Вплив дифузії атомів на форму лінії темного резонансу у просторово обмежених лазерних полях
Запропоновано дифузiйну модель для недавно вiдкритого дифузiйно-iндукованого звуження Рамзея, що виникає пiд час дифузiї атомiв у комiрцi з буферним газом у полi лазерного випромiнювання. Рiвняння дифузiї для когерентностi метастабiльних станiв, пов’язаних зi збудженим станом лазерним випромiнювання...
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| Date: | 2010 |
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| Format: | Article |
| Language: | Ukrainian |
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Відділення фізики і астрономії НАН України
2010
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/13428 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Вплив дифузії атомів на форму лінії темного резонансу у просторово обмежених лазерних полях / В.І. Романенко, О.В. Романенко, Л.П. Яценко // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 394-404. — Бібліогр.: 22 назв. — укр. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859574965703016448 |
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| author | Романенко, В.І. Романенко, О.В. Яценко, Л.П. |
| author_facet | Романенко, В.І. Романенко, О.В. Яценко, Л.П. |
| citation_txt | Вплив дифузії атомів на форму лінії темного резонансу у просторово обмежених лазерних полях / В.І. Романенко, О.В. Романенко, Л.П. Яценко // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 394-404. — Бібліогр.: 22 назв. — укр. |
| collection | DSpace DC |
| description | Запропоновано дифузiйну модель для недавно вiдкритого дифузiйно-iндукованого звуження Рамзея, що виникає пiд час дифузiї атомiв у комiрцi з буферним газом у полi лазерного випромiнювання. Рiвняння дифузiї для когерентностi метастабiльних станiв, пов’язаних зi збудженим станом лазерним випромiнюванням рiзної частоти у трирiвневiй схемi взаємодiї атома з полем, отримано у наближеннi сильних зiткнень Дослiджено залежнiсть форми лiнiї поглинання поблизу максимуму пропускання випромiнювання однiєї з частот вiд вiдстроювання вiд двофотонного резонансу для рiзних геометричних конфiгурацiй комiрки.
Предложена диффузионная модель для недавно открытого диффузионно-индуцированного сужения Рамзея, возникающего при диффузии атомов в ячейке с буферным газом в поле лазерного излучения. Уравнение диффузии для когерентности метастабильных состояний, связанных с возбужденным состоянием лазерным излучением разной частоты в трехуровневой схеме взаимодействия атома с полем, получено в приближении сильных столкновений. Исследована зависимость формы линии поглощения вблизи максимума пропускания излучения одной из частот от отстройки от двухфотонного резонанса для разных геометрических конфигураций ячейки.
We propose a diffusion model for the recently discovered diffusioninduced Ramsey narrowing arising when atoms diffuse in a buffergas cell in the laser radiation field. The diffusion equation for the coherence of metastable states coupled with an excited state by laser radiation of different frequencies in a three-level scheme of the atom-field interaction is obtained in the strong-collision approximation. The dependence of the shape of an absorption line near the transmission maximum of one of the frequencies on the two-photon resonance detuning for various geometries of the cell is investigated.
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| first_indexed | 2025-11-27T01:33:16Z |
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| fulltext |
INFLUENCE OF DIFFUSION OF ATOMS
INFLUENCE OF DIFFUSION OF ATOMS ON THE DARK
RESONANCE LINESHAPE IN SPATIALLY BOUNDED
LASER FIELDS
V.I. ROMANENKO,1 A.V. ROMANENKO,2 L.P. YATSENKO1
1Institute of Physics, Nat. Acad. of Sci. of Ukraine
(46, Nauky Ave., Kyiv 03680, Ukraine; e-mail: vr@ iop. kiev. ua )
2Taras Shevchenko National University of Kyiv
(2, Academician Glushkov Ave., Kyiv 03022, Ukraine)
PACS 42.50.Gy, 42.50.Hz;
32.80.Qk, 33.80.Be
c©2010
We propose a diffusion model for the recently discovered diffusion-
induced Ramsey narrowing arising when atoms diffuse in a buffer-
gas cell in the laser radiation field. The diffusion equation for the
coherence of metastable states coupled with an excited state by
laser radiation of different frequencies in a three-level scheme of
the atom-field interaction is obtained in the strong-collision ap-
proximation. The dependence of the shape of an absorption line
near the transmission maximum of one of the frequencies on the
two-photon resonance detuning for various geometries of the cell
is investigated.
1. Introduction
In a three-level system subjected to two laser fields
coupling two metastable states (or a metastable state
and a stable one) with an excited one, a dark or light-
nonabsorbing state, namely a coherent superposition of
two metastable states, can be formed. This phenomenon
is called a coherent population trapping (CPT). The
condition of the formation of a CPT state is the two-
photon resonance under interaction of an atom with
light, where the difference between the frequencies of two
laser fields is equal to the frequency of a transition be-
tween metastable states. If this condition is realized, one
observes an abrupt decrease of the fluorescence intensity
of the atom in the laser radiation field [1–3]. Due to the
CPT phenomenon, it is possible to create a window in
the absorption spectrum. As a result, light can propa-
gate almost without losses through a medium that ab-
sorbs light under usual conditions, which represents the
well-known phenomenon of electromagnetically induced
transparency (EIT) [4]. In addition, dark resonances
are used for the light slowing down [5] and in the con-
struction of compact laser frequency standards [6] and
underlie the effective method of population transfer be-
tween different states of atoms or molecules – stimulated
Raman adiabatic passage (STIRAP) [7]. The resonance
width depends on both the coherence decrease rate for
the lower states and on other factors, particularly on the
pattern of atomic motion in a buffer-gas cell. The latter
aspect will be the focus of our attention in this work.
In the case where atoms move through a laser beam
of finite width, the role of the coherence time is played
by the residence time of an atom in the field. If the cell
contains a buffer gas in addition to active atoms, their
residence time in the laser beam increases. As a result,
narrow resonances with a width of the order of tens of
hertzs are registered in buffer-gas cells [9]. The authors
of works [8–10] emphasize the role of the buffer gas in the
experiments on coherent population trapping in a three-
level system, though it is considered that, after atoms
have left the region of interaction with radiation, they
do not return there anymore.
A more detailed description of the process of atom-
field interaction must take into account that, having left
the region of interaction with the field, the atom can
return there again [11, 12], so that atoms can interact
with radiation several times before the loss of coher-
ence. Thus, diffusion of atoms in a buffer gas essen-
tially affects their response to the resonance excitation
by the laser field. If the coherence relaxation time of
the metastable states considerably exceeds the time till
the repeated atom-field interaction, the atom can get
back without loss of coherence after having spent some
time in a dark region (beyond the beam). As a result, it
is worth expecting the narrowing of the resonance line.
This phenomenon was called a diffusion-induced Ram-
sey narrowing [13] (by analogy with the Ramsey method
of separated oscillating fields [14, 15]).
The diffusion-induced Ramsey narrowing of the trans-
mission spectrum in the case of EIT observations was
investigated in [13] adducing the results of experiments
with rubidium vapor and neon serving as a buffer gas,
as well as the results of theoretical calculations. Accord-
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 393
V.I. ROMANENKO, A.V. ROMANENKO, L.P. YATSENKO
Ωa
Ωb
1
2
3
∆a
∆a
Fig. 1. Three-level system: Ωa and Ωb are the Rabi frequencies
ing to [13], the necessity of taking the diffusion process
into account depends on the laser-beam diameter – the
contribution made into the signal by atoms that had
spent some time beyond the beam and got back be-
comes determinative with decrease in the beam diame-
ter. The same authors subsequently published a detailed
description of the developed theory [16]. Experimental
results and theoretical calculations unambiguously con-
firm the physical interpretation of the phenomenon of
diffusion-induced Ramsey narrowing of the transmission
spectrum observed in [13]. In particular, with increase in
the laser-beam diameter or the buffer gas pressure, the
shape of the spectral line changes from the non-Lorentz
to Lorentz one in accordance with a decrease of the con-
tribution made by atoms that have come back from the
region beyond the laser field. In the cited works, the
motion of atoms is described in the form of the Ramsey
sequences: in each of them, an atom spends the time tin1
in the radiation field, moves in the dark region during
the time tout
2 , returns back to the radiation region for
the time tin3 , and so on. In order to find the transmission
spectrum, the density matrix describing the Ramsey se-
quence was integrated with the probability distribution,
i.e. one performed averaging over all possible trajecto-
ries. The probability distribution was obtained in [16]
from the diffusion equation.
We propose an alternative approach to the descrip-
tion of the diffusion-induced Ramsey narrowing. It is
based on the diffusion equation obtained from the ini-
tial motion equations for the density matrix on the basis
of the strong-collision approximation [8, 17]. The trans-
mission spectrum will be directly obtained in the equi-
librium approach instead of averaging over trajectories.
As compared to works [13] and [16], we consider the re-
laxation at cell walls (supposing that the coherence is
broken due to a collision of an atom with the wall) and
take a realistic (Gaussian) intensity distribution in the
radial direction. In addition, we also take the finiteness
of the gas cell in the direction parallel to the laser beam
into account. In order to understand how the dimension
of the problem influences the result, we also consider the
one-dimensional case where the cell is infinite in the di-
rection of the laser beam, while the beam itself is infinite
along one of the transverse coordinates.
2. Basic Equations
Let us consider a gas of three-level atoms with the
excited state |2〉 and the metastable lower states |1〉
and |3〉. A field with the frequency ωa couples the states
|1〉 and |2〉, while that with the frequency ωb couples
the states |3〉 and |2〉 (see Fig. 1). The interaction of
these fields with atoms is described by the Rabi frequen-
cies Ωa = µ12 ·Ea/~ and Ωb = µ32 ·Eb/~, respectively.
As the beams a and b are spatially bounded, the Rabi
frequencies depend on the position r of an atom in space.
The wave vectors will be considered close in magnitude:
ka ' kb ' k.
The equations for nondiagonal elements of the den-
sity matrix in the rotating-wave approximation have the
form
ρ̇12 = i(Δa − kv)ρ12 +
iΩ∗a
2
(ρ22 − ρ11)−
− iΩ
∗
b
2
ρ13 +
(
∂ρ12
∂t
)
relax
+
(
∂ρ12
∂t
)
coll
, (1)
ρ̇23 = −i(Δb − kv)ρ23 +
iΩa
2
ρ13+
+
iΩb
2
(ρ33 − ρ22) +
(
∂ρ23
∂t
)
relax
+
(
∂ρ23
∂t
)
coll
, (2)
ρ̇31 = −i(Δa −Δb)ρ31 −
iΩa
2
ρ∗23 +
iΩ∗b
2
ρ∗12+
+
(
∂ρ31
∂t
)
relax
+
(
∂ρ31
∂t
)
coll
. (3)
394 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
INFLUENCE OF DIFFUSION OF ATOMS
Here, the terms with the index “coll” describe the re-
laxation processes due to collisions of active atoms (in-
teracting with the field) with atoms of the buffer gas
resulting in a change of the velocity, while the coher-
ence is conserved. The terms with the index “relax” de-
scribe the rest of relaxation processes, Δa = ω21 − ωa
and Δb = ω23 − ωb are the one-photon detunings, and
Δω = Δa−Δb is the two-photon detuning. It is assumed
that(
∂ρ12
∂t
)
relax
= −Γ12ρ12,
(
∂ρ23
∂t
)
relax
= −Γ23ρ23,
where Γ12 and Γ23 stand for the coherence relaxation
rates for the transitions 2 → 1 and 2 → 3, respectively.
If Γ13 and Γ23 are sufficiently large, the collision terms
in Eqs. (1) and (2) can be neglected. The relaxation rate
for the forbidden transition |1〉 → |3〉 is non-zero due to
collisions between atoms,(
∂ρ31
∂t
)
relax
= −γ13ρ31. (4)
In the general case, the expression for the collision
term [18–20] can be presented as(
∂ρij(v,v ′, t)
∂t
)
coll
= −νρij(v,v ′, t)+
+
∫
Kij(v ′,v)ρij(r,v ′, t) dv ′ , (5)
where Kij(v ′,v) is the collision kernel, and ν is, in the
general case, a complex-valued quantity with the fre-
quency dimension. It can be interpreted as the collision
frequency in the case where the scattering amplitudes in
both states i and j are identical (see [18]).
The collision term can be simplified using the strong-
collision approximation (light atoms are scattered by
heavy particles [17]) and assuming that ν and Kij are
real, while the kernel Kij(v ′,v) does not depend on v ′,
i.e. the velocity of an atom v after a collision does not
depend on its velocity v ′ before it. In this case, the ve-
locity distribution (arbitrary in the general case) turns
into the Maxwellian distribution after only several colli-
sions, that is, an atom quickly forgets its initial velocity.
As was shown in [17], Kij(v) = νW (v), where W (v) is
the Maxwellian distribution. Thus [20], expression (5)
takes the form(
∂ρij
∂t
)
coll
= −ν
[
ρij −W (v)Nij
]
,
where Nij(r, t) =
∫
dv ρij(r,v, t) (6)
can be interpreted as the number of atoms with values
of ρij lying in the unit volume in the neighborhood of
the point r, and ν denotes the collision frequency [18].
In this approximation, collisions with the buffer gas
change only the external degrees of freedom of atoms.
Using the approximation
∣∣ d
dt
∣∣� Γ12,Γ23 and neglect-
ing the collision terms in (1) and (2), one can find the
stationary solutions for ρ12 and ρ23. After that, with
regard for the fact that the nondiagonal elements ρii are
close to the equilibrium values ρ(0)
ii (they differ from the
latter by a small quantity of the second order in the field
intensity), one obtains the equation for ρ31:
ρ̇31 = −(γ13 + iΔω)ρ31−
−
[
Ω∗aΩa
Γ23 − i(Δb − kv)
+
Ω∗bΩb
Γ12 + i(Δa − kv)
]
ρ31
−ΩaΩ∗b
4
[
ρ
(0)
11 − ρ
(0)
22
Γ12 + i(Δa − kv)
+
ρ
(0)
33 − ρ
(0)
22
Γ23 − i(Δb − kv)
]
+
+
(
∂ρ31
∂t
)
coll
. (7)
The second term on the right-hand side describes the
field broadening. It can be neglected in the case of weak
fields and large relaxation rates Γij .
In order to simplify the equation, we also neglect the
Doppler broadening in the third term considering that
Γij � Δa,b,Δω, kv.
Taking into account that the equilibrium elements of
the density matrix ρii are proportional to the distribu-
tion function W (v), we obtain a kinetic equation of the
Boltzmann type for ρ31.
In the stationary case of interest, it has the following
form:
(v ·∇)ρ(r,v) = −(ν + γ + iΔω)ρ(r,v)
+W (v)
[
λ(r) + νN(r)
]
. (8)
Here and below, we use the notations
ρ(r,v) = ρ31(r,v), γ = γ13,
N = N31, W (v) = W0e
−v 2/v20 ,
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 395
V.I. ROMANENKO, A.V. ROMANENKO, L.P. YATSENKO
where N31 is determined by (6), and W0 is the normal-
ization constant of the Maxwellian distribution,
λ(r) =
1
4
(
ρ
(0)
11
Γ12
+
ρ
(0)
33
Γ13
)
ΩaΩ∗b . (9)
Here, we took into account that ρ(0)
22 � ρ
(0)
11 , ρ(0)
22 � ρ
(0)
33 .
As one can see from (9), the function λ(r) ∼ Ωa(r)Ω∗b(r)
describes the transverse profile of the beams.
For the sake of simplicity, we introduce the substitu-
tion
α0 = γ + iΔω , α = ν + γ + iΔω = ν + α0 .
It is assumed that the collisions of atoms with walls re-
sult in the failure of the coherence between the lower
states, so that
ρ(r,v)|r∈S = 0,
where S denotes the surface confining the buffer-gas cell.
The shape of the spectral line is determined by the
function T (Δω) = Re [S(Δω)/S(0)], where S(Δω) has
the form
S(Δω) =
∫∫
dr dv λ(r)ρ(r,v) =
∫
λ(r)N(r) dr . (10)
In order to find it, it is necessary to obtain the function
ρ.
In the case of a laser beam with the Gaussian intensity
distribution in the plane normal to the direction of its
propagation, the expression for λ(r) in the cylindrical
coordinates takes the form
λ(r, ϕ, z) = λ0e
−r2/a2
, (11)
where λ0 is determined by (9), and r lies at the beam
axis.
In the strong-collision approximation, the time be-
tween collisions τν = 1/ν is small as compared to the
characteristic time of flight of an atom through the in-
teraction region τa = a/v0. Therefore, we consider that
ντa � 1.
In what follows, we investigate the effect of the beam
size and the distance to the walls of the gas cell on the
lineshape specified by the function ReS(Δω). The form
of this function depending on the dimension of the prob-
lem will be considered as well.
3. Solution for Infinite Region
For an infinite region, Eq. (8) can be solved with the
help of the Fourier transformation with respect to the
argument r:
ρ̂(k,v) =
∫
ρ(r,v) e−ik·r dr , N̂(k) =
∫
ρ̂(k,v) dv .
Hence,
N̂(k) =
λ̂(k)F̂ (k)
1− νF̂ (k)
, (12)
where the function
F̂ (k) =
∫
W (v) dv
α+ ikv
=
√
π
|k|v0
eα
2/k2v20 erfc
(
α
|k|v0
)
,
describes the Voigt profile. Let us find the signal S∞
using the properties of the Fourier transformation:
S∞(Δω) =
∫
drN(r)λ(r) =
1
(2π)n
∫
dk N̂(k)λ̂(k) ,
(13)
where n denotes the space dimension of the region (1, 2,
or 3). Substituting N from (12), we obtain
S∞(Δω) =
1
(2π)n
∫
λ̂2(k)F̂ (k)
1− νF̂ (k)
dk.
In some cases, the value of S(Δω) is mainly determined
by small k due to the factor λ̂2(k). That is why the func-
tion F̂ (k) can be replaced by the asymptotic expansion
in the quadratic approximation (see [21]):
S∞(Δω) ' 1
(2π)n
∫
λ̂2(k) dk
α0 + k2v20
2α
. (14)
It is valid for sufficiently small k, for which |k|v0/ν � 1.
3.1. One-dimensional case
In the given case, λ(x) = λ0e
−x2/a2
, and (14) has the
form
S(1)
∞ (Δω) =
λ2
0a
2
∞∫
−∞
e−u
2/2
α0 + u2/τD
du =
=
πλ2
0a
3
2
β
α0
eβ
2a2/2 erfc
(
βa√
2
)
, (15)
where
β2 =
2αα0
v2
0
, τD =
2αa2
v2
0
, β2a2 = τDα0
396 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
INFLUENCE OF DIFFUSION OF ATOMS
(the quantity τD will be interpreted below). In our ap-
proximation, ν � γ,Δω, that is why one can consider
τD = const. The right-hand side of (15) can be pre-
sented in the form of a superposition of the profiles
S(1)
∞ (Δω) ∼
∞∫
−∞
s(Δω, u)g(1)
∞ (u) du , (16)
where
s(Δω, u) =
γeff(u)
γeff(u) + iΔω
, γeff(u) = γ + u2/τD
with the weighting factor (independent of ω)
g(1)
∞ (u) =
1
γeff(u)
e−u
2/2 . (17)
The combination τ(u) = τD/u
2 can be called the effec-
tive diffusion time of an atom; τ(u) for “fast” atoms is
larger than that for “slow” ones.
The profile S(1)
∞ (Δω) can be described as the effective
Lorentzian with the center at the origin of coordinates
SL(Δω) =
Γ0
Γ0 + iΔω
.
Its width Γ0 is determined by the relation
Γ2
0 = −2S(0)
S′′(0)
.
Simple calculations for γτD � 1 yield
Γ2
0 =
8
3
γ2 ·
(
1−
√
2
π
√
γτD + . . .
)
. (18)
Therefore, we obtain Γ0 '
√
8
3 γ for small γ.
3.2. Two-dimensional case
Using the similar procedure for the function λ(r) =
λ0e
−r2/a2
in the polar coordinates, we obtain
S(2)
∞ (Δω) =
πλ2
0a
2
2
∞∫
0
ue−u
2/2 du
α0 + u2/τD
=
=
πλ2
0a
4
4α0
eβ
2a2/2 Ei1
(
β2a2
2
)
. (19)
Here, Ei1 denotes the integral first-order exponent1.
The obtained expression is similar to (15) and has the
same interpretation, though with the weighting function
g(2)
∞ (u) =
1
γeff(u)
ue−u
2/2 . (20)
The width of the effective Lorentzian for γτD � 1 has
the form
Γ2
0 = γ22
[
ln 2− γE − ln(γτD)
]
+ o(γτD) . (21)
4. Effective Diffusion Equation
4.1. Derivation of diffusion equation
According to (10), the complex signal S(Δω) can be ex-
pressed with the help of the zero-order momentum N(r)
of the distribution ρ(r) with respect to v. Let us find
the equation for the moments of higher orders N (k)(r).
First, we consider the one-dimensional case. Here,
N (k)(x) =
∞∫
−∞
vkρ(x, v) dv , N (0)(x) = N(x) .
Equation (8) presented in the one-dimensional case as
v
∂ρ(x, v)
∂x
= −αρ(x, v) +W (v)
[
λ(x) + νN (0)(x)
]
(22)
will be multiplied by vk and integrated over v. This
procedure yields a chain of equations for the moments
N (k)(x).
Writing down the obtained equations separately for
even and odd k and successively substituting the follow-
ing equation into the previous one m times, we obtain
the following expression for N (0):
αN (0) =
m∑
k=0
〈v2k〉
α2k
d2k
dx2k
[
λ(x) + νN (0)(x)
]
+
+
1
α2m+1
d2m+2N (2m+2)
dx2m+2
. (23)
The result for higher space dimensions will be similar,
though more complicated as the moments will be speci-
fied by tensors. A similar procedure yields
αN (0)(r) =
m∑
k=0
〈vi1 . . . vi2k
〉
α2k
×
1 Notation: Ei1(x) =
∞∫
x
e−t
t
dt .
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 397
V.I. ROMANENKO, A.V. ROMANENKO, L.P. YATSENKO
×
(
∇i1 . . .∇i2k
)[
λ(r) + νN (0)(r)
]
+ . . . (24)
Expressions (23) and (24) represent asymptotic expan-
sions, where one can leave only the first terms in the
case of sufficiently large ν (and α). In particular, the
second-order terms result in the diffusion equation.
Using the known expressions for the averages 〈vivj〉,
one obtains
N (0) =
(
1 +
〈v2〉
nα2
Δ
)
(λ+ νN (0)) + . . .
in the quadratic approximation, where n stands for the
space dimension. In view of the equality 〈v2〉 = n
2 v
2
0 ,
one derives
α0N(r) =
νv2
0
2α2
ΔN(r) + λ(r) , N(∞) = 0 (25)
for the arbitrary n (accurate to terms of the order
of 1/ν). This equation can be interpreted as the diffusion
equation with the absorption coefficient α0 = γ + iΔω
and the complex-valued diffusion coefficient
D̃ =
νv2
0
2α2
. (26)
The diffusion coefficient is identical for all space dimen-
sions.
Expression (26) at large ν becomes real and turns into
D =
v2
0
2ν
. (27)
Proceeding from the formula a =
√
DτD, we obtain
the characteristic diffusion time (approximate time, for
which an atom leaves the beam)
τD =
a2
D
=
2a2ν
v2
0
. (28)
For the further consideration, it is convenient to put
down the diffusion equation (25) in the form
ΔN(r)− β2N(r) = −f(r) , (29)
where f(r) = β2
α0
λ(r), β2 = 2α0α
2
νv20
' α0τD
a2 . Solving it
with the help of the Green function method, one can
see that the expression for the signal obtained by solv-
ing (29) in the case of large ν will be the same as that
obtained earlier by direct calculations (14). One can
also see that the number of terms used in the asymp-
totic expansion F̂ (k) of expression (14) correlates with
the number of terms of Eq. (24) that must be kept in
order to obtain the diffusion equation.
In the case of a finite cell, the kinetic equation (8)
can be solved formally, by interpreting the last term on
the right-hand side as a nonuniform one (see [22]). The
result will be the same as that derived from the solution
of (25) accurate to the terms ν−2 (to which the diffusion
equation is actually valid).
The atomic motion is characterized by five character-
istic times:
τa =
a
v0
, τR =
R
v0
, τγ =
1
γ
,
τν =
1
ν
, τD =
a2
D
= ντ2
a . (30)
According to the accepted approximations, they satisfy
the following conditions:
τν < τa < τR , τD < τγ , τν � τγ , τa < τD . (31)
The first condition results from the strong-collision ap-
proximation and the geometric configuration R > a for
small ν and (or) large R. The approximation used for
the derivation of the diffusion equation will be valid for
a finite-size cell. The second and third conditions corre-
spond to the slowness of relaxation processes, while the
last condition is evident.
Let us introduce the dimensionless time and space
scales t̂ = γt and r̂ = r/a and denote the dimension-
less velocity by v̂0 = v0/(γa), the diffusion coefficient
by D̂ = D/(γa2), the collision frequency by ν̂ = ν/γ,
and the cell size by R̂ = R/a. The dimensionless char-
acteristic times will have the form
τ̂γ = 1 , τ̂ν = γτν =
1
ν̂
, τ̂D = γτD =
2ν̂
v̂2
0
,
τ̂a = γτa =
1
v̂0
, τ̂R = γτR =
R̂
v̂0
. (32)
As will be seen from the further solution, the shape of
the transmission line in the infinite case is completely
determined by two dimensionless characteristic times τ̂ν
and τ̂D, whereas, in the case of a finite cell, it depends
on its size – R̂ (width) and l̂ (length, in the three-
dimensional case).
The diffusion equation in the form (29) will be solved
for different space dimensions. The most effective ap-
proach is that based on the Green functions. In order
to simplify the comparison of the cases of finite R and
R→∞, we present the result of investigating the effect
of space confinements on the signal in the form most
similar to the solution for an infinite cell.
398 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
INFLUENCE OF DIFFUSION OF ATOMS
4.2. One-dimensional case
Using the general solution of the one-dimensional diffu-
sion equation with the boundary conditions N(±R) = 0,
we obtain
S
(1)
R (Δω) =
1
2πa
+∞∫
−∞
λ̂(u/a)λ̂R(u/a) du
α0 + u2/τD
, (33)
where λ̂R(u/a) = λ̂(u/R)b(1)(u,R), and b(1)(u,R) is a
factor depending on R:
b(1)(u,R) = B(1)(u/a,R) ,
B(1)(k,R) =
1
λ̂(k)
R∫
−R
λ(x)
[
e−ikx − e−ikR ch(βx)
ch(βR)
]
dx.
(34)
This expression means that, similar to the infinite
case (16), S(1)
R can be written down as a superposition
of Lorentzians:
S
(1)
R (Δω) ∼
+∞∫
−∞
s(Δω, u)g(1)
R (u,Δω) du ,
though, in contrast to (17), the weighting function in the
case of finite R depends on R and Δω:
g
(1)
R (u,Δω) = g(1)
∞ (u) b(1)(u,Δω,R) ;
moreover, limR→∞ g
(1)
R (u,Δω) = g
(1)
∞ (u). In the case of
R → ∞, the dependence on Δω disappears. The factor
g
(1)
R (u,Δω) cannot be interpreted as a weighing one due
to the dependence on Δω. In addition, the Lorentzian
s(Δω, u) cannot be replaced by the more complicated
profile sR(Δω, u) = s(Δω, u) · b(1)(u,Δω,R) in order to
separate out the weight, as it was done in the infinite
case, because this “profile” sR becomes singular for some
values of u (in particular, for u→∞), though this singu-
larity is compensated by the other factor g(1)
∞ (u). Thus,
the interpretation of S(1)
R (Δω) as a weighted superposi-
tion is impossible here.
For the Gaussian intensity distribution, the function
b(1)(k,R) can be expressed in terms of the error func-
tions:
B(1,g)(k,R) = 1−
erfc
(
R
a + ika2
)
+ erfc
(
R
a − i
ka
2
)
2
−
−e(k
2+β2)/4 cos kR
chβR
erf
(
R
a + βa
2
)
+ erfc
(
R
a −
βa
2
)
2
.
(35)
In the case R� a, Ra �
βa
2 , one can obtain the asymp-
totic behavior of B(1)(k,R) for small k:
B(1,g)(k,R) ' 1− e(k
2+β2)a2/4 cos kR
chβR
. (36)
The graphs of ReSR(Δω) and the associated param-
eters for the typical values
γ = 1.0× 102 Hz , ν = 1.0× 106 Hz ,
a = 1.0× 10−3 m , v0 = 2.95× 102 m/s,
and τD = 2.3 × 10−5 s with the corresponding dimen-
sionless values
τ̂D = 2.3× 10−3 , τ̂ν = 1.0× 10−4
are given in Fig. 2.
One can see that the line will be narrower for larger R
(where the probability to return without loss of co-
herence is higher). Formally, the geometric factor
g
(1)
R (u,Δω) for finite R is larger than g
(1)
∞ (u). There-
fore, the profile ReSR(Δω) will be wider.
4.3. Two-dimensional case
Using the axially symmetric solution of the two-
dimensional diffusion equation in the region r < R with
the boundary condition N(r)
∣∣
r=R
= 0, one obtains
S
(2)
R (Δω) =
1
2π
∞∫
0
uλ̂(u/a)λ̂R(u/a) du
α0 + u2/τD
b(2)(u,R) , (37)
where λ̂R(u/a) = λ̂(u/a)b(2)(u,R) and
b(2)(u,R) = B(2)(u/a,R) ,
B(2)(k,R)=
2π
λ̂(k)
R∫
0
rλ(r)
[
J0(kr)−J0(kR)
I0(βr)
I0(βR)
]
dr ;
(38)
moreover, limR→∞ b
(2)
R (u,R) = 1.
The same way as in the one-dimensional case,
the weighting factor of the Lorentzian has the form
g
(2)
R (u,Δω) = g
(2)
∞ (u)b(2)(u,R,Δω). The interpretation
of (38) and the comparison with the infinite case (19)
are the same as those in the one-dimensional one (35).
The graphs of ReS(2)
R (Δω) in Fig. 3 are plotted for the
same values of the parameters as in Fig. 2.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 399
V.I. ROMANENKO, A.V. ROMANENKO, L.P. YATSENKO
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0 R/a = infinity
R/a = 10
R/a = 20
R/a = 30
∆ω/γ
T
a
0 5 10 15 20 25
0
20
40
60
80
100
120
140
H
al
fw
id
th
R/a
b
Fig. 2. One-dimensional case: normalized transmission T =
ReS(Δω)/S(0) as a function of Δω/γ for different R (a) and the
transmission spectrum halfwidth as a function of R/a (b). The
values of the parameters are given in text
4.4. Three-dimensional case
In the three-dimensional case where N(r) is zero on
the cylindrical surface of radius r = R confined by the
planes z = ±l, the diffusion equation is identical to that
obtained in the two-dimensional case, whereas the signal
has the following form (modified two-dimensional pro-
file):
S
(3)
R (Δω) =
l
16πa2
β2
α0
×
×
∞∫
0
du
uλ̂2(u/a)
α0 + u2/τD
b(2)(u,R)b(3)(u, l) , (39)
where λ̂(k) denotes the two-dimensional Fourier trans-
form of λ(r), b(2)(u,R) is the factor from the two-
0 5 10 15 20
0.4
0.5
0.6
0.7
0.8
0.9
1.0
T
∆ω/γ
R/a = infinity
R/a = 10
R/a = 20
R/a = 30
a
0 10 20 30 40 50
0
50
100
150
200
250
300
350
H
al
fw
id
th
R/a
b
Fig. 3. Two-dimensional case: normalized transmission T =
ReSR(Δω)/SR(0) as a function of Δω/γ for R =∞ and finite R
(a) and the transmission spectrum halfwidth as a function of R/a
(b). The values of the parameters are the same as in Fig. 2
dimensional case depending on R (see (38)), and
b(3)(u, l,Δω) = 1− tanh ξ(u,Δω)
ξ(u,Δω)
,
ξ(u,Δω) =
l
a
√
u2 + β2a2 . (40)
The weighting factor of the Lorentzians in the three-
dimensional case (39) has the form
g
(3)
R,l(u,Δω) = g
(2)
R (u)b(3)(u, l) =
= g(2)
∞ (u)b(2)(u,R,Δω)b(3)(u, l,Δω).
Its properties are similar to those in the one- and two-
dimensional cases. It is worth noting that its dependence
400 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
INFLUENCE OF DIFFUSION OF ATOMS
0 5 10 15 20
0.5
0.6
0.7
0.8
0.9
1.0
1.1
∆ω/γ
T
R/a = infinity, l/a = 20
R/a = 10, l/a = 20
R/a = infinity, l/a = 50
R/a = 10, l/a = 50
Fig. 4. Three-dimensional case. T = ReS
(3)
R (Δω)/S
(3)
R (0) as a
function of Δω/γ for different space parameters
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
1.2
T
n
∆ω/γ
R = infinity
n=1
n=2
n=3, l/a = 20
n=3, l/a = 50
Fig. 5. Comparison of the functions Tn = ReS
(n)
∞ (Δω)/S
(n)
∞ (0)
for n = 1, 2, 3
on the cell dimensions R and l appears in the form of
independent factors.
The transmission spectra normalized to one at the
maximum for different values of the space parameters
are given in Fig. 4. As expected, a decrease of l re-
sults in the broadening of the spectrum. Figures 5 and 6
present the transmission spectra for all dimensions in the
case of an infinite cell and a cell with the size R/a = 10.
One can see that the narrowest spectrum is obtained in
the one-dimensional case and the widest – in the three-
dimensional one. The values of the parameters used for
plotting the graphs in Figs. 4–6 are the same as in Fig. 2.
5. Comparison with Experiment
Figure 7 shows the comparison of the results of numeri-
cal calculations with experimental data [13]. The calcu-
0 2 4 6 8 10 12 14 16 18 20
0.4
0.6
0.8
1.0
1.2
T
n
∆ω/γ
R/a = 10
n=1
n=2
n=3, l /a = 20
Fig. 6. Functions Tn = ReS
(n)
R (Δω)/S
(n)
R (0), n = 1, 2, 3, for
R/a = 10
0 20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
∆ω/γ
Experiment
Theory
T
Fig. 7. Normalized transmission T = ReS
(3)
R (Δω)/S
(3)
R (0) — com-
parison with experiment
lations were performed for the dimensionless parameters
τ̂a = 3.39× 10−4 , τ̂r = 1.69× 10−3 ,
τν = 1.00× 10−4 , τ̂D = 2.30× 10−3 ,
v̂0 = 2.95× 103 , D̂ = 4.35× 102 ,
R̂ = 5 , l̂ = 5
close to the experimental conditions. One can see that
our theoretical results are in rather good agreement with
those of experimental measurements.
It is worth noting that the experimental data pre-
sented in Fig. 7 also agree well with the theoretical cal-
culation of the transmission spectrum obtained by aver-
aging over atom trajectories. This testifies to the fact
that the proposed model is also in good agreement with
that based on the averaging over trajectories [13, 16],
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 401
V.I. ROMANENKO, A.V. ROMANENKO, L.P. YATSENKO
at least in the cases where these theories can be consis-
tently compared. The proposed model has a wider range
of applications, since the model with averaging over tra-
jectories can be used only for the case of an infinite cell
(at least in its present form).
6. Conclusions
We have constructed a model for the description of the
phenomenon of Ramsey diffusion-induced narrowing of
the transmission spectrum of atoms in a buffer-gas cell
for the case of weak fields in the strong-collision approx-
imation. This model can be used for an arbitrary inten-
sity distribution of laser beams in the direction normal
to their propagation. The general theory is illustrated
by calculations for the Gaussian intensity distribution.
The analytical expressions for transmission spectra
obtained with the help of the effective diffusion equa-
tion qualitatively agree with experimental data and the
results obtained by averaging over atom trajectories.
The proposed model gives a possibility to investigate the
shape of the transmission spectrum as a function of not
only the intensity distribution in the plane of the laser
beam but also the size of the buffer-gas cell. We have
considered different geometric configurations (one-, two-,
and three-dimensional). Comparing the spectra for one-,
two-, and three-dimensional models, one can see that the
line becomes wider with increase in the dimension.
The work is carried out in the framework of Projects
F28.2/035 and RFFD/1-09-25.
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Received 15.10.09
Translated from Ukrainian by H.G. Kalyuzhna
ВПЛИВ ДИФУЗIЇ АТОМIВ НА ФОРМУ ЛIНIЇ ТЕМНОГО
РЕЗОНАНСУ У ПРОСТОРОВО ОБМЕЖЕНИХ
ЛАЗЕРНИХ ПОЛЯХ
В.I. Романенко, О.В. Романенко, Л.П. Яценко
Р е з ю м е
Запропоновано дифузiйну модель для недавно вiдкритого
дифузiйно-iндукованого звуження Рамзея, що виникає пiд час
дифузiї атомiв у комiрцi з буферним газом у полi лазерно-
го випромiнювання. Рiвняння дифузiї для когерентностi ме-
тастабiльних станiв, пов’язаних зi збудженим станом лазерним
випромiнюванням рiзної частоти у трирiвневiй схемi взаємодiї
атома з полем, отримано у наближеннi сильних зiткнень. До-
слiджено залежнiсть форми лiнiї поглинання поблизу макси-
муму пропускання випромiнювання однiєї з частот вiд вiдстро-
ювання вiд двофотонного резонансу для рiзних геометричних
конфiгурацiй комiрки.
402 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
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| last_indexed | 2025-11-27T01:33:16Z |
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| publisher | Відділення фізики і астрономії НАН України |
| record_format | dspace |
| spelling | Романенко, В.І. Романенко, О.В. Яценко, Л.П. 2010-11-08T17:05:31Z 2010-11-08T17:05:31Z 2010 Вплив дифузії атомів на форму лінії темного резонансу у просторово обмежених лазерних полях / В.І. Романенко, О.В. Романенко, Л.П. Яценко // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 394-404. — Бібліогр.: 22 назв. — укр. 2071-0194 PACS 42.50.Gy, 42.50.Hz; 32.80.Qk, 33.80.Be https://nasplib.isofts.kiev.ua/handle/123456789/13428 535.372 Запропоновано дифузiйну модель для недавно вiдкритого дифузiйно-iндукованого звуження Рамзея, що виникає пiд час дифузiї атомiв у комiрцi з буферним газом у полi лазерного випромiнювання. Рiвняння дифузiї для когерентностi метастабiльних станiв, пов’язаних зi збудженим станом лазерним випромiнюванням рiзної частоти у трирiвневiй схемi взаємодiї атома з полем, отримано у наближеннi сильних зiткнень Дослiджено залежнiсть форми лiнiї поглинання поблизу максимуму пропускання випромiнювання однiєї з частот вiд вiдстроювання вiд двофотонного резонансу для рiзних геометричних конфiгурацiй комiрки. Предложена диффузионная модель для недавно открытого диффузионно-индуцированного сужения Рамзея, возникающего при диффузии атомов в ячейке с буферным газом в поле лазерного излучения. Уравнение диффузии для когерентности метастабильных состояний, связанных с возбужденным состоянием лазерным излучением разной частоты в трехуровневой схеме взаимодействия атома с полем, получено в приближении сильных столкновений. Исследована зависимость формы линии поглощения вблизи максимума пропускания излучения одной из частот от отстройки от двухфотонного резонанса для разных геометрических конфигураций ячейки. We propose a diffusion model for the recently discovered diffusioninduced Ramsey narrowing arising when atoms diffuse in a buffergas cell in the laser radiation field. The diffusion equation for the coherence of metastable states coupled with an excited state by laser radiation of different frequencies in a three-level scheme of the atom-field interaction is obtained in the strong-collision approximation. The dependence of the shape of an absorption line near the transmission maximum of one of the frequencies on the two-photon resonance detuning for various geometries of the cell is investigated. Роботу виконано за проектами Ф28.2/035 та РФФД/1-09-25. uk Відділення фізики і астрономії НАН України Оптика, лазери, квантова електроніка Вплив дифузії атомів на форму лінії темного резонансу у просторово обмежених лазерних полях Влияние диффузии атомов на форму линии темного резонанса в пространственно ограниченных лазерных полях Influence of Diffusion of Atoms on the Dark Resonance Lineshape in Spatially Bounded Laser Fields Article published earlier |
| spellingShingle | Вплив дифузії атомів на форму лінії темного резонансу у просторово обмежених лазерних полях Романенко, В.І. Романенко, О.В. Яценко, Л.П. Оптика, лазери, квантова електроніка |
| title | Вплив дифузії атомів на форму лінії темного резонансу у просторово обмежених лазерних полях |
| title_alt | Влияние диффузии атомов на форму линии темного резонанса в пространственно ограниченных лазерных полях Influence of Diffusion of Atoms on the Dark Resonance Lineshape in Spatially Bounded Laser Fields |
| title_full | Вплив дифузії атомів на форму лінії темного резонансу у просторово обмежених лазерних полях |
| title_fullStr | Вплив дифузії атомів на форму лінії темного резонансу у просторово обмежених лазерних полях |
| title_full_unstemmed | Вплив дифузії атомів на форму лінії темного резонансу у просторово обмежених лазерних полях |
| title_short | Вплив дифузії атомів на форму лінії темного резонансу у просторово обмежених лазерних полях |
| title_sort | вплив дифузії атомів на форму лінії темного резонансу у просторово обмежених лазерних полях |
| topic | Оптика, лазери, квантова електроніка |
| topic_facet | Оптика, лазери, квантова електроніка |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/13428 |
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