PHYSICAL REVIEW A 83, 042706 (2011)

Predicting and verifying transition strengths from weakly bound molecules
K. Aikawa,1,* D. Akamatsu,2,† M. Hayashi,1,† J. Kobayashi,2 M. Ueda,3,4 and S. Inouye2,3
12Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Institute of Engineering Innovation, The University of Tokyo, Yayoi, Bunkyo-ku, Tokyo 113-8656, Japan
3
JST, ERATO, Yayoi, Bunkyo-ku, Tokyo 113-8656, Japan
4
Department of Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan(Received 3 November 2010; published 11 April 2011)

We investigated transition strengths from ultracold weakly bound 41 K87 Rb molecules produced via thephotoassociation of laser-cooled atoms. An accurate potential energy curve of the excited state (3) 1  + wasconstructed by carrying out direct potential fit analysis of rotational spectra obtained via depletion spectroscopy.
Vibrational energies and rotational constants extracted from the depletion spectra of v  = 41–50 levels werecombined with the results of the previous spectroscopic study and they were used for modifying an ab initio
potential. An accuracy of 0.14% in vibrational level spacing and 0.3% in rotational constants was sufficient topredict the large observed variation in transition strengths among the vibrational levels. Our results show thattransition strengths from weakly bound molecules are a good measure of the accuracy of an excited state potential.
DOI: 10.1103/PhysRevA.83.042706

PACS number(s): 34.20.Cf, 33.20.Sn, 37.10.Mn, 82.80.Ms

I. INTRODUCTION

Ultracold molecular gases are prominent candidates for realizing novel distinctive applications in physics and chemistry,including precision measurements, quantum computation,
ultracold chemistry, and novel quantum phases [1,2]. Thus
far the production of ultracold molecules in the vibrational
ground state has been dependent on the optical transfer
of weakly bound molecules formed via either photoassociation [3] or magnetoassociation [4]. Among the previously proposed methods [5–10], stimulated Raman adiabatic
passage (STIRAP) [11–13] of weakly bound molecules is themost efficient method for preparing a molecular sample in asingle quantum state.
In the STIRAP transfer an excited state is used as anintermediate state. It is important to select an excited state
having large transition strengths from both weakly bound anddeeply bound levels. In general, the strengths of moleculartransitions from low-vibrational levels are readily predictedon the basis of the Franck-Condon factors (FCFs) calculated
from a potential energy curve or molecular constants [14]. The
small number of nodes in the radial wave function for a low
vibrational level indicates a low sensitivity to deviations in theradial direction, thereby enabling us to explain the intensity
distribution over vibrational levels from molecular constants.
However, it is difficult to predict transition strengths fromweakly bound molecular levels because weakly bound levels
have a large number of nodes in their wave functions; hence the
FCFs are quite sensitive to the wave functions of excited states.
In this article we show that transition strengths from weakly
bound levels can be predicted on the basis of an accurate
potential energy curve of the excited state, constructed via
direct potential fit (DPF) analysis [15] of both vibrationalenergies and rotational constants.

*

ka cypridina@atomtrap.t.u-tokyo.ac.jp
Present address: National Metrology Institute of Japan, Tsukuba
305-8563, Japan.
‡
Present address: NIKON Corporation, Kanagawa 252-0328, Japan.
†

1050-2947/2011/83(4)/042706(7)We focus on the (3) 1  + state of KRb in the range 11 400–
11 800 cm−1 with respect to the ground atomic threshold
4S + 5S which was proposed as a potential candidate for
the STIRAP transfer of weakly bound molecules to therovibrational ground state [16] (see Fig. 1). An RydbergKlein-Rees (RKR) potential curve of the (3) 1  + state wasreported from the potential minimum up to 10 400 cm−1 [17].
An ionization spectrum for 39 K87 Rb obtained by using a pulsed
laser and a depletion spectrum near 11 700 cm−1 were provided
in Ref. [16]. Recently we have realized the STIRAP transfer
of weakly bound molecules (v  = 91, J  = 0 of X 1  + ) to
the rovibrational ground state (v  = 0, J  = 0 of X1  + ),mediated by the v  = 41 level of the (3) 1  + state [19].
In this article we report on the spectroscopic studies on the(3) 1  + state carried out in the range 11 400–11 800 cm−1 ,which reveals 10 vibrational levels (v  = 41–50). We findthat the width of the observed spectra is highly dependent
on the vibrational levels of the (3) 1  + state. By introducing
an analytical representation for power broadening, we extract
the transition strengths for each vibrational level. The otherelectronic states that correlate with the (3) 1  + state via
spin-orbit interaction are far away; hence we assume thatmost of the perturbations from these electronic states are
negligible. Thus we can analyze the experimentally obtained
spectra on the basis of a single potential curve. In addition
to the previous spectroscopic work near the bottom of thepotential [17], vibrational energies and rotational constants
extracted from the spectra are used to construct an accurate
potential via DPF analysis. Using the modified potential, it ispossible to explain the variation in transition strengths among
the vibrational levels of the (3) 1  + state in terms of the FCFs.

II. DEPLETION SPECTROSCOPY

Previously our experimental setup for the spectroscopy ofultracold 41 K87 Rb molecules was described in detail [19]. We
provide a brief summary here. We start with a dual-species
magneto-optical trap (MOT) of 1 × 108 41 K atoms and 2 × 108
87
Rb atoms. A compressed MOT (C-MOT) procedure is042706-1©2011 American Physical Society

PHYSICAL REVIEW A 83, 042706 (2011)
Ion counts in v"=91, X Σ (arb.units)

K. AIKAWA et al.

30

1 +

Σ

Π

Σ

Σ

Π
Σ

J' = 1←J" = 0
20

J' = 3←J" = 2
J' = 1←J" = 2

10011428.911429.011429.1
11429.2-1
Depletion wavenumber (cm )11429.3

FIG. 2. (Color online) Depletion spectrum of v  = 41 level of(3) 1  + taken with molecules in v  = 91 level of X 1  + . The
spectrum gives rotational constants of both ground and excited states.
In addition, the width of the spectrum enables us to estimate thetransition strength.

where  is the Rabi frequency,  is the natural width of theexcited state, and τ is the interaction time. Roughly speaking,the width increases not only with the light intensity but alsowith the interaction time. In our case τ was estimated asa few milliseconds on the basis of the temperature of themolecules and the beam diameter of the depletion laser. The
natural width of the (3) 1  + state has not been precisely
3.0-1

applied for 40 ms to compress and cool the 41 K and 87 Rb
atoms. The typical densities and temperatures of 41 K and87
Rb at the end of C-MOT are 2 × 1011 cm−3 and 400 µK
and 4 × 1011 cm−3 and 100 µK for 87 Rb, respectively. A
photoassociation (PA) laser (wave number 12 570.13 cm−1 ,intensity 1 × 103 W/cm2 ) is applied for 10 ms at the endof the C-MOT process. The produced molecules are detectedusing microchannel plates (MCP) after they are ionized via
resonance enhanced multiphoton ionization (REMPI) using
a pulsed dye laser (wave number 16 543 cm−1 , intensity
3 × 106 W/cm2 ).
Depletion spectra are obtained by monitoring ion counts
in the v  = 91 level of X 1  + , while a CW Ti:sapphire laser(Sirah Matisse TX, intensity 50 W/cm2 , beam waist 350 µm)
is continuously applied and scanned. In the present study weanalyze spectra for the v  = 91, J  = 2 level, whose binding
energy with respect to the atomic threshold FK = 1 + FRb = 1
was measured as −12.454(1) cm−1 . The frequency of theTi:sapphire laser is monitored using a Fabry-Perot cavity
which is locked to the 87 Rb D2 line. The cavity transmission
signal is used to calibrate the variation in the scanning speed
and to measure the relative frequency of the laser with respect
to the cavity transmission peak with a precision of 3 MHz.
The absolute frequency is measured using a commercial wave
meter (Wavelength WS-7, accuracy 60 MHz). The accuracy ofmeasurements for rotational constants is limited by the spectral
width, which is of the order of 100 MHz, whereas that forvibrational energies is limited by both the spectral width andthe wave meter.
Figure 2 shows a depletion spectrum for the v  = 41 level ofthe (3) 1  + state, which was used as an intermediate state for
the STIRAP transfer from the v  = 91, J  = 0 level of X1  +to the v  = 0, J  = 0 level [18]. We can extract transition
strengths as well as rotational constants for both ground andexcited states from the spectra. In the Appendix we showthat an approximate representation of the full width at half
maximum (FWHM) of a depletion spectrum is given by√(1)2 ∼ 0.79 τ ,-2

FIG. 1. (Color) Relevant potential energy curves of KRb. Weakly
bound molecules were produced by the photoassociation of lasercooled 41 K and 87 Rb atoms. The (3) 1  + state was investigated via
the spectroscopy of molecules in the v  = 91 level of X1  + . Theintermediate state for REMPI ionization was the (4) 1  + state.

Rotational constants (10 cm )

Σ

2.8

1

3

(1) Π

(1) Π

1 +

1 +

(3) Σ

3 +

(2) Σ

(2) Σ

Observed

2.6
2.4
2.2
2.0
1.8
1.6
114 00115 00116 00117 00118 00
1 +-1
Transition wavenumber from v"=91, X Σ (cm )FIG. 3. (Color online) Comparison between calculated and observed rotational constants. Calculated rotational constants of five
molecular states in this range are plotted against energy levels fromthe ground atomic threshold. The observed rotational constants arein good agreement with those obtained from the ab initio potential ofthe (3) 1  + state. There is a systematic deviation of ∼1% between theobserved values and ab initio values, which indicates the inaccuracyof the ab initio potential.

042706-2

PREDICTING AND VERIFYING TRANSITION STRENGTHS . . .

PHYSICAL REVIEW A 83, 042706 (2011)

TABLE I. Parameters for EMO potential obtained via DPF
analysis of the observed spectra. The units are cm−1 for Vmin and
De , Å for Re , and Å−1 for φi .

an analytical extended Morse oscillator (EMO) potential of the(3) 1  + state on the basis of (1) the RKR curve, (2) a few points
from the inner curve of the ab initio potential, and (3) a few
points around 12 500 cm−1 from the outer curve of the abinitio potential. The potential curve of the ground state X 1  +is also required for the calculations. We use an EMO potential
fitted to an accurate, experimentally determined potential [23].
Then the analytical potential of the (3) 1  + state is modified
to reproduce our data by using the DPotFit program code [24].
A good convergence is achieved when we modify the potential
significantly by manually moving the points from the ab initio
potential. This procedure is repeated until the eigenvalues ofthe potential are within 0.05 cm−1 of the observed levels, thatis, only 0.14% of the vibrational level spacing. The remaining
deviations are presumably due to the incomplete analytical
function used to represent the potential, although perturbations
from other states can also be the origin of the deviations forsome levels. Table I lists the final potential parameters. These
parameters are used to represent the potential energy curve inthe following form:

Parameter
Vmin
De
Re
φ0
φ1
φ2
φ3
φ4

Value

Parameter

Value

9777.6963
3246.0363
5.259 041 19
0.449 794 066
0.200 265 883
0.406 840 126
0.207 884 795−0.476 349 301

φ5
φ6
φ7
φ8
φ9
φ10
φ11
φ12

−0.804 673 587
3.004 740 08
11.387 732 4−2.140 717 462−39.731 105 86−17.014 003 11
50.932 796 47
44.831 374 19

known, but it was estimated as 2π × 300 kHz from an
ab initio calculation [20]. In the following discussion we derive
the Rabi frequency from the observed√ spectrum by assumingτ = 2 ms,  = 2π × 300 kHz, and τ = 60.
III. ANALYSIS

Although the ab initio potential of Ref. [21] enabled us
to identify the symmetry of the observed spectra without
any ambiguity, it gave rotational constants that exceeded theexperimental ones by approximately 1% (Fig. 3). There are
two reasons for this deviation. First, the potential minimum
exceeds the minimum of the RKR curve by 97 cm−1 . Second,the outer turning point is at a shorter internuclear separation
than the RKR potential. We find that the variation in theobserved transition strengths cannot be understood with the
ab initio potential. The observations cannot be attributed tothe small change in the vibrational quantum number, whichcorresponds to the energy difference between the minima of the
RKR and the ab initio potentials. These facts indicate that the
ab initio potential is not sufficiently accurate. In order to obtainan accurate potential, we carried out a DPF analysis, whereby apotential is iteratively modified until its eigenvalues coincidewith those determined from the experimental spectra [15].
With the aid of the phiFIT program code [22], we first construct

V (R) = Vmin + De (1 − e−φ(R)(R−Re ) )2 ,φ(R) =12

φi y(R,Re )i ,

(2)

i=0

y(R,Re ) =R 3 − Re3.
R 3 + Re3

Figure 4 shows the RKR potential, ab initio potential, andmodified potential. With the new potential, rotational constants
are reproduced within 0.3% of the observed values (Table II).
We excluded the v  = 44 level from the analysis because thecharacteristics of this level are anomalous; a much larger
transition strength, a much smaller rotational constant, anda much larger deviation in a vibrational energy were observedat this level. These features indicate that this level is coupled
to the (2) 1  + state which was observed at ∼1 cm−1 above thev  = 44 level.

TABLE II. Comparison between observed, ab initio, and modified values for vibrational energies (E) and rotational constants (B). Vibrationalenergies are energy levels of the J  = 1 level. Error bars in vibrational energies are 2 × 10−3 cm−1 , whereas those in rotational constants are
5 × 10−5 cm−1 . Observed vibrational energies are reproduced within 5 × 10−2 cm−1 , whereas observed rotational constants are reproduced
within 7 × 10−5 cm−1 . The vibrational numbering for ab initio values is shifted by 2 because the potential minimum lies below that of thecorrect potential.
E (cm−1 )B (10−2 cm−1 )

v

Obs.

ab initio−Obs.

Mod.−Obs. (units of 10−3 )Obs.

ab initio−Obs. (units of 10−2 )Mod.−Obs. (units of 10−3 )

41
42
43
44
45
46
47
48
49
50

11 428.965
11 466.644
11 504.193
11 541.484
11 578.919
11 616.036
11 652.982
11 689.751
11 726.308
11 762.685

−1.5
−1.6
−1.8
−1.9
−2.2
−2.4
−2.6
−2.7
−2.7
−2.7

2.5
29.9
42.6
164.2
−11.4
−28.0
−37.3
−37.5
1.2
43.0

2.067
2.056
2.05
1.99
2.038
2.037
2.027
2.017
2.008
2.005

1.9
2.3
2.4
7.8
2.4
1.8
2.2
2.5
2.9
2.5

−7.0
−1.7
−1.5
52.6
−1.5
−6.8
−3.2
0.3
2.5
−1.4

042706-3

PHYSICAL REVIEW A 83, 042706 (2011)

Observed Rabi frequency (2π MHz)K. AIKAWA et al.

15

10

5

0

The most important implication of this analysis is that thevariation in the transition strengths with respect to the vibrational levels of the excited state can be accurately explained on
the basis of the new potential. Figure 5 shows a plot of the Rabi
frequencies derived from the observed spectra against those
calculated from the corrected potential. For comparison, thesame plots based on the ab initio potential are also shown. The
calculated values are calibrated on the basis of our recent measurement of the transition dipole moment between (3) 1  + ,v  = 41 and X1  + , v  = 91, 0.035(2)ea0 , determined via
dark resonance spectroscopy of the rovibrational ground-state
molecules. Weakly bound molecules have more than 90 nodes
in their wave function; hence the FCFs are highly dependent
on the wave function of the excited state. In other words,
the FCFs can serve as a sensitive measure of the accuracy ofwave functions. Our results show that accuracies of 0.14% in
vibrational level spacings and 0.3% in rotational constants are
sufficient to predict the FCFs from weakly bound levels; thesevalues are justified by considering the typical size of nodes in
the radial wave function. On one hand, the weakly bound levelv  = 91 in the ground state X1  + has an outer turning point of∼10 Å and an inner turning point of ∼2 Å in the internuclear
distance. Within these two points there are 91 nodes; hence
each node has a typical size of ∼0.1 Å. Therefore, the required
accuracy for representing the wave function is ∼10−2 Å. On
the other hand, in the present analysis an accuracy of ∼0.3%
in rotational constants or ∼0.15% in internuclear distance isobtained for the (3) 1  + state because the relation between
the rotational constant B and the internuclear distance R isgiven by B ∝ R −2 . Assuming the typical size of moleculesin the v  = 41–50 levels of (3) 1  + as 6 Å, we can derive theaccuracy of the modified potential in the radial direction as∼10−2 Å; this value is in good agreement with the required
accuracy for representing the weakly bound level.
Now that we obtained an accurate potential curve as wellas the absolute values of the transition dipole moment of thev  = 41, (3) 1  + level with the v  = 0 and v  = 91 levels of
X1  + , we can predict the transition dipole moment for each

Observed Rabi frequency (2π MHz)

42

FIG. 4. (Color) Potential energy curves of (3) 1  + state. The
modified potential obtained in the present study is compared with the
ab initio potential and the experimental RKR potential. The shaded
area denotes the region where depletion spectra were obtained.

(a)

500 mW
104 mW
11 mW (v'=44)
26 mW (v'=47)

15

44
46
Vibrational number

15

48

50

(b)

10
5

10

0
0

5

5

10

15

500 mW
104 mW
11 mW (v'=44)
26 mW (v'=47)

0
0

5
10
15
Calculated Rabi frequency (2π MHz)

FIG. 5. (Color online) Observed transition strengths. (a) Variation
among the vibrational levels. (b) Comparison between the observed
values and the calculated values. Observed values are determinedfrom the spectra obtained via a depletion laser at 104 mW (red circle)
and 500 mW (blue rectangle). For the v  = 44 and the v  = 47 levels,
the spectra are taken with low powers of 11 mW (v  = 44) and 26 mW(v  = 47) in order to extract the peaks correctly. Calculations based on
the modified potential can explain the variation in transition strengths
with respect to the vibrational levels in terms of the Franck-Condon
factors. The v  = 44 level is expected to be mixed with other states,
and thereby shows a significantly larger transition strength than otherlevels. The inset in (b) shows the same plot based on the ab initio
potential.

transition. Figure 6 shows our prediction for the transition
dipole moments of the (3) 1  + state with the least-bound stateand for those with the lowest rovibrational level (v  = 0).
The v  = 41 level used in Ref. [18] has favorable wave
function overlaps with both the weakly bound and the lowest
rovibrational levels; however other levels such as v  = 38
and v  = 39 can potentially serve as an intermediate state
for the STIRAP transfer of weakly bound molecules to therovibrational ground state. The potential presented can enable
an accurate prediction for other isotopic combinations of KRb.
Furthermore, the present method for achieving an accurate
potential and verifying its accuracy can be extended to othermolecular states that exhibit significant spin-orbit mixing
by evaluating eigenvalues via coupled channel calculations
including spin-orbit interaction.

042706-4

PREDICTING AND VERIFYING TRANSITION STRENGTHS . . .

PHYSICAL REVIEW A 83, 042706 (2011)

Transition dipole moment (units of ea0)

6
5

∆

4
3

Ω
2

0.01

Γ Γ1

Γ

9
8
7
6

Least bound level
Lowest vibrational level

5
4

36

38

40
42
44
46
Vibrational number

48

50

FIG. 6. (Color online) Transition dipole moments of the (3) 1  +state with the X1  + state calculated from the experimentally obtained
accurate potential. The absolute values for weakly bound levels arecalibrated by dark resonance spectroscopy for the transition v  =41,(3) 1  + ← v  = 91,X1  + , whereas those for the rovibrational
ground state are calibrated by dark resonance spectroscopy for thetransition v  = 41,(3) 1  + ← v  = 0,X1  + .
IV. CONCLUSION
1+The (3)  state of KRb was investigated via the depletion spectroscopy of ultracold molecules formed by thephotoassociation of laser-cooled 41 K and 87 Rb atoms. The
spin-orbit mixing of other electronic states with the (3) 1  +state was negligible; hence we could assume this state is welldescribed by a single potential. The simplicity of the (3) 1  +state enabled us to modify the potential to reproduce ourobservations as well as to assign the spectra. We observed
10 vibrational levels in the range 11 400–11 800 cm−1 withrespect to the 4S + 5S atomic threshold. We developed atheoretical model that related the spectral width with the Rabi
frequency, which was used to compare the transition strengths
for each vibrational level. Rotational constants extracted fromthe observed spectra showed a systematic 1% deviation fromthose calculated using an ab initio potential. By carryingout a DPF analysis we constructed an accurate potentialthat reproduced energy levels with an accuracy of 0.14% in
vibrational level spacing and 0.3% in rotational constants. The
variation in transition strengths among vibrational levels could
be understood in terms of FCFs calculated with the modified
potential. Our results indicate that the transition strengths
from weakly bound levels serve as a sensitive measure ofwave functions, which can be used to test the accuracy ofthe potential curve. In general, the proposed procedure can be
adopted for constructing an accurate potential and verifying its
accuracy on the basis of rotational spectra for weakly boundmolecules.
ACKNOWLEDGMENTS

FIG. 7. Energy levels relevant to depletion spectroscopy ofphotoassociated molecules. Weakly bound molecules are formed at
a rate of APA . A laser having Rabi frequency of  detuned from theresonance by  excites ground state molecules. Excited molecules
spontaneously decay to the initial level and other levels at rates of1 and 2 − 1 , respectively. We mainly consider a situation where2 > 1 is satisfied.

assistance with the experiment. K.A. and D.A. acknowledge
the support of the Japan Society for the Promotion of
Science.

APPENDIX: ANALYTICAL EXPRESSION FOR THE LINE
SHAPE OF A DEPLETION SPECTRUM

We consider a general situation (as shown in Fig. 7) and
we assume that radiative transitions occur much faster than
the time variation of the population in each molecular level
because of the photoassociative creation of molecules. We
first consider the evolution of the population in the ground andexcited states of a single molecule. The optical Bloch equation
for this system can be written asdss−w= −(2 − 1 )
,
dt
2
dw
s−w
= (2 + 1 )− 2Im(ρeg ),
dt
2


ρ
2
i
eg
=−
− i ρeg + w,
dt
2
2

(A1)

−it
ρ
,
eg ≡ ρeg e

where s = ρgg + ρee , w = ρgg − ρee , and |e and |g denote
an excited state and a weakly bound level, respectively. and  denote the detuning frequency and the Rabi frequency,
respectively. The decay rates from the excited state to theinitial weakly bound level and to other levels are given by 1and 2 , respectively. The photoassociation rate is denoted by
APA . The width is much larger than the Rabi frequency in theexperiment; hence we can assume that the time evolution ofs and w is much slower than that of ρeg . Thus, we can setd ρ
eg /dt = 0 and obtain the following expression for ρeg :We thank P. Naidon and T. Kishimoto for insightful
discussions, and K. Oasa, Y. Tanooka, and K. Mori for their
042706-5

ρ
eg =

iw
.
2 − 2i

(A2)

K. AIKAWA et al.

PHYSICAL REVIEW A 83, 042706 (2011)

Substituting Eq. (A2) into Eqs. (A1) we obtain alternativeequations for s and w asdss−w
= −(2 − 1 ),dt
2
22 2 w
s−w
dw= (2 + 1 )
− 2
.
dt
2
2 + 42

Ng (τ ) → APA τ.
(A3)

Here we assume that the time taken by the mean value ofthe population ratio w/s to attain a constant value z after isgreater than the typical time for radiative transitions. By using
the relation w = zs, Eqs. (A3) give the following equations:ds
1−z= −(2 − 1 )
s,
dt
2
ds
22 21−zz= (2 + 1 )
s− 2
zs.
dt
2
2 + 42

on resonance ( = 0 and w → 0), whereas Ng at an infinite
detuning ( → ∞ and w → 1) is given as(A4)

Ng at an infinite detuning is much larger than Ng on resonance.
Thus the width of the spectrum is determined by finding w suchthat it satisfies
Ng (τ ) = 12 APA τ.

An analytical solution of this equation is given by(A5)

This equation is readily solved, and it gives the following
expression for z:(1 + k 2 )22 + 21 2 k − (k2 + 1 ),z=
2 − 1
(A6)
22
k= 2
.
2 + 42This expression is used in the following discussion. Next, wederive rate equations for the population in the ground andexcited molecular levels for the number of molecules:dN= APA − (2 − 1 )Ne ,
dt
1−z
Ne =
N,
(A7)
2
1+z
N,
Ng =
2
where Ng and Ne denote the number of molecules in the groundand excited states, respectively, and N = Ng + Ne is the total
number of molecules. In these rate equations a typical timescale is of the order of 1 ms and it is governed by APA . This ismuch longer than the typical time scale for radiative transitions
in most cases (1 µs). The time evolution of Ng is given by
dNg
1+z
1−z
=
APA −
(2 − 1 )Ng .
(A8)
dt
2
2
Thus, the solution for Ng is given by



1−z
APA 1 + z
1 − exp −Ng (τ ) =.
(2 − 1 )τ
2 − 1 1 − z
2(A9)This expression gives the line shape of a depletion spectrum
for an interaction time τ . Assuming (2 − 1 )τ  1, Eq. (A9)gives
Ng (τ ) →APA2 − 1

4 + 2W (−2/e2 )(2 − 1 )τ
1
× 3.187 24 . . . ,=(2 − 1 )τ

x=

(A10)

(A14)

where W is the Lambert W function. Thus the parameter k in
Eqs. (A6) is given byk=

3.187 24(1 − 1.593 62/ 2 τ ).
(2 − 1 )τ − 3.187 24(A15)

Assuming (2 − 1 )τ  1, we obtain the following expression for the FWHM:
2 ≈ 0.79 (2 − 1 )τ .

(A16)

When decays from the excited state to the initial state
are negligible (2  1 ), we obtain a simple relation (1).
Expression (1) was experimentally confirmed by observing
the relation between the width and the intensity√ (Fig. 8). It
is difficult to evaluate the numerical factor τ precisely.
Therefore the width of a depletion spectrum cannot serveas an accurate measure of the transition strength. However,expression (1) enables us to systematically compare transition
strengths for different vibrational levels.
500
400
Width (MHz)42 2z.
22 + 42

(A12)

Substituting Eq. (A9) into Eq. (A12) and rewriting the equation
with a new variable x ≡ 1 − z ( 1), we obtain the following
equation:x2 − 12
1 − exp − (2 − 1 )τ
τ. (A13)=x
2
2

Substituting the first equation in the second equation we obtaina time-independent equation for z:(1 − z)[2 (1 + z) + 1 (1 − z)] =

(A11)

300
200
100
0
0

10

20
30
2
Peak intensity (W/cm )40

50

FIG. 8. (Color online) Width of the depletion spectrum vsintensity. The widths of the depletion spectra for the v  = 42 level areplotted with respect to the intensity of the depletion laser. The errors
indicate standard deviations in the fits to the spectra.

042706-6

PREDICTING AND VERIFYING TRANSITION STRENGTHS . . .

PHYSICAL REVIEW A 83, 042706 (2011)[1] R. V. Krems, W. C. Stwalley, and B. Friedrich, Cold Molecules:
Theory, Experiment, Applications (CRC, Boca Raton,
FL, 2009).
[2] L. D. Carr, D. DeMille, R. V. Krems, and J. Ye, New J. Phys.
11, 055049 (2009).
[3] K. M. Jones, E. Tiesinga, P. D. Lett, and P. S. Julienne, Rev.
Mod. Phys. 78, 483 (2006).
[4] T. Kohler, K. Goral, and P. S. Julienne, Rev. Mod. Phys. 78,
1311 (2006).
[5] J. M. Sage, S. Sainis, T. Bergeman, and D. DeMille, Phys. Rev.
Lett. 94, 203001 (2005).
[6] M. Viteau et al., Science 321, 232 (2008).
[7] J. Deiglmayr, A. Grochola, M. Repp, K. Mortlbauer, C. Gluck,
J. Lange, O. Dulieu, R. Wester, and M. Weidemuller, Phys. Rev.
Lett. 101, 133004 (2008).
[8] K.-K. Ni et al., Science 322, 231 (2008).
[9] F. Lang, K. Winkler, C. Strauss, R. Grimm, and
J. HeckerDenschlag, Phys. Rev. Lett. 101, 133005 (2008).
[10] J. G. Danzl et al., Nat. Phys. 6, 265 (2010).
[11] K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. 70,
1003 (1998).
[12] N. V. Vitanov, M. Fleischhauer, B. W. Shore, and K. Bergmann,Adv. At. Mol. Opt. Phys. 46, 55 (2001).
[13] P. Kral, I. Thanopulos, and M. Shapiro, Rev. Mod. Phys. 79, 53(2007).

[14] G. Herzberg, Molecular Spectra and Molecular Structure.
1. Spectra of Diatomic Molecules (Van Nostrand, New York,1950).
[15] J. Seto, R. Le Roy, J. Vergès, and C. Amiot, J. Chem. Phys. 113,3067 (2000).
[16] D. Wang, J. T. Kim, C. Ashbaugh, E. E. Eyler, P. L. Gould, andW. C. Stwalley, Phys. Rev. A 75, 032511 (2007).
[17] C. Amiot, J. Vergès, J. d’Incan, and C. Effantin, Chem. Phys.
Lett. 315, 55 (1999).
[18] C. Amiot, Chem. Phys. Lett. 318, 289 (2000).
[19] K. Aikawa et al., Phys. Rev. Lett. 105, 203001 (2010).
[20] K. Aikawa, D. Akamatsu, J. Kobayashi, M. Ueda, T. Kishimoto,and S. Inouye, New J. Phys. 11, 055035 (2009).
[21] R. Beuc et al., J. Phys. B 39, S1191 (2006).
[22] S. Rousseau, A. R. Allouche, and M. Aubert-Frecon, J. Mol.
Spectrosc. 203, 235 (2000).
[23] R. J. Le Roy, “A Computer Program to Fit Pointwise Potentials to
Selected Analytic Functions,” (2007), current version is betaFIT
2.0, available at [http://leroy.uwaterloo.ca/programs.html].
[24] A. Pashov, O. Docenko, M. Tamanis, R. Ferber, H. Knockel,
and E. Tiemann, Phys. Rev. A 76, 22511 (2007).
[25] R. J. Le Roy, J. Y. Seto, and Y. Huang, “A Computer Program for Fitting Diatomic Molecule Spectral
Data to Potential Energy Functions,” (2006), available at[http://leroy.uwaterloo.ca/programs.html].

042706-7