Journal of Modern Physics, 2013, 4, 48-53
doi:10.4236/jmp.2013.45B010 Published Online May 2013 (http://www.scirp.org/journal/jmp)
Coherent Resonance of Saturated Absorption in
Spectroscopy of Counterpropagating Waves
A. A. Chernenko1, E. G. Saprykin2, A. M. Shalagin2,3
1Institute of Semiconductor Physics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
2Institute of Automation and Electrometry, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia
3Novosibirsk State University, Novosibirsk, Russia
Email: chernen@isp.nsc.ru
Received 2013
ABSTRACT
Results of theoretical researches of the saturated absorption resonance shape in a method of the probing field on V-type
of transitions are represented. It is shown that in case of opposite circulary polarized optical fields the resonance is
shown in the form of cross, and its form strongly depends on relaxation constants of levels and it can be represented as
in the form of a dip, and absorption peak. Thus, the peak form has exclusively coherent character. Atomic transitions
are offered, on which observation of the given effect is possible.
Keywords: Nonlinear Spectroscopy; Atomic Transition; Saturation Absorption; Cross Resonance; Light Wave
1. Introduction
It is known that action of the strong light fields on a
resonant medium leads to nonlinear effects which are
shown in the form of change of the density populations
of resonant levels (effect of saturation), splitting of levels
and an interference of levels – so-called nonlinear inter-
ferential effect (NIEF) as result of simultaneous absorp-
tion some photons [1]. The effect of saturation has exclu-
sively incoherent character, and two others are caused by
coherent processes. The method of a probing field [1,2]
allows investigating the specified nonlinear effects. The
resonance registered in this method in the absorption
spectrum of the probing field by the resonant gas me-
dium is shown in the form of a dip on the Doppler line
contour and carries generally incoherent (populational)
character. The coherent processes are shown in a form of
the field broadening of the resonance spectrum (effect of
level splitting) and as an additive in the dip amplitude
(contribution of NIEF). And, the value of the NIEF con-
tribution depends on the direction of propagation of the
strong and probing light wave; effect is maximum at uni-
directional waves, and in case of opposite directed waves
the contribution of effect to resonance amplitude is strongly
suppressed (in relation of the uniform width to the Dop-
pler line width of atomic transition). So for the two-level
medium under optimum conditions the share of NIEF in
amplitude of a resonance doesn’t exceed some percent
[2]. In spite of the fact that researches on nonlinear spec-
troscopy of multilevel quantum systems are conducted
already for a long time, a situation when the nonlinear
resonance in opposite directed waves would be defined
by only coherent processes, as far as we know, isn’t
found until now. There are cases, as in paper [3] where to
the observed spectrum features of the nonlinear reso-
nance, having the incoherent nature, NIEF manifestation
was mistakenly attributed.
In this work we consider quantum system in which in
case of the opposite directed light waves the nonlinear
resonance having exclusively coherent nature (without a
contribution of populated effects) with characteristic for
coherent processes by an interferential form of a spec-
trum can be observed. Such situation is realized in the
V-type of transitions the upper state of which has to
breaks up only in the bottom state, and light waves have
to be circular polarized with the counter direction of ro-
tation of polarization vectors. A number of atomic transi-
tions on which probably experimental observation of a
resonance of coherent type is offered.
2. Theoretical Model
Let us consider the problem about the absorption spec-
trum of a probing field in a medium with V-type of tran-
sition in the presence of a strong field with the same fre-
quency and opposite propagation direction. The scheme
of transitions is presented in Figure 1. The strong wave
is assumed to be plane (with the frequency
, wave vec-
tor k, and electric field strength Е) and resonant to the
atomic transition mn (with the transition frequency
mn). Polarization of the strong wave is either linear or
circular (separate case). The probe wave is also mono-
Copyright © 2013 SciRes. JMP
A. A. CHERNENKO ET AL. 49
chromatic (with the frequency
=
, the wave vector k
= – k, and the electric field strength Е) with circular
polarization. When solving the problem, the saturation of
a medium by the probing wave is taking into account
under the assumption of its weakness as compared to the
strong wave. The gas is assumed to be rarefied enough to
neglect collisions. The medium is assumed to be opti-
cally thin.
We consider the problem in a coordinate system with
the quantization exis along the direction of wave vector k
of the strong field. In this coordinate system light fields
induces transitions between magnetic sublevels (Figure
1) with variation М = ± 1 (at linear polarization), or
transitions with М = +1 or М = –1 (at circular polari-
zation depending on a direction of its rotation).
In solving the problem we start from kinetic equations
for a density matrix of the atomic system. According to
[1], the dynamics of diagonal elements of the density
matrix (
i =
ii) in the model of relaxation constants is
described by the system of equations:
2Re()
2Re()
 

i
i iiki kijji
kj
ij ji
j
Qi
t
iV
V
 
(1)
For off-diagonal elements of the density matrix the
next system is valid:
()[,][,
 
ik
ikik ikikik
iiViV
t
]
 
(2)
Here Гi and Гik are the widths of the levels and transi-
tions, Qi is the number of exitation acts of the i-th level
per unit time, the summand
ki k
k
in system of
equations (1) determines the spontaneous decay of the
upper state m to the lower state n and is absent in the
equations for the populations of upper levels, V and V
are operators of the interaction between the atom and the
strong and probing fields. The operators are defined as V
= – Gexp(i(krt)) +H.с. and V
= – G
exp(i(k
r
t))
+H.с., where the operators G = dE2, and G
= dE
2,
and d is the operator of the reduced dipole moment of
atomic transition. Let's mark, that the given statement of
the problem and the obtained solutions are fair both for
transitions between exited states, and for a case, when the
lower state n is the ground state of atom. In this case the
width of the lower state is determined by mean value of
the interaction time of particles with the light fields.
 
Following [1], we seek solutions of the system of
equations for the density matrix in the following form
(the first harmonic approximation by the difference of
wave frequencies
=
): the diagonal elements are
found as
i =
i
0 +
i
+exp(i
t) +
i
-exp(–i
t); off-diagonal
cordingly are found as
ik = Rik exp(–it) + Rik
exp(–i
t)
+ Rik
s exp(–ist) and
ik = rik
0 + rik
+exp(i
t) + rik
-exp(–i
t),
where
s = 2
. The substantiation of the given kind
of solutions and their accuracies for the V-, Λ- and J =
1-J =1 types of transitions are given by us in works [4,5].
In the approximation of rotating optical fields the sys-
elements on the allowed and forbidden transitions ac-
te
lations of
th
m of equations for the density matrix in the considered
transition scheme in the stationary case is reduced to a
system of algebraic equations for
i
0,
i
, Rik, Rik
, R
ik
s,
rik
0, and rik
. Taking into account the hermiticity of these
coefficients, we write only independent equations for a
case of the linear polarized strong field. The equations
for a case of the circular polarized strong field are easily
received from reduced below, equating to zero one of the
circular strength component of strong field.
In a case of the V- scheme (Figure 1) popu
e lower (0) and the upper (1 and 2) levels are described
by a following system of equations:
00
2R
00 0 00
1, 21, 2
02 20
e( )
2Re()

QA

nkk
kk
iGR
iG R

kk
(3a)
)
0001 100220
1,2
02 2020 0210 01
() (
()

 

ss
nkk
k
iAiGRG
iGRGRG R

 
R
(3б)
(3в)
0
0
1110 01
2Re( )
mQiGR
11001011
()(
)
s
miiGRGR

(3г)
0
2220 0220 02
2Re()2Re() 
mQiGR iGR
(3д)
22002 02200220
()(
 s
miiGRGRGR


)
The system of equations for polarizations
lo
(3е)
at the al-
wed (1– 0 and 2– 0 ) and at the forbidden 2– 1 transi-
tions has the next form:
0
1
2
n
m
2
Figure 1. Scheme of interaction of light fields with V-type of
atomic transitions, - shift of levels.
Copyright © 2013 SciRes. JMP
A. A. CHERNENKO ET AL.
50
00 0
1001011002 2102 21
()()
 
mn iRiGiG riG r

(4a)
1001011002 21
(( ))()

 
s
mn iRiGiG r

(4б)
0
2 21
r (4в)
10 01101002 210
()()
 
 
mn iRiGiG riG
 

00
20 2020 2020 20
0
10 21
() ()(

 
mn i RiGiG
iG r
)

(4г)
202020201021
(( ))()

 
s
mn iRiGiG r

(4д)
00
20 2020 2020 20
10 21
() ()(

 
mn i RiGiG
iG r
 
)

)
(4е)
0
2121200101 2020 01
()( 
miriGRGRGR
)
(5а)
2121200101 20
(( ))(
 
s
miriGRG

R
(5б)
2121200101 20
(( ))(
 2001 )
s
miriGRGR


GR (5в)
In equations (3) - (5) Гn and Гm are the widths
lower and upper levels, Гmn is the width of the tran
lin
anges in the equations:
atom. The situation of
co
ollowing relation:
erical Solution of the
Initial Equations
b-
taits of the density matrix were
of the
sition
e, 21 is the width of the forbidden transition between
magnetic sublevels of the upper state; ik =
ik and
ik =
ik are the values of frequency detuning of
the strong and probing fields from the frequencies ik of
transitions between the magnetic sublevels of the m and n
states.
The motion of atoms is taken into account by the fol-
lowing ch
ik ik kv,
ik
ik k
v, аnd
– (k
k)v, where v is the velocity of
unter propagation of waves with equal frequencies was
then analyzed:
=
, and k
= k.
The shape of the absorption line of the probing field
(per atom) was determined by the f
/0 = – mn Re(i(R20
))/G
, where the designation
<...> means the averaging by the Maxwell velocity dis-
tribution of particles, and
0 = 4
mnd2/cmn is the
resonant absorption cross section. In calculations the
probabilities Aki of the decay of magnetic sublevels by
each of the spontaneous channels were set to be similar
and equal to Amn/2.
3. Results of Num
Stationary system of equations (3) - (5) that were o
ned above for elemen
solved numerically upon varying the values of width of
the upper Гm and lower Гn atom levels, the parameter of
the radiation branching from the upper level а0 (a0 =
Amn/Γm) and the intensities of optical fields using the
relaxation characteristics of the 1s2 - 2p8 transition of the
neon atom (Аmn = 1.88 × 107 с-1, Гm = 5.5 × 107 с-1, Гn =
105 106 с–1, Гmn = (Гm + Гn)/2). The values of level
widths varied from the aforesaid values to Гm = Гn = Гmn,
at the same time, the transition width Гmn remained con-
stant and the value of branching parameter а0 varied
within the range 0.34 1. The line width of the forbidden
transition between the sublevels of the upper state relied
to width of the upper level. The Doppler line width is
taken to be equal kvT = 5.2×109 с-1, and the variation
range of particle velocities at integration was 3kvT with
a step kvT = (10-3 ÷ 10-4)kvT. The saturation parameters
of the strong (κs) and probing (κp) fields were chosen in
the form
2
2(/2)/ ()

s
mn n
dE
and 2
2(/ 2)/ ()dE
,
pm
nn
where E and E
are the hs of the circular compo
nents of the strong and ng fields and d is the re-
tion spec-
ion in the
the cross resonance can
m
strengt-
probi
duced dipole moment of transition. The values of satura-
tion parameters varied within the limits κs 50 and κp
κs. Below, results are presented for small value of the
saturation parameter of the probing field κp = 10-3.
3.1. Strong Field of Linear Polarization
Results of numerical simulation of the absorp
trum of the probing wave of circular polarizat
presence of a strong saturating field of linear polarization
are presented in Figures 2 and 3. When the upper state is
degenerated the shape of the nonlinear resonance is rep-
resented as a dip in the center of the Doppler contour of
the absorption line. Its amplitude and width depend on
the saturation parameters of optical fields, on width of
transition level and on the branching parameter а0. With
elimination of the degeneracy (splitting) of the upper
levels one can observe a dip of the saturated absorption
at the shifted transition frequency in the shape of the ab-
sorption line of the circular polarized probing wave
(named as РR, the sign of its shift is defined by the sign
of circular polarization) and a cross resonance (CR) at
the unshifted frequency тn. Amplitude and sign of the
cross resonance depend essentially on the relation be-
tween the level widths Гm and Гn and on the branching
parameter а0, at the same time, the effect of these char-
acteristics on the parameters of the parent (PR) resonance
is expressed much weaker.
It is seen from Figure 2 that, depending on the value
of the branching parameter a0,
anifest itself not only as a dip, but also as an absorption
peak. The maximal value of the peak amplitude is reached
for the closed transition (for parameter а0 = 1) both when
the values of the level widths are similar (Гm Гn) and
when they are considerably different (Гm >> Гn). Thus it
appears, that at a similar values of the saturation parame-
ters κs of the strong field, the peak amplitude in case of
similar values of level width is considerably larger than
the peak amplitude in case of a strong difference between
the widths (curves 1 and 5).
Copyright © 2013 SciRes. JMP
A. A. CHERNENKO ET AL.
Copyright © 2013 SciRes. JMP
51
-0.25 00.5 /kv
T
0.002
0.006
0.01
0.014
0.018
α α
0
6
7
8
5
1
2
3
4
Figure 2. Resonance shapes at different values а0 and Γn: = 40Γmn;
s=50,
p =10-3; Γn = Γmn (1 - 4), 0.04 Γmn (5 - 8); a0=1 (1,
5), 0.98 (6), 0.75 (2, 7), 0.6 (3); 0.34 (4, 8).
-0.25 0 0.5 /kv
T
0.01
0.012
0.014
0.016
0.018 α/α
0
1
4
4
3
3
2
Figure 3. Behavior of the resonance shapes from intensity of the strong field: = 40 Γmn; a0 = 1;p = 10-3, s = 0.2 (1); 1.0 (2)
5.0 (3) 50 (4); Γn = 0.04 Γmn (continuous lines), Γmn (dashed lines).
ross resonance is transformed to a dip from the peak form.
M
Taking the approximation of a weak probing field and
neglecting the polarization at the combination frequency
polarization at the probing
fie
With a decrease in the value of the branching parameter
c
oreover, in the case of similar values of level widths Гm
= Гn the flip is observed for values of the branching pa-
rameter а0 0.6; for the relation between the level
widths Гm >> Гn the change in the sign of the resonance
amplitude occurs near the value а0 0.98. Note also that,
in case of similar values of the level widths the dip ampli-
tude of the cross resonance is considerably less and the
width is considerably larger than in case of when Гm >>
Гn (compare curves 4 and 8).
2 - one can obtain from the system of equations (3) -
(5) an analytical solution for
ld frequency 20 ()
R
as:
2
10
20 20
21 21
00
()
()
(

 
mn
G
iR
i


2
01
10
00
20 02
10 2121
)
(( ))
()(())


mn
G
iG ii

(6)
A. A. CHERNENKO ET AL.
52
Here, the differences of level populations are deter-
mined via population of the lower level, branching pa-
rameter а0 and rates of stimulated transitions Wik as:
00 0
02 020
00 0
01 010
22 2
00
0
()
;
,
 

m
ik ns mnmnik
W
N
aW aW
(7)
10 20
10 20
(1) ;
(1);
(1 0)(1 0)
111
 
 

 
m
nn
mm
W
W
WW
 
 
where 00

n
NQ
bsence of the st
is the population of the lower level
in the arong field.
In the velocity (balanced) approximation the presence
of a cross resonance is associated with amplitudes of the
Bennett dips or peaks in the population of the common
level. The specificity of the V-scheme is that the common
the lower le
vel to the third levels
(b
shape of
form and for the dip. However, the functional rela-
tio
o-
na
ilar,1 only the parent resonance will take
on the
of split-
oving atoms. The graphs cor-
re
level isvel and, in the absence of the sponta-
neous radiation from the upper le
ranching parameter а0 = 1), a dip in the population of
the common level is absent (it follows from (7) that
0
00
N
). It is compensated by the arrival of the spon-
taneously emitted particles. In this case the parent reso-
nance is caused exclusively by the non-equilibrium
population of the upper level. Under these conditions, it
follows from solution (6) and averaging over the velocity
distribution of particles that the populational part in the
the CR (the first summand in solution (6)) is
represented by the Doppler contour and the peaks (Fig-
ure 2, curves 1 and 5) are caused only by coherent proc-
esses (the second summand in (6)), leading to an increase
in the absorption coefficient. The performed calculations
show that, in the given scheme of transitions, the contri-
bution of coherent processes to the CR shape is maximal
at equal values of level widths, and for case of Гm >> Гn
their contribution is less almost by an order of magnitude.
This is illustrated in Figure 2 by the dependences for two
relations between values of level widths both for the CR
peak shape (curves 1 and 5) and for the dip (curves 4 and
8).
The effect of the saturating field intensity on the CR
peak shape is shown in Figure 3. Here, for all relations
between level widths and any values of the branching
parameter, an increase in the strong field intensity leads
to an increase in the CR amplitude and width both for the
peak
n of amplitudes for the peak and for the dip form of
the CR in the range of the used intensities of the saturat-
ing field appears different. If the CR amplitude in the
form of a dip increases by a law close to the square root
dependence, the relation of the amplitude of the coherent
CR peak under these conditions is of linear character,
with the square root dependence for the PR amplitude.
Let us mark the character of influence of the probing
field intensity: an increase of the probing field intensity
results in decreasing in the amplitude of the Doppler lin-
ing contour of the absorption line and in the amplitudes
of the cross (for the dip and peak shapes) and parent
resonances, thus an increase of the widths of these res
nces is observed. However, here we did not take into
account a capability of appearance of the spectrum fea-
tures stipulated by spatial modulation of the nonlinear
medium susceptibility in a field of standing waves. These
problems were considered earlier (see, for example, ac-
tivity [6,7] and reduced links in them). The nonlinear
interference effects at the account of saturation by the
probing field were calculated also in activity [8], but only
for a case of unidirectional waves, in absence of the cross
resonances.
3.2. Strong Field of Circular Polarization
The case of circular polarizations of the strong and prob-
ing wave is of interest. If the signs of rotation of electric
field vectors of the strong and probing waves in a me-
dium are sim
place, its shift in the frequency scale will depend
direction of rotation of field vectors and a value
ting of the upper levels.
In case of the counter circular polarized vectors of
fields only a cross resonance will be observed in the ab-
sorption line spectrum of the probing wave. The CR is
located at the half sum of transition frequencies in the
region of the interaction of counter propagating waves
with the same group of m
sponding to this case are presented in Figure 4. Here
the saturated absorption line shapes, calculated for two
values of splitting of the upper levels and a number of
values of the atomic transition parameters are shown. It
is visible, that for any value of level splitting within the
limits of the Doppler line width the resonance is exhib-
ited on the same light wave frequency. And, in the ab-
sence of branching parameter (а0 = 1) resonance is rep-
resented as the peak of absorption (curves 1 - 5). As it
was marked above, in this case the population part of the
CR is absent, and its shape is caused exclusively by the
nonlinear coherent process (NIEF agrees with [1]). The
presented relations (the most obvious are curves 2 and 4)
demonstrate the alternating shape which is typical for the
NIEF and a decrease in the effect with an increase in the
difference between the level lifetimes. A decrease in the
branching parameter а0 leads to appearance of the popu-
lational contribution and to transformation of the peak
form into a dip (curves 5 - 8).
Thus, in case of the counter propagating and counter cir-
cular polarized waves the observed resonance of saturated
1In practice, these are experiments in which the probing wave is ob-
tained as result of reflection from the mirror behind the cell with a gas.
Copyright © 2013 SciRes. JMP
A. A. CHERNENKO ET AL.
Copyright © 2013 SciRes. JMP
53
-0.2 -0.1
0 0.1 0.2 kv
T
0.012
0.014
0.016
0.018
0.02
αα
0
4
3
1
8
7
5
6
Figure 4. Shapes of resonances at orthogonally circular polarization of fields: κs = 50; κp =10-3; = 0 (1, 2), = 40 Гmn (3-8); a0=1(1-4), 0.9(5),
0.75 (6), 0.6 (7), 0.34 (8); Гn = Гmn (2, 4 – 8), 0.04 Гmn (1, 3).
absorption is a cross resonance actually. It is located in
e line (in particular, in absence of splitting) and has
between
st
scopy and Its
Application Program (project 9.5) of the Branc
Physical Sciences, Russian Academences.
REFERENCES
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2
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the center of the unshifted transition line at any value of
splitting of the upper levels within the Doppler width of
th
under certain conditions extremely coherent nature. Ac-
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the case of counter propagating waves that are absent in
the approximation of first nonlinear corrections can ap-
pear in the next order with respect to saturation. This fact
explains the aforementioned linear character of the in-
crease of the coherent CR amplitude in Figure 2 with the
square rooted dependence of the PR amplitude.
In conclusion we shall indicate on a capability of ex-
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absorption resonance. The given resonance will be real-
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“On the Shape of Cross Resonances in Counterpropagat-
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ates with the full moments J = 0 and J = 1, thus the up-
per state of transition should breaks up mainly on the
lower state to supply the value of branching parameter a0
~ 1. Transitions 1S01,3P1 from the ground state of the
alkaline-earth metal atoms, and of the Hg, Cd, Zn and Yb
atoms satisfy to these requirements. The wave lengths of
these transitions are located in spectral area of generation
of existing power tunable lasers (in the visible range), or
their second harmonics (in the UV range).
4. Acknowledgements
This work was supported by the NSh-2979.2012.2 pro-
ject and the Fundamental Optical Spectro
113, No. 5, 2012, pp. 585-597.
doi:10.1134/S0030400X12080152
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