Broadening and intensity redistribution in the Na(3p) hyperfine excitation spectra due to optical pumping in the weak excitation limit

Abstract

Detailed analysis of spectral line broadening and variations in relative intensities of hyperfine spectral components due to optical pumping is presented. Hyperfine levels of sodium 3p 1/2 and 3p 3/2 levels are selectively excited in a supersonic beam at various laser intensities under the conditions when optical pumping time is shorter than transit time of atoms through the laser beam. The excitation spectra exhibit significant line broadening at laser intensities well below the saturation intensity, and redistribution of intensities of hyperfine spectral components is observed, which in some cases is contradicting with intuitive expectations. Theoretical analysis of the dynamics of optical pumping shows that spectral line broadening sensitively depends on the branching coefficient of the laser-driven transition. Analytical expressions for branching ratio dependent criti-cal Rabi frequency and critical laser intensity are derived, which give the threshold for onset of noticeable line broadening by optical pumping. The critical laser intensity has its smallest value for transitions with branching coefficient equal to 0.5, and it can be much smaller than the saturation intensity. Transitions with larger and smaller branching coefficients are relatively less affected. The theoretical excitation spectra were calculated numerically by solving density matrix equations of motion using the split propagation technique, and they well-reproduce the observed effects of line broadening and peak intensity variations. The calculations also show that presence of dark i.e., not laser coupled Zeeman sublevels in the lower state results in effective branching coefficients which vary with laser intensity and differ from those implied by the sum rules, and this can lead to peculiar changes in peak ratios of hyperfine components of the spectra.

Full text

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Broadening and intensity redistribution in the Na(3p) hyperfine excitation spectra
due to optical pumping in the weak excitation limit
I. Sydoryk, N. N. Bezuglov,*I. I. Beterov,†K. Miculis, E. Saks, A. Janovs, P. Spels, and A. Ekers
Laser Centre, University of Latvia, LV-1002 Riga, Latvia
?Received 6 February 2008; published 30 April 2008?
Detailed analysis of spectral line broadening and variations in relative intensities of hyperfine spectral
components due to optical pumping is presented. Hyperfine levels of sodium 3p1/2and 3p3/2levels are
selectively excited in a supersonic beam at various laser intensities under the conditions when optical pumping
time is shorter than transit time of atoms through the laser beam. The excitation spectra exhibit significant line
broadening at laser intensities well below the saturation intensity, and redistribution of intensities of hyperfine
spectral components is observed, which in some cases is contradicting with intuitive expectations. Theoretical
analysis of the dynamics of optical pumping shows that spectral line broadening sensitively depends on the
branching coefficient of the laser-driven transition. Analytical expressions for branching ratio dependent criti-
cal Rabi frequency and critical laser intensity are derived, which give the threshold for onset of noticeable line
broadening by optical pumping. The critical laser intensity has its smallest value for transitions with branching
coefficient equal to 0.5, and it can be much smaller than the saturation intensity. Transitions with larger and
smaller branching coefficients are relatively less affected. The theoretical excitation spectra were calculated
numerically by solving density matrix equations of motion using the split propagation technique, and they
well-reproduce the observed effects of line broadening and peak intensity variations. The calculations also
show that presence of dark ?i.e., not laser coupled? Zeeman sublevels in the lower state results in effective
branching coefficients which vary with laser intensity and differ from those implied by the sum rules, and this
can lead to peculiar changes in peak ratios of hyperfine components of the spectra.
DOI: 10.1103/PhysRevA.77.042511PACS number?s?: 32.70.Jz, 32.80.Xx
I. INTRODUCTION
Optical pumping is the well-known phenomenon that is
usually associated with redistribution of population within
hyperfine ?HF? components or Zeeman sublevels of the
ground state due to coupling by resonant light fields ?1?.
Optical pumping is being exploited in various applications,
like cooling below the Doppler limit ?2,3?, vibrational exci-
tation of molecules in the electronic ground state ?4?, orien-
tation and alignment of atomic and molecular ground states
?5?, etc. When optical pumping is involved in the control of
quantum states it is usually associated with large laser inten-
sities exceeding the saturation limit ?3?. Therefore the popu-
lations of quantum states depend nonlinearly on laser inten-
sities and the excitation spectra are affected by power
broadening ?6?.
In the present study we are concerned with line-shape
effects due to optical pumping in the weak excitation limit.
Specifically, we measure laser excitation spectra of the 3p1/2
and 3p3/2states of Na in a supersonic beam. Coupling of the
F?=1 and F?=2 levels of the ground state with different HF
components of the upper states allows us to study two-level
systems with different branching coefficients. The smaller
the branching coefficient, the more population irreversibly
leaves the two-level system, and vice versa.At very low laser
intensities the excitation spectra do not reveal any abnor-
malities. When laser intensity is increased but still below the
saturation intensity, essential modification of the excitation
spectra is observed: most of the hyperfine spectral compo-
nents exhibit additional broadening while their intensity ra-
tios cease to obey the line strengths rules. Note that usually
line broadening is considered to be a strong-field effect due
to power broadening at laser intensities above the saturation
intensity ?3,6?.
Since the experiments were performed at low number
densities of sodium atoms ?n3s?1010cm−3?, line-shape
modifications by radiation trapping can be disregarded ?7,8?.
We attribute the observed line-shape effects to optical pump-
ing, which leads to depletion broadening of spectral lines
?9,10?. If transit time ?trof atoms through the laser beam is
much larger than lifetime ?natof the 3p state, populations of
levels and the associated fluorescence signals can become
nonlinear on laser intensity Ilaslong before the saturation
limit is reached ?i.e., at Ilas?Isat?. Due to interaction with
laser field the ground state g has a finite width ?2,11?
?? = ?2
?nat
+ 4?2,
?nat
2
?1?
where ?nat=2???nat=1/?natis the natural width ?in units of
angular frequency ?s−1?? of the excited state e, ? is the laser
detuning from the line center, and ? is the Rabi frequency of
the transition. The width ?? is equal to the rate of photons
spontaneously emitted from state e. If the level system is
partially open, only the fraction ? of the spontaneous tran-
sitions will return the population to the initial state g; the
fraction 1−? associated with decay to levels other than g
will be lost from the ?g,e?-system during each excitation-
*Also at: Fock Institute of Physics, St. Petersburg State Univer-
sity, 198904 St. Petersburg, Russia.
†Permanent address: Institute of Semiconductor Physics SB RAS,
630090, Novosibirsk, Russia.
PHYSICAL REVIEW A 77, 042511 ?2008?
1050-2947/2008/77?4?/042511?10?
©2008 The American Physical Society042511-1

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emission cycle. The rate of such pumping is obviously
?pump=?1−????. Hence the pumping time can be written as
?pump??? =
1
?pump
=
?nat
2
+ 4?2
?nat?2?1 − ??.
?2?
If the transit time ?tris long, such that ?tr??pump
=0?, the population of the ?g,e?-system will be fully de-
pleted during interaction with the laser field. In terms of Rabi
frequencies the condition for population depletion can be
rewritten as ???cr??sat?2?nat/??tr?1−???, where the
saturation Rabi frequency is ?sat=?nat/?2 ?3?. Note that the
parameter 1/??nat?tr, which was considered in ?5,12? as the
parameter associated with saturation due to optical pumping
in the case of open level systems ?i.e., no population return
from state e to state g?, is identical to our critical Rabi fre-
quency ?crin the limiting case of ?=0.
If the weak excitation limit is combined with long inter-
action times of atoms with the laser field, such that ?tr
??nat, the value of critical Rabi frequency is small ??cr
??sat? and broadening and saturation of spectral lines can
be observed at laser intensities well below the saturation
limit, long before power broadening starts affecting the line
shapes.
?0?
??pump??
II. EXPERIMENT
The experiment was performed in a supersonic beam of
Na atoms ?see Fig. 1?. Two skimmers and an entrance aper-
ture of the excitation zone collimate the beam with flow
velocity vfto a small divergence angle ?, thus reducing the
Doppler width for excitation perpendicular to the beam axis
to ??D?vf?/?2??, where ? is the laser wavelength. The
divergence angle ? was set to either 0.67° or 0.92° by using
the entrance aperture b of either 2 or 3 mm diameter. The
laser beam crosses the atomic beam at right angles, and it is
linearly polarized parallel to the molecular beam axis z,
which is also the quantization axis. Only Zeeman sublevels
with identical quantum numbers mFare coupled by the laser
field due to the selection rule ?mF=0, while for transitions
between levels with the same F the transition mF?=0↔mF?
=0 is forbidden. The number density of atoms in the beam
was chosen sufficiently low ??1010cm−3?, thus ensuring
that the beam is not optically thick and effects of radiation
trapping and photon reabsorption ?8? can be safely neglected.
An important consequence of optical transparency of the
beam is that absorption P???L? ?the total number of photons
absorbed per second? and excitation J???L? ?the integrated
over frequencies flux of emitted photons in the direction of
observation? profiles as a function of the laser detuning ??L
do not vary in the interaction volume defined by the crossing
atomic and laser beams. Both profiles are proportional to the
integral ?over the interaction volume and HF sublevels?
population of the excited state.
The 3s→3p transition was excited using a single mode
cw radiation source ?Coherent CR-699–21 dye laser? with a
linewidth of 1 MHz. The fluorescence emitted by Na atoms
was collected into two fiber bundles at the angles of 90° and
45° with respect to the directions of the axis of the molecular
beam, laser beam, and laser polarization. The fluorescence
light was guided via the fiber bundles to two photomultipli-
ers, and the signals proportional to J???L? were registered
using photon counters. The resulting excitation spectra of the
3p state were recorded as a function of laser detuning ??L.
The arrangement with two different simultaneous detection
geometries allowed us to verify that radiation trapping,
which is strongly anisotropic with respect to the direction of
the observation, does not affect the measured spectra. It also
allowed us to rule out the influence of polarization effects on
variations in line shapes and relative line intensities.
The mean flow velocity vfof atoms in the beam was
measured to be 1160 m/s. For excitation perpendicular to the
atomic beam axis the apertures of the excitation zone of b
=2 and 3 mm correspond to the residual Doppler width
??D=11.2 and 15.9 MHz ?full width at half maximum
?FWHM??, respectively. These should be compared to the
excitation perpendicular to the natural width of ??natof 9.8
MHz ??nat=16.23 ns for the 3p3/2?13??.
Figure 2 shows the hyperfine energy levels of the 3s and
3p states. The excitation spectra were obtained by scanning
the laser frequency across the 3s1/2→3p1/2 and 3s1/2
→3p3/2transitions. The HF splittings are larger than both the
Doppler width and the natural width for all but one pair of
components ?3p3/2F?=0 and F?=1?. The measurements for
the D1line ??=589.593 nm? were performed with the aper-
ture b=2 mm. In the case of the D2line ??=588.996 nm?,
b=3 mm was used. The radius of the laser beam was rlas
=1.5 mm, which corresponds to the transit time ?tr
=2rlas/vf=2.65 ?s at vf=1160 m/s. Thus the transit time is
by more than two orders of magnitude larger than the natural
lifetime of the 3p state.
2
0
8 21
23 25
Nozzle
d = 0.4 mm
Skimmer 1
= 2.0 mm
?
Aperture
b = 2 mm
Skimmer 2
= 1.5 mm
?
L cm
[]
?
Z
FIG. 1. Collimation of the sodium beam by skimmers and aper-
tures. The laser beam crosses the atomic beam at right angles 23 cm
downstream from the nozzle.
23Na(3s)
23Na(3s)
23Na(3p)
23Na(3p)
2S1/2
2P1/2
2P3/2
3
2
1
0
2
11
22
11
190 MHz190 MHz
16 MHz 16 MHz
34 MHz 34 MHz
59 MHz59 MHz
1772 MHz1772 MHz
F
3
2
1
0
2
2S1/2
2P1/2
2P3/2
F
FIG. 2. Hyperfine energy levels of the 3s and 3p states of
Na.
SYDORYK et al.
PHYSICAL REVIEW A 77, 042511 ?2008?
042511-2

Page 3

III. SPECTRA
Figure 3 shows the measured excitation spectra of the
3p1/2state from the F?=2 sublevel of the ground state at
various laser intensities. The spectra exhibit two peaks cor-
responding to the excitation of the 3s1/2, F?=2→3p1/2, F?
=1,2 HF transitions. The spectra of Fig. 3 were measured at
the divergence angle of the atomic beam of ?=0.67°, which
corresponds to residual Doppler width of ??D=11.2 MHz
?see Sec. VI for details on Doppler line shape?. The spectrum
of Fig. 3?a? was measured at a very low laser intensity of
21 ?W/cm2. Both HF components appear equally strong,
which is obviously due to equal line strengths of both HF
transitions ?14?. The line shapes are determined by a com-
bined effect of natural ???nat=9.8 MHz? and Doppler
broadening. In Fig. 3?b? the laser intensity has been in-
creased by a factor of about 70 compared to Fig. 3?a? to the
value of 1 mW/cm2, which is still much smaller than the
saturation intensity of both HF components ?7.5 and
12.5 mW/cm2?. One can observe that the F?=2→F?=2
?right-hand side ?rhs?? component has become somewhat
smaller than the F?=2→F?=1 ?left-hand side ?lhs?? compo-
nent. Peculiarly, when the laser intensity is further increased
to 25 mW/cm2?Fig. 3?c??, the rhs component becomes
somewhat larger than the lhs component, while the widths of
the peaks ???=75 MHz? are substantially larger than the
width ??sat=16.5 MHz expected from saturation broaden-
ing at this laser intensity.
The 3s1/2, F?=2→3p3/2, F?=1,2,3 excitation spectra are
shown in Fig. 4. The relative peak intensities match the the-
oretical line strengths of individual HF transitions when laser
intensity is very small ?15 ?W/cm2, Fig. 4?a??. When laser
intensity is increased to 1.5 mW/cm2, which is still below
the saturation intensity, the relative intensities of the compo-
nents corresponding to the excitation of the F?=1 and F?
=2 HF levels are smaller than expected from the theoretical
line strengths ?Fig. 4?b??. When laser intensity is close to
saturation intensity, the F?=1 and F?=2 peaks are so weak
compared to the F?=3 peak that it is ambiguous to attempt
analysis of their linewidth.
The 3s1/2, F?=1→3p1/2, F?=1,2 excitation spectra are
shown in Fig. 5. Like in the case of Fig. 3, also here a
significant broadening is observed at laser intensities below
the saturation limit. In contrast to Fig. 3?b?, however, the lhs
peak corresponding to the excitation of the F?=1 component
of the upper state grows monotonically as compared to the
rhs peak.
IV. THEORETICAL LINE STRENGTHS AND SATURATION
INTENSITY
At very small laser intensities the strengths of individual
peaks in the excitation spectra shown in Figs. 3–5 corre-
spond to the respective theoretical line strengths Si
vidual HF transitions i=?F?→F?? within the D1?j=1/2? or
?j?of indi-
-100
(b)
0100200300
0.0
0.2
0.4
0.6
0.8
1.0
Theory:
Ilas=24?W/cm2
Experiment
Ilas~21?W/cm2
?=1/2
?=5/6
(a)
0.0
0.2
0.4
0.6
0.8
1.0
Theory
Ilas=0.93mW/cm2
Experiment
Ilas~1.0mW/cm2
?=1/2
?=5/6
-1000100200300
0.0
0.2
0.4
0.6
0.8
1.0
Theory:Ilas=23mW/cm2
Laserdetuning[MHz]
(c)
?=1/2
?=5/6
Experiment
Ilas~25mW/cm2
Fluorescence intensity [arb.units]
FIG. 3. Excitation spectra of the 3s1/2, F?=2→3p1/2, F?=1,2
transitions in Na. Residual Doppler width due to finite collimation
angle is ??D=11.2 MHz ?at b=2 mm?. The expected peak ratio is
1:1. Saturation intensities of the lhs and rhs components are 7.5 and
12.5 mW/cm2, respectively.
0.0
0.2
0.4
0.6
0.8
1.0
Theory:
Ilas=17?W/cm2
(a)
Experiment
Ilas~15?W/cm2
?=1
?=1/2
?=1/6
-100
Laserdetuning[MHz]
-50050
0.0
0.2
0.4
0.6
0.8
1.0
Theory:
(b)
Experiment
Ilas~1.5mW/cm2
?=1/6
?=1
?=1/2
1- Ilas=17?W/cm2
2- Ilas=0.63mW/cm2
3- Ilas=1.7mW/cm2
1
2
3
Fluorescence intensity [arb.units]
FIG. 4. Excitation spectra of the 3s1/2, F?=2→3p3/2, F?
=1,2,3 transitions in Na. Residual Doppler width due to finite col-
limation angle is ??D=15.9 MHz ?at b=3 mm?. The expected
peak ratio is 1:5:14. Saturation intensities of the three components
are 37.4, 12.5, and 6.2 mW/cm2, respectively.
BROADENING AND INTENSITY REDISTRIBUTION IN …
PHYSICAL REVIEW A 77, 042511 ?2008?
042511-3

Page 4

D2?j=3/2? lines ?14?. The values of Si
to the reduced matrix elements of the transitions and the
partial natural width of the respective transition:
?j?are directly related
Si
?j?= ??1/2,F??D?j,F???2;
?nat
?i?=
4?i
3?c3
3
1
2F? + 1Si
?j?.
?3?
Intensity of the component i is proportional to its line
strength because the product ?nat
ton flux of this component under the conditions of thermo-
dynamic equilibrium ?14?. Figure 6 shows the theoretical line
strengths S˜i
element ??3s?D?3p??2of the unresolved 3s→3p transition,
i.e., Si
such that ?j,iS˜i
ratios can be directly taken from Fig. 6, and they agree with
?i??2F?+1? regulates the pho-
?j??square frames? in units of the reduced matrix
?j?=S˜i
?j???3s?D?3p??2. The values of S˜i
?j?=8 ?14?. Thus the theoretical values of peak
?j?are normalized
the experimental observations at very small laser intensities
?see Figs. 3?a? and 4?a??.
Saturation intensity of each hyperfine transition depends
on the natural width of the transition ??natand the branching
ratio ?i?3? ?see also in Sec. V?:
Isat
?i?=4?3?c
3?i
3
??nat
?i
.
?4?
The values of the hyperfine branching coefficients ?i?circu-
lar frames in Fig. 6? are easily obtained from the reduced line
strengths S˜i
natural width of ??nat=9.8 MHz.
The saturation intensity given by Eq. ?4? gives the limit-
ing laser intensity after which stimulated transitions start
transforming the excitation spectra ?3?. The lowest laser in-
tensities used in measurements of the spectra shown in Figs.
3–5 are much smaller than saturation intensity Isat
the HF transitions. Therefore one naturally expects the peak
ratios to be in accordance with the line strengths of Fig. 6
and the linewidth to correspond to the residual Doppler
width determined by beam divergence. This agrees with the
observations made for the smallest laser intensities.
When laser intensity is increased by a factor of about 100,
it is still well below the saturation intensity Isat
less, a curious transformation of the spectra is observed:
widths of the peaks increase, and peak ratios of the HF com-
ponents change. Interestingly, the strongest F?=2→F?=3
component of the D2line ?Fig. 4? is not affected by broad-
eningatallalthoughits
=6.2 mW/cm2is the smallest. Intuitively, one would expect
the transition F?=2→F?=3 to be the first one that is affected
by broadening when laser intensity is increased.
Not only the widths are affected. Relative intensities of
the HF peaks change as laser intensity is increased. At first
glance it seems that relative intensities of components with
smaller branching ratios ?ishould decrease when optical
pumping becomes non-negligible, as less population returns
to the lower laser coupled level than it does for levels with
larger ?i. This is clearly the case in Figs. 3?b?, but not in the
case of Fig. 5. Moreover, Fig. 3 shows another unexpected
feature: after the intensity of the peak with smaller ?ihas
initially decreased with respect to the peak with larger ?i
?cf., Figs. 3?a? and 3?b??, a further increase of laser intensity
leads to an increase of the peak with smaller ?i?cf., Figs.
3?b? and 3?c??. Explanation of these observations requires a
detailed analysis of the dynamics of optical pumping.
?j?. Note that all HF transitions have the same
?i?of any of
?i?. Neverthe-
saturationintensity
Isat
?i?
V. DYNAMICS OF OPTICAL PUMPING AND ITS EFFECT
ON THE FLUORESCENCE SIGNALS
The measured fluorescence signals are affected by various
factors like the detection efficiency and geometry. The spec-
tral components are excited and detected at very close wave-
lengths under identical conditions, therefore it can be safely
assumed that the detection efficiency is equal for all of them.
Since we are interested in relative intensities and widths of
the components, it is sufficient to consider the fluorescence
signals that are proportional to the total number of photons
-1000 100200 300
0.0
0.2
0.4
0.6
0.8
1.0
E xperim ent:1
~5.7?W /cm
~0.94m W /cm
~7.1m W /cm
2
2
3
2
2
Theory:
1-Ilas=4.9?W/cm2
2-Ilas=1.1mW/cm2
3-Ilas=7.8mW/cm2
Fluorescen. intensity[arb.units]
Laserdetuning[MHz]
2
3
1
?=1/6
?=1/2
FIG. 5. Excitation spectra of the 3s1/2, F?=1→3p1/2, F?=1,2
transitions in Na. Residual Doppler width due to finite collimation
angle is ??D=11.2 MHz ?at b=2 mm?. The expected peak ratio is
1:5. Saturation intensities of the lhs and rhs components are 37.4
and 12.5 mW/cm2, respectively.
(a)
(b)
F’=2
F’=1
F”=2
F”=1
190MHz
1772MHz
3P3/2
3S1/2
1/6 1/2
5/6 1/2
1
1/2 1/2
5/6
1/6 1/21
F’=3
F’=2
F’=1
F’=0
F”=2
F”=1
59MHz
34MHz
16MHz
1772MHz
3P1/2
3S1/2
FIG. 6. Line strengths S˜i
?i?circular frames? for ?a? 3s1/2→3p1/2and ?b? 3s1/2→3p3/2hy-
perfine transitions.
?j??square frames? and branching ratios
SYDORYK et al.
PHYSICAL REVIEW A 77, 042511 ?2008?
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Page 5

emitted by atoms at all times in all directions.
We consider the following model problem. Two-level at-
oms with the ground state g and the excited state e propagate
along the z axis with the flow velocity of the beam vf. The
atoms cross the laser beam with radius rlas, frequency ?, and
Gaussian intensity distribution
I?z? = Ilasexp?− z2/rlas
2?;
?tr= 2rlas/vf.
?5?
Such distribution corresponds to Gaussian switching of Rabi
frequency ? of the g-e transition:
??t? = ?0exp?− 2t2/?tr
2?;
t = z/vf.
?6?
The value ?0=E0?g?dz?e? is the Rabi frequency of the g-e
coupling in the center of the laser beam which is linearly
polarized parallel to the z axis.
A. Evaluation of the fluorescence signal
In what follows we shall assume that the transit time is
much larger than the lifetime of the upper state, ?tr??nat,
which is true for the parameters of our experiment. This
allows us to use the adiabatic elimination for the nondiagonal
density matrix element ?eg?15?:
?eg?t? =
i??t?
?e− i2??ng?t? − ne?t??.
?7?
The above equation relates ?eg?t? to the populations ng?t?
=?gg?t? and ne?t?=?ee?t?. The decay rate ?e=1/?natgives the
natural width of the level e, while ?=2???Lis the laser
detuning. With ?egdefined by Eq. ?7?, the time evolution of
the populations is given by simple balance equations:
d
dtne= − ?ene+ r?t??ng− ne?;
?8?
d
dtng= ??ene+ r?t??ne− ng?.
?9?
The first equation describes the population loss from level e
via two processes: ?i? spontaneous decay at the rate ?e, and
?ii? stimulated emission at the rate equal to the optical pump-
ing rate r?t?:
r?t? = ?e
?2?t?
4?2+ ?e
2.
?10?
The population of level g is affected by three competing
processes: ?i? photon absorption at the rate r?t? resulting in
the population of level e, ?ii? return of population from level
e to level g due to stimulated emission, and ?iii? return of
population from level e to level g due to spontaneous emis-
sion. The rate of the latter is determined by the branching
coefficient ? of the given HF transition. The branching co-
efficients are normalized such that ?=0 for an entirely open
system ?no spontaneous return from level e to level g? and
?=1 for a closed system ?no transitions outside the g-e sys-
tem?.
The initial conditions of Eqs. ?8? and ?9? follow from the
requirement that initially all the population is in level g while
level e is not populated: ng?t=−??=1; ne?t=−??=0. The as-
sumption ?tr??natleads to a further simplification of Eqs.
?8? and ?9? in the weak excitation limit, when r?t??0.5?e.As
weak excitation we understand excitation at laser intensities
smaller than the saturation intensity given by Eq. ?4?, i.e.,
when Rabi frequency of the transition does not exceed the
saturated value, ?0??sat, where ?sat??e/?2 ?3?. In that
case, the adiabatic elimination implies that dne/dt=0 ?15?,
and Eq. ?8? immediately yields
− ?ene+ r?t??ng− ne? = 0 ⇒ ne?t? =r?t?
?e
?ng?t? − ne?t??.
?11?
Equation ?9? can then be transformed into the form
dt??1 +
d
r
?e?n−?= − r?t??1 − ??n−?t?;
n−?t? ? ng?t? − ne?t?.
?12?
The above equation can be comfortably used for the evalua-
tion of the fluorescence signal J. Integration of both sides of
Eq. ?8? yields the total number of spontaneous photons emit-
ted by the excited atoms:
J = ?e?
−?
?
dt ne?t? =?
−?
?
dt r?t?n−.
?13?
Integration of Eq. ?12? and combination of the result with
Eq. ?13? yields
?1 − ??J = 1 − n−?t = ?? = 1 − ng?t = ??.
?14?
The above expression has a straightforward physical mean-
ing: the number of spontaneously emitted photons on transi-
tions outside the g-e system is equal to the total loss of
ground state population during interaction with the laser
field.
Using Eqs. ?12? and ?14?, we can derive the fluorescence
signal in an explicit analytical form:
J =
1
?1 − ???1 − exp?− ?1 − ??R??;
R =?
−?
?
dt
r?t?
1 + r?t?/?e
.
?15?
Since we consider the case of weak excitation when r?t?
?0.5?e, the integral R in Eq. ?15? further simplifies to the
form
R ??
−?
?
dtr?t? =???e?tr
2
?0
2
4?2+ ?e
2;
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?0? ?sat? ?e/?2.
?16?
Dependence of the fluorescence signal on the laser detuning
?=2???Lcan now be rewritten as
J??? =???0
2?e
2?tr
1
Ppump?1 − exp?−
Ppump
1 + 4??/?e?2??;
Ppump=
?tr
?0?
?pump
;
?pump
?0?
=
2?e
2?1 − ??.
???0
?17?
The above equation shows that the excitation spectrum
strongly depends on the pumping parameter Ppump, which is
given by the ratio of transit time ?trand pumping time ?pump
The latter was already discussed in Sec. I ?Eq. ?2??, and it has
the meaning of optical pumping time at resonant excitation
??=0?. Importantly, the parameter Ppumpcan be large even at
laser intensities well below the saturation limit: Ppump?1
when ?tr??pum
?0?
.
?0?and ?0??sat.
B. Line broadening by optical pumping
When the pumping parameter is small ?Ppump?1?, Eq.
?17? simplifies to yield the ordinary Lorentz line shapes:
JL??? = R =???0
2?e
When Ppumpis increased, Eq. ?18? no longer holds and Eq.
?17? must be used. An almost tenfold increase of the line-
width is observed as Ppumpis increased from 0.1 to 50 ?see
Fig. 7?. Such broadening has a simple explanation. Consider
a near resonant case, when ??L=?/2??0. Starting from
values Ppump?1 the atoms spend sufficient time in the laser
field for the population of the ground state to be depleted,
ng?t=???0. Depletion of level g is associated with the
emission of a fixed number of photons 1/?1−?? ?see Eq.
?14??. Hence optical pumping saturates the observed signal
I???0? via depletion saturation, provided that ?pump???
??tr. Further increase of Ppumpcannot increase the number
2?tr
1
1 + 4??/?e?2.
?18?
of photons emitted upon excitation at the line center. At the
same time, the number of photons emitted upon excitation in
the wings continues increasing with Ppumpuntil depletion
saturation is reached at consecutively larger laser detunings
?. Therefore linewidths in the excitation spectra will increase
with Ppump, and line shapes will exhibit the characteristic
flat-top peaks at large Ppump.
The relative increase of the width ??OP=?OP/2?
?FWHM? of the line profile affected by optical pumping as
compared to the natural width can be easily obtained from
Eq. ?17?:
=?
??OP
??nat
Ppump
ln 2 − ln?1 + exp?− Ppump??− 1;
??nat= ?e/2?.
?19?
Variation of the width with Ppumpis shown on Fig. 8. As can
be seen, the broadening becomes noticeable at about Ppump
=1, and further increase of the width scales as the square
root of Ppumpfor large values of Ppump.
The condition Ppump?1 for broadening by optical pump-
ing can be reformulated in terms of Rabi frequencies, i.e.,
Rabi frequency of the laser-driven transition must be larger
than some critical value ?cr:
? ? ?cr=?
2
?1/2?nat?tr?1 − ??;
?20?
?cr= ?sat?
2?nat
?tr?1 − ??.
?21?
Inequality ?20? generalizes the results obtained in ?5,12? for
the limit of entirely open level systems with ?=0, which is
often used as an approximation in the case of molecules with
many possible rovibronic transitions. Broadening by optical
pumping turns out to be sensitively dependent on the branch-
ing ratio ?. In the limit of a closed level system ??=1? ?cr
is formally equal to infinity. It is not surprising: if there are
two isolated quantum states, then no pumping can occur re-
gardless of how strong the exciting laser field is.
-10 -8-6-4
Reduceddetuning
-20246810
0.0
0.2
0.4
0.6
0.8
1.0
Fluorescen. intensity[arb.units.]
50
30
15
10
5
2
0.1
FIG. 7. Signal J ?Eq. ?17?? as a function of the reduced detuning
?/?efor different values of the pumping parameter Ppump?shown as
labels of the curves?.
02
Pumping parameter Ppum
46810
0
1
2
3
4
Reduced width
FIG. 8. The reduced width ??pum/??natgiven by Eq. ?19? as a
function of the pumping parameter Ppump.
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It is important to note that the above described broadening
mechanism differs from the classical textbook examples of
saturation and power broadening ?6?. In ?6?, the saturation
broadening is attributed to the strong field effects, when
light-induced pumping rate becomes comparable to the re-
laxation rates, while the power broadening is attributed to a
considerable Rabi flopping frequency. The saturation param-
eter in both cases is given by the ratio of pumping rate to
relaxation rate. In our considered case, in contrast, the pump-
ing rate is very small, laser intensity is well below the tradi-
tional saturation intensity given by Eq. ?4?, yet a notable
depletion of level g is reached via optical pumping in a par-
tially open two-level system due to long interaction time
with the laser field; the line broadening thus occurs in the
weak excitation limit, when ne?t??ng?t?.
Line broadening by optical pumping is thus the dominat-
ing line broadening mechanism when ?crit????sat. At
???satthe rate of laser-induced transitions exceeds the
spontaneous transition rate. When the splitting ???? of laser
dressed states exceeds their widths, the population is equally
shared between the levels e and g ?2,16?. The pumping time
?pump?0? then stabilizes at 2/??e?1−??? and becomes inde-
pendent of further increases of laser intensity.
C. Critical laser intensity
It is useful to rewrite Eq. ?20? in terms of laser intensities,
since those are usually used by experimentalists. We shall
therefore express the laser intensity in terms of the transition
Rabi frequency as follows:
Ilas=4?2?c
3?3
?2
?e?.
?22?
The above equation is well-known for closed systems with
?=1 ?3?. In the case of a partially open system we have
replaced the natural width ?eby the partial natural width
?e? of the given transition. This can be done because Rabi
frequency ?=Ed/? involves the dipole element d=?g?dz?e?
associatedwith thepartial
=4?3d2/3?c3?14?. Since Ilas=E2c/8?, we obtain Eq. ?22?.
Using Eq. ?22?, Eq. ?20? can be rewritten as
natural broadening
?e?
Icr=8?2?c
??3?3
1
?tr??1 − ??=
4?nat
???tr?1 − ??Isat.
?23?
Simultaneously, the pumping parameter can now be rewritten
in terms of the ratio of the actual laser intensity to the critical
laser intensity:
Ppum= Ilas/Icr.
?24?
Equation ?23? shows the relation between the critical laser
intensity and the traditional saturation intensity ?Eq. ?4??,
which gives the limit for onset of power broadening.
Note that formally Icr→? when ?→0 ?completely open
system?. The limit of ?→0 corresponds to d→0, i.e., to
forbidden optical transitions. Therefore even a very small
transfer of population from level g to level e will require an
extremely large laser intensity.
VI. RESIDUAL DOPPLER BROADENING
Besides the broadening due to optical pumping, the spec-
tral lines are also affected by a small but non-negligible Dop-
pler broadening due to finite divergence angle ? of the
atomic beam. Such divergence is associated with nonzero
velocity components in the direction of the laser beam for
atoms moving not exactly parallel to the atomic beam axis.
For atoms experiencing the Doppler shift ?? the laser de-
tuning ? will transform into the detuning ?+??. A corre-
sponding replacement ?→?+?? should therefore be done
in Eq. ?17?. The resultant profile of a line in the excitation
spectrum is thus given by a sum of profiles ?17? resulting
from absorption of laser photons by atoms of different veloc-
ity groups. If the probability of atoms to have a velocity
leading to the Doppler shift ?? is PD????, then the resultant
line profile is given by the integral
Jres??? =?
−?
?
d??PD????J?? + ???.
?25?
The analytical form of the function PD???? for effusive
beams has been derived in ?17?, while for the case of super-
sonic beams it will be analyzed in detail in ?18?. This analy-
sis builds on the following assumptions: ?i? nozzle diameter
d is small compared to the diameter b of the entrance aper-
ture and distance L from the nozzle to the excitation zone
?see Fig. 1?; ?ii? divergence angle ? of the atomic beam is
small; ?iii? size of the excitation zone ?rlasis small com-
pared to the distance L, and the distribution of atoms within
?rlasis uniform; and ?iv? the velocity distribution in the
direction perpendicular to the atomic beam axis is due to the
divergence of the beam with the axial velocity distribution
F?v?. The distribution functions F?v? for various kinds of
beams can be found in ?19?.
Leaving the somewhat lengthy detailed derivation of the
function PD???? to the forthcoming paper ?18?, we shall
give here the final form of the most essential core ?????
???D/1.5? part of the distribution function for the super-
sonic beam:
PD
?cor????? =
2
???D?1 − ??2/??D
2,
?26?
with
??D?vf
?
?
2;
? =b + d
L
.
?27?
The values of the parameters vf, ?, b, d, and L are given in
Sec. II. Note that the function ?26? deviates strongly from the
Gaussian function, which is usually associated with Doppler
profiles. The frequency dependence of PDin the wings of the
spectral line ????????D/1.5? differs from that given by Eq.
?26?. Nevertheless, in our case it is sufficient to use only the
core part of PD. Since ??Dis comparable with ??nat, the
natural broadening outcompetes the exponentially small
wings of the Doppler profile at large ?? ?18?.
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VII. RESULTS AND DISCUSSION
Calculations of the theoretical spectra are performed in
two steps: ?i? solution of the evolution problem for an indi-
vidual atom excited by linearly polarized laser field detuned
by ??=?/2?, and ?ii? calculation of the resultant line profile
by performing the convolution ?25?. The first step is per-
formed by modeling quantum dynamics of individual pairs
of Zeeman sublevels mFwithin the F?mF?→F?mF?HF tran-
sition during coupling of the levels by the electrical field of
laser light distributed as ?E??=E0exp?−z2/2rlas
Correspondingly, the spatial distribution of Rabi frequencies
of individual HF transitions also follows the Gaussian distri-
bution:
2??e ?zcos??t??.
??m?= E0exp?− z2/2rlas
2??F?mF??dz?F?mF??.
?28?
It is convenient to introduce the reduced Rabi frequency ?red
associated with the unresolved 3s-3p transition:
?red?E0
????3s?D?3p??.
?29?
Rabi frequencies of individual Zeeman components can then
be calculated from ?redusing the known line strengths S˜i
given in Fig. 6 and the 6j symbols ?14?:
?j??
?j?
?0
?m?= ?red?S˜i
F?
1 F?
− mF?0 mF??,
?30?
where indexes i and j indicate the chosen HF component and
the chosen level of the upper state, respectively.
The values of ?redused in the calculations may be ob-
tained from their relation to the laser intensity Ilas?see Eq.
?22??:
Ilas=4?2?c
3?3
?3p?red
g3p
2
,
?31?
where g3p=3 is the statistical weight of the 3p state. Note
that in the calculations of theoretical spectra we used ?redas
the only fitting parameter. The theoretical values of Ilasgiven
in Figs. 3–5 were calculated from the fitted ?redvalues using
Eq. ?31?, and they are in a good agreement with the experi-
mental values calculated from measured laser power and ra-
dius of the laser beam.
For a qualitative interpretation of the experimental results
it is helpful to consider a simplified model in which the
populations of Zeeman components evolve independently
and the resulting signal J is simply a sum of individual sig-
nals Jm with the same mF?=mF?=m: J????=?mJm????.
However, in reality Zeeman sublevels are subject to sponta-
neous emission on transitions with ?mF=?1. As will be
shown below, such cascading can significantly change the
excitation spectrum as compared to the simplified treatment
when couplings with ?mF=?1 are neglected.
In order to account for cascading, we have elaborated an
accurate numerical algorithm allowing the integration of
equations of motion for the density matrix ?11,20?
d?
dt= −i
??H,?? −1
2??? + ??? + L???,
?32?
whereby Zeeman structure of all sublevels of the system de-
picted in Fig. 2 is taken into account. In Eq. ?32?, the Hamil-
tonian H describes the system “atom+laser field,” the matrix
? describes the spontaneous emission, and L??? describes the
cascade effects and has a simple explicit form in the repre-
sentation of polarization moments ?5?. In order to achieve a
fast and efficient solution of Eq. ?32?, we employ the split
propagation technique ?21,22?.
A. Regular changes of line profiles
The calculated excitation spectra J???? in the case of very
small laser intensities ?Ilas?Icr? are shown in Figs. 3?a? and
4?a?, whereby the residual Doppler broadening has been
taken into account by performing the convolution ?25?. An
excellent agreement with the experimental results is ob-
served. One can also see that the relative peak intensities
correspond to those expected from the theoretical line
strengths given in Fig. 6.
The theoretical spectra in the case when Icr?Ilas?Isatare
shown in Figs. 3?b? and 4?b?. An interesting observation can
be made in Fig. 3?b?. Intuitively one would expect that opti-
cal pumping is manifested more strongly and at smaller laser
intensities for lines with smaller values of branching ratio ?i,
when only a small fraction of population spontaneously re-
turns to the initial level. However, this is not the case. In fact,
Eq. ?23? implies that the critical laser intensity Icrhas a mini-
mum at ?=0.5. Therefore nonlinear effects associated with
optical pumping are more pronounced for HF transitions
with branching coefficients ? close to 0.5. Note that all the
excited HF levels considered here have equal lifetimes ?16.2
ns?. Low values of ?i
of both the line strengths Si
quencies ?0
cient for transitions with small ?i
out to be most pronounced for transitions with ?=0.5. This
can be best seen in Fig. 5: the relative intensity of the smaller
peak with ?i=1/6 increases with respect to the stronger
peak with ?i=1/2 as the laser intensity is increased in the
range Icr?Ilas?Isat. This is because the lower level in the
case of the component with ?i=1/2 is depleted more
quickly than it is in the case of the component with ?i
=1/6.
Another important consequence of optical pumping is line
broadening, which can be observed when laser intensity Ilas
is close to the critical value Icr?Eq. ?23?? of the given tran-
sition, or when Rabi frequency ? is close to the critical Rabi
frequency ?crgiven by Eq. ?21?. Since Icrhas a minimum at
?=0.5, the spectral lines with such a branching ratio are
most strongly affected by broadening due to optical pump-
ing. Dependence of the linewidth on laser intensity is illus-
trated in Fig. 9?a? for the 3s1/2, F?=2→3p1/2, F?=1 transi-
tion with ?i=5/6 ?lhs peak in Fig. 3? and the 3s1/2, F?=2
→3p1/2, F?=2 transition with ?i=1/2 ?rhs peak in Fig. 3?.
One can see that remarkable broadening takes place at laser
intensities below the saturation intensity Isat?marked in Fig.
?j?are thus associated with low values
?j?and the individual Rabi fre-
?m?, such that interaction with laser light is ineffi-
?j?. Optical pumping turns
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Page 9

9?a? with arrows?, while the component with ?i=1/2 ?F?
=2? exhibits broadening at smaller intensities than the other
transition. For comparison, dashed curves in Fig. 9?a? show
the intensity dependence of the power broadened linewidths
calculated as ??pow=??nat?1+I2/Isat
obvious that in the laser intensity range considered here the
power broadening is much smaller than broadening due to
optical pumping even at intensities exceeding the saturation
intensity.
?i?2?3?. It is immediately
B. Irregular changes of line profiles and Zeeman structure
Variations of the excitation spectrum of the 3s1/2, F?=2
→3p1/2, F?=1,2 transitions with laser intensity ?Fig. 3? are
significantly different from those observed for the 3s1/2, F?
=1→3p1/2, F?=1,2 transition ?Fig. 5? in two ways. ?i? The
relative intensity of the peak with ?i=1/2 first decreases
slightly and then increases as the laser intensity is increased.
?ii? Broadening of the peak with ?i=1/2 is actually smaller
than broadening of the peak with ?i=5/6. The key to under-
standing such striking differences is in the different Zeeman
sublevel structures in both cases. Numerical simulations us-
ing Eq. ?32?, which include cascade transitions with ?mF
=?1, yield a ratio R between the peak with ?i=1/2 and
the peak with ?i=5/6, which initially decreases with in-
creasing laser intensity and reaches a minimum at Ilas
?1.3 mW/cm2?see Fig. 9?b??. As laser intensity is further
increased, the ratio starts growing, reaches unity at Ilas
?3.2 mW/cm2, and grows to values slightly larger than
one. If the cascade transitions with ?mF=?1 are ignored,
the calculations yield a monotonously decreasing ratio R
?lower curve in Fig. 9?a?? without any “abnormalities.”
The effect of ?mF=?1 transitions becomes obvious at
closer inspection of Zeeman sublevels involved in the 3s1/2,
F?=2→3p1/2, F?=1 ?Fig. 10?a?? and 3s1/2, F?=2→3p1/2,
F?=2 ?Fig. 10?b?? transitions. Since the laser field is linearly
polarized, only the levels with the same mFare coupled by it.
The presence of “dark” levels becomes immediately obvious.
In the case of the F?=2→F?=1 transition, the mF=?2 sub-
levels of the lower level are not coupled by the laser field and
thus act as dark states, which accumulate population chan-
neled to them via optical pumping from the mF=?1 sublev-
els of the upper level. As a result, the branching coefficient
?i=5/6 should be replaced by a smaller effective branching
coefficient ?eff?F?=2,F?=1??5/6, which accounts for the
population loss to dark states. In the case of the F?=2
→F?=2 transition the dark state is mF=0 ?due to the selec-
tion rule ?mF?0 for F?=F??, therefore ?eff?2,2??1/2.
Increase of laser intensity leads to a larger population of
the dark states, and, consequently, to a monotonous decrease
of ?eff?F?,F??. At very weak laser fields ?eff?2,1?=5/6,
?eff?2,2?=1/2, and Icr?2,1??Icr?2,2? ?see Eq. ?23??. Hence
the transition F?=2→F?=2 is more strongly affected by op-
tical pumping than the other transition. Correspondingly, the
ratio R decreases with increasing laser intensity. As laser
intensity is further increased, both ?eff?2,1? and ?eff?2,2?
decrease. At some value of Ilasboth effective branching ra-
tiossatisfy the equality
=?eff?2,2??1−?eff?2,2??. In that case, Icr?2,1?=Icr?2,2?,
and both HF transitions are equally strongly affected by op-
tical pumping. This corresponds to the ratio R=1 at Ilas
=Ieq=3.2 mW/cm2in Fig. 9?b?. At Ilas?Ieqthe value of
?eff?2,2? becomes larger than ?eff?2,1?, such that Icr?2,1?
?Icr?2,2?, and the ratio R becomes larger than one. At large
laser intensities the ratio R asymptotically approaches the
value of 1.09. This is because the populations of the dark mF?
levels reach their maximum possible values when other mF?
levels are fully depleted. In the large intensity limit the val-
ues ?eff?2,2? and ?eff?2,1? differ by only 9%, therefore
both HF transitions exhibit similar broadening due to optical
?eff?2,1??1−?eff?2,1??
0
10
20
30
40
50
60
70
80
90
Isat
m'=2Isat
m'=1
F'=2
F'=1
(a)
Experiment
F'=1
F'=2
F'=1
F'=2
Full width[MHz]
0510 15 2025 3035
0.4
0.6
0.8
1.0
1.2
Without ?m=-1,1cascades
With?m=-1,0,1cascades
(b)
Peak ratio
LaserIntensityIlas[mW/cm2]
FIG. 9. ?a? The FWHM widths of the lhs ?F?=1? and rhs ?F?
=2? peaks of Fig. 3 as a function of laser intensity Ilas. Solid
squares, experiment, F?=1; open triangles, experiment, F?=2; solid
curves, theory; and dashed curves, pure power broadening neglect-
ing the broadening due to optical pumping. Arrows indicate satura-
tion intensities of both transitions. ?b? Calculated peak ratio R of
the rhs ?F?=2? and the lhs ?F?=1? peaks of Fig. 3 as a function of
laser intensity Ilas. The calculation was performed with and without
taking the ?mF=?1 cascades into account.
0
0
01
1
12
2
2
-2
-2
-2-1
-1
-1
00001111-1-1 -1-1
1/6
1/6
1/6
1/6
1/6
1/6
+++
F’’=2F’’=2
F’=1 F’=1
F’=2
F’=2
F’=2
+++
000011112222
-2-2-2-2
-1 -1 -1-1
000011112222
-2 -2-2 -2-1-1-1-1
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
F’’=2 F’’=2
(a)
(b)
FIG. 10. ?Color online? Zeeman sublevels involved in ?a? the
3s1/2, F?=2→3p1/2, F?=1 transition, and ?b? the 3s1/2, F?=2
→3p1/2, F?=2 transition. Spontaneous emission leads to the popu-
lation loss to the F?=1 level of the ground state, and to the dark
mF?=?2 levels in ?a? and to mF?=0 in ?b?. The effective branching
coefficient in case ?a? is therefore ?eff?2,1??5/6 and in case ?b? it
is ?eff?2,2??1/2.
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Page 10

pumping ?see Fig. 3?. In contrast, in the case of the 3s1/2,
F?=1→3p1/2, F?=1,2 transitions the component with ?i
=1/2 is apparently more strongly broadened than the com-
ponent with ?i=1/6, which could be expected ?see Fig. 5?.
VIII. SUMMARY
We have analyzed the effects of line broadening and re-
distribution of relative peak intensities in the hyperfine exci-
tation spectra of Na atoms due to optical pumping in the
weak excitation limit, when interaction times of atoms with
the laser field are long compared to the characteristic optical
pumping time. The study was motivated by the lack of avail-
ability of detailed theoretical models describing such effects
in partially open level systems at laser intensities below the
saturation limit. A number of significant results were ob-
tained. ?i? It is shown that spectral lines can be significantly
broadened at laser intensities well below the saturation inten-
sity, which is usually regarded as a threshold for onset of
broadening effects. ?ii? It is shown that the presence of dark
mFsublevels can vary the effective branching coefficients of
the transitions, and this variation depends on laser intensity.
Changes in the effective branching coefficients lead to ir-
regular changes of peak ratios. For example, the minimum in
the intensity dependence of the peak ratio deviates from the
ratio expected from the given original branching coefficients.
?iii? Analytical expressions are derived, which allow for the
calculation of critical values for the laser intensity and Rabi
frequency, above which linewidths and peak ratios are nota-
bly affected by optical pumping. ?iv? It is shown that the
critical laser intensity and critical Rabi frequency depend on
the branching coefficient ? of the transition and that they
have a minimum at ?=1/2.
Accurate theoretical simulations of the density matrix
equations of motion using the split propagation technique
yielded a good agreement with the experimental observa-
tions. In this study we have explored the limiting case of
long interaction times of atoms with a laser field, which jus-
tified the use of the adiabatic elimination approach. It is pos-
sible, however, to obtain explicit formulas for the excitation
spectra in the weak excitation limit also without the limita-
tion of adiabaticity in switching Gaussian laser pulses. In a
forthcoming publication we shall discuss some unexpected
effects related to transit time broadening in the other limiting
case, when the transit time is much smaller than the natural
lifetime.
ACKNOWLEDGMENTS
This work was supported by the EU FP6 TOK Project
LAMOL ?Contract No. MTKD-CT-2004-014228?, NATO
Grant No. EAP.RIG.981378, INTAS, Latvian Science Coun-
cil, and European Social Fund. We thank Professor K. Berg-
mann, Professor H. Metcalf, and Professor M. Auzinsh for
helpful discussions.
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