Force of light on a two-level atom near an ultrathin optical fiber

We study the force of light on a two-level atom near an ultrathin optical fiber using the mode function method and the Green tensor technique. We show that the total force consists of the driving-field force, the spontaneous-emission recoil force, and the fiber-induced van der Waals potential force. Due to the existence of a nonzero axial component of the field in a guided mode, the Rabi frequency and, hence, the magnitude of the force of the guided driving field may depend on the propagation direction. When the atomic dipole rotates in the meridional plane, the spontaneous-emission recoil force may arise as a result of the asymmetric spontaneous emission with respect to opposite propagation directions. The van der Waals potential for the atom in the ground state is off-resonant and opposite to the off-resonant part of the van der Waals potential for the atom in the excited state. Unlike the potential for the ground state, the potential for the excited state may oscillate depending on the distance from the atom to the fiber surface.


Introduction
It is known that the interaction between light and an atom leads to an optical force. Exerting controllable optical forces on atoms finds important applications in many areas of physics, in particular in laser cooling and trapping. A large number of schemes for such phenomena have been proposed, studied, and implemented [1,2]. A common feature of the cooling and trapping schemes for atoms in free space is that the average of the recoil over many spontaneous emission events results in a zero net effect on the momentum transfer. Thence, the optical forces on atoms in free space are determined by only the absorption and stimulated emission of light and the light shifts of the ground and excited states [1,2].
An atom near a material object undergoes a dispersion force, which can be called the van der Waals force or the Casimir-Polder force in the nonretarded or retarded interaction regime [3,4,5,6]. The van der Waals interactions between atoms and cylinders have been studied [7,8,9,10,11,12,13,14]. In most of the previous work, the atoms were considered as point-like polarizable particles. When an atom is driven by an external field near an object, the van der Waals interaction depends on the atomic excitation. In addition, the atom undergoes a radiation force, which depends on the field intensity, the field polarization, and the atomic dipole orientation. Moreover, due to the presence of the object, a nonzero spontaneous emission recoil force may appear.
Indeed, for atoms near a nanofiber [15,16,17,18,19,20], a flat surface [20,21,22], a photonic topological material [23,24], a photonic crystal waveguide [25], or a nonreciprocal medium [26], spontaneous emission may become asymmetric with respect to opposite directions. This directional effect is due to spin-orbit coupling of light carrying transverse spin angular momentum [27,28,29,30,31,32,33]. Asymmetric spontaneous emission may lead to a nonzero average spontaneous emission recoil and, hence, may contribute to the optical force on the atoms. In particular, a lateral spontaneous emission recoil force may arise for an initially excited atom near a nanofiber [19,20], a flat surface [20,22], or a photonic topological material [23,24]. Such a lateral force appears because, in the presence of a material object, the interaction between the radiation field and the atom is chiral [15,16,17,18,19,20,21,22,23,24,25,26]. For an atom driven by a guided field, the spontaneous emission rate and the Rabi frequency may depend on the field propagation direction. The effects of the directional dependencies of the spontaneous emission rate and the Rabi frequency on the optical force for an atom near an ultrathin optical fiber have recently been studied [34]. The Casimir-Polder potential of an atom driven by a laser field near a flat surface has been calculated [35]. It is worth noting that asymmetric coupling not only allows one to selectively excite modes in a preferential direction but also leads to effects like modified superradiance and subradiance [36,37], nonreciprocal transmission [38], and modified strong-coupling regime [39].
The aim of this paper is to present a significant extension and comprehensive treatment for the force of light on a two-level atom near an ultrathin optical fiber.
We calculate analytically and numerically all the components of the force of light. Furthermore, in this paper we use the mode function method as well as the Green function technique and show the connection between them. This gives us access to more details and broader insights. In particular, we compute the van der Waals potentials for the atom in the ground and excited states.
The paper is organized as follows. In Sec. 2 we describe the model system. Section 3 is devoted to deriving the expressions for the force in terms of the mode functions and the Green tensor. In Sec. 4 we present numerical results. Our conclusions are given in Sec. 5.

Model
We consider a two-level atom driven by a classical field in a guided mode of a vacuumclad ultrathin optical fiber (see figure 1). The atom has an upper energy level |e and a lower energy level |g , with energies ω e and ω g , respectively. The atomic transition frequency is ω 0 = ω e −ω g . The fiber is a dielectric cylinder of radius a and refractive index n 1 > 1 and is surrounded by an infinite background vacuum or air medium of refractive index n 2 = 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. In addition to the classical guided driving field, the quantum electromagnetic field interacts with the atom leading to spontaneous emission and energy level shift.

Quantum electromagnetic field
The positive-frequency part E (+) of the electric component of the field can be decomposed into the contributions E (+) g and E (+) r from guided and radiation modes, respectively, as In view of the very low losses of silica in the wavelength range of interest, we neglect material absorption.
Regarding the guided modes, we assume that the fiber supports the fundamental HE 11 mode and a few higher-order modes [40] in a finite bandwidth around the atomic transition frequency ω 0 . We label each guided mode in this bandwidth by an index µ = (ωNf 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, . . . being the azimuthal order and m = 1, 2, . . . being the radial mode order, the index f = +1 or −1 denotes 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 and, hence, the mode polarization is single and the polarization index p can take an arbitrary value. For convenience, we assign the value p = 0 to the polarization index p for TE 0m and TM 0m modes. In the interaction picture, the quantum expression for the positive-frequency part E (+) g of the electric component of the field in guided modes is [18] Here, e (µ) = e (µ) (r, ϕ) is the profile function of the guided mode µ in the classical problem, a µ is the corresponding photon annihilation operator, µ = N f p ∞ 0 dω is the generalized summation over the guided modes, β is the longitudinal propagation constant, and β ′ is the derivative of β with respect to ω. The constant β is determined by the fiber eigenvalue equation [40]. The operators a µ and a † µ satisfy the continuousmode bosonic commutation rules [a µ , a † The normalization condition for the guided mode profile function e (µ) is where n ref (r) = n 1 for r < a and n 2 for r > a. The explicit expressions for the profile functions e (µ) of guided modes are given in [40,41]. 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 .
For radiation modes, the longitudinal propagation constant β for each value of the frequency ω can vary continuously, from −kn 2 to kn 2 (with k = ω/c). We label each radiation mode by an index ν = (ωβlp), where l = 0, ±1, ±2, . . . is the mode order and p = +, − is the mode polarization. In the interaction picture, the quantum expression for the positive-frequency part E (+) r of the electric component of the field in radiation modes is [18] Here, e (ν) = e (ν) (r, ϕ) is the profile function of the radiation mode ν in the classical problem, a ν is the corresponding photon annihilation operator, and ν = lp ∞ 0 dω kn 2 −kn 2 dβ is the generalized summation over the radiation modes. The operators a ν and a † ν satisfy the continuous-mode bosonic commutation rules [a ν , a † The explicit expressions for the mode functions e (ν) are given in [18,40].

Classical guided driving field
We describe the classical guided driving field. We assume that the driving field is prepared in a hybrid HE or EH mode, a TE mode, or a TM mode. Let ω L be the frequency of the field. For a quasicircularly hybrid HE lm or EH lm mode with propagation direction f L and phase circulation direction p L , the field amplitude is where A is a constant. For a TE 0m mode with propagation direction f L , the field amplitude is For a TM mode with propagation direction f L , the field amplitude is 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 values ϕ pol = 0 and π/2 define two orthogonal polarization profiles, called even and odd, respectively. In equations (6)-(9), the mode profile function components e r , e ϕ , and e z are evaluated at ω = ω L and β = β L .

Atom-field interaction
We introduce the atomic operators σ ij = |i j|, where i, j = e, g. The operators σ eg = |e g| and σ ge = |g e| describe the upward and downward transitions, respectively. The operators σ ee = |e e| and σ gg = |g g| describe the populations of the upper and lower levels, respectively. We denote the position of the atom as (r, ϕ, z).
The Hamiltonian for the atom-field interaction in the dipole approximation is given by where Ω = d · E/ is the Rabi frequency, the notations α = µ, ν and α = µ + ν stand for the general mode index and the full mode summation, respectively, and the coefficients characterize the coupling of the atom with the guided mode µ and the radiation mode ν. Here, d = e|D|g is the matrix element of the atomic dipole operator D. The coefficient G α characterizes the coupling of the atom with mode α via the corotating term σ eg a α . The coefficientG α describes the coupling of the atom with mode α via the counterrotating term σ ge a α . In deriving the Hamiltonian (10) we have used the rotating-wave approximation for the driving field but not for the quantum field.

Radiation force on an atom
The interaction between an atom and the light field affects the internal state of the atom and leads to a radiation force.

Excitation of an atom
We consider the excitation of an atom. We call ρ (I) the density operator of the atomic internal state in the interaction picture. We introduce the phase-shifted density operator ρ with the matrix elements ρ ee = ρ ge e −i(ω L −ω 0 )t , and ρ eg = ρ (I) eg e i(ω L −ω 0 )t . We obtain the generalized Bloch equations [1] ρ ee = i 2 (Ωρ ge − Ω * ρ eg ) − Γρ ee , Here, ∆ = ω L −ω 0 is the detuning of the frequency ω L of the driving field from the frequencyω 0 = ω 0 + δω 0 of the atomic transition between the shifted levels, with [42] The parameter Γ = γ g + γ r is the rate of spontaneous emission, with [18] and being the contributions from the resonant guided and radiation modes, respectively. We consider the regime where the atom is at rest and in the steady state. In this regime, we can set the derivatives in equations (13) to zero. Then, we obtain [1] ρ ee = 1 2 where is the saturation parameter.

Force on an atom in terms of the mode functions
We consider the center-of-mass motion of the atom and perform a semiclassical treatment for this motion. In such a treatment, the center-of-mass motion is governed by the force calculated from the quantum internal state of the atom. The force of the light field on the atom is defined by the formula We use the interaction picture. Inserting equation (10) into equation (19) gives the following expression for the force: Meanwhile, the Heisenberg equation for the photon operator a α isȧ α = G * α σ ge e i(ω−ω 0 )t + G * α σ eg e i(ω+ω 0 )t . Integrating this equation, we find where t 0 is the initial time. In deriving equation (21), we have neglected the time dependence of the position of the atom. We consider the situation where the quantum electromagnetic field is initially in the vacuum state. We assume that the evolution time t − t 0 and the characteristic atomic lifetime τ are large as compared to the characteristic optical period T = 2π/ω 0 . Under these conditions, since the continuum of the field modes is broadband and the interaction between the atom and the field is weak, the Born-Markov approximation σ ge (t ′ ) = σ ge (t) can be applied to describe the back action of the second and third terms in equation (21) on the atom [42]. Under the condition t − t 0 ≫ T , we calculate the integral with respect to t ′ in the limit t − t 0 → ∞. With the above approximations, we obtain where the notation P stands for the principal value. We substitute equation (22) into equation (20) and neglect fast-oscillating terms. With the use of the relations ρ ee = σ ee , where is the force resulting from the interaction with the driving field, is the force resulting from the recoil of spontaneous emission of the atom in the excited state [19], and and are the forces resulting from the van der Waals potentials for the excited and ground states, respectively. In equation (25), the notation α 0 is the label of a resonant guided mode µ 0 = (ω 0 Nf p) or a resonant radiation mode ν 0 = (ω 0 βlp), and the generalized summation α 0 is defined as α 0 = µ 0 + ν 0 with µ 0 = N f p and ν 0 = lp k 0 n 2 −k 0 n 2 dβ. We note that F (spon) and F (vdW)e enter equation (23) with the weight factor ρ ee , while F (vdW)g enters with the weight factor ρ gg . The term F (scatt) ≡ ρ ee F (spon) is the force produced by the recoil of the photons that are scattered from the atom with the excited-state population ρ ee . In deriving equation (27) we have used the symmetry property |G α | 2 = |Gα| 2 , whereα =μ = (ω, N, −f, −p) for α = µ = (ω, N, f, p) in the case of guided modes andα =ν = (ω, −β, −l, p) for α = ν = (ω, β, l, p) in the case of radiation modes [18].
The force F (drv) of the driving field includes the effects of the momentum transfers in the competing elementary absorption and stimulated emission processes. This force also includes the effect of the AC-Stark shifts of the atomic energy levels.
The forces F (vdW)e and F (vdW)g are produced by the van der Waals potentials U e and U g [43], that is, F (vdW)e = −∇U e and F (vdW)g = −∇U g . These body-induced potentials are given as where δE (vac) e and δE (vac) g are the energy level shifts induced by the vacuum field in free space (in the absence of the fiber). Note that δE is the Lamb shift of the transition frequency of the atom in free space. The detuning of the field from the atom near the fiber can be written as is the detuning of the field from the atom in the absence of the fiber.
We now calculate the individual components of the force. When we use the symmetry of the mode profile functions, we find F (vdW)e z = F (vdW)g z = 0. Then, the axial component F z of the total force is found to be where Here, we have introduced the notation for the rate of spontaneous emission into the guided modes of type N with the propagation direction f = ±, and the notation for the rate of spontaneous emission into the radiation modes with the axial component β of the wave vector. It is clear that F is a light pressure force [1]. We can show that F For the atom in the steady-state regime, we find the expression F In this case, we have Making use of the symmetry properties of the mode functions, we can show that F (spon) r = 0. Then, the radial component F r of the total force is found to be where Due to the evanescent-wave behavior of guided modes in the transverse plane, the radial component F (drv) r of the force of the driving field in a guided mode is a gradient force (dipole force) [1].
Finally, we calculate the azimuthal component F ϕ of the total force. The result is where It is clear that the azimuthal component F (drv) ϕ of the driving-field force is determined by the gradient of the Rabi frequency of the driving field with respect to the azimuthal angle ϕ. This component is, in general, a combination of the pressure and gradient forces in the azimuthal direction [1]. We can show that F . This result means that F (spon) ϕ is nonzero when Im [d * r d ϕ ] = 0, that is, when the atomic dipole rotates in the fiber transverse plane xy.

Force in terms of the Green tensor
Expressions (25)-(27) describe the spontaneous-emission recoil force F (spon) and the van der Waals forces F (vdW)e and F (vdW)g in terms of the mode functions. These forces can also be presented in terms of the Green tensor [43,44]. The explicit expression for the Green tensor G of a two-layer fiber is given in [45,46,47]. The connection between the Green tensor and the mode functions is given in Appendix A.
With the help of equations (A.4) and (11), we can rewrite equation (25) for the spontaneous-emission recoil force F (spon) as where G (R) is the reflected part of the Green tensor. The equivalence of equations (38) and (25) can be easily verified by substituting equations (A.4) into equation (38) and making use of equations (11). It is clear from equation (38) that, when d is a real vector, that is, when the dipole of the atom is linearly polarized, we have F (spon) = 0. However, when d is a complex vector, that is, when the dipole of the atom is elliptically polarized, we may obtain F (spon) = 0. Similarly, with the help of equations (A.4) and (11), we can rewrite equations (28) for the van der Waals potentials U e and U g as We can easily verify the equivalence of equations (39) and (28) by substituting equations (A.4) into equation (39) and making use of equations (11). It follows from the reciprocity property G ji (R ′ , R; ω) that U e and U g are real functions. We use the contour integral technique to change the integrals in equations (39) to the imaginary frequency. Then, we obtain [43] The first and second terms in the expression for U e in equations (40) The potential U g for the ground state |g does not contain a resonant part. Note that U g is opposite to the off-resonant part U (off ) e of U e , that is, U g = −U (off ) e . Thus, expression (23) for the total radiation force F can be rewritten as Equation (42) is in agreement with the results of [43], where multilevel atoms were considered. When we neglect the second term in equation (42), which corresponds to the off-resonant part of the van der Waals force, and assume the weak excitation regime, we can reduce equation (42) to where ℘ = αE is the positive frequency component of the induced dipole, with being the fiber-enhanced atomic polarizability tensor. Under the condition ω L ≃ ω 0 , equation (43) is in agreement with the results of [44] for classical point dipoles.

Numerical calculations
We calculate numerically the force acting on the atom in the case where it is at rest and in the steady state. We use the wavelength λ 0 = 780 nm and the natural linewidth γ 0 /2π = 6.065 MHz, which correspond to the transitions in the D 2 line of a 87 Rb atom. The atomic dipole matrix element d is calculated from the formula γ 0 = d 2 ω 3 0 /3πǫ 0 c 3 for the natural linewidth of a two-level atom. We assume that the fiber radius is a = 350 nm, and the refractive indices of the fiber and the vacuum cladding are n 1 = 1.4537 and n 2 = 1, respectively. The fiber can support the HE 11 , TE 01 , TM 01 , and HE 21 modes. The atom is positioned on the x axis if not otherwise specified. The driving field is prepared in a quasilinearly polarized hybrid HE mode, a TE mode, or a TM mode. In the case of HE modes, we choose the x polarization, which leads to a maximal longitudinal component of the field at the position of the atom.

Driving-field force
We first calculate the driving-field force F (drv) . We plot in figure 2 the radial dependence of the axial component F (drv) z of the driving-field force in the cases where the driving field is at exact resonance with the atom (∆ = 0) and the dipole orientation vector d ≡ d/d coincides with one of the unit basis vectorsx,ŷ, andẑ of the Cartesian coordinate system. As already mentioned in the previous section, F (drv) z is a pressure force. Figure 2 shows that F (drv) z depends on the mode type and the orientation of the dipole vector. We note that for the parameters of figure 2, the radial component F of the driving-field force in the case where the driving field is at exact resonance with the atom (∆ = 0) and the dipole orientation vector isd = (x +ŷ +ẑ)/ √ 3. We observe from figure 3(b) that, when the dipole is not strictly oriented along the x, y, or z direction, the radial force component F      In order to get insight into the origin of the dependence of the driving-field force on the propagation direction, we perform a simple analysis. For an x-polarized hybrid mode or a TM mode with the propagation direction f L , the field at the position of the atom is E(ϕ = 0) ∝ e rx + f L e zẑ . For d ∝ ix −ẑ, the Rabi frequency is Ω ∝ ie r − f L e z . Since the relative phase between the complex amplitudes e r and e z is π/2, the magnitude of Ω is proportional to |e r | − |e z | or |e r | + |e z | depending on f L . The direction dependence of Ω leads to the direction dependence of the excited-state population ρ ee , which is proportional to |Ω| 2 in the non-saturation regime. The corresponding difference between the excited-state populations ρ ee ∝ |e r ||e z |. This difference is proportional to the electric transverse spin density ρ e−spin y ∝ Im[E * × E] ·ŷ ∝ f L |e r ||e z | of the driving field [41]. Due to spin-orbit coupling of light [27,28,29,30,31,32,33], the sign of ρ e−spin y depends on f L . The direction dependence of ρ ee leads to that of the absolute value of the force component F  In general, the driving-field force F (drv) depends on the azimuthal position ϕ of the atom. We plot in figure 6 the azimuthal dependence of the axial component F (drv) z in the case where the driving field is at exact resonance with the atom and the dipole orientation vector isd = (ix −ẑ)/ √ 2. In order to get a broader view, we plot in figure  7 the spatial profile of F (drv) z in the fiber transverse plane for an x-polarized HE 21 mode. We observe from figures 6 and 7 that F

Spontaneous-emission recoil force
In this subsection, we study the spontaneous-emission recoil force F (spon) . This force appears when the atomic dipole rotates in the meridional plane containing the atomic position, that is, when the dipole orientation vectord is a complex vector in the zx plane [19,34]. We plot in figure 8 the radial dependence of the axial component F oscillates with increasing r and can be negative and positive, depending on the radial position r of the atom [19,34]. The oscillations of F (spon) z with varying r are due to the oscillations of the decay rate into radiation modes [18]. Such oscillations result from the interference due to reflections from the fiber surface.
Radial distance r/a F z (spon) (zN) In general, the spontaneous-emission recoil force F (spon) depends on the azimuthal position ϕ of the atom. We plot in figure 9 the azimuthal dependence of the axial component F  According to equation (23), the spontaneous-emission recoil force F (spon) enters the expression for the total force F with the weight factor ρ ee . It is clear that the force produced by the recoil of the scattered photons is F (scatt) = ρ ee F (spon) . We depict in figure 11 the radial dependence of the axial component F

Fiber-induced van der Waals potential and force
In this subsection, we calculate the fiber-induced van der Waals potentials U g and U e for the atom in the ground and excited states. We plot in figures 13 and 14(a) the radial dependencies of the potentials U g and U e , respectively. We show in figure 14(b) the resonant part U (res) e of the potential U e for the excited state. We observe from the figures that both U g and U e depend on the orientation of the atomic dipole. We also observe that U g varies monotonically while U e oscillates with increasing r. The magnitude of U e is substantially larger than that of U g . We recall that the off-resonant part of U e is U the dipole orientation vector isd =x andẑ. The corresponding spatial profiles of the potentials in the fiber transverse plane are shown in figure 16. We note that the dependencies of U e and U g on ϕ lead to the azimuthal components F  shows that the force F (vdW)e r for the excited state oscillates with increasing r, and can take not only negative but also positive values depending on the distance r.
Depending on the orientation of the dipole matrix-element vector d and the position (r, ϕ) of the atom in the fiber transverse plane, the azimuthal components F    We note that, in figures 13, 14, 16, 17, the van der Waals potentials and the corresponding forces are divergent when r/a is 1. This divergence is a consequence of the fact that, when the distance r − a from the atom to the fiber surface is very small, the van der Waals potential of the atom near the fiber can be approximated by the van der Waals potential of an atom near a flat dielectric surface, which is proportional to −1/(r − a) 3 .

Total force
Finally, we compute the total force F of the field on the atom. We plot in figure 19 the radial dependencies of the axial component F z and the radial component F r of the total force F of the field on the atom in the case where the dipole orientation vector isd =x. In these numerical calculations, we take into account the effect of the fiber-induced van der Waals potentials on the detuning of the driving field from the atomic transition frequency. Since d is a real vector, we have F (spon) = 0 and, hence, F z = F of the fiber-induced van der Waals forces with the weight factors ρ ee and ρ gg , respectively. For the parameters of figure 19, the azimuthal component F ϕ of the total force is zero and is therefore not shown. We observe from figure 19(a) that the axial component F z of the force of the HE 21 mode is larger than that of the other modes. Figure 19(b) shows that the radial component F r of the total force can be positive or negative depending on the position r of the atom. The repulsive feature of the force in the region of large r is mainly due to the fact that a positive detuning ∆ 0 was used in the calculations.
Radial distance r/a Figure 19.
Radial dependencies of the axial component F z (a) and the radial component F r (b) of the total force F of the field on the atom with the dipole orientation vectord =x. The detuning of the driving field from the atom in the absence of the fiber is ∆ 0 /2π = 10 MHz. The power of the driving field is P = 100 pW. The fiber-induced shift of the atomic transition frequency is taken into account. Other parameters are as for figure 2.
When the atomic dipole matrix-element vector is a complex vector, the propagation direction dependence of the Rabi frequency and the asymmetric spontaneous emission may occur as shown earlier. In this case, the absolute value of the total force may depend on the propagation direction of the probe field. We plot in figures 20 and 21 the radial and azimuthal dependencies of the components of the total force F in the case where the dipole orientation vector is a complex vectord = (ix −ẑ)/ √ 2. The incident light field is in an x-polarized HE 21 mode. The figure shows that the absolute values of components of the force depend on the propagation direction f L . For the parameters of figure 20, where ϕ = 0, the azimuthal component F ϕ vanishes and is therefore not shown in this figure. However, in the case of figure 21, where ϕ is arbitrary, F ϕ may become nonzero.

Summary
In this work, we have calculated analytically and numerically the force of light on a twolevel atom near an ultrathin optical fiber. We have derived the expressions for the force in terms of the mode functions and the Green tensor. We have shown that the total force consists of the driving-field force, the spontaneous-emission recoil force, and the fiberinduced van der Waals potential force. The axial component of the driving-field force is a light pressure force, while the radial component is a gradient force. The azimuthal component of the driving-field force may also appear and is, in general, a combination of the pressure and gradient forces in the azimuthal direction. Due to the existence of   a nonzero axial component of the field in a guided mode, the Rabi frequency and hence the magnitude of the force of the guided driving field may depend on the propagation direction. When the atomic dipole rotates in the meridional plane, the spontaneousemission recoil force may arise as a result of the asymmetric spontaneous emission with respect to opposite propagation directions. The spontaneous-emission recoil force has a nonvanishing axial or azimuthal component when the atomic dipole rotates in the meridional or cross-sectional plane, respectively. The van der Waals potential for the atom in the ground state is off-resonant and opposite to the off-resonant part of the van der Waals potential for the atom in the excited state. Unlike the potential for the ground state, the potential for the excited state has a resonant part, which is dominant with respect to the off-resonant part, and may oscillate depending on the distance from the atom to the fiber surface.
Our results are fundamental, as they quantify a new physical behavior of the force of light. They can also be envisioned to have significant influence on ongoing and future experiments in quantum and atom optics. Having a controllable force of a structured light field on atoms can help to develop near-field optics, break the existing limits, and reach new dynamical regimes.