Optical trap for an atom around the midpoint between two coupled identical parallel optical nanofibers

We study the trapping of a ground-state cesium atom in a small region around the midpoint between two coupled identical parallel optical nanofibers. We suggest to use a blue-detuned guided light field in the odd $\mathcal{E}_z$-sine array mode to produce an optical potential with a local minimum of exact zero at the midpoint between the nanofibers. We find that the effects of the van der Waals potential on the total trapping potential around the minimum point are not significant when the fiber separation distance and the power of the guided light field are large. For appropriate realistic parameters, a net trapping potential with a significant depth of about 1 mK, a large coherence time of several seconds, and a large recoil-heating-limited trap lifetime of several hours can be obtained. We investigate the dependencies of the trapping potential on the power of the guided light field, the fiber radius, the wavelength of light, and the fiber separation distance.

A successful technique for trapping atoms near a nanofiber is to combine optical dipole forces of a blueand a red-detuned guided light field [5,6]. This two-color trap scheme has been experimentally realized for lasercooled alkali-metal atoms at about 200 nm above the nanofiber surface [7,8,[10][11][12][13]. Other nanofiber-based atom traps have been proposed and investigated, such as a combination of the attractive potential of a red-detuned guided field and the repulsive potential of the centrifugal force [14], interference of higher-order modes [15,16], a diffracted laser field impinging perpendicularly to the fiber [17], a helical two-color trapping potential [18], a combination of fictitious and real magnetic fields [19], and a nanofiber with a slot [20,21].
Coupled waveguides play an important role in numerous optical devices such as multicore fibers, optical directional couplers, polarization splitters, ring resonators, and interferometers [22][23][24]. Recently, optical devices comprising of twisted or knotted nanofibers have been fabricated [25]. Coupling between two nanofibers has been studied [25,26] in the framework of the coupled mode theory [22][23][24]. It has been shown that the guided normal (array) modes of two coupled dielectric rods can be calculated by using the circular harmonics expansion [27]. This method has been extended to multicore fibers [28][29][30][31]. The propagation constant and the flux density of the field in a guided normal mode have been studied [27,31,32]. The polarization patterns [31] and the mode cutoffs [33] have been investigated. In optomechanics, forces arising from internal illumination by light traveling in coupled waveguides have been studied [34], and light-guiding arrays of mechanically compliant glass nanospikes have been fabricated [35].
Recently, the spatial distributions of the fields in the guided array modes of two coupled parallel nanofibers have been examined [36]. It has been shown that the distribution of the field intensity in the odd E z -sine array mode has a local minimum of exact zero at the midpoint between the nanofibers [36]. This feature can be used to trap atoms with a single blue-detuned light field. In order to realize an optical trap for atoms in a small region around the midpoint between the nanofibers, we need to produce an optical potential that can dominate the surface-induced van der Waals potential in the trapping region. The parameters for the system must be realistic, while the characteristics of the obtained trapping potential must be appropriate for applications. The possibility of trapping atoms between two parallel nanofibers may open up new applications in nonlinear optics, quantum optics, and quantum information. While a setup with infinite, parallel nanofibers might be currently hard to realise, the configuration is also somewhat similar to a slotted fiber, as proposed for atom trapping in [20] and demonstrated for nanoparticle trapping in [21]. Therefore, it is desirable to study this issue in detail.
In this work, we study the possibility to trap a groundstate cesium atom in a small region around the midpoint between two coupled identical parallel optical nanofibers. We investigate the dependencies of the trapping potential on the power of the guided light field, the fiber radius, the wavelength of light, and the fiber separation distance. We show that a blue-detuned far-off-resonance field in the odd E z -sine array mode can produce a trapping potential around the midpoint between the nanofibers with a significant depth, a large coherence time, and a large trap lifetime.
The paper is organized as follows. In Sec. II we describe the model of two coupled identical parallel optical nanofibers and discuss the properties of the spatial field distribution in the odd E z -sine array mode. In Sec. III, we calculate the trapping potential of an atom outside the nanofibers. Finally, we conclude in Sec. IV. We study two identical vacuum-clad, optical nanofibers that are aligned parallel to each other in the direction of the fiber axis z (see Fig. 1). The fibers are labeled by the indices j = 1, 2. Each nanofiber j can be treated as a dielectric cylinder with a radius a j = a and a refractive index n j = n f > 1, surrounded by an infinite background of vacuum or air with a refractive index n 0 = 1. The nanofiber diameters are a few hundreds of nanometers. Depending on the fiber size parameter V = ka n 2 f − n 2 0 , an individual nanofiber j can support either a single or multiple modes. Here, k = ω/c is the wave number of light with optical frequency ω in free space. We are interested in the normal modes of the two-fiber system. We assume that the fibers are fixed in space and are, therefore, not interested in the van der Waals interaction between them.
We introduce the global Cartesian coordinate system {x, y, z}. Here, the z axis is parallel to the z 1 and z 2 axes of the fibers, the x axis is perpendicular to the z axis and connects the centers O 1 and O 2 of the fibers, and the y axis is perpendicular to the x and z axes (see Fig. 1). The plane xy is the transverse (cross-sectional) plane of the fibers. The x and y axes are called the radial and tangential axes, respectively, of the two-fiber system [see Fig. 1(b)]. The positions of the fiber centers O 1 and O 2 on the x axis are O 1 = −(a + d/2) and O 2 = a + d/2, where d is the fiber separation distance. We introduce the polar coordinate system {r, ϕ} associated with the central Cartesian coordinate system {x, y}. For each individual fiber j, we introduce the local polar coordinate system {r j , ϕ j }.
We consider a light field with an optical frequency ω which propagates in the +z direction with a propagation constant β. The electric and magnetic components of the field can be written as E = [Ee −i(ωt−βz) + c.c.]/2 and H = [He −i(ωt−βz)t + c.c.]/2, respectively, where E and H are the slowly varying complex envelopes.
The normal modes of the coupled fibers are called array modes. The exact theory for the guided normal modes of two parallel dielectric cylinders has been formulated in [27]. According to [27], there are four kinds of normal modes, denoted as even E z -cosine, odd E z -cosine, even E z -sine, and odd E z -sine array modes. We are interested in the case where the fiber radii are small enough that no more than one normal mode of each of the four kinds can be supported. It has been shown that, for an odd E z -sine array mode of two coupled identical parallel fibers, the electric intensity distribution attains a local minimum of exactly zero at the two-fiber center (the midpoint between the nanofibers) [36].
We employ the theory of [27] to calculate the propagation constant and the spatial distribution of the field in an odd E z -sine array mode of two identical parallel vacuum-clad silica nanofibers [36]. The key results of [27] for this mode are summarized in Appendix A. We solve equations (A6) and use the expressions (A1), (A2), and (A8) to calculate the components of the field. In our numerical calculations, the infinite number of circular harmonics is truncated at a finite number N max in the range from 9 to 19. The value of N max is chosen such that the propagation constant converges and the boundary conditions are satisfied with a reasonable accuracy [27]. To calculate the refractive index n f of the silica nanofibers, we use the four-term Sellmeier formula for fused silica [37,38].
We plot in Fig. 2 the propagation constant β of the odd E z -sine array mode as functions of the fiber radius a, the light field wavelength λ, and the fiber separation distance d. We observe from Figs. 2(a) and 2(b) that the odd E z -sine array mode has cutoffs in the dependencies of β on a and λ. The position of a cutoff is determined by the solution to the equation β/k = 1 [27].
We display in Fig. 3 the cross-sectional profile and the axial profiles of the electric intensity distribution |E| 2 of the field in the odd E z -sine array mode of two identical parallel nanofibers. We see from the figure that |E| 2 is symmetric with respect to the principal axes x and y. Figures 3(b) and 3(c) show that the electric field E of the odd E z -sine array mode is exactly equal to zero at the midpoint (x, y) = (0, 0) between the nanofibers. This feature of the odd E z -sine array mode can be used to trap ground-state atoms with a blue-detuned light field [46][47][48]. The existence of a local minimum of exact zero at the two-fiber center is due to the destructive interference between the fields of the individual fibers in the odd E z -sine array mode, and occurs for any fiber separation distance d.

III. TRAPPING POTENTIAL OF AN ATOM OUTSIDE THE NANOFIBERS
We consider an alkali-metal atom moving in the optical potential generated by an off-resonant guided light field in the odd E z -sine array mode outside the two nanofibers.

A. Optical potential
We assume that the atom is in the ground state and the field is off resonance with the atom. The optical potential U opt of the atom in the field is then given by [39] where α = α(ω) is the real part of the scalar dynamical polarizability of the atom at the optical frequency ω. The factor 1/4 in Eq. (1) results from the fact that the dipole of the atom is not a permanent dipole but is induced by the field, giving 1/2, and from the fact that the intensity is averaged over optical oscillations, giving another 1/2.
The function α(ω) for a ground-state alkali-metal atom is given by [39] Here, g j and g a are the statistical weights of the excited level |j and the ground-state level |a , respectively, ω ja and A ja are the frequency and emission transition probability, respectively, of the spectral line ja, and γ j is the lifetime of the excited level |j . We note that the vector polarizability is neglected in Eqs. (1) and (2) and the tensor polarizability is vanishing for the ground-state alkali-metal atom [40,41].
x (nm) y (nm) We plot in Fig. 4 the spatial profile of the optical potential U opt of a ground-state cesium atom in the area {r 1 , r 2 > a; |x| < a + d/2; |y| < a}, which lies between the two fibers. The inset of the figure shows U opt in a small vicinity of the two-fiber center O = (0, 0). We observe that U opt has a local minimum of exact zero at O. This feature can be used to trap atoms. Note that, due to the geometry of the system, the optical potential is not cylindrically symmetric.
The rate of spontaneous scattering caused by a light field E is given by where κ = κ(ω) is the imaginary part of the scalar dynamical polarizability of the atom. The function κ(ω) for a ground-state alkali-metal atom is given by [39] κ(ω) = 2πǫ 0 c 3 For atoms spending time in a motional quantum state |ψ , the average scattering rate is Γ sc ≡ ψ|Γ sc |ψ = (κ/4h) |E| 2 with |E| 2 ≡ ψ|(|E| 2 )|ψ . The characteristic coherence time of the trap is A scattered photon imparts a recoil energy E rec = (hk) 2 /2M to the atom, where M is the mass of the atom. Therefore, the absorption of incident photons and the emission of other photons result in a loss of atoms from the trapping potential. For a trap depth U D , the trap lifetime due to recoil heating is given by [43] When the light field frequency is far from the atomic resonances and the field at the trapping potential minimum is weak, the scattering rate is small and, therefore, the coherence time and the trap lifetime are large.

B. van der Waals potential
An atom near the surface of a medium undergoes a van der Waals force. The van der Waals potential of an atom at a radial position r near the surface of a cylindrical dielectric rod of radius a, with r > a, is given by [44] where Here ǫ(iξ) is the dynamical dielectric function of the medium for the imaginary frequency iξ, and I n and K n are the modified Bessel functions of the first and second kinds, respectively. The van der Waals potential of a ground-state cesium atom near a silica fiber has been calculated numerically [6].
For an atom near two identical parallel optical nanofibers, the van der Waals potential is given as Here, r 1 and r 2 are the distances from the atom to the fiber axes z 1 and z 2 , respectively.

C. Total potential
The total potential of the atom is given as We plot in Fig. 5(a) the cross-sectional profile of the total potential U of a ground-state cesium atom in the vicinity of the two-fiber center O = (0, 0). In Figs. 5(b) and 5(c), we depict the axial profiles of the total potential U in the region between the nanofibers. We observe from the figures that U (x, y) has a negative local minimum U min ≡ U (0, 0) = U vdW (0, 0) < 0 at O. Comparison between the solid red and dashed blue lines of Figs. 5(b) and 5(c) shows that the total potential U is mainly determined by the optical potential U opt in the vicinity of the two-fiber center O, which is positioned at a distance of 150 nm from the fiber surfaces in the case of the figures. This feature is a consequence of the fact that the effects of the van der Waals potential on the total trapping potential around the minimum point are not significant when the fiber separation distance and the power of the guided light field are large enough. Like the optical potential U opt , the total potential U is not cylindrically symmetric due to the geometry of the system. Figures 5(b) and 5(c) and the scales of their vertical axes show that the depth of the profile of the potential along the x axis is larger than that of the profile of the potential along the y axis. Therefore, the depth of the potential profile along the axis y is the effective depth of the trapping potential.
The one-dimensional motion of the atom along the x or y axis can be treated as the motion of a particle in the potential U x (x) = U (x, 0) or U y (y) = U (0, y). In Fig. 6, we plot the wave functions of the first five levels of the one-dimensional motion of the atom in the potentials U x and U y for the case of Fig. 5. We find that the trapping frequencies are ω x /2π ∼ = 1441 kHz and ω y /2π ∼ = 438 kHz, comparable to the characteristic values for the singlenanofiber two-color traps [7,8,[10][11][12][13]. The spacing between the the energies of the ground state and the first excited state is roughly equal tohω x ∼ = k B × 69 µK for the motion along the x axis andhω y ∼ = k B × 21 µK for the motion along the y axis. The characteristic sizes of the corresponding motional ground states are ∆x = h/2M ω x ∼ = 5.1 nm and ∆y = h/2M ω y ∼ = 9.3 nm.
The scattering rates for the ground states of the atomic motions along the x and y axes in the case of Fig. 5 are found to be Γ sc x ∼ = 0.17 s −1 and Γ sc y ∼ = 0.05 s −1 . The scattering rate Γ sc of the atom in the trap is estimated as the maximal value of Γ sc x and Γ sc y , and is, in the case of Fig. 5, given as Γ sc ∼ = 0.17 s −1 . The corresponding coherence time is τ coh ≡ 1/ Γ sc ∼ = 5.8 s. The trap depth is estimated as the minimal value of the trap depths for the motions along the x and y axes, and is, in the case of Fig. 5(c), equal to U D ∼ = 0.7 mK, comparable to the depths of the single-nanofiber two-color traps [7,8,[10][11][12][13]. The corresponding recoillimited trap lifetime is τ trap ∼ = 4.8 h. This value is much larger than the recoil-limited lifetimes of 100 s and 30 s estimated for the single-nanofiber two-color traps in the experiments [7] and [8], respectively.

D. Dependencies of the trapping potential on the parameters of the fibers and the light field
The trapping potential U depends on the light field power P , the fiber radius a, the light field wavelength λ, and the fiber separation distance d.
The power of the guided light is given by P = (1/2) Re [E × H * ] z dr, where r = (x, y) and dr = ∞ −∞ dx ∞ −∞ dy. We plot in Fig. 7 the trapping potential U for different values of the power P of the guided light. It is clear from the figure that the magnitude and the depth of the trapping potential are almost linearly proportional to the power. This feature is a result of the fact that, when the position r is not too close to the fiber surfaces, the total potential U (r) is mainly deter- We plot in Fig. 8 the trapping potential U for different values of the fiber radius a. We observe from the figure that, when a is small (see the dotted red lines) or large (see the dashed green lines), U is shallow. These features are consequences of the wide spread of the guided field outside a thin fiber and the tight confinement of the guided field inside a thick fiber.
We plot in Fig. 9  that, when λ is closer to the resonance with the atom (852 nm for the atomic cesium D 2 line), U is deeper. These features appear as the consequences of the dependence of the atomic polarizability on the wavelength of a detuned light field. Note that the dependence of the trapping potential on the wavelength of light occurs through not only the atomic polarizability but also the mode profile and, consequently, the field intensity distribution. We plot in Fig. 10  . We also observe that an increase in d leads to an almost linear increase in the width of the trapping potential along the x axis but does not affect much the width of the trapping potential along the y axis.

IV. SUMMARY
In this work, we have studied the trapping potential of a ground-state cesium atom in a small region around the midpoint between two coupled identical parallel optical nanofibers. We have suggested to use a blue-detuned guided light field in the odd E z -sine array mode of the two nanofibers to produce an optical potential with a local minimum at the midpoint between the nanofibers. The vanishing of the field at the two-fiber center is a result of the destructive interference between the coupled modes of the individual nanofibers. We have demonstrated that the effects of the van der Waals potential on the total trapping potential around the minimum point are not significant when the fiber separation distance and the power of the guided light field are large. We have shown that, for appropriate realistic parameters, a net potential with a significant depth, a large coherence time, and a large trap lifetime can be obtained. We have demonstrated, for example, that a pair of 200-nm-radius silica fibers carrying 100 mW of 780-nm-wavelength light in the odd E z -sine array mode gives for cesium atoms a trap depth of 0.7 mK, a coherence time of 5.8 s, and a recoil-heatinglimited trap lifetime of 4.8 h. Due to the geometry of the system, the trapping potential is not cylindrically symmetric. The depth of the trapping potential profile along the tangential direction y is smaller than that of the trapping potential profile along the radial direction x and is, hence, the effective trap depth. The dependencies of the trapping potential on the power of the guided light field, the fiber radius, the wavelength of light, and the fiber separation distance have been investigated. It has been shown that, at a position that is not too close to the fiber surfaces, the total potential is mainly determined by the optical potential and, hence, is linearly proportional to the power of light. When the fiber radius is small or large enough, the potential at the midpoint between the nanofibers is shallow due to the wide spread of the guided field outside a thin fiber and the tight confinement of the guided field inside a thick fiber. We have observed that the wavelength of light affects the trapping potential through the atomic polarizability and the field intensity distribution. We have found that the depth of the trapping potential decreases with increasing fiber separation distance.

ACKNOWLEDGMENTS
This work was supported by the Okinawa Institute of Science and Technology Graduate University and by the Grant-in-Aid for Scientific Research (C) 20K03795.
Appendix A: Odd Ez-sine array mode According to [27], for an odd E z -sine guided normal (array) mode, the longitudinal components E z and H z of the electric and magnetic parts, respectively, of the field are given, inside fiber j = 1, 2, as E z = ∞ n=0 E nj J n (hr j ) sin nϕ j , and, outside the two fibers, as G nj K n (qr j ) sin nϕ j , H nj K n (qr j ) cos nϕ j .
Here, we have introduced the fiber parameters h = k 2 n 2 f − β 2 , q = β 2 − k 2 n 2 0 , which determine the scales of the spatial variations of the field both inside and outside the fibers. In Eqs. (A1) and (A2), E nj , F nj , G nj , and H nj are the mode expansion coefficients, J n represents the Bessel functions of the first kind, and K n represents the modified Bessel functions of the second kind. The symmetry relations for the coefficients E mj , F mj , G mj , and H mj with j = 1, 2 are [27] The coefficients E n1 and F n1 are given by the equations [27] J n (u)E n1 = K n (w)G n1 + I n (w) ∞ m=0 g nm G m1 , The coefficients G n1 and H n1 are nonzero solutions of the equations [27] n 1 u 2 +