafter.

4 The model could be further extended by adding several families of harmonic oscillators; each one is resonant at its own frequency (Sellmeier equation

for material dispersion).

124 P. Ginzburg / Reviews in Physics 1 (2016) 120–139

2.4. Green’s functions (Langevin) quantization

In order to address complex geometries (nontrivial-shaped dispersive and absorptive material bodies) other techniques

are better to be applied. One of the most widely used approaches is based on so called local Langevin quantization [49] .

The intuitive description of the method with the rigorous basis behind it relies on the Fluctuation–Dissipation Theorem

[9] . Recall, that absorption in an infinitesimal space volume should necessarily lead to random fluctuations of charges and

currents, which in turn could be considered as sources of vacuum fluctuations. Those local excitations give rise to electro-

magnetic fields, which are given by a kernel integral between the sources and classical electromagnetic Green’s functions.

The quantum operator of the field in the Schrödinger picture is given by:

−→

E (−→

r , ω

)= i

√

h

πε 0

∫ ∞

0

d ω

(ω

c

)2 ∫ d 3

−→

r ′ ·√

Im ε (−→

r ′ , ω

)←→

G

(−→

r , −→

r ′ , ω

) f

(−→

r ′ , ω

)+ c.c., (4)

where f ( −→

r ′ , ω ) is the local annihilation operator of the field excitation, ←→

G ( −→

r , −→

r ′ , ω ) is a classical electromagnetic Green’s

dyad (calculated with classical material parameters of a system), and the rest are widely used parameters and constants.

First, it should be noted that the field operator in this representation is strictly nonlocal because of the kernel integral

structure. Furthermore, commonly used field modes are replaced by local excitation operators, obeying canonical commuta-

tion relations [ f k ( −→

r , ω ) , f †

k ′ ( −→

r ′ , ω

′ ) ] = δkk ′ δ( −→

r − −→

r ′ ) δ( ω − ω

′ ) , as guaranteed by the Fluctuation-Dissipation Theorem. The

drawback of this type of quantization is that those local operators are not directly related to the modal structure of the

field, but an unambiguous strength is that both absorption and dispersion of material components are included via rigorous

quantum-mechanical treatment. This Langevin model could be further extended to address material nonlocalities and spatial

dispersion [50] . The local quantization is extremely useful for calculating lifetimes and Purcell factors by adopting the Fermi

Golden Rule. The local density of states (LDOS) of electromagnetic modes is directly proportional to the imaginary part of

the corresponding electromagnetic Green’s function, and the resulting rate of spontaneous emission is given by:

�spont =

2 ω

2 0

h ε 0 c 2 � μ∗ ↔

G ( � r 0 , � r 0 , ω 0 ) � μ, (5)

where � μ is the emitter’s dipole moment, ω 0 transition angular frequency, and

↔

G ( � r 0 , � r 0 , ω 0 ) dyadic Green’s function calcu-

lated at emitter’s location

� r 0 . It is worth noting, that Mode Decomposition Method provides exactly the same result as the

Local Langevin Quantization, but if and only if materials’ losses and dispersion vanish. In this case the Green’s function could

be spanned with the help of Eigen modes of Eq. (1) [9] .

The appearance of classical Green’s functions in expressions, used for lifetimes calculations ( Eq. (5) ), is rather expected. In

fact, classical-quantum correspondence principle ensures relations between theories and, in particular, rates of spontaneous

emission could be calculated by employing purely classical approaches, such as radiation reaction forces [9] . Specifically,

quantum emitters could be represented by classical point dipoles. In this description, the strength of light-matter interaction

depends on two key parameters – amplitude of electromagnetic field at a point of an emitter and the dipole moment of the

emitter itself. In the frame of Antennae Theory [51] this procedure describes an impedance matching of a localized source

to a far-field. This approach, bridging the gap between classical (antenna) and quantum (vacuum fluctuations) theories,

introduces additional damping force on a dipole in order to balance the outflow of radiated energy to the far-field with

the one, stored in oscillations. This simple energy conservation principle enables deriving exactly the same decay rates, as

could be calculated by employing second quantization of electromagnetic field and solving for quantum equations of motion.

This classical approach is the cornerstone for majority of numerical simulations, done in the field. A complementary classical

alternative is the calculation of so-called S-parameters of a small antenna, e.g. [52] . The full quantum-mechanical formula for

the radiation reaction force in arbitrary electromagnetic environment was recently derived, bringing this phenomenological

description to the solid quantum-mechanical ground [53] . It is worth noting, that the above discussion is related to the

weak coupling regime of light-matter interaction, while the strong coupling analysis requires more sophisticated approaches

and solutions of time-dependent equations of motion. Those equations of motion, however, could be also written with

the help of the Green’s functions and subsequently solved by employing numerical techniques [49] . In this case damped

Rabi oscillations and even spectrum splitting could be recovered. Classical circuit model with introducing characteristic

impedances was recently employed for description of plasmon-enhanced fluorescence [54] , which is especially relevant to

structures, having analytical expressions of Green’s functions.

2.5. Discussion

Majority of recent investigations in the field of quantum plasmonics and metamaterials are based on the approaches

(or their combinations), reviewed above – namely Mode Decomposition Method, Hopfield, Huttner and Barnett Models,

and Local (Green Functions-based) Langevin quantization ( Fig. 1 summarizes the approaches in a graphical way). Direct

applications of those methods and mesoscopic models, based on combinations of certain concepts will be descried hereafter.

Nevertheless, there are many quantum aspects of light-matter interactions with plasmonic and metamaterial structures that

cannot be described within the above mentioned formalism. Tunneling over small gaps, nonlocal material responses and

P. Ginzburg / Reviews in Physics 1 (2016) 120–139 125

Fig. 1. (Color Online). Summary of quantization methods – Hopfield model, introducing material degrees of freedom explicitly [40] ; Mode Decomposition,

spanning electromagnetic field into orthonormal basis of Eigen modes [34] (two-dimensional modes’ slices in XY plane are presented); and Green’s function

(Langevin) quantization, attributing local stochastic sources at each point of space [49] . (For interpretation of the references to color in this figure legend,

the reader is referred to the web version of this article).

few others require development of special tools, capable of addressing a specific problem. Few representative examples will

be reviewed at corresponding sections.

As an additional remark it is worth noting, that all the beforehand described methods concentrate on electromagnetic

enhancement of lifetimes, including both bright and dark modes contributions with the ability of separating their relative

contributions. However, the total decay time ( τtot ) is defined by both radiative ( τrad ) and nonradiative ( τnonrad ) processes.

The later could be related to both internal structure of and emitter and its interaction with an environment. For example,

internal quantum yield and various mechanisms, e.g. collision, oxygen quenching could have a substantial impact on the

decay dynamics [55] . In the week coupling regime, where the exponential decay dynamics is formed by non-competing

depopulation mechanisms, the rate is given by:

1

τtot =

1

τnonrad

+

1

τrad

. (6.1)

The radiative decay channel could be indeed accelerated by the Purcell factor, while the nonradiative one, having non-

electromagnetic origin, could be assumed unchanged, as a first approximation (with few exceptions available, e.g. [56] ).

Within this approximation, the modified rate ( τ−1 tot ) is given by:

1

˜ τtot =

1

τnonrad

+

P

τrad

, (6.2)

where P is the Purcell factor. It might be seen from Eq. (6.2) , that emitters with initially high internal quantum yield

( τnonrad << τrad ) will benefit from Purcell enhancement, while the effect is less significant in the other case. It should be

emphasized, that Eq. (6.2) addresses only a weak coupling regime, where depopulation channels do not compete. In order

to investigate the full dynamics of the process, including the strong coupling, time evolution of quantum operators should

be solved (e.g. [38,39] ).

3. Strucutres for PQCED

Nano-plasmonic structures enable achieving high filed concentrations in small volumes. This unique property is beneficial

for controlling light-matter interactions strength, which could be even sufficient for reaching strong coupling regimes [57–

63] . Negative permittivities of metals make them attractive for controlling near fields and LDOS. It is convenient to consider

the phenomenon in separate sub-sections: one- and two-dimensional field confinement of propagating waves, 3D localization

of subwavelength modes [9] , and metamaterials ( Fig. 2 for the overview).

126 P. Ginzburg / Reviews in Physics 1 (2016) 120–139

Fig. 2. (Color Online). The scope of plasmon cavity quantum electrodynamics (PCQED) - interaction scenarios between emitters and plasmonic structures.

(a) Surface. (b) Waveguide. (c) Particle. (d) Metamaterial.

3.1. Plasmonic surfaces

Surface plasmon waves, propagating on metal dielectric boundaries, could serve as an additional energy decay channel

for emitters, situated at their near fields. Those modes have high DOS due to their ‘slow wave’ nature [64] and high filed

concentration next to boundaries. However, in order to obtain a full physical picture of the interaction, several nonradia-

tive quenching processes causing excited state depopulation due to scattering on phonons, and other probable solid state

processes should be taken into account [65] . Those effects are especially relevant for emitters, situated next to metal struc-

tures as close as 10 nm and less, e.g. [66,67] . Being a subject for numerous studies (e.g. [68] ), modification of spontaneous

emission rates next to plane surfaces was analyzed by means of competitive decay channels for the most. Radiative decay

and nonradiative quenching (coupling to dark modes) could be separated by integrating the near- and far-field contribu-

tion of a classical dipole by enclosing it by a pair of small and large integration surfaces. Flux of a corresponding Poynting

vector should be calculated. In the case of a planar surface, radiative rates have relatively simple and intuitive expressions,

derived with the help of Lorentz reciprocity theorem [69] . Radiation to an upper half plane, containing a dipole, polarized

perpendicularly to a surface is enhanced by a factor of [69] :

�⊥ =

3

4

∫ π2

0

∣∣1 + r p ( θ ) e 2 i k 0 d cos θ∣∣2

sin

3 θdθ, (7)

where r p is the Fresnel reflection coefficient for TM-polarized waves (magnetic field parallel to the surface), k 0 the wavevec-

tor in the material above the pane, d the distance between the dipole and the surface, and θ the angle of incidence with

respect to the surface’s normal, over which the integration is performed. Similar expression for �‖ could be written for a

dipole, parallel to the surface [69] . Eq. (7) describes the radiative part of the rate, which could be calculated with the help

of Eq. (5) . It could be seen by setting an upper limit on the integrand of Eq. (7) that the radiative rate enhancement (to the

upper half space, including the dipole) cannot exceed the factor of 2. The maximal enhancement could be obtained with

a perfectly reflecting mirror, while the fundamental limitation on both �⊥ and �‖ comes from the conservation law of a

wave’s energy, reflected from a surface. It is worth underlining several important issues. First, Fresnel reflection coefficients

could be defined for any flat geometry having translational symmetry along the surface. Hence, the above-mentioned lim-

itation does not applicable for irregularly patterned surfaces, such as gradient metasurfaces, for example [70,71] . Second,

P. Ginzburg / Reviews in Physics 1 (2016) 120–139 127

the overall emission rate could be enhanced with planar geometries, e.g. layered hyperbolic metamaterials [72] , but the ma-

jority of the radiation will be trapped in the substrate, while the field, emitted to the upper half-space will not gain any

significant enhancement. Finally, the rate enhancement factor has a physical meaning only in the case of the weak cou-

pling regime of interaction, when the Fermi Golden Rule could be applied. Furthermore, this approach relies on the hard

assumption of Markovian exponential dynamics (standard decay process without a memory) [32] , predefined by the fastest

decay mechanism. The immediate implication of this assumption is that the fast nonradiative quenching will speed up the

radiation to the far-field on the expense of the overall quantum yield (ratio between number of photons, emitted to the far-

field and emitter’s excitations; aspects of internal dynamics of an emitter, including its absorption crossections are excluded

from the discussion, see Section 2.5 for the details). While this interpretation enables capturing basic effects in the case of

the weak coupling regime, non-exponential behaviors, related to the strong coupling and complex decay dynamics could

emerge. For example, rigorous quantum analysis of coupling and dissipation mechanisms in open nano-scale resonators is

required in order to address the interplay between radiative and nonradioactive channels theoretically. Coupling between

quantum emitters and surface plasmon modes was addressed in e.g. [73,74] (modified Mode Decomposition Method) and

was shown to be in a good agreement with Green’s Functions Approach. Recently, stochastic Hamiltonian approach was

applied for description of competitive decay channels in open plasmonic resonators. In this typical case both direct sponta-

neous emission to the free space and nonradiative quenching, resulting from electromagnetic coupling to dark modes of a

structure (nano-resonator), could be obtained simultaneously [38] .

Investigations of light-matter interactions, tailored by metasurfaces, is one of the promising directions to pursuit and

examine validities of various homogenization schemes (defining effective material parameter to a structured media with

subwavelength features) (e.g. [75] ) in application to excitations with localized quantum emitters.

3.2. Plasmonic waveguides

One of the major advantages of tailoring spontaneous emission properties with plasmonic waveguides ( Fig. 2 (b)) is the

ability to improve photon collection efficiencies. While spontaneous radiation is almost uniformly distributed over 4 π angle

in free space scenarios 5 , high-DOS propagating modes could serve as a preferable radiation channel. All the information, yet

in classical permittivity description of material components, is encapsulated in electromagnetic Green’s function. It’s imagi-

nary part is capable for describing the rate of spontaneous emission, while two-point Green’s function (so-called mutual or

cross density of states [76] ) hold the information on the signal’s propagation from an emitter to a detection point – all this

in a full correspondence with the classical antenna theory. For example, wire waveguide geometry was studied to a great

extent by means of Green’s functions [77] . This approach utilizes classical permittivity description of plasmonic materials

and is in a good agreement with majority of experimental demonstrations, available nowadays. Brief survey of works on

this topic includes reports on collection efficiency of photons, emitted from a CdSe quantum dot source, and coupled to a

silver wire. The values of efficiencies as high as 60% (with propagation losses factored out) were claimed and antibunching

properties of photons, guided through the wire and scattered from its edge, were demonstrated [78] . Subsequently, those

propagating single plasmons were directly detected at the near-field without a need for a free-space out coupling [79] . Ra-

diative Purcell enhancement in wire geometries of this kind could be as large as 10 0 0, yet without resolving single emitter

behavior [80] . Plasmonic excitations in holes arrays were shown to mediate quantum entanglement [81] with further ex-

tension to plasmonic waveguides [82] . Plasmonic version of the Hong–Ou–Mandel interference was reported at [83] , while

other quantum plasmonic effects are discussed in the review manuscript [84] . Plasmonic waveguides were also proposed to

assist in generation of high degree of entanglement between distant qubits owing to their improved collection efficiencies

and high dissipation losses [85] .

As it was previously underlined, theoretical description of beforehand mentioned effects is performed in the frame of

Mode Decomposition and Green’s functions methods for the most [86] . In this sense, plasmonic structures are not much

different from conventional dielectric devices with an additional and predominating decay channel. This classical dissipa-

tion, however, do not introduce qualitatively new quantum effects. Nonradiative quenching mechanisms, however, should be

properly addressed for extracting a relevant data. In order to obtain information on a true radiative Purcell enhancements

experimentally, two steps should be made. The first one is the time-correlated single photon counting (TCSPC) experiment

[55] , which provides the total relaxation time of excited carriers. This lifetime, however, could be predefined by nonradioac-

tive processes, even though emitted photons are measured (Eq. (6)). In order to resolve several decay channels, theoretical

models should be employed. Challenges in this niche are related, in part, to obtaining clear separation between decay mech-

anisms by employing solid-state models of material components beyond their classical optical susceptibilities together with

considering internal quantum dynamics of emitters themselves.

3.3. Plasmonic nanolasers

Investigation of competitive decay channels and maximization of directive spontaneous emission into pre-defined modes

is required for designing active nano-scale light-emitting devices, e.g. lasers. While the smallest available all-semiconductor

5 Improving collection efficiencies/achieving directional emission with structuring are of primary importance for lasers. For example, threshold less laser

utilizes close to 100% efficiency of spontaneous photon to be emitted into a lasing mode [159,193] .

128 P. Ginzburg / Reviews in Physics 1 (2016) 120–139

lasers (VECSEL) (e.g. [87] ) have modal volumes of a cubic wavelength in a material and limited by the classical diffraction,

plasmonic confinement enables achieving true lasing on nanoscale. The feedback loop, required for coherence buildup of

laser oscillations, in the case of plasmonic nanolasers is still implemented via propagating phase accumulation in the third

dimension. Few plasmonic lasers were demonstrated since 2007 [88] and are based on several designs of plasmonic cavities

(e.g. [89–91] and few others - for additional discussion [92,93] ). Majority of theoretical studies of plasmonic lasers adopt

semi-classical approaches, utilizing rate equations framework, e.g. [94] . The relevance of quantum aspects of laser phenom-

ena [32] to nano-plasmonic scenarios is yet not fully explored. For example, analysis of noises in nanolasers will require

employment of quantum tools.

3.4. Plasmonic nano-antennas

Localized plasmon resonance (LPR) phenomenon, being electromagnetic charges oscillation on a boundaries of subwave-

length objects, enables achieving small modal volumes with true 3D confinement of energy [9] . Properly designed metal

particles could concentrate incident illumination into sub-diffraction hot spots via coupling to LPRs. The reciprocal effect is

the impedance matching between localized sources and far-field radiation – that’s the concept of optical antennas [95,96] .

A difference between deep sub-wavelength plasmonic and classical antennas is in the effect’s interpretation. While classical

antenna design requires constructive interference between phase-lagged elements, LPR phenomenon in small particles is of

a pure quasistatic nature with almost no retardation involved

6 . In this case, a small dipole (emitter) is coupled to a bigger

one (plasmonic structure) via near-field interactions. Since the bigger object has larger radiation efficiency (multi-polar mo-

ments could be also involved [97] ), the overall emission rates are accelerated. It is worth underlining, that optical properties

of plasmonic particles could be tuned to emitter’s resonant frequencies by changing local geometries, for example by con-

trolling concavity parameters [98] , on demand responses engineering [99] , and many others (for the review [100] ). Among

many applications of hybrid plasmonic light sources it is worth mentioning their abilities to serve as efficient bio-markers

[101,102] .

The natural approach for analyzing emission from optical antennas is to utilize Green’s function quantization. While most

trivial geometries have analytical solutions (e.g. spheres [49] ), others could be addressed by means of numerical modeling,

e.g. numerical solution of the master equation [103] . Poynting vector flux through a small virtual sphere, enclosing a clas-

sical dipole, will provide an information on the total decay rate via radiation reaction force approach [9,53] . Imaginary part

of the Green’s function could be extracted directly from a numerical calculation. In this case it is worth taking values from

a nearby grid point, as the value of the real part at the position of an emitter diverges. The flux though a large sphere,

containing the entire geometry, is proportional to the total radiative decay. The difference between those two quantities is

the amount of energy, quenched in lossy materials (coupling to dark modes of a structure). Green’s function analysis also

enables addressing weak and strong coupling regimes of interactions by encapsulating numerically evaluated constants into

operators’ equations of motion [104] . At certain situations, however, where a separation of background fields and emission

is required (e.g. structures with Fano resonances), other techniques could be applied, e.g. [105] . It is worth noting, that the

presence of competitive decay channels (e.g. excited state depopulation due to phonon scattering and radiation) could sub-

stantially influence the dynamics and requires investigations of time-dependent models. Stochastic Hamiltonian approach

enables investigations of operators’ time evolution and is a preferable chose for solving problems of this kind, as was re-

cently reported [39] . Strong coupling effects in emitter-nanoparticle hybrid systems could lead to quite remarkable optical

properties, such as appearance of Fano resonances in absorption spectra (e.g. [106,107] ) and others (e.g. for the review [108] ).

Plasmonic antennas could also assist in entanglement generation (e.g. [109,110] ).

Majority of beforehand mentioned approaches adopt classical material susceptibilities and, as the result, could overlook

situations, were emitters situated in a close proximity to particle’s boundaries. Landau damping, direct electron–hole exci-

tations and other effects are not captured by the model.

3.5. Spasers

Nanoscale resonators and antennas could be also employed for controlling stimulated emission regimes, where the near-

filed feedback replaces traveling phase in a cavity. Spaser concept [111] was experimentally demonstrated in few geometries.

The main deference between spaser and plasmonic nanolaser is the degree of confinement – while the first one operates in

completely quasistatic regime ( [112] with a debates on whenever spasing or random lasing was observed), the later has at

least one dimension where retardation effects take place (recall the differences with Section 3.3 ). Apart from being coherent

near-field sources, spaser were also proposed for biomedical applications [113] . Additional advantage of nanoscale resonators

is their potential ability to support ultra-fast interaction regimes [114,115] , associated with fast relaxation of electrons in

a metal, carriers’ injection mechanisms and others. Majority of experimental demonstrations and theoretical predictions

in this niche are described with either semi-classical tools (without quantizing electromagnetic fields) or by applying the

Fermi Golden Rule, assuming the weak coupling regime of interaction. Furthermore, nano-resonators, being inherently open

6 Bigger plasmonic particles could also serve as antennas. In this case, retardation effects play an important role. Furthermore, several plasmonic particles

could form an antenna; here phase accumulation in propagation between different particles will result in antenna’s gain and directivity [95] .

P. Ginzburg / Reviews in Physics 1 (2016) 120–139 129

in majority of cases, enable pathways to a direct spontaneous emission to the far-field, without being coupled to localized

modes. Those direct radiative channels have several advantages over conventional cavities [38] . The open nature of nano-

resonators is rather related to technological aspects and, in principle, is not fundamental. Investigation of quantum aspect

of spasers requires development of mesoscopic tools – one of the examples, related to linewith enhancement [116] , will be

described hereafter.

3.6. Hyperbolic metamaterials

Subwavelength plasmonic particles with pre-designed hierarchy of resonances could serve as building blocks of larger

scale structures – metamaterials. For example, gold nanorods, closely packed in to arrays could exhibit collective responses

due to modes’ hybridization of individual structures (e.g. [117,118] ). Furthermore, since experimental studies of light-mater

interactions on atomic scales are still quite challenging, metamaterials, having unit cells of tenth of nanometers in size,

already facilitates addressing few important aspects of related phenomena.

One of the very important class of structured media is the Hyperbolic Metamaterial – artificially created uniaxial crystal

with epsilon ordinary and extraordinary having opposite signs (positive and negative, or vice versa) [119] . The unusual

dispersion of electromagnetic waves, allowed to propagate inside this material, results in infinitely high density of photonic

states and, as the result, promises infinitely fast spontaneous emission rates. Few experiments with various metamaterials

realizations were reported (e.g. [120,121,25] ), and suggested for quantum applications [122] . Furthermore, additional nano-

patterning of layered metamaterial was show to enhance emission rates many folds [123] . However, fundamental limits of

emission rates in Hyperbolic Metamaterials should be addressed from a theoretical standpoint and it will be demonstrated

hereafter.

4. Mesoscopic models

One of the challenges and key questions in studies of light-matter interactions in the presence of nanostructured mate-

rials is whenever classical permittivity/permeability ( ε/ μ) descriptions of homogeneous materials could be adopted. Those

aspects are especially important when emitters approaching sharp material boundaries or even situated inside crystals. In

both of the cases, internal structure of an emitter could become comparable with lattice constants, questioning the validity

of homogenized classical descriptions. Those aspects of relevant dimension parameters are also related to tunneling across

small gaps and nonlocal material responses. Several mesoscopic approaches, treating material components beyond their clas-

sical permittivity description, will be discussed in this section. While phenomenological classical models could be applied

to few of the examples below, quantum mechanical treatment gives a deeper insight into phenomena.

4.1. Memory effects in LPR decays due to near-field feedback

The knowledge of relaxation dynamics of LPRs is of primary importance for analyzing behavior of hybrid light sources.

Light emitters, coupled to localized modes of plasmonic structures, undergo relaxation oscillations owing to both far-field

radiation and dissipation losses. The later mechanism play a key role for small particles, since their radiation efficiency is

small [124] . While laws of LPRs decays are usually assumed to be exponential, any physical system with bounded from be-

low energy spectrum cannot relax to its ground state exponentially [125] . Consequently, LPRs, having discrete and bounded

spectrum, could exhibit unusual dynamics of relaxation. Exponential decay laws result from absence of memory effects

(Markov approximation) and a next state of a physical system is defined only by a present without any relation to previous

history of interactions. The origin of the memory effect in LPRs decay relies on strong near-field feedback and, to some

extent, could be interpreted as an analog of Rabi oscillations with an additional damping. Localized resonant modes are

coupled to dissipative degrees of freedom of plasmonic materials, and this stored energy has nonzero probability to make a

revival back to the optical excitation. In this interpretation, energy is exchanged between the optical mode and dissipation

degrees of freedom and it finally decays because of the continuous nature of the latter. It is worth noting two important

aspects – first, losses play the key role and lead to non-exponential decays; second, the effect is much more pronounced for

higher order (dark modes) because of the suppressed far-field radiation channel and stronger near-field feedback. The rigor-

ous quantum theory of the effect was developed in [48] . The key idea behind this mesoscopic model is to keep track after

time-evolution of an Eigen mode and introduce losses at the same time. Straightforward utilization of Mode Decomposition

Method is unable addressing this scenario because of the losses and, even if it will be occasionally applied, it will lead to

an exponential law of decay. On the other hand, the standard theory of dissipation, employed for leaky cavities description

in Quantum Optics, contains no information on cavity’s geometry, which might be important in the case of LPRs. As the

result, neither of those methods standalone could address the nature of non-exponential decays. By adopting the combina-

tion on the above mentioned methods it could be shown, that the following integral form (Laplace transformation) gives

the time-evolution of the mode’s annihilation operator [48] :

a ( t ) =

1

2 π

∫ ∞

−∞

a ( 0 )

is +

�C

2

(ω 2

0 + γ 2

( ω 0 −s ) 2 + γ 2

)· ω

0

( ω 0 −s )

e ist ds , (8)

130 P. Ginzburg / Reviews in Physics 1 (2016) 120–139

where ω 0 is the resonant frequency of the mode, γ the electron collision rate determining the material (Ohmic) losses, and

�C the classical exponential rate of decay, which could be calculated either analytically or numerically with widely available

numerical tools. In fact, Eq. (8) is a hybrid form of Weisskopf–Wigner approximation [32] with explicit introduction of the

dissipation bath, which enables fulfilling canonical commutation relations between mode’s creation/annihilation operators.

Finally, it was shown on an example of small spherical particles, that steep dispersion of the nano-cavity Eigen modes

together with strong near-field feedback lead to considerable deviations from generally assumed conventional exponential

decay law with up to 40% changes in the relaxation time. This effect could have quite an important impact on performances

of light sources and spasers. It should be emphasized, that the non-exponential nature of the effect has nothing to do with

non-exponential behavior of many fluorescent dyes owing to their nontrivial internal level hierarchy [55] or averaging of

position-dependent Purcell enhancement in structures lacking translational symmetries, e.g. [26] .

4.2. Spasers and nanolasers linewidth enhancement

Radiation properties of spasers and nanolasers are under extensive studies owing to a progress in their fabrication and

characterization [93] . Linewidth of a laser represents the measure of its temporal coherence and is the one of the most

important characteristics. The spectrum of the radiation undergoes narrowing once the laser passes from the regime of

spontaneous to the predominant stimulated emission and lasing. While the natural Schawlow–Townes limit of the band-

width is inversely proportional to the output power of the radiation [126] , almost all realistic laser systems, in particular

those having semiconductor gain media, cannot reach an optimal proportionality constant due to additional noise factors.

Fluctuations of a resonator, being one of the major reasons, could result from inhomogeneities, mechanical instabilities and

other factors. The important parameter, characterizing this phenomenon, is a linewidth enhancement factor ( α) resulting

from fluctuations of the refractive index of the laser cavity [127] . The actual linewidth is larger than the predicted by the

Schawlow-Townes theory by a factor of α2 + 1. Existent theoretical descriptions of nanolasers and spasers are based on the

notion of LPRs. However, the rigorous treatment of the emission linewidth in those cases cannot be considered with the

same formalism as for conventional lasers, since the later involves time-dependent treatment of the laser cavity and absent

in the quasistatic approximation. First, the quasistatic description of LSP resonances, defining the near-field cavity, does not

include radiation losses, as material losses are predominant mechanism for small particles [17] . Secondly, the quasistatic

approach does not explicitly include the time derivatives of the field. A theoretical framework for linewidth enhancement

in plasmonic nanolasers and spasers was developed at [116] . Here, in order to investigate coherence properties, the time

dependence of quasistatic modes was introduced via the dispersion of spectral parameter, defining the LPRs frequency. For

example, linewidth enhancement of GaAs-based spaser was shown to be of order of 3–6, strongly depending on carrier

density in the active layer. Modified Mode Decomposition Method with explicit introduction of dispersion characteristics

was utilized for addressing linewidth enhancement phenomenon in quasi-static spasers. Full understanding of microscopic

mechanisms in spasers and laser nanocavities is required for analyzing their properties and valuable for their future designs.

4.3. Hyperbolic metamaterials

Hyperbolic metamaterials support modes with special type of dispersion, leading to an infinitely high density of photonic

states and, as the result, infinitely fast spontaneous emission rates (see Section 3.5 ). A basic parameter, enabling treatment

of periodic structured media as homogeneous, is a physical size of forming unit cells. This finite structuring should be

small-compared with an excitation wavelength. Metamaterials’ electromagnetic properties could be homogenized under cer-

tain circumstances [128] and, at the same time, their yet finite size unit cells enables testing various mesoscopic aspects

of light-matter interactions beyond classical permittivity description of material components. In order to describe emission

rates, tailored by hyperbolic type of dispersion in a crystal, Mode Decomposition Method could be adopted. Another option

to address the problem is to calculate Purcell factors relying on Local Quantization and the LDOS, which is proportional

to the imaginary part of electromagnetic Green’s function ( Eq. (5) ). Calculations of LDOS for homogeneous hyperbolic ma-

terial, performed in this way, will lead to divergent values too [129] in a full correspondence between two methods. This

correspondence is the direct consequence of vanishing losses assumption [9] . The major issue of divergent emission rates

in Hyperbolic Metamaterials could be resolved by creating a corresponding mesoscopic model, keeping key aspects of the

interaction. The first one is the material dispersion and absorption, suggesting going beyond the simple Mode Decompo-

sition approach. The second one is related to solid state aspects, as homogeneous properties of an embedding material in

vicinity of an emitter cannot be utilized for describing the interaction. Local field correction model relies on exclusion of a

small depolarization volume around an emitter in order to bypass the problem of material’s granularity [130] . However, ge-

ometries of excluded volumes have nontrivial impacts on resulting Green’s functions, making predictions on emission rates

to be ambiguous. Local quantization approach was applied on the case of spherical depolarization volume in homogeneous

materials, taking into account losses and dispersion [131] . Local field correction formalism for hyperbolic metamaterials, in-

cluding aspects of spatial dispersion, was recently developed by adopting transfer matrix method. Depolarization volume,

in this case was taken to be a thin layer of a homogeneous material with the same effective real part of permittivity as

a layered structure. Losses in this layer were suppressed [132] . Filing the depolarization volume with a material instead of

vacuum insures faster convergence of emission rates.

P. Ginzburg / Reviews in Physics 1 (2016) 120–139 131

While emission rates in a particular hyperbolic metamaterial realization could be straightforwardly addressed with vari-

ous numerical tools, a model for underlining fundamental limits of spontaneous emission rates in hyperbolic metamaterials

will be presented hereafter and it utilizes a structure of simple cubic crystal [133] ( Fig. 3 (a)). In order to achieve strong

anisotropic effect, atoms forming the crystal are assumed to be strongly polarized along one predefined direction ( z -axis)

and unpolarized in a perpendicular plane. Effective parameters of the crystal, calculated by averaging single dipole po-

larizability over the unit cell volume result in three different dispersion regimes: (i) elliptic - Re ( ε zz ) > 0 , (ii) hyperbolic

- Re ( ε zz ) < 0 , (iii) and epsilon near zero (ENZ) - Re ( ε zz ) ≈ 0 . The permittivities in the transverse plane are ε xx = ε yy ≈ 1 ( Fig.

3 (b)). An emitter is situated inside the unit cell and interacts with the entire crystal. While the Mode Decomposition Method,

relying on DOS calculations, will provide a divergently high rate of spontaneous emission, the solid state mesoscopic model,

described above, treats the interaction in a completely different way. Here, the emitter interacts with the whole (infinite)

set of lattice dipoles, which, in turn, interact with each other, creating a collective excitation of the entire structure. The rate

of spontaneous emission in this case is calculated by adopting the Green’s functions approach. Fig. 3 (c) shows the disper-

sion regimes of collective modes (k-vectors of modes, allowed to propagate inside the crystal), calculated by evaluating lat-

tice sums, while the insets show corresponding Green’s functions, obtained either for the effective medium (homogeneous)

crystal or for the discrete lattice [133] . The comparison between Green’s functions in both cases underlines inapplicability

of using effective medium approximation in the case of hyperbolic dispersion. Furthermore, lattice sums enable addressing

spatial dispersion effects, emer ging at ENZ regime. Here, the both hyperbolic and elliptic dispersions (so-called additional

waves [134] ) emerge [135] and cannot be described in the frame of simple homogenized effective medium. As the result, the

discrete lattice model of hyperbolic metamaterial suggest still high but finite rates of spontaneous emission, limited by the

size of the unit cell. The discrete lattice model, adopted for describing the interaction is, in fact, the extension of the Hop-

field approach, while the emission rates calculations utilize Langevin quantization technique (Green’s functions and related

LDOS). Combination of those two approaches is the basic idea behind the developed mesoscopic model. The intuitive expla-

nation of this result, beyond the rigorous derivation of [133] , is as follows: a localized dipolar emitter could be decomposed

into a collection of plane waves with an infinite spectrum in a momentum space (k-space). K-vectors, that are shorter than

the reciprocal lattice of the crystal, will experience a homogeneous structure (similar to the criteria of ‘excitation wavelength

is larger than a metamaterial period’), while long k-vectors of the point dipole decomposition will propagate in a periodic

(photonic crystal-like) medium. The finite size of the unite sell or, more generally a characteristic length of a structure, pro-

vides a fundamental limit for emission rates inside hyperbolic metamaterials. For example, hydrodynamic type nonlocalities

of free electron gas could also serve as a rate limitation [136] – here the Fermi wavelength (so-called nonlocality radius

[137] ) is the characteristic length-scale.

Recent realizations of hyperbolic metamaterials are commonly based on layered metal-dielectric structures [24] or on

arrays of vertically aligned nanorods, [117] . In the later case the effective medium approximation was demonstrated to be

in a full correspondence with experimental data, obtained with is with ellipsometric techniques (k-vectors, shorter than

the reciprocal lattice, were involved) [118] . However, investigations of emission inside the nanorods arrays shows a quali-

tatively different behavior, favoring contribution of Fabry–Perot modes in finite size structures over bulk properties [138] .

Near-field interactions between rods and objects usually dominate collective phenomena, as was shown for opto-mechanical

interactions, for example [139] .

4.4. Nonlocalities, tunneling and Ab initio approaches

One of very important aspects, affecting light-matter interactions with nanostructures, is the presence of very small fea-

tures, which could respond to electromagnetic field differently from bulk materials. Many probable mechanisms could affect

the phenomena, among them nonlocalities and spatial dispersion, matter granularity, electron spill out and tunneling, and

few others. Careful consideration of all probable effects require ab initio modeling of material components by considering

quantum potentials of each constitutive element (e.g. [140] and many others since then). Being very heavy by means of

computations, various multi-scale methods are under extensive studies nowadays 7 . Few plasmonic structures were recently

analyzed with various ab initio approaches, but are still capable of addressing only very small structures and clusters (e.g.

[141–145] ). However, in order to investigate intermediate scenarios, where structures are too small for being addressed clas-

sically, but too big for atomistic simulations, mesoscopic approaches are required. Hereafter, few of them will be mentioned

in brief.

Optical properties of plasmonic nanostructures could be described, within a good approximation, by considering dynam-

ics of conduction electrons in noble metals. Hydrodynamic model, approaching this dynamics with equations of charged

fluids driven by electromagnetic fields, enables going beyond classical Drude description of simple harmonic oscillator [146] .

The added value of this description is in the ability to account for nonlocalities (e.g. [147] ), owing to collective electron–

electron interactions [148] , and nonlinear effects [149,150,151] . While there are few quantum hydrodynamic models available

(e.g. [152] ), light-matter interactions with nonlocal responses of this kind are not fully explored [153] .

Another important effect, that could be faced while dealing with very small gaps, is quantum tunneling. Recent experi-

ments reveal the presence of this effect in plasmonic structures [154] . Quantum-corrected model of tunneling in plasmonic

7 http://www.nobelprize.org/nobel _ prizes/chemistry/laureates/2013/

132 P. Ginzburg / Reviews in Physics 1 (2016) 120–139

Fig. 3. (Color Online). Mesoscopic model for description of light emission in Hyperbolic metamaterials. (a) Unit cell of anisotropic crystal (simple cubic).

Light emitter (red sphere) is situated inside the unit cell. (b) Effective dielectric tensor components ( Re ( ε zz ) - blue line, ( Im ( ε zz ) - red line) of the crystal

as the function of frequency, obtained by means of averaging dipolar responses over the unit cell. (c) Central graph – dispersion relation of polaritons,

allowed propagating in the lattice. Axis – components of k-vectors of polaritons. Insets – intensities of dipolar emission (emitter’s position inside the unit

cell is in the lower-left inset). Radiation patterns, calculated in the frame of effective medium theory are on the right-hand side, while results of lattice

summations (true polaritons modes) are on the left. Upper set of insets correspond to the elliptic regime of dispersion, while the second row of insets

associated with the hyperbolic. ENZ regime shows mixed dispersion (both elliptic and hyperbolic (additional wave)). Arrows assist associating field patterns

with relevant polaritons dispersions. The mesoscopic model was developed in [133] . (For interpretation of the references to color in this figure legend, the

reader is referred to the web version of this article).

P. Ginzburg / Reviews in Physics 1 (2016) 120–139 133

particles was reported and discusses many aspects, related to this type of interaction [155] . Density functional theory was

employed for describing tunneling effects over small gaps under intense external illumination [156] . The topical review on

the subject of quantum mechanical effects in structures with very small gaps could be found in [28] .

4.5. Summary

There is a great variety of other mesoscopic models in the field. The current section briefly introduced some of them,

emphasizing impacts quantization approaches, discussed in this review. It is worth noting, that the meaning of ‘quantum’

in application to beforehand mentioned examples varies from model to model, as each one requires a unique mesoscopic

approach. In those cases different aspects of quantum nature should be captured, while others, less important, to be sup-

pressed in order to simplify theoretical descriptions. Future development of related approaches promises to bridge the gap

between macroscopic and microscopic descriptions of light-matter interactions phenomena with nanostructures involved.

5. Emulation of complex light-matter interaction phenomena

It is quite common occurrence that completely unrelated physical phenomena are described with the same, or at least

very similar, set of mathematical equations. For example, gravity effects on light could be emulated by considering light-

matter interactions in a lab (e.g. [157] ), waveguide modes could be an analogue of electrons in quantum wells (e.g. [158] ),

along with many others available examples. Furthermore, bringing entire conceptions from one scientific discipline to an-

other has a potential of delivering enormous impacts, e.g. as in the case of photonic crystals [159,160] .

In the context of PCQED scalability of Maxwell’s equations in respect to an operation frequency provide very powerful

tools for detailed studies of various complex scenarios. Scaling up of physical dimensions makes fabrications and measure-

ments processes being straightforward and gives enormous advantages for experimental investigations. For example, the

mature field of microwave technology provides many techniques, developed over the last century. It is worth underlining

that quantum effects at the GHz range are smeared by thermal noise at room temperature conditions and their observation

requires substantial cooling, e.g. [161] . The emulation concept, described hereafter, relies on classical-quantum correspon-

dence and aims to provide a tool for analog solution of complex electromagnetic problems. In particular, as it was underlined

at the beginning of this review, electromagnetic fields could be quantized by means of classical Green’s functions [49] . Scal-

ability properties enable efficient calculations (physical measurements at this case) of those impulse responses. Results of

those experiments could serve as an input into operators’ equations of motion, describing full quantum dynamics of a pro-

cess. One of the representative examples is related to the process of spontaneous emission next to plasmonic particles. Fig.

4 (a) shows scanning electron microscope (SEM) image of plasmonic counterpart of Yagi-Uda antenna, loaded by a quantum

dot [162] . Being fairly complicated experiment in optics, this scenario could be relatively straightforwardly realized at the

microwave regime 8 . For example, array of carefully designed ceramic balls of different radii in conjugation with a feeding

elements ( Fig. 4 (b)) provide emulation for the relevant phenomena 9 . In this case both all components of electromagnetic

field could be measured.

In order to achieve a full correspondence between wave phenomena at optical and microwave spectral ranges material

properties should be scaled accordingly. Temporal dispersion of dielectric constants prevents the direct replication. How-

ever, well developed concepts of microwave metamaterials enable achieving quite a broad span of dielectric parameters at

relatively narrow, but yet sufficient, frequency bandwidth [163] . One of the examples, that will be reviewed here, is related

to the effect of Near-field Interference, providing the possibility to control photon propagation direction with polarization

handedness of the later [164] . This effect, being hard for a direct experimental observation in optics, was emulated at MHz

frequency range [165] . The key idea here relies on several steps with the major one, related to replication of Hyperbolic

metamaterial at low frequencies with the help of lumped circuits ( Fig. 4 (c)). By bringing the analogy between Kirchhoff’s

equations on a discrete grid and Maxwell’s wave equation for transverse magnetic (TM) waves in a continuum media, it

was shown that the flat periodic array of capacitors and inductors ( Fig. 4 (c, inset)) could serve as a replica of homogeneous

anisotropic crystals. The following set of equations underlines the analogy. Kirchhoff’s law of currents, for one node, marked

with x and y subscripts, is given by:

Y x (U x −1 ,y + U x +1 ,y

)+ Y y

(U x,y −1 + U x,y +1

)− 2 U x,y ( Y x + Y y ) = U x,y Y z , (8)

where U i, j indicates a voltage at the node and Y k is a relevant admittance. By identifying the discrete differences form in Eq.

(8) , the latter could be rewritten in the form of continuous derivatives as:

Y x ∂ xx U + Y y ∂ yy U =

1

S Y z U , (9)

where ˜ U is the continuous counterpart of the voltage and S is the area of the unit sell. The validity of the passage between

Eqs. (8) and ( 9 ) is directly related to material granularity aspects, whenever they are important. Eq. (9) has the exactly the

8 In this particular example the historical occurrence was reversed - Yagi-Uda antenna was demonstrated for radio waves first, of course [51] . 9 The major goal of this work was to bring an analogy with all-dielectric photonic components.

134 P. Ginzburg / Reviews in Physics 1 (2016) 120–139

Fig. 4. (Color Online). The concept of optical emulation with microwaves. (a) SEM image of Yagi-Uda nanoantenna with, fed by a quantum dot [186] . (b)

Yagi-Uda microwave antenna, fed by a classical source [187] . (c) Experimental setup for emulating Near-field interference effect – Hyperbolic metamaterial

is realized as a planar periodic array of lumped elements. The structure is excited by a loop antennas and the field is scanned by a wire probe. Inset in a

red-dashed frame – zoom on the excitation area + lumped circuits’ implementation of the metamaterial. (d) Simulation of the near-field in the metamaterial,

excited by linearly-polarized dipole. (e) Experimental data, corresponding to panel (d). Figures on panels (c)–(e) are adopted from [165] . (For interpretation

of the references to color in this figure legend, the reader is referred to the web version of this article).

same structure, as the wave equation for TM modes in uniaxial crystal:

1

ε z ∂ xx H y +

1

ε x ∂ zz H y = ω

2 μ0 H y , (10)

where H y is magnetic field component of the wave, ε’s tensor components of crystal’s permittivity, and ω operation fre-

quency. The similarity between Eqs. (9) and ( 10 ) is the basis for the emulation experiment. Figs. 4 (c and d) show the

intensity of a field (theory and experiment), radiated by linearly polarized dipole, interacting with the metamaterial. As

a side remark, experimental data and electromagnetic simulation of the phenomenon in continuous media show quite a

remarkable difference, related to a discrete nature of the metamaterial. This experimental setup could be further utilized

for investigating impact of material granularity and nonlocality on spontaneous emission processes. The effect of Near-field

interference in optical domain was recently demonstrated with cesium atoms in a magnetic field [166] and with hyperbolic

metasurfaces [167] and both required quite a substantial experimental efforts.

The impact of emulation experiments on the field of PCQED is not yet fully understood and currently is under extensive

studies, already showing very promising directions.

6. Higher-order effects in PQCED

The ability of plasmonic structures to manipulate near-fields is beneficial for controlling higher order interactions, where

several light quanta are involved. The scope of relevant problems could be divided in to three sub-sets – spontaneous pro-

cesses, stimulated and their combinations. It is worth noting, that nanostructures could simultaneously provide field en-

hancement for both real and virtual photons (vacuum fluctuations). In terms of quantum-mechanical description the prob-

ability amplitude of an interaction is proportional to photon number-state (n) with an addition of unity (n + 1) in the case

of spontaneous process. Quantum aspects of nonlinear interactions could be treated with the help of effective Hamiltonians,

derived in the frame of Green’s functions quantization scheme [168] .

One of the spontaneous multi-photon interaction examples is the two-photon emission (spontaneous two-photon emis-

sion – STPE). In this process electrons undergo spontaneous transition from an excited state to a ground one by emitting

photon pairs. Regular one-photon transitions could be either suppressed by certain selection rules (e.g. 2s −1 s transitions in

hydrogen atoms [169] ), or just spectrally filtered out in the case of semiconductors [170] . Simultaneous emission of photon

pairs ensures a presence of energy-time entanglement states [171,172] , beneficial for quantum information processing appli-

cations [6] . STPE could serve as an alternative to spontaneous parametric down conversion (SPDC) [173] , as both could be

employed for achieving heralded single photon sources. However, STPE is inherently weak process, as it involves combina-

tion of two vacuum fields [174,175] . Similar problem in SPDC is solved by enforcing phase-matching conditions and making

P. Ginzburg / Reviews in Physics 1 (2016) 120–139 135

an entire crystal to operate as a whole unit. STPE, on the other hand, could be substantially enhanced by plasmonic struc-

turing, as the later could serve as an optimal tool for increasing efficiencies of inherently weak photonic processes [176] .

STPE from GaAs bulk semiconductors, being more than five orders of magnitude weaker than one-photon emission [170] ,

was shown to be enhanced by almost tree orders of magnitude by arrays of plasmonic bow-tie antennas, deposited on top

of the bulk-air interface [177] . In this case the process of plasmonic enhancement is different from a regular one-photon

Purcell effect. Since each pair of photons with energies adding up to an overall energy of the electron transition could be

emitted, extremely broad-band enhancement is required. In contrary to high-Q microcavities, plasmonic resonators are in-

herently broadband and, in the case of semiconductor STPE, are an ultimate solution for manipulating emission rates and

spectra. Two-photon Purcell enhancement was derived by employing Green’s function quantization scheme and is based on

a spectral convolution between one-photon LDOS [178] .

Nonlinear processes in plasmonic structures are under extensive studies nowadays [179] , as they hold a promise to de-

liver improved performances of nonlinear generation on smaller scales. However, spontaneous higher-order processes, such

as SPDC, still require fulfilling phase matching conditions. In this case, coherence build up should predominate ohmic losses,

and this condition is still hard to achieve with lossy plasmonic structures. As a theoretical challenge it is worth noting, that

phased-matched nonlinear conversion is usually takes place between modes of a crystal. However, this mode decomposition

technique fails is the case of lossy dispersive metals. On the other hand, local Green’s function quantization is a cumber-

some tool for addressing collective effects of this kind. Furthermore, processes with external illumination involved, require

a seemingly different treatment for incident fields and locally created spontaneous light quanta. This issue was addressed

in part at [180] . Another promising direction in studies of down conversion processes, inspired by plasmonic structuring,

is an adaptation of mesoscopic plasmon models for conduction electrons. One of the widely used models, encapsulating

collective effects of electrons beyond their classical Drude description, is the Hydrodynamic model. It could capture effects

of nonlocality in small metal structures [148] and provide an information on nonlinear harmonic generation of any order

[151] . Nonlinear scattering of a photon by a two-level system ultrastrongly coupled to a waveguide was analyzed [181] . This

model utilizes a chain of coupled bound states and has a relation to the model, described in Sections 4.3 and 5 . Among

other higher-order effects, that could be substantially enhanced by plasmonic structuring, is a resonant energy transfer,

where a donor molecule conveys its excitation to an acceptor without radiating a photon. This process could be described

by classical dipole-dipole interaction (up to some extend), or quantum-mechanically (rigorously) by considering virtual pho-

ton exchange. Since a pair of photons is involved in the process, it might be considered as a higher-order [55] . Plasmonic

nanostructures hold a potential of increasing the interaction range and plasmon-assisted energy transfer was studied both

theoretically [182] and experimentally [183,184] , with still debatable conclusions. This process was recently emulated in

microwave spectral range by utilizing high-Q ceramic resonators [185] .

7. Outlook and conclusions

Nanostructured media provide vast range of opportunities for tailoring and controlling dynamics of light-matter interac-

tions. General complexity of experimental setups and related (to a certain extend) ambiguous experimental data motivates

the development of rigorous theoretical frameworks, leading to deeper understanding of basic natural phenomena. Several

theoretical approaches, which enable analysis of quantum phenomena in structured media, were reviewed above and their

applicability to certain scenarios was discussed. In particular, Mode Decomposition Method, Hopfield, Huttner and Barnet

Models and Green’s functions (Langevin) local quantization were revised. Limitations of those methods are sometimes over-

looked, since transitions between micro- and nano-structures involved in interactions, are not always result in a straightfor-

ward scaling. In many cases, complex light-matter interaction phenomena, tailored by nanostructures, cannot be rigorously

analyzed with those conventional tools and require the development of new mesoscopic models. The key idea behind those

models is to capture major contributing effects and factor out the rest, greatly reducing the complexity of the analysis. How-

ever, there are no general guidelines for building those models and each particular case requires a unique approach. Among

recent challenges in PCQED is the development of a set of reliable tools that enables addressing the majority of light-matter

interaction scenarios with subsequent investigations of applicability ranges. While several examples, utilizing certain combi-

nations between above mentioned methods, were presented, many important phenomena, such as electron tunneling over

small gaps, material nonlocality, temperature and non-equilibrium effects and several others are not yet developed to the

level of being able to reliably reproduce experimental data and predict new effects. The com plementary and relatively new

approach, aiming to address complex interaction behavior by means of analog simulation, is an emulation of optics by mi-

crowaves. Several recent experiments underline a possible strength of the method, while its’ impact on predicting quantum

aspects of interactions is due to come.

Acknowledgments

The author acknowledges support from TAU Rector Grant, German-Israeli Foundation (GIF, grant number 2399 ). The work

was supported in part by the Russian Fund for Basic Research (project #15-02-01344 ), Government of the Russian Federation

(No. 074-U01 ). Chapter 2 has been supported by the Russian Science Foundation Grant No. 14-12-01227 .

136 P. Ginzburg / Reviews in Physics 1 (2016) 120–139

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