Download Spontaneous emission of a cesium atom near a nanofiber: Efficient

PHYSICAL REVIEW A 72, 032509 共2005兲
Spontaneous emission of a cesium atom near a nanofiber: Efficient coupling of light
to guided modes
1
Fam Le Kien,1,* S. Dutta Gupta,1,2 V. I. Balykin,1,3 and K. Hakuta1
Department of Applied Physics and Chemistry, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
2
School of Physics, University of Hyderabad, Hyderabad, India
3
Institute of Spectroscopy, Troitsk, Moscow Region, 142092, Russia
共Received 16 June 2005; published 28 September 2005; publisher error corrected 3 October 2005兲
We study the spontaneous emission of a cesium atom in the vicinity of a subwavelength-diameter fiber. We
show that the confinement of the guided modes and the degeneracy of the excited and ground states substantially affect the spontaneous emission process. We demonstrate that different magnetic sublevels have different
decay rates. When the fiber radius is about 200 nm, a significant fraction 共up to 28%兲 of spontaneous emission
by the atom can be channeled into guided modes. Our results may find applications for developing nanoprobes
for atoms and efficient couplers for subwavelength-diameter fibers.
DOI: 10.1103/PhysRevA.72.032509
PACS number共s兲: 32.30.Jc, 32.70.Jz, 32.80.Pj, 03.75.Be
I. INTRODUCTION
Coupling of light to subwavelength structures and its control pose one of the greatest challenges of recent research 关1兴.
In this paper, we show how such coupling with efficiency of
up to 28% can be achieved in a realistic system of a cesium
atom near a subwavelength-diameter fiber. Note that modification of the vacuum near the fiber and its effect on the
spontaneous emission has been studied in the context of twolevel atoms 关2–4兴. Many other studies exist involving other
geometries 关5,6兴. Most of these investigations, to the best of
our knowledge, do not go beyond the two-level approximation for the atom. The inclusion of hyperfine structure of the
atom can significantly affect the actual rate of spontaneous
decay. Some parameters that describe the decay of crosslevel coherences arise only in the framework of a multilevelatom model. The knowledge of both diagonal and offdiagonal types of decay characteristics is required for the
studies of absorption and emission properties, optical response, and dynamical behavior of realistic atoms 关7兴. We
show that the confinement of the guided modes and the degeneracy of the excited and ground states substantially affect
the spontaneous emission process. We find that different
magnetic sublevels have different decay rates. We demonstrate that the thin fiber can indeed act as a subwavelength
probe, since about one fourth of the spontaneous emission
from the atom can be channeled into guided modes. The
knowledge of spontaneous emission characteristics is important yet from the angle of atom optics 关8兴, in particular, from
design considerations of atom traps 关9兴. A recent proposal for
microscopic trapping of individual atoms involves the use of
a subwavelength-diameter silica fiber with a single 共reddetuned兲 关10兴 or two 共red- and blue-detuned兲 light beams
关11兴 launched into it.
The paper is organized as follows. In Sec. II we describe
the model. In Sec. III we derive the basic characteristics of
*Also at Institute of Physics and Electronics, Vietnamese Academy of Science and Technology, Hanoi, Vietnam.
1050-2947/2005/72共3兲/032509共7兲/$23.00
spontaneous emission for the model. In Sec. IV we present
numerical results. Our conclusions are given in Sec. V.
II. MODEL
We consider a cesium atom trapped in the vicinity of a
subwavelength-diameter silica fiber 共see the upper part of
Fig. 1兲. We use the fiber axis z as the quantization axis for
atomic states. For atoms trapped in a magneto-optical trap,
the quantization axis can be specified and hence controlled
by the direction of the magnetic field in the trap. We study
the cesium D2 line, which occurs at the wavelength ␭0
= 852 nm and corresponds to the transition from the excited
state 6P3/2 to the ground state 6S1/2 共see the lower part of
Fig. 1兲. We assume that the atom is initially prepared in the
hyperfine-structure 共hfs兲 level F⬘ = 5 of the state 6P3/2. We
FIG. 1. 共a兲 An atom interacting with guided and radiation modes
in the vicinity of a thin optical fiber. 共b兲 Schematic of the 6P3/2F⬘
= 5 and 6S1/2F = 4 hfs levels of a cesium atom.
032509-1
©2005 The American Physical Society
PHYSICAL REVIEW A 72, 032509 共2005兲
LE KIEN et al.
introduce the notation e M ⬘ and g M for the magnetic sublevels
F⬘M ⬘ and FM of the hfs levels 6P3/2F⬘ = 5 and 6S1/2F = 4,
respectively. Due to the selection rule, spontaneous emission
from the hfs level 6P3/2F⬘ = 5 to the ground state is always to
the hfs level 6S1/2F = 4, not to the hfs level 6S1/2F = 3. Therefore, the magnetic sublevels e M ⬘ and g M of the hfs levels
6P3/2F⬘ = 5 and 6S1/2F = 4, respectively, form a closed set,
which is used for laser cooling in magneto-optical traps 关12兴.
The coupling between e M ⬘ and g M by spontaneous emission
is illustrated in the lower part of Fig. 1.
In the interaction picture, the atomic dipole operator is
given by
*
† i␻0t
␴gee−i␻0t + deg␴ge
e 兲.
D = 兺 共deg
共1兲
eg
†
= ␴eg = 兩e典具g兩 describe
Here, the operators ␴ge = 兩g典具e兩 and ␴ge
the downward and upward transitions, respectively, and
deg = 具e兩D兩g典e−i␻0t is the dipole matrix element. We
introduce the notation d共−1兲 = 共dx − idy兲 / 冑2, d共0兲 = dz, and
d共1兲 = −共dx + idy兲 / 冑2 for the spherical tensor components of
the dipole vector d. We label the spherical tensor components by the index l = 0 , ± 1. The l spherical component of the
dipole moment for the transition between e M ⬘ and g M is
given by 关13兴
d共l兲
e
M ⬘g M
= 共− 1兲I+J⬘−M ⬘具J⬘储D储J典冑共2F + 1兲共2F⬘ + 1兲
⫻
再
J⬘ F⬘ I
F J 1
冎冉
F 1
F⬘
M l − M⬘
冊
,
共2兲
where the array in the curly braces is a 6j symbol, the array
in the parentheses is a 3j symbol, J is the total electronic
angular momentum, I is the nuclear spin, F is total atomic
angular momentum, M is the magnetic quantum number, and
具J⬘储D储J典 is the reduced electric-dipole matrix element in the
J basis.
We assume that the fiber has a cylindrical silica core of
radius a and refractive index n1 and an infinite vacuum clad
of refractive index n2 = 1. We retain the silica dispersion and
at the frequency of the cesium D2 line the refractive index n1
of the fiber is taken as 1.45. The positive-frequency part E共+兲
of the electric component of the field can be decomposed
into the contributions from the guided and radiation modes
as
共+兲
共+兲
+ Erad
.
E共+兲 = Eguided
quantize the field in the guided modes, we obtain the follow共+兲
ing expression for Eguided
in the interaction picture:
冕 冑
⬁
共+兲
= i兺
Eguided
fp
d␻
0
ប␻␤⬘
a␮e共␮兲e−i共␻t−f ␤z−p␸兲 .
4␲⑀0
共4兲
Here ␤ is the longitudinal propagation constant, ␤⬘ is the
derivative of ␤ with respect to ␻, a␮ is the respective photon
annihilation operator, and e共␮兲 = e共␮兲共r , ␸兲 is the electric-field
profile function of the guided mode ␮ in the classical problem. The constant ␤ is determined by the fiber eigenvalue
equation 关15兴. The operators a␮ and a␮† satisfy the
continuous-mode bosonic commutation rules 关a␮ , a␮† 兴
⬘
= ␦共␻ − ␻⬘兲␦ f f ⬘␦ pp⬘. The mode function e共␮兲 is given in Ref.
关15兴 共see Appendix A兲. The normalization of e共␮兲 is given by
冕 冕
2␲
d␸
⬁
nrf2兩e共␮兲兩2rdr = 1.
共5兲
0
0
Here nrf共r兲 = n1 for r ⬍ a, and nrf共r兲 = n2 for r ⬎ a.
Unlike the case of guided modes, in the case of radiation
modes, the longitudinal propagation constant ␤ for each
value of ␻ can vary continuously, from −kn2 to kn2 共with
k = ␻ / c兲. We label each radiation mode by the index
␯ = 共␻ , ␤ , m , p兲, where m is the mode order and p is the mode
polarization. When we quantize the field in the radiation
共+兲
in the
modes, we obtain the following expression for Erad
interaction picture:
共+兲
= i 兺 mp
Erad
冕 冕 冑
⬁
d␻
0
kn2
d␤
−kn2
ប␻
a␯e共␯兲e−i共␻t−␤z−m␸兲 .
4␲⑀0
共6兲
Here a␯ is the respective photon annihilation operator, and
e共␯兲 = e共␯兲共r , ␸兲 is the electric-field profile function of the radiation mode ␯ in the classical problem. The operators a␯ and
a␯† satisfy the continuous-mode bosonic commutation rules
关a␯ , a␯† 兴
⬘
= ␦共␻ − ␻⬘兲␦共␤ − ␤⬘兲␦mm⬘␦ pp⬘. The mode function e共␯兲 is given
in Ref. 关15兴 共see Appendix B兲. The normalization of e共␯兲 is
given by
冕 冕
2␲
d␸
⬁
0
0
nrf2关e共␯兲e共␯⬘兲*兴␤=␤⬘,m=m⬘,p=p⬘rdr = ␦共␻ − ␻⬘兲.
共3兲
共7兲
We do not take into account the evanescent modes, which do
not contribute to the spontaneous emission process 关3兴. We
use the cylindrical coordinates 共r , ␸ , z兲 with z as the axis of
the fiber. In view of the very low losses of silica in the
wavelength range of interest, we neglect material absorption.
The continuum field quantization follows the procedures
presented in Ref. 关14兴. Regarding the guided modes, we assume that the single-mode condition 关15兴 is satisfied for a
finite bandwidth of the field frequency ␻ around the cesium
D2-line frequency ␻0. We label each guided mode by an
index ␮ = 共␻ , f , p兲, where f = + , − denotes forward or backward propagation direction, and p = + , − denotes the counterclockwise or clockwise rotation of polarization. When we
Assume that the atom is located at a point 共r , ␸ , z兲. The
Hamiltonian for the atom-field interaction in the dipole and
rotating-wave approximations is given by
冕
兺冕 冕
Hint = − iប 兺
fpeg
⬁
†
d␻ G␮eg␴ge
a␮e−i共␻−␻0兲t
0
⬁
− iប
mpeg
0
d␻
kn2
†
d␤ G␯eg␴ge
a␯e−i共␻−␻0兲t + H.c..
−kn2
共8兲
Here the coefficients G␮eg and G␯eg characterize the coupling
of the atomic transition e ↔ g with the guided mode
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PHYSICAL REVIEW A 72, 032509 共2005兲
SPONTANEOUS EMISSION OF A CESIUM ATOM NEAR…
␮ = 共␻ , f , p兲 and the radiation mode ␯ = 共␻ , ␤ , m , p兲, respectively. Their expressions are
G␮eg =
G␯eg =
冑
冑
共r兲
␥ee
=
␥共r兲 .
⬘ 兺 ee⬘gg
共l兲
, e共l ␮0兲,
In terms of the spherical tensor components 关13兴 deg
共␯0兲
共␮0兲
共␯0兲
and el of the vectors deg, e , and e , respectively, we
obtain
␻␤⬘
共deg · e共␮兲兲ei共f ␤z+p␸兲 ,
4 ␲ ⑀ 0ប
␻
共deg · e共␯兲兲ei共␤z+m␸兲 .
4 ␲ ⑀ 0ប
共g兲
␥ee
=
⬘gg⬘
共9兲
共r兲
␥ee
=
⬘gg⬘
III. CHARACTERISTICS OF SPONTANEOUS
EMISSION
The solutions to the Heisenberg equations for the photon
operators a␮ and a␯ can be written as
a␮共t兲 = a␮共t0兲 + 兺 G␮* eg
eg
a␯共t兲 = a␯共t0兲 + 兺 G␯*eg
eg
冕
t
冕
Ull⬘ =
共10兲
t0
1
兺 ⌫ee⬘␴ge⬘ + ␰ˆ ge ,
2e
共l兲 ⬘
共− 1兲l+l⬘deg
de⬘g⬘ U−l,−l⬘ ,
l,l⬘=0,±1
␻0
兺
2⑀0ប mp
共11兲
共g兲
␥ee
=
共r兲
共r兲
␥ee
=
共r兲
⌫ee⬘gg⬘ = ␥ee⬘gg⬘ + ␥ee⬘gg⬘
共12兲
characterize the total spontaneous emission and ␰ij are the
共g兲
共g兲
noise operators. The coefficients ␥ee and ␥ee gg describe
⬘
⬘ ⬘
spontaneous emission into guided modes. They are given by
共g兲
共g兲
␥ee = 2␲兺 fpgG␮0egG␮* e⬘g and ␥ee gg = 2␲兺 fpG␮0egG␮* e⬘g⬘,
共r兲
0
where ␮0 = 共␻0 , f , p兲. The coefficients ␥ee and ␥ee gg de⬘
⬘ ⬘
scribe spontaneous emission into radiation modes. They are
共r兲
共r兲
k0n2
given by ␥ee = 2␲兺mpg兰−k
d␤ G␯0egG␯* e⬘g and ␥ee gg
0n2
⬘
⬘ ⬘
0
k0n2
= 2␲兺mp兰−k
d␤ G␯0egG␯* e⬘g⬘, where ␯0 = 共␻0 , ␤ , m , p兲. We
0n2
0
find the relations
共g兲
␥ee
=
␥共g兲 ,
⬘ 兺 ee⬘gg
g
k0n2
−k0n2
共␯0兲*
d␤ e共l ␯0兲el
⬘
共15兲
.
共g兲
⬘
guided and radiation modes, respectively. When we use the
symmetry properties of the mode functions, we find
⌫ee⬘ = ␥ee⬘ + ␥ee⬘ ,
共r兲
冕
⬘
2␻0␤0⬘
兺 兩共deg · e共␻0,+,+兲兲兩2 ,
⑀ 0ប g
Here the coefficients
⬘ ⬘
␻0␤0⬘
兺 e共␮0兲e共l⬘␮0兲* ,
2⑀0ប fp l
Equations 共12兲–共15兲 together with the mode functions e共␮兲
and e共␯兲, given in Appendixes A and B, respectively, constitute explicit expressions for the decay parameters of atomic
populations as well as atomic coherences. These equations as
well as the Heisenberg-Langevin equations 共11兲 are the key
analytical results of our paper.
Note that, due to the selection rules, the spherical tensor
共l兲
is nonzero only for l = M e − M g. On the other
component deg
共g兲
hand, due to the cylindrical symmetry, the functions Ull and
ee⬘
0
共14兲
共r兲
and ␥ee are nonzero only for M e⬘ = M e , M e ± 2. The param⬘
共g兲
共r兲
and ␥ee
describe the spontaneous emission from the
eters ␥ee
excited magnetic sublevel 兩e典 = 兩F⬘ = 5M ⬘典 with M ⬘ = M e into
␴˙ gg⬘ = 兺 ⌫ee⬘gg⬘␴ee⬘ + ␰ˆ gg⬘ .
⬘
共r兲
⬘
⬙
共g兲
共l 兲*
共r兲
1
兺 共⌫e⬘e⬙␴ee⬙ + ⌫e⬙e␴e⬙e⬘兲 + ␰ˆ ee⬘ ,
2e
共g兲
共g兲
Ull are zero for 共l = 0 , l⬘ = ± 1兲 and 共l = ± 1 , l⬘ = 0兲. Hence, ␥ee
⬘
␴˙ ee⬘ = −
兺
共l 兲*
l,l⬘=0,±1
共g兲
共r兲
Assume that the field is initially in the vacuum state. Since
the continuum of the guided and radiation modes is broadband around the atomic frequency, the Markoff approximation ␴ge共t⬘兲 = ␴ge共t兲 can be applied to describe the back action
of the second terms in Eqs. 共10兲 on the atom. We neglect the
vacuum frequency shifts. Then, for the atomic operators
␴ij = 兩i典具j兩, we obtain the Heisenberg-Langevin equations
␴˙ ge = −
共l兲 ⬘
共− 1兲l+l⬘deg
de⬘g⬘ U−l,−l⬘ ,
Ull⬘ =
dt⬘␴ge共t⬘兲ei共␻−␻0兲t⬘ ,
dt⬘␴ge共t⬘兲ei共␻−␻0兲t⬘ .
兺
where
t0
t
共13兲
g
2␻0
兺兺
⑀ 0ប g m
冕
k0n2
d␤ 兩共deg · e共␻0,␤,m,+兲兲兩2 .
共16兲
0
Here e共␻0,+,+兲 is the mode function of the fundamental guided
mode ␮ = 共␻ , f , p兲 with ␻ = ␻0 共atomic resonant frequency兲,
f = + 共forward direction兲, and p = + 共counterclockwise polarization兲, while e共␻0,␤,m,+兲 is the mode function of the radiation
mode ␯ = 共␻ , ␤ , m , p兲 with the frequency ␻ = ␻0 and the polarization p = +. The total decay rate of the population of 兩e典
is
共g兲
共r兲
+ ␥ee
.
⌫ee = ␥ee
共17兲
The summation over the lower-sublevel index g in Eqs.
共16兲 is a consequence of the fact that the decay rate of the
population of an arbitrary level is the sum of the decay rates
of the associated downward transitions. The magnitudes and
orientations of the dipoles of the partial transitions are determined by the quantum numbers. They can not be arbitrary.
032509-3
PHYSICAL REVIEW A 72, 032509 共2005兲
LE KIEN et al.
They are different from each other. To calculate the population decay rate of a level of a specific realistic atom, we need
to calculate and sum up the decay rates of the associated
downward transitions using the quantum numbers of the specific atomic level structure. Due to the split into and summation over the partial transitions, whose dipoles are complex
and differing in magnitude and orientation, the typical absolute and relative magnitudes of the population decay rate of a
realistic multilevel atom can be substantially different from
that for the case of a two-level atom. This is the essence of
our treatment as well as most other treatments for the population decay rates of realistic multilevel atoms. In addition,
off-diagonal elements such as ⌫ee⬘ with e ⫽ e⬘ and ⌫ee⬘gg⬘
with e ⫽ e⬘ or g ⫽ g⬘ do not exist in the case of two-level
atoms. They describe the decay of cross-level coherences
and arise only in the framework of a multilevel-atom model.
The knowledge of both diagonal and off-diagonal types of
decay characteristics is important for the studies of absorption and emission properties of the multilevel atom 关7兴.
Our formalism allows the atom to be inside or outside the
fiber. It also allows the atom to have, in principle, an arbitrary level structure. Our results, when reduced to the case of
a two-level atom inside or outside a passive fiber, are in
perfect agreement with other studies 关3,4兴. The agreement
with the results of Ref. 关3兴 has been confirmed analytically
and numerically. The agreement with the results of Ref. 关4兴
has been checked numerically. For a multilevel atom in free
space, we again have agreement with the previous results 关7兴.
Henceforth, as a typical example, we focus our attention only
to the case of a cesium atom in the outside of a thin fiber.
IV. NUMERICAL RESULTS
In what follows, we demonstrate the results of our numerical calculations for the decay characteristics of magnetic
sublevels of a cesium atom in the presence of a thin fiber. For
simplicity, we show only the numerical results for the diagonal decay coefficients, which describe the rates of population decay. We do not present the numerical results for the
off-diagonal decay coefficients because we cannot gain much
insight from showing them graphically. However, we emphasize that these parameters can be easily calculated from Eqs.
共12兲–共15兲 and that the knowledge of these parameters is important for investigating the dynamics of cesium atoms interacting with guided fields.
We plot in Fig. 2 the spatial dependence of the spontaneous emission rates for various magnetic sublevels
6P3/2F⬘ = 5M ⬘ into guided modes, radiation modes, and both
types of modes. The fiber radius a is chosen to be 200 nm,
which is in the optimal range for producing optical trapping
potentials 关10,11兴. According to Fig. 2, different magnetic
sublevels of the same state 6P3/2 have different decay rates in
the vicinity of the fiber surface, unlike the case of atomic
cesium in free space. The presence of the fiber produces
substantial decay rates into guided modes. Indeed, when the
atom is positioned on the fiber surface, the decay rates into
共g兲
guided modes ␥ee
vary from 0.31⌫0 for M e = 0 to 0.48⌫0 for
M e = ± 5 关see Fig. 2共a兲兴. Here ⌫0 = 33⫻ 106 s−1 is the decay
rate of the cesium state 6P3/2 in free space. In addition, the
FIG. 2. Spontaneous emission rates for various magnetic sublevels 6P3/2F⬘ = 5M ⬘ into 共a兲 guided modes, 共b兲 radiation modes,
and 共c兲 both types of modes as functions of the position of the atom.
Different lines in each plot correspond to different values 兩M ⬘兩 = 0,
1, 2, 3, 4, and 5. The fiber radius is a = 200 nm. The wavelength of
the atomic transition is ␭0 = 852 nm. The refractive indices of the
fiber and the vacuum clad are n1 = 1.45 and n2 = 1, respectively. The
rates are normalized to the free-space decay rate ⌫0.
共r兲
decay rates into radiation modes ␥ee
and the total decay rates
⌫ee are enhanced from the free-space rate ⌫0 by small fac共r兲
tors. The maximal values of ␥ee
/ ⌫0 and ⌫ee / ⌫0 are around
1.2 and 1.6, respectively 关see Figs. 2共b兲 and 2共c兲兴. As expected, the effect of the fiber on the decay rates is largest for
the atom on the fiber surface. When the atom is far away
共g兲
共r兲
from the fiber, ␥ee
reduces to zero while ␥ee
and ⌫ee approach the free-space value ⌫0. The small oscillations around
the value of unity in Fig. 2共b兲 can be ascribed to the
constructive/destructive interference due to reflections from
the fiber surface 关3兴. Note that the decay rates of the sublevels M e and −M e are the same. Therefore, the maximum number of lines in each plot is six. However, since the difference
between the decay rates for M e = 0 and 兩M e兩 = 1 is very small,
we can distinguish only five lines in each plot of Fig. 2.
In Fig. 3, we plot the spontaneous emission rates for various magnetic sublevels as functions of the fiber size parameter k0a for the atom on the fiber surface. The separation
between the curves in the figure confirms again that different
magnetic sublevels have different decay rates due to the
presence of the fiber. We observe that, in the shown range of
共g兲
have a maximum
k0a, the decay rates into guided modes ␥ee
关see Fig. 3共a兲兴, while, depending on the magnetic sublevels,
共r兲
the decay rates into radiation modes ␥ee
have one or two
minima 关see Fig. 3共b兲兴. Such a behavior indicates that there
can exist an optimal value of the fiber size parameter for
which the fraction of decay due to guided modes reaches its
largest value 关see also Fig. 4共b兲 below兴. The overall behavior
of the curves in Fig. 3共c兲 implies that the total decay is less
sensitive to k0a as compared to that for the guided modes.
In order to get deeper insight into the decay into guided
共g兲
modes, we plot the ratio ␥ee
/ ⌫ee for various magnetic sublevels in Figs. 4共a兲 and 4共b兲 as a function of r / a and k0a,
respectively. We observe from Fig. 4共a兲 that, in the close
vicinity of the fiber surface, the fractional decay rates into
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PHYSICAL REVIEW A 72, 032509 共2005兲
SPONTANEOUS EMISSION OF A CESIUM ATOM NEAR…
to the results for a two-level atom with a specific real radial,
axial, or tangential dipole.
V. SUMMARY
FIG. 3. Spontaneous emission rates for various magnetic sublevels into 共a兲 guided modes, 共b兲 radiation modes, and 共c兲 both
types of modes as functions of the fiber radius. The atom is located
on the fiber surface. Other parameters are as in Fig. 2.
guided modes are substantial, in the range from 0.2 共for
M e = 0兲 to 0.28 共for M e = ± 5兲. With increasing distance of the
共g兲
atom from the surface, ␥ee
/ ⌫ee quickly reduces to zero, as
共g兲
expected. As can be seen from Fig. 4共b兲, ␥ee
/ ⌫ee is sensitive
to k0a, reaching a maximum at around k0a = 1.45. This size
parameter corresponds to a fiber radius of about 200 nm,
which is in the optimal range for producing optical trapping
potentials 关10,11兴. For such a parameter, a significant fraction 共up to 28% for M e = ± 5兲 of spontaneous emission by the
atom can be channeled into guided modes. The above maximal value, obtained for a cesium atom, is substantially
smaller than the corresponding value for a two-level atom
with radial dipole 关4兴. Such a reduction is a consequence of
the degeneracy of the ground state of cesium as well as the
complexity of its dipole. Indeed, the dipoles of the partial
transitions are complex and differing in magnitude and orientation. Therefore, the results for cesium do not correspond
In conclusion, we have studied the spontaneous emission
by a cesium atom in the vicinity of a subwavelengthdiameter fiber. We have shown that the confinement of the
guided modes and the degeneracy of the magnetic sublevels
play an important role in deciding the spontaneous emission
from the cesium atom. When the fiber radius is about
200 nm, the fractional decay rates into guided modes can be
significant 共up to 28%兲. Our results for this realistic “atom
+ fiber” system can be used for future experiments on generating and guiding a few photons along a thin fiber. They may
also find applications as atom nanoprobes as well as efficient
subwavelength fiber couplers.
ACKNOWLEDGMENTS
This work was carried out under the 21st Century COE
program on “Coherent Optical Science.”
APPENDIX A: MODE FUNCTIONS OF THE
FUNDAMENTAL GUIDED MODES
For the guided modes, the longitudinal propagation constant ␤ is determined by the fiber eigenvalue equation 关15兴
n2 + n2 K⬘共qa兲
1
J0共ha兲
=− 1 2 2 1
+ 2 2
haJ1共ha兲
2n1 qaK1共qa兲 h a
−
+
再冋
冉
1
1
␤2
2 2
2 2 + 2 2
q
a
h
a
n 1k
册
冊冎
2
2
1/2
.
共A1兲
Here the parameters h = 共n21k2 − ␤2兲1/2 and q = 共␤2 − n22k2兲1/2
characterize the fields inside and outside the fiber, respectively. The notation Jn and Kn stand for the Bessel functions
of the first kind and the modified Bessel functions of the
second kind, respectively.
The mode functions of the electric parts of the fundamental guided modes 关15兴 are given, for r ⬍ a, by
er共␮兲 = iA
q K1共qa兲
关共1 − s兲J0共hr兲 − 共1 + s兲J2共hr兲兴,
h J1共ha兲
e␸共␮兲 = − pA
q K1共qa兲
关共1 − s兲J0共hr兲 + 共1 + s兲J2共hr兲兴,
h J1共ha兲
ez共␮兲 = fA
FIG. 4. Fractional decay rates for various magnetic sublevels
into guided modes as functions of 共a兲 the atomic position and 共b兲
the fiber radius with the atom on the fiber surface. Other parameters
are as in Fig. 2. Note that the peaks in 共b兲 occur around k0a = 1.45,
which correspond to a = 200 nm for the cesium D2 line.
n21 − n22 K1⬘共qa兲
2n21 qaK1共qa兲
2q K1共qa兲
J 共hr兲,
␤ J1共ha兲 1
and, for r ⬎ a, by
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er共␮兲 = iA关共1 − s兲K0共qr兲 + 共1 + s兲K2共qr兲兴,
e␸共␮兲 = − pA关共1 − s兲K0共qr兲 − 共1 + s兲K2共qr兲兴,
共A2兲
PHYSICAL REVIEW A 72, 032509 共2005兲
LE KIEN et al.
ez共␮兲 = fA
2q
K 共qr兲.
␤ 1
共A3兲
Here, the parameter s is defined as s = 共1 / q2a2
+ 1 / h2a2兲 / 关J1⬘共ha兲 / haJ1共ha兲 + K1⬘共qa兲 / qaK1共qa兲兴, and the coefficient A is determined from the normalization condition.
To normalize the guided mode functions, we need to calculate the constant
冕 冕
2␲
N␮ =
d␸
0
⬁
nrf2兩e共␮兲兩2rdr.
共A4兲
0
We find
N␮ = 2␲A2a2共n21 P1 + n22 P2兲,
where
P1 =
q2K21共qa兲
h2J21共ha兲
再
冋
兺冋
␻␮0
i
共j兲
D jHm
␤qC jHm共j兲⬘共qr兲 + im
共qr兲 ,
2 兺
r
q j=1,2
e␸共␯兲 =
␤
i
共j兲
共j兲
⬘共qr兲 ,
im C jHm
共qr兲 − q␻␮0D jHm
r
q2 j=1,2
ez共␯兲 =
C j = 共− 1兲 j
共A6兲
Vj =
共A7兲
APPENDIX B: MODE FUNCTIONS OF THE RADIATION
MODES
For the radiation modes, we have −kn2 ⬍ ␤ ⬍ kn2. The
characteristic parameters for the field in the inside and outside of the fiber are h = 冑k2n21 − ␤2 and q = 冑k2n22 − ␤2, respectively. The mode functions of the electric parts of the radiation modes 关15兴 are given, for r ⬍ a, by
冋
冋
册
册
␻␮
i
⬘ 共hr兲 + im 0 BJm共hr兲 ,
2 ␤hAJm
r
h
= AJm共hr兲,
mk␤ 2
共j兲*
共n − n21兲Jm共ha兲Hm
共qa兲,
ah2q2 2
n21
n2
⬘ 共ha兲Hm共j兲*共qa兲 − 2 Jm共ha兲Hm共j兲*⬘共qa兲.
Jm
h
q
Lj =
冕 冕
2␲
0
d␸
⬁
0
nrf2关e共␯兲e共␯⬘兲*兴␤=␤⬘,m=m⬘rdr = N␯␦ pp⬘␦共␻ − ␻⬘兲.
共B5兲
This leads to
␩ = ⑀ 0c
冑
n22兩V j兩2 + 兩L j兩2
兩V j兩2 + n22兩M j兩2
共B1兲
N␯ =
and, for r ⬎ a, by
冉
共B6兲
.
The normalization constant N␯ is given by
关1兴 S. A. Maier et al., Nat. Mater. 2, 229 共2003兲; T. W. Ebbesen et
al., Nature 共London兲 391, 667 共1998兲.
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关3兴 T. Søndergaard and B. Tromborg, Phys. Rev. A 64, 033812
共2001兲.
关4兴 V. V. Klimov and M. Ducloy, Phys. Rev. A 69, 013812 共2004兲.
关5兴 See, for example, Cavity Quantum Electrodynamics, edited by
共B4兲
We specify two polarizations by choosing B = i␩A and
B = −i␩A for p = + and p = −, respectively. The orthogonality
of the modes requires
␤
i
⬘ 共hr兲 ,
e␸共␯兲 = 2 im AJm共hr兲 − h␻␮0BJm
r
h
ez共␯兲
共B3兲
1
1
⬘ 共ha兲Hm共j兲*共qa兲 − Jm共ha兲Hm共j兲*⬘共qa兲,
M j = Jm
h
q
2
er共␯兲 =
i ␲ q 2a
共i⑀0cAV j − BM j兲,
4
where
P2 = 共1 − s兲2关K21共qa兲 − K20共qa兲兴 + 共1 + s兲2关K1共qa兲K3共qa兲
q
关K 共qa兲K2共qa兲 − K21共qa兲兴.
␤2 0
i ␲ q 2a
共AL j + i␮0cBV j兲,
4n22
D j = 共− 1兲 j−1
and
− K22共qa兲兴 + 2
共B2兲
The coefficients C j and D j are related to the coefficients A
and B as 关3兴
共1 − s兲2关J20共ha兲 + J21共ha兲兴 + 共1 + s兲2关J22共ha兲
冎
兺 C jHm共j兲共qr兲.
j=1,2
共A5兲
h2
− J1共ha兲J3共ha兲兴 + 2 2 关J21共ha兲 − J0共ha兲J2共ha兲兴
␤
册
册
er共␯兲 =
冊
8␲␻ 2 2 ␮0
n2兩C j兩 + 兩D j兩2 .
q2
⑀0
共B7兲
P. R. Berman 共Academic, New York, 1994兲; Optical Processes
in Microcavities, edited by R. Chang and A. Campillo 共World
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关6兴 V. V. Klimov, M. Ducloy, and V. S. Letokhov, Eur. Phys. J. D
20, 133 共2002兲.
关7兴 See S. Chang and V. Minogin, Phys. Rep. 365, 65 共2002兲, and
the references therein.
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SPONTANEOUS EMISSION OF A CESIUM ATOM NEAR…
关8兴 S. Chu, Rev. Mod. Phys. 70, 685 共1998兲; C. Cohen-Tannoudji,
ibid. 70, 707 共1998兲; W. D. Phillips, ibid. 70, 721 共1998兲.
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Laser Light, Vol. 18 of Laser Science and Technology 共Harwood Academic, New York, 1995兲, p. 115; P. Meystre, Atom
Optics 共Springer, New York, 2001兲, p. 311.
关10兴 V. I. Balykin, K. Hakuta, F. L. Kien, J. Q. Liang, and M.
Morinaga, Phys. Rev. A 70, 011401共R兲 共2004兲.
关11兴 Fam Le Kien, V. I. Balykin, and K. Hakuta, Phys. Rev. A 70,
063403 共2004兲.
关12兴 H. J. Metcalf and P. van der Straten, Laser Cooling and Trap-
ping 共Springer, New York, 1999兲.
关13兴 See, for example, B. W. Shore, The Theory of Coherent Atomic
Excitation 共Wiley, New York, 1990兲.
关14兴 C. M. Caves and D. D. Crouch, J. Opt. Soc. Am. B 4, 1535
共1987兲; K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J.
Shepherd, Phys. Rev. A 42, 4102 共1990兲; P. Domokos, P.
Horak, and H. Ritsch, ibid. 65, 033832 共2002兲.
关15兴 See, for example, D. Marcuse, Light Transmission Optics
共Krieger, Malabar, FL, 1989兲; A. W. Snyder and J. D. Love,
Optical Waveguide Theory 共Chapman and Hall, New York,
1983兲.
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