Enhancement of the quadrupole interaction of an atom with guided light of an ultrathin optical fiber

We investigate the electric quadrupole interaction of an alkali-metal atom with guided light in the fundamental and higher-order modes of a vacuum-clad ultrathin optical fiber. We calculate the quadrupole Rabi frequency, the quadrupole oscillator strength, and their enhancement factors. In the example of a rubidium-87 atom, we study the dependencies of the quadrupole Rabi frequency on the quantum numbers of the transition, the mode type, the phase circulation direction, the propagation direction, the orientation of the quantization axis, the position of the atom, and the fiber radius. We find that the root-mean-square (rms) quadrupole Rabi frequency reduces quickly but the quadrupole oscillator strength varies slowly with increasing radial distance. We show that the enhancement factors of the rms Rabi frequency and the oscillator strength do not depend on any characteristics of the internal atomic states except for the atomic transition frequency. The enhancement factor of the oscillator strength can be significant even when the atom is far away from the fiber. We show that, in the case where the atom is positioned on the fiber surface, the oscillator strength for the quasicircularly polarized fundamental mode HE$_{11}$ has a local minimum at the fiber radius $a\simeq 107$ nm, and is larger than that for quasicircularly polarized higher-order hybrid modes, TE modes, and TM modes in the region $a<498.2$ nm.


I. INTRODUCTION
Dipole-allowed optical transitions in atoms, ions, and molecules plays a key role in modern atomic, molecular, and optical physics [1]. The corresponding Rabi frequency is proportional to the intensity of the light field. Energy levels that are not connected to lower energy levels by dipole-allowed transitions are metastable states and have many applications ranging from precision clocks [2] to quantum gates [3]. Electric quadrupole transitions, on the other hand, are proportional to the gradient of the electric field and are less explored. Techniques to investigate non-dipole transitions have been explored theoretically and experimentally for atoms in free space [4][5][6][7][8][9][10][11][12], in evanescent fields [13][14][15], near a dielectric microsphere [16], near an ideally conducting cylinder [17], and near plasmonic nanostructures [18,19]. However, the difficulty in achieving large electric field gradients over a long distance makes the study of quadrupole transitions in an extended medium a challenging task.
Ultrathin optical fibers [20][21][22] allow tightly radially confined light to propagate over a long distance. Apart from a high intensity, the evanescent field that extends radially beyond the physical boundary of an ultrathin fiber also offers a large intensity gradient in the radial direction [23,24]. The corresponding intensity gradient can be used to confine atoms near the surface of an ultrathin fiber [25][26][27]. Furthermore, the higher-order modes of an ultrathin fiber [28][29][30] may also offer an azimuthal phase gradient.
The aim of the present paper is to investigate the electric quadrupole interaction of an alkali-metal atom with guided light in the fundamental and higher-order modes of a vacuum-clad ultrathin optical fiber. We calculate the quadrupole Rabi frequency, the quadrupole oscillator strength, and their enhancement factors. In the example of a rubidium-87 atom, we study the dependencies of these characteristics on the quantum numbers of the transition, the mode type, the phase circulation direction, the propagation direction, the orientation of the quantization axis, the position of the atom, and the fiber radius.
The paper is organized as follows. In Sec. II we study the electric quadrupole interaction of an alkalimetal atom with an arbitrary monochromatic light field. In Sec. III we examine the interaction of the atom with guided light of an ultrathin optical fiber and derive an expression for the enhancement factor of the quadrupole oscillator strength in terms of the fiber mode functions. In Sec. IV we present numerical results. Our conclusions are given in Sec. V.

II. QUADRUPOLE INTERACTION OF AN ATOM WITH AN ARBITRARY LIGHT FIELD
Consider an atom with a single valence electron interacting with an external optical field E through an electric quadrupole transition. We use Cartesian coordinates {x 1 , x 2 , x 3 } to describe the electric quadrupole and the internal states of the atom [see Fig. 1(a)]. We assume that the center of mass of the atom is located at the origin x = 0 of this coordinate system. The electric quadrupole moment tensor Q ij of the atom, with i, j = 1, 2, 3, is defined as where x i is the ith coordinate of the valence electron of the atom and R = x 2 1 + x 2 2 + x 2 3 is the distance from the electron to the center of mass of the atom. The electric quadrupole interaction energy is [31] where the spatial derivatives of the field components E j with respect to the coordinates x i are evaluated at the position x = 0 of the atom. For simplicity, we neglect the effect of the surface-induced potential on the atomic energy levels. This approximation is good when the atom is not too close to the fiber surface [32]. We represent the field as E = (Ee −iωt + E * e iωt )/2, where E is the field amplitude and ω the field frequency. Let |e and |g be upper and lower states of the atom, with energieshω e andhω g , respectively. In the interaction picture and the rotating wave approximation, the interaction Hamiltonian of the system can be written as where ω eg = ω e − ω g is the atomic transition frequency and is the Rabi frequency for the quadrupole transition between the states |g and |e . Consider the case of an alkali-metal atom with degenerate transitions between the magnetic sublevels |g = |nF M and |e = |n ′ F ′ M ′ [see Fig. 1(b)]. Here, n denotes the principal quantum number and also all additional quantum numbers not shown explicitly, F is the quantum number for the total angular momentum of the atom, and M is the magnetic quantum number. The matrix elements n ′ F ′ M ′ |Q ij |nF M of the quadrupole tensor operators Q ij are, as shown in Appendix A, given as [7] where the matrices u (q) ij with q = −2, −1, 0, 1, 2 are given by Eqs. (A12), the array in the parentheses is a 3j symbol, and the invariant factor n ′ F ′ T (2) nF is the reduced matrix element of the tensor operators T Here, Y lq is a spherical harmonic function of degree l and order q, and θ and ϕ are spherical angles in the spherical coordinates {R, θ, ϕ} associated with the Cartesian coordinates {x 1 , x 2 , x 3 }.
It is clear from Eq. (5) that the selection rules for F and F ′ are |F ′ − F | ≤ 2 ≤ F ′ + F , and the selection rule for M and M ′ is |M ′ − M | ≤ 2. Since the tensor (a) Local quantization coordinate system {x1, x2, x3} for an atom. (b) Schematic of the hfs levels of the 4D 5/2 and 5S 1/2 states of a rubidium-87 atom. (c) Atom in the vicinity of an ultrathin optical fiber with the fiber-based Cartesian coordinate system {x, y, z} and the corresponding cylindrical coordinate system {r, ϕ, z}.

operators T
(2) q do not act on the nuclear spin degree of freedom, the dependence of the reduced matrix element n ′ F ′ T (2) nF on F and F ′ may be factored out as [33] n ′ F ′ T (2) where J is the quantum number for the total angular momentum of the electrons, I is the nuclear spin quantum number, and the array in the curly braces is a 6j symbol. The selection rules for J and J ′ are |J ′ − J| ≤ 2 ≤ J ′ + J. Furthermore, since the tensor operators T (2) q do not act on the electron spin degree of freedom, we have [33] where L is the quantum number for the total orbital angular momentum of the electrons and S the quantum number for the total spin of the electrons. It follows from the addition of angular momenta that the quadrupole matrix elements may be nonzero only if |L ′ − L| ≤ 2 ≤ L ′ + L. On the other hand, the parity of the tensor T (2) q ∝ Y 2q is even. Therefore, the quadrupole matrix elements may be nonzero only if L and L ′ have the same parity. Thus, the electric quadrupole transition selection rules for L and L ′ are |L ′ − L| = 0, 2 and L ′ + L ≥ 2. We note that, in the special case where L = 0 and L ′ = 2, we have n ′ , L ′ = 2 T (2) n, L = 0 = 2/3 n ′ , L ′ = 2|R 2 |n, L = 0 .
We now calculate the quadrupole Rabi frequency Ω ge = Ω F MF ′ M ′ , defined by Eq. (4). When we insert Eq. (5) into Eq. (4), we obtain In general, the Rabi frequency Ω F MF ′ M ′ for the transition between the atomic states |nF M and |n ′ F ′ M ′ depends on the relative orientation of the quantization axis x 3 with respect to the electric field vector E. The root-mean-square (rms) Rabi frequencyΩ F F ′ is given by the rule [34] We insert Eq. (8) into Eq. (9) and perform the summations over M and M ′ . Then, we obtain We note that Eqs. (8) and (10) can be used for a monochromatic light field with an arbitrary spacedependent amplitude E. In the particular case of standing-wave laser fields, Eqs. (8) and (10) reduce to the results of Ref. [7].
We assume that the field is near to resonance with the atom, that is, ω ≃ ω 0 , where ω 0 ≡ ω eg . The oscillator strength f F F ′ can be calculated from the rms Rabi frequencyΩ F F ′ by using the relation [34] where m e is the mass of an electron. This yields Equation (12) can be used for a monochromatic light field with an arbitrary space-dependent amplitude E. In the particular case where E = E 0 e iK·x with E 0 and K being constant real or complex vectors, Eq. (12) reduces to an expression that is in agreement with Refs. [13][14][15].
With the help of Eqs. (10) and (11), we find (13) We emphasize that Eq. (13) can be used for an arbitrary monochromatic light field. Due to the summation over M and M ′ in Eq. (9), the rms Rabi frequencyΩ F F ′ and, consequently, the oscillator strength f F F ′ do not depend on the orientation of the quantization axis x 3 . The quadrupole oscillator strength f F F ′ , given by Eq. (13), is a measure that characterizes the proportionality of the rms Rabi frequencyΩ F F ′ to the field magnitude E through Eq. (11). This measure depends on not only the quadrupole of the atom but also the normalized gradients of the field components. We note that, for atoms in free space, the oscillator strength can be interpreted as the ratio between the quantum-mechanical transition rate and the classical absorption rate of a single-electron oscillator with the same frequency [31,34]. However, this interpretation may not be valid for atoms in the vicinity of an object because the modifications of the transition rate are much more complicated than that of the Rabi frequency.
According to expressions (10) and (13), the dependencies ofΩ 2 F F ′ and f F F ′ on F and F ′ are included only in the factors | n ′ F ′ T (2) nF | 2 and | n ′ F ′ T (2) nF | 2 /(2F + 1). These factors are determined by the internal atomic states. They do not depend on the center-of-mass position of the atom and the parameters of the fiber. They act as scaling factors for the dependencies on different F and F ′ . Consequently, the shapes of the dependencies ofΩ 2 F F ′ and f F F ′ on the position of the atom and the radius of the fiber do not depend on the quantum numbers F and F ′ . We introduce the notationsΩ (0) F F ′ and f (0) F F ′ for the rms Rabi frequency and oscillator strength of an atom interacting with a plane-wave light field in free space via an electric quadrupole transition. According to [6,7,15], we haveΩ and The enhancements of the rms Rabi frequency and oscillator strength in arbitrary light are characterized by the factors We find It is clear from Eq. (17) that η Rabi and η osc are independent of the quantum numbers F and F ′ . Moreover, these factors do not depend on any characteristics of the atomic states except for the atomic transition frequency ω 0 . They are determined by the normalized spatial variations of the mode profile function E at the frequency ω 0 . We note that the oscillator strength f JJ ′ of the transition from a lower fine-structure level |nJ to an upper fine-structure level |n ′ J ′ of the atom may be obtained by summing up f F F ′ over all values of F ′ . The result is (18) In the case of an atom interacting with a plane-wave light field in free space, we have [6,7,15] The relation between f F F ′ and f JJ ′ is [13,14,35]

III. QUADRUPOLE INTERACTION OF AN ATOM WITH GUIDED LIGHT
We consider the electric quadrupole interaction between the atom and a guided light field of a vacuum-clad ultrathin optical fiber [see Fig. 1(c)]. We assume that the fiber is a dielectric cylinder of radius a and refractive index n 1 and is surrounded by an infinite background medium of refractive index n 2 , where n 2 < n 1 . We use Cartesian coordinates {x, y, z}, where z is the coordinate along the fiber axis, and also cylindrical coordinates {r, ϕ, z}, where r and ϕ are the polar coordinates in the fiber transverse plane xy.
We assume that the fiber supports the fundamental HE 11 mode and a few higher-order modes [36] in a finite bandwidth around the central frequency ω 0 = ω e − ω g of the atom. The theory of fiber guided modes is given in Ref. [36] and is summarized in Appendix C. The propagation constant β of a guided mode is determined by Eq. (C1). We consider the class of quasicircularly polarized hybrid HE and EH modes, TE modes, and TM modes. A guided mode in this class can be labeled by an index µ = (ω, N, f, p). Here, ω is the mode frequency, the notation N = HE lm , EH lm , TE 0m , or TM 0m stands for the mode type, with l = 1, 2, . . . and m = 1, 2, . . . being the azimuthal and radial mode orders, respectively, the index f = +1 or −1 denotes respectively the forward or backward propagation direction along the fiber axis z, and p is the polarization index. The HE lm and EH lm modes are hybrid modes. For these modes, the azimuthal order is l = 0, and the index p is equal to +1 or −1, indicating the counterclockwise or clockwise circulation direction of the helical phasefront. The TE 0m and TM 0m modes are transverse electric and magnetic modes. For these modes, the azimuthal mode order is l = 0, the mode polarization is single, and the polarization index p can be dropped.
For a quasicircularly hybrid HE lm or EH lm mode with the propagation direction f and the phase circulation direction p, the field amplitude is [29,36] where e r , e ϕ , and e z are given by Eqs. (C10) and (C11) for β > 0 and l > 0. For a TE 0m mode with the propagation direction f , the field amplitude is [29,36] where the only nonzero cylindrical component e ϕ is given by Eqs. (C12) and (C13). For a TM mode with the propagation direction f , the field amplitude is [29,36] where the components e r and e z are given by Eqs. (C14) and (C15) for β > 0. An important property of the mode functions of hybrid and TM modes is that the longitudinal component e z is nonvanishing and in quadrature (π/2 out of phase) with the radial component e r .
Quasilinearly polarized hybrid modes are linear superpositions of counterclockwise and clockwise quasicircularly polarized hybrid modes. The amplitude of the guided field in a quasilinearly polarized hybrid mode can be written in the form where the phase angle ϕ pol determines the orientation of the symmetry axes of the mode profile in the fiber transverse plane. In particular, the specific phase angle values ϕ pol = 0 and π/2 define two orthogonal polarization profiles, one being symmetric with respect to the x axis and the other being the result of the rotation of the first one by an angle of π/2l in the fiber transverse plane xy.
In order to calculate the quadrupole Rabi frequency Ω F MF ′ M ′ and the quadrupole oscillator strength f F F ′ , we need to transform the position and the field from the coordinate system {x, y, z} to the coordinate system {x 1 , x 2 , x 3 }. For this purpose, we translate the local coordinate system {x 1 , x 2 , x 3 } from the position of the atom to the origin of the fiber-based coordinate system {x, y, z}. We denote the new coordinate system as Let θ Q be the angle between the quantization axis x ′ 3 and the fiber axis z [see Fig. 1(c)]. Assume that the plane (z, x ′ 3 ) intersects with the fiber transverse plane xy at a line ζ. Let ϕ Q be the azimuthal angle between ζ and x. Without loss of generality, we choose the axes x 1 and x 2 such that x ′ 1 is in the plane (z, x ′ 3 ) and x ′ 2 is in the plane (x, y). Then, the transformation from the Cartesian coordinates {x, y, z} to the Cartesian The inverse transformation reads The relations between Meanwhile, the transformation from the components of the field vector E in the Cartesian coordinate system {x, y, z} to that in the Cartesian coordinate system {x 1 , x 2 , x 3 } is given by the equations The relations between the Cartesian-coordinate vector components E x and E y and the cylindrical-coordinate vector components E r and E ϕ are E x = E r cos ϕ−E ϕ sin ϕ and E y = E r sin ϕ+E ϕ cos ϕ. With the help of the above transformations, we can easily calculate the quadrupole Rabi frequency Ω F MF ′ M ′ , the quadrupole oscillator strength f F F ′ , and their enhancement factors η osc and η Rabi = η 1/2 osc for guided light. We now derive a simple analytical expression for the enhancement factor η osc for quasicircularly hybrid HE and EH modes, TE modes, and TM modes. Since η osc does not depend on the orientation of the quantization axis x 3 , we use, without loss of generality, the fiber coordinate system {x, y, z} as the quantization coordinate system, that is, we take In addition, we assume that the atom is positioned on the positive side of the x axis, that is, we set ϕ = z = 0. Then, for a quasicircularly hybrid HE or EH mode, a TE mode, or a TM mode, we have where e ′ r,ϕ,z = ∂e r,ϕ,z /∂r. When we insert Eqs. (28) into Eq. (17) and use Eqs. (A12), we find where β 0 = β(ω 0 ). We can decompose η osc as η osc = η r + η ϕ + η z + η mix , where are the contributions from the field gradients in the r, φ, and z directions, respectively, and is a mixed term. In Eqs. (29)-(31), the mode functions and their spatial derivatives must be evaluated at the atomic transition frequency ω 0 . The expression for η z in Eqs. (30) indicates that η z is quadratically proportional to the propagation constant β 0 and increases with increasing magnitude |e z |/|e| of the axial component of the polarization vector e/|e|. Meanwhile, the expression for η ϕ in Eqs. (30) contains the factor 1/r 2 . Due to this factor, η ϕ is small when r and a are large. However, η ϕ may become significant when r and a are small. It is clear that the expression for η ϕ in Eqs. (30) contains some terms with the coefficients proportional to l. However, this expression also contains some terms with the coefficients independent of l. In addition, the mode function components e r , e ϕ , and e z depend on l implicitly. Consequently, the dependence of η ϕ on l is complicated.
In particular, η ϕ is not zero even for l = 0, which corresponds to TE and TM modes. We emphasize that, due to the summation over transitions with different magnetic quantum numbers and the cylindrical symmetry of the field in a quasicircularly hybrid HE or EH mode, a TE mode, or a TM mode, Eqs.

IV. NUMERICAL RESULTS
In this section, we demonstrate the results of numerical calculations for the characteristics of an electric quadrupole transition of an atom interacting with a guided light field of an ultrathin optical fiber. As an example, we study the electric quadrupole transition between the ground state 5S 1/2 and the excited state 4D 5/2 of a rubidium-87 atom. For this transition, we have L ′ = 2, J ′ = 5/2, L = 0, J = 1/2, S = 1/2, and I = 3/2. The wavelength of the transition is λ 0 = 516.5 nm. The experimentally measured oscillator strength of the transition 5S 1/2 → 4D 5/2 in free space is f (0) JJ ′ = 8.06 × 10 −7 [5]. In our numerical calculations, we assume that the field is at exact resonance with the atom (ω = ω 0 ).
We plot in Fig. 2 the absolute value of the Rabi frequency Ω F MF ′ M ′ as a function of the radial distance r for the transitions between a lower sublevel |F M and different upper sublevels |F ′ M ′ via the interaction with different guided modes N = HE 11 , TE 01 , TM 01 , and HE 21 . For the calculations of this figure, we choose the quantization axis x 3 = z. We observe that |Ω F MF ′ M ′ | reduces quickly with increasing r. The steep slope in the radial dependence of |Ω F MF ′ M ′ | is a manifestation of the evanescent-wave behavior of the guided field outside the fiber. It is clear from Fig. 2 that |Ω F MF ′ M ′ | depends on the magnetic quantum numbers and the guided mode type. The dotted blue curve in Fig. 2(b), which stands for the case of the upper sublevel M ′ = 2 and the TE mode, is zero. This means that the TE mode does not interact with the quadrupole transition between the sublevels |5S 1/2 , F = 2, M = 2 and |4D 5/2 , F ′ = 4, M ′ = 2 for the quantization axis x 3 = z. The vanishing of this interaction is a consequence of the properties of the TE mode, the quadrupole operator Q ij , and the internal atomic states.
The Rabi frequency Ω F MF ′ M ′ for the transition be- of the quantization axis, the mode type, and the transition type, |Ω F MF ′ M ′ | may depend on p and f . The dependence of |Ω F MF ′ M ′ | on f is related to the spin-orbit coupling of light [37][38][39][40][41][42][43]. It has been shown that, due to the spin-orbit coupling of light, spontaneous emission and scattering from an atom with a circular dipole near a nanofiber can be asymmetric with respect to the opposite propagation directions along the fiber axis [44][45][46][47][48][49][50]. We note that we have |Ω F MF ′ M ′ | = 0 for both directions f = ±1 in Figs. 5(c) and 5(f). The vanishing of the quadrupole transitions in the cases of these figures is a consequence of the properties of the guided field, the quadrupole operator, and the internal atomic states. We plot in Figs. 6 and 7 the radial dependencies of the rms Rabi frequencyΩ F F ′ and the oscillator strength f F F ′ of the atom. As already pointed out in Sec. II, due to the summation over transitions with different magnetic quantum numbers,Ω F F ′ and f F F ′ do not depend on the relative orientation of the quantization axis x 3 with respect to fiber axis z. Figures 6 and 7 show thatΩ F F ′ and f F F ′ achieve their largest values at r/a = 1. We observe thatΩ F F ′ reduces quickly and f F F ′ decreases slowly with increasing r. We note that the shapes of the curves in Figs. 6(a) and 7(a), where F = 2 and F ′ = 4, are the same as the shapes of the corresponding curves in Figs. 6(b) and 7(b), where F = 2 and F ′ = 3. The difference between these curves is given by a scaling factor [see Eqs. (10) and (13)]. Figures 6 and 7 show that the rms Rabi frequency r/a Ω F F ′ and the oscillator strength f F F ′ depend on the mode type. Comparison between the curves for different modes shows that, for the parameters of the figures, the oscillator strength f F F ′ for the fundamental mode HE 11 (see the solid black curve in Fig. 7) is the largest, while the corresponding rms Rabi frequencyΩ F F ′ (see the solid black curve in Fig. 6) is the smallest or the second smallest. The contrast between these relations is due to the fact that the rms Rabi frequencyΩ F F ′ is proportional to the product of the oscillator strength f F F ′ and the electric field intensity |E| 2 [see Eq. (11)]. Outside the fiber, for a given power, the magnitude of the intensity of the field in the fundamental mode is smaller than that in other modes [29].  We show in Figs. 8 and 9 the radial dependencies of the rms Rabi frequencyΩ F F ′ and the oscillator strength f F F ′ of the atom interacting with the fundamental mode HE 11 via the quadrupole transitions between different pairs of hfs levels F and F ′ of the ground state 5S 1/2 and the excited stated 4D 5/2 . We observe from the figures that the curves for different pairs of F and F ′ have the same shape, that is, the curves for different pairs of F and F ′ are different from each other just by a scaling factor [see Eqs. (10) and (13)]. Comparison between the curves show that the rms Rabi frequency and the oscillator strength are largest and smallest for the transitions between levels F = 2 and F ′ = 4 and between levels F = 2 and F ′ = 1, respectively. We note from Figs. 8(a) and 9(a) that the transitions between levels F = 1 and F ′ = 1 and between levels F = 1 and F ′ = 3 have almost the sameΩ F F ′ and the same f F F ′ . We show in Figs. 10 and 11 the rms Rabi frequencȳ Ω F F ′ and the oscillator strength f F F ′ as functions of the fiber radius a. We observe from Fig. 10 that the rms Rabi frequencyΩ F F ′ first increases and then decreases with increasing a. It is clear from this figure thatΩ F F ′ for different guided modes have different maxima at different values of a. We observe from Fig. 11 that, for the fundamental mode HE 11 , the oscillator strength f F F ′ has a local minimum at a ≃ 107 nm. Meanwhile, for the higher-order modes, f F F ′ increases with increasing a. In the region a < 498.2 nm, f F F ′ for the HE 11 mode is larger than that for higher-order modes. When a is in the region from 498.2 nm to 1000 nm, f F F ′ for the TM 01 mode is lager than that for other modes.
The increase of f F F ′ for the HE 11 and higher-order modes with increasing a in the region of large a is a consequence of the fact that expression (13) for f F F ′ contains the terms that are proportional to the gradients ∂E x,y,z /∂z of the field amplitudes E x,y,z in the direction of the fiber axis z. These gradients are proportional to the propagation constant β, which increases with increasing fiber radius a [29,36]. The decrease of f F F ′ with increasing a in the region of small a for the HE 11 mode (see the solid black curve in Fig. 11) is a result of the changes in the structure of the field. The initial decrease and the subsequent increase lead to the occurrence of a minimum in the dependence of f F F ′ on a in the case of the HE 11 mode (see the solid black curve in Fig. 11). We plot in Fig. 12 the radial dependencies of the oscillator-strength enhancement factor η osc for different guided modes. It is clear from the figure that η osc achieves its largest values at r/a = 1. We see that η osc reduces slowly with increasing radial distance r. This result means that, despite the evanescent wave behavior, the enhancement factor η osc can be significant even when the atom is far away from the fiber. The reason is that the oscillator strength f F F ′ and consequently the enhancement factor η osc are determined by not the field amplitude but the ratio between the field gradient and the field amplitude. We show in Fig. 13 the oscillator-strength enhancement factor η osc as a function of the fiber radius a for different guided modes. Similar to the oscillator strength f F F ′ , the enhancement factor η osc for the fundamental mode HE 11 has a local minimum at the fiber radius a ≃ 107 nm, and is larger than that for higher-order modes in the region a < 498.2 nm. Meanwhile, the enhancement factor η osc for higher-order modes monotonically increases with increasing a. When a is in the region from 498.2 nm to 1000 nm, the factor η osc for the TM 01 mode is lager than that for other modes.
According to Eq. (30), the oscillator-strength enhancement factor η osc can be decomposed into the sum of the components η r , η ϕ , η z , and η mix , which characterize the contributions of the field gradients in the radial, azimuthal, and axial directions as well as the interference between them. We plot these components in Figs. 14 and 15 as functions of the radial distance r and the fiber radius a for different guided modes. We observe from these figures that η z (dash-dotted blue curves), η r (dashed red curves), and η mix (dash-dot-dotted magenta curves) are significant. Meanwhile, η ϕ (dotted green curves) is small except for the case of the HE 11 mode of a fiber with a small radius a. This feature is consistent with the fact that the expression for η ϕ in Eqs. (30) contains the factor 1/r 2 , which is small when r and a are large. With the help of an additional careful inspection of the dotted green curves in Figs. 14 and 15, we find that, among the contributions η HE11 that η HE11 ϕ , which corresponds to l = 1, is smaller than η TE01 ϕ and η TM01 ϕ , which correspond to l = 0. The explanation is that the azimuthal gradient of the transverse component of the field in a quasicircularly polarized HE 11 mode is proportional to ie r − e ϕ [see Eqs. (28)], which is small because ie r and e ϕ are real and have the same sign and comparable magnitudes [see Eqs. (C11)].
Due to the summation over transitions with different magnetic quantum numbers and the cylindrical symmetry of the field in a quasicircularly polarized hybrid mode, the oscillator strength f F F ′ and the enhancement factor η osc for such a mode do not depend on the azimuthal position ϕ of the atom in the fiber transverse plane. For the field in a quasilinearly polarized hybrid mode, since the cylindrical symmetry is broken, f F F ′ and η osc vary with varying ϕ. We plot in Figs. 16 and 17 the dependencies of η osc for the quasilinearly polarized HE 11 and HE 21 modes on the radial distance r and the fiber radius a for different azimuthal angles ϕ. We observe from the figures that, depending on ϕ, the factor η osc for a quasilinearly polarized hybrid mode may decrease or increase with increasing distance r, may be larger or smaller than that for the corresponding quasicircularly polarized hybrid mode, and may have a minimum in the dependence a (nm) η osc and its components HE11 (a) on the fiber radius a. Figure 16 shows that η osc varies slowly in the radial direction. Comparison between the curves for different azimuthal angles in Figs. 16 and 17 indicates that η osc for quasilinearly polarized modes varies significantly in the azimuthal direction.
In order to get a better view of the spatial profiles of the enhancement factor η osc for quasilinearly polarized hybrid modes, we plot in Figs. 18 and 19 this factor as a function of the azimuthal angle ϕ and as a function of the Cartesian coordinates x and y of the position of the atom in the fiber cross-sectional plane. The figures show that η osc for quasilinearly polarized hybrid modes varies significantly in the azimuthal direction but slightly in the radial direction, and is relatively large or small along the major or minor symmetry axes of the modes, respectively.

V. CONCLUSION AND DISCUSSION
In this work, we have studied the electric quadrupole interaction of an alkali-metal atom with guided light in the fundamental and higher-order modes of a vacuumclad ultrathin optical fiber. We have calculated the quadrupole Rabi frequency, the quadrupole oscillator strength, and their enhancement factors. In the example of a rubidium-87 atom, we have studied the dependencies of the Rabi frequency on the quantum numbers of the transition, the mode type, the phase circulation direction, the propagation direction, the orientation of the quantization axis, the position of the atom, and the fiber radius. We have found that the rms quadrupole Rabi frequency and the quadrupole oscillator strength are enhanced by the effect of the fiber on the gradient of the field amplitude. With increasing radial distance, the rms Rabi frequency reduces quickly but the oscillator strength varies slowly. The enhancement factors of the rms Rabi frequency and the oscillator strength do not depend on any characteristics of the internal atomic states except for the atomic transition frequency. These factors are determined by the normalized spatial variations of the mode profile function at the atomic transition frequency. Like the oscillator strength, its enhancement factor η osc varies slowly with increasing distance from the atom to the fiber surface. Due to this fact, the factor η osc can be significant even when the atom is far away from the fiber. We have found that, in the case where the atom is positioned on the fiber surface, the oscillator strength for the quasicircularly polarized fundamental mode HE 11 has a local minimum at the fiber radius a ≃ 107 nm. Meanwhile, for quasicircularly polarized higher-order hybrid modes, TE modes, and TM modes, the oscillator strength monotonically increases with increasing a. In the region a < 498.2 nm, the oscillator strength for the quasicircularly polarized HE 11 mode is larger than that for quasicircularly polarized higher-order hybrid modes, TE modes, a (nm) and TM modes. We have shown that, depending on the azimuthal position of the atom, the enhancement factor η osc for a quasilinearly polarized hybrid mode may decrease or increase with increasing distance, and may be larger or smaller than that for the corresponding quasicircularly polarized hybrid mode. We have found that the factor η osc for quasilinearly polarized hybrid modes varies significantly in the azimuthal direction, and is relatively large or small along the major or minor symmetry axes of the modes, respectively.
Our results may find application in future research on probing electric quadrupole transitions of atoms, molecules, and particles using the fundamental and higher-order modes of ultrathin optical fibers. Direct access to electric quadrupole transitions might be beneficial for fiber-based optical clocks [51]. A photon in a higher-order hybrid mode may have significant orbital angular momentum in addition to spin angular momentum. Therefore, our results on the enhanced electric quadrupole interaction of an atom with guided light might lead to an efficient way for transferring more than one quantum of angular momentum per photon to the internal degrees of freedom of the atom [8,9]. Furthermore, the particular atomic transition addressed in this article allows one to prepare a rubidium atom in the excited state 4D 5/2 . The only dipole-allowed decay of this state to the ground state is via the intermediate level 5P develop a fiber-based source of entangled photon pairs at wavelengths relevant to telecom and atomic references.

ACKNOWLEDGMENTS
We acknowledge support for this work from the Okinawa Institute of Science and Technology Graduate University.
Appendix A: Matrix elements of the quadrupole tensor operators We introduce the notations for the spherical tensor components of the position vector x = (x 1 , x 2 , x 3 ). In terms of these components, we have 0 .
(A2) We can write where u (q) i with i = 1, 2, 3 are the components of the spherical basis vectors u (q) in the Cartesian coordinate system {x 1 , x 2 , x 3 }. The expressions for the vectors u (q) are We note that u (q) * = (−1) q u (−q) , u (q) · u (q ′ ) * = δ qq ′ , and q u (q) i u (q) * j = δ ij . It follows from Eq. (A3) that In order to calculate the direct product x M2 , we use the formula [33] x (1) with q = −K, . . . , K are the tensor elements of the irreducible tensor products We can show that and where Y lq (θ, ϕ) are spherical harmonics with θ and ϕ being spherical angles.
We insert Eq. (A6) into Eq. (A5) and use Eq. (1). Then, we obtain where u (q) 14 The explicit expressions for the tensors u ii = 0. The matrix elements of the tensor T (2) q can be calculated using the Wigner-Eckart theorem [33] n ′ F ′ M ′ |T (2) q |nF M = The invariant factor n ′ F ′ T (2) nF is a reduced matrix element. The selection rules for F and F ′ are |F ′ − F | ≤ 2 ≤ F ′ + F . The selection rules for M and M ′ are |M ′ − M | ≤ 2 and M ′ − M = q. When we use Eqs. (A10) and (A13), we obtain [7] n ′ F ′ M ′ |Q ij |nF M = 3eu Appendix B: Quadrupole interaction of an atom with a plane-wave light field in free space Assume that the field is a plane wave E = Eεe ik·x in free space, where E is the amplitude, k is the wave vector, and ε is the polarization vector. In this case, the rms Rabi frequencyΩ (B1) Without loss of generality, we assume that the field propagates along the x 3 direction and is linearly polarized along the x 1 direction. Then, we have k = (0, 0, k) and ε = (1, 0, 0) in the Cartesian coordinate system {x 1 , x 2 , x 3 }. These expressions lead to k i = kδ i,3 and ε j = δ j,1 . Then, Eq. (B1) gives From Eqs. (A12), we find q |u (q) 31 | 2 = 1/2. Hence, we obtainΩ The oscillator strength f F F ′ is related to the rms Rabi frequencyΩ (0) F F ′ via the formula (11). With the help of this formula, we find The oscillator strength f The rate γ F ′ F of quadrupole spontaneous emission from an upper hyperfine-structure level |n ′ F ′ to a lower hyperfine-structure level |nF of the atom in free space is related to the oscillator strength f The rate γ (0) J ′ J of quadrupole spontaneous emission from an upper fine-structure level |n ′ J ′ to a lower finestructure level |nJ of the atom in free space may be obtained by summing up γ (B10)

Quasicircularly polarized hybrid modes
We consider quasicircularly polarized hybrid modes N = HE lm or EH lm . It is convenient to introduce the parameter s = l 1 h 2 a 2 + 1 q 2 a 2 J ′ l (ha) haJ l (ha) + K ′ l (qa) qaK l (qa) Here, the parameter A is a constant that can be determined from the propagating power of the field.

TE modes
We consider transverse electric modes N = TE 0m . For r < a, we have