Chiral force of guided light on an atom

We calculate the force of a near-resonant guided light field of an ultrathin optical fiber on a two-level atom. We show that, if the atomic dipole rotates in the meridional plane, the magnitude of the force of the guided light depends on the field propagation direction. The chirality of the force arises as a consequence of the directional dependencies of the Rabi frequency of the guided driving field and the spontaneous emission from the atom. This provides a unique method for controlling atomic motion in the vicinity of an ultrathin fiber.

Applying controllable optical forces to atoms plays a central role in many areas of physics, in particular in laser cooling and trapping [1]. Various sophisticated schemes for exerting optical forces on atoms have been developed [1,2]. A common feature of many cooling and trapping schemes for atoms in free space is that, since spontaneous emission is in a random direction and symmetric with respect to two opposite propagation directions, the average of the recoil over many spontaneous emission events results in a zero net effect on the atomic momentum.
However, it has recently been shown that, for atoms near a nanofiber [3][4][5][6] or a flat surface [7], spontaneous emission may become asymmetric with respect to opposite propagation directions. Such directional spontaneous emission can modify the optical force on atoms. In particular, a resonant lateral Casimir-Polder force may arise for an initially excited atom with a rotating dipole near a nanofiber [8]. Such a force appears because, in the presence of a nanofiber, the interaction between the field and the atom with a rotating dipole is chiral [3][4][5][6]. Chiral optical forces have been studied extensively for chiral molecules and nanoparticles in free space [9][10][11][12], in optical lattices [13], and near optical nanofibers [14]. However, under normal conditions, an atom is essentially achiral because it has the high degree of symmetry associated with a sphere [15].
The possibility of creating chiral forces acting on atoms holds significant potential in many area of physics, in particular in laser cooling, quantum metrology, and atomic state preparation. It enables, for example, the transfer of photonic superposition states to atomic center-of-mass superposition states, opening the possibility of a new way of constructing atomic interferometers. In these, the absorption of a photon superposed in different directions would lead to an atomic motion superposed in the same degree of freedom. Furthermore, directional forces could help with sorting atoms to achieve optical lattices with unit filling factors [16,17], or lead to new laser cooling schemes that can exceed the recoil limit.
In this Letter, we calculate the force of a near-resonant guided light field of an ultrathin optical fiber on a twolevel atom. We show that directional absorption and emission of guided photons leads to a significant chiral optical force. We study a two-level atom driven by a near-resonant classical field with optical frequency ω L and slowly varying envelope E near a vacuum-clad ultrathin optical fiber (see Fig. 1). The atom has an upper energy level |e and a lower energy level |g , with energieshω e andhω 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 , where n 2 = 1. For the atomic position and the field distribution, 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. The atom interacts with the classical driving field E and the quantum electromagnetic field via the spontaneous emission process. The positive-frequency part E of the electric component of the quantum field can be decomposed into the contributions from the guided modes, E g , and the radiation modes, E r . In view of the very low losses of silica in the wavelength range of interest, we neglect material absorption.
The Hamiltonian for the atom-field interaction in the dipole approximation is given by where σ ij = |i j| with i, j = e, g are the atomic operators, a α and a † α are the photon operators, Ω = d eg · E/h is the Rabi frequency produced by the driving field, with d eg = e|D|g being the matrix element of the atomic dipole operator D, and G α andG α are the coupling coefficients for the interaction between the atom and the quantum field. The notations α = µ, ν and α = µ + ν stand for the general mode index and the mode summation, respectively. The index µ = (ωN f p) labels guided modes, where ω is the mode frequency, N = HE lm , EH lm , TE 0m , or TM 0m is the mode type, with l = 1, 2, . . . and m = 1, 2, . . . being the azimuthal and radial mode orders, respectively, f = ±1 denotes the forward or backward propagation direction along the fiber axis z, and p is the polarization index. The index ν = (ωβlp) labels radiation modes, where β is the longitudinal propagation constant, l = 0, ±1, ±2, . . . is the mode order, and p = +, − is the mode polarization. The notations dβ are the generalized summations over the guided and radiation modes, respectively.
The expressions for the coupling coefficients G α and G α with α = µ, ν are given as where e (µ) and e (ν) are the normalized mode functions given in Refs. [6,18]. An important property of the mode functions of hybrid HE and EH modes 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 (µ) r . We note that in deriving the Hamiltonian (1) we have used the rotating-wave approximation for the driving field E.
In a semiclassical treatment, the center-of-mass motion of the atom is governed by the force that is defined by the formula F = − ∇H int [1,2]. In the framework of the Born-Markov approximation, we find Here, F drv is the force produced by the interaction between the driving field and the atom (the recoil of absorption and the dynamical Stark shifts of the energy levels), F spon is the force produced by spontaneous emission from the excited state, and F vdW are the forces associated with the surface-induced van der Waals potentials for the excited and ground states, respectively. The notation ρ stands for the density operator of the internal atomic state in the coordinate frame rotating with the frequency ω L , the notation α 0 = µ 0 , ν 0 labels resonant guided modes µ 0 = (ω 0 N f p) or resonant radiation modes ν 0 = (ω 0 βlp), the generalized summation α0 is α0 = µ0 + ν0 with µ0 = N f p and ν0 = lp k0n2 −k0n2 dβ, and the notation P stands for the principal value of the integral over ω.
where δE is the Lamb shift. The environment-induced shift of atomic transition frequency is δω 0 = δω (vac) 0 + (U e − U g )/h. The shifted atomic transition frequency is ω A = ω 0 + δω 0 . When the atom is not too close to the fiber, we have |δω 0 | ≪ ω 0 , which leads to ω A ≃ ω 0 . We formally incorporate δω 0 into ω 0 .
Equation (4) is valid for an arbitrary driving field, which includes the incident field and the scattered field. When the atom is in free space, we have F spon = F (e) vdW = F (g) vdW = 0, which leads to F = F drv , that is, the force on the atom is just the conventional radiation force [1,2]. We note that F (g) vdW and F spon + F (e) vdW are the total surface-induced forces for the ground and excited states, respectively. These forces have previously been calculated using the Green function approach [20]. When the excitation of the atom is weak, our results reduce to those of Ref. [21] for a point dipole near an interface.
We now assume that the driving field is in a guided mode propagating along the fiber axis z with the propagation constant β L in the f L direction, that is, E = E 0 (r, ϕ)e ifL βLz . We are interested in the axial component F z of the force. Due to the symmetry of the system, the potentials U e and U g do not depend on z. Therefore, we find where β (N ) 0 is the propagation constant of the guided modes N at the frequency ω 0 , γ (f ) gN = 2π p |G ω0N f p | 2 is the rate of spontaneous emission into the guided modes N with the propagation direction f and γ (β) r = 2π lp |G ω0βlp | 2 is the rate of spontaneous emission into the radiation modes with the axial wave-vector component β [6]. Note that the first term in Eq. (6) is the recoil of the absorption, while the second and third terms are the recoils of spontaneous emission into guided and radiation modes.
To calculate the axial force F z in detail, we first assume that the atom is at rest and in the steady state. We can then use the steady-state solution for the internal state of the atom and find where Here, ∆ = ω L − ω 0 is the detuning of the driving-field frequency ω L from the atomic transition frequency ω 0 , and Γ = γ g + γ r is the rate of spontaneous emission, with γ g = N (γ r dβ being the rate of emission into radiation modes [6].
For an atom with a circular dipole near a nanofiber, the spontaneous emission rates γ can be asymmetric with respect to the opposite axial propagation directions [3][4][5][6][7][8]. These directional effects are the signatures of spin-orbit coupling of light carrying transverse spin angular momentum [22][23][24][25][26][27][28]. They are due to the existence of a nonzero longitudinal component of the field in the presence of the nanofiber. This component oscillates in phase quadrature with respect to the radial transverse component and, hence, makes the field chiral. The effect occurs when the atom has a dipole rotating in the meridional plane, that is, when the atom is chiral and the ellipticity vector of the dipole overlaps with the ellipticity vector of the field [3][4][5][6][7][8]. As a consequence, the absolute value of the force F z , given by Eq. (7), can also be asymmetric with respect to the opposite propagation directions f L = ± of the driving field. The asymmetry of the force can be characterized by the parameter η = (|F is the force when the driving field propagates in the direction ±z. We now calculate numerically the dependence of the magnitude of the force F z on the propagation direction f L of the driving field. We assume that the atom is positioned on the x axis and the dipole matrix element d is a complex vector in the meridional plane zx (see Fig. 1). To be concrete, we take d eg = d(ix −ẑ)/ √ 2, which corresponds to the σ + transition with respect to the quantization y axis. The results for the σ − transition can be obtained from the results for the σ + by replacing F , respectively. We assume that the driving field is prepared in a quasilinearly polarized hybrid HE or EH mode or a TM mode. In the case of HE and EH modes, we choose the x polarization, which leads to a maximal longitudinal component of the field at the position of the atom. We do not consider the case of a TE mode because of the vanishing of the interaction between such a mode and the chosen atomic dipole. For an x-polarized hybrid HE or EH mode or a TM mode with the propagation direction f L , the field amplitude at the position of the atom is E(r, ϕ = 0, z = 0) = A(e rx + f L e zẑ ), where A is determined by the power of the driving field [18,19]. The corresponding Rabi frequency is Ω = (dA/h √ 2)(ie r −f L e z ). Since the relative phase between the complex functions e r and e z is π/2 [18,19], the absolute value |Ω| of the Rabi frequency depends on the propagation direction f L . This leads to a direction dependence of the excited-state population ρ ee and, hence, contributes to the asymmetry of the force F z . Thus, both excitation and spontaneous emission can contribute to the dependence of the force F z on the propagation direction of the driving field. Note that the effects of excitation and spontaneous emission on the asymmetry of the force F z may enhance or partially compensate each other.
The radial dependencies of the absolute value |F z | of the force for the cases where the driving field is in an x-polarized HE 11 mode, a TM 01 mode, or a x-polarized HE 21 mode with the propagation direction f L = ± are shown in Fig. 2. One can see that the absolute value |F z | of the force has different magnitudes for different propagation directions f L of the driving field. This chiral effect occurs not only for the fundamental mode HE 11 but also for higher-order hybrid HE and EH modes and TM modes. Figure 2(a) shows that the force of the HE 11 mode on the atom is almost fully chiral.
While the absolute value |F z | of the force reduces quickly with increasing radial distance r, the asymmetry parameter η can be seen in Fig. 3 to vary slowly. Moreover, in the limit of large distances, η approaches a nonzero limiting value. This result means that, despite the evanescent wave behavior of the force, the asymmetry parameter η can be significant even when the atom is far away from the fiber. The reason is that η is determined by not the field amplitude but the ratio between the axial and radial components of the guided field. Indeed, in the limit of large r, we have |F z | ∝ ρ ee ∝ |Ω| 2 . This leads to η ≃ 2Im(e r e * z )/(|e r | 2 + |e z | 2 ) for d ∝ ix −ẑ and E ∝ e rx + f L e zẑ . We can show that e z /e r → −iq L /β L Radial distance r/a for r → ∞, where q L is the evanescent-wave penetration parameter for the driving field [18,19]. Hence, we find η → η ∞ = 2β L q L /(β 2 L + q 2 L ) for r → ∞. Thus, the limiting value of η is nonzero and is determined by the fiber guided mode parameters β L and q L . Since q L = β 2 L − n 2 2 k 2 L < β L ≤ n 1 k L , we have η ∞ ≤ 2n 1 n 2 1 − n 2 2 /(2n 2 1 − n 2 2 ) < 1. It is clear that one can enhance the limiting value η ∞ by increasing the refractive index n 1 of the fiber.
It is interesting to note that the asymptotic value η ≃ 2Im(e r e * z )/(|e r | 2 + |e z | 2 ) for large r is proportional to the electric transverse spin density ρ e-spin z ) of the guided driving field [19]. A simple explanation is that, for the atom with a dipole rotating in the meridional plane zx, the axis y is the quantization axis and, hence, the selection rule corresponds to the transverse spin angular momentum conservation. Due to this fact, the Rabi frequency is determined by the field spherical tensor component which rotates in the same direction as that of the dipole in the plane zx. When the propagation direction of light is reversed, the rotation direction of the spin angular momentum is also reversed in accordance with the spin-orbit coupling of light [22][23][24][25][26][27][28]. Therefore, the difference between the squared absolute values of the Rabi frequencies for the opposite propagation directions is proportional to the difference between the squared absolute values of the opposite spherical tensor components of the guided driving field in the plane zx. Meanwhile, the first difference is proportional to the difference between the excitations and, hence, to the asymptotic difference between the forces for large r, and the second difference is proportional to the electric transverse spin density. This explains why the asymptotic value of the asymmetry parameter η is proportional to the electric transverse spin density ρ e-spin y . Thus, the asymmetry of the forces is a signature of spin-orbit coupling of light [22][23][24][25][26][27][28].