Diffusion, thermalization and optical pumping of YbF molecules in a cold buffer gas cell
ABSTRACT We produce YbF molecules with a density of 10^18 m^3 using laser ablation
inside a cryogenicallycooled cell filled with a helium buffer gas. Using
absorption imaging and absorption spectroscopy we study the formation,
diffusion, thermalization and optical pumping of the molecules. The absorption
images show an initial rapid expansion of molecules away from the ablation
target followed by a much slower diffusion to the cell walls. We study how the
time constant for diffusion depends on the helium density and temperature, and
obtain values for the YbFHe diffusion crosssection at two different
temperatures. We measure the translational and rotational temperatures of the
molecules as a function of time since formation, obtain the characteristic time
constant for the molecules to thermalize with the cell walls, and elucidate the
process responsible for limiting this thermalization rate. Finally, we make a
detailed study of how the absorption of the probe laser saturates as its
intensity increases, showing that the saturation intensity is proportional to
the helium density. We use this to estimate collision rates and the density of
molecules in the cell.
 [Show abstract] [Hide abstract]
ABSTRACT: We use cryogenic helium buffergas cooling to form large densities of lithium atoms in a highdensity helium gas, from which LiHe molecules form by threebody recombination. These weakly bound van der Waals molecules are detected spectroscopically, and their binding energy is measured from their equilibrium thermodynamic properties.Physical Review Letters 04/2013; · 7.73 Impact Factor  SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: Cryogenic buffergas beams are a promising method for producing bright sources of cold molecular radicals for cold collision and chemical reaction experiments. In order to use these beams in studies of reactions with controlled collision energies, or in trapping experiments, one needs a method of controlling the forward velocity of the beam. A Stark decelerator can be an effective tool for controlling the mean speed of molecules produced by supersonic jets, but efficient deceleration of buffergas beams presents new challenges due to longer pulse lengths. Travelingwave decelerators are uniquely suited to meet these challenges because of their ability to confine molecules in three dimensions during deceleration and the versatility afforded by the analog control of the electrodes. We have created ground state CH($X^2\Pi$) radicals in a cryogenic buffergas cell with the potential to produce a cold molecular beam of $10^{11}$ mol./pulse. We present a general protocol for Stark deceleration of beams with large position and velocity spreads for use with a travelingwave decelerator. Our method involves confining molecules transversely with a hexapole for an optimized distance before deceleration. This rotates the phasespace distribution of the molecular packet so that the packet is matched to the time varying phasespace acceptance of the decelerator. We demonstrate with simulations that this method can decelerate a significant fraction of the molecules in successive wells of a travelingwave decelerator to produce energytuned beams for cold and controlled molecule experiments.Physical Review A 12/2013; 90(3). · 2.99 Impact Factor  Xiujuan Zhuang, Anh Le, Timothy C. Steimle, N. E. Bulleid, I. J. Smallman, R. J. Hendricks, S. M. Skoff, J. J. Hudson, B E. Sauer, E. A. Hinds, M. R. Tarbutt[Show abstract] [Hide abstract]
ABSTRACT: The fluorescence spectrum resulting from laser excitation of the A^{2}\Pi_{1/2}  X^{2}\Sigma^{+} (0,0) band of ytterbium monofluoride, YbF, has been recorded and analyzed to determine the FranckCondon factors. The measured values are compared with those predicted from RydbergKleinRees (RKR) potential energy curves. From the fluorescence decay curve the radiative lifetime of the A^{2}\Pi_{1/2} state is measured to be 28\pm2 ns, and the corresponding transition dipole moment is 4.39\pm0.16 D. The implications for laser cooling YbF are discussed.10/2011;
Page 1
Diffusion, thermalization and optical pumping of YbF molecules in a cold buffer gas
cell
S. M. Skoff, R. J. Hendricks, C. D. J. Sinclair, J. J. Hudson, D. M. Segal, B. E. Sauer, E. A. Hinds, and M. R. Tarbutt∗
Centre for Cold Matter, Blackett Laboratory, Imperial College London,
Prince Consort Road, London, SW7 2AZ. United Kingdom.
We produce YbF molecules with a density of 1018m−3using laser ablation inside a cryogenically
cooled cell filled with a helium buffer gas. Using absorption imaging and absorption spectroscopy we
study the formation, diffusion, thermalization and optical pumping of the molecules. The absorption
images show an initial rapid expansion of molecules away from the ablation target followed by a
much slower diffusion to the cell walls. We study how the time constant for diffusion depends on the
helium density and temperature, and obtain values for the YbFHe diffusion crosssection at two
different temperatures. We measure the translational and rotational temperatures of the molecules
as a function of time since formation, obtain the characteristic time constant for the molecules to
thermalize with the cell walls, and elucidate the process responsible for limiting this thermalization
rate. Finally, we make a detailed study of how the absorption of the probe laser saturates as its
intensity increases, showing that the saturation intensity is proportional to the helium density. We
use this to estimate collision rates and the density of molecules in the cell.
I.INTRODUCTION
Cold molecules are useful for a wide variety of ap
plications in physics and chemistry. A review of these
applications and the current status of the field is given
in reference [1]. An important application is the use of
cold molecules for testing fundamental laws of physics
[2]. Cold molecules are being used to test the standard
model of particle physics and its extensions via a mea
surement of the electron’s electric dipole moment [3–5].
They can also be used to test the time invariance of the
fine structure constant and of the protontoelectron mass
ratio [6–10], test Lorentz invariance [11], and measure
parityviolating interactions in nuclei [12] and in chiral
molecules [13].
Buffer gas cooling is a very versatile way of producing
high densities of atoms and molecules at temperatures
down to a few hundred milliKelvin [14]. The molecules
are introduced into a cryogenicallycooled cell where they
thermalize with a cold buffer gas. The buffer gas is usu
ally helium, though neon has also been used [15]. The
molecules can be brought into the cell in a number of
ways, depending on their vapour pressure and reactivity.
Methods that have been demonstrated include laser ab
lation of a precursor target e.g. [16], injection through
a capillary e.g. [17], loading from a molecular beam e.g.
[18], and direct gas flow loading e.g. [15].
Once loaded, there are three main options available.
In the simplest case the molecules are allowed to diffuse
freely to the cell walls, where they will stick, and mea
surements are made on the molecules in the interval of
time between loading and sticking. This method has been
used to measure molecular parameters by millimetre and
submillimetre spectroscopic techniques [17, 19, 20], and
∗Electronic address: m.tarbutt@imperial.ac.uk
more recently by laser spectroscopy, e.g. [21, 22]. Elastic
and inelastic collision crosssections can also be measured
this way, e.g. [23, 24]. A second approach is to trap the
molecules magnetically by placing the cell inside a strong
magnetic trapping field and pumping away the helium as
soon as the molecules are cold [25]. The coldest molecules
are then confined in the magnetic trap. Several molecu
lar species have been trapped this way, [26, 27], allowing
elastic collision and Zeeman relaxation rates to be mea
sured at low temperatures, [28]. Trapping times of 20s
have been demonstrated for NH molecules [29]. A third
approach is to extract the molecules from the cell to make
a high intensity cold beam [30] which could then be used
in a molecular beam experiment or for trapping. The
molecules that form the beam can be guided away from
the cell and separated from the helium using a magnetic
guide [31], an electrostatic guide [32] or an alternating
gradient electric guide [33].
It is important to have a good understanding of the
processes that occur inside the buffer gas cell, since this
is the starting point for all experiments whether they be
done directly inside the cell, in a magnetic trap loaded
from the cell, or in a beam extracted from the cell. In
this paper we focus on the formation, diffusion and ther
malization of YbF molecules produced by laser ablation
of a target inside a closed helium buffer gas cell. These
molecules are of particular interest for the measurement
of the electron’s electric dipole moment [3, 4] and, for that
application, the formation of a high density of YbF is one
of our aims. The work presented here will be applicable
to many other buffer gas cooling experiments, particu
larly those where laser ablation is the loading method or
where laser spectroscopy is used to make measurements
inside the cell.
arXiv:1009.5124v2 [physics.atomph] 25 Feb 2011
Page 2
2
Target
Gas inlet
Ablation laser
Probe laser(s) for
imaging and
spectroscopy
x
y
FIG. 1: (Color online) An illustration of the cell showing the
position of the target and the gas inlet and the propagation
directions of the ablation and probe lasers. The coordinate
system we use is also shown and has its origin at the centre
of the cell.
II. EXPERIMENTAL DETAILS
Figure 1 shows a few details of the experiment. To
make the cell we took a solid aluminium cube of side
40mm and bored three orthogonal 20mm diameter holes,
two passing all the way through the cell and the other
half way through. The face without a hole mounts onto
the cold plate of a liquid helium cryostat (Infrared Lab
oratories, model HD3). Vacuumtight indium seals close
the five open faces, three with windows and two with
blanks. One window is used for laser ablation of a target
placed on the opposite side of the cell and the other two
are used for laser absorption spectroscopy and imaging
of the molecules formed inside. The cryostat is pumped
by a turbomolecular pump to a pressure of approximately
10−6mbar. A radiation shield, in thermal contact with a
liquid nitrogen dewar, surrounds the setup and prevents
room temperature radiation from reaching the cold sur
faces. Windows in this shield, coated with indium tin ox
ide, provide optical access at visible wavelengths, whilst
reflecting thermal radiation.
YbF molecules are created in the cell by laser ablation
of a vacuum hotpressed target consisting of 70% Yb and
30% AlF3by mass. The ablation pulses have a duration
of 8ns, an energy of approximately 50mJ, a wavelength
of 1064nm, and a spot size at the target of 2mm.
Helium gas enters the cell through a hole beneath the
target, via a thinwalled tube of inner diameter 2.5mm.
This is in two parts: a stainless steel tube for most of the
length is connected to a copper tube near the cold plate
of the cryostat. To ensure that the gas is cold when it
enters the cell, the copper tubing winds around, and is
silversoldered to, a copper bobbin connected to the cold
plate. At the room temperature end, outside the cryo
stat, the feed tube is connected to a chamber where the
helium pressure is measured and controlled. The helium
enters this chamber through a leak valve and is pumped
0.02
0.05
0.10
0.20
0.50
1.0
2.0
0.02 0.05
Inlet pressure (mbar)
0.100.20 0.501.0 2.0
Cell pressure (mbar)
FIG. 2: (Color online) Pressure measured at the cell versus
pressure measured at the inlet tube, all at room temperature.
Red squares: during ablation. Black open circles: pressure
settled after ablation. Blue filled circles: after pumping on
the cell through the gas inlet tube for about 12 hours. The
errors are about 1% of the reading. The line has zero offset
and a gradient of 1. Note the logarithmic scale.
away to a rotary pump through a second valve. These
two valves provide fine control over the helium pressure,
which is measured by a Pirani gauge. When the cell is
cold, there is a thermomolecular pressure drop between
the two ends of the tube, and we account for this using
the empirical formula given in [34]. Even when the cell is
at room temperature, it is not immediately clear whether
the pressure measured at the inlet end of the feed tube is
the same as the pressure in the cell. To investigate this,
we temporarily connected a second Pirani gauge directly
to the cell, via a port in the cryostat usually used as a
window. Both pressure gauges were calibrated to a third
gauge which, for helium, is accurate to 0.5%.
Figure 2 shows the pressure in the cell versus the pres
sure at the inlet. One set of measurements was taken
during ablation of the target at 10Hz, another set a few
minutes after stopping the ablation, and a final set after
pumping the cell through the gas feed tube for about 12
hours. The figure shows that the cell pressure is equal to
the inlet pressure plus a constant offset. The offset is due
to outgassing of the target which raises the total pressure
in the cell. This outgassing is greatest during ablation of
the target, when the offset is 0.17 ± 0.05mbar. When
the ablation is stopped the offset falls with two sepa
rate timescales. First it falls quickly, reaching a steady
value of 0.06 ± 0.02mbar in a few minutes, and then it
falls much more slowly, reaching 0.015 ± 0.003mbar af
ter about 12 hours. We could only make these direct
pressure measurements at room temperature, so we do
not know how the offset changes with temperature. The
pressure offset has implications for some of our measure
ments, and we will highlight these where relevant.
The temperature is measured using a silicon diode tem
perature sensor attached to the cold plate of the cryostat.
At the lowest temperatures however, temperature mea
surements of the fully thermalized molecules (Sec.V) in
Page 3
3
dicate that the cell is about 10K hotter than the cold
plate, presumably due to insufficient thermal conduction
between the two. We have also used the Doppler broad
ening of the atomic Yb spectrum to determine the tem
perature, and this agrees with the results obtained from
the molecules. Using a new experimental setup, where
the thermal conduction is improved, we find that the
molecules do indeed reach the same temperature as the
4K cell. All the measurements reported here were done
with the earlier apparatus and we use the temperature
of the molecules as the most reliable measure of the cell
temperature.
All the measurements presented in this paper are based
on absorption of a cw probe laser beam. Light from a
dye laser is used to address individual rotational lines
within the (0–0) band of the X2Σ+–A2Π1/2transition
of YbF, at a wavelength of 552nm. The laser frequency
is measured with an accuracy of approximately 600MHz
using a wavemeter (HighFinesse WS6). Absorption im
ages reveal the spatial distribution of the molecules as
a function of time. For these measurements, the laser
beam passes through a 100MHz acoustooptic modulator
(AOM) acting as a fast optical shutter, and is then colli
mated and expanded to a diameter of 20mm. This beam
passes through the buffer gas cell and onto a ccd camera
(Marlin F033B). The shortest exposure time offered by
this camera is 34µs. To take images with higher time
resolution, we use the AOM to turn on the absorption
imaging beam for a period of 10µs. For timeresolved
measurements of the YbF density, translational temper
ature and rotational temperature at a specific point in
the cell, we use a combination of Dopplerlimited and
Dopplerfree absorption spectroscopy, using 1mm diam
eter pump, probe and reference beams, as described in
detail in [22].
Data are taken at a repetition rate of 10Hz, synchro
nised to the 50Hz line frequency to suppress the effects
of line noise in the experiment. In the imaging experi
ments the laser frequency is fixed and each shot yields
one image from the ccd camera. In the spectroscopy ex
periments, the outputs of the photodiodes are recorded
for 8ms with a sample rate of 250kHz, and the laser fre
quency is stepped between one shot and the next. As de
scribed in [22], the absorption of a fixed frequency probe
beam from a second dye laser is used for normalization
purposes, factoring out the slow drift in ablation yield
that occurs as the target degrades.
III.ABSORPTION IMAGING
Figure 3 shows a sequence of absorption images for
three different buffer gas densities at a cell temperature
of 20K. To obtain each image, we take the difference
between two pictures, one where the ablation laser fires
and one where it does not. The field of view is 20mm
wide and the colour represents the fractional absorption
of the probe laser. The target is on the left side of the
image, just outside the field of view. The probe laser
is tuned near the Pbranch bandhead where there is a
high density of overlapping spectral lines from rotational
states in the range N = 3−6. The camera is exposed to
the probe for 10µs with a variable delay relative to the Q
switch of the ablation laser. The images therefore show
how the density of YbF molecules in lowlying rotational
states evolves with time, and how this depends on the
buffer gas density.
In Fig.3(a), the helium density is relatively low, nHe=
0.9 × 1022m−3. In this case, the molecules fill the entire
field of view within 30µs. Their distribution is roughly
uniform in all the images.
creases for the first 180µs, and then slowly decreases
after that. We propose that the molecules are formed
close to the target, then expand ballistically, and because
their mean free path is relatively long at this density, they
rapidly fill the entire cell. They then diffuse slowly to the
walls, where they stick. The increase in absorption over
the first 180µs suggests either that the formation of new
molecules continues for this length of time, or that the
rotational temperature starts out high and cools on this
timescale, thereby increasing the population of the low
lying rotational states being probed in the experiment.
The measurements presented in Sec.V confirm that the
rotational temperature does indeed fall considerably over
this initial period. In the first frame of the sequence of
images hardly any molecules are visible because they are
too hot during this initial 10µs period to be observed. In
Fig.3(b) the helium density is increased to 17×1022m−3,
and the evolution has changed considerably. The initial
ballistic expansion is arrested before the molecules have
filled the cell, so that in the first frame we see the pro
jection of a hemispherical distribution with a radius of
about 8mm. This distribution then diffuses slowly into
the rest of the cell. The absorption peaks after about
60µs, indicating that the rotational cooling is faster than
before. We might expect the decay of the molecule den
sity to be slower than before because the diffusion will be
slower at higher helium density, but in fact the density
appears to decay more rapidly. This is even more evi
dent in Fig.3(c) where the helium density is increased to
94 × 1022m−3. Here, the ballistic expansion is arrested
even closer to the target, the peak absorption occurs at
even earlier times, and the propagation into the rest of
the cell is even slower. After 540µs the molecules have
still not filled the entire field of view, and yet most of the
molecules have vanished. The rapid decay of the signal
is partly due to the proximity of the molecules to the
walls of the cell. The target is close to the wall, and the
molecules are stopped close to the target, so the distance
that they have to diffuse before they reach a wall is small.
This tends to lower their survival time in the cell, even
though the mean free path is small. Other loss processes
may also be contributing to the decay of the molecular
density. The YbF molecules and other ablation products
are confined at high density and the YbF radicals may
react with these other atoms and molecules to form more
The overall absorption in
Page 4
4
0 µs 30 µs 540 µs 450 µs 360 µs 270 µs 180 µs 90 µs 60 µs
0
1
Fractional
absorption
(a) 0.9 x 1022 m3
(b) 17 x 1022 m3
(c) 94 x 1022 m3
FIG. 3: (Color online) Absorption images of YbF at various times since ablation and for three different helium densities at a
cell temperature of 20K.
stable products. We investigate the decay times of the
molecules more quantitatively in the next section.
IV. DIFFUSION
We have studied the diffusion of the YbF molecules
through the helium buffer gas. In order to obtain an
estimate of the YbFHe diffusion crosssection, without
needing an accurate measure of the helium density in
the cell, we produce and probe lithium atoms in the cell
simultaneously with the YbF molecules. The LiHe dif
fusion crosssection can be calculated accurately, and so
the Li diffusion times provide a good reference against
which the YbF diffusion times can be compared.
A. Theory
Following ablation, there is an initial ballistic expan
sion of molecules away from the target. Over a certain
distance from the target, depending inversely on the he
lium pressures, this ballistic expansion is arrested and
the molecules then diffuse towards the walls. The flux
of diffusing particles, J, is proportional to the gradient
of the density n, J = D∇n. For diffusion of one gas
into another at temperature T, the relationship between
the diffusion coefficient, D, and the thermally averaged
diffusion crosssection, ¯ σD, is given, with sufficient accu
racy, by the first ChapmanEnskog approximation [35–
37]. Since the number density of helium, nHe, is many
orders of magnitude larger than that of our diffusing gas,
the result is
D =
3
16¯ σDnHe
?
2πkBT
µ
, (1)
where µ = mM/(m + M) is the reduced mass related to
the masses of the helium, m, and the diffusing molecules,
M. Although this expression is only the first term of
a series approximation for the diffusion coefficient, it is
expected to be accurate to better than 2% for the two
cases, LiHe and YbFHe, that we consider here [37].
The density of molecules in their journey through time
and space is described by the time dependent diffusion
equationdn
the form n(r,t) =?
neous diffusion coefficient is a solution of the differential
equation
dt= ∇2(Dn). The solution can be written in
kckfk(r)e−t/τk, where ckis the am
plitude of the diffusion mode fk(r), which for a homoge
fk
Dτk
+ ∇2fk= 0(2)
subject to the boundary conditions imposed by the walls.
Since the molecules stick to the walls with high proba
bility, we take n = 0 at the walls. When there is no
particular symmetry in the problem, three indices are
needed to label the diffusion modes, represented here by
the single label k. In all cases, the product of the diffusion
coefficient and the time constant τkdepends only on the
cell geometry and on the indices labelling the diffusion
mode. The higher order diffusion modes have smaller
time constants.
If the interaction potential between the two colliding
species is known, the thermallyaveraged diffusion cross
section ¯ σD can be calculated. For all temperatures, T,
of interest here, the calculation can be done classically
Page 5
5
using the following set of equations [37]:
¯ σD=1
2
?∞
?∞
0
x2e−xσD(E)dx,(3)
σD(E) = 2π
0
[1 − cosχ(E,b)]bdb,
?∞
(4)
χ(E,b) = π − 2b
rc
r−2dr
?1 − V (r)/E − b2/r2. (5)
Here, σD(E) is the diffusion crosssection at centreof
mass collision energy E, x = E/(kT), χ(E,b) is the
deflection angle for a collision with energy E and im
pact parameter b, V (r) is the interaction potential as
a function of the particle separation r, and rc is the
distance of closest approach in the collision, given by
1 − V (rc)/E − b2/r2
The interaction potential for LiHe is well established,
there being a long history of theoretical and experimen
tal work on this system (see [38] and references therein).
We have calculated ¯ σD,LiHefor a range of temperatures
using the potential given in reference [38]. The results at
293, 80 and 20K are given in Table I on page 9. There
are two sources of error in this calculation. The first is
due to our neglect of quantum effects. Using the tabu
lation in [37], we estimate that quantum effects increase
the crosssection by about 2% at 20K, and are negligible
at the higher temperatures. The second source of error
is due to the uncertainty in the interaction potential. To
estimate this we repeated the calculations using the alter
native potential given in [39]. The crosssections obtained
from this latter potential are smaller by 0.3% at 293K,
5% at 80K and 6% at 20K. Taking these differences as
indicative of the likely accuracy, we assign a fractional
uncertainty of 5% to the crosssections calculated at all
temperatures. The interaction potential for YbFHe has
also been calculated [40] and we use this to obtain theo
retical values for ¯ σD,YbFHeat these same temperatures.
In this case, the potential is not only a function of the
YbFHe separation but also of the angle between the YbF
internuclear axis and the incident velocity vector of the
collision. We calculate an approximate diffusion cross
section by calculating the crosssection as a function of
this angle and then averaging over all angles. The results
are given in Table I.
c= 0.
B.Model
Since our cell geometry is not simple, we model the dif
fusion using finite element software1. This model uses the
same cell geometry as the experiment and only accounts
for diffusion. It does not include the initial momentum
away from the target that the molecules have following
1Comsol Multiphysics 3.2
0 µs90 µs 540 µs 1500 µs 3000 µs
(a)
(b)
0
1
Fractional
absorption
FIG. 4:
diffusion coefficient of 0.005m2s−1. This corresponds, for ex
ample, to a cell temperature of 20K, a diffusion crosssection
of 80×10−20m2and a helium density of 3×1022m−3. (a) Uni
form initial distribution throughout the cell. (b) Distribution
localized near the target.
(Color online) Simulated absorption images for a
ablation. For low helium densities, this initial momen
tum helps to distribute the molecules throughout the cell
faster than diffusion would be able to do. To understand
the effect of this, we study two extreme cases for the ini
tial condition in the model. In one case, the molecules
start out with a uniform distribution over the whole cell.
This is intended to model the situation at low helium
density where the initial ballistic expansion completely
fills the cell with molecules. In the other extreme case
the initial density is taken to be e−r/wwhere r is the
distance away from the target and w = 1mm. This is
intended to model the situation at high helium density
where the ballistic expansion is rapidly arrested with the
molecules localized close to the target.
Figure 4 shows the predictions of these simulations for
the two extreme initial conditions. The behaviour seen
in these simulated images is qualitatively similar to that
of the experimental absorption images shown in Fig.3.
In Fig.4(a) the initial density distribution is a constant.
This evolves to a distribution with a maximum at the
middle, and that distribution then decays away slowly.
In Fig.4(b) the initial density is localized close to the
target, and we see that as time progresses the distribution
expands slowly away from the target but decays away on
a faster timescale than in case (a). This is because the
molecules are localized close to the target and so can
diffuse back onto the target and the nearby wall. The
diffusion equation tells us that high spatial gradients in
density decay away rapidly. The highly localized initial
distribution is represented by highorder diffusion modes
localized close to the target, and these decay away quickly
with a large fraction of the molecules quickly reaching
the target and nearby wall, leaving only loworder modes
whose amplitudes are so small that they are barely visible
in the images.
As well as taking absorption images, we measure the
absorption locally using a small probe laser. To simulate
these measurements, we integrate the simulated density
over the volume of the probe laser. Doing this, we find
that the time evolution of the density is sensitive to the
Page 6
6
position of the probe, and to the initial condition, as
shown in Fig.5. In this figure, we plot the simulated
time evolution of the density probed at three different
distances from the target, as well as the evolution of the
density averaged over the whole cell. With reference to
the coordinate system shown in Fig.1, the probe laser
is in the xyplane, displaced along y. In each case, the
density is normalized to the average density in the cell at
t = 0, and is plotted on a logarithmic scale. Our choice
of time axis in this plot requires some explanation. Con
sider again the diffusion equation and introduce scaled
coordinates r?related to the original coordinates r by
r?= r/L where L is a characteristic length scale for the
cell. Its precise value is unimportant and we will take
L = 1cm. Taking derivatives with respect to the scaled
spatial coordinates the diffusion equation is
where t?= Dt/L2is a dimensionless time. This is the
time axis we use in Fig.5 because this makes it obvi
ous how the results scale with the overall size of the cell
and with the diffusion coefficient (and hence the diffusion
crosssection, helium density and temperature).
dn
dt? = ∇2
r?n
In the case where the initial density distribution is
uniform [Fig.5(a)], the time evolution does not depend
strongly on position. There are some differences in the
time evolution for different beam position at very early
times, but after that the evolution is well described by
a single exponential decay with a time constant which
is the same at all positions. For our cell, this time con
stant is found to be τ = 0.22 × 10−4/D, where D is the
diffusion coefficient in m2s−1and τ is in s. For exam
ple, for a helium density of 1022m−3, a YbFHe diffusion
crosssection of 80×10−20m2and a temperature of 20K,
the diffusion coefficient is D = 0.015m2s−1, and the ex
pected diffusion lifetime is τ = 1.5ms. For diffusion in
a cube of side a, the time constant associated with the
lowest order diffusion mode is τ = a2/(3π2D). It follows
that the diffusion time constant for our cell is the same
as that of a cubic cell with a = 2.6cm. This is reassur
ingly similar to the size scale of our cell. The position
sensitivity is far stronger when the molecules are initially
concentrated near the target, as in Fig.5(b). While the
initial density is largest close to the source of molecules,
there is a rapid decay of the density at this position. As
time goes on, the rate of decay slows down, and even
tually the time evolution becomes a single exponential
with the same time constant as in the constant density
case. However, that does not occur until the density has
fallen by two orders of magnitude. Further away from
the source, the density first increases with time as the
molecules arrive at the probe, and then it decreases again
with the same time constant as in the constant density
case. The peak density decreases as we move further
from the target, and the time taken to reach this peak
increases with increasing distance. We conclude that at
long times the time evolution is insensitive to either the
initial condition or the position of the probe, but that at
short times it is sensitive to probe position particularly
when the initial density is strongly localised.
D t / L2
Normalized density
Whole cell
2mm from target
10mm from target
18mm from target
(a)
(b)
0.00.2 0.40.60.8 1.0
0.01
0.10
1
10
0.0 0.2 0.4 0.60.8 1.0
0.01
0.10
1
10
FIG. 5: (Color online) Simulated evolution of the molecule
density at different positions in the cell. The density is nor
malised to the initial density averaged over the whole cell.
The time axis is expressed in the dimensionless units Dt/L2,
where D is the diffusion coefficient and L = 1cm is a char
acteristic size scale for the cell. (a) Uniform initial density
throughout the cell. (b) Initial density localised near the tar
get.
These observations can be explained by expressing
the density distribution as a sum over a set of diffu
sion modes, the higher order modes representing more
rapid spatial variations in density and decaying away
more quickly. When the cell is uniformly filled, the den
sity rapidly settles into the lowest order diffusion mode,
which gives us the longest diffusion time. To represent
an initial distribution that is strongly localized we need
to put large amplitude into highorder modes, and these
then decay away rapidly. Eventually, only the loworder
modes, which persist for much longer, will remain, but
the amplitude of the loworder mode may be very small.
In an experiment, it may be too small to observe above
the noise.
C. Experimental results
We compare the YbF diffusion times to the diffusion
time of lithium atoms introduced into the cell at the same
time. If we assume that the initial distribution is the
same for the YbF and the Li, the product of the diffusion
coefficient and the time constant is the same for both
Page 7
7
species, since this product is only a function of the cell
geometry. It follows that
¯ σD,2= ¯ σD,1
?µ1
µ2
τ2
τ1
(6)
where subscript 1 refers to LiHe and subscript 2 to YbF
He. To introduce Li into the cell, a piece of lithium
wire was mounted into the target holder along with the
AlF3/Yb target. The two materials were ablated simul
taneously using a single Nd:YAG laser spot. The lithium
density was monitored by absorption of light from a diode
laser tuned into resonance with the7Li D1 line at 671nm.
This probe laser was overlapped with the YbF probe
beam and the absorption signals for the two species were
measured simultaneously on two separate photodiodes.
In this way, both species were measured at the same place
and at the same time, and both were produced in the
same place on the target. The probe lasers were in the
xyplane (see Fig.1), about 7mm from the target. It was
possible to obtain extremely high optical depths for Li,
but through judicious positioning of the ablation spot on
the target we were able to obtain optical depths of the
same order of magnitude for the two species. Because
the Li and YbF densities are many orders of magnitude
lower than the He density collisions between them are
not important.
Figure 6 shows the time evolution of the YbF and Li
absorption signals for a helium density of 2.2×1022m−3.
At this density we expect the molecules to fill the en
tire cell through ballistic expansion. We plot the absorp
tion coefficient, α, which is directly proportional to the
molecule density, the interaction length and the absorp
tion crosssection, and is related to the fractional absorp
tion A via α = −ln(1 − A) (see appendix A). Transients
due to optical pumping and collisional redistribution of
the population are too fast to be observable in the ex
periment. The YbF density is probed near the AX(00)
bandhead where there is a dense cluster of overlapping
Pbranch lines. For Li, there is an initial fast rise in
the density during the first 20µs as the atoms fill the
cell. After that, the signal is described well by a double
exponential decay, one exponential having a short time
constant that dominates during the first 150µs, and the
second having a longer time constant that dominates at
later times. This same behaviour is observed over a wide
range of buffer gas densities. We find that as the helium
density increases the short time constant decreases from
about 40µs when the density is 0.25×1022m−3to about
20µs when the density is 2.5 × 1022m−3. We suppose
that this fast decay is the decay of higherorder diffu
sion modes, the result being sensitive to the initial con
dition and the position of the probe laser as discussed
above. The long time constant on the other hand in
creases with buffer gas density, and we take this to be
the characteristic diffusion time for Li in the cell. The
YbF absorption curves are influenced by both diffusion
and by rotational cooling which, at early times, increases
the relative population in the lowlying rotational states
0 200400 600
Time (µs)
800 10001200 1400
0.001
0.005
0.010
0.050
0.100
0.500
Absorption coefficient, α
YbF
Li
FIG. 6: (Color online) Absorption coefficient on a logarithmic
scale plotted as a function of time for Li (red) and YbF (blue)
at a helium density of 2.2 × 1022m−3and a cell temperature
of 293K.
probed in the experiment, particularly for low cell tem
peratures. In Sec.V we measure how the rotational tem
perature changes with time. A model that includes both
diffusion and the rotational cooling that we have mea
sured predicts that, for a cell temperature of 20K, the
change in absorption is strongly influenced by rotational
cooling at early times, but is dominated by diffusion at
later times. At higher temperatures the influence of rota
tional cooling is far weaker. To obtain the YbF diffusion
time constants we fit single exponential decays to the late
part of the absorption curves (t > 0.32ms for the 20K
data and t > 0.14ms for 293K). Our model shows that
we will obtain accurate diffusion times using this proce
dure. We have also verified that, at 20K, fitting only to
data beyond 0.6ms makes a negligible difference to our
results.
Figure 7 shows the characteristic diffusion time con
stants for Li and YbF as a function of helium density,
for cell temperatures of 293K, 80K and 20K. At 80K
no useful YbF data was obtained, so only Li data are
shown. The error bars are determined from the scatter
of repeated measurements taken under the same nominal
conditions. They therefore appear as errors in the time
constant though we believe the major source of the scat
ter to be due to the difficulty of setting the buffer gas
density, which was very slow to reach equilibrium partic
ularly at low temperatures. For low buffer gas densities,
all the measured time constants increase linearly with
density and both species diffuse more slowly at lower tem
peratures as expected in a simple model of diffusion. At
both 293K and 20K the time constant for YbF increases
more rapidly with increasing helium density than for Li.
At higher densities however, the diffusion times stop in
creasing and become roughly independent of the helium
density. This behaviour is displayed by both YbF and Li
at 293K, by Li at 80K and YbF at 20K. The data for Li
at 20K appear to show a more complicated behaviour
at high density. For Li, the lifetimes level off at ap
proximately 400µs at 293K, 600µs at 80K, and roughly
Page 8
8
02468 10 12
0
200
400
600
800
1000
02468
0
200
400
600
800
0.00.5
Helium density, nHe (1022 m3)
1.0 1.52.02.5
0
500
1000
1500
Diffusion lifetime (µs)
(a)
(b)
(c)
FIG. 7: (Color online) Diffusion time constants as a function
of buffer gas density for Li (open circles) and YbF (filled cir
cles) for buffer gas temperatures of (a) 293K, (b) 80K (Li
only) and (c) 20K. The lines (solid for Li, dashed for YbF)
are linear fits to the data at low buffer gas density. Note the
different scales on both axes for the three graphs.
500µs at 20K. For YbF, the corresponding values are
approximately 800µs at 293K and 1000µs at 20K. The
density at which the levelling off occurs is about the same
for the two species, and shifts to lower values as the tem
perature is lowered.
At low density, the diffusion times show the expected
linear increase with helium density, so we fit straight lines
to the low density parts of the five datasets. The fit re
sults are shown by the lines in Fig.7. For the 293K and
80K data, the first 10 data points were used for the lin
ear fit. For the 20K data it is unclear whether to fit
to the first 4 or first 5 data points (the latter is shown
in Fig.7(c)), and so we do both. The gradients of these
two fits are consistent within errors, and we take the
mean value as our best estimate. The gradients of the
linear fits, together with Eq.(6) and the calculated Li
He diffusion crosssection, provide a measurement of the
YbFHe diffusion crosssection, ¯ σD,YbFHe. The results
are given in the fourth column of Table I. The uncer
tainty is derived by adding in quadrature the statistical
errors in the gradients of the linear fits, the error in the
LiHe crosssection estimated above, and in the case of
the 20K result, half the difference between the gradients
of the 4point and 5point fits. Neither the helium den
sity or the cell temperature enter in the determination
of the crosssection. Errors arising due to a difference in
the initial spatial distribution of the Li and YbF or small
displacements of the two probes are deemed to be negligi
ble because at these helium densities the initial ballistic
expansion fills the cell, as in Fig.3(a), and our simula
tion results indicate that the diffusion time is insensitive
to the exact initial distribution or probe position in this
case. Errors arising due to a difference in temperature be
tween the two species are also negligible because the time
for thermalization with the helium is much smaller than
the characteristic diffusion time (see Sec.V). Our exper
imental values for the YbFHe diffusion crosssection are
consistent with our theoretical values. In particular, the
values are in agreement at 293K where the experimental
uncertainty is smallest.
Extrapolating the data to zero helium density as shown
by the lines in Fig.7 we observe an offset in all of the
lifetimes. This suggests that the ablation process itself
puts a significant density of material in the cell which
the YbF and Li then have to diffuse through. This is
consistent with the offset observed in Fig.2. The offset
is smaller for the Li data at 80K. We do not know the
reason for this. We speculate that, when the cell is warm,
vacuum contaminants such as water and hydrocarbons
reach the cell and are absorbed into the target, and when
the cell is very cold the target absorbs helium just like a
cryosorb. At 80K neither occur because this is not cold
enough to absorb helium but is too cold for water and
hydrocarbons to reach the cell.
D. Discussion
We have used the Li data to calibrate the YbF data
and so our determination of the YbFHe diffusion cross
section does not rely on the diffusion model or on know
ing the helium density. It is nevertheless interesting to
compare the experimental results at low density with the
predictions of the diffusion model for Li. We assume that
at low helium densities, the cell is immediately filled with
a uniform density of particles. Fitting a sum of two ex
ponentials to the results of the model with this initial
condition gives the longer of the two time constants as
τ = 0.22 × 10−4/D, where D is the diffusion coefficient
in m2s−1and τ is in s. As noted above, the long time
constant is insensitive to the initial condition, so the er
Page 9
9
T¯ σD,LiHe (theory)
(10−20m2)
35.8
51.9
71.0
¯ σD,YbFHe (theory)
(10−20m2)
40.9
55.8
79.6
¯ σD,YbFHe (experiment)
(10−20m2)
41 ± 4

203 ± 75
τLi/nHe (theory)
(10−22µs/m−3)
171
482
1300
τLi/nHe (experiment)
(10−22µs/m−3)
79 ± 2
489 ± 48
1022 ± 226
(K)
293
80
20
TABLE I: Diffusion crosssections and lifetimes for Li and YbF at three temperatures. The second and third columns give
theoretical values for the LiHe and YbFHe diffusion crosssections, calculated using the potentials of [38] and [40] respectively.
The fourth column gives experimental values for the YbFHe diffusion crosssection obtained from the data shown in Fig.7,
calibrated by the LiHe crosssections. The fifth and sixth columns give theoretical and experimental values for the Li diffusion
lifetime divided by the helium density. The theoretical values are determined from the theoretical crosssections and a numerical
model of diffusion in the cell. The experimental values are the slopes of the straight line fits shown in Fig.7.
ror arising from the uncertainty in the initial distribution
is small. Using the definition of D given in Eq.(1) and
the calculation of the LiHe diffusion crosssections, we
obtain a theoretical prediction for the gradient τLi/nHe.
The theoretical results obtained at 293, 80 and 20K are
given in Table I where they are compared with the exper
imental gradients obtained from the fits shown in Fig.7.
The error bars given here are the fitting errors, and do
not include a possible systematic error in the determina
tion of the helium density. The experimental and theo
retical results are in good agreement at 80K and 20K,
but disagree by a factor of approximately 2 at 293K. We
do not have an explanation for this discrepancy. The
temperature only enters the model through the diffusion
coefficient, so if the discrepancy is due to an error in
the model it must be that the calculated diffusion cross
section is accurate at 80K and 20K, but not at 293K.
This seems unlikely. The alternative is that there is a sys
tematic error in the experiment which is large at 293K,
but small at the lower temperatures. The most obvious
source of systematic error is in the determination of the
helium density, but here we would expect the error to be
smallest at 293K since the measurements were carefully
calibrated at this temperature. We note that it is quite
common to determine diffusion crosssections from diffu
sion times in a buffer gas cell. To do so without using a
reference atom requires a careful calibration of the buffer
gas density and, unless the shape of the cell is particu
larly simple, a numerical model of diffusion in the actual
cell geometry being used.
Next, we turn to the question of why the diffusion
times flatten off at higher helium density. A similar be
haviour, suggestive of an additional loss mechanism, has
been noted in several other papers e.g. [16, 23, 24, 41]. In
reference [41] the decay times were measured as a func
tion of helium density for lithium, rubidium, silver and
gold. In all cases, the decay times first increased linearly
with helium density, but then turned over and gradually
decreased with increasing density. The authors suggest
three possible loss channels that might be responsible for
the shortening of the lifetime at high density: the for
mation of dimers, capture of atoms by clusters formed
during the ablation process, or atom loss on impurities
present in the buffer gas or ablated off the target. One of
these mechanisms might be responsible for the behaviour
we observe. Here, we suggest a fourth mechanism, based
only on diffusion and on the way the initial molecule dis
tribution depends on helium density. The absorption im
ages, Fig.3, show that there is a rapid initial ballistic ex
pansion of molecules into the cell which sets the initial
condition for the subsequent diffusion to the walls. When
the density is low, the molecules rapidly fill the entire cell
and so the initial distribution is set by the cell geometry
and is independent of density. In this regime the diffu
sion time constant is proportional to the helium density.
At higher densities however, an initial cloud of molecules
is produced whose size depends on the helium density
rather than on the cell size. The higher the density, the
smaller the size of the initial cloud. This localization
of the cloud results in a more rapid initial decay, as in
dicated in Fig.5, particularly if the probe is moved to a
place where the signal is initially high, which it is natural
to do experimentally. We see that at high density there
are two competing densitydependent effects: increasing
the density decreases the diffusion coefficient which slows
down the diffusion, but increasing the density also re
duces the initial size of the distribution which in turn
drives faster diffusion, at least at the beginning. The
turnover from the proportional to the weaklydependent
regime occurs once the initial ballistic expansion fails to
fill the cell, and this will depend on the cell’s size and
on the energy of the ablation pulse. Ideally, it would
be possible to observe the multiexponential decay that
will occur at high density, and to find the longest decay
time which will be independent of the initial condition.
In practice the amplitude of the lowest order mode may
be too small to observe, in which case the initial condi
tion will inevitably affect the measurement of the time
constant in the way we have described.
V. THERMALIZATION
The translational and rotational temperatures can be
determined by recording Dopplerbroadened spectra of
the YbF molecules. The translational temperature is
found from the Gaussian widths of the lines, and the
rotational temperature is determined from the relative
Page 10
10
strengths of transitions originating from different rota
tional states. Since we record the timeevolution of the
absorption at each laser frequency, we can determine the
translational and rotational temperatures as a function
of time.
A.Translational temperature
The measurement of the width and the area of the
Dopplerbroadened lines is complicated by hyperfine
splittings that are comparable to the Doppler widths of
the features themselves, and by the existence of several
YbF isotopic species. The use of Dopplerfree saturated
absorption spectroscopy enables us to identify the cen
tre frequencies of the underlying components, and hence
to fit a combination of multiple Voigt profiles to the
Dopplerbroadened spectra [22]. Figure 8 shows typical
Dopplerbroadened spectra in an experiment where the
cell temperature was approximately 20K and the helium
density was approximately 0.9 × 1022m−3. Two spectra
are shown, one averaged over a 200µs wide timewindow
centred 200µs after the ablation pulse, and the other
averaged over a window of the same width but centred
2000µs after the the ablation pulse. In the early time
window the presence of multiple hyperfine components is
hidden by the large Doppler width, but this is revealed
in the later time window when the translational tem
perature is lower. Underneath the Dopplerbroadened
profiles is shown the saturated absorption spectrum that
was recorded simultaneously. There are four hyperfine
components, though two of them remain unresolved in
the Dopplerfree spectrum so we only see three resolved
spectral lines. Knowing the spacing of the hyperfine
components from the Dopplerfree spectrum, we fit to
the Dopplerbroadened spectra using a sum of Voigt pro
files, where the only free parameters are a single Gaus
sian width and the amplitudes of the various components.
The width of the Lorentzian in the Voigt profile is fixed at
30MHz which is the typical width obtained from fitting
Lorentzians to a large number of lines in the Dopplerfree
spectrum. The fitted widths give translational temper
atures of 76±4K and 23±1K for the two sets of data
shown, clearly indicating the translational cooling of the
YbF towards the temperature of the cell walls.
B. Rotational temperature
The rotational temperature of the molecules is deter
mined from the relative strengths of absorption lines orig
inating from different rotational states N. We obtain the
relative line strengths from the areas of the Voigt profiles
fitted to the Dopplerbroadened spectra. Figure 9 shows
the relative line strengths of the Qbranch transitions
with N between 4 and 13, determined at two different
times, 400µs and 2000µs after the ablation pulse. The
width of the time window is 200µs in both cases. The fits
0 200
Relative laser frequency (MHz)
400600800 1000
0.0
0.2
0.4
0.6
0.8
1.0
Absorption coefficient, α
FIG. 8: (Color online) Upper traces: Dopplerbroadened ab
sorption spectra of the Q(7) transition with a cell temper
ature of approximately 20K. Various hyperfine components
and YbF isotopologues contribute to the absorption.
dark red markers correspond to a spectrum recorded 200µs af
ter the ablation pulse that produces the molecules. The light
blue markers show data recorded 2000µs after ablation. The
solid lines are multicomponent Voigt fits to the data. The
dashed lines isolate the components in the fit corresponding to
the174YbF isotopologue. Lower trace: Saturated absorption
spectrum recorded simultaneously allowing the line centres of
the hyperfine components to be determined and fixed in the
fits to the Dopplerbroadened spectra.
The
shown in the figure are Boltzmann distributions, modi
fied by the slight dependence of the transition matrix
elements on the rotational state. These fits determine
the rotational temperature to be 37±3K at 400µs and
20±1K at 2000µs indicating that the rotational temper
ature of the YbF also cools over this timescale.
C. Thermalization models
Figure 10 shows the evolution of the translational and
rotational temperatures plotted at intervals of 100 and
200µs, respectively. The error bars indicate the uncer
tainties in the parameters obtained from fits such as those
shown in figures 8 and 9. The figure shows that the trans
lational and rotational degrees of freedom are in close
equilibrium throughout the cooling process. The data
also indicate that the molecules initially cool rapidly dur
ing the first 200µs, and then continue to cool but on a
much longer time scale.
We fit this data to a model in which the temperature,
T, of the molecules evolves with time according to
T = Tc+ Ti1e−t/τ1+ Ti2e−t/τ2. (7)
This model fits very well to our data, as indicated by the
Page 11
11
05 1015 20
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Rotational state, N
Relative line strength
FIG. 9: (Color online) Relative strengths of Qbranch transi
tions originating from different rotational states N, at 400µs
after ablation (filled points) and 2000µs after ablation (open
points). The line strengths are determined from the areas
of the Dopplerbroadened spectral lines. The cell tempera
ture was approximately 20K and the helium density approx
imately 0.9×1022m−3. The solid and dashed lines are fits to
the data, with the rotational temperature as a free parameter
in the fit.
best fit lines shown in Fig.10. Indeed, the quality of the
fits to the data tend to be better than would be expected
if the error bars were purely statistical. This is because
the data at different times are obtained from the same
experimental sequence, with the absorption integrated
over different time windows, and so there is a high degree
of correlation in the noise at different times. For the
translational motion the best fit parameters are Tc =
21±1K, Ti1= 329±117K, Ti2= 28±3K, τ1= 84±14µs
and τ2= 880±160µs. For the rotational motion, the fit
parameters are less precise but are all consistent with the
above values.
We now explain why we choose to fit to this model, and
we interpret the two time constants. In a simple ther
malization model [42], the temperature of the molecules
is given by
dT
dt= −R(T − Tbg)/κ,(8)
where R is the collision rate, Tbgis the buffer gas tem
perature and κ = (M + m)2/(2Mm). The collision rate
is given by the product of the helium density, nHe, the
collision crosssection, σ, and the thermallyaveraged rel
ative speed ¯ v, R = nHeσ¯ v. The helium atoms are much
lighter than the YbF molecules and so, over the temper
ature range of interest here, ¯ v is dominated by the speed
of the helium and has little dependence on the tempera
ture of the molecules. In this case, provided the collision
crosssection depends only weakly on the temperature,
the collision rate is approximately constant. Then the
solution is simply
T = (T0− Tbg)exp[−Rt/κ] + Tbg,
where T0is the initial temperature of the molecules. This
single exponential decay does not fit well to the data.
Modifying this model to take into account the depen
dence of ¯ v on the molecule temperature does not improve
the fit.
This leads us to propose a more complicated model in
which the molecules thermalise with the buffer gas as be
fore, but the buffer gas temperature is initially raised by
the ablation process to Tbg,0, and then cools with a time
constant τbgto the temperature of the cell walls, Tc. The
function that describes this model is the same as Eq.(9),
but with the buffer gas temperature now a function of
time, Tbg= (Tbg,0− Tc).exp[−t/τbg] + Tc. This model
fits the data very well. For the translational motion, this
fit suggests that the crosssection for thermalization of
the YbF with the helium is σ = 7.1 ± 1.0 × 10−20m2,
that the buffer gas is initially at Tbg,0 = 48±4K and
that it cools to Tc = 20±1K with a time constant
of τbg = 915±180µs.
rotational temperature data, we obtain similar fit pa
rameters: σ = 6.8 ± 6.0 × 10−20m2, Tbg,0 = 49±36K,
Tc = 19±1K, and τbg = 460±250µs. The energy re
quired to explain the 30K increase in buffer gas tem
perature corresponds to roughly 0.5% of the pulse en
ergy of the ablation laser being rapidly transferred to
the helium buffer gas. A time constant of about 1ms
for rethermalization of the buffer gas with the cold walls
of the cell is as expected given the size of the cell and
the thermal diffusivity of helium at these temperatures
and pressures. However, an atommolecule thermaliza
tion crosssection of about 7 × 10−20m2is smaller than
we would expect in light of the measurements presented
in Sec.IV. It also seems unlikely that the time constant
for rotational thermalization would be so similar to that
of translational thermalization. Therefore, although the
model fits to the data, it only does so with parameters
that we find unrealistic.
These observations suggest that the rotational and
translational motions thermalize with the helium on a
timescale that is shorter than the time resolution of the
experiment, and the temperature evolution we observe in
Fig.10 is entirely that of the helium. This implies that
the heat of ablation deposited into the helium is dissi
pated with two different time constants, one being about
10 times longer than the other. We suppose that the
heat is initially deposited into a localized region close to
the target. This hot helium then mixes with the cold
helium so that, on the timescale τ1, the temperature be
comes roughly uniform throughout the cell, but higher
than that of the cell walls. Finally, the entire bulk of
helium cools to the cell temperature with a longer time
constant τ2. In terms of diffusion modes, the initial tem
perature distribution is represented by a sum of modes,
the higherorder ones representing the more rapid varia
tions in temperature and decaying away rapidly, leaving a
(9)
Fitting the same model to the
Page 12
12
01234
0
50
100
150
Time (ms)
Temperature (K)
FIG. 10: (Color online) Measured temperatures of the trans
lational motion (open points, red) and rotational motion
(filled points, blue) when the buffer gas cell is maintained at
approximately 20K and the helium density is approximately
0.9 × 1022m−3. Some of the error bars are smaller than the
size of the data points. The solid and dashed lines are double
exponential fits to the translational and rotational data, as
described in the text.
small amplitude lowestorder mode which decays slowly.
This model for the diffusion of heat is exactly analogous
to the diffusion of the molecules in the cell, giving the
double exponential decay that we have fit to the data,
Eq.(7). Modelling the diffusion of heat to the cell walls
using the thermal diffusivity of helium at this tempera
ture and density, we find that the time constants found
from the fit are reasonable. We conclude that this is the
most likely explanation for the cooling we observe.
VI. MOLECULE DENSITY AND SATURATION
OF THE ABSORPTION
All of the measurements presented in this paper are
based on absorption of a probe laser. In this section, and
in Appendix A, we consider this absorption in more de
tail. We are particularly concerned to obtain an accurate
value for the molecule density from the observed absorp
tion, and to understand how the fractional absorption
depends on the probe intensity and on the collision rates
between the molecules and the helium.
Figure11(a) shows how the fractional absorption of the
probe laser depends on the laser intensity for three dif
ferent values of the buffer gas density. In these experi
ments, the temperature was 80K. As the laser intensity
increases, the fractional absorption decreases. This fall in
fractional absorption is less rapid when the helium den
sity is high than when it is low. The data show that there
is a saturation of the absorption, and that the intensity
at which this saturation sets in increases with increasing
buffer gas density. It is interesting to explore the mech
anism responsible for this saturation.
In a two level system, the absorption saturates once
Intensity at centre of probe laser (W/m2)
Fractional absorption
(a)
(b)
0.66 x 1022 m3
5.1 x 1022 m3
1.5 x 1022 m3
0 5001000150020002500 3000 3500
0.00
0.05
0.10
0.15
0.20
0.25
0.30
05
Helium density (1022 m3)
10 1520
0
500
1000
1500
2000
Saturation intensity (W/m2)
FIG. 11: (Color online) (a) Fractional absorption versus probe
intensity at three different helium densities. The temperature
is 80K. (b) Saturation intensity versus helium density at 80K.
the rate of excitation from ground to excited state ex
ceeds the rate of decay of the excited state. Once this
occurs the rate at which photons are absorbed is limited
by the decay rate, and an increase in the incident power
does not produce a proportional increase in the absorbed
power. Collisions increase the decay rate, resulting in
an increase in the intensity required to reach saturation,
accompanied by broadening of the spectral line.
saturation data show a strong dependence on the helium
density, but we observed no pressure broadening of the
spectral lines in our saturated absorption spectra for den
sities up to 20 × 1022m−3. We conclude that this is not
the mechanism responsible for the saturation observed
here.
The molecules are not well modelled by a two level
system, but they can be fairly well modelled using three
levels. The probe laser connects levels 1 and 2, while
level 3 represents all other levels in the molecule and is
not coupled at all to the laser. Since level 2 can decay
to level 3, the molecules are optically pumped into level
3 by the probe laser. The intensity needed to saturate
the absorption is set by a competition between the optical
pumping of molecules out of level 1, and any process that
puts population back into this level. We identify three
such processes. Firstly, diffusion brings new molecules
into the interaction region. Secondly, inelastic collisions
transfer molecules from level 3 back to level 1. Thirdly,
Our
Page 13
13
when the Doppler width is much larger than the natural
linewidth, optical pumping into level 3 occurs only in the
velocity group that is resonant with the laser. Elastic
collisions then repopulate that velocity group from the
reservoir of molecules with all other velocities. The rate
for the first process is inversely proportional to the helium
density and is very small compared to all other relevant
rates over the entire range of densities used here. The
rates for the second and third processes are proportional
to the helium density and, together, are responsible for
the saturation of the absorption that we observe.
The expected fractional absorption is calculated in Ap
pendix A. To model our data, we first make the small ab
sorption approximation which simplifies the results con
siderably. In this approximation we assume that the
laserdriven excitation rate is constant throughout the
length of the absorbing column of gas, because the change
in the intensity as the laser propagates through the col
umn is small. This allows us to replace the variable in
tensity I with the constant value Iinon the right hand
side of Eq.(A24). The maximum fractional absorption
is 30% for the data analyzed here and shown in Fig.11.
This gives a rough indication of the maximum error that
we make by assuming a constant excitation rate through
out the sample. Next we make the assumption, which
we will verify later, that all the relevant collision rates
are considerably smaller than the natural decay rate of
the upper level, Γ. In this case, the fractional absorp
tion is given by Eq.(A40) with the saturation intensity
Isreplaced by I?
of the Doppler width to the natural linewidth, w, enters
the expression for the fractional absorption, and its value
given by Eq.(A26) is w = 15 in this experiment. This be
ing much larger than 1, the fractional absorption varies
with intensity as (1 + Iin/I?
sumes a constant intensity across the laser beam but the
probe beam used in these experiments had a Gaussian
intensity distribution. We need to account for this in our
analysis by integrating over the distribution. The result
for the fractional absorbed power, which is the quantity
measured in the experiment, is then
swhich is given by Eq.(A41). The ratio
s)−1/2. Equation (A40) as
¯ A=
?∞
0
√πα0e−ρ2e
1
4w2
?
1+I0
I?se−ρ2?
w
?
?
1−erf
1+I0
I?
?
se−ρ2
×
1
2w
?
1+I0
I?s
e−ρ2
??
ρdρ,(10)
where I0is the intensity at the centre of the probe beam.
The quantity I?
is given by Eq.(A41), while α0 is the optical thickness
of the sample for laser light resonant with the 1 → 2
transition and is given by Eq.(A30). We fit the data
shown in Fig.11(a), along with similar datasets taken
at other helium densities, to Eq.(10) with α0and I?
the only free parameters in the fit. The solid lines in
Fig.11(a) are the results of this fit for the three datasets
shown there.
sis a modified saturation intensity and
sas
Figure 11(b) shows how the value of I?
these fits depends on the helium density, along with a
weighted linear fit to these data. The linear dependence
of the saturation intensity on the helium density supports
our assumption that the collision rates are small com
pared with Γ, at least for the lower density data which has
the greatest weight in the fit. When the collision rates
become comparable with Γ, we expect terms of higher
order in the density to appear in the expression for the
saturation intensity, as shown by Eq.(A19). A weighted
fit to quadratic and cubic polynomials shows the coeffi
cients of these higherorder terms to be zero within the
error of the fit. The intercept of the linear fit is also zero
within the fit error, as we would expect, and the values
of α0 are independent of helium density. We conclude
that our model is the correct one. We see from Eq.(A41)
that the slope of the straight line fit gives us the ratio
(γ13+ γc)/nHe, where γ13is the rate at which collisions
transfer population from level 1 to level 3, and γcis the
rate for velocitychanging collisions. We do not know
which of these two rates, γ13or γc, is the larger, and our
data does not help us to distinguish between them, so we
simply take them together. Both rates are responsible
for refilling the hole produced by the laser in the com
bined velocity and internalstate distribution. We define
the thermallyaveraged crosssections, σ13and σc, via
sobtained from
γX= nσX
?
8kT
MHeπ.
(11)
The sum of the two crosssections is obtained from the
gradient g = 1.0 × 10−20(Wm−2)/m−3of the straight
line fit:
σ13+ σc=3λ3r(1 − r)pg
2πhc
?
MHeπ
8kT
. (12)
Here, p is a factor that depends on the number of
magnetic substates in levels 1 and 2, and is given by
Eq.(A15). For this experiment its value is p = 1/3. The
branching ratio r is the probability that the excited state
decays back to the state resonant with the laser. For our
experiment, it is the product of the FranckCondon fac
tor, Z, for the 0−0 vibrational transition, and an angular
factor that accounts for the probability of returning to
the same rotational and hyperfine state. The calculation
of this angular factor is similar to the one presented in
the appendix of reference [43].
Consider driving a transition from a positiveparity
ground state X2Σ+(v,N,J = N ± 1/2) to a negative
parity excited state A2Π1/2(v?,J?= N + 1/2), where
the spinorbit splitting of the excited state is large com
pared to the rotational splitting, there are no inter
mediate electronic levels, and we neglect for the mo
ment the hyperfine structure arising from any nuclear
spin. The only possible decay routes are to the states
X2Σ+(v??,N??,J??) where v??can take any value with
weight given by the FranckCondon factor for v?→ v??,
but (N??,J??) are constrained to one of three possibilities,
Page 14
14
(1) (N??= N,J??= N+1/2) (2) (N??= N,J??= N−1/2)
or (3) (N??= N + 2,J??= N + 3/2). The corresponding
transitions are usually called Q11, R12, and P12, respec
tively. We find that the relative branching ratios between
these three depends on the value of J?. When J?= 1/2
the three branches have relative weights of 2/3,0,1/3, and
as J?increases these tend to the asymptotic values 1/2,
1/4, 1/4. For the data analyzed here the excitation was
to J?= 27/2, and the ratios are 0.50, 0.24, 0.26, very
close to their asymptotic values. In excitation, the two
transitions with J = N ± 1/2, Q11 and R12, are unre
solved in our Dopplerlimited experiment, but are sepa
rated by more than the natural linewidth. Both transi
tions are excited, but the velocity group is different for
the two excitation routes. In this case, we should take
the mean value of r for the two branches. The situation
is further complicated by the fact that each transition
consists of a pair of hyperfine components which are un
resolved in the Dopplerlimited spectrum. In the case of
the Q11line the components are resolved in the Doppler
free spectrum and we should take the two hyperfine states
as distinct. This halves the hyperfineaveraged value of
r for this line. For the R12line, the situation is different
because the hyperfine components are unresolved in the
Dopplerfree spectrum. In this case it is better to treat
the two hyperfine levels as a single level and the value of
r is unchanged. These considerations lead us to an aver
aged branching ratio of r ≈ 0.25Z. We do not know of
any direct measurements or calculations of the Franck
Condon factor for the A(v?= 0)X(v??= 0) transition of
YbF. However, we know that when the molecule is ex
cited strongly on the F = 1 component of the Q11(1/2)
line, the mean number of photons scattered is 1.9 ± 0.1
[44]. This information, together with the table of branch
ing ratios given in [43] provides an estimated value of
Z = 0.91 ± 0.05. A high value for Z is to be expected
since YbF closely resembles the alkalineearth monofluo
rides which all have this property [45]. Finally then, we
take r ≈ 0.23 for this experiment.
Putting these values into Eq.(12) we obtain σ13+σc=
37 × 10−20m2. There are a number of uncertainties in
determining this value. The fractional error on the gra
dient of the straight line fit is 15%. Using the spread
in the diffusion lifetimes measured at the same nominal
helium density we estimate the fractional uncertainty in
our measurement of the helium density to be 20%. The
fractional uncertainty in calibrating the laser intensity is
also 20%. The fractional uncertainty in the branching
ratio r is estimated to be 24%, partly from the 5% un
certainty in the FranckCondon factor, but mostly from
the uncertainty of how best to treat the four unresolved
components of the line as discussed above. This leads to
a 19% error in the crosssection which depends on the
value of r(1 − r). Other possible errors come from the
approximations we have made in our model. First, we
have assumed that the absorption is small; this does not
contribute a significant error at this level. Second, our
analysis relies on the approximation that the collision
rates are much smaller than Γ. We have previously esti
mated Γ to be approximately 6×107rads−1[22]. When
we use our crosssection to calculate the collision rate,
we obtain γ13+ γc= 5 × 107s−1at n = 2 × 1023m−3.
This implies that the collision rate is approaching Γ at
the upper end of the plot. However, it is the points at
low helium densities that primarily fix the gradient of
the fit, since these have far more weight in the fit, and
for these values the collision rates are much smaller than
Γ, consistent with our original assumption. Third, we
have used a fitting function that is appropriate when the
laser is resonant for molecules at rest, but this cannot be
satisfied for all four of the unresolved components. We
find that setting δ?
3%, and setting δ?
12%. Considering the mean displacement of the compo
nents from the line centre at 80K, we conclude that the
resulting change in the crosssection, and the associated
error, are both negligible for this measurement. The final
result is σ13+ σc= 37 ± 14 × 10−20m2. This result is
close to our calculated YbFHe diffusion crosssection at
80K, given in Table I, as one would expect if velocity
changing collisions dominate over inelastic collisions in
determining the saturation intensity.
The values of α0 obtained from the fits to the data
shown in Fig.11(a), along with similar datasets taken
at other helium densities, are independent of helium
density and are consistent with one another across the
whole range of helium densities.
α0= 18.1 ± 0.8. The fact that δL?= 0 for the four unre
solved components has a much larger effect on α0than
on I?
corrected value of α0= f1nσL = 23±4. Using the above
values of r and p, the value for the optical absorption
crosssection given by Eq.(A14) is σ = 1.1 × 10−14m2.
Taking an absorption length of L = 3cm and the frac
tion of all molecules in N = 13 to be f1 = 0.05 at
80K, we find the total density of YbF molecules to be
n ≈ 1.3×1018m−3in these experiments. The total num
ber of YbF molecules produced is approximately 3×1013.
Of course, the YbF density depends on the energy and
spot size of the ablation laser, the target quality and the
buffer gas density. Nevertheless, we consistently produce
this density of molecules once the experimental condi
tions are optimized.
L= w/2 decreases the crosssection by
L= w decreases the crosssection by
The mean value is
s, and by considering the effect of this we obtain a
VII. CONCLUSIONS
We have investigated the dynamics of YbF molecules
produced inside a helium buffer gas cell using a com
bination of absorption imaging, absorption spectroscopy
and saturated absorption spectroscopy. While we have
focussed on this particular molecule, we expect our re
sults to carry over to a wide range of other polar radicals
produced using similar methods.
The absorption images show that, upon ablation of the
target, the molecules that are formed expand ballistically
Page 15
15
into the cell on a µs timescale. When the helium den
sity is low, this initial ballistic expansion completely fills
the cell, but at higher helium density the expansion is
arrested with the molecules confined to a region around
the target. The molecules then diffuse slowly to the walls
of the cell. We have measured how the diffusion time de
pends on the helium density and find the expected linear
dependence at low density. At higher densities the dif
fusion times no longer increase with density but instead
reach a roughly constant value. Our modelling of the ex
periment shows that the diffusion is more rapid when the
molecules are localized near the target than when they
fill the cell, and that in the former case the decay curves
depend strongly on the probe position, with the density
decaying much faster when the probe is close to the tar
get. This observation, along with the insights obtained
from the absorption imaging, suggest that the levelling
off of the lifetimes is due, at least in part, to the initial
confinement of the molecules to a region close to the tar
get. By comparing the diffusion of YbF and Li at low
helium densities, and using values for the Li diffusion
crosssection calculated from a wellestablished interac
tion potential, we have measured the YbFHe diffusion
crosssection at two temperatures. The experimental val
ues agree with the theoretical values we calculate from
the YbFHe potential given in [40]. This is a first ex
perimental test of that potential. We have compared our
experimental results for the Li diffusion times with the
results of a numerical model of diffusion in this cell. The
results agree at 80K and 20K, but differ by a factor
of two at 293K where the results are most precise. This
discrepancy is not understood but suggests that diffusion
crosssections determined by measuring lifetimes in a cell
may not be accurate unless a reference atom is used.
We have measured how the translational and rotational
temperatures of the molecules change as a function of the
time since their formation, for typical helium densities of
about 1022m−3, and have compared our experimental
results with those of various thermalization models. Our
conclusion is that the YbF molecules thermalize with the
helium more rapidly than the 50µs time resolution of
the experiment. Instead of observing that thermalization
process, we see the cooling of the helium buffer gas which
heats up when the target is ablated and then slowly cools
back down to the temperature of the cell walls. We see
that the cooling occurs on two separate timescales which
differ by a factor of 10, and we suppose that the heat of
ablation is deposited in a local region close to the target,
and that this heat first dissipates quickly into the rest of
the gas, and then more slowly as it diffuses to the walls.
Finally, we have made a detailed study of the absorp
tion of a probe laser by molecules in a buffer gas cell,
elucidating the main dynamical processes that are im
portant. As the power of the probe laser is increased
the absorbed power increases at first, but eventually sat
urates because the population in the resonant state is
optically pumped into a multitude of other states that
are not resonant with the laser. The resulting hole in the
combined velocity and internal state distribution is re
populated by collisions, and so saturation of the absorp
tion sets in when the rate of optical pumping is equal to
the relevant collision rates. We have measured the frac
tional absorption as a function of laser intensity over a
range of helium densities and find that the data fits well
to our model. The intensity needed to saturate the ab
sorption is directly proportional to the helium density for
densities up to 1022m−3. The crosssection for the “hole
filling” collisions is consistent with our theoretical value
for the YbFHe diffusion crosssection, as we would ex
pect if velocitychange is the dominant hole filling mech
anism. The same data are used to estimate the total
YbF density, which is approximately 1018m−3. The to
tal number of YbF molecules produced is approximately
3 × 1013.
Using an improved setup, we have now reduced the
cell temperature to 3.5K. At this temperature there will
be about 3 × 1012molecules in the rotational ground
state. Using a hydrodynamic flow of helium to sweep
the molecules out of the cell, we expect that at least
10% can be extracted into a beam with a divergence of
about 0.1steradian [31], resulting in a cold, slowmoving
molecular beam with a flux of 3 × 1012ground state
YbF molecules per steradian per pulse.
times more intense than the flux produced in a super
sonic source of the same molecules [44], and provides
molecules at considerably lower speed, and so promises a
large improvement in the sensitivity of molecular beam
experiments such as the measurement of the electron’s
electric dipole moment.
This is 2000
Acknowledgments
We are grateful to Jon Dyne, Steve Maine and Va
lerijus Gerulis for their expert technical assistance. We
thank Roman Krems for sending us the YbFHe inter
action potential. This work was supported by the EP
SRC, the STFC, and the Royal Society. The research
leading to these results has received funding from the
European Community’s Seventh Framework Programme
FP7/20072013 under the grant agreement 216774.
Appendix A: Laser absorption by molecules in the
presence of a buffer gas
To understand the interaction of the probe laser with
the molecules in the buffer gas, we use a 3 level model of
the molecule. In this model, a plane polarized laser field,
E = E0ˆ zcos(ωt), is detuned from resonance with the
transition from level 1 to level 2 by an angular frequency
δ. Level 3 represents all other relevant molecular levels
and is not coupled to the laser field. Level 2 can decay
by spontaneous emission, either to level 1 at rate rΓ,
or to level 3 at rate (1 − r)Γ. Inelastic collisions with
helium atoms also cause level 2 molecules to decay to
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