## Abstract

We propose a new scheme to guide cold atoms (or molecules) using a blue-detuned TE_{01} doughnut mode in a hollow metallic waveguide (HMW), and analyze the electromagnetic field distributions of various modes in the HMW. We calculate the optical potentials of the TE_{01} doughnut mode for three-level atoms using dressed-atom approach, and find that the optical potential of the TE_{01} mode is high enough to guide cold atoms released from a standard magneto-optical trap. Our study shows that when the input laser power is 0.5W and its detuning is 3GHz, the guiding efficiency of cold atoms in the straight HMW with a hollow radius of 15 μm can reach 98%, and this guiding efficiency will be almost unchanged with the change of curvature radius *R* of the bent HMW as *R* > 2cm, which is a desirable scheme to do some atom-optics experiments or realize a computer-controlled atom lithography with an arbitrary pattern. We also analyze the losses of the guided atoms in the HMW due to the spontaneous emission and background thermal collisions and briefly discuss some potential applications of our guiding scheme in atom and molecule optics.

©2005 Optical Society of America

## 1. Introduction

In atom optics, the guiding of cold atoms is one of the basic techniques to manipulate and control neutral atoms, in which the laser guiding of cold atoms is an important and popular candidate. There are two kinds of laser guiding schemes for cold atoms, one is the atomic guiding using a blue-detuned dark hollow beam (DHB) [1–4] or a red-detuned Gaussian beam [5,6]. Owing to the straight propagating property of the laser beams, they can only be used to straightly guide cold atoms, and it is difficult to realize a bent atomic guiding. The other is the atomic guiding using a red-detuned Gaussian mode [7–9] or a blue-detuned LP_{01} mode evanescent wave (EW) [10–12] in the hollow optical fiber (HOF). Due to the flexibility of the HOF, it can be used to realize the bent guiding of cold atoms, and then to realize atom lithography. However, there is a higher heating from the spontaneous-emission of the guided atoms in the red-detuned Gaussian mode, which will cause a serious loss of atomic coherence, even result in a considerable drop of atomic guiding efficiency. Generally speaking, the hollow diameter of the HMW is larger than one of the HOF used for the atomic guiding using a red-detuned Gaussian-mode since which is the lowest order mode and far larger than one of the HOF used for the EW atomic guiding with a blue-detuned LP_{01}-mode, so it is difficult to obtain a higher vacuum in both the HOF atomic guides. Also, it cannot be used to realize the evanescent-wave cooling of the guided atoms in the HOF because a weak repumping beam cannot be coupled efficiently into the micro-sized hollow region of the HOF.

Since the HMW has a larger complex refractive index in an optical frequency region [13,14], it is far less sensitive to the curvature of the guide axis. If a blue-detuned TE_{01} doughnut mode in the HMW with a hollow radius of a few 10 - μm can be used to guide cold atoms, it cannot only retains the flexibility of the HOF guiding, but also reserves some advantages of the atomic guiding using a blue-detuned DHB. In this case, cold atoms cannot only be guided in the dark region of the TE_{01} mode in the HMW (which will suffer the minimal light shift and the lowest coherence loss, and obtain a better vacuum), but also be cooled by the intensity-gradient-induced Sisyphus effect from the TE_{01} doughnut mode, which can be used to realize the computer-controlled atomic lithograph or generate an ultracold atomic beam and so on. So it would be interesting and worthwhile to study and realize a novel atomic guiding by using the blue-detuned TE_{01} mode in the HMW.

This paper is organized as follows. In Sec.2, we propose a new scheme to guide cold atoms, and analyze the mode structures of the electromagnetic fields in the HMW and their propagation losses, and calculate the absolute intensity distributions of the TE_{01} mode in the HMW. In Sec.3, the optical potentials for a Λ -configuration three-level atom and its spontaneous-emission rates are derived and calculated. In Sec.4, the guiding efficiency of cold atoms in the HMW is calculated by using the atom flux probability based on a classical model, and the corresponding loss mechanisms are discussed. In Sec.5, some potential applications of our guiding scheme in the fields of atom and molecule optics are briefly discussed. Some main results and conclusions are summarized in the final Section.

## 2. Guiding scheme and electromagnetic fields in HMW

#### 2.1. Guiding scheme of cold atoms

A proposed scheme to guide cold atoms using a blue-detuned TE_{01} doughnut mode in the cylindrical HMW is shown in Fig. 1. A collimated Gaussian beam with a right-rotated (σ^{+}) circular polarization passes through a 2*π* phase plate and is focused by a positive lens with a focal length *f* , and a focused hollow beam (FHB) will be generated [15] and coupled into the hollow region of the HMW. The HMW [see Fig. 1(a)] is composed of a hollow region with a radius *a*, a thin metallic layer with a negligible thickness and a cladding with an infinite outer radius.

Since there is the largest dark spot size at the - *f*/2 position of the FHB, our magneto-optical trap (MOT) is prepared at this -*f*/2 position. To obtain cold atoms and realize an efficiently loading of cold atoms from the MOT into the TE_{01} mode in the hollow region of the HMW, a two-dimensional (2D) moving optical molasses is used to prepare the MOT and generate a cold atomic beam along the propagating direction of the FHB. When the incident Gaussian beam is blue-detuned, the cold atomic beam extracted from the MOT is focused and loaded into the dark central region of the TE_{01} mode in the HMW by using the blue-detuned FHB as an atomic funnel, and then to realize laser guiding of cold atoms in the HMW.

#### 2.2. Mode analysis of the electromagnetic fields in the HMW

We assume that the wavelength *λ* of the coupling FHB in free space is much smaller than the hollow radius *a* of the HMW, and then the laser energy will nearly not be propagated in the metallic layer due to almost zero penetrating depth into the metallic layer of the HMW, but essentially propagated in the hollow region, and there is almost no energy loss in the HMW due to reflection of the metal layer [13]. In the cylindrically symmetric coordinates (*r,ϕ,z*), the electromagnetic field modes *E*_{r}
, *E*_{ϕ}
, *H*_{r}
and *H*_{ϕ}
in the hollow region of the HMW can be written as

where *ω* is the angular frequency, *γ* is the axial propagation constant, *m* is the azimuthal index, and ${k}_{i}^{2}$ - *μ*
_{0}
*ε*
_{0}
*ω*
^{2} - *γ*
^{2} . *E*_{z}
and *H*_{z}
obey Helmholtz equations and can be further written as

where *J*_{m}
(*k*_{i}*r*) is the first kind Bessel function and *N*_{m}
(*k*_{i}*r*) is the second kind Bessel function. Considering the finite value of *E*_{z}
(*r*)and *H*_{z}
(*r*) at *r* = 0 , we obtain *C*
_{2} = *C*
_{4} = 0. Therefore, substituting Eq. (2) into Eq. (1), we derive the expressions of *E*_{r}
and *E*_{ϕ}
as follows

Considering the boundary condition of *r* = *a* and under the approximation of |*η*_{m}
|*u*_{mq}
<< *ka* , we obtain *k*_{i}*a* ≈ *u*_{mq}
(1 - *iη*_{m}
/*ka*) ≈ *u*_{mq}
[13], here *k* = 2*π*/*λ* and *u*_{mq}
is the *q*th root of Bessel function of the first kind: *J*
_{m-1} (*u*_{mq}
) = 0, *η*_{m}
= 1/(${\eta}_{d}^{2}$ - 1)^{1/2}, and *η*_{d}
= *n* + *iκ* is the complex refractive index of the metallic layer in the HMW.

From the above analysis, we know that there are three types of modes in the hollow region of the HMW [13]: transverse circular electric modes TE_{0q}, transverse circular magnetic modes TM_{0q} and hybrid modes EH_{mq}. First, the lines of electric field of the TE_{0q} modes are transverse concentric circles centered on the *z* axis, and their magnetic-field lines are in the planes containing the *z* axis. In particular, the first-order TE_{01} mode has a circular polarization and a doughnut-shaped intensity distribution, which will be analyzed in some detail below. Second, the lines of electric field of the TM_{0q} modes are contained in radial planes, while their magnetic-field lines are transverse concentric circles centered on the *z* axis. The first-order TM_{01} mode has a linear polarization and a doughnut-shaped intensity distribution. In final, the EH_{mq} modes are hybrid, and the electric and magnetic field are almost transverse. In fact, the first-order EH_{11} mode is a Gaussian one, it has a linear polarization and the Gaussian intensity distribution in the HMW.

Since a focused doughnut beam with a circular polarization is coupled into the hollow region in the HMW, a TE_{01} mode that has same polarization and similar intensity distribution will be excited selectively in the hollow regime of the cylindrical HMW, while the TM_{0q} and EH_{mq} modes with a linear polarization cannot be effectively excited [16]. So we only consider the electric field distribution of the TE_{01} mode in the HMW due to the magnetic field distribution of the TE_{01} mode is far smaller than its electric field one.

For TE_{0q} mode, we have *m* = 0 and *E*_{z}
= 0 . From Eqs. (1)–(3), when *q*=1, the electric field of the TE_{01} mode in the hollow region of the HMW can be expressed as

From Eq. (4b), we calculate the normalized electric field distribution of the TE_{01} mode in the HMW for three different hollow radiuses, and the results are shown in Fig. 2(a). It is clear from Fig. 2(a) that the TE_{01} mode in the HMW is a doughnut-shaped one, which can be used to guide cold atoms as the incident laser beam is blue-detuned.

#### 2.3. Mode coupling and the absolute intensity distribution of the TE_{01} mode

In the focal plane of *z*=0, the electric field profile of the FHB can be approximately described by the TE${\mathrm{M}}_{01}^{*}$ doughnut beam mode as follows [15]

where *P*
_{in} is the input power of the FHB, *w*
_{0} is the beam waist of the FHB in the focal plane, *k*
_{1} is the fitting parameter. If such a FHB with a circular polarization is coupled into the HMW, the TE_{01} mode can be selectively excited, and the coupling efficiency *A*, i.e., the ratio of the power *P* of the TE_{01} mode to the input power *P*
_{in} of the FHB, is given by [17]

From Eq. (6), we calculate the dependence of the coupling efficiency *A* for the TE_{01} mode on the beam waist *w*
_{0} of the incident FHB and find that there is a maximum coupling efficiency for a given *a*. If we choose the beam waist of the FHB as *w*
_{0} = 8.46μm and *a* = 15 μm , the coupling efficiency *A* for the TE_{01} mode can be reached the maximum value 96.7%, i.e., *P* ≈ 0.967*P*_{in}
, while the coupling efficiency for the TE_{02} mode is 0.94%, and the coupling efficiency for other modes (including a EH_{11} Gaussian mode) is only ~ 2.36%.

The absolute radial intensity distribution of the TE_{01} mode in the HMW with a radius *a* is set as *I*(r), and supposing that the propagating loss of the TE_{01} mode can be neglected (see below), the propagating power *P* of the TE_{01} mode in the HMW can be calculated by
*P* = ${\int}_{0}^{a}$ 2*πrI*(*r*)*dr*. So we can obtain the absolute intensity distribution of the TE_{01} mode in the HMW as follows

From Eq. (7) and considering *P* ≈ 0.967*P*_{in}
, we calculate the intensity distribution of the TE_{01} mode in the HMW as the hollow radius *a* = 15μm and *P*
_{in} = 0.5 W, and show that the maximum intensity *I*
_{0} of the TE_{01} mode at the radial position *r*
_{0} = 7.2μm can reach 1.43×10^{8}mW/cm^{2} , which is far greater than the maximum evanescent-wave intensity (1.64×10^{6}mW/cm^{2}) of the LP_{01} mode in the HOF with a hollow radius *a* = 3.5μm and the same power [2].

#### 2.4. Propagation losses of various modes in the HMW

In the HMW, the TE_{0q} mode for the straight waveguide has a loss coefficient
[13], where *η*
_{0} = 1/(${\eta}_{d}^{2}$ - 1)^{1/2} from which we can see that the losses for the higher order TE_{0q} modes are increased by a factor of 3 for each higher order mode.

To know and compare the straight propagation losses of the EH_{11}, TM_{01} and TE_{01} modes in the different HMWs [13,14], we calculate the dependences of the electric field amplitudes of the EH_{11}, TM_{01} and TE_{01} modes in two HMWs (Al [14] and Ni [18]) on the propagation distance *z* for *a* = 15μm, *λ* = 0.78μm, and the results are shown in Fig. 2(b), here the electric field amplitude of the EH_{11} or TM_{01} mode is normalized by that of the TE_{01} mode. We can see from Fig. 2(b) that the EH_{11} (or TM_{01}) mode in the HMWs will be attenuated rapidly with the propagation distance *z*, and the greater the complex refractive index of the metallic layer is, the larger the propagation loss of the EH_{11} (or TM_{01}) mode in the HMW is. In particular, when the metallic Al hollow waveguide is used, the effective straight propagation distance of the EH_{11} (or TM_{01}) mode determined by 1/e of the field amplitude is only about 3 mm (or 0.7mm). While the loss of the TE_{01} mode in the HMWs is very small so that it can be neglected in several 10-cm propagating distance, and it is nearly independent of the complex refractive index of the metallic layer on the propagation distance *z* within several 10-cm region.

Moreover, the propagation loss of the TE_{01} mode in the bending HMW with *a* = 15μm is given by ${\alpha}_{01}^{b}$ ≈ *α*
_{01}(1 + 2.97×10^{-4}/*R*
^{2}) [13,14], where *R* is the curvature radius of the HMW. Therefore, the electric field amplitude of the TE_{01} mode is almost unchanged for both strait and bent (*R* > 2cm) HMW with a length of several 10-cm.

From the above analysis, we find that an aluminum HMW with a smaller hollow radius (such as *a* = 15μm) should be desirable to selectively excite and propagate a TE_{01} doughnut mode in the HMW by using a circular polarized FHB, and to efficiently restrain the excitation and propagation of the HE_{11} (TM_{01} and other higher-order) mode in the HMW.

## 3. Optical potentials and spontaneous emissions

#### 3.1. Optical potentials for a three-level atom

We consider a Λ -configuration three-level atom with one excited state |*e*〉 and two hyperfine ground states |*g*
_{1}〉 and |*g*
_{2}〉, which is interacted with an intense laser field that has a detuning *δ* from the atomic resonant frequency *ω*
_{0} between the lower hyperfine ground state |*g*
_{1}〉 and the excited state |*e*〉.

The equation of eigenstates of the atom-light interaction system in the dressed-state picture can be written as

where *ω*_{L}
is the laser frequency, *δ*_{hfs}
is the level splitting between two hyperfine ground states, and *G*
_{1} (*G*
_{2}) is a real coupling parameter corresponding to Rabi frequency Ω_{1} (Ω_{2}) between |*g*
_{1},*n* + 1〉 (|*g*
_{2},*n* + 1) and |*e,n*〉 . In Eq.(8), *A*_{i}
, *B*_{i}
and *C*_{i}
are three probability
amplitudes respectively in |*g*
_{2},*n* + 1〉 , |*g*
_{1},*n* + 1〉 and |*e,n*〉 corresponding to the dressed-eigenstate |*i,n*〉, here *i*=1,2,3.

Since the exact solutions of Eq. (8) are very complex and cannot give clear analytic relationships between the eigenenergies (or optical potentials) and the parameters (*δ*,*δ*_{hfs}
,Ω_{1},Ω_{2}) of atom-light interaction system, we substitute a trial solution ${E}_{\mathit{\text{Dr}}1}^{t}$ =(*n* + 1)*ħω*_{L}
- *ħδ*/2±ħ(*δ*
^{2} + ${\mathrm{\Omega}}_{1}^{2}$)/2 into Eq.(8) and derive two other solutions ${E}_{\mathit{\text{Dr}}2}^{t}$ and ${E}_{\mathit{\text{Dr}}3}^{t}$. Here the trial solution ${E}_{\mathit{\text{Dr}}1}^{t}$ is the eigenenergy of the dressed-eigenstate |1,*n*〉 coming from the two-level atom model [19–21], and ${E}_{\mathit{\text{Dr}}2}^{t}$, ${E}_{\mathit{\text{Dr}}3}^{t}$ correspond to dressed-eigenstates |2,*n*〉, |3,*n*〉 respectively. Then substituting ${E}_{\mathit{\text{Dr}}2}^{t}$ into Eq. (8) and considering the conservation of energy, we derive two other eigenenergies, by which we can obtain three approximate expressions of the optical potentials corresponding to the dressed-eigenstates |1,*n*〉, |2,*n*〉, |3,*n*〉 as follows

where Ω'_{1} = (*δ*
^{2} + ${\mathrm{\Omega}}_{1}^{2}$)^{1/2}. In Eq. (9), “+” and “-” in “±” or “-”and “+” in “+” represent the case of *δ* > 0 and *δ* < 0 respectively, “sgn” is equal to 1 as *δ* > - *δ*_{hfs}
and -1 as *δ* < -*δ*_{hfs}
, which will be the same meaning in all of the following equations. If we choose the small saturation parameter approximation [22] and the zero-order approximation of Eq. (9), the optical potentials *U*
_{1}, *U*
_{2} and *U*
_{3} can be reduced as the same as the expressions of the optical potentials in Refs. [19–21], which can be further simplified as the same as the expressions of the optical potentials in Refs. [3,22–27] by using the small saturation parameter approximation again and taking the first-order approximation.

For the alkali-metal atom, Ω_{j} = (*I*/2*I*_{sat}
)^{1/2}Γ${f}_{j}^{1/2}$, *I* is the laser intensity, *I*_{sat}
is the saturation intensity of the atom, *f*_{j}
= 2/3 is the relative transition strength from |*e*〉 to |*g*_{j}
〉, j=1,2, T is the spontaneous emission rate of the excited state |*e*〉 [26]. From Eq. (9), we calculate the optical potentials for a three-level ^{85}Rb atom and obtain the dependences of the optical potentials on the detuning *δ*/2*π* when *δ*_{hfs}
/2*π* = 3.04GHz , Γ/2*π* = 6MHz , *I*_{sat}
= 1.6mW/cm^{2} and *I* = 500 mW/cm^{2}, and the results are shown in Fig. 3(a). We find from Fig. 3(a) that there is a pair of non-resonant peaks of the optical potential at the detuning *δ*/2*π* = 0 for *U*
_{1} and at the detuning *δ*/2*π* = -*δ*_{hfs}
/2*π* for *U*
_{2}. From Eq.(9), we also calculate the dependences of the optical potentials *U*
_{1} on the radius *r* for *a* = 15 μm, *P*
_{in}=0.5 W and *δ*/2*π* = 0.5GHz, 3GHz, 5GHz, as shown in Fig. 3(b), and find that the optical potential is greater than 450 mK as *δ*/2*π* < 5GHz, which is high enough to guide cold atoms (~ 120μK) from a standard MOT.

#### 3.2. Spontaneous emissions of a three-level atom

When a three-level atom moves in the light field of the blue-detuned TE_{01} mode in the HMW, it will experience spontaneous emission and its heating, which will cause some atoms to be escaped from the trap. To obtain the spontaneous emission rates of the three-level atom in the intense laser field, we substitute three dressed-eigenenergies into Eq.(8) respectively and solve them, and then three eigenstates |1,*n*〉, |2,*n*〉 and |3,*n*〉 in the dressed-atomic picture can be given by

where *i* = 1,2,3, the probability amplitudes *A*_{i}
, *B*_{i}
and *C*_{i}
are given by

where

$${a}_{12}=\frac{{\Omega}_{2}}{{-\frac{\delta}{2}\mp \frac{{\Omega}_{1}^{\text{'}}}{2}\pm \left[{\left(\pm \frac{{\Omega}_{1}^{\text{'}}}{2}+\frac{\delta}{2}\right)}^{2}+{\Omega}_{1}^{2}\right]}^{\frac{1}{2}}-{2\delta}_{\mathit{hfs}}},$$

$${a}_{21}=\frac{-{\Omega}_{1}}{{\frac{\delta}{2}\pm \frac{{\Omega}_{1}^{\text{'}}}{2}-{\delta}_{\mathit{hfs}}-\mathrm{sgn}\times \left[{\left(\pm \frac{{\Omega}_{1}^{\text{'}}}{2}+\frac{\delta}{2}+{\delta}_{\mathit{hfs}}\right)}^{2}+{\Omega}_{2}^{2}\right]}^{\frac{1}{2}}},$$

$${a}_{22}=\frac{{-\Omega}_{2}}{{\frac{\delta}{2}\pm \frac{{\Omega}_{1}^{\text{'}}}{2}+{\delta}_{\mathit{hfs}}-\mathrm{sgn}\times \left[{\left(\pm \frac{{\Omega}_{1}^{\text{'}}}{2}+\frac{\delta}{2}+{\delta}_{\mathit{hfs}}\right)}^{2}+{\Omega}_{2}^{2}\right]}^{\frac{1}{2}}},$$

$${a}_{31}=\frac{-{\Omega}_{1}}{{\delta \mp {\Omega}_{1}^{\text{'}}-{\delta}_{\mathit{hfs}}\pm \left[{\left(\pm \frac{{\Omega}_{1}^{\text{'}}}{2}+\frac{\delta}{2}\right)}^{2}+{\Omega}_{1}^{2}\right]}^{\frac{1}{2}}+\mathrm{sgn}\times {\left[{\left(\pm \frac{{\Omega}_{1}^{\text{'}}}{2}+\frac{\delta}{2}+{\delta}_{\mathit{hfs}}\right)}^{2}+{\Omega}_{2}^{2}\right]}^{\frac{1}{2}}},$$

$${a}_{32}=\frac{-{\Omega}_{2}}{{\delta \mp {\Omega}_{1}^{\text{'}}+{\delta}_{\mathit{hfs}}\pm \left[{\left(\pm \frac{{\Omega}_{1}^{\text{'}}}{2}+\frac{\delta}{2}\right)}^{2}+{\Omega}_{1}^{2}\right]}^{\frac{1}{2}}+\mathrm{sgn}\times {\left[{\left(\pm \frac{{\Omega}_{1}^{\text{'}}}{2}+\frac{\delta}{2}+{\delta}_{\mathit{hfs}}\right)}^{2}+{\Omega}_{2}^{2}\right]}^{\frac{1}{2}}}.$$

In consideration of the coupling of the dressed atom with the vacuum modes, the guided atoms will spontaneously emit fluorescent photons, and the corresponding spontaneous emission rates Γ_{ij} from |*j,n*〉 to |*i,n*-1〉 can be given by [26,28]

where Γ_{1} and Γ_{2} are the partial spontaneous emission rates given by Γ_{1} = *q*
_{1}Γ and Γ_{2} = *q*
_{2}Γ, *q*
_{1} (*q*
_{2}) is the relative branching ratio of the spontaneous emission from |*e*〉 to |*g*
_{1}〉 (or |*g*
_{2}〉). For a ^{85}Rb atom, *q*
_{1} = 0.74, *q*
_{2} = 0.26 [26], and the dependences of the spontaneous-emission rates on the detuning *δ*/2*π* for *I* = 500mW/cm^{2} are shown in Fig.4(a). We can see from Fig. 4(a) that there are the maximum spontaneous emission rates at the detuning *δ*/2*π* = 0 for Γ_{11}, Γ_{21}, Γ_{31} and at the detuning *δ*/2*π* = -*δ*_{hfs}
/2*π* for Γ_{12}, Γ_{22}, Γ_{32}, which shows a resonant property of the spontaneous emission coming from two resonant effects between |*g*
_{1}〉 → |*e*〉 and between |*g*
_{2}〉 → |*e*〉, respectively.

Since the dressed-state |1,*n*〉 contains a small mixture of the excited state [see Eq. (10)], it may spontaneously decay to a lower dressed-state |1, *n* - 1〉, |2, *n* - 1〉 or |3, *n* - 1〉 with the corresponding rate Γ_{i1} (*i* = 1,2,3), that is,

Substituting Eqs. (11) and (12) into Eq. (14), we can obtain the expressions of the spontaneous-emission rates Γ_{11} , Γ_{21} and Γ_{31} . Moreover, under the approximation of small saturation parameters [22], we can obtain the approximate expressions of Eq. (14), which are the same as the results in Ref. [27].

From Eq. (14), we calculate the dependences of the spontaneous emission rates Γ_{11},Γ_{21}, Γ_{31} on the laser intensity for *δ*/2*π* = 3GHz, and the results are shown in Fig. 4(b). We can find from Fig. 4(b) that the spontaneous emission rates Γ_{11}, Γ_{21} and Γ_{31} will be increased with the intensity increasing, and have Γ_{11} > Γ_{21} > Γ_{31}.

## 4. Guiding efficiency and loss mechanisms

#### 4.1. Straight guide of cold atoms

In the energy conservative limit, we evaluate the guiding efficiency of cold atoms in the blue-detuned TE_{01} mode in the HMW by calculating the atom flux probability [9]. Let us make several assumptions to simplify the atom-flux calculation. First, we assume that the cold atoms at the HMW inlet have a uniform spatial distribution and a Maxwell-Boltzmann velocity distribution. Second, cold atoms are regarded as classical particles in the course of guiding. Third, we neglect the intensity decaying of the TE_{01} doughnut mode along the guiding axis, which is valid for the straight HMW with a length of several 10-cm. In addition, we think that the motion of the guided cold atoms in the TE_{01} mode satisfies the adiabatic approximate condition and we choose |1,*n*〉 as the guided state.

According to the first assumption mentioned above, atoms at the input cross section of the HMW have a uniform position distribution 1/*V* in the coordinates (*x, y,z*) and a velocity distribution

where *M* is the atomic mass and *T* is the temperature. The input flux can then be derived by [9]

where *r*
_{0} is the radial position of the maximum intensity of the TE_{01} mode. If the guided cold atoms do not hit the metal core of the HMW, they are not lost from the guide, and the atoms entering the TE_{01} mode in the HMW will emerge at the outlet of the HMW. Thus, in the absence of an optical guiding potential, the atoms must follow a ballistic trajectory through the hollow region, and the guiding efficiency will be very small when the length of the HMW is far larger than the radius of the hollow region. In the presence of the optical field, the transverse atomic motion is bound by the optical potential and the guiding efficiency can be increased dramatically. The output flux can be calculated by the following integration in the cylindrically symmetric coordinates (*r, ϕ, z*) in the straight HMW

where the motion of the guided atoms in the velocity region S must satisfy the energy conservation

where *U*(*r*) [or *U*(*r*
_{0})] is the optical potential given by Eq.(9a) at the radial position *r* = [*x*
^{2} + *y*
^{2}]^{1/2} (or *r* = *r*
_{0}). In Eq. (18), since the optical potential is cylindrically symmetric, the angular momentum is conserved and can be expressed as *v*_{ϕ}*r* = const, where *v*_{ϕ}
is the azimuthal component of the atom’s velocity. In order to calculate the integration of Eq. (17), we make the following transformations

Using Eq. (19), the integration of Eq. (17) becomes

where

Then the guiding efficiency can be defined by *η* = *J*
_{0}/ *J*_{i}
, and given by

From Eq.(22), we calculate the dependence of the guiding efficiency of the straight HMW guiding (*a* = 15 μm) for cold ^{85}Rb atoms in |1,*n*〉 on the input power of the coupling laser with different detuning, and the results are shown in Fig. 5(a). We know from Fig. 5(a) that the guiding efficiency is increased with the increasing of the input laser power and decreased with the increasing of the laser detuning. When *δ*/2*π* - 0.5GHz (3 GHz) and *P*
_{in}= 0.1 mW, the guiding efficiency can reach 89% (78%), while the input laser power *P*
_{in} is increased to 0.5W, the guiding efficiency can be increased to 99% (98%).

#### 4.2. Bent guide of cold atoms

When the HMW is bent, some hotter atoms, with a kinetic energy larger than the maximum transverse guiding potential, will be lost from the guiding channel. For the bent HMW with a curvature radius of *R* ≤ 2 cm, the intensity decaying of the TE_{01} doughnut mode along the guiding axis should be considered. If the HMW is bent in the (*y, z*) plane with a curvature radius of *R*, we introduce a curvilinear coordinates (*x’, y’, z’*) instead of the initial coordinates (*x, y, z*), with the axis *z’* following the bent HMW [9]

The unit vectors (**e**
_{x}, **e**
_{y}, **e**
_{z}) for the old coordinate system are related to the new ones (**e**
_{x}', **e**
_{y}', **e**
_{z}') by the following equations

From Eqs. (23) and (24), the transformation of the acceleration of the atomic motion between such two coordinate systems can be expressed as

Since *R* >> *r*
_{0}, the region of the coordinates and velocities of the guided atoms can be derived by integrating Eq. (25)

This shows that the angular momentum is relative to the curvature center of the bent HMW. Eq. (26) presents the energy conservation when the atoms move inside the TE_{01} mode in the HMW, and shows that the motion of the cold atoms in the bent HMW is analogous to that in a gravity field with the acceleration constant ${v}_{z\text{'}}^{2}$ /*R* . For simplicity [9], we assume that the cross section of the HMW is a square -*r*
_{0} < *x*' < *r*
_{0}, - *r*
_{0} < *y*' < *r*
_{0}, and the optical potential can be rewritten as

If we assume that the velocity components in *x*' and *y*' axis are separate, the trap condition can be expressed as

Therefore, the guiding efficiency of the cold atoms in the bent HMW with a square cross section can be given by

$$\phantom{\rule{13.2em}{0ex}}\times {\int}_{-1}^{1}\frac{\mathrm{exp}\left[-\frac{U\left({r}_{0},z\text{'}\right)}{{k}_{\mathit{B}}T}\frac{R}{{2r}_{0}\left(1+l\right)}\right]}{{\left[\frac{R}{{2r}_{0}\left(1+l\right)}-1\right]}^{\frac{1}{2}}}\times \mathrm{erfi}\left({\left\{\frac{U\left({r}_{0},z\text{'}\right)}{{k}_{B}T}\left[\frac{R}{{2r}_{0}\left(1+l\right)}-1\right]\right\}}^{\frac{1}{2}}\mathit{\right)}\mathit{dl},$$

where is the error function, and erfi(*x*) = erf(*ix*
**)/ i is the imaginary error function. In Eq.(29), the first term of the right-hand side of the equation is the guiding efficiency when R → ∞ (that is, the straightly guiding efficiency of cold atoms) and the second term is the effect of the curvature to the guiding efficiency. For the TE_{01} mode in the bending HMW, I
_{0}(z
^{'})=I
_{0}(0)×exp(-2${\alpha}_{01}^{b}$
z
^{'}), and U(r
_{0},z
^{'}) (=U
_{1}⌊I
_{0}(z
^{'})⌋ can be written as U(r
_{0},0)f(z
^{'}), where U
_{1} is given by Eq. (9a) and U(r
_{0},0) is the maximum optical potential at z
^{'} =0 . If the length of the HMW is 30cm , we can replace U(r
_{0},z
^{'}) with U(r
_{0}, z
^{'})_{z'=0.3} to estimate the guiding efficiency using Eq. (29). From Eq. (29), we calculate the dependence of the guiding efficiency of the bent HMW for cold ^{85}Rb atoms in |1, n〉 on the input laser power as a = 15 μm, δ/2π = 3 GHz and different curvature radius R, and the results are shown in Fig. 5(b). We find from Fig. 5(b) that the guiding efficiency will be increased with the increasing of the input laser power and decreased with the decreasing of the curvature radius R. If the curvature radius R is larger than 2cm, there is almost no bent effect on the guiding efficiency. This is because the optical propagation loss of the TE_{01} mode in the HMW can almost be neglected as R ≥ 2cm. For a smaller R (such as R= 1cm, 0.8cm), the guiding efficiency can arrive at the same value for both straight and bent HMW when the input laser power is increased to an enough large value.**

**4.3. Atomic loss mechanisms and their loss rates**

**4.3.1 Atomic loss from background thermal collisions**

**For the guided Rb atoms, we assume that the collision loss from the non-rubidium atoms can be negligible, the atomic collision rate from the background thermal Rb atoms can be estimated by [20]**

**$${\mathit{\gamma}}_{\mathit{ac}}=\frac{1}{{\tau}_{\mathit{ac}}}\approx 100n{\sigma}_{\mathrm{Rb}}\left(\frac{{3k}_{B}{T}_{\mathit{ther}}}{M}\right),$$**

**where n = 0.1333×10^{-3}
p/(k_{B}T_{ther}
) is the density of the background Rb atoms in the vacuum chamber (cm^{-3}). Here p is the pressure (Torr) in the vacuum chamber, M is the Rb atomic mass (kg), T
_{ther} is the temperature of the background Rb vapor (K), and σ_{Rb} is the collision cross section of the Rb atoms (cm^{2}). If taking p=10^{-9} Torr, σ_{Rb} ~10^{-13}cm^{2} and T
_{ther} =300K, we obtain γ_{ac}
= 9.55% from Eq. (30). In this case, when a cold atomic beam with an average longitudinal velocity of 14m/s [29] is loaded into our HMW guiding channel with a length L of 10–30 cm, the corresponding collision loss is 0.07%–0.21%. If taking p=10^{-8} Torr, the corresponding collision loss is about 0.7%–2.1%. So the atomic loss from the background thermal collisions can be neglected as p less than 10^{-8} Torr.**

**4.3.2. Atomic loss from spontaneous emission of the guided atoms**

**For the guided atoms with a temperature of 120μK , the mean intensity Ī within the mean atomic penetrating depth r_{APD}
can be calculated by Ī = |∫^{rAPD}
_{0}
I(r) dr/r_{APD}
. When P
_{in}= 0.5 W
and δ/2π = 3GHz, the penetrating depth r_{APD}
is 1.96×10^{-2}μm, and the mean intensity Ī is ~ 881.1mW/cm^{2}, and then we estimate that the mean spontaneous emission rate $\overline{\Gamma}$
_{1} = $\overline{\Gamma}$
_{11} + $\overline{\Gamma}$
_{21} + $\overline{\Gamma}$
_{31} ≈ 1.1×10^{3} s^{-1}. In this case, when a cold atomic beam with an average longitudinal velocity of 14m/s [29] is loaded into our HMW guiding channel with a length of 10 – 30 cm, the total mean spontaneous emission is 7.86 – 23.5 times, which corresponds to a spontaneous-emission heating of 1.45 – 4.36 μK. This shows that the spontaneous-emission heating is very small, even can be neglected as compared to the temperature of the guided cold atoms (120μK) in the HMW.**

**4.3.3. Atomic loss from non-adiabatic transition of the guided atoms**

**In the blue-detuned TE _{01} mode in the HMW, it can see from Eq.(9) that the atoms in the states |1,n〉 and |2,n〉 are pushed to the minimum intensity position and can be trapped and guided by the TE_{01} doughnut mode in the HMW, whereas the atoms in the state |3,n〉 are attracted to the maximum intensity position and may be lost due to the huge spontaneous emission rate at the maximum intensity position. The course of the atoms from the guided state |1,n〉 (or |2,n〉) to the non-guided state |3,n〉 is defined as the non-adiabatic transition, which can be described by the spontaneous emission rates Γ_{31} and Γ_{32}. When P
_{in}= 0.5 W and δ/2π = 0.5GHz(3GHz), the mean non-adiabatic transition rate is 5.76 s^{-1} (0.2s^{-1}). In this case, when a cold atomic beam with an average longitudinal velocity of 14m/s [29] is loaded into our HMW guiding channel with a length of 10 – 30 cm, the total mean non-adiabatic transition is 0.04–0.12 (0.0014–0.0042) times. It is clear that when δ/2π = 3GHz, the total mean non-adiabatic transition is so small that can be neglected in the atomic guiding.**

**5. Potential applications**

**Since the blue-detuned TE _{01} mode in the HMW has a doughnut-like intensity profile, and it cannot only be used to realize straight guiding of cold atoms (or molecules), but also to realize bent guiding of cold atoms (or molecules), our proposed guiding scheme has some new potential applications in the fields of atom and molecule optics. Such as:**

*Computer-controlled atom lithography*: In order to fabricate two dimensional (2D) submicron-scale structures, the precise control of a cold atomic beam is required in the atom-optical deposition or lithography. In recent years, the guiding technique of the cold atomic beam using a HOF with a hollow radius of a few 1 - μm escorted by the evanescent-light wave was proposed to realize a novel atom-optical lithography [30]. Due to a small hollow radius of such a HOF, it is difficult to obtain a higher vacuum in the hollow region of the HOF, and then it is difficult to realize an atomic guiding with a higher efficiency. However, the hollow radius of our HMW can reach to a few 10 - μm, and it is easy to obtain a higher vacuum in the hollow radius of the HMW, which can also be used to high-effectively guide cold atoms and then to realize a computer-controlled atom lithography with an arbitrary pattern, even to form an atom-fiber gyroscope.*Generation of dark hollow beam*: From Fresnel and Fraunhofer diffraction theory, we calculate the near- and far-field distributions of the TE_{01}mode output beam at the outlet of the HMW, and find that the near field intensity distribution of the TE_{01}mode output beam from the HMW is a focused hollow beam and the far field intensity distribution of the TE_{01}mode output beam is a divergent hollow beam, which can be used to form a novel atomic lens and a simple atomic funnel, respectively. If a micro-collimation technique [31] is used to collimate the TE_{01}mode output beam, a collimated-well hollow laser beam will be generated from the HMW.*Realization of single mode atomic waveguide*: We can see from Fig. 3(b) that when*P*_{in}= 0.5W and*δ*/2*π*= 3 GHz, the guiding potential for cold^{85}Rb atoms is higher than 465 mK, which is far higher than the temperature (~ 120μK) of the guided cold^{85}Rb atoms loaded from the Standard MOT. In this case, transverse motion region of the guided cold atoms is smaller than 19.6 nm, which is about equal to the mean de Broglie wavelength*λ*_{dB}= [2*πħ*^{2}/(*mkT*)]^{1/2}≈ 17.3nm of the guided cold atoms. So our guiding scheme can also be used to realize a single-mode atomic waveguide in the blue-detuned TE_{01}mode in the HMW.*Intensity-gradient cooling of the guided atoms*: Sisyphus cooling of atoms in the standing-wave field or the evanescent-wave one, originating from the intensity gradient of the light field, is usually called “intensity-gradient cooling” (IGC) [1]. Neutral atoms moving in these light fields with a high intensity-gradient will be cooled down to near the recoil temperature. Since the average intensity gradient (*I*_{max}(*r*_{0})/*r*_{0}=1.9×10^{11}mW/cm^{3}) of the TE_{01}mode in the HMW, as a = 15μm and*P*_{in}= 0.5W, is far greater than that (2.98×10^{10}mW/cm^{3}) of the LP_{01}mode in the HOF [2] with a hollow radius*a*= 3.5μm and the same laser power, and also far greater than that (1.61×10^{9}mW/cm^{3}) of the evanescent-wave surface trap with the same power [23], the blue-detuned TE_{01}mode in the HMW can be used to cool the guided atoms to near the recoil temperature by adding a weak red-detuned repumping beam in the HMW [21].*Generation of a continuous cold molecular beam*: From the above analysis, our HMW guiding scheme can be used to realize the bent guiding of cold atoms. Also, our guiding scheme can be used to realize the straight and bent guiding of cold molecules according to ac Stark effect. Since the velocity and flux of the guided cold molecular beam can be controlled by adjusting the power and detuning of the coupling laser and the curvature radius of the HMW, a blue-detuned TE_{01}mode in the bent HMW with a curvature radius of*R*, as an energy low-pass filter, can be used to realize the generation of a continuous-wave (CW) cold molecular beam, which is similar to the scheme of bent electrostatic guiding for cold polar molecules [32]. It is an important method to generate a CW cold molecular beam for those molecules without an electric or magnetic dipole moment, such as I_{2}molecules and so on.

**6. Conclusion**

**In this paper, we have proposed a new scheme to guide cold atoms (or molecules) using a blue-detuned TE _{01} mode in the HMW, and calculated the electric field distribution of the TE_{01} mode in the HMW and its optical potential for ^{85}Rb atoms, and estimated the spontaneous-emission rate of the guided atoms and atomic loss rates as well as the guiding efficiency, and found that the optical potential of the blue-detuned TE_{01} mode for ^{85}Rb atoms is high enough to guide cold atoms from a standard MOT. For the given optimal parameters, such as a = 15 μm, P
_{in}= 0.5W, δ/2π = 3 GHz and L = 10 – 30cm, the straight guiding efficiency of cold atoms can reach 98% , and the bending of the HMW with a curvature radius of R > 2cm will only result in a slight reduction of the guiding efficiency. Our study also shows that: (1) the spontaneous-emission induced heating is about 1.45–4.36 μK; (2) The total mean non-adiabatic transition is about 0.0014–0.0042 times; (3) When p=10^{-9} Torr, the atomic loss from the background thermal collisions is about 0.07% – 0.21% ; While p=10^{-8} Torr, the corresponding collision loss is about 0.7% – 2.1%. It is clear that these atomic losses are very small, even can be neglected.**

**In final, we have briefly discussed some potential applications of our proposed guiding scheme and found that it has some new and important applications in atom and molecule optics, such as, computer-controlled atom lithography with an arbitrary pattern, the generation of dark hollow beam, realization of single-mode atomic waveguide, intensity-gradient cooling of the guided atoms, production of a CW cold molecular beam, even formation of an atom-fiber gyroscope, and so on.**

**Acknowledgments**

**This work was supported by the National Natural Science Foundation of China under Grant Nos.10174050, 10374029 and 10434060, Shanghai Priority Academic Discipline and the 211 Foundation of the Educational Ministry of China.**

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