Transfer of angular momentum of guided light to an atom with an electric quadrupole transition near an optical nanofiber

We study the transfer of angular momentum of guided photons to a two-level atom with an electric quadrupole transition near an optical nanofiber. We show that the generation of the axial orbital torque of the driving guided field on the atom is governed by the internal-state selection rules for the quadrupole transition and by the angular momentum conservation law with the photon angular momentum given in the Minkowski formulation. We find that the torque depends on the photon angular momentum, the change in the angular momentum of the atomic internal state, and the quadrupole-transition Rabi frequency. We calculate numerically the torques for the quadrupole transitions between the sublevel $M=2$ of the hyperfine-structure level $5S_{1/2}F=2$ and the sublevels $M'=0$, 1, 2, 3, and 4 of the hyperfine-structure level $4D_{5/2}F'=4$ of a $^{87}$Rb atom. We show that the absolute value of the torque for the higher-order mode HE$_{21}$ is larger than that of the torque for the fundamental mode HE$_{11}$ except for the case $M'-M=2$, where the torque for the mode HE$_{21}$ is vanishing.

Excitations of electric quadrupole transitions of atoms using the guided light fields of optical nanofibers have been experimentally realized [28]. Unlike electric dipole transitions, electric quadrupole transitions of atoms depend on the gradient of the field amplitude, which can be steep in the case of near fields. In addition, the atomicinternal-state selection rules for quadrupole transitions are more complicated than those for dipole transitions. Consequently, the exchange of angular momentum between a nanofiber-guided light field and an atom for quadrupole transitions is not simple and deeper insight into the processes involved is desirable.
The aim of this paper is to study the transfer of angular momentum of nanofiber-guided photons to a two-level atom via an electric quadrupole transition. We show that the generation of the axial orbital torque of the driving guided field on the atom is governed by the internal-state selection rules for the quadrupole transition and by the angular momentum conservation law with the photon angular momentum given in the Minkowski formulation.
The paper is organized as follows. In Sec. II, we de-scribe the model of a two-level atom with an electric quadrupole transition driven by the guided light field of an optical nanofiber. In Sec. III, we calculate analytically the azimuthal force and axial orbital torque of the guided light on the atom. In Sec. IV, we present the results of numerical calculations for the torque. Our conclusions are given in Sec. V.

II. MODEL
We consider a two-level atom with an electric quadrupole transition interacting with the guided light field of an optical nanofiber (see Fig. 1). We review the descriptions of the atomic electric quadrupole and the guided light field below.

A. Electric quadrupole transition of the atom
We assume that the atom has a single valence electron. To describe the electric quadrupole and the internal state of the atom, we use the local Cartesian coordinate system {x 1 , x 2 , x 3 }, where the center of mass of the atom is located at the origin x = 0 [see Fig. 1(a)]. The components Q ij with i, j = 1, 2, 3 of the electric quadrupole moment tensor of the atom are given as where x i and x j are the ith and jth coordinates of the valence electron and R = x 2 1 + x 2 2 + x 2 3 is the distance from this electron to the center of mass of the atom.
Let E be the electric component of the optical driving field. The energy of the electric quadrupole interaction between the atom and the field is W = −(1/6) ij Q ij (∂E j /∂x i )| x=0 , where the spatial derivatives of the field components E j with respect to the coordinates x i are calculated at the position x = 0 of the center of mass of the atom [29].
(a) Atom with the local quantization coordinate system {x1, x2, x3} in the vicinity of an optical nanofiber with the fiber-based Cartesian coordinate system {x, y, z} and the corresponding cylindrical coordinate system {r, ϕ, z}. (b) Schematic of a two-level atom with an electric quadrupole transition. The upper level |n ′ F ′ M ′ and the lower level |nF M of the atom are the magnetic sublevels of an alkalimetal atom. The transition between the two levels is characterized by the electric quadrupole tensor Qij with i, j = 1, 2, 3. The population of the upper level |n ′ F ′ M ′ may decay with the rate Γ into other levels that are not shown in the figure.
We assume that the driving field is near to resonance with a quadrupole transition between two atomic internal states, namely the upper state |e with the energȳ hω e and the lower state |g with the energyhω g . To be concrete, we consider the quadrupole transition between the magnetic sublevels |e = |n ′ F ′ M ′ and |g = |nF M of an alkali-metal atom [see Fig. 1(b)]. Here, n ′ and n denote the principal quantum numbers and also all additional quantum numbers not shown explicitly, F ′ and F are the quantum numbers for the total internalstate angular momenta of the atom, and M ′ and M are the magnetic quantum numbers. The matrix elements n ′ F ′ M ′ |Q ij |nF M of the quadrupole tensor operators Q ij are [30] where the matrices u (q) ij with q = M ′ −M = −2, −1, 0, 1, 2 characterize the tensor structures of the spherical components of Q ij and are given as In Eq. (2), 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 (2) q = 2(2π/15) 1/2 R 2 Y 2q (ϑ, φ). 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 local Cartesian coordinates ij represents the tensor structure of the electric quadrupole operator Q ij for the transition between the magnetic sublevels |nF M and We write the electric component of the optical field as E = (Ee −iωt + E * e iωt )/2, where E is the field amplitude and ω the field frequency. The interaction Hamiltonian of the system in the interaction picture and the rotatingwave approximation reads where ω 0 = ω e − ω g is the atomic transition frequency, σ ge = |g e| is the atomic transition operator, and is the Rabi frequency for the quadrupole transition. We insert Eq. (2) into Eq. (5). Then, we obtain [30] The electric quadrupole transition selection rules for F and F ′ and for M and For the quantum numbers J and J ′ of the total electronic angular momenta, the selection rules are |J ′ − J| ≤ 2 ≤ J ′ + J. For the quantum numbers L and L ′ of the orbital electronic angular momenta, the selection rules read |L ′ − L| = 0, 2 and L ′ + L ≥ 2.
Note that the electric dipole transition selection rule for L and L ′ is |L ′ − L| = 1. Consequently, when electric quadrupole transitions are allowed, electric dipole transitions are forbidden. We also note that the change in the angular momentum of the atomic internal state due to an upward transition ish(M ′ −M ). The selection rules for the quadrupole transitions do not require the equality between this change and the angular momentum of the absorbed photon. For example, a quadrupole transition between the magnetic sublevels with |M ′ − M | = 2 can be caused by a linearly polarized plane-wave light field.

B. Guided light of the optical nanofiber
We consider the case where the external field interacting with the atom is the guided light field of a nearby vacuum-clad optical nanofiber [see Fig. 1(a)] [23][24][25][26]. 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 . To describe the guided field, we use Cartesian coordinates {x, y, z}, where z lies along the fiber axis, and also cylindrical coordinates {r, ϕ, z}, where r and ϕ are the polar coordinates in the cross-sectional plane xy.
We examine the vacuum-clad nanofiber whose radius is small enough so that it can support just the fundamental HE 11 mode and possibly a few higher-order modes in a finite bandwidth around the central frequency ω 0 = ω e − ω g of the atom [23][24][25][26]. The theory of guided modes of cylindrical fibers is described in Ref. [31] and is summarized and analyzed in detail for nanofibers in Ref. [32].
The field amplitude of a quasicircularly polarized hybrid mode HE lm or EH lm is [31,32] Here, β with the convention β > 0 is the longitudinal propagation constant determined by the fiber eigenvalue equation, l = 1, 2, . . . and m = 1, 2, . . . are the azimuthal and radial mode orders, f = +1 or −1 denotes the forward or backward propagation direction along the fiber axis z, and p = +1 or −1 is the polarization circulation direction index. The functions e r = e r (r), e ϕ = e ϕ (r), and e z = e z (r) correspond to the cylindrical components of the quasicircularly polarized hybrid mode with f = +1 and p = +1 and are given in [31,32]. Equation (7) can be used for not only quasicircularly polarized hybrid modes but also transverse electric and magnetic modes. For the transverse electric and magnetic modes TE 0m and TM 0m , the azimuthal mode order is l = 0, the mode polarization is single, and the polarization index p can be omitted [31,32]. The field amplitude of a mode TE 0m is given by Eq. (7) with l = 0 and e r = e z = 0. The field amplitude of a mode TM 0m is given by Eq. (7) with l = 0 and e ϕ = 0.

III. AZIMUTHAL FORCE AND AXIAL ORBITAL TORQUE ON THE ATOM
We assume that the field is in a quasicircularly polarized hybrid HE lm or EH lm mode, a TE 0m mode, or a TM 0m mode, that is, the field amplitude is given by Eq. (7). Let the atom be at a position {x, y, z} in the fiber-based Cartesian coordinates or {r, ϕ, z} in the corresponding cylindrical coordinates. For the local coordinate system {x 1 , x 2 , x 3 }, we take x 1 x, x 2 y, and x 3 z. The relation x 3 z means that we use the fiber axis z as the quantization axis for the atomic internal states.
In a semiclassical treatment, the motion of the center of mass of the atom is governed by the force F = − ∇H I [36][37][38][39] of the driving field. It follows from the interaction Hamiltonian (4) that the force is Here, we have introduced the notations ρ ij = i|ρ|j with i, j = e, g for the matrix elements of the density operator ρ for the atomic internal state.
The field amplitude E [see Eq. (7)] depends on the azimuthal angle ϕ for the position of the atom, and so does the quadrupole-transition Rabi frequency Ω [see Eq. (6)]. This dependence leads to the azimuthal component of the force F, which is responsible for the rotational motion of the atom around the fiber axis. The axial component of the orbital (center-of-mass-motion) torque on the atom is It characterizes the rate of the change of the axial component of the orbital (center-of-mass-motion) angular momentum of the atomic system. Equation (6) indicates that the Rabi frequency Ω depends on the sum ij u With the help of expressions (3) and (7), we find where Here, the notations e ′ r,ϕ,z = ∂e r,ϕ,z /∂r have been introduced. Equations (12) show that the factors V q (r) depend on r but not on ϕ and z. Then, it follows from Eq. (11) that the dependence of the sum Then, Eq. (10) yields Note that the time evolution equation for the population ρ ee of the atomic upper state |e reads [36] Here, Γ is the total rate of decay of the excited state |e , which includes not only the rate of decay to the ground state |g but also the rate of decay to other states. Hence, the axial component T z of the orbital torque of the driving field on the atom satisfies the equation Equation (16) is an expression of the angular momentum conservation law. It governs the exchange of angular momentum between the guided driving field and the two-level atom with a quadrupole transition. According to this equation, the magnitude and sign of the axial torque T z depend on the factorh(pl − M ′ + M ), wherehpl stands for the canonical angular momentum of a photon in the guided driving field in the Minkowski formulation [27,[33][34][35], andh(M ′ − M ) stands for the change of the total internal-state (spin) angular momentum of the atom due to an upward transition. The factor Γρ ee +ρ ee on the right-hand side of Eq. (16) is the absorption rate, where Γρ ee is the scattering rate andρ ee is the atomic excitation rate [36]. Equation (16) indicates that the angular momentum of absorbed guided photons is converted into the orbital and spin angular momenta of the atomic system. In addition, we see that the angular momentum of the guided photon imparted on an atom near a nanofiber is of the Minkowski form. This is in agreement with the results of Refs. [33,[40][41][42][43][44][45][46].
According to Eqs. (14) and (16), the torque T z is vanishing when pl = M ′ − M , that is, when the Minkowski angular momentumhpl of an absorbed guided photon is equal to the changeh(M ′ − M ) of the angular momentum of the atomic internal state. When pl = M ′ − M , a nonzero axial torque T z can appear. It is interesting to note that Eqs. (14) and (16) are in agreement with the results for the torque of guided light on a two-level atom with an electric dipole transition [27].
Equation (14) shows that the torque T z depends on the quadrupole-transition Rabi frequency Ω and the atomic coherence ρ ge . The time evolution of ρ ge is governed by the equation [36] where ∆ = ω − ω 0 is the detuning of the field frequency. In the weak-field limit, where the condition |Ω| ≪ Γ is satisfied, we can use the approximations ρ ee ∼ = 0, ρ gg ∼ = 1, andρ ge ∼ = 0. In this case, Eq. (17) yields ρ ge = Ω * /(iΓ − 2∆). Inserting this expression for ρ ge into Eq. (14), we obtain It is clear that if Ω = 0 then we have T z < 0 or T z > 0 for pl < M ′ − M or pl > M ′ − M , respectively. Like the absorption of light, the scattering of light also changes the angular momentum of the atom. The description of the scattering is beyond the framework of the model Hamiltonian (4). Similar to the case of atoms with dipole transitions [27], the axial orbital torque of scattering of light due to the quadrupole transition between the levels M ′ and M is found to be T is the axial orbital torque due to quadrupole spontaneous emission and is given as In Eq. (19), γ µ0 and γ ν0 are the rates of quadrupole spontaneous emission into the resonant guided mode µ 0 = (ω 0 N f p) and the resonant radiation mode ν 0 = (ω 0 βlp), and Γ M ′ M = µ0 γ µ0 + ν0 γ ν0 is the total quadrupole decay rate. The index µ = (ωN f p) labels the guided modes, where N = HE lm , EH lm , TE 0m , or TM 0m is the mode type. Here, l = 1, 2, . . . for HE and EH modes or 0 for TE and TM modes and m = 1, 2, . . . are the azimuthal and radial mode orders [31]. The index ν = (ωβlp) labels the radiation modes, where l = 0, ±1, ±2, . . . is the mode order and p = +, − is the mode polarization index [31].

IV. NUMERICAL RESULTS
In this section, we present the results of numerical calculations for the axial torque T z of the guided light field on an atom with an electric quadrupole transition. 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 87 Rb atom. For this transition, we have L = 0, J = 1/2, L ′ = 2, J ′ = 5/2, and I = 3/2. The wavelength of the transition is λ 0 = 516.5 nm. It is known that the experimentally measured oscillator strength of the quadrupole transition 5S 1/2 → 4D 5/2 in free space is [47]. The reduced quadrupole matrix element n ′ J ′ T (2) nJ is calculated from f (0) JJ ′ by using the relation [30,48] where m e is the mass of an electron. In our numerical calculations, we assume that the driving field is at exact resonance with the atom (ω = ω 0 ). The axial torque T z depends on the Rabi frequency Ω. We plot in Fig. 2 the absolute value |Ω| of the Rabi frequency for the quadrupole transition between the sublevel M = 2 of the hyperfine (hfs) level 5S 1/2 F = 2 and a sublevel M ′ of the hfs level 4D 5/2 F ′ = 4 as a function of the radial distance r for different magnetic quantum numbers M ′ = 0, 1, 2, 3, and 4 and different guided mode types HE 11 , TE 01 , TM 01 , and HE 21 . Figure 2 shows that |Ω| decreases almost exponentially with increasing radial distance r. The steep slope in the radial dependence of |Ω| is a consequence of the evanescent-wave behavior of the guided field outside the fiber. We observe from Fig. 2 that |Ω| depends on the type of the guided mode and the magnetic quantum numbers of the atomic transition. Note that the dashed green curve in Fig. 2(c), which stands for the case of the atom with the levels M ′ = M = 2 interacting with the field in the TE mode, is zero [49]. This is a consequence of the specific properties of the TE mode and the quadrupole operator Q ij for the transition |F = 2, M = 2 → |F ′ = 4, M ′ = 2 with the quantization axis x 3 z.
The axial torque T z also depends on the decay rate Γ of the excited state. For the level 4D 5/2 of atomic rubidium, Γ is mainly determined by the dipole transition from this level to the level 5P 3/2 with the wavelength 1528.95 nm r/a FIG. 2. Absolute value of the Rabi frequency Ω for the quadrupole transition between the sublevel M = 2 of the hfs level 5S 1/2 F = 2 and a sublevel M ′ of the hfs level 4D 5/2 F ′ = 4 of a 87 Rb atom as a function of the radial distance r for different magnetic quantum numbers M ′ = 0, 1, 2, 3, and 4 and different guided mode types HE11, TE01, TM01, and HE21. The fiber radius is a = 280 nm. The wavelength of the atomic transition is λ0 = 516.5 nm. The refractive indices of the fiber and the vacuum cladding are n1 = 1.4615 and n2 = 1, respectively. The power of the guided light field is 1 nW. The field propagates in the +z direction, and the hybrid modes HE11 and HE21 are counterclockwise quasicircularly polarized. The quantization axis is x3 z. [50]. When the atom is in free space, the decay rate is Γ = Γ 0 = 1.119 × 10 7 s −1 [51]. When the atom is in the vicinity of a nanofiber, Γ is modified [52]. We use the technique of Ref. [52] to calculate Γ. We plot in Fig. 3 the radial dependencies of Γ for different magnetic sublevels M ′ of the hfs level 4D 5/2 F ′ = 4. The figure shows that Γ is enhanced and depends on the magnetic sublevel M ′ and the radial distance r. It is interesting to note that all the curves for different M ′ cross each other at a radial point r/a ∼ = 2.12. The reason is that at this point the fiber-modified decay rates for the σ ± and π transitions are equal to each other and, hence, the decay rate of the magnetic sublevel M ′ does not depend on M ′ .
We use Eq. (18)  of the hfs level 5S 1/2 F = 2 and a sublevel M ′ of the hfs level 4D 5/2 F ′ = 4 of a 87 Rb atom. We plot in Fig. 4 the torque T z as a function of the radial distance r for different magnetic quantum numbers M ′ = 0, 1, 2, 3, and 4 and different guided mode types HE 11 , TE 01 , TM 01 , and HE 21 . We observe that the torque depends on the atomic transition and the field mode. The dashed green and dotted blue curves in Fig. 4(c) show that T z is vanishing for the TE and TM modes (with l = 0) in the case M ′ − M = 0. Similarly, the solid red curve in Fig. 4(d) and the dash-dotted magenta curve in Fig. 4(e) indicate that T z = 0 for the HE 11 mode (with l = 1) in the case M ′ − M = 1 and for the HE 21 mode (with l = 2) in the case M ′ − M = 2. Such a vanishing of the torque T z occurs when pl = M ′ − M [see Eq. (18)], that is, when the angular momentum per photon is equal to the change in the angular momentum of the atomic internal state per transition. When pl = M ′ − M , the torque T z is nonzero and is governed by the conservation law expression (16). It is interesting to note from Fig. 4 that the sign of T z can be positive or negative depending on the sign of the factor pl − M ′ + M [see Eq. (18)]. Comparison between the solid red and dash-dotted magenta curves of Fig. 4 shows that the absolute value of the torque for the higher-order mode HE 21 (see the dash-dotted magenta curves) is larger than that of the torque for the fundamental mode HE 11 (see the solid red curves) except for the case of Fig. 4(e), where the torque for the mode HE 21 is vanishing for pl = M ′ − M = 2.
We note that the maximal values of the axial torque T z in Fig. 4  driving guided field is used in our numerical calculations. For the radial distance r = 300 nm from the fiber center, the corresponding azimuthal force F ϕ is on the order of 0.002 zN. Such an optical quadrupole force is significantly weaker than the typical dipole forces (∼ 10 zN) on single atoms in laser cooling and trapping techniques [36]. By increasing the power of the guided driving field, we can achieve larger forces and torques on atoms with quadrupole transitions. Note that the power of a few µW for the driving guided field was used in the experiment [28]. For such a power, an azimuthal force on the order of 1 zN and an axial torque on the order of 1000 zN nm can be achieved. We do not calculate numerically the torque of scattering (re-emission) of light from an atom in a magnetic sublevel M ′ of the hfs state 4D 5/2 F ′ = 4. The scattering from this state is mainly determined by the dipole transition between it and the state 5P 3/2 F = 3. The numerical calculations for the torque produced by this scattering process would involve the multilevel structure of the atom and are beyond the scope of this paper. In the framework of the model of a two-level atom with a dipole transition, the torque of scattering of nanofiberguided light has been studied analytically and numerically [27].

V. SUMMARY
In conclusion, we have studied the transfer of angular momentum of guided photons to a two-level atom with an electric quadrupole transition near an optical nanofiber. We have shown that the generation of the axial orbital torque of the driving guided field on the atom is governed by the internal-state selection rules for the quadrupole transition and by the angular momentum conservation law with the photon angular momentum given in the Minkowski formulation. We have found that the torque depends on the photon angular momen-tum, the change in the angular momentum of the atomic internal state, and the quadrupole-transition Rabi frequency. We have calculated numerically the torques for the quadrupole transitions between the sublevel M = 2 of the hfs level 5S 1/2 F = 2 and the sublevels M ′ = 0, 1, 2, 3, and 4 of the hfs level 4D 5/2 F ′ = 4 of a 87 Rb atom. We have shown that the absolute value of the torque for the higher-order mode HE 21 is larger than that of the torque for the fundamental mode HE 11 except for the case M ′ − M = 2, where the torque for the mode HE 21 is vanishing. Our results are important for generation, control, and manipulation of orbital angular momenta of atoms using nanofiber guided light.