The influence of chiral spherical particles on the radiation of optically active molecules

In the framework of the perturbation theory of the nonrelativistic quantum electrodynamics, a theory of spontaneous emission of a chiral molecule located near a chiral (bi-isotropic) spherical particle is developed. It is shown that the structure of photons in the presence of chiral spherical particles differs significantly from the structure of TE or TM photons. Exact analytical expressions for the spontaneous emission radiative decay rate of a chiral molecule with arbitrary electric and magnetic dipole moments of transition located near a chiral spherical particle with arbitrary parameters are obtained and analyzed in details. Simple asymptotes for the case of a nanoparticle are obtained. Substantial influence of even small chirality on a dielectric or"left-handed"sphere is found. It is shown that by using chiral spherical particles it is possible to control effectively the radiation of enantiomers of optically active molecules.


I. INTRODUCTION
Chirality is a geometric property of a three-dimensional body not to coincide with its reflection in a mirror in any shifts and turns. Such a property, for example, belongs to a human hand or a spring. The term "chirality" was proposed by Lord Kelvin in 1873 to explain some special properties of molecules [1]. The chirality plays an important role in biology and pharmacy, because many complex organic compounds (amino acids, proteins, and sugars) have chiral properties. For this reason, a body can react differently with different enantiomers of the same substance. For example, the same drug, depending on what type of molecules it contains, may have a different taste and smell, or effect differently. In physics, there is a considerable interest to chiral media [2][3][4][5], in which there is a difference between the distributions of left-and right-handed circularly polarized electromagnetic waves. The simplest example of such an optically active medium is an aqueous solution of sugar. At the present time, the study of the chiral properties is again under special attention because of the possibility to create metamaterials based on chiral objects [6][7][8][9][10][11][12].
The spontaneous emission of atoms and molecules in the vicinity of material bodies is studied in a large number of works. The influence of dielectric microspheres on the spontaneous emission of atoms is considered in [13][14][15][16], spontaneous emission of an atom located near a microsphere made of material with negative refraction was considered in [17] for first time. Optical properties of chiral spherical particles have also been studied quite extensively. At the present time, the scattering of plane electromagnetic waves on homogeneous [3][4][5] and inhomogeneous [18] chiral microspheres, including multilayered chiral microspheres [19,20], is already considered. The scattering of a Hermitian laser beam on a chiral microsphere is studied in [21], the scattering of a plane electromagnetic wave by a chiral sphere located in a chiral medium is considered in [22]. The radiation pressure force acting on a chiral spherical particle placed in circularly polarized wave is examined in [23].
Finally, the radiation of a point dipole source located inside a chiral spherical particle was regarded in [24]. At the same time, as far as we know, there is no investigation devoted to the influence of chiral spherical particles on the radiation of chiral (optically active) molecules.
However, such processes become more and more important because they always appear when different biomolecules and drug molecules are investigated and modified by optical methods.
So, in this paper we will present the detailed investigation of this problem The goal of the present work is to develop an approach that allows describing quantitatively the influence of a chiral spherical particle made of an arbitrary metamaterial on the spontaneous emission of a chiral molecule. All results will be obtained in the framework of nonrelativistic quantum electrodynamics, in the assumption that the radiative linewidth of the molecule is much smaller than the frequency of radiation. The last assumption allows us to use the perturbation theory. The rest part of the paper is organized as follows. In the Section II, the canonical quantization of the electromagnetic field in the presence of a chiral spherical particle with arbitrary permittivity and permeability is performed. In the Section III, we obtain general expressions for the spontaneous emission radiative decay rate of a chiral molecule located near a chiral sphere. In the Section IV, the graphical illustration of the results obtained and their discussion are presented. The geometry of the problem is shown in Fig. 1.

ENCE OF A CHIRAL SPHERICAL PARTICLE
To solve the problem of quantization of the electromagnetic field in the presence of a chiral spherical particle, let us consider a spherical cavity of an infinite radius Λ → ∞ with a perfectly conducting wall and with the a chiral particle located in its center. For simplicity, we assume that the nanoparticle is surrounded by vacuum (see Fig. 1). The usual procedure of quantization of the electromagnetic field in spherical geometry [25][26][27] is inapplicable in the case of chiral particles, and below we will develop a new approach that is valid in this case.
In the description of classical fields in the presence of chiral spherical particles, we will use the method proposed in [4,28]. At the same time, to describe the chiral medium, we use the constitutive equations in the Fedorov's form [29]: where D, E and B, H are the inductions and the strengths of electric and magnetic fields, correspondingly, ε, µ are the dielectric permittivity and the magnetic permeability of the chiral medium, and η is the dimensional parameter of chirality. Substituting equations (1) into Maxwell's equations, one can obtain where k 0 = ω/c is the wave number in vacuum, and χ = k 0 η is the dimensionless parameter of chirality. In Eq. (2) and everywhere further, the time dependence exp (−iωt) is assumed.
If we rewrite the system (2) in the matrix form: then, with the help of the Bohren's transformation for the electric and magnetic fields [28]: the matrix K can be diagonalized: where are the wave numbers of the left-(L) and right-polarized (R) waves. The components of the transformed field (4) satisfy the equations [4,28]: Thus, the fundamental solutions of the Maxwell's equations in a chiral medium are the left-polarized and right-polarized waves with different wave numbers.
In the spherical geometry under consideration, in the case of an arbitrary electromagnetic field, the expressions for Q L and Q R can be written down as follows (we will omit a multiplier exp (−iωt) here and below): where A In accordance with the general rules of quantization of the electromagnetic field, its expansion in the complete system of eigenfunctions of the classical problem (standing spherical waves) can be represented as follows (the indices are omitted): where a s and a † s are the operators of annihilation and creation of a photon in s-th mode respectively, with usual rules of the commutation, e (s, r) and h (s, r) are the modes of the electric and the magnetic field, the vector index s = (n, m, ν) denotes a chosen set of quantum numbers: the orbital (n), the azimuthal (m), and the radial (ν), the asterisk denotes the operation of the complex conjugation. In accordance with Eqs. (4) and (8), the fields inside the chiral sphere can be written as Outside of the chiral particle, all waves have the same propagation velocity and can be represented as where C (1) mn and D (1) mn , D mn are some coefficients, and expressions for the spherical vector harmonics N ζ   8)). For the coefficients that describe fields outside of the sphere, one can obtain the following relations from the boundary conditions: in which In Eq. (13), we have used the abbreviations (J = L, R): where P = √ εµ/µ, ψ n (x) and ζ (1) n (x) are the Riccati-Bessel functions (see the Appendix), and the prime near the function means the derivative with respect to its argument.
To study the interaction between a two-level molecule and the continuum of electromagnetic modes modified by the presence of a chiral spherical particle, it is necessary to know the density of final states. To find it, one should use the condition of disappearance of tangential components of electric field on the inner surface of the cavity. As a result we have where ζ (2) n (x) is the Riccati-Bessel function (see the Appendix), and the prime near the function means its derivative.
Substituting Eq. (12) into Eq. (15) and using asymptotic expressions for the Hankel function at Λ → ∞ [30], we obtain The system (16) has nontrivial solutions only if its determinant equals to zero. As a result, we have a quadratic equation, which determines two different relationships between the coefficients C (2) mn and D The relations (17) define two types of electromagnetic modes (photons) occurring in the presence of a chiral spherical particle, which are analogous to TM and TE modes in the case of a nonchiral spherical particle. The upper sign (−) in Eq. (17) corresponds to modes which we will call A-type photons, while another sign (+) gives us B-type photons. If the chirality parameter tends to zero (χ → 0), then, as it follows from Eqs. (13) and (14), the right side of Eq. (17) tends to zero or to infinity, which leads to a decoupling between TE and TM modes (vector spherical harmonics N ζ (2) mn and M ζ (2) mn in Eq. (11)), and as a result leads to the appearance of usual TM or TE polarized photons.
Thus, in the case of a chiral spherical particle we have two types of polarization (A and B), which are different from the TM and TE polarizations.
Substituting Eq. (17) into Eq. (16), we find the next asymptotic expressions (Λ → ∞): where ω s is the mode eigenfrequency, and c is the speed of light in vacuum, and From Eq. (18), it follows immediately that the density of final states for both A-type and B-type photons has the same value: and does not dependent on the presence of a spherical particle of a final volume in the cavity, which is consistent with the Courant's theorem [31].
For the normalization of modes, we will assume that there is only one photon in the quantization volume. As a result, we obtain the condition: where δ ss ′ is the Kronecker's delta, and the inductions d and b of photonic modes can be expressed via the operators of field strengths e and h with the help of the constitutive equations (1). To calculate the integral in Eq. (21), we note that the main contribution to it comes from the region r ∼ Λ outside the spherical particle, where there is no chirality and where it is possible to use asymptotic expressions for the Hankel functions. As a result, we obtain (n = 1, 2, 3, . . .; m = 0, ±1, ±2, . . ., ±n): Thus, expression (11), (12), and (22), together with the expression for the density of the final states (20), fully describe the quantized electromagnetic field in the presence of a chiral spherical particle, and allow to describe the interaction of this field with arbitrary atoms and molecules. In the next section, we will apply these expressions to find the radiative decay rate of the spontaneous emission of a chiral molecule.

MOLECULE LOCATED NEAR A CHIRAL SPHERICAL PARTICLE
The radiative decay rate of the spontaneous emission (the radiative linewidth γ) of a chiral molecule located near a chiral spherical particle can be found by making use of the Fermi's "golden rule" [32]: where summation is performed over all possible final states, which include both types of photons (A and B, see Eq. (17)), ρ (ω) is the density of the final states (20), and H int is the interaction Hamiltonian of a chiral molecules with the electromagnetic field. In this work, we assume that the Hamiltonian is Hermitian, i.e. there are no losses in the material of a particle. If there are losses, there is a possibility of nonradiative transition from the excited state to the ground state. The expression for the nonradiative part of the spontaneous decay rate can be obtained within the framework of the classical approach [33] or by using the nonstandard quantization scheme [34][35][36][37]. In further calculations, for brevity we will consider only one channel of the decay of the initial state of the molecule (transition e → g), i.e. the two-level molecule. To take into account the possibility of transition of the molecule into several states, it is enough to sum the partial linewidths.
The Hamiltonian of interaction of a chiral molecule with the electromagnetic field in Eq. (23) can be written down in the next form [38]: where r 0 is the radius vector of the molecule position,d andm are operators of the electric and the magnetic dipole moments of the molecule, and field operatorsÊ andĤ are defined in Eq. (9). We suppose that the initial state corresponds to the vacuum state of the field and to the molecule in the excited state, initial| = e| vac|, and that the final field state correspond the single photon (of A-type or B-type) and the molecule in the ground state |f inal = |1 n,m,ν |g . Besides, for definiteness we will consider a spiral model of a molecule in which electrons are constrained to move along a helical path. Substituting Eq. (24) into Eq. (23), we obtain the following expression for the total radiative decay rate of spontaneous emission of the molecule: where, for example, in the case of photons of A-type where e d g = d 0 and e|m|g = −im 0 are dipole moments of the considered transition of the molecule on the frequency ω ≈ ω s (see Eq. (18)). The chosen phase difference between the electric and magnetic dipoles is due to the fact that the magnetic moment operator is purely imaginary. This definition also agrees with the adopted spiral model of a molecule.
For more complicated chiral molecules, there can be a different phase relation between electric and magnetic dipole matrix elements.
Let us, for clarity, call the molecules with parallel d 0 and m 0 the "right" molecules, while the molecules with antiparallel d 0 and m 0 will be referred to as the "left" molecules. Explicit Let us assume for definiteness that the molecule is located on the z-axis of a Cartesian coordinate system at the point r 0 > a. In this case, only components with m = 0, ±1 are nonzero, and the explicit expressions for Eq. (25) will have the following form: where and In the case of photons mode of B-type, one can obtain where and In Eqs. (28)-(30), (32)-(34), we have used the following notations: In the case when the chirality parameter of a spherical particle is equal to zero, Eqs. (28)-(30), (32)-(34) are simplified and have the following form: and where Eqs. (36)-(41) correspond to components of the radiative decay rate of spontaneous emission of a chiral molecule located near the dielectric sphere. If we put m 0 = 0 or d 0 = 0 in these expressions, it will be possible to find known relations for the radiative decay rate of spontaneous emission of an atom located near a dielectric spherical particle, and which has nonzero electric or magnetic dipole moment, respectively [39].
Note, that despite of the fact, that this derivation was performed in assumption of absence of losses in the material of the particle, the expressions obtained will also exactly describe radiation losses for nanoparticles made of real materials, i.e. for complex ε and µ. It follows from the fact that classical and quantum calculations give the same analytical expressions for the radiative decay rate of spontaneous emission through the Green's function of a classical problem [40,41]. Since the classical approach is valid in the case of both absorbing and nonabsorbing bodies, this means that for a generalization of the expressions obtained for the case of a real (absorbing) material of a spherical particle, it is only necessary to substitute appropriate complex values of dielectric permittivity and magnetic permeability into obtained expressions.
To calculate the relative radiative decay rate of spontaneous emission of chiral molecules located near a chiral spherical particle, one should normalize the expression (25) to the value of the radiative decay rate of spontaneous emission of the molecule in the absence of the particle. For this purpose, we use Eqs. (36)-(41) where we put T A n = L B n = 0. By substituting the expressions obtained in such a way into Eqs. (27), (31) and (25), and performing summation, it is possible to find the radiative decay rate of spontaneous emission of a chiral molecules in free space: This expression, of course, coincides with the sum of the known expressions for the linewidths of electric and magnetic dipoles [39], because in the absence of chirality the interference of radiation does not occur.

IV. ANALYSIS OF THE RESULTS OBTAINED AND ILLUSTRATIONS
Cumbersome formulas obtained in the previous section cannot fully reveal the features of the spontaneous decay of a chiral molecule in the system under consideration. In this section, we will explore the expressions found in the typical special cases.
In a very important case of a chiral nanoparticle, i.e. a particle having a size much smaller than the wavelength of the radiation, the main contribution to Eqs.
The condition that the denominator of Eq. (44) will be a zero we will apply a different approach used in [43]. If the nanoparticle and the molecule are placed close enough, then their total radiation will be determined by their total electric and magnetic dipole moments: where δd and δm are the electric and the magnetic dipole moments induced in the nanoparticle while d 0 and −im 0 are the oscillating dipole moments of the molecule. Explicitly, these values can be written as follows [43]: where the polarizability of a chiral spherical nanoparticle in a homogeneous field is The fields in Eq. (47) have the form: where n is the unit vector in the direction from the center of the particle to the observation point.
In practice, the orientation of the molecule can be arbitrary with respect to the nanoparticle surface, therefore to get an effective radiative decay rate of spontaneous emission one should average Eq. (46) on the orientations of molecules, or, equivalently, on the unit vector n. As a result we obtain where ξ is defined by the relation m 0 = ξd 0 .
As a rule, the magnetic dipole moment of the molecule is much smaller than the electric dipole moment |m 0 | ≪ |d 0 |. The chirality parameter, even in hypothetical metamaterials, is also small (χ ≪ 1). This fact determines that the second term in Eq. (50), corresponding to the induced electric dipole moment, is usually greater than the term corresponding to the induced magnetic dipole moment. Thus, the effective interference between the electric and magnetic fields is possible only if the following two conditions take place: 1. In the system under consideration, a chiral-plasmon resonance must be present, i.e.
the condition (45) must be satisfied. Under this condition, the contribution of a magnetic radiation increases.
2. The electric dipole moment induced in the nanoparticle should be zero, i.e. the next condition must be satisfied (see Eq. (50)) The solution of the system of Eqs. (45) and (51) determines the values of dielectric permittivity and magnetic permeability of the nanoparticle, which correspond to minimal values of the radiative decay rate of a chiral molecule. It means that the interference between the electric dipole and magnetic dipole radiation becomes maximal and destructive if  [44,45]. On the other, the chiral DNG-metamaterials can be also synthesized [12].
The above expressions give a good description of properties of arbitrary chiral molecules radiation of near nanoparticles, i.e., when the retardation can be ignored. In the case when retardation effects are significant, it is necessary to use the full expression for the decay rates given in the previous section.
In any case, the process of spontaneous decay of a chiral molecule located near a chiral spherical particle is very complex and below for clarity we will give graphical illustrations of some possible regimes of interaction of a chiral molecule with nanoparticles of different composition but neglecting losses in them.
The Figure 2 shows the distribution of absolute values of the coefficient T A 4 (see Eq. (35)) depending on the dielectric permittivity and the magnetic permeability of the material of nonchiral (Fig. 2(a)) and chiral (Fig. 2(b)) spherical particles for different values of k 0 a.
Large values of this coefficient correspond to a resonance. This figure clearly shows that there is a significant difference between these cases, as reflected in the fact that nonzero chirality of the particle leads both to an increase in the number of observed modes (whispering gallery (WG) modes in dielectric "right-handed" (RH) material and in "left-handed" (LH) material, LH surfaces modes [17]), and to the change of their structure as compared with a nonchiral particle. This fact is due to the simultaneous presence of both left-and right-polarized waves in a chiral particle, which in turn leads to excitation of resonances analogous to TM and TE resonances of nonchiral sphere, simultaneously. The Figure 2 is very important because it allows one to qualitative understand the structure of the radiative decay rates of spontaneous emission of a chiral molecule placed near the different nanoparticles.
It is very interesting that even for a dielectric sphere with a small admixture of chirality, strong and unexpected effects occur. In Fig. 3, the radiative decay rate of spontaneous emission of a chiral molecule placed near a chiral spherical particle with positive values of permittivity and permeability is shown as a function of the size parameter. As it is seen in this figure, the presence of chirality in a spherical particle leads to a shift of the maxima of the spontaneous emission radiative decay rate in comparison with the case of a nonchiral particle. Indeed, as it is known, a condition that allows to estimate the position of the resonance is given by k 0 a √ εµ ≈ n + 1/2, where n is the orbital quantum number. In the case of chiral spherical particles, the left-and the right-polarized waves exist simultaneously, and k L a > k 0 a √ εµ if χ > 0. Hence the above condition should be changed to k L a ≈ n+ 1/2.
As a result, this leads to smaller resonant values of k 0 a than for a nonchiral particle (see Fig. 2). Even more interesting feature of ordinary dielectric particles with a small admixture of chirality is a substantial increase in the quality factor of whispering gallery modes. This figure clearly shows that the corresponding linewidth can be increased by the factor 9 or even more.
Note also that in Fig. 3, only a slight difference between the radiative decay rates of spontaneous emission corresponding to the different orientations of the transition magnetic dipole moment of a chiral molecule at a fixed electric dipole moment is observed. However, one should expect a growth of this difference in the case of a gradual increasing of absolute values of the magnetic dipole momentum. Figure 4 shows the radiative decay rate of spontaneous emission of a chiral molecule located near chiral spherical nanoparticles (k 0 a = 1) as a function of the chirality χ = k 0 η.
As it is clearly seen in this figure, changing the chirality of the sphere has the greatest impact on the rate of spontaneous decay of the molecules near dielectric and left-handed spherical particles, in which high-Q modes can be excited. When the chirality parameter approaches to the critical value χ crit = 1/ √ εµ, then the number of oscillations in the radiative decay rate increases rapidly, and k L → ∞ (see Eq. (6)). In the case of metallic particles, the dependence of the radiative decay rate of spontaneous emission on the parameter χ, which is shown in Fig. 4, is weak, due to imaginary values of √ εµ, and due to the absence of propagating waves.
Chiral molecules play an especially important role in biology and pharmacy. Therefore, it is extremely important to arrange the separation of "right" and "left" enantiomers of molecules in the racemic mixtures. This can be done in various ways, for example by using radiation pressure forces in the electromagnetic field of left-or right-handed circularly polarized electromagnetic waves [23], or by using spiral optical beams [46]. In addition, such selection can be efficiently performed with the help of chiral nanoparticles [43]. and in the framework of a simplified quasistatic calculations presented at the beginning of this section (see also [42]) yield almost identical results, confirming the correctness of both approaches.
The Figure 6 shows the ratio of the effective radiative decay rate of spontaneous emission of the "left" molecule and the radiative decay rate of the "right" molecule and vice versa.
From this figure, it follows that if the condition (52) is satisfied, then the decay rates of "left" and "right" molecules differ by the factor 15 or 60 or even more, depending on the chirality of the molecule considered as a reference one. In other words, nanoparticles with the parameter given by Eq. (52) will enhance the radiation of the "right" molecules and slow down the radiation of the "left" molecules and vice versa. Let us stress that "left-handedness" of a chiral sphere or its negative µ are of crucial importance for such discrimination. Possible applications of the effect of discrimination of the radiation one can find in [43].

V. CONCLUSION
Thus, in the present work the analytical expressions for the radiative decay rate of spontaneous emission of a chiral molecule located near an arbitrary spherical particle with chiral properties were obtained and investigated within full QED theory. Using this approach, the spontaneous emission decay rates of chiral molecule placed near a spherical particle made of different materials (dielectrics, metals, "left-handed" metamaterials etc.) were investigated in details. It is shown that nonzero chirality of a spherical particle leads to an increase in the number of excited modes in comparison with nonchiral one due to coupling between left-and right-polarized waves. Quasistatic expressions for the radiative decay rate of spontaneous emission of chiral molecule located near a chiral nanosphere were obtained and investigated.
These expressions are in good agreement with the exact QED results and allow one to estimate the parameters of nanoparticles when the radiation of "right" or "left" molecules is suppressed. It was found that for the suppression of the radiation of the "left" molecules the sphere should be made of MNG metamaterials, while for the suppression of radiation the "right" molecules the DNG materials should be used.
The results obtained can be used to calculate the radiative decay rate of spontaneous emission of chiral molecules in the vicinity of the chiral spherical particles, for interpretation of experimental data on interaction of chiral molecules and particles, and for detection and selection of chiral molecules with the help of chiral particles.

Appendix: Spherical vector harmonics
Spherical vector harmonics that describe the electromagnetic field inside a chiral spherical particle have the form: N ψ (J) mn =n (n + 1) ψ n (k J r) (k J r) 2 P m n (cos θ) e imφ e r + ψ ′ n (k J r) k J r ∂P m n (cos θ) ∂θ e imφ e θ +i ψ ′ n (k J r) k J r mP m n (cos θ) sin θ e imφ e φ , (A.1) where 0 ≤ r < ∞, 0 ≤ θ < π and 0 ≤ φ < 2π are spherical coordinates, e r , e θ , e φ are unit vectors of the spherical coordinate system, ψ n (k J r) = (πk J r/2) 1/2 J n+1/2 (k J r) is the Riccati-Bessel function [30], where J n+1/2 (k J r) is the Bessel function [30], the prime near the function means the derivative with respect to its argument, P m n (cos θ) is the associated Legendre function [30], and the index J = L, R.
Spherical vector harmonics that describe the electromagnetic field outside a chiral spherical particle have the following form: n+1/2 (k 0 r) is the Riccati-Bessel function [30], where H (j) n+1/2 (k 0 r) is the Hankel function of the first (j = 1) and second (j = 2) kind, respectively, and the prime near the function means the derivative with respect to its argument.
More information about the properties of spherical vector harmonics can be found, for example, in [47]. The electric and magnetic dipole moments of the transition of the molecule are oriented along the z-axis (normally to the particle surface). The dashed line shows the asymptotic expression (46).
The nanoparticle is placed in vacuum.