Spontaneous emission and the operation of invisibility cloaks: Can the invisibility cloaks render objects invisible in quantum mechanic domain?

# Spontaneous emission and the operation of invisibility cloaks: Can the invisibility cloaks render objects invisible in quantum mechanic domain?

## Abstract

As a probe to explore the ability of invisibility cloaks to conceal objects in the quantum mechanics domain, we study the spontaneous emission rate of an excited two-level atom in the vicinity of an ideal invisibility cloaking. On this base, first, a canonical quantization scheme is presented for the electromagnetic field interacting with atomic systems in an anisotropic, inhomogeneous and absorbing magnetodielectric medium which can suitably be used for studying the influence of arbitrary invisibility cloak on the atomic radiative properties. The time dependence of the atomic subsystem is obtained in the Schrodinger picture. By introducing a modified set of the spherical wave vector functions, the Green tensor of the system is calculated via the continuous and discrete methods. In this formalism, the decay rate and as well the emission pattern of the aforementioned atom are computed analytically for both weak and strong coupling interaction, and then numerically calculations are done to demonstrate the performances of cloaking in the quantum mechanics domain. Special attention is paid to different possible orientations and locations of atomic system near the spherical invisibility cloaking. Results in the presence and the absence of the invisibility cloak are compared. We find that the cloak works very well far from its resonance frequency to conceal a macroscopic object, whereas at near the resonance frequency the object is more visible than the situation that the object is not covered by the cloak.

Canonical quantization, Spontaneous emission rate, Green tensor, spherical wave vector functions, Spherical invisibility cloaking

## I Introduction

As a result of the implement of many intriguing features that have not yet been found in nature, metamaterial has attracted a great deal of attention in the field of optics. These features prepare new opportunities for realizing exotic phenomena such as invisibility devices Pendry 2006 ()Schurig 2006 (), superlenses Pendry 2000 (); Schurig 2007 (), field rotators Chen 2007 (), optical analogues of black holes Narimanova 2009 (); Genov 2009 (), Schwarzschild spacetime Chen 2010 (), wormholes Greenleaf 2007 () and the ”Big Bang” and cosmological in ation Smolyaninov 2011 (); Smolyaninov 2012 (). The theoretical basis for some of these phenomena is coordinate transformation, which stems from the formal invariance of Maxwells equations. This enable both physics and engineering societies to manipulate electromagnetic waves in almost any fashion. In this paper, we focus on the invisibility cloaking, and attempt to gain some physical insight regarding the quantum electrodynamics of this topic.

Based on a coordinate transformation, Pendry et al. first proposed an invisibility cloak, which can protect the cloaked object of arbitrary shape from electromagnetic radiation. The external observer got therefore unaware of the presence of the cloak and the object. This idea has been verified numerically by full-wave simulations Cummer 2006 () and experimentally by using metamaterial at the microwave frequency Schurig 2006 (), and even at the optical frequencies Cai 2007 (). However, these invisibility cloaks were encountered a serious limitation: they required extreme values of material properties and can only work within a narrow-band frequency. The first issue is circumvented by using simplified constitutive parameters Schurig 2006 (). To overcome the bandwidth limitation, it was proposed carpet cloak to conceal an object that is placed under a bulging reflecting surface by imitating the reflection of a flat surface Li 2008 (). Such cloaks were experimentally demonstrated in both microwave Liu 2009 (); Ma 2010 () and optical frequencies Valentine 2009 ()-Ergin 2010 () using metamaterial structures with feature sizes in the centimeter to nanometer scale. It inevitably requires complicated nanofabrication processes which restrict the size of hiding objects in the visible frequencies to a few wavelengths. So, this carpet cloaking has not enabled to hide a large object at least as experimentally. Furthermore, the scattered waves are suffered a lateral shift, which makes the object to be detectable. To bypass these limitations, macroscopic invisibility cloaking has been introduced by using the birefringence property of a natural crystal such as calcite at broadband visible wavelengths Chen 2010 (); Zhang 2011 ().

As mentioned above briefly, all researches on cloaking reported so far have been focused on the case in which the electromagnetic field is taken into account classically. However, the performances of cloaking in the quantum mechanics domain, which can be an important topic, have been rarely considered Kamp 2013 (). The question that naturally arises in this context is whether such invisibility clocks work in the quantum mechanics domain as well as in the classical regime. The present paper is intended to respond to this question. Without loss the generality of our approach, we restrict our attention to the special case at Pendry cloaking which was relatively easy to be constructed, simulated and analyzed.

It is well known that the spontaneous decay of an atom is influenced by the geometry and the optical properties of the material body. We therefore expect that if an atom is located near the ideal invisibility cloaking, the electromagnetic interaction between them will lead to drastically modification of the density of radiation modes and subsequently the spontaneously decay rate. In this sense, it is useful to look at the decay rate of an excited atom as a probe that allows us to examine the operation of this type of cloaks in the quantum mechanic regime. On this base, we study the spontaneous emission of an exited two-level atom and as well the spatial distribution of its radiation intensity as a function of the atomic transition frequency and the distance between the atom and the invisibility cloaking.

On one hand, the spontaneous emission is a phenomena which corresponds to the conventional framework of quantum electrodynamics. On the other hand, such invisibility cloak with position-dependent and anisotropic optical parameters was demonstrated for microwave frequencies by utilizing concentric layers of split-ring resonators. Due to the metallic nature of resonator structure, such invisibility cloaks are usually associated with a high loss factor and accompanied by a strong dispersion to fulfill causality. Therefore, a fully quantum mechanical treatment is needed to consider the dissipative and dispersive effects of the invisibility cloak along with their inhomogeneous and anisotropic features on the spontaneous emission rate.

There are two approaches to quantize the electromagnetic field in the presence of a dissipative and dispersive media, in general: canonical and phenomenological method. In this paper, we consider the rigorous canonical approach. For this purpose, as a lateral purpose of the present paper, we extend the canonical quantization scheme in Kheirandish 2010 ()-Amooghorban 2016 () to a more general case that the medium is described in terms of a spatially varying and anisotropic permittivity and permeability, and as well the composed field-medium system interacting with atomic systems. This derivation helps us to gain a physical insight into the influence of arbitrary cloak on the atomic radiative properties.

The paper is organized as follows. In Sec. II we present a canonical quantization of the electromagnetic field interacting with charged particles in presence of an anisotropic, inhomogeneous and absorbing magnetodielectric medium. In Sec. III, the spontaneous emission rate of an exited atom near a spherical invisibility cloak is expressed in terms of the imaginary part of the classical Green tensor at the position of the atom. By expanding the Green tensor of system into a modified set of the spherical wave vector functions, the decay rate is computed analytically in both weak and strong coupling regime. As an application of this formalism, the numerical evaluations are performed for the spherical cloak whose material absorption and dispersion is of the Lorentz type. Then, we discuss the role of the orientation of the dipole moment of atomic system. In Sec. IV the spatially intensity of the spontaneously emitted light is calculated as functions of the atomic transition frequency and the distance between the atom and the hidden object which is covered by the spherical invisibility cloaking. A summary and conclusions are presented in Sec. V. The derivation of the Green tensor via two methods, exact and discrete, is provided in Appendix A.

## Ii Canonical quantization of electromagnetic field

Our analysis of the spontaneous decay of an excited atom placed in vicinity of a Pendry cloaking is based on a generalization of a canonical scheme for quantization of the electromagnetic field in an isotropic magnetodielectrics medium developed in Refs Kheirandish 2010 ()-Amooghorban 2016 (). We give only the bare essentials needed for an appreciation of the present paper.

Based on an inspiration of the microscopic Hopfield model Hopfield 1958 (), quantum electrodynamic in an inhomogeneous, anisotropic, dissipative and dispersive magnetodielectric medium can be accomplished by modeling the medium by two independent reservoirs comprised of a continuum of three dimensional harmonic oscillators. These two independent sets of harmonic oscillators are characterized by means of two harmonic oscillator fields and which interact with the electric and the magnetic fields through a dipole interaction term. Hereby, we can describe the polarizability and the magnetizability characters of the magnetodielectric medium, as well as its dissipative behavior. Bearing these in mind, let us start with the total Lagrangian density of the system composed of the electromagnetic field, the external charged particles and the medium including the dissipative behavior

 L=LEM+Le+Lm+Lq+Lint, (1)

where the electromagnetic part has the standard form that the electric field and magnetic field are written in terms of the vector potential and scalar potential . The electric and the magnetic parts of the material Lagrangian density and , which are modeled by a continuum of harmonic oscillators, are given by and , respectively. The polarization and magnetization fields of the medium in term of two harmonic oscillator fields and are defined as

 P(r,ω)=∫∞0dω¯¯ge(r,ω)⋅Xω(r,ω), (2a) M(r,ω)=∫∞0dω¯¯gm(r,ω)⋅Yω(r,ω), (2b)

where the interaction with the material is described via the coupling tensors, and , which are assumed to be analytic functions of in the upper half plane. It is worth noting that the permittivity and the permeability of medium will be determined in term of these coupling tensors. So, we take here the coupling tensor as a function of position to be the second order tensor, since the Pendry clocks under consideration are nothing but an inhomogeneous and anisotropic metamaterial.
The forth term in Eq. (1) is the Lagrangian density of free charged particles with particles mass and position , which is written as

 Lq=12∑αmα˙r2α. (3)

Finally, is the interaction Lagrangian density which includes the linear interaction between the medium and the charged particles with the electromagnetic field. It is found that such interaction is given by

 Lint = J(rα,t)⋅A(rα,t)−ρ(rα)φ(rα) (4) + P(r,t)⋅E(r,t)+∇×A(r,t)⋅M(r,t)

where is the current density of charged particles. An analysis of the Lagrangian density (1) shows that the scalar potential does not appear. Therefore, the scalar potential is not a proper dynamical variable and the corresponding equation of motion can be treated as a constraint. It enables us to eliminate the scalar potential, and then get a reduced Lagrangian where only the vector potential A, the material fields and , and their time derivatives are appeared. To do this, we apply Euler-Lagrange equations to the scalar potential. It leads to

 φ = φA+φp (5) = 14πε0∫d3r′ρA(r′)|r−r′|+14πε0∫d3r′ρp(r′)|r−r′|,

where and are the charge density and polarization-charge density, respectively, and subsequently and are the corresponding scalar potentials arisen from these charge distributions. By substituting Eq.(5) into Lagrangian (1), the total Lagrangian can be recast into the reduced form

 L = 12∑αmα˙r2α+12ε0˙A2(r,t)−12μ0(∇×A(r,t))2 (6) + 12∫∞0dω{˙X2ω(r,t)−ω2X2ω(r,t)} + 12∫∞0dω{˙Y2ω(r,t)−ω2Y2ω(r,t)} + ∑αeα˙rα⋅A(rα,t)+A⋅˙P(r,t)+M⋅∇×A(r,t) − Wcoul,

where is Coulomb energy of the charged particles, the polarization-charge and their interactions which in term of and is defined as

 Wcoul = 12∫d3rρA(r)φA(r)+∫d3rρA(r)φP(r) (7) + 12∫d3rρP(r)φP(r).

The Lagrangian (6) can now be used to obtain the corresponding canonical conjugate variables for the fields

 −ε0E⊥(r,t)=δLδ˙A(r,t)=ε0˙A(r,t), (8a) Qω(r,t)=δLδ˙Xω(r,t)=¯¯ge(r,ω)A(r,t)+˙Xω(r,t), (8b) Πω(r,t)=δLδ˙Yω(r,t)=˙Yω(r,t), (8c) pα(rα,t)=∂L∂˙rα=mαrα+eαA(rα,t). (8d)

Now, the transition from the classic to the quantum domain can be accomplished in a standard fashion by applying commutation relation on the variables and their corresponding conjugates. For the electromagnetic field, we have

 [^A(r,t),−ε0^E⊥(r′,t)] = iℏδ⊥(r−r′), (9)

and for the material fields and the dynamical variable of charged particles

 [^Xω(r,t),^Qω′(r′,t)] = iℏδ(r−r′)δ(ω−ω′), (10a) [^Yω(r,t),Πω′(r′,t)] = iℏδ(r−r′)δ(ω−ω′), (10b) [qα,^pβ(r,t)] = iℏδαβ. (10c)

By applying the Lagrangian (1) and the expressions for canonical conjugate variables in (8), we can form the Hamiltonian density as,

 H = ∑α12mα[pα(rα,t)−eαA(rα,t)]2+12ε0E⊥2(r,t) (11) + B2(r,t)2μ0+12∫∞0dω{Qω(r,t)+ω2˙X2ω(r,t)} + 12∫∞0dω{Πω(r,t)+ω2˙Y2ω(r,t)} − ∇×A(r,t)⋅M(r,t)−˙P(r,t).A(r,t) − 12∫∞0dω(¯¯ge(r,ω)⋅A(r,t))2+Wcoul.

By using the Hamiltonian density (11) and recalling the commutation relations (9) and (10), it is straightforward to prove that the Heisenberg equations for the vector potential, the transverse electric field and the particle coordinates yield the correct Maxwell equations and the Newtonian equation of motion in the quantum domain. Let us begin with the Heisenberg equations for the vector potential and the transverse electric field. Thus, the time derivative of  and  in a straightforward manner is given by

 ˙A(r,t) = 1iℏ[A(r,t),H]=−E⊥(r,t), (12a) ε0˙E⊥(r,t) = 1iℏ[ε0E⊥(r,t),H]=∇×∇×A(r,t)μ0 (12b) − ∇×M(r,t)−˙P⊥(r,t)−J⊥(r,t)

By using the constitutive equations of the displacement field and the magnetic field strength , Eqs.(12) lead to and as expected, where is the transverse displacement field and is transverse component of current density. In the presence of charged particles, the longitudinal components of electric and displacement fields can be written respectively as

 E∥(r,t) = −P∥(r,t)ε0−∇φA, (13a) D∥(r,t) = ε0E∥(r,t)+P∥(r,t)=−ε0∇φA. (13b)

The Heisenberg equation of the charged particles in presence of  and  leads to the quantum mechanical version of Lorentz force, namely,

 m¨rα=1iℏ[m˙rα,H]=eαE(rα,t)+eα˙rα×B(rα,t).

Calculations analogous to those of (12) give the following Heisenberg equations for the dynamical variables and , respectively, as

 ¨Xω(r,t) = −ω2Xω(r,t)+¯¯ge(r,t)⋅E(r,t), (15a) ¨Yω(r,t) = −ω2Yω(r,t)+¯¯gm(r,t)⋅B(r,t). (15b)

The formal solution of Eq.(15a) is obtained as

 Xω(r,t) = (˙Xω(r,0)sinωtω+Xω(r,0)cosωt) (16) + ¯¯ge(r,ω)⋅∫t0dt′E(r,t′)sinω(t−t′)ω.

A similar relation also holds for . To facilitate the calculations, let us introduce the following annihilation operators:

 fe(r,ω,t)=1√2ℏω[−iωXω(r,t)+Qω(r,t)], (17a) fm(r,ω,t)=1√2ℏω[ωYω(r,t)+iΠω(r,t)], (17b)

where and denote two independent infinite sets of bosonic operators, which associated with the electric and magnetic excitations of the system. By making use of Eqs. (10a) and (10b), it is easily seen that the bosonic operators have the commutation relations of the form

 [fej(r,ω,t),f†ej′(r′,ω′,t)]=δjj′δ(ω−ω′)δ(r−r′), (18a) [fmj(r,ω,t),f†mj′(r′,ω′,t)]=δjj′δ(ω−ω′)δ(r−r′). (18b)

We can now invert Eq. (17) to obtain the material field and in term of the bosonic operators and . With this in mind, the polarization and magnetization fields of the medium (2) in term of the bosonic operators are written as

 P(r,t)=ε0∫∞0dt′¯¯χe(r,t−t′)⋅E(r,t′)+PN(r,t), (19a) M(r,t)=μ−10∫∞0dt′¯¯χm(r,t−t′)⋅B(r,t′)+MN(r,t),

where electric and magnetic susceptibilities tensors of the medium are respectively defined as

 ¯¯χe(r,t)=Θ(t)ε−10∫∞0dω¯¯gte⋅¯¯ge(r,ω)sinωtω, (20a) ¯¯χm(r,t)=Θ(t)μ0∫∞0dω¯¯gtm⋅¯¯gm(r,ω)sinωtω. (20b)

Here, the superscript indicates the transpose of a tensor.

Let and , respectively, be the electric and the magnetic susceptibilities tensors in frequency space. Then the electric permittivity and the magnetic permeability tensors of the medium in term of the susceptibilities tensors are written as and , where is the identity tensor. These are complex tensors of frequency which their real and imaginary parts satisfy Kramers-Kronig relations and their dependence on coupling tensors, and , are given through the susceptibilities tensors as:

 ¯¯χe(r,ω) = ε−10∫∞0dω′¯¯gte⋅¯¯ge(r,ω′)ω′2−ω2+i0+, (21a) ¯¯χm(r,ω) = μ0∫∞0dω′¯¯gtm⋅¯¯gm(r,ω′)ω′2−ω2+i0+. (21b)

Given the electric permittivity and the magnetic permeability tensors of medium , we can inverse the relations (21) and obtain the coupling tensors in term of these response tensors. Therefore, we find

 ¯¯gte⋅¯¯ge(r,ω) = 2ε0ωπIm[¯¯ε(r,ω)], (22a) ¯¯gtm⋅¯¯gm(r,ω) = −2ωπμ0Im[¯¯μ−1(r,ω)]. (22b)

The fields and in Eqs. (19) are, respectively, the noise polarization and the noise magnetization operators which associated to the dissipation effects within medium. As in the phenomenological method, we can separate the positive and negative parts of fields, like , where is the conjugate of the negative part (analogously for ) and yields

 PN(+)(r,t) = i∫∞0dω√ℏ2ω¯¯ge(r,ω)⋅fe(r,ω,0)e−iωt, (23a) MN(+)(r,t) = ∫∞0dω√ℏ2ω¯¯gm(r,ω)⋅fm(r,ω,0)e−iωt,

By taking the time derivative of Maxwell’s equations Eq. (12b) and using Eq. (19), we obtain the frequency-domain wave equation for the positive-frequency part of the vector potential as,

 ∇×¯¯μ−1∇×E(+)(r,ω)−ω2c2¯¯ε(r,ω)E(+)(r,ω) =μ0ω2PN(+)(r,ω)+iμ0ω∇×MN(+)(r,ω). (24)

The formal solution of the above equation may be obtained through finding an appropriate Green tensor. We thereby arrive at the following expression for the electric field

 E(+)(r,t)=(iωμ0)∫∞0%dω∫d3r′¯¯G(r,r′,ω)⋅ [−iωPN(+)(r′,ω)+∇×MN(+)(r′,ω)]e−iωt, (25)

where is the classical Green tensor that satisfying the inhomogeneous Helmholtz equation with the space- and frequency-dependent complex permittivity and permeability of medium,

 ∇×[¯¯μ−1(r,ω)∇×¯¯G(r,r′,ω)]− ω2¯¯ε(r,ω)c2¯¯G(r,r′,ω)=δ3(r−r′)¯¯I, (26)

The set of Eqs. (23), (II), and (II) together with the commutation relations (12), provide us with the electromagnetic field quantization in an anisotropic, dissipative and dispersive magnetodielectric medium. It is easily seen that these relations are the same relations which were obtained via the phenomenological quantization method Matloob 1995 ()Dung 2003 (). Thus, based on a rigorous quantization scheme, we arrive at the identical results.

## Iii spontaneous decay of an excited two-level atom

### iii.1 The model

Let us consider an excited two-level atom with transition frequency and the dipole moment placed at the point in the vacuum near an ideal invisibility cloaking. To simplify the treatment of the problem, we consider a spherical hidden object which is covered by an invisibility spherical shell. Furthermore, with regard to the symmetry of this cloak, we assume that the atom located on axis. Our approach can be simply extended to other type of cloaks and other locations. In this case, the cloak in the annular region is a kind of rotationally uniaxial media characterized by Pendry 2006 ()

 ¯¯ε(r,ω)=[(εr−εt)]^r^r+εt¯¯I, (27)
 ¯¯μ(r,ω)=[(μr−μt)]^r^r+μt¯¯I, (28)

where is the unit dyad, the subscripts and denote the parameters along radial and tangential direction or , respectively, and the permittivity and permeability tensor components for the cloak shell are given by

 εt(ω) = μt(ω)=bb−aκL(ω), (29a) εr(r,ω) = (29b)

Here, is the inner radius (radius of hidden object) and is the outer radius of the cloaking shell [see Fig. 1(a)]. As mentioned before in the introduction, to achieve above material parameters in experimental, the most of the cloak device are constructed with metamaterials consisting of resonating structures. This structure inevitably shows high loss and dispersion. For this purpose, we adopt a single-resonance Lorentz models for both the permittivity and permeability Jackson 1999 (). Thus, the permittivity and permeability tensor components of the cloak have been multiplied by a lorentzian factor, , to consider the material absorbtion and dispersion of metamaterial structures. Here, and are respectively the plasma frequency and the resonant frequency and is the absorption coefficient of the cloaking. Without loss of generality, we assume a homogenous and isotropic object as a hidden object placed in the central region with the material parameters where is a constant.

The results obtained in the previous section can now be used to study the spontaneous emission of an excited two-level atom near such cloaked object. By inserting Eqs. (23) and (II) into (11), the Hamiltonian of the whole system under the electric-dipole approximation and the rotating wave approximation is recast to the following convenient form (refer to Kheirandish 2010 ())

 ^H = ∑λ=e,m∫d3r∫∞0dωℏω^f�†λ(r,ω)⋅^fλ(r,ω) (30) + ℏωA^σ†^σ−[^σ†dA⋅∫∞0dω^E(+)(rA,ω)+H.c.],

Here, and are respectively the atomic lowering and raising operators where () is the upper (lower) state of the atom whose energy is (zero). Furthermore, is transition dipole moment which defined as . Since the spontaneous decay of an initially excited atom is studied here, the state of the whole of system at time can be expanded into the ground and excited states of the composite system including electromagnetic field and the cloak, and , and the unperturbed atomic states as

 |ψ(t)⟩=Cu(t)e−i~ωAt|{0}⟩|u⟩ +∑λ=e,m∫d3r∫∞0dωe−iωtCλl(r,ω,t)⋅|1λ(r,ω)⟩|l⟩,

The population probability amplitudes of the upper and lower states of the whole system, and , can be easily calculated from the Schrödinger equation. In the case of , by making use of the Green tensor integral relationship that was proven in Knöll 2001 (); Dung 2003 (), and inserting the initial conditions and , the following time evolution is yield:

 ˙Cu(t)=−iδωCu(t)+∫t0dt′K(t−t′)Cu(t′), (32)

where the kernel function is determined by the Green tensor of the system in the position of the atom

 K(t−t′) = −1ℏπε0∫∞0dωω2c2e−i(ω−~ωA)(t−t′) (33) × dA⋅Im[¯¯G(rA,dA,ωA)]⋅dA,

in which is the shifted transition frequency of the atom in the presence of the cloak and is the Lamb shift. It is easily seen that the coupled integro-differential equation (32) cannot be solved analytically. But, we can gain a physical insight into the spontaneous emission process of the atomic system near the cloak on base of the analytical solutions. So, we concentrate our attention to the limiting cases of weak and strong atom-field coupling.

### iii.2 Weak and strong coupling regimes

Let us first consider the weak coupling where the atom is only slightly perturbed by the vacuum fields, then the Markov approximation applies. In this regime, the coefficient in Eq. (32) can be replaced by and the time integral in Eq.(33) can be approximated by the zeta function where . By applying these approximations, the probability amplitude is written as the form

 Cu(t)=exp[(−12Γ+iδω)t], (34)

where the Lamb shift and the decay rate are given respectively as

 δω = 1ℏπε0P∫∞0dωω2c2dA⋅Im¯¯G(rA,rA,ω)⋅dAω−~ωA, (35a) Γ = 2~ω2Aℏε0c2dA⋅Im¯¯G(rA,rA,~ωA)⋅dA. (35b)

It is obviously seen that the effect of cloak and their properties are contained in the Green tensor. Therefore, in order to calculate the spontaneous emission rate of the excited atom, we first need to compute the Green tensor of the system.

As seen from Eq. (29), the cloak shell requires the material parameters with radius-dependent and anisotropic characteristics. At the first glance, it seems that the calculation of the Green tensor of such system, which is needed for substitution in Eq. (35b), is impossible. However, in the special case in which the cloak has a symmetric geometry shape such as the spherical cloak, the evaluation of the Green tensor, though lengthy, is rather straightforward. We call this Green tensor extraction exact method against another one which is approximately called discrete method. The latter one is important at least from one aspect: It offers the possibility of realizing such cloak by layered structures in experimental Schurig 2006 (), since the evaluation of the Green tensor is done by a discrete model of layered structure.

Similar to the practical realization of an anisotropic and inhomogeneous cylindrical cloak by concentric layered structure consisting of anisotropic structure Schurig 2006 (), we can imagine a layered structure of homogeneous and anisotropic materials to mimic the ideal cloak by enforcing the tangential components of the permittivity and the permeability of different layers to vary with the radius according to Eq. (29). The cloak that is modeled in this manner [see Fig.1(b)], is an example of a spherically layered magnetodielectric medium for which its dyadic Green function is known Tai 1994 (); Qiu 2007 (). The details of these calculation are omitted here for the sake of brevity. The complete descriptions of both methods are given in Appendix A.

Of course, a cloaking structure was proposed in Huang 2007 (); Qiu 2009 (), that does not require metamaterials to realize the anisotropy or inhomogeneity of the material parameters. This allows us to realize the cloak through natural materials by using a layered structure of alternating homogeneous and isotropic materials. We do not consider this proposed method here.

Now, we assume that the imaginary part of the Green tensor in the resonance region of atom-field coupling has a Lorentzian shape with the central frequency and the half width at half maximum . Unlike to the weak coupling, the zeta function no longer act as the function. In this case, the frequency integral in Eq. (33) can be done by expanding the limit of integration to which leads to Dung 2000 ()

 ¯¯K(t−t′) = −12Γδωce−i(ωc−ωA)(t−t′)e−δωc∣∣t−t′∣∣. (36)

By substituting the above expression into Eq. (32) and making the differentiation of both sides of the resulting equation over time, one arrives at a second order homogeneous differential equation

 ¨Cu(t)+[i(ωc−ωA)+δωc]˙Cu(t)+(Ω/2)2Cu(t)=0,

where . This is in the form of the damped harmonic oscillator equation with the formal solution given for by

 Cu(t) = 12(1+δωc√δωc2−Ω2)e(−δωc+√δωc2−Ω2)t2 + 12(1−δωc√δωc2−Ω2)e−(δωc+√δωc2−Ω2)t2.

It is seen that when , the decay probability amplitude of the upper atomic state shows the well-known phenomenon of damped Rabi oscillations

 Cu(t)=e−δωm2tcos(Ωt/2). (39)

This is the principal signature of strong atom-field coupling. In the opposite case, when , we recover the weak coupling result (34) which obtained within the Markovian approximation.

### iii.3 Analytical and numerical results

Equation (35b) can now be applied to the atom which its atomic dipole moment may be perpendicular to the interface along the z axis, and/or parallel to the interface along the y axis. We denote the former case by the superscript and the latter one by the superscript . With the Greens tensor in hand given in [Appendix A] and use of the symmetry of our system, the scattering part of the Green tensor for the two special cases of radial and tangential direction are simplified as follows:

 ¯¯G(11)s(rA,rA,ω) = ik1μ14π∞∑n=0n(n+1)(2n+1) (40a) × ¯¯G(11)s(rA,rA,ω) = ik1μ18π∞∑n=0(2n+1)B11M(z(1)n(kr))2 (40b) + B11N⎛⎜ ⎜⎝1krd[rz(1)n(kr)]kr⎞⎟ ⎟⎠2.

It is worth noting that in the above derivation of Eq. (40) seams only one of the electromagnetic Green tensors of our system which have been obtained in Eqs. (64a) and (A.2) to be used here. In fact, the exact Green tensor (64a) has exactly the same form as the discrete Green tensor (A.2) for with one exception for the coefficients which given by Eqs. (66) and (77), respectively. This follows from the assumption that the field point and source point in our case are located out of the cloak.

By substituting the expressions above into Eq. (35b), we arrive at the following formulas for the spontaneous decay rate of the aforementioned atom at arbitrary position

 Γ⊥Γ0 = 1+6πωIm[ik1μ14π∞∑n=0n(n+1)(2n+1) (41a) ×B11N⎛⎝z(1)n(kr)kr⎞⎠2⎤⎥⎦,