Evanescent atom guiding in metal-coated hollow-core optical fibers

A new type of atom guiding structure has been analyzed. It consists of a low-core optical fiber (step-index) which is not clad, but instead has a metal coating on its outer lateral surface. It will be shown that this structure produces the maximum evanescent field in the hollow region of the fiber and guiding can be accomplished with lower power lasers. Both the dipole and the vander Waals potentials have been combined and the resulting barrier height was maximized as a function of both .DELTA., the detuning, and r, the position. An optimized potential having a barrier height of 1 K has been determined by iteratively solving for the required laser intensity. The probability of atoms tunneling through this barrier to the inner wall has been calculated and is expected to be unimportant. Centripetal effects due to a bending of the fiber have also been estimated and are small for the barrier considered here. Compared to other structures, this new-type of guide provides bigger barriers for the same laser power, and therefore enhanced atom guiding.

BACKGROUND OF THE INVENTION 
1. Technical Field 
The invention relates generally to a method and apparatus for guiding atoms 
by blue-detuned evanescent waves in a hollow-core optical fiber. More 
particularly, the invention relates to the use of a metal-coated 
hollow-core optical fiber as the wave guide to maximize the evanescent 
guiding field in the hollow region of the fiber. 
2. Background Art 
The polarizability of an atom is almost always positive, but it can be 
negative and some unusual effects can then be observed. This can occur 
when a laser or other monochromatic source is tuned slightly above or to 
the "blue" of an atomic resonance. The interaction of the external field 
on the atom through its negative polarizability produces a gradient dipole 
force which tends to drive the atom to regions of minimum intensity. Cook 
and Hill suggested using an evanescent wave to produce an atom mirror 
outside of a dielectric. Reference: R. J. Cook, R. K. Hill, An 
Electromagnetic Mirror for Neutral Atoms, Optics Comm. 43 (1982) 258. 
Zoller, et. al. analyzed the case for a clad, hollow fiber in which the 
external field was confined to the annular region and used the resulting 
evanescent field in the hollow region to guide atoms. Reference: S. 
Marksteiner, C. M. Savage, P. Zoller, S. L. Rolston, Coherent Atomic 
Waveguides from Hollow Optical Fibers: Quantized Atomic Motion, Phys. Rev. 
A 50 (1994) 2680. In what has become known as "blue-guiding", Renn and Ito 
have experimentally demonstrated evanescent wave guiding of rubidium atoms 
in hollow optical fibers. See: M. J. Renn, E. A. Donley, E. A. Cornell, C. 
E. Wiemann, D. Z. Anderson, Evanscent-wave Guiding of Atoms in Hollow 
Optical Fibers, Phys. Rev. A 53 (1996) 648A; and H. Ito, T. Nakata, K. 
Sakaki, M. Ohtsu, K. I. Lee, W. Jhe, Laser Spectroscopy of Atoms Guided by 
Evanscent Waves in Micron-sized Hollow Optical Fibers, Phys. Rev. Lett. 76 
(1996) 4500. 
SUMMARY OF THE INVENTION 
It is an object of the invention to provide an improved method for guiding 
atoms by evanescent laser light through hollow-core optical fibers, which 
maximizes the evanescent guiding field in the hollow region of the fiber. 
It is another object of the invention to provide a new type of atom guiding 
structure which can be used in existing systems for guiding atoms through 
hollow-core optical fibers, which maximizes the guiding barrier for a 
given laser power. 
It is a further object of the invention to provide such a new type of atom 
guiding structure which also minimizes the loss of atoms to the inner wall 
of the fiber from quantum tunneling. 
It is still another object of the invention to provide such a new type of 
atom guiding structure which also minimizes the loss of atoms to the inner 
wall of the fiber from centripetal force due to physical bending of the 
fiber. 
The atom guiding structure, according to the invention, comprises a 
hollow-core optical fiber (step index) which has a coating on its outer 
lateral surface of a material, such as a metal, which has a high optical 
reflectivity. Typically, this coating comprises a metal having very high 
electrical conductivity, such as silver, gold, chromium, and aluminum.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
A new type of enhanced, atom guiding structure has been analyzed. It 
consists of a hollow-core, step index optical fiber which is not clad but 
instead has a metal coating on its outer lateral surface. It will be shown 
that the longitudinal electric fields of the lowest order TM.sub.On modes 
of this structure have a global maximum at the inner wall of the hollow 
fiber. This produces the maximum barrier and therefore, maximum guiding in 
the hollow region of the fiber. 
A perfect, hollow dielectric cylinder of inner radius, a, outer radius, b, 
and longitudinal axis in the z direction, is analyzed as an atom guide. 
The outer lateral surface at radius, b, is assumed to be coated with a 
perfect conductor, This coating not only produces a structure having 
significant advantages as an atom guide but also greatly simplifies the 
boundary conditions. The lowest order TM modes of this infinitely long 
wave guide are calculated in the standard way, which is set forth in the 
textbook Fields and Waves in Communication Electronics, Third 
Edition,(Wiley, N.Y., 1993), which is incorporated herein by reference. 
The Helmholtz equation for E.sub.z1 in the hollow region (r.ltoreq.a) is 
given by: 
##EQU1## 
where 
EQU w.sub.1.sup.2 =(.beta..sup.2 -k.sup.2) a.sup.2 &gt;0, (2) 
and E.sub.z2 in the annular dielectric (a.ltoreq.,r.ltoreq.b) is 
##EQU2## 
where 
EQU u.sub.2.sup.2 =(k.sub.2.sup.2 -.beta..sup.2)a.sup.2 &gt;0. (4) 
Here w.sub.1 and u.sub.2 are the eigenvalues of equations (1) and (3), 
respectively, .beta. is the propagation constant in the z direction, 
k=2.pi./.lambda. is the wave number (free space) where .lambda. is the 
wavelength, and k.sub.2 =n.sub.2 k is the wave number in the dielectric of 
index of refraction n.sub.2. For zero azimuthal dependence, the general 
solution of equation (1) is 
##EQU3## 
and for equation (3) is 
##EQU4## 
where J.sub..nu., Y.sub..nu., I.sub..nu., and K.sub..nu., are Bessel 
functions of order .nu., as set forth in the Handbook of Mathematical 
Functions, M. Abramowitz and I. Stegun, Editors, (Dover, N.Y., 1965) pages 
355-430, incorporated herein by reference. A, B, C, and D are constants 
determined by the boundary conditions. The zero azimuthal dependence 
automatically permits the exclusion of hybrid modes as the solutions 
separate into TM and TE sets. 
Equations (5 and 6) are solved simultaneously subject to the following 
boundary conditions: E.sub.z1 (r=0) is finite, E.sub.z1 (r=a)=E.sub.z2 
(r=a)=1 (normalizes solutions to unity), E.sub.z2 (r=b)=0, and 
H.sub..phi.1 (r=a)=H.sub..phi.2 (r=a). The solution of these equations 
gives the first of the two determinental equations, which for zero 
azimuthal dependence, reduces to 
##EQU5## 
where .di-elect cons..sub.1 =.di-elect cons..sub.0, the vacuum 
permittivity, and .di-elect cons..sub.2 =n.sub.2.sup.2 .di-elect 
cons..sub.0. For the case to be considered here where b=3a, this reduces 
to 
##EQU6## 
The second determinental equation is obtained from (2) and (4) as 
EQU u.sub.2.sup.2 +w.sub.1.sup.2 =a.sup.2 k.sup.2 (n.sub.2.sup.2 -1).(9) 
Equations (8) and (9) can be solved either graphically or numerically for 
the eigenvalues, w.sub.1 or u.sub.2. For the parameters of the guide 
described here: a=2.5 .mu.m, .lambda.=0.5 .mu.m, and n.sub.2 =.sqroot.5, 
the results for the five lowest-order TM modes (TM.sub.01 . . . TM.sub.05) 
have been calculated and the corresponding eigenvectors E.sub.z2 are shown 
in FIG. 1. It is significant that all of the E.sub.z2 shown here have 
their global maximum value at r=a. This gives the maximum possible values 
for the evanescent fields, E.sub.r1 at r=a, because the fields in the 
hollow region have unique solutions which do not depend on the solutions 
in the annular region except in so far as they are connected by the 
boundary conditions at the interface (r=a). 
The potential ener for the interaction of an atomic dipole in an 
oscillating electric field is treated classically as 
##EQU7## 
where .alpha. is the polarizability. In the hollow part of the guide, E is 
the evanescent field E.sub.r1, and neglecting the phase factor, 
##EQU8## 
where, for the guide considered here, .beta.=2.81.times.10.sup.7 m.sup.-1, 
w.sub.1 =62.8, u.sub.2 =0.960, and E.sub.L is the laser field strength in 
the dielectric at the interface r=a. The required laser intensity, 
I.sub.L, is given by I.sub.L =1/2n.sub.2 .di-elect cons..sub.0 c 
E.sub.L.sup.2. While the classical description of the dipole potential 
provides a simple understanding of the geometric aspects of this 
interaction because the induced dipole moment is co-linear with the 
evanescent field and this field must be in the radial direction in order 
to guide atoms, the quantum representation is necessary in order to 
analyze this problem in more detail. The quantum dipole potential, 
U.sub.dip-qm, is given by 
##EQU9## 
where h=h/2.pi.=1.05.times.10.sup.-34 J.multidot.s, the detuning 
.DELTA.=.omega.-.omega..sub.0, .gamma. is the decay rate of the upper 
level, d is the transition dipole moment between levels 1 and 2, and E is 
the electric field amplitude given by equation (10). See: A. Ashkin, Phys. 
Rev. Lett. 40 (1978) 729; and J. Dalibard and C. Cohen-Tannoudji, J. Opt. 
Soc. Am. B 2 (1985) 1707. The van der Waals potential for an atom in 
proximity to an infinite dielectric slab is given by 
##EQU10## 
where .di-elect cons. is the dielectric constant, .mu..sup.2.sub..SIGMA. 
is the sum of the squares of all the transition dipole moments, and x is 
the distance from the atom to the dielectric surface. See: M. J. Renn, E. 
A. Donley, E. A. Cornell, C. E. Wiemann, D. Z. Anderson, Evanscent-wave 
Guiding of Atoms in Hollow Optical Fibers, Phys. Rev. A 53 (1996) 648A, 
cited above. This represents an approximation of the attractive potential 
tending to draw the atom to the inner wall of the cylinder, Changing 
variables from x to r, the distance from the center of the cylinder, 
substituting .di-elect cons.=n.sub.2.sup.2, and adding a constant offset 
term to make U.sub.vdw =0 at r=0 gives 
##EQU11## 
The total potential, U, in the hollow region is obtained by adding (11 and 
12) where 
##EQU12## 
A synthetic two level atom has been assumed for U.sub.dip-qm with the 
following properties: d=2.10.times.10.sup.-29 Cm and .gamma.=10.sup.8 
s.sup.-1. For U.sub.vdw, a multilevel atom was assumed where 
.mu..sup.2.sub..SIGMA. =9.times.10.sup.-58 C.sup.2 m.sup.2 and is 
equivalent to a two-level atom whose transition dipole moment is 
1.50.times.d. Substituting these values together with the solution for the 
TM.sub.01 mode into (13) gives 
##EQU13## 
Equation (14) was simultaneously maximized in both r and .DELTA.. For 
E.sub.L =2.68.times.10.sup.6 V/m (corresponding to a laser input intensity 
of 2.13.times.10.sup.6 w/cm.sup.2), U.sub.max, the maximum value for U, 
was found to be 1.00 K for .DELTA.=1.76.times.10.sup.11 s.sup.-1 and 
r=2.4925 .mu.m. The 1 Kelvin barrier is shown in FIG. 2. This barrier is 
representative of conditions for which the loss of atoms via tunneling to 
the inner wall of the fiber is of interest. Near the inner wall, the van 
der Waals potential is strongly attractive and dominates the total 
potential there. The tunneling can be calculated using the WKB 
approximation, described in Quantum Theory, by D. Bohm (Prentice-Hall, 
N.Y., 1951) and in textbfQuantum Mechanics, by D. H. Rapp (Holt, Rinehart 
and Winston, N.Y., 1971), which yields 
##EQU14## 
where T is the tunneling probability per bounce, tp1 and tp2 are the 
turning points for the initial energy, m is the mass of the atom and is 
taken here to be 4.00.times.10.sup.-26 kg, U is given by equation(14), 
.function. is the fraction of the total barrier height corresponding to 
the initial energy. This is at least a useful approximation as the 
tunneling is small in the region of interest, .function..ltoreq.0.98, and 
the de Broglie wavelength, .lambda..sub.dB =0.901 nm for 
(.function.=0.98), is small compared to the minimum thickness, (tp.sub.2 
-tp.sub.1).gtoreq.3.58 nm, of the barrier in this region. The probability 
per bounce for an atom to tunnel through a 1 K barrier to the wall is 
shown in FIG. 3 where the fraction of initial energy ranges from 0.80 to 
1.00 of the barrier height. For this barrier, the probability of tunneling 
per bounce is T.ltoreq.10.sup.-3 for .function..ltoreq.0.98 of the total 
barrier height. From FIG. 3, it can be seen that for 
.function..apprxeq.0.90, quantum tunneling is expected to be unimportant. 
A curved or bent fiber will generate a centripetal force on the atom as it 
moves in an arc around the bend. If this force is greater than the inward 
repulsive force of the potential, then the atom will penetrate the 
barrier, hit the wall, and be lost. For the 1 K barrier considered above, 
the force (which includes both the dipole and the van der Wall forces) is 
shown in FIG. 4. An estimate of the minimum bending radius, 
##EQU15## 
is derived in a publication by J. P. Dowling and J. Gea-Banaeloche, Adv. 
At. Mol. Opt. Phys. 36 (1996) 1, and has been calculated in two references 
cited above, namely, S. Marksteiner, C. M. Savage, P. Zoller, S. L. 
Rolston, Coherent Atomic Waveguides from Hollow Optical Fibers: Quantized 
Atomic Motion, Phys. Rev. A 50 (1994) 2680; and M. J. Renn, E. A. Donley, 
E. A. Cornell, C. E. Wiemann, D. Z. Anderson, Evanscent-wave Guiding of 
Atoms in Hollow Optical Fibers, Phys. Rev. A 53 (1996) 648. Here a is the 
radius of the fiber, v.sub..vertline..vertline. is the longitudinal 
velocity, and v.sub..perp. is the maximum allowed trapped transverse 
velocity. For a 1 K barrier and a longitudinal velocity of the beam 
appropriate to 1000 K, R.sub.min =0.5 cm., and centripetal effects are 
small for the potential used here. 
Zoller, et. al. have reported a comprehensive analysis of evanescent 
blue-guiding of atoms. Refer to: S. Marksteiner, C. M. Savage, P. Zoller, 
S. L. Rolston, "Coherent Atomic Waveguides from Hollow Optical Fibers: 
Quantized Atomic Motion", Phys. Rev. A 50 (1994) 2680, cited above and 
incorporated herein by reference. They analyzed a clad, hollow dielectric 
fiber and primarily considered the hybrid HE.sub.11 mode because they were 
mainly interested in a single mode fiber. The enhancement provided by the 
metal-coated fiber can be estimated by comparing the guiding produced by 
the HE.sub.11 mode in their clad fiber with that in a TM.sub.01 mode in an 
identical fiber except that the cladding has been removed and the outer 
dielectric surface is coated with a perfect conductor which is a perfect 
reflector. The radial dependence of the longitudinal electric field, 
E.sub.z from the above-cited reference (S. Maxksteiner, C. M. Savage, P. 
Zoller, S. L. Rolston, "Coherent Atomic Waveguides from Hollow Optical 
Fibers: Quantized Atomic Motion", Phys. Rev. A 50 (1994) 2680) is shown 
with permission in FIG. 5. It is seen that the maximum electric field 
occurs well inside the core and is about 81/2 times larger than the field 
at the hole-core boundary (r/.rho..sub.1 =1) where .rho..sub.1 =a in their 
notation. (As expected, a calculation using their parameters (a=1.65 
.mu.m, b=3.3 .mu.m, n.sub.2 =1.5, and .lambda.=0.57 .mu.m) for the 
TM.sub.01 mode shows that the maximum value of E.sub.z occurs at (r=a)). 
The radial electric fields in the hollow region have been calculated for 
the TM.sub.01 and HE.sub.11 modes as E.sub.r1TM and E.sub.r1HE, 
respectively. For the case where E.sub.z2TM (r=a)=E.sub.z2HE (r=a), 
E.sub.r1TM and E.sub.Er1HE appear to be identical. An amplitude scaling 
factor is defined as S=[E.sub.z2 (r=a)/E.sub.z2 (max)] and S.sub.TM01 
=1.00 (FIG. 1) and S.sub.HE11 =.about.(81/2).sup.-1 (FIG. 5). From the 
boundary conditions, E.sub.z1 (r=a)=E.sub.z2 (r=a) and H.sub.z1 
(r=a)=H.sub.z2 (r=a), it can be shown that E.sub.r1 scales directly with 
S. The ratio of the radial electric fields can be written as [E.sub.r1TM 
(r=a)/E.sub.r1HE (r=a)]=S.sub.TM01 /S.sub.HE11 =.about.81/2. Because the 
dipole potential, equation (13), depends on E.sup.2.sub.r1 in a somewhat 
complicated way..sup.1, and the enhanced guiding is due to the fact that 
one mode is more effective than another in producing a larger E.sub.r1 
(r=a) for the same maximum value of E.sub.z2, the enhancement will be 
calculated in terms of the increased laser intensity required to satisfy 
the condition: E.sub.r1HE (r=a)=E.sub.r1TM (r=a). The result is that the 
laser intensity must be increased by the factor [S.sub.TM01 /S.sub.HE11 
].sup.2 =.about.(81/2).sup.2 .about.72 in order for the dielectric-clad 
fiber to provide the same degree of guiding as in the metal-coated fiber. 
FNT .sup.1 For small E, U.sub.dip-qm .about.E.sup.2, but for 
E=2.68.times.10.sup.6 V/m, U.sub.dip-qm has dropped by nearly a factor of 
3 from that predicted by the over-simplified .about.E.sup.2 dependence. 
Even so, atom guiding is significantly enhanced. 
The use of the TM.sub.01 mode in the metal-coated guide proposed here 
offers several advantages over structures considered previously. First, 
E.sub.z2 has a global maximum at r=a and provides for maximum guiding in 
the hollow region. While FIG. 1 shows that the lowest five modes have a 
global maximum at r=a, it has not been proven that this is the case for 
all of the allowed TM.sub.0n modes. These modes are expected to be 
partially coherent and their travelling waves will average over the 
relatively slow motion of the atoms, and should only result in a modest 
correction to the effective laser field in the expression for 
U.sub.dip-qm. 
The use of the metal coated "atom guide" described herein will enhance the 
performance of, as well as permit the minaturization of devices such as 
atomic clocks, and atom interfer-ometers and their applications such as 
rotational and gravitational sensors.