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 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 YbF-He diffusion cross-section 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.

Full text

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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 YbF-He diffusion cross-section 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 proton-to-electron mass
ratio [6–10], test Lorentz invariance [11], and measure
parity-violating 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 cryogenically-cooled 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 cross-sections 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.atom-ph] 25 Feb 2011

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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). Vacuum-tight 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 hot-pressed 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 thin-walled 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
silver-soldered 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
0.020.05
Inlet pressure (mbar)
0.100.200.50 1.02.0
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-

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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 acousto-optic 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 time-resolved
measurements of the YbF density, translational temper-
ature and rotational temperature at a specific point in
the cell, we use a combination of Doppler-limited and
Doppler-free 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 P-branch 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 low-lying 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-

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0 µs 30 µs 540 µs 450 µs 360 µs 270 µs180 µs 90 µs 60 µs
0
1
Fractional
absorption
(a) 0.9 x 1022 m-3
(b) 17 x 1022 m-3
(c) 94 x 1022 m-3
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 YbF-He diffusion cross-section, 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 Li-He dif-
fusion cross-section 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 cross-section, ¯ σD, is given, with sufficient accu-
racy, by the first Chapman-Enskog 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, Li-He and YbF-He, 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 thermally-averaged diffusion cross-
section ¯ σD can be calculated. For all temperatures, T,
of interest here, the calculation can be done classically

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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 cross-section at centre-of-
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 Li-He 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,Li-Hefor 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 cross-section 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 cross-sections 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 cross-sections calculated at all
temperatures. The interaction potential for YbF-He has
also been calculated [40] and we use this to obtain theo-
retical values for ¯ σD,YbF-Heat these same temperatures.
In this case, the potential is not only a function of the
YbF-He 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 cross-section 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 µs 90 µs 540 µs 1500 µs3000 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 cross-section
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 high-order 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 low-order 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