A state-insensitive, compensated nanofiber trap

Laser trapping and interfacing of laser-cooled atoms in an optical fiber network is an important capability for quantum information science. Following the pioneering work of Balykin et al. and Vetsch et al., we propose a robust method of trapping single Cesium atoms with a two-color state-insensitive evanescent wave around a dielectric nanofiber. Specifically, we show that vector light shifts (i.e., effective inhomogeneous Zeeman broadening of the ground states) induced by the inherent ellipticity of the forward-propagating evanescent wave can be effectively canceled by a backward-propagating evanescent wave. Furthermore, by operating the trapping lasers at the magic wavelengths, we remove the differential scalar light shift between ground and excited states, thereby allowing for resonant driving of the optical D2 transition. This scheme provides a promising approach to trap and probe neutral atoms with long trap and coherence lifetimes with realistic experimental parameters.


Introduction
The development of a matter-light quantum interface using cold atoms and optical fibers has been an active field of research over the past years [1]. Recent advances towards this goal include the observation of electromagnetically induced transparency and the loading of ultracold atoms in hollow-core optical fibers [2,3,4,5], as well as the trapping and probing of atomic ensembles via the evanescent fields surrounding tapered nanofibers [6,7,8,9]. While prominent examples of off-resonant interaction between evanescent waves and matter have used a plane dielectric geometry for atom optics and interferometry [10,11] as well as for surface traps of quantum degenerate gases [12,13,14], recent progress of atom-light interactions with optical waveguides [4,5,7,8,9] sets the stage for the fiber integration of free-space quantum systems in a quantum network via quantum-state transfer between matter and light [15,16,17,18] and for strong coupling of single atoms and photons trapped near microcavities [19,20,21,22,23,24]. Furthermore, these effective 1-dimensional systems may be applied for investigating quantum many-body phenomena in low dimensions with longrange interactions mediated by the waveguide [25,26,27,28].
One major drawback of many optical traps is that spatially inhomogeneous energy shifts U (r) generally depend on the atomic electronic state, limiting long-lived trap and coherence times necessary for repeated coherent operations [29]. This is traditionally alleviated by constructing a state-insensitive optical trap designed to decouple atomic transition frequencies from the spatially varying potential of each electronic state [30]. Specifically, at the "magic" wavelength conditions, the differential response of the dynamic scalar polarizabilities α (0) (ω) for the ground and excited states α |e . This leads to a vanishing differential atomic level shift δU scalar = 0 [30,31,32,33,34]. Differential shifts for the hyperfine ground states can be minimized by using far off-resonant beams, whereas Zeeman coherence can be conveniently protected by using linearly polarized light in which the vector light shifts are zero.
Although such magic wavelengths can be used for nanofiber traps [35], the strongly guiding nature of the waveguide inevitably leads to non-negligible longitudinal electric fields E z in the evanescent region, which are out of phase with the transverse field E ⊥ = (E x , E y ). Here, z refers to the direction parallel to the fiber axis, while x and y are the coordinates perpendicular to the fiber axis. The resulting local polarization at location r is in general elliptical even for linearly polarized input beams, and induces vector shifts U vector . The differential vector shift δU vector in turn manifests itself as a "fictitious magnetic field", leading to inhomogeneous Zeeman broadening [36]. Furthermore, the spatially varying elliptical polarization of the evanescent field on a scale δr < λ renders it difficult to cancel δU vector using bias fields, resulting in increased heating rate [29] and limited coherence time [37].
Building upon the recent realization of a nanofiber trap as proposed in Ref. [38] and demonstrated in Refs. [6,39], we propose a promising strategy for a state-insensitive evanescent field trap. Differential scalar shifts δU scalar between |g and |e are canceled using "magic" wavelength conditions. The inhomogeneous Zeeman broadening δU vector caused by a forward propagating blue-detuned field E (fwd) is canceled by a backward propagating field E (bwd) with a small relative frequency detuning δ f b . Thus, our scheme can compensate for the light shifts of the strongly guided evanescent waves to the first order in the space external to the dielectric fiber, leading to favorable parameters for the realization of a long-lived fiber-integrated quantum memory and resonant coupling to ultra-high quality micro-cavities based on optically trapped atoms.

A state-insensitive nanofiber trap
In this section, we discuss an ab initio calculation of the optical nanofiber trap for atomic Cesium. We show that the light shifts caused by the elliptically polarized components of the fiber's evanescent field are not negligible. We then propose a scheme to cancel these shifts and generate a two-color, state-insensitive, three-dimensional trap for Cs atoms along the nanofiber.

ac Stark shift Hamiltonian
We start by considering the Hamiltonian for an atom interacting with an electric field E in the dipole approximation:Ĥ whered is the electric dipole operator andÊ is the electric field operator. Taking into account the atomic hyperfine structure, this Hamiltonian can be decomposed into its spherical tensor components parameterized by the dynamic polarizability α(ω) [40,41,42]:Ĥ where α (0) , α (1) and α (2) are the scalar, vector and tensor atomic dynamic polarizabilities, E (+) andÊ (−) are the positive and negative frequency components of the electric field, F =Î +Ĵ is the atomic total angular momentum operator, withÎ andĴ the nuclear and electronic angular momentum operators, µ, ν ∈ {−1, 0, 1} are components in the spherical tensor basis, andĤ 0 ,Ĥ 1 , andĤ 2 are the terms associated with the scalar, vector, and tensor light shifts, respectively. The light shifts U scalar , U vector , and U tensor arising from each term have been expressed explicitly in Refs. [40,41].
For two-level atoms with ground and excited states |g , |e , the scalar shift U scalar can be approximated by U scalar ∝ |E| 2 /δ for detunings δ = ω − ω a large compared to the excited state decay rate Γ, where ω is the electric field angular frequency and ω a is the |g → |e transition frequency. The ground state will experience a repulsive potential for blue-detuned (δ > 0) electric fields, and an attractive potential for red-detuned (δ < 0) electric fields. The scalar dynamic polarizability α (0) is in general different for the states |g and |e resulting in a differential scalar shift and a mismatch of the ground and excited state potentials. For the typically anti-trapped excited state, nearresonant driving of the transition by an additional beam with frequency ω 2 ω a can cause significant heating of a trapped atom [29]. This situation can be remedied by the use of "magic" wavelengths for which α [30,31,32,34]. The vector termĤ 1 of Eq. (2) induces a Zeeman-like splitting proportional to a projection of the total atomic angular momentum F and arises from a so-called "fictitious magnetic field" proportional to the ellipticity of the electric field [36]. In the case of a free-space plane wave propagating along the z axis,Ĥ 1 can be expressed in terms of the Stokes operatorsŜ = (Ŝ 0 ,Ŝ x ,Ŝ y ,Ŝ z ) as [41]: is the ellipticity of the electric field. For an elliptically polarized beam, the vector shift can be as large as the scalar shift, and can, for example, be used to cancel the differential light shifts of Rubidium atoms confined in a 3D optical lattice [43].
The last termĤ 2 in Eq. (2) represents the tensor shift. It vanishes for atoms with total angular momentum F = 1/2 [41]. In the case of the D 2 transition of Cs, that we consider here, it will depend only on the electronic angular momentumĴ for detunings large compared to the 6P 3/2 excited state hyperfine structure, and vanish for J = 1 2 [40,41]. It will therefore only act on the excited state of the Cs D 2 transition, inducing shifts on the Zeeman m F sublevels proportional to m 2 F .

Evanescent optical traps using the fundamental mode of the waveguide
When the radius a of an optical fiber is reduced well below the propagating field wavelength λ, the resulting cladding-to-air waveguide supports only the "hybrid" fundamental mode HE 11 [7,44]. In this strongly guiding regime, a significant fraction of energy of the HE 11 mode is carried in the form of an evanescent wave outside of the nanofiber. The evanescent field intensity is azimuthally asymmetric when the input polarization is linear [7,44]. Fig. 1a shows the electric field intensity |E| 2 = |E x | 2 + |E y | 2 + |E z | 2 in a plane transverse to the fiber for a single, linearlypolarized beam. The unit vectors (e x , e y , e z ) form the basis of the (x,y,z) frame, and (r, φ) are the cylindrical coordinates in the transverse plane (x,y). By appropriately combining blue-detuned and red-detuned fields E red and E blue in an optical nanofiber, an atomic trapping potential can be engineered from the evanescent electric fields [38]. Different configurations can be used for obtaining 3D confinement. Here, we consider the schemes illustrated in Fig. 1, for an infinite SiO 2 cylindrical waveguide of radius a = 250 nm. In all configurations, the beams are linearly polarized at the waveguide input to ensure azimuthal confinement for trapped atoms. A pair of x-polarized red-detuned beams generates a 1D lattice along the fiber axis for longitudinal confinement. Fig. 1b shows the configuration used in Ref. [6], namely a pair of red-detuned, x-polarized beams, and a single blue-detuned, y-polarized beam. An alternative scheme would be to use the three beams with parallel polarizations, as illustrated in Fig. 1c. This scheme allows for the use of lower power for the blue-detuned beam, by about a factor of three for the parameters that we will consider in section 3, but results in larger vector shifts as we will discuss in the next sections. The scheme that we propose makes use of four beams with parallel linear input polarizations, as shown in Fig. 1d. The additional blue-detuned beam compensates for the vector shifts of its companion blue-detuned beam, as we will show. Fig. 2 illustrates a trap generated using the configuration of Fig. 1b and the parameters in Ref. [6]. About 2000 atoms were trapped in a 1-D lattice with a 50 ms lifetime. This advance in the group of A. Rauschenbeutel represents an important milestone towards the micro-manipulation of ultra-cold atoms using evanescent field traps.

HE 11 mode -electric field polarization
The fundamental mode HE 11 is often referred to as "quasi-linear" when excited with a linearly polarized input beam. However, for a dielectric waveguide in the strong-guiding regime with indices of refraction n 1 ≈ 1.5 inside the waveguide and n 2 ≈ 1.0 outside, the HE 11 mode actually exhibits a significant ellipticity for a λ/2, leading to vector shifts of the Zeeman sublevels. Formally, for a linearly polarized input, the evanescent field E = (E x , E y , E z ) of the fundamental mode propagating in the fiber can be expressed as follows for r ≥ a [7,44,45]: x y z c) x y z d) Figure 1. Trapping schemes. a) Field intensity |E| 2 in the plane transverse to the fiber for a single, x-polarized beam at λ = 937 nm. |E| 2 is normalized to the intensity just outside the fiber I 0 = |E(r = a + , φ = 0)| 2 , with a = 250 nm and a + = a + |r − a| , r → a. The red dashed arrow indicates the input polarization. b) Trapping scheme used in Ref. [6]. Red(blue)-detuned beams are shown by red(blue) thick arrows. Input polarizations are shown by the thin arrows. A single, y-polarized blue-detuned beam is used. c) Three-beam scheme with parallel x-polarizations. All beams have an intensity maximum in the x − y plane along the direction of the input polarization. d) A second x-polarized blue-detuned beam is added to compensate for the vector shifts, as discussed in the main text.
Here, φ denotes the azimuthal position in the transverse plane ( Fig. 1a), ϕ 0 indicates the polarization axis for the input polarization relative to the x axis, n 1 and n 2 are the indices of refraction inside and outside the waveguide, β 11 is the mode propagation constant, 1/h 11 is the characteristic decay length for the guided mode inside <-0.4mK >0mK Figure 2. A ground state trapping potential U trap is generated by two orthogonally polarized evanescent fields, confining Cs atoms outside a 500 nm diameter optical fiber. Input polarizations are denoted by the arrows. a) x-y plane. b) x-z plane. U trap results from two counter-propagating red-detuned beams (1064 nm, 2 × 2.2 mW), and a single blue-detuned beam (780 nm, 25 mW), as shown in Fig. 1b [6]. The standing wave structure of the attractive red-detuned field and the repulsive force from the bluedetuned beam enable 3D confinement of Cs atoms at each minimum of U trap near the dielectric waveguide. U trap diverges as the surface is approached due to the attractive van der Walls force.
the fiber, 1/q 11 is the characteristic decay length for the guided mode outside the fiber, A lin is the real-valued amplitude for the linearly polarized input, J l is the l-th Bessel function of the first kind, and K l is the l-th modified Bessel function of the second kind. It is clear from Eq. (4) that the electric field intensity is not azimuthally symmetric. For a beam polarized along e x , i.e. ϕ 0 = 0, the intensity at the fiber's outer surface is maximum for φ = 0, π and minimum for φ = ±π/2.
Notably, the evanescent modes of the nanofiber have a significant longitudinal component E z along the fiber propagation direction, which is π/2 out-of-phase with the transverse components (E x , E y ) (Eq. (4c)). At the outer fiber surface, E z is maximum for φ = ϕ 0 , ϕ 0 + π (i.e., along the input polarization axis), and vanishes for φ = ϕ 0 ±π/2. For an x-polarized input at 937 nm and a nanofiber of radius a = 250 nm, |Ez| 2 |E| 2 (r = a + , φ = 0) 20%. As a consequence, the polarization of a single propagating beam will be elliptical everywhere except for φ = ϕ 0 ± π/2. The ellipticity of the beam will be maximum for φ = ϕ 0 , ϕ 0 + π as is illustrated in Fig. 3, giving rise to significant vector shifts, which we describe in section 3.
We can re-write Eqs. (4) as follows: where A, B, and C are real functions of r and φ. In particular, if one combines a forward-propagating beam E (fwd) expressed as Eq. (6) with a backward-propagating beam of same amplitude and input polarization E (bwd) = Ae i(ωt+β 11 z) e x +Be i(ωt+β 11 z) e y − iCe i(ωt+β 11 z) e z , the total field can be expressed as: The resulting electric field E (tot) = E (fwd) + E (bwd) forms an optical lattice with spatially rotating linear polarization as illustrated in Fig. 4. In particular, the polarization state of the field rotates between the pure linear x and z-polarizations along z at φ = 0, as illustrated in Fig. 5.

Cancellation of the vector shifts
In Ref. [6], the properties shown in Figs. 4 and 5 were used to cancel the vector shifts of ground and excited m F states in Cs for the pair of red-detuned trapping beams for the configuration in Fig. 1b. However, vector shifts due to the single blue-detuned beam were zero only for φ = ϕ 0 , ϕ 0 + π. Although the atoms are trapped at φ = ϕ 0 , ϕ 0 + π, inevitable fluctuations of the atom position will lead to non-zero vector light shifts of both ground and excited states. The scheme in Fig. 1c allows for the use of reduced power for the blue-detuned beams as compared to Fig. 1b but with the consequence of large vector shifts from the ellipticity of the electric field even for φ = 0. We will therefore not consider this scheme in the next sections. . Electric field amplitude after interference E (tot) = E (fwd) + E (bwd) of two 937 nm beams with δ f b = 0, at t = 0 and r = a + as in Fig. 1d (i.e., x-polarized inputs with ϕ 0 = 0). The fields are normalized to the intensity I 0 at r = a + , φ = 0, z = 0. a) axial direction z (at φ = 0). b), azimuthal direction φ (at z = 0). In particular, E (tot) has a fixed linear polarization at any given point r which rotates as r is varied.
By contrast, the vector shifts of both the ground and excited states can be canceled for both the red and blue-detuned fields by using pairs of counter-propagating beams, as shown in Fig. 1d. In the x − z plane, the vector shift for each pair becomeŝ where ω (fwd,bwd) are the angular frequencies for the forward and backward propagating beams, and δ ± δ f b /2 are their detunings from the atomic transition frequency ω a , with two-photon detuning δ f b = ω (fwd) − ω (bwd) . For an atom in the x − z plane, the total electric field is also contained in the x − z plane, such that the scalar product Ê (−) ×Ê (+) ·F in Eq. (2) is proportional toF y .
In the case of the red-detuned lattice, ω = 0, precisely as in Ref. [6]. Adding a blue-detuned lattice with δ f b = 0 would result in two superimposed lattices with unmatched spatial periods 2π/β red 11 , 2π/β blue 11 . To avoid this effect, the interference between the counter-propagating blue-detuned fields E (fwd) blue and E (bwd) blue can be averaged over times short compared to the time scale of the motional and internal dynamics of a trapped atom by offsetting the frequencies of the two fields by δ f b (ω trap , δ hfs ), where ω trap and δ hfs are the trap angular frequency and the hyperfine splitting for the ground state, respectively. This will also suppress spurious two-photon processes (e.g., two-photon Stark shift [46]) as well as parametric heating due to intensity modulation [47].

Magic wavelengths for an evanescent field trap
To make the nanofiber trap state-insensitive, it is necessary to cancel the differential scalar shift δU scalar by operating the trap at the magic wavelengths, as proposed in Ref. [35], in which only the effects of the scalar and tensor shifts were considered.
Here we deal with the full complexity of the vector field E(r) and the resulting vector light shifts. We numerically determine the red-detuned and blue-detuned magic wavelengths, following the procedure described in Refs. [30,34,48,49]. We include all the hyperfine levels F and Zeeman sublevels m F of the electronic states {6S 1/2 , · · · , 15S 1/2 }, {6P 1/2 , · · · , 11P 1/2 }, {6P 3/2 , · · · , 11P 3/2 }, {6D 3/2 , · · · , 11D 3/2 }, and {6D 5/2 , · · · , 11D 5/2 }. The effect of the tensor shifts on the excited state is manifest in the quadratic splitting of the m F sublevels (Fig. 6). We find a red-detuned magicwavelength located around 935nm, in accordance with the previously published values [34,49]. In the next sections, we will use the value λ red = 937 nm, that cancels δU scalar for the 6P 3/2 excited state |F = 4, m F = 0 . We choose F = 4 due to its relevance to coherent two-photons processes [15,16,17]. There are several blue-detuned magic wavelengths [35,49]. For our trap, we use the magic wavelength λ blue at approximately 687 nm [35]. Since this is the second closest blue-detuned magic wavelength to 852 nm, it has the second highest ground-state polarizability and therefore requires the second lowest optical intensity to generate the required trapping potential (we do not consider the magic wavelength at 792 nm, as it is too close to the 8S 1/2 to 6P 3/2 transition at 794 nm). We have neglected higher order processes in our analysis, including two-photon and electric quadrupole transitions, near 687 nm [50].  Figure 6. Magic wavelengths of the Cs D 2 line. We display the light shift U ls for a linearly polarized beam with constant intensity 2.9 × 10 9 W/m 2 around a), the bluedetuned magic wavelength at λ blue 687 nm, and b), red-detuned magic wavelength at λ red 937 nm.

Numerical results: trapping potentials
Using the atomic interaction Hamiltonian in the dipole approximation with the actual polarization profile of the evanescent field, we proceed to analyze the adiabatic potentials for the nanofiber trap for a Cs atom in its 6S 1/2 ground and 6P 3/2 excited states.

Total potential
For a specific atomic state of Cs, the total atomic trap potential U trap consists of the total light-shift potential U ls calculated from the full Stark shift Hamiltonian (Eq. (2)), as well as the surface interaction potential of an atom with the dielectric waveguide U surface , namely The Casimir-Polder interaction between the atom and dielectric surface has a significant effect on the atomic motion at distance scales near 100 nm [20,51,52,53,54]. The surface potential of a ground state Cs atom near a planar dielectric surface can be reasonably approximated by the van der Waals potential which decays as d −3 , where d = r − a: where we use C 3 /h = 1.2 kHz µm 3 [21]. Because the retarded Casimir-Polder forces decrease faster away from the surface than the van der Waals forces, using U surface overestimates the surface interaction at the trap location d ≈ 200 nm. Additionally, the curvature of the nanofiber cylindrical geometry reduces the potential strength even further [21,55]. The d −3 scaling of the van der Waals expression for a planar surface is therefore an overestimate of the actual surface potential, and is used for simplicity in the calculations presented here, with more complete expressions for Cs presented in Ref. [21]. Furthermore, we neglect any dependence on the m F sublevels of the excited state 6P 3/2 , and simply approximate U [56]. We calculate the adiabatic potential of Eq. (9) by diagonalizing the total interaction HamiltonianĤ =Ĥ ls +Ĥ surface at each point in space, whereĤ surface is the scalar surface Hamiltonian. At each point r(r, φ, z), we obtain a set of eigenstates and the corresponding eigenenergies. These eigenstates are superpositions of the |F, m F bare Zeeman sublevels. Due to the complex polarization of the trapping fields, the energy eigenstates are not necessarily eigenstates of any projection of the angular momentum operator.

Effect of the light shifts in a "non-magic" trap
First, we consider the trapping parameters for the experiment by Vetsch et al. [6]. Despite its impressive experimental success, the ground-state levels exhibit splittings that impair the ground state coherence. In the realization of Ref. [6], the two-color evanescent trap is constructed using a pair of counter-propagating x-polarized (ϕ 0 = 0) red-detuned beams E red = E (fwd) red + E (bwd) red (P red = 2 × 2.2 mW) at λ red = 1064 nm, forming an optical lattice, and a single repulsive y-polarized (ϕ 0 = π/2) blue-detuned beam E blue (P blue = 25 mW) at λ blue = 780 nm. The SiO 2 tapered optical fiber has a radius a = 250 nm in the trapping region. ؆ ( r a d . ) Figure 8. Azimuthal dependence of the trapping potential of the ground and excited states for the scheme in Fig. 1b and for the parameters used in Fig. 7. r − a = 230 nm, z = 0. a) The ground state splitting is minimum for φ = 0 and φ = π. Everywhere else, the polarization of the blue-detuned field induces large vector shifts. b) Expanded view of a) near a trap minimum at φ = π. The Zeeman-like splitting of the ground states is large even for small azimuthal angles. The excited-state level structures are greatly altered by the combined vector and tensor shifts. Fig. 7 shows the radial trapping potential U trap (r, φ, z) of the ground states F = 3 and F = 4 of 6S 1/2 and excited states F = 4 of 6P 3/2 , for z = 0 and φ = 0 (x-axis) (Fig.  7a) and for φ = π/2 (y-axis) (Fig. 7b). The energy sublevels of the ground states at the trap location (φ = 0) are degenerate, as both trapping fields are linearly polarized as illustrated in Fig. 4. The excited state energy sublevels are shifted due to the vector and tensor shifts. The trap depth for the ground state is U depth = −0.4 mK, located at r − a 230 nm and φ = 0, whereas the excited states are not trapped at all.
The azimuthal dependence of the trap potential reveals a significant inhomogeneous broadening of the energy sublevels due to the ellipticity of E blue for φ = 0, π (Fig. 8).
To estimate this broadening, we assume that the potential is harmonic around the trap minimum. By fitting the ground state F = 3 potential with a harmonic potential around φ = π, we obtain an azimuthal trapping frequency ν trap φ 150 kHz. For an atom in its azimuthal motional ground state |n φ = |0 φ in such a potential, the halfwidth σ r φ of the corresponding single-atom distribution is given by σ r φ = (rφ) 2 . This leads to fast decoherence of the hyperfine and Zeeman levels, even with ground state cooling. Specifically, we estimate a spin-wave coherence time τ m = 1/δν φ 5 µs, derived from the δν φ = 200 kHz splitting between the sublevels of the F = 4 atomic ground state 16 nm away from the trap minimum. This is significantly limited compared to the quantum memory performances of atomic ensembles in optical lattices (see, e.g., [57]). Finally, we plot in Fig. 9 a cross-section of the axial potentials showing the axial confinement of the ground and excited states.
The excited states are untrapped in all directions except along the fiber axis z for the . Axial dependence of the trapping potential for the ground and excited states for the scheme in Fig. 1b and for the parameters used in Fig. 7. a) Longitudinal potential along φ = 0. The distance from the fiber surface is set to r − a = 230 nm at the trap minimum. b) Longitudinal potential along φ = π/2. The distance from the fiber surface is again set to 230 nm.
parameters of Ref. [6]. An atom excited to these untrapped potentials will experience dipole-force fluctuations, leading to heating [29] and preventing near-resonant driving of the optical transition [58].

State-insensitive trapping potential
Now, we analyze our proposed "magic compensation" scheme (as illustrated in Fig. 1d), demonstrating how using magic wavelength beams and compensating the trap ellipticity can reduce inhomogeneous broadening of the Zeeman sublevels in a nanofiber trap. For this trap, we use a pair of counter-propagating x-polarized (ϕ 0 = 0) red-detuned beams (P red = 2 × 0.95 mW) at the "magic" wavelength λ red = 937 nm, forming a 1-D optical lattice. Counter-propagating, x-polarized blue-detuned beams at the second "magic" wavelength λ blue = 687 nm are used with a power P blue = 2×16 mW. The resulting interference is averaged out by detuning the beams by δ f b = 30 GHz, as explained in section 2.4, leading to a first-order cancellation of vector light shifts as expressed by Eq. (8). The beam intensities are chosen to generate a trap of similar depth as the one demonstrated in Ref. [6]. The resulting adiabatic potential U trap allows for state-insensitive 3D confinement of cold Cs atoms around a SiO 2 nanofiber of radius a = 250 nm.
In Fig. 10, we show the radial trapping potential U trap (r, φ, z) of the ground and excited states for z = 0, φ = 0 (x-axis) (Fig. 10a) and for z = 0, φ = π/2 (y-axis) (Fig. 10b). Because the trapping fields are now effectively linearly polarized, the ground states are degenerate at both φ = 0 and φ = π/2. In contrast to a non-magic wavelength trap, the excited states are trapped with gradients that closely map that of the ground states. The sublevels of 6P 3/2 are still non-degenerate due to the tensor shifts. For P red , P blue specified above, we find that the trap depth is U depth = −0.4 mK, located at r − a r trap − a = 210 nm and φ = 0, π. The azimuthal confinement of the atoms is shown by Fig. 11. In contrast to the configurations shown in Figs. 1b and 1c, the inhomogeneous Zeeman broadening from the ellipticity of E blue is greatly reduced thanks to the compensation scheme of Fig. 1d. It is non-zero, however, as expressed by Eq. (8). The remaining splitting of the F = 4 ground state is δ ν ≈ 700 Hz, limiting the coherence time to τ 1/δ ν = 1.4 ms.
In the case of perfect cancellation of the vector shift with δ f b = 0, a residual non-zero ground state splitting δν φ would still arise from the different scalar dynamic polarizabilities of the 6S 1/2 F = 3 and F = 4 ground states [59]. For atoms in their azimuthal motional ground state |n φ = |0 φ , the single-atom distribution half-width is σ r φ 30 nm (or σ φ 4 • ) with azimuthal trap frequency ν trap φ 44 kHz obtained from a harmonic fit of the potential around φ = π. We estimate a spin-wave coherence time τ m = 1/∆ (δν φ ) ≤ 30 ms, coming from the spread ∆ (δν φ ) = δν φ (φ = π) − ؆ ( r a d . ) Figure 11. a) Azimuthal dependence of the trapping potential of the ground and excited state for the "magic compensated" trap ( Fig. 1d) with the parameters used in Fig. 10, for r − a = 210 nm. b) Expanded view of a) around a trap minimum. The effect of the compensation beam in Fig. 1d is to suppress the vector shifts and to reduce the ground-state splittings δν φ in the transverse plane for φ = 0, π.
We note that the longest achievable coherence time in the "magic compensated" adiabatic potential in the absence of ground-state splitting δν φ would be limited by spontaneous Raman scattering driven by the trapping beams [34]. ‫=|؆‬ π/2 Figure 12. Axial dependence of the trapping potentials of the ground and excited states for the "magic compensated" trap ( Fig. 1d) with the parameters used in Fig. 10. a) Longitudinal potential along φ = 0 for r − a = 210 nm. b) Longitudinal potential along φ = π/2 for r − a = 210 nm.
Finally, we also plot the axial potentials in Fig. 12, showing the confinement for both the ground and excited states.
Thanks to the use of the magic wavelengths, the excited states are trapped in all directions. This results in greatly suppressed dipole-force fluctuations, allowing for on-resonance driving of the optical transition.

Conclusion
We have proposed and analyzed in detail a scheme for a state-insensitive optical nanofiber trap which utilizes realistic experimental parameters. The "magic" trapping wavelengths of 937 nm and 687 nm for Cs atoms are readily available using semiconductor laser sources, and require a reasonable power for trapping. Tapered optical fibers can be made with sub-wavelength diameters and high transmission, as has been demonstrated experimentally [60,61]. In Ref. [62] we describe an experiment for trapping Cs using Fig. 1d configuration and explicitly demonstrate features of the "compensated magic" trap for a nanofiber.
Furthermore, extension to other nanoscopic dielectric waveguides [63] would make evanescent optical trapping possible in integrated hybrid quantum devices [64]. It is worthwhile to note that the compensation scheme of the vector shift would work at any wavelength, increasing the ground state coherence time in a straightforward manner.
The proposed "magic compensated" scheme allows for in-trap resonant processes, and leads to increased ground state coherence time relative to uncompensated schemes. These properties will make quantum-state engineering more feasible in such a trap, allowing for a wide range of experiments including creating quantum memories, coupling of single atoms and ensembles to optical or mechanical resonators, and studying 1-D spin chains.

Acknowledgment
We gratefully acknowledge interactions with E. S. Polzik and M. Pototschnig. Funding is provided by the Institute for Quantum Information and Matter, an NSF Physics Frontier Center with support of the Gordon and Betty Moore Foundation, by NSF Grant # PHY0652914, by the DoD NSSEFF program, by the AFOSR MURI for Quantum Memories, and by Northrop Grumman Aerospace Systems. Akihisa Goban is supported by the Nakajima Foundation.