Alternating gradient guiding of cold, polar molecules
T.E. Wall, S. Armitage, J.J. Hudson, B.E. Sauer, J.M. Dyne, E.A. Hinds and M.R. Tarbutt
Centre for Cold Matter, Blackett Laboratory, Imperial College London, SW7 2AZ, United Kingdom
Introduction
Cold polar molecules are useful for many applications in physics and chemistry, including high resolution spectroscopy, tests of fundamental physics, studying chemcial reactions and quantum
information processing.
For these applications it can be extremely useful to transport the molecules from the source to the experiment, while maintaing the high phase-space density of the source.
We have built an alternating gradient (AG) guide and used it to transport ground-state CaFmolecules. We have characterized it both experimentally and with analytical and numerical simulations.
1) Guiding strong-field seekers
2) Experimental setup
The Stark shift experienced by a molecule in an applied electric field can either increase or decrease the
molecule’s total energy.
The AG guide consists of four 1m-long steel rods, with radius 3mm, arranged on the vertices of a square
of side 7.87 mm. The polarity of two of the electrodes can be switched with a bipolar HV switch.
If the Stark shift is negative, the molecule will be drawn to regions of strong field, and is known as a
strong-field seeker. Conversely, a positive Stark shift leads to a weak-field seeking molecule.
x
Guiding a beam of weak-field seeking molecules is a simple task. All that is required is an electrode
arrangement along the beam-line that creates a minimum of field along the axis. However, it is often
desirable to transport molecules in their ground state, which is always strong-field seeking.
+V
State 2
+V
-V
+V
State 1
Figure 1. Transverse
electric field
distribution created
by four long parallel
rods (shown in grey).
V = 10 kV. The
distrbution can be
rotated through 90o
by switching the
polarity of two
diagonal rods.
120 kV/cm
State 2
0 kV/cm
+V
-V
-V
-V
Probe laser
z
Support
rods
Static guiding of a strong-field seeker would require a maximum of electric field along the beam-line.
However, this is impossible to create. Instead, the next best approach is to create a saddle-shaped field
orthogonal to the beam-line. Molecules will be focussed in one transverse direction, while being
defocussed in the other (”State 1” in fig. 1).
State 1
Lens
y
Mirror
AG guide
Guide electrodes
Skimmer
Target
Figure 2.
Experimental set-up,
and photograph
looking down the
guide
Ablation laser
Solenoid valve
The molecules used in this experiment (CaF) were created in a separate source chamber by supersonic
expansion [1]. The CaF radicals were entrained in a 600 m/s Ar beam, with longitudinal temperature ~ 3 K.
Having traversed the guide, the molecules were then detected by laser induced fluorescence at 606.3 nm.
After the molecules have traversed the guide for a while in State 1, the orientation of the field is rotated
by 90o (”State 2”), which alternates the direction of the focussing and defocussing forces. This method of
alternating molecular lenses can be used to achieve net focussing along the length of the guide.
The properties of the guide were investigated by observing the fluorescence signal as a function of applied
voltage and lens duration. For every guiding measurement there was a corresponding measurement with
no applied voltage. The resulting fluorescence signal ratio gives a measure of the strength of the guiding,
as well as normalizing out fluctuations in the molecular flux.
3) Modelling the guide
4) Results
E(X,Y) = E0(1 + a3(X -Y ) + (2a3 - 6a5) X Y + a5(X + Y )),
2
2
2
2
2
4
4
where a3 and a5 are expansion coefficients representing the strength of the hexapole and decapole terms
in the potential expansion.
The equations of motion for a molecule in this field are:
X’’ = 2(X + Y2X + X3),
Y’’ = 2(-Y + X2Y + Y3)
where differentiation is with respect to time,  is a measure of the Stark shift, and  and  represent the
strengths of the coupling and cubic terms in the force.
Figure 3 (a) show example molecular trajectories. They consist of a high frequency oscillation (the
micromotion) which has the period of the switching fields, superimposed on a larger, slower oscillation
caused by the imbalance of the trajectory-averaged force along the guide. Figure 3 (b) shows the
acceptance (a measure of the number of stable trajectories that can be guided) as a function of lens
switching time, for an applied voltage of 8 kV.
0.0
0.5
0
200
400
600
z (mm)
800
1000
-2
2
250
1.5
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Lens duration (s)
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300
400
Figure 4. Signal ratio data (points) and simulations
(dashed) versus lens durations for three voltages: 5.5 kV
(red), 8 kV (blue) and 9.5 kV (purple).
1.0
(b)
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0.8
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200
150
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100
50
0
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Figure 5 (a) shows the unexpected result that the signal persists as the DC voltage is increased, and even
increases and peaks around 5.5 kV.
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0
50
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100
150
200
Lens duration (s)
Signal ratio
x (mm)
0.5
300
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It was found that the signal ratio remained
non-zero for much longer lens durations
than expected. To investigate this the guide
was used in “DC mode” - in which the fields
were static and not switched.
(a)
(b) 350
Acceptance (mm ms )
(a)
Figure 4 shows the signal ratio versus lens
duration for three applied voltages (5.5 kV, 8
kV and 9.5 kV) [3]. Also shown are the
results of numerical simulations. The shapes
of the simulations fit very well to the data.
However, the simulated ratios have been
scaled down by 3.2, 2.8 and 3.4 respectively.
This discrepancy is thought to have arisen
from a misalignment of the probe laser
beam.
Signal ratio
The electric potential of the guide can be considered as a multipole expansion. Ignoring higher order terms
the resulting field can be written in terms of scaled transverse coordinates (X,Y) [2]:
2.0
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250
�
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300
It can be seen that the anharmonic terms lengthen the period of the macromotion and drastically reduce
the acceptance, particularly at low lens durations.
We have built a cryogenic buffer-gas cell for making cold YbF molecules.
A hole in the cell provides a very high flux, but divergent beam of cold
molecules. We will use the guide to focus ground-state YbF molecules,
and transport them away from the cryogenic environment, while
maintaining the very high density available at the source.
Our goal is to use the buffer gas cell and AG guide as an intense source of
molecules for a future EDM measurement.
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0.4
0.0
0
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0.2
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Figure 3: (a) Example trajectories along the guide; (b) Acceptance as a function of lens duration for a guide
working at 8 kV. In both figures the red curve is based on a purely harmonic force, the green curve includes
the cubic force, the blue curve includes the cubic and coupling terms, and the purple curve is a complete
numerical simulation using the exact field.
Outlook
�
0.6
�
2
4
6
8
10
Voltage (kV)
Figure 5. (a) DC signal ratio versus voltage: data (points) and simulations (lines). The blue line shows a
simulation that accounts for probe beam misalignment. (b) The trajectory of a DC guided molecule.
The explanation for this behaviour is that in the centre of the guide is a region of strong electric field. A
molecule that leaves the guide in one transverse direction will feel a force towards the centre of the guide.
This force can focus such molecules. Simulations show that the observed peak in signal around 5.5 kV
corresponds to molecules that leave the centre of the guide and are focussed towards the axis, into the
detection region. An example of such a trajectory is shown in figure 5 (b).
Chemical Precursor
Target
Helium Buffer
Gas at 4K
Output
Molecular
Beam
YAG ablation
beam
Heavy Polar
Molecules
References
[1] T.E. Wall et al., Phys. Rev. A 78, 062509 (2008)
[2] M.R. Tarbutt and E.A. Hinds, New J. Phys. 10, 073011 (2008)
[3] T.E. Wall et al., submitted (arXiv: 0905.2082)