F eature A rticle
Evaporative Cooling in Linear Paul T raps
by Lohrasp S e if y a n d Robert T hompson
ooling trapped ions is essential to high-precision
studies in many fields including atomic and
molecular spectroscopy[1], quantum informa­
tion [2], quantum computing[3], time standards[4]
and antimatter physics[5,6]. Different temperature regimes
are required for different fields and are attained by a range
of methods, which include but are not limited to laser
cooling[7], sympathetic cooling[8] and evaporative cool­
ing. This computational study explored the possibility of
evaporative cooling of an ensemble of ions trapped in a
Linear Paul Trap (LPT), and optimized its effectiveness by
maximizing the temperature drop per particle evaporated.
interact and equilibrate to achieve a typical Maxwell
Boltzmann energy distribution such as that shown in Fig. 1,
and ii) one of the highest energy particles must escape so
that the average energy is lowered after the evaporation. As
the temperature of the system is reduced, the evaporation
events Calgary, AB T2N 1N4 become less frequent and the
cooling rate is reduced. To counter this effect, the trap
depth can be reduced to allow particles to continue
escaping at similar rates and hence, for the system to reach
lower temperatures. An example simulation where eva­
porative cooling is applied to a system to reduce its
temperature is shown in Fig. 2.
EVAPORATIVE COOLING
LINEAR PAUL TRAPS
By applying evaporative cooling to trapped neutral
particles, the Bose-Einstein condensate state[9] was
achieved for the first time. Later, applying the same
cooling method was one of the key components in
allowing the production and trapping of anti-hydrogen
by the ALPHA group[10]. Evaporative cooling does not
require any additional tools such as lasers, and so it is a
non-invasive cooling method that can be applied to any
type of trap without any extra cost.
Linear Paul Traps (LPT) are widely used in many areas of
physics such as mass spectrometry[11] and contain
particles by electric quadrupole fields that have an
oscillating saddle shape in the radial direction and a
harmonic potential well in the axial direction. These
oscillating fields and the ion-ion interactions cause the
system to be non-conservative and can increase the
energy of the system over time. Any cooling method
applied to the trapped ensemble is required to compete
with this heating and over come it. in the case of
evaporative cooling, the system requires time to re­
equilibrate for the evaporations to reduce the average
energy of the system. However, the longer the system is
allowed to interact and equilibrate, the more the heating
will take over. Therefore, there is a delicate balance
C
Evaporative cooling can be illustrated by the example of
the cooling of hot coffee. Even when the temperature of
the coffee is below its boiling point, there are particles
that have high enough energy to overcome the surface
tension of the liquid and evaporate. These particles have
an energy higher than that of the average particle in the
fluid. Hence, when they leave and the system is allowed
to re-equilibrate, the average energy and temperature of
the coffee will be reduced.
in the case of a trapped ion ensemble, particles require
suffcient energy to overcome the trapping potentials to
evaporate. However, for evaporation to cool the system,
two conditions must be met: i) The system needs to
Summary
Computational studies are used to demon­
strate and quantify the effectiveness of
evaporative cooling of ions contained in a
linear paul trap.
Lohrasp Seify
<lseify@ucalgary.
ca >
Fig. 1
The energy distribution of a sample ensemble
before and after evaporative cooling. On the left,
the particle(s) with the highest energy which are
shown by the arrow have enough energy to
evaporate. In the middle, the particles have
escaped the trap, however the system has not
re-equilibrated yet. On the right, the system after
it re-equilibrates is shown. The new energy
distribution will have a lower mean and so the
temperature of the system is reduced.
and
Dr. Robert
Thompson
<rthompso@
ucalgary.ca >
Department of
Physics and
Astronomy,
University of
Calgary, Calgary,
AB T2N 1N4
La P hysique au Canada / Vol. 71, No. 3 (2015 ) ·
147
E vaporative Cooling
in
L inear Paul T raps (S eify/T hompson)
0_
5000
10 000
15 000
20 000
25 000
Radial Potential Reduction Rate
Fig. 2
Temperature as particles are evaporated. After a drop in
temperature due to the expansion of the system, as
particles evaporate and leave the system, the temperature
is further reduced to the point where the system reaches
temperature of less than 20 K.
Fig. 3
Temperature drop per particle as RPRR is varied for
q = 0.64 and a = 5.95 x 10-3 . There is a clear
increase at RPRR : 10000 s-1 . When the value of
RPRR is too low, the system has too much time to
interact, and so heating takes over, lowering the
temperature lost per particle. As RPRR gets too large,
particles are allowed to escape the system no matter
their energy, and so evaporations might heat the
ensemble.
between how long the system is allowed to interact with itself,
and the rate of evaporation.
To be able to model the trap computationally, parameters a and
q are defined [12], such that
UD^C
a/ —
X2
(1)
q- ^
(2)
where Ω is the oscillation frequency of the fields, and UDC and
UAC are the applied DC and AC potentials respectively that
create the trapping fields. These parameters would control the
trapping potential amplitudes and hence, could be reduced to
allow for evaporations to occur even when the temperature of
the system is lowered. in a lab setting, the values of the a and q
parameters can be reduced by reducing the applied voltage
responsible for creating the trapping fields. Hence, the rate of
this reduction can be modeled by parameters RPRR and APRR
which represent the Radial potential Reduction Rate and Axial
potential Reduction Rate respectively. To allow the system to
equilibrate, the potential amplitude follows a stepwise function
where the potential is dropped and then kept constant to allow
for the system to equilibrate with itself.
shapes such as spherical, cigar shaped, or cylinder shaped
systems and different interaction rates. From there, different
RPRR and APRR parameters were randomly picked using a
Monte-Carlo script. Then, simulations were run to evolve the
system using these values and the temperature and the number
of particles lost during the simulations were recorded at the end
of each run. By keeping the initial trapping parameters a0 and
q0 constant, and varying the RPRR and APRR, the optimum
cooling parameters were determined. An example of such
a run can be seen in Fig. 3, where a system of 256 particles
at 900 K were modeled and a maximum in temperature drop
per particle can be seen when the RPRR : 10000 s -1 for
q = 0.64 and a = 5.95 x 10-3 .
This would indicate that if the cooling parameters are set
correctly, the temperature of the ensemble can be reduced by
850 K (almost 95% of the initial temperature) after only
20 particles evaporated (less than 10% of initial particles).
This suggests that if tuned correctly, any study that makes use
of LPT s can greatly reduce the temperature of the trapped
system by sacrificing a very small portion of their sample, and
allow for more precise experiments to be designed and
performed.
RESULTS
This computational study was conducted by running simula­
tions that model and solve equations of motion of a trapped
ensemble of ions inside an LPT using the RK4 integrating
method. parameters a and q were picked systematically to
represent different trapped systems with different general
148 ·
Physics in Canada / V ol. 71, No. 3 (2015)
ACKNOWLEDGEMENTS
The author would like to thank Dr. Michael Wieser for his
support, Dr. Michael Cummings for providing the base code,
WestGrid for providing the computation time and NSERC for
providing the financial support.
Evaporative C ooling
in
L inear Paul T raps (Seify/T hompson)
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