Atomic (or molecular) guiding using a blue-detuned doughnut mode in a hollow metallic waveguide

doi:10.1364/OPEX.13.008406

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Atomic (or molecular) guiding using a blue-detuned doughnut mode in a hollow metallic waveguide

Zhengling Wang, Meng Dai, and Jianping Yin »View Author Affiliations

Optics Express, Vol. 13, Issue 21, pp. 8406-8423 (2005)
http://dx.doi.org/10.1364/OPEX.13.008406

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Abstract

We propose a new scheme to guide cold atoms (or molecules) using a blue-detuned TE₀₁ 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 TE₀₁ doughnut mode for three-level atoms using dressed-atom approach, and find that the optical potential of the TE₀₁ 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.

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

J. Yin , W. Gao , and Y. Zhu , “ Generation of dark hollow beams and their applications ,” Prog. Opt. 45 , 119 – 204 ( 2003 ). [CrossRef]

] or a red-detuned Gaussian beam [5

L. Pruvost , D. Marescaux , O. Houde , and H. T. Duong , “ Guiding and cooling of cold atoms in dipole guide ,” Opt. Commun. 166 , 199 – 209 ( 1999 ). [CrossRef]

B. T. Wolschrijn , R. A. Cornelussen , R. J. C. Spreeuw , and H. B. van Linden van den Heuvell , “ Guiding of cold atoms by a red-detuned laser beam of moderate power ,” New J. Phys. 4 , 69.1 – 69.10 ( 2002 ). [CrossRef]

]. 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

M. A. Ol’Shanii , Yu. B. Ovchinnikov , and V. S. Letkhov , “ Laser guiding of atoms in a hollow optical fiber ,” Opt. Commun. 98 , 77 – 79 ( 1993 ). [CrossRef]

] or a blue-detuned LP₀₁ mode evanescent wave (EW) [10–12

M. J. Renn , E. A. Donley , E. A. Cornell , C. E. Wieman , and D. Z. Anderson , “ Evanescent-wave guiding of atoms in hollow optical fibers ,” Phys. Rev. A 53 , R648 – R651 ( 1996 ). [CrossRef] [PubMed]

] 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 LP₀₁-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

E. A. J. Marcatili and R. A. Schmeltzer , “ Hollow metallic and dielectric waveguides for long distance optical transition and lasers ,” Bell Syst. Tech. J. 43 , 1783 – 1809 ( 1964 ).

,14

E. Garmire , T. McMahon , and M. Bass , “ Propagation of infrared light in flexible hollow waveguide ,” Appl. Opt. 15 , 145 – 150 ( 1976 ). [CrossRef] [PubMed]

], it is far less sensitive to the curvature of the guide axis. If a blue-detuned TE₀₁ 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 TE₀₁ 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 TE₀₁ 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 TE₀₁ 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 TE₀₁ 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 TE₀₁ 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 2π phase plate and is focused by a positive lens with a focal length f , and a focused hollow beam (FHB) will be generated [15

Y. Xia and J. Yin , “ Generation of a focused hollow beam by an 2 Pi-phase plates and its applications in atom or molecule optics ,” J. Opt. Soc. Am. B 22 , 529 – 536 ( 2005 ). [CrossRef]

] 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.

Fig. 1. (a) The structure of the HMW; (b) schematic diagram of atomic guiding. HMW, BDGB, MOT and 2π PP stand for hollow metallic waveguide, blue-detuned Gaussian beam, magneto-optic trap and 2π -phase plate.

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 TE₀₁ 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 TE₀₁ 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

E. A. J. Marcatili and R. A. Schmeltzer , “ Hollow metallic and dielectric waveguides for long distance optical transition and lasers ,” Bell Syst. Tech. J. 43 , 1783 – 1809 ( 1964 ).

]. 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

E_{r} = - \frac{1}{k_{i}^{2}} (iγ \frac{\partial E_{s}}{\partial r} + \frac{m}{r} μ_{0} {ωH}_{z}),

(1a)

E_{ϕ} = - \frac{1}{k_{i}^{2}} (- {iμ}_{0} ω \frac{\partial H_{z}}{\partial r} + \frac{m}{r} {γE}_{z}),

(1b)

H_{r} = - \frac{1}{k_{i}^{2}} (iγ \frac{\partial H_{z}}{\partial r} - \frac{m}{r} ε_{0} {ωE}_{z}),

(1c)

H_{ϕ} = - \frac{1}{k_{i}^{2}} ({iε}_{0} ω \frac{\partial E_{z}}{\partial r} + \frac{m}{r} {γH}_{z}),

(1d)

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

E_{z} (r) = C_{1} J_{m} (k_{i} r) + C_{2} N_{m} (k_{i} r),

(2a)

H_{z} (r) = C_{3} J_{m} (k_{i} r) + C_{4} N_{m} (k_{i} r),

(2b)

where J_m (k_ir) is the first kind Bessel function and N_m (k_ir) 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

J. Yin , W. Gao , and Y. Zhu , “ Generation of dark hollow beams and their applications ,” Prog. Opt. 45 , 119 – 204 ( 2003 ). [CrossRef]

), we derive the expressions of E_r and E_ϕ as follows

E_{r} = - \frac{1}{{k_{i}}^{2}} [- i γ_{0} C_{1} J_{m}' (k_{i} r) + \frac{m}{r} μ_{0} ω C_{3} J_{m} (k_{i} r)],

(3a)

E_{ϕ} = - \frac{1}{{k_{i}}^{2}} [- i μ_{0} ω C_{3} J_{m}' (k_{i} r) + \frac{m}{r} {γC}_{1} J_{m} (k_{i} r)] .

(3b)

Considering the boundary condition of r = a and under the approximation of |η_m |u_mq << ka , we obtain k_ia ≈ u_mq (1 - iη_m /ka) ≈ u_mq [13

E. A. J. Marcatili and R. A. Schmeltzer , “ Hollow metallic and dielectric waveguides for long distance optical transition and lasers ,” Bell Syst. Tech. J. 43 , 1783 – 1809 ( 1964 ).

], here k = 2π/λ and u_mq is the qth root of Bessel function of the first kind: J _m-1 (u_mq ) = 0, η_m = 1/(

η_{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

E. A. J. Marcatili and R. A. Schmeltzer , “ Hollow metallic and dielectric waveguides for long distance optical transition and lasers ,” Bell Syst. Tech. J. 43 , 1783 – 1809 ( 1964 ).

]: 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 TE₀₁ 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 TM₀₁ 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 EH₁₁ 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 TE₀₁ 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

R. L. Abrams and A. N. Chester , “ Resonator theory for hollow waveguide lasers ,” Appl. Opt. 13 , 2117 – 2125 ( 1974 ). [CrossRef] [PubMed]

]. So we only consider the electric field distribution of the TE₀₁ mode in the HMW due to the magnetic field distribution of the TE₀₁ 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 TE₀₁ mode in the hollow region of the HMW can be expressed as

E_{r} = 0,

(4a)

E_{ϕ} (r) = \frac{- {iμ}_{0} {ωC}_{3}}{{k_{i}}^{2}} J_{1} (u_{01} \frac{r}{a}) .

(4b)

Fig. 2. Normalized electric field distribution: (a) against the radial position r for the TE₀₁ mode; (b) against the propagation distance z.

From Eq. (4b), we calculate the normalized electric field distribution of the TE₀₁ 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 TE₀₁ 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 TE₀₁ mode

In the focal plane of z=0, the electric field profile of the FHB can be approximately described by the TE

M_{01}^{*}

doughnut beam mode as follows [15

Y. Xia and J. Yin , “ Generation of a focused hollow beam by an 2 Pi-phase plates and its applications in atom or molecule optics ,” J. Opt. Soc. Am. B 22 , 529 – 536 ( 2005 ). [CrossRef]

]

E^{FHB} (r) = {(\frac{{4 k}_{1} P_{in}}{π})}^{\frac{1}{2}} \times \frac{r}{{w_{0}}^{2}} \times \exp (- \frac{r^{2}}{{w_{0}}^{2}}),

(5)

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 TE₀₁ mode can be selectively excited, and the coupling efficiency A, i.e., the ratio of the power P of the TE₀₁ mode to the input power P _in of the FHB, is given by [17

M. Saito , S. Sato , and M. Miyagi , “ Loss characteristics of infrared hollow waveguides in multimode transmission ,” J. Opt. Soc. Am. A 10 , 277 – 282 ( 1993 ). [CrossRef]

]

A = \frac{{∣ \int_{0}^{a} E^{FHB} (r) E_{ϕ} (r) r dr ∣}^{2}}{\int_{0}^{\infty} {∣ E^{FHB} (r) ∣}^{2} r dr \int_{0}^{a} {∣ E_{ϕ} (r) ∣}^{2} r dr} .

(6)

From Eq. (6), we calculate the dependence of the coupling efficiency A for the TE₀₁ 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 TE₀₁ mode can be reached the maximum value 96.7%, i.e., P ≈ 0.967P_in , while the coupling efficiency for the TE₀₂ mode is 0.94%, and the coupling efficiency for other modes (including a EH₁₁ Gaussian mode) is only ~ 2.36%.

The absolute radial intensity distribution of the TE₀₁ mode in the HMW with a radius a is set as I(r), and supposing that the propagating loss of the TE₀₁ mode can be neglected (see below), the propagating power P of the TE₀₁ mode in the HMW can be calculated by P =

\int_{0}^{a}

2πrI(r)dr. So we can obtain the absolute intensity distribution of the TE₀₁ mode in the HMW as follows

I (r) = \frac{P J_{1}^{2} (\frac{u_{01} r}{a})}{2 π \int_{0}^{a} J_{1}^{2} (\frac{u_{01} r}{a}) r dr} = \frac{P J_{1}^{2} (\frac{u_{01} r}{a})}{- a^{2} π J_{0} (μ_{01}) J_{2} (μ_{01})} .

(7)

From Eq. (7) and considering P ≈ 0.967P_in , we calculate the intensity distribution of the TE₀₁ 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 TE₀₁ mode at the radial position r ₀ = 7.2μm can reach 1.43×10⁸mW/cm² , which is far greater than the maximum evanescent-wave intensity (1.64×10⁶mW/cm²) of the LP₀₁ mode in the HOF with a hollow radius a = 3.5μm and the same power [2

J. Yin , Y. Zhu , W. Wang , Y. Wang , and W. Jhe , “ Optical potential for atom guidance in a dark hollow laser beam ,” J. Opt. Soc. Am. B 15 , 25 – 33 ( 1998 ). [CrossRef]

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

E. A. J. Marcatili and R. A. Schmeltzer , “ Hollow metallic and dielectric waveguides for long distance optical transition and lasers ,” Bell Syst. Tech. J. 43 , 1783 – 1809 ( 1964 ).

], where η ₀ = 1/(

η_{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 EH₁₁, TM₀₁ and TE₀₁ modes in the different HMWs [13

E. A. J. Marcatili and R. A. Schmeltzer , “ Hollow metallic and dielectric waveguides for long distance optical transition and lasers ,” Bell Syst. Tech. J. 43 , 1783 – 1809 ( 1964 ).

,14

E. Garmire , T. McMahon , and M. Bass , “ Propagation of infrared light in flexible hollow waveguide ,” Appl. Opt. 15 , 145 – 150 ( 1976 ). [CrossRef] [PubMed]

], we calculate the dependences of the electric field amplitudes of the EH₁₁, TM₀₁ and TE₀₁ modes in two HMWs (Al [14

E. Garmire , T. McMahon , and M. Bass , “ Propagation of infrared light in flexible hollow waveguide ,” Appl. Opt. 15 , 145 – 150 ( 1976 ). [CrossRef] [PubMed]

] and Ni [18

M. Miyagi and S. Kawakami , “ Design theory of dielectric-coated circular metallic waveguide for infrared transition ,” IEEE J. Lightwave Technol. 2 , 116 – 126 ( 1984 ). [CrossRef]

]) 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 EH₁₁ or TM₀₁ mode is normalized by that of the TE₀₁ mode. We can see from Fig. 2(b) that the EH₁₁ (or TM₀₁) 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 EH₁₁ (or TM₀₁) mode in the HMW is. In particular, when the metallic Al hollow waveguide is used, the effective straight propagation distance of the EH₁₁ (or TM₀₁) mode determined by 1/e of the field amplitude is only about 3 mm (or 0.7mm). While the loss of the TE₀₁ 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 TE₀₁ mode in the bending HMW with a = 15μm is given by

α_{01}^{b}

≈ α ₀₁(1 + 2.97×10^-4/R ²) [13

E. A. J. Marcatili and R. A. Schmeltzer , “ Hollow metallic and dielectric waveguides for long distance optical transition and lasers ,” Bell Syst. Tech. J. 43 , 1783 – 1809 ( 1964 ).

,14

E. Garmire , T. McMahon , and M. Bass , “ Propagation of infrared light in flexible hollow waveguide ,” Appl. Opt. 15 , 145 – 150 ( 1976 ). [CrossRef] [PubMed]

], where R is the curvature radius of the HMW. Therefore, the electric field amplitude of the TE₀₁ 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 TE₀₁ doughnut mode in the HMW by using a circular polarized FHB, and to efficiently restrain the excitation and propagation of the HE₁₁ (TM₀₁ 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

ħ (\begin{matrix} (n + 1) ω_{L} + δ_{hfs} & 0 & G_{2} {(n + 1)}^{\frac{1}{2}} \\ 0 & (n + 1) ω_{L} & G_{1} {(n + 1)}^{\frac{1}{2}} \\ G_{2} {(n + 1)}^{\frac{1}{2}} & G_{1} {(n + 1)}^{\frac{1}{2}} & {nω}_{L} + ω_{0} \end{matrix}) (\begin{matrix} A_{i} \\ B_{i} \\ C_{i} \end{matrix}) = E_{Dr i} (\begin{matrix} A_{i} \\ B_{i} \\ C_{i} \end{matrix}),

(8)

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 Ω₁ (Ω₂) 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 ,Ω₁,Ω₂) of atom-light interaction system, we substitute a trial solution

E_{Dr 1}^{t}

=(n + 1)ħω_L - ħδ/2±ħ(δ ² +

Ω_{1}^{2}

)/2 into Eq.(8) and derive two other solutions

E_{Dr 2}^{t}

and

E_{Dr 3}^{t}

. Here the trial solution

E_{Dr 1}^{t}

is the eigenenergy of the dressed-eigenstate |1,n〉 coming from the two-level atom model [19–21

V. I. Balykin , D. V. Laryushin , M. V. Subbotin , and V. S.L etokhov , “ Increase of the atomic phase density in a hollow laser waveguide ,” JETP Lett. 63 , 802 – 807 ( 1996 ). [CrossRef]

], and

E_{Dr 2}^{t}

E_{Dr 3}^{t}

correspond to dressed-eigenstates |2,n〉, |3,n〉 respectively. Then substituting

E_{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

U_{1} = - \frac{ħδ}{4} \mp \frac{ħ Ω_{1}^{'}}{4} \pm \frac{ħ {[{(\pm \frac{Ω_{1}^{'}}{2} + \frac{δ}{2})}^{2} + Ω_{1}^{2}]}^{\frac{1}{2}}}{2},

(9a)

U_{2} = - \frac{ħ δ}{4} \mp \frac{{ħ Ω}_{1}^{'}}{4} - \frac{{ħδ}_{hfs}}{2} + sgn \times \frac{ħ {[{(\pm \frac{Ω_{1}^{'}}{2} + \frac{δ}{2} + δ_{hfs})}^{2} + Ω_{2}^{2}]}^{\frac{1}{2}}}{2},

(9b)

U_{3} = \frac{ħ δ}{4} \pm \frac{{ħ Ω}_{1}^{'}}{4} + \frac{{ħδ}_{hfs}}{2} \mp \frac{ħ {[{(\pm \frac{Ω_{1}^{'}}{2} + \frac{δ}{2})}^{2} + Ω_{1}^{2}]}^{\frac{1}{2}}}{2} - sgn \times \frac{ħ {[{(\pm \frac{Ω_{1}^{'}}{2} + \frac{δ}{2} + δ_{hfs})}^{2} + Ω_{2}^{2}]}^{\frac{1}{2}}}{2},

(9c)

where Ω'₁ = (δ ² +

Ω_{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

J. Söding , R. Grimm , and Yu. B. Ovchinnikov , “ Gravitational laser trap for atoms with evanescent-wave cooling ,” Opt. Commun. 119 , 652 – 662 ( 1995 ). [CrossRef]

] 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

V. I. Balykin , D. V. Laryushin , M. V. Subbotin , and V. S.L etokhov , “ Increase of the atomic phase density in a hollow laser waveguide ,” JETP Lett. 63 , 802 – 807 ( 1996 ). [CrossRef]

], which can be further simplified as the same as the expressions of the optical potentials in Refs. [3

J. Yin , Y. Zhu , W. Jhe , and Y. Wang , “ Atom guiding and cooling in a dark hollow laser beam ,” Phys. Rev. A 58 , 509 – 513 ( 1998 ). [CrossRef]

,22–27

J. Söding , R. Grimm , and Yu. B. Ovchinnikov , “ Gravitational laser trap for atoms with evanescent-wave cooling ,” Opt. Commun. 119 , 652 – 662 ( 1995 ). [CrossRef]

] by using the small saturation parameter approximation again and taking the first-order approximation.

For the alkali-metal atom, Ω_j = (I/2I_sat )^1/2Γ

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

X. Xu , Y. Wang , and W. Jhe , “ Theory of atom guidance in a hollow laser beam: dressed-atom approach ,” J. Opt. Soc. Am. B 17 , 1039 – 1050 ( 2000 ). [CrossRef]

]. From Eq. (9), we calculate the optical potentials for a three-level ⁸⁵Rb atom and obtain the dependences of the optical potentials on the detuning δ/2π when δ_hfs /2π = 3.04GHz , Γ/2π = 6MHz , I_sat = 1.6mW/cm² and I = 500 mW/cm², 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/2π = 0 for U ₁ and at the detuning δ/2π = -δ_hfs /2π 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/2π = 0.5GHz, 3GHz, 5GHz, as shown in Fig. 3(b), and find that the optical potential is greater than 450 mK as δ/2π < 5GHz, which is high enough to guide cold atoms (~ 120μK) from a standard MOT.

Fig. 3. Dependences of the optical potentials: (a) on the detuning; (b) on the radial position r.

3.2. Spontaneous emissions of a three-level atom

When a three-level atom moves in the light field of the blue-detuned TE₀₁ 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

∣ i, n 〉 = A_{i} ∣ g_{2}, n + 1 〉 + B_{i} ∣ g_{2}, n + 1 〉 + C_{i} ∣ e, n 〉,

(10)

where i = 1,2,3, the probability amplitudes A_i , B_i and C_i are given by

A_{i} = \frac{∣ a_{i 2} ∣}{{(1 + {a_{i 1}}^{2} + {a_{i 2}}^{2})}^{\frac{1}{2}}}, B_{i} = \frac{∣ a_{i 1} ∣}{{(1 + {a_{i 1}}^{2} + {a_{i 2}}^{2})}^{\frac{1}{2}}}, C_{i} = \frac{1}{{(1 + {a_{i 1}}^{2} + {a_{i 2}}^{2})}^{\frac{1}{2}}},

(11)

where

a_{11} = \frac{Ω_{1}}{{- \frac{δ}{2} \mp \frac{Ω_{1}^{'}}{2} \pm [{(\pm \frac{Ω_{1}^{'}}{2} + \frac{δ}{2})}^{2} + Ω_{1}^{2}]}^{\frac{1}{2}}},

a_{12} = \frac{Ω_{2}}{{- \frac{δ}{2} \mp \frac{Ω_{1}^{'}}{2} \pm [{(\pm \frac{Ω_{1}^{'}}{2} + \frac{δ}{2})}^{2} + Ω_{1}^{2}]}^{\frac{1}{2}} - {2 δ}_{hfs}},

a_{21} = \frac{- Ω_{1}}{{\frac{δ}{2} \pm \frac{Ω_{1}^{'}}{2} - δ_{hfs} - sgn \times [{(\pm \frac{Ω_{1}^{'}}{2} + \frac{δ}{2} + δ_{hfs})}^{2} + Ω_{2}^{2}]}^{\frac{1}{2}}},

a_{22} = \frac{{- Ω}_{2}}{{\frac{δ}{2} \pm \frac{Ω_{1}^{'}}{2} + δ_{hfs} - sgn \times [{(\pm \frac{Ω_{1}^{'}}{2} + \frac{δ}{2} + δ_{hfs})}^{2} + Ω_{2}^{2}]}^{\frac{1}{2}}},

a_{31} = \frac{- Ω_{1}}{{δ \mp Ω_{1}^{'} - δ_{hfs} \pm [{(\pm \frac{Ω_{1}^{'}}{2} + \frac{δ}{2})}^{2} + Ω_{1}^{2}]}^{\frac{1}{2}} + sgn \times {[{(\pm \frac{Ω_{1}^{'}}{2} + \frac{δ}{2} + δ_{hfs})}^{2} + Ω_{2}^{2}]}^{\frac{1}{2}}},

a_{32} = \frac{- Ω_{2}}{{δ \mp Ω_{1}^{'} + δ_{hfs} \pm [{(\pm \frac{Ω_{1}^{'}}{2} + \frac{δ}{2})}^{2} + Ω_{1}^{2}]}^{\frac{1}{2}} + sgn \times {[{(\pm \frac{Ω_{1}^{'}}{2} + \frac{δ}{2} + δ_{hfs})}^{2} + Ω_{2}^{2}]}^{\frac{1}{2}}} .

(12)

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

X. Xu , Y. Wang , and W. Jhe , “ Theory of atom guidance in a hollow laser beam: dressed-atom approach ,” J. Opt. Soc. Am. B 17 , 1039 – 1050 ( 2000 ). [CrossRef]

,28

J. Dalibard and C. Cohen-Tannoudji , “ Dressed-atom approach to atomic motion in laser light: the dipole force revisited ,” J. Opt. Soc. Am. B 2 , 1707 – 1720 ( 1985 ). [CrossRef]

]

Γ_{ij} = {B_{i}}^{2} {C_{j}}^{2} Γ_{1} + {A_{i}}^{2} {C_{j}}^{2} Γ_{2},

(13)

where Γ₁ and Γ₂ are the partial spontaneous emission rates given by Γ₁ = q ₁Γ and Γ₂ = q ₂Γ, q ₁ (q ₂) is the relative branching ratio of the spontaneous emission from |e〉 to |g ₁〉 (or |g ₂〉). For a ⁸⁵Rb atom, q ₁ = 0.74, q ₂ = 0.26 [26

X. Xu , Y. Wang , and W. Jhe , “ Theory of atom guidance in a hollow laser beam: dressed-atom approach ,” J. Opt. Soc. Am. B 17 , 1039 – 1050 ( 2000 ). [CrossRef]

], and the dependences of the spontaneous-emission rates on the detuning δ/2π for I = 500mW/cm² 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/2π = 0 for Γ₁₁, Γ₂₁, Γ₃₁ and at the detuning δ/2π = -δ_hfs /2π for Γ₁₂, Γ₂₂, Γ₃₂, which shows a resonant property of the spontaneous emission coming from two resonant effects between |g ₁〉 → |e〉 and between |g ₂〉 → |e〉, respectively.

Fig. 4. Dependences of the spontaneous emission rates: (a) on the detuning δ/2π ; (b) on the intensity.

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,

Γ_{i 1} = {B_{i}}^{2} {C_{1}}^{2} Γ_{1} + {A_{i}}^{2} {C_{1}}^{2} Γ_{2} .

(14)

Substituting Eqs. (11) and (12) into Eq. (14), we can obtain the expressions of the spontaneous-emission rates Γ₁₁ , Γ₂₁ and Γ₃₁ . Moreover, under the approximation of small saturation parameters [22

J. Söding , R. Grimm , and Yu. B. Ovchinnikov , “ Gravitational laser trap for atoms with evanescent-wave cooling ,” Opt. Commun. 119 , 652 – 662 ( 1995 ). [CrossRef]

], we can obtain the approximate expressions of Eq. (14), which are the same as the results in Ref. [27

H. Nha and W. Jhe , “ Sisphus cooling on the surface of a hollow-mirror atom trap ,” Phys. Rev. A 56 , 729 – 736 ( 1997 ). [CrossRef]

From Eq. (14), we calculate the dependences of the spontaneous emission rates Γ₁₁,Γ₂₁, Γ₃₁ on the laser intensity for δ/2π = 3GHz, and the results are shown in Fig. 4(b). We can find from Fig. 4(b) that the spontaneous emission rates Γ₁₁, Γ₂₁ and Γ₃₁ will be increased with the intensity increasing, and have Γ₁₁ > Γ₂₁ > Γ₃₁.

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 TE₀₁ mode in the HMW by calculating the atom flux probability [9

M. J. Renn , A. A. Zozulya , E. A. Donley , E. A. Cornell , and D. Z. Anderson , “ Optical-dipole-force fiber guiding and heating of atoms ,” Phys. Rev. A 55 , 3684 – 3693 ( 1997 ). [CrossRef]

]. 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 TE₀₁ 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 TE₀₁ 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

f (v_{x}, v_{y}, v_{z}) = {(\frac{M}{{2 πk}_{B} T})}^{\frac{3}{2}} \exp [- \frac{M}{{2 k}_{B} T} ({v_{x}}^{2} + {v_{y}}^{2} + {v_{z}}^{2})],

(15)

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

M. J. Renn , A. A. Zozulya , E. A. Donley , E. A. Cornell , and D. Z. Anderson , “ Optical-dipole-force fiber guiding and heating of atoms ,” Phys. Rev. A 55 , 3684 – 3693 ( 1997 ). [CrossRef]

]

J_{i} = {\int\int}_{x^{2} + y^{2} < {r_{0}}^{2}} \frac{1}{V} dxdy {\int\int\int\int}_{v_{z} > 0} v_{z} f (v_{x}, v_{y}, v_{z}) {dv}_{x} {dv}_{y} {dv}_{z} = {(\frac{π}{2})}^{\frac{1}{2}} {(\frac{k_{B} T}{M})}^{\frac{1}{2}} \frac{{r_{0}}^{2}}{V},

(16)

where r ₀ is the radial position of the maximum intensity of the TE₀₁ 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 TE₀₁ 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

J_{0} (δ, P_{in}) = {\int\int}_{r^{} < r_{0}} \frac{1}{V} rdrd ϕ {\int\int\int}_{{S, v}_{z} > 0} v_{z} f (v_{x}, v_{ϕ}, v_{z}) {dv}_{r} {dv}_{ϕ} {dv}_{z},

(17)

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

S = {r, v_{r}, v_{ϕ} : \frac{1}{2} {Mv}_{r}^{2} + \frac{1}{2} {Mv}_{ϕ}^{2} + U (r) < \frac{1}{2} M \frac{r^{2}}{{r_{0}}^{2}} {v_{ϕ}}^{2} + U (r_{0})},

(18)

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_ϕr = 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

r = {ρr}_{0}, v_{r} = u_{r} {(\frac{{2 k}_{B} T}{M})}^{\frac{1}{2}}, v_{ϕ} = u_{ϕ} {(\frac{{2 k}_{B} T}{M})}^{\frac{1}{2}}, v_{z} = u_{z} {(\frac{{2 k}_{B} T}{M})}^{\frac{1}{2}} .

(19)

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

J_{0} (δ, P_{in}) = \frac{{r_{0}}^{2}}{V} {(\frac{{2 k}_{B} T}{M})}^{\frac{1}{2}} \int_{0}^{1} ρ {\underset{S' (ρ)}{\int\int} \exp [- ({u_{r}}^{2} + {u_{ϕ}}^{2})] {du}_{r} {du}_{ϕ}} dρ,

(20)

where

S^{'} (ρ) = {ρ, u_{r}, u_{ϕ} : {u_{r}}^{2} + k_{B} T (1 - ρ^{2}) {u_{ϕ}}^{2} < U (r_{0}) - U (r_{0} ρ)} .

(21)

Then the guiding efficiency can be defined by η = J ₀/ J_i , and given by

η = \frac{π}{2} \int_{0}^{1} ρ {\underset{S^{'} (ρ)}{\int\int} \exp [- ({u_{r}}^{2} + {u_{ϕ}}^{2})] {du}_{r} {du}_{ϕ}} dρ .

(22)

Fig. 5. Dependences of the guiding efficiency on the input laser power: (a) for the straight HMW; (b) for the bent HMW.

From Eq.(22), we calculate the dependence of the guiding efficiency of the straight HMW guiding (a = 15 μm) for cold ⁸⁵Rb 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/2π - 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 TE₀₁ 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

M. J. Renn , A. A. Zozulya , E. A. Donley , E. A. Cornell , and D. Z. Anderson , “ Optical-dipole-force fiber guiding and heating of atoms ,” Phys. Rev. A 55 , 3684 – 3693 ( 1997 ). [CrossRef]

]

x = x^{'},

(23a)

y = R ⌊ 1 - \cos (\frac{z^{'}}{R}) ⌋ + y^{'} \cos (\frac{z^{'}}{R}),

(23b)

z = (R - y^{'}) \sin (\frac{z^{'}}{R}) .

(23c)

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

e_{x} = {e_{x}}^{'},

(24a)

e_{y} = {e_{y}}^{'} \cos (\frac{z^{'}}{R}) + {e_{z}}^{'} \sin (\frac{z^{'}}{R})

(24b)

e_{z} = - {e_{y}}^{'} \sin (\frac{z^{'}}{R}) + {e_{z}}^{'} \cos (\frac{z^{'}}{R}) .

(24c)

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

a = a_{x^{'}} e_{x^{'}} + [a_{y^{'}} + \frac{{v_{z^{'}}}^{2}}{R} (1 - \frac{y^{'}}{R})] e_{y^{'}} + [a_{z^{'}} (1 - \frac{y^{'}}{R}) - \frac{{2 v}_{y^{'}} {v_{z}}^{'}}{R}] e_{z^{'}} .

(25)

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

\frac{1}{2} {Mv}_{x^{'}}^{2} + \frac{1}{2} {Mv}_{y^{'}}^{2} + M \frac{{v_{z^{'}}}^{2}}{R} (y^{'} + r_{0}) + U (x^{'}, y^{'}, z^{'}) \leq U (r_{0}, z^{'}) .

(26)

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 TE₀₁ 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'}^{2}

/R . For simplicity [9

M. J. Renn , A. A. Zozulya , E. A. Donley , E. A. Cornell , and D. Z. Anderson , “ Optical-dipole-force fiber guiding and heating of atoms ,” Phys. Rev. A 55 , 3684 – 3693 ( 1997 ). [CrossRef]

], 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

U (x^{'}, y^{'}, z^{'}) = {\begin{matrix} 0, ∣ x^{'} ∣ < r_{0} and ∣ y^{'} ∣ < r_{0}, \\ U (r_{0}, z^{'}), ∣ x^{'} ∣ = r_{0} or ∣ y^{'} ∣ = r_{0}, \end{matrix}

(27)

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

\frac{1}{2} {Mv}_{x^{'}}^{2} \leq U (r_{0}, z^{'})

(28a)

\frac{1}{2} {Mv}_{y^{'}}^{2} + M \frac{{v_{z^{'}}}^{2}}{R} (y^{'} + r_{0}) \leq U (r_{0}, z^{'}) .

(28b)

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

η = {(\erf {{[\frac{U (r_{0}, z^{'})}{k_{B} T}]}^{\frac{1}{2}}})}^{2} - \frac{1}{2} \erf {{[\frac{U (r_{0}, z')}{k_{B} T}]}^{\frac{1}{2}}}

\times \int_{- 1}^{1} \frac{\exp [- \frac{U (r_{0}, z')}{k_{B} T} \frac{R}{{2 r}_{0} (1 + l)}]}{{[\frac{R}{{2 r}_{0} (1 + l)} - 1]}^{\frac{1}{2}}} \times erfi ({\frac{U (r_{0}, z')}{k_{B} T} [\frac{R}{{2 r}_{0} (1 + l)} - 1]}^{\frac{1}{2}}) dl,

(29)

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 TE₀₁ mode in the bending HMW, I ₀(z ^')=I ₀(0)×exp(-2

α_{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 ⁸⁵Rb 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 TE₀₁ 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

J. Yin , Y. Zhu , and Y. Wang , “ Evanescent light-wave atomic funnel: A tanden hollow-fiber, hollow-beam approach ,” Phys. Rev. A 57 , 1957 – 1966 ( 1998 ). [CrossRef]

]

γ_{ac} = \frac{1}{τ_{ac}} \approx 100 n σ_{Rb} (\frac{{3 k}_{B} T_{ther}}{M}),

(30)

where n = 0.1333×10^-3 p/(k_BT_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 (cm²). If taking p=10^-9 Torr, σ_Rb ~10^-13cm² 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

Z. T. Lu , K. L. Corwin , M. J. Renn , M. H. Anderson , E. A. Cornell , and C. E. Wieman , “ Low-velocity intense source of atoms from a magneto-optical trap ,” Phys. Rev. Lett. 77 , 3331 – 3334 ( 1996 ). [CrossRef] [PubMed]

] 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 = |∫^r_APD 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^-2μm, and the mean intensity Ī is ~ 881.1mW/cm², and then we estimate that the mean spontaneous emission rate

\bar{Γ}

₁ =

\bar{Γ}

₁₁ +

\bar{Γ}

₂₁ +

\bar{Γ}

₃₁ ≈ 1.1×10³ 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 TE₀₁ 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 TE₀₁ 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 Γ₃₁ and Γ₃₂. 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 TE₀₁ 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
H. Ito , K. Sakaki , M. Ohtsu , and W. Jhe , “ Evanescent-light guiding of atoms throught hollow optical fiber for optically controlled atomic deposition ,” Appl. Phys. Lett. 70 2496 – 2498 ( 1997 ). [CrossRef]
]. 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 TE₀₁ mode output beam at the outlet of the HMW, and find that the near field intensity distribution of the TE₀₁ mode output beam from the HMW is a focused hollow beam and the far field intensity distribution of the TE₀₁ 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
J. Yin , H. Noh , K. Lee , K. Kim , Y. Wang , and W. Jhe , “ Generation of a dark hollow beam by a small hollow fiber ,” Opt. Commun. 138 , 287 – 292 ( 1997 ). [CrossRef]
] is used to collimate the TE₀₁ 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/2π = 3 GHz, the guiding potential for cold ⁸⁵Rb atoms is higher than 465 mK, which is far higher than the temperature (~ 120μK) of the guided cold ⁸⁵Rb 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 = [2πħ 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 TE₀₁ 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
J. Yin , W. Gao , and Y. Zhu , “ Generation of dark hollow beams and their applications ,” Prog. Opt. 45 , 119 – 204 ( 2003 ). [CrossRef]
]. 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 =1.9×10¹¹mW/cm³) of the TE₀₁ mode in the HMW, as a = 15μm and P _in = 0.5W, is far greater than that (2.98×10¹⁰mW/cm³) of the LP₀₁ mode in the HOF [2
J. Yin , Y. Zhu , W. Wang , Y. Wang , and W. Jhe , “ Optical potential for atom guidance in a dark hollow laser beam ,” J. Opt. Soc. Am. B 15 , 25 – 33 ( 1998 ). [CrossRef]
] with a hollow radius a = 3.5μm and the same laser power, and also far greater than that (1.61×10⁹mW/cm³) of the evanescent-wave surface trap with the same power [23
Yu.B. Ovchinnikov , I. Manek , and R. Grimm , “ Surface trap for Cs atoms based on evanescent-wave cooling ,” Phys. Rev. Lett. 79 , 2225 – 2228 ( 1997 ). [CrossRef]
], the blue-detuned TE₀₁ 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
J. Yin , Y. Zhu , and Y. Wang , “ Gravito-optical trap for cold atoms with doughnut-hollow-beam cooling ,” Phys. Lett. A 248 , 309 – 318 ( 1998 ). [CrossRef]
].
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 TE₀₁ 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
T. Junglen , T. Rieger , S. A. Rangwala , P. W. H. Pinkse , and G. Rempe , “ Slow ammonia molecules in an electrostatic quadrupole guide ,” Eur. Phys. J. D 31 , 365 – 373 ( 2004 ). [CrossRef]
]. It is an important method to generate a CW cold molecular beam for those molecules without an electric or magnetic dipole moment, such as I₂ 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 TE₀₁ mode in the HMW, and calculated the electric field distribution of the TE₀₁ mode in the HMW and its optical potential for ⁸⁵Rb 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 TE₀₁ mode for ⁸⁵Rb 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.

References and links

1.	J. Yin , W. Gao , and Y. Zhu , “ Generation of dark hollow beams and their applications ,” Prog. Opt. 45 , 119 – 204 ( 2003 ). [CrossRef]
2.	J. Yin , Y. Zhu , W. Wang , Y. Wang , and W. Jhe , “ Optical potential for atom guidance in a dark hollow laser beam ,” J. Opt. Soc. Am. B 15 , 25 – 33 ( 1998 ). [CrossRef]
3.	J. Yin , Y. Zhu , W. Jhe , and Y. Wang , “ Atom guiding and cooling in a dark hollow laser beam ,” Phys. Rev. A 58 , 509 – 513 ( 1998 ). [CrossRef]
4.	X. Xu , K. Kim , W. Jhe , and N. Kwon , “ Efficient optical guiding of trapped cold atoms by a hollow laser beam ,” Phys. Rev. A 63 , 063401 ( 2001 ). [CrossRef]
5.	L. Pruvost , D. Marescaux , O. Houde , and H. T. Duong , “ Guiding and cooling of cold atoms in dipole guide ,” Opt. Commun. 166 , 199 – 209 ( 1999 ). [CrossRef]
6.	B. T. Wolschrijn , R. A. Cornelussen , R. J. C. Spreeuw , and H. B. van Linden van den Heuvell , “ Guiding of cold atoms by a red-detuned laser beam of moderate power ,” New J. Phys. 4 , 69.1 – 69.10 ( 2002 ). [CrossRef]
7.	M. A. Ol’Shanii , Yu. B. Ovchinnikov , and V. S. Letkhov , “ Laser guiding of atoms in a hollow optical fiber ,” Opt. Commun. 98 , 77 – 79 ( 1993 ). [CrossRef]
8.	M. J. Renn , D. Montgomery , O. Vdovin , D. Z. Anderson , C. E. Wieman , and E. A. Cornell , “ Laser-guided atoms in hollow-core optical fibers ,” Phys. Rev. Lett. 75 , 3253 – 3256 ( 1995 ). [CrossRef] [PubMed]
9.	M. J. Renn , A. A. Zozulya , E. A. Donley , E. A. Cornell , and D. Z. Anderson , “ Optical-dipole-force fiber guiding and heating of atoms ,” Phys. Rev. A 55 , 3684 – 3693 ( 1997 ). [CrossRef]
10.	M. J. Renn , E. A. Donley , E. A. Cornell , C. E. Wieman , and D. Z. Anderson , “ Evanescent-wave guiding of atoms in hollow optical fibers ,” Phys. Rev. A 53 , R648 – R651 ( 1996 ). [CrossRef] [PubMed]
11.	H. Ito , T. Nakata , K. Sakaki , M. Ohtsu , K. I. Lee , and W. Jhe , “ Laser spectroscopy of atoms guiding by evanescent waves in micron-sided hollow optical fibers ,” Phys. Rev. Lett. 76 , 4500 – 4503 ( 1996 ). [CrossRef] [PubMed]
12.	H. Ito , K. Sakaki , W. Jhe , and M. Ohtsu , “ Evanescent-light induced atom-guidance using a hollow optical fiber with light coupled sideways ,” Opt. Commun. 141 , 43 – 47 ( 1997 ). [CrossRef]
13.	E. A. J. Marcatili and R. A. Schmeltzer , “ Hollow metallic and dielectric waveguides for long distance optical transition and lasers ,” Bell Syst. Tech. J. 43 , 1783 – 1809 ( 1964 ).
14.	E. Garmire , T. McMahon , and M. Bass , “ Propagation of infrared light in flexible hollow waveguide ,” Appl. Opt. 15 , 145 – 150 ( 1976 ). [CrossRef] [PubMed]
15.	Y. Xia and J. Yin , “ Generation of a focused hollow beam by an 2 Pi-phase plates and its applications in atom or molecule optics ,” J. Opt. Soc. Am. B 22 , 529 – 536 ( 2005 ). [CrossRef]
16.	R. L. Abrams and A. N. Chester , “ Resonator theory for hollow waveguide lasers ,” Appl. Opt. 13 , 2117 – 2125 ( 1974 ). [CrossRef] [PubMed]
17.	M. Saito , S. Sato , and M. Miyagi , “ Loss characteristics of infrared hollow waveguides in multimode transmission ,” J. Opt. Soc. Am. A 10 , 277 – 282 ( 1993 ). [CrossRef]
18.	M. Miyagi and S. Kawakami , “ Design theory of dielectric-coated circular metallic waveguide for infrared transition ,” IEEE J. Lightwave Technol. 2 , 116 – 126 ( 1984 ). [CrossRef]
19.	V. I. Balykin , D. V. Laryushin , M. V. Subbotin , and V. S.L etokhov , “ Increase of the atomic phase density in a hollow laser waveguide ,” JETP Lett. 63 , 802 – 807 ( 1996 ). [CrossRef]
20.	J. Yin , Y. Zhu , and Y. Wang , “ Evanescent light-wave atomic funnel: A tanden hollow-fiber, hollow-beam approach ,” Phys. Rev. A 57 , 1957 – 1966 ( 1998 ). [CrossRef]
21.	J. Yin , Y. Zhu , and Y. Wang , “ Gravito-optical trap for cold atoms with doughnut-hollow-beam cooling ,” Phys. Lett. A 248 , 309 – 318 ( 1998 ). [CrossRef]
22.	J. Söding , R. Grimm , and Yu. B. Ovchinnikov , “ Gravitational laser trap for atoms with evanescent-wave cooling ,” Opt. Commun. 119 , 652 – 662 ( 1995 ). [CrossRef]
23.	Yu.B. Ovchinnikov , I. Manek , and R. Grimm , “ Surface trap for Cs atoms based on evanescent-wave cooling ,” Phys. Rev. Lett. 79 , 2225 – 2228 ( 1997 ). [CrossRef]
24.	X. Xu , V. G. Minogin , K. Lee , Y. Wang , and W. Jhe , “ Guiding cold atoms in a hollow laser beam ,” Phys. Rev. A 60 , 4796 – 4804 ( 1999 ). [CrossRef]
25.	J. Yin and Y. Zhu , “ Dark-hollow-beam gravito-optical atom trap above an apex of a hollow optical fibre ,” Opt. Commun. 152 , 421 – 428 ( 1998 ). [CrossRef]
26.	X. Xu , Y. Wang , and W. Jhe , “ Theory of atom guidance in a hollow laser beam: dressed-atom approach ,” J. Opt. Soc. Am. B 17 , 1039 – 1050 ( 2000 ). [CrossRef]
27.	H. Nha and W. Jhe , “ Sisphus cooling on the surface of a hollow-mirror atom trap ,” Phys. Rev. A 56 , 729 – 736 ( 1997 ). [CrossRef]
28.	J. Dalibard and C. Cohen-Tannoudji , “ Dressed-atom approach to atomic motion in laser light: the dipole force revisited ,” J. Opt. Soc. Am. B 2 , 1707 – 1720 ( 1985 ). [CrossRef]
29.	Z. T. Lu , K. L. Corwin , M. J. Renn , M. H. Anderson , E. A. Cornell , and C. E. Wieman , “ Low-velocity intense source of atoms from a magneto-optical trap ,” Phys. Rev. Lett. 77 , 3331 – 3334 ( 1996 ). [CrossRef] [PubMed]
30.	H. Ito , K. Sakaki , M. Ohtsu , and W. Jhe , “ Evanescent-light guiding of atoms throught hollow optical fiber for optically controlled atomic deposition ,” Appl. Phys. Lett. 70 2496 – 2498 ( 1997 ). [CrossRef]
31.	J. Yin , H. Noh , K. Lee , K. Kim , Y. Wang , and W. Jhe , “ Generation of a dark hollow beam by a small hollow fiber ,” Opt. Commun. 138 , 287 – 292 ( 1997 ). [CrossRef]
32.	T. Junglen , T. Rieger , S. A. Rangwala , P. W. H. Pinkse , and G. Rempe , “ Slow ammonia molecules in an electrostatic quadrupole guide ,” Eur. Phys. J. D 31 , 365 – 373 ( 2004 ). [CrossRef]

OCIS Codes
(020.0020) Atomic and molecular physics : Atomic and molecular physics
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.7010) Lasers and laser optics : Laser trapping

ToC Category:
Research Papers

History
Original Manuscript: July 22, 2005
Revised Manuscript: September 30, 2005
Published: October 17, 2005

Citation
Zhengling Wang, Meng Dai, and Jianping Yin, "Atomic (or molecular) guiding using a blue-detuned doughnut mode in a hollow metallic waveguide," Opt. Express 13, 8406-8423 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-21-8406

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References

J. Yin, W. Gao, and Y. Zhu, �??Generation of dark hollow beams and their applications,�?? Prog. Opt. 45, 119-204 (2003). [CrossRef]
J. Yin, Y. Zhu, W. Wang, Y. Wang, and W. Jhe, �??Optical potential for atom guidance in a dark hollow laser beam,�?? J. Opt. Soc. Am. B 15, 25-33 (1998). [CrossRef]
. J. Yin, Y. Zhu, W. Jhe, and Y. Wang, �??Atom guiding and cooling in a dark hollow laser beam,�?? Phys. Rev. A 58, 509-513 (1998). [CrossRef]
X. Xu, K. Kim, W. Jhe, N. Kwon, �??Efficient optical guiding of trapped cold atoms by a hollow laser beam,�?? Phys. Rev. A 63, 063401 (2001). [CrossRef]
L. Pruvost, D. Marescaux, O. Houde, and H. T. Duong, �?? Guiding and cooling of cold atoms in dipole guide,�?? Opt. Commun. 166, 199-209 (1999). [CrossRef]
B. T. Wolschrijn, R. A. Cornelussen, R. J. C. Spreeuw, and H. B. van Linden van den Heuvell, �??Guiding of cold atoms by a red-detuned laser beam of moderate power,�?? New J. Phys. 4, 69.1-69.10 (2002). [CrossRef]
M. A. Ol�??Shanii, Yu. B. Ovchinnikov, and V. S. Letkhov, �??Laser guiding of atoms in a hollow optical fiber,�?? Opt. Commun. 98, 77-79 (1993). [CrossRef]
M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, �??Laser-guided atoms in hollow-core optical fibers,�?? Phys. Rev. Lett. 75, 3253-3256 (1995). [CrossRef] [PubMed]
M. J. Renn, A. A. Zozulya, E. A. Donley, E. A. Cornell, and D. Z. Anderson, �??Optical-dipole-force fiber guiding and heating of atoms,�?? Phys. Rev. A 55, 3684-3693 (1997). [CrossRef]
M. J. Renn, E. A. Donley, E. A. Cornell, C. E. Wieman, and D. Z. Anderson, �??Evanescent-wave guiding of atoms in hollow optical fibers,�?? Phys. Rev. A 53, R648-R651 (1996). [CrossRef] [PubMed]
H. Ito, T. Nakata, K. Sakaki, M. Ohtsu, K. I. Lee, and W. Jhe, �??Laser spectroscopy of atoms guiding by evanescent waves in micron-sided hollow optical fibers,�?? Phys. Rev. Lett. 76, 4500-4503 (1996). [CrossRef] [PubMed]
H. Ito, K. Sakaki, W. Jhe, and M. Ohtsu, �??Evanescent-light induced atom-guidance using a hollow optical fiber with light coupled sideways,�?? Opt. Commun. 141, 43-47 (1997). [CrossRef]
E. A. J. Marcatili, and R. A. Schmeltzer, �??Hollow metallic and dielectric waveguides for long distance optical transition and lasers,�?? Bell Syst. Tech. J. 43, 1783-1809 (1964).
E. Garmire, T. McMahon, and M. Bass, �??Propagation of infrared light in flexible hollow waveguide,�?? Appl. Opt. 15, 145-150 (1976). [CrossRef] [PubMed]
Y. Xia, and J. Yin, �??Generation of a focused hollow beam by an 2 Pi-phase plates and its applications in atom or molecule optics,�??J. Opt. Soc. Am. B 22, 529-536 (2005). [CrossRef]
R. L. Abrams, and A. N. Chester, �??Resonator theory for hollow waveguide lasers,�?? Appl. Opt. 13, 2117-2125 (1974). [CrossRef] [PubMed]
M. Saito, S. Sato, and M. Miyagi, �??Loss characteristics of infrared hollow waveguides in multimode transmission,�?? J. Opt. Soc. Am. A 10, 277-282 (1993). [CrossRef]
M. Miyagi, and S. Kawakami, �??Design theory of dielectric-coated circular metallic waveguide for infrared transition,�?? IEEE J. Lightwave Technol. 2, 116-126 (1984). [CrossRef]
V. I. Balykin, D. V. Laryushin, M. V. Subbotin, and V. S.L etokhov, �??Increase of the atomic phase density in a hollow laser waveguide,�?? JETP Lett. 63, 802-807 (1996). [CrossRef]
J. Yin, Y. Zhu, and Y. Wang, �??Evanescent light�??wave atomic funnel: A tanden hollow-fiber, hollow-beam approach,�?? Phys. Rev. A 57, 1957-1966 (1998). [CrossRef]
J. Yin, Y.Zhu, and Y. Wang, �??Gravito-optical trap for cold atoms with doughnut-hollow-beam cooling,�?? Phys. Lett. A 248, 309-318 (1998). [CrossRef]
J. Söding, R. Grimm, and Yu. B. Ovchinnikov, �??Gravitational laser trap for atoms with evanescent-wave cooling,�?? Opt. Commun. 119, 652-662 (1995). [CrossRef]
Yu.B. Ovchinnikov, I. Manek, and R. Grimm, �??Surface trap for Cs atoms based on evanescent-wave cooling,�?? Phys. Rev. Lett.79, 2225-2228 (1997). [CrossRef]
X. Xu, V. G. Minogin, K. Lee, Y. Wang, and W. Jhe, �??Guiding cold atoms in a hollow laser beam,�?? Phys. Rev. A 60, 4796-4804 (1999). [CrossRef]
J. Yin, and Y. Zhu, �??Dark-hollow-beam gravito-optical atom trap above an apex of a hollow optical fibre,�?? Opt. Commun. 152, 421-428 (1998). [CrossRef]
X. Xu, Y. Wang, and W. Jhe, �??Theory of atom guidance in a hollow laser beam: dressed-atom approach,�?? J. Opt. Soc. Am. B 17, 1039-1050 (2000). [CrossRef]
H. Nha, and W. Jhe, �??Sisphus cooling on the surface of a hollow-mirror atom trap,�?? Phys. Rev. A 56, 729-736 (1997). [CrossRef]
J. Dalibard, and C. Cohen-Tannoudji, �??Dressed-atom approach to atomic motion in laser light: the dipole force revisited,�?? J. Opt. Soc. Am. B 2, 1707-1720 (1985). [CrossRef]
Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, �??Low-velocity intense source of atoms from a magneto-optical trap,�?? Phys. Rev. Lett. 77, 3331-3334 (1996). [CrossRef] [PubMed]
H. Ito, K. Sakaki, M. Ohtsu, and W. Jhe, �??Evanescent-light guiding of atoms throught hollow optical fiber for optically controlled atomic deposition,�?? Appl. Phys. Lett. 70 2496-2498 (1997). [CrossRef]
J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, �??Generation of a dark hollow beam by a small hollow fiber,�?? Opt. Commun. 138, 287-292 (1997). [CrossRef]

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Atomic (or molecular) guiding using a blue-detuned doughnut mode in a hollow metallic waveguide

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Abstract

1. Introduction

2. Guiding scheme and electromagnetic fields in HMW

2.1. Guiding scheme of cold atoms

2.2. Mode analysis of the electromagnetic fields in the HMW

2.3. Mode coupling and the absolute intensity distribution of the TE01 mode

2.4. Propagation losses of various modes in the HMW

3. Optical potentials and spontaneous emissions

3.1. Optical potentials for a three-level atom

3.2. Spontaneous emissions of a three-level atom

4. Guiding efficiency and loss mechanisms

4.1. Straight guide of cold atoms

4.2. Bent guide of cold atoms

4.3. Atomic loss mechanisms and their loss rates

4.3.1 Atomic loss from background thermal collisions

4.3.2. Atomic loss from spontaneous emission of the guided atoms

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

5. Potential applications

6. Conclusion

Acknowledgments

References and links

References

Appl. Opt.

Appl. Phys. Lett.

Bell Syst. Tech. J.

IEEE J. Lightwave Technol.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

JEPT

New J. Phys.

Opt. Commun.

Phys. Lett. A

Phys. Rev. A

Phys. Rev. Lett.

Prog. Opt.

Cited By

Figures

2.3. Mode coupling and the absolute intensity distribution of the TE₀₁ mode