INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS
J. Opt. B: Quantum Semiclass. Opt. 5 (2003) S65–S74
PII: S1464-4266(03)56003-7
Magnetic trapping and evaporative
cooling of metastable triplet helium
Norbert Herschbach, Paul Tol, Andrey Tychkov, Wim Hogervorst
and Wim Vassen
Laser Centre Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
Received 11 November 2002, in final form 23 January 2003
Published 2 April 2003
Online at stacks.iop.org/JOptB/5/S65
Abstract
A cloverleaf magnetic trap is loaded from a magneto-optical trap containing
2 × 109 helium atoms in the metastable 2 3 S1 state. With optical molasses
and a spin polarization pulse up to 1.5 × 109 atoms are magnetically
trapped. The vacuum limited trap lifetime is ∼12 s. Compression in 0.5 s
yields a density increase to 8 × 109 cm−3 . With rf-forced evaporation using
an exponentially shaped frequency sweep with a 5 s time constant the phase
space density is increased from 1 × 10−7 to 4 × 10−3 . The central density
increases to 3 × 1011 cm−3 , while an increase in elastic collision rate from
40 to 220 s−1 indicates runaway evaporation.
Keywords: Magnetic trapping, evaporative cooling, metastable helium
1. Introduction
Bose–Einstein condensation (BEC) of metastable triplet
helium has recently been achieved by Robert et al [1] and
Pereira Dos Santos et al [2] applying different techniques
to observe the Bose condensate. After evaporative cooling
in a cloverleaf magnetic trap Robert et al observed the
transition to BEC in the time-of-flight (TOF) spectrum of
the atomic cloud measured with a microchannel plate (MCP)
detector. The large internal energy (19.8 eV) of metastable
helium atoms in the 2 3 S1 state (He*) allows the use of
this highly efficient detection technique for neutral atoms.
Pereira Dos Santos et al achieved quantum degeneracy of
the gas in a QUIC (quadrupole Ioffe configuration) trap and
measured the characteristic expansion of the Bose condensate
by absorption imaging, more commonly used in alkali systems.
Both detection techniques have their own specific difficulties.
Absorption imaging using a low-intensity laser beam nearresonant with the cooling transition at 1083 nm is problematic
as the detection efficiency of silicon-based CCD cameras is
low (∼1%) at this wavelength. Thus a dedicated camera
is required to observe the BEC transition with 106 atoms
left after evaporative cooling. Zeeman shifts produced by
residual magnetic fields may hamper accurate determination of
atom numbers and possibly cloud dimensions from absorption
imaging. Since helium is a light atom and the atomic
cloud expands relatively fast, it is important to minimize the
switching time of the magnetic field. MCP detectors allow
close to 100% detection efficiency, but in TOF measurements
the atom trajectories can be perturbed by residual magnetic
field gradients present in the experimental set-up after the
trap is switched off. This can render the interpretation of the
measured TOF spectra difficult and less reliable.
In this paper we present results obtained with our
experimental set-up, previously described [3] and now
extended to allow magnetic trapping and evaporative cooling.
In brief, a beam of metastable helium atoms produced in a dc
discharge source is collimated and deflected by transverse laser
cooling over a length of 20 cm. The intensified He* beam is
decelerated in a 2 m long Zeeman slower. At the end of the
Zeeman slower the atoms are trapped in a magneto-optical trap
(MOT). Experiments characterizing the performance of these
parts of our set-up have been reported [4–6]. For detection we
use both absorption imaging and TOF measurements with an
MCP detector. The MCP detector is positioned 18 cm below
the trap centre, 4 cm off-axis. Results regarding the loading
of our cloverleaf magnetic trap, compression of the trapped
cloud of atoms as well as evaporative cooling will be presented
in this work. Optimization of loading and compression of
the magnetic trap is described in section 3. After choosing
rf sweep parameters for efficient forced evaporative cooling,
as presented in section 4, evaporation entered the runaway
regime. However, the experimental conditions were not
sufficiently favourable to reach the transition to quantum
degeneracy.
1464-4266/03/020065+10$30.00 © 2003 IOP Publishing Ltd Printed in the UK
S65
N Herschbach et al
2. MOT cloud, optical molasses and spin
polarization
Since the measurements described in [6] the loading rate of
the MOT has been improved. Full loading is now attained
within 250 ms. Absorption imaging of the MOT cloud yields
typically N ≈ 2 × 109 for the number of atoms; the rms radii
of the trapped cloud in the horizontal and vertical directions
are σρ = 0.25 cm and σz = 0.33 cm. The use of very small
probe laser intensities (∼0.05ISat , ISat = 0.167 mW cm−2 )
and a short imaging pulse duration of 50 µs as well as the
narrow linewidth of the probe laser (∼0.5 MHz) allow for a
determination of the number of atoms in the MOT within a
relative uncertainty of 20%. The temperature of the MOT
cloud obtained from TOF measurements with the MCP is
1.2 mK. From the central density n 0 and the rate coefficient
for Penning ionization βPI we
√ find for the improved loading
rate of the MOT βPI n 0 N/(2 2) ≈ 1010 atoms s−1 , two times
larger than described in [6].
Before loading of the magnetic trap the atomic cloud is
cooled by an optical molasses pulse. Laser detuning, intensity
and pulse duration for molasses are chosen experimentally
such that the central density after capturing in the cloverleaf
trap is optimized. Best results are obtained for detunings
between −3 and −2, intensities from 0.2ISat to 0.7ISat per
beam, and a pulse duration of 1 ms. Ballistic expansion of
the cloud observed in absorption imaging indicates a positiondependent velocity distribution after molasses, which can be
explained as an effect of the optical thickness of the atomic
cloud. Nevertheless a temperature of ∼0.4 mK can be
deduced from both TOF measurement on the MCP and ballistic
expansion observed in absorption imaging.
After molasses the atoms are spin-polarized by optical
pumping in a magnetic field of a few gauss using a retroreflected laser beam of ∼10ISat intensity for 0.1 ms. Almost
three times more atoms are trapped with this spin polarization
pulse as compared with the situation without.
3. Loading and compression of the cloverleaf trap
3.1. Theoretical considerations
To load the cloverleaf trap the currents would ideally be chosen
such that the potential is harmonic with trap frequencies ωρ,z ,
2
)]1/2 of the
matching equilibrium radii σρ,z = [kB T /(mωρ,z
trapped cloud with those measured before the trap is switched
on. Under this condition the phase space density is kept at
a maximum. Adiabatic compression of the trapped cloud
then yields the lowest temperature. In this case also an
optimum density as well as an optimum elastic collision rate are
obtained, provided atom losses are negligible during loading
and compression.
The phase space density is the quantity that has to be
increased by evaporative
cooling. It is given by D = n3 ,
√
where = h/ 2πmkB T is the thermal de Broglie wavelength
and n is the density of the gas. In the trap the energy
distribution of the atoms is truncated at t determined by the
finite trap depth or the rf knife. Using the truncated Boltzmann
energy distribution Luiten et al [7] obtained expressions for
important thermodynamic quantities. The phase space density
S66
is conveniently expressed by D = N/ζ with the number of
atoms N and the single particle partition function ζ , which
can be calculated from the trap parameters (radial gradient α,
axial curvature β and the central bias field B0 ), temperature T
and truncation parameter η = t /kB T .
For compression of the trapped cloud α and β are increased
to maximum values and B0 is decreased to a minimum. In its
radial position dependence the trapping potential is thereby
altering from harmonic to linear. The energy density of states
changes accordingly. Under adiabatic conditions the phase
space density increases by the factor [7, 8]
exp N kEBf Tf
Df
=
,
(1)
Di
exp N kEBi Ti
where E i,f denotes the internal energy of the gas and the indices
i and f mark the initial and final situation. The temperature
Tf after adiabatic compression can be found by solving
equation (1). When the compression is done instantaneously
instead of adiabatically
the internal energy of the cloud E i
increases by E = n i (ρ, z)[Vf (ρ, z) − Vi (ρ, z)]2πρ dρ dz,
where n i (ρ, z) is the density distribution before compression
and Vi,f (ρ, z) the trapping potential in the uncompressed
and compressed situation respectively. The temperature after
compression Tf follows from solving E i + E = E f .
3.2. Design of the cloverleaf trap
A schematic view of the cloverleaf magnetic trap is given in
figure 1. The cloverleaf coils have a radius of 2 cm and are
positioned with their centre 5 cm away from the axis with the
two cloverleaf planes 3 cm above and below the trap centre.
The pinch coils have a radius of 3 cm and lie 5 cm away from
the centre of the trap. The compensation coils, positioned at
the same distance, have a radius of 7 cm. This geometry allows
4 cm diameter laser beams for slowing and trapping. All coils
are composed of 14 windings each and are wound with coated
rectangular (2 mm × 3 mm) copper wire. The coils, drenched
with varnish after winding to become mechanically stable
compounds, are glued inside a paramagnetic stainless steel
structure through which the cooling fluid is circulated. After
passing isolating ceramic feedthroughs the coil wire endings
are welded to copper tubing. These carry the current and bring
the cooling water to the coils. Another set of ceramic vacuum
feedthroughs is necessary to connect the current supplies and
cooling circuitry outside the vacuum chamber.
Two 200 A current sources supply the current for axial
and cloverleaf coils respectively. At maximum current the
coils only heat up by a few degrees. With an additional current
delivered to the pinch coils by a 50 A supply the central bias
field B0 is set. The compensation coils are usually connected in
series with the pinch coils such that fluctuations in the current
cancel out for the central field in the fully compressed trap.
The currents through the pinch coils can be adjusted with
bypass resistors connected in parallel to each of them. They
consist of 0.5 mm thick copper wire in a plastic tube which
also guides cooling water. With the choice of the length of
the wires the values of the resistances are set such that B0 is
small for small currents in the 50 A supply. This minimizes
the influence of current fluctuations of this supply, which are
Magnetic trapping and evaporative cooling of metastable triplet helium
z
cloverleaf coil
pinch coil
compensation coil
Figure 1. Schematic view of the cloverleaf magnetic trap. Arrows
indicate the direction of the currents in the different coils.
not compensated. The current switches are each composed
of two power MOSFETs connected in parallel, which have an
on-resistance of 7 m.
For the MOT the pinch coils are used as an anti-Helmholtz
pair.
3.3. Loading of the cloverleaf trap
At the moment of loading the cloud radii were larger than
anticipated when designing the magnetic trap. Attempts
to compress the cloud in the MOT were not successful.
Delays caused by the finite switching time of the magnetic
field result in expansion of the cloud before and after the
molasses pulse. With a temperature of 0.4 mK and typical
cloud radii at the moment of loading (σρ ≈ 3.1(3) mm and
σz ≈ 3.8(4) mm) adiabatic trapping requires trap frequencies
around ωρ /2π ≈ 46 Hz and ωz /2π ≈ 38 Hz. The trap
currents necessary to achieve these frequencies are already
relatively high, leaving little room for further compression of
the cloud in the magnetic trap. In addition, under conditions of
adiabatic capturing, the trap depth is relatively small (around
2.5 mK) and TOF measurements on the MCP yield a capture
efficiency of only 30% and a temperature of the cloud in the
cloverleaf trap of ∼0.45 mK. At a temperature T ≈ 0.4 mK,
obtained in optical molasses, one would expect a larger capture
efficiency at this trap depth. The position dependence of
the velocity distribution after molasses as well as oscillations
of the trap currents shortly after switching-on of the trap,
however, may lead to additional non-adiabaticity and heating
during the capture process. By increasing the radial gradient
α and varying B0 , the trap depth was gradually improved to
3.4 mK, which enhanced the captured fraction to 75% while
the temperature increase could be kept small. A temperature
increase to T ≈ 0.65 mK appeared unavoidable due to already
existing non-adiabaticities as well as the use of a larger radial
trap frequency ωρ /2π ≈ 100 Hz and a smaller harmonic radius
ρH = α/B0 ≈ 2.4 mm. The optimum trap parameters for
capturing are α = 37 G cm−1 , β = 22 G cm−2 and B0 =
8.8 G. TOF measurement on the MCP yields a temperature
T = 0.68(2) mK for the cloud after loading of the magnetic
trap. The number of atoms in the magnetic trap is also extracted
from the TOF signal using the calibration of the MCP current
by absorption imaging in the MOT, where the absorption
rate is not significantly affected by residual magnetic fields.
We obtain a large number of magnetically trapped atoms:
N = 1.4(3) × 109 . For comparison, the MOT contained
N = 1.9(4) × 109 atoms during these experiments. These
numbers compare very well with the French results [1, 2].
Due to the persistence of a residual magnetic field in the
axial direction of ∼2 G for several milliseconds after magnetic
trapping, the absorption rate is reduced and absorption imaging
of the cloud yields a number of atoms smaller by a factor
of ∼2 compared with the TOF measurement on the MCP.
Around the trap centre residual magnetic field gradients were
measured to decay fast: 0.5 ms after the trap is switched off
gradients are smaller than 0.2 G cm−1 . Thus we can assume
the reduction of the absorption rate to occur homogeneously
over the atomic cloud such that shape and size of the cloud
are still determined accurately by absorption imaging. Also
the atom trajectories are only affected in a minor way by these
small magnetic field gradients, such that the TOF signal on
the MCP gives a more reliable measurement of the number
of atoms than absorption imaging. Fitting a Gaussian density
distribution to the absorption image of the cloud yields the radii
of the cloud. With the temperature from the TOF measurement
on the MCP, cloud radii in the trap σρ = 2.7(3) mm and
σz = 5.1(5) mm are obtained. Radially the potential is strictly
harmonic only in the central region. Since the harmonic radius
is still sufficiently large in the uncompressed trap (ρH ≈ σρ ) the
error introduced by the Gaussian approximation is only about
10%. The rms radii calculated with the theoretically expected
density distribution are σρ = 2.3 mm and σz = 4.6 mm.
The experimental values are 10% respectively 15% larger than
the theoretical values, which may be caused by the position
dependence of the velocity distribution after molasses and
insufficient thermalization after 1 s in the trap. Temperature
values for expansion in the axial and radial directions have
been obtained by fitting to cloud radii determined from
measurements at different ballistic expansion times. The
temperature obtained from the vertical expansion coincides
closely with the temperature obtained from the TOF signal on
the MCP. A slight anisotropy in the expansion corresponding
to a 20% difference in the temperature extracted in the axial
and radial directions was noticeable after trapping for 1 s,
indicating that thermalization may be insufficient. Repeating
this measurement with a trapping time of 5 s revealed no
anisotropy in the expansion.
With temperature and number of atoms extracted from
the TOF measurement on the MCP the central density n 0 =
3.8(8)×109 cm−3 is calculated using the truncated distribution
approach [7]. Assuming a Gaussian density distribution with
the radii obtained from absorption imaging the central density
becomes n 0 = N/[(2π)3/2 σρ2 σz ] = 2.4(7) × 109 cm −3 . With
σ vT = 3 × 10−9 cm3 s−1 [9] an elastic collision rate per
atom nσ vT = 7(2) s−1 is obtained in the centre of the
trap. The phase space density in the uncompressed trap is
n 0 3 = 1.0(3) × 10−7 .
3.4. Compression
In view of evaporative cooling, compression of a magnetic
trap is preferentially done by optimizing the elastic collision
rate nσ vT in the fully compressed trap. In our case the
S67
temperature after compression is around 1 mK, where the
rate constant of elastic collisions σ vT is expected to take
a flat maximum [9]. Thus the temperature dependence of the
elastic collision rate is almost entirely contained in the density
n, which scales as N/ T 5/2 in a compressed cloverleaf trap.
One can therefore choose to optimize the density in order to
find the largest elastic collision rate.
As long as the trap is harmonic, in order to stay adiabatic
it is sufficient that the compression occurs on a time scale
that is long compared with 1/ωz . Outside the harmonic
regime elastic collisions are required to keep the gas in thermal
equilibrium and the time scale for adiabatic compression rather
is determined by the inverse elastic collision rate. The central
bias field B0 must therefore be lowered after compression
in the harmonic regime, when the elastic collision rate is
already enhanced. In our case, however, the rms radius of the
cloud after loading is already close to the harmonic radius, so
that only small improvements are expected from this strategy.
Hence we chose to increase α and β gradually to their final
values of 68 G cm−1 and 28 G cm−2 and simultaneously to
decrease B0 to a value around 1 G.
With a trap lifetime limited to ∼10 s by collisions with
background gas molecules, a strictly adiabatic compression is
unlikely to lead to the best result, as during the longer time
required to compress the cloud adiabatically loss of atoms
may significantly counteract increase in density. To find
experimentally the best timing for compression, the density
of the trapped cloud is measured by absorption imaging for
different durations of compression, ranging from the rise time
limit of the current source to 4 s. After compression the cloud
stays trapped for 2 s in order to equilibrate. The highest
density is measured with compression starting shortly after
loading and lasting for 0.5 s. With a 4 s compression the
density is measured to be 30% below and with 0.2 s it is 10%
below the best result. These density variations are not large
and temperatures obtained from the TOF signals on the MCP
decreased only slightly with longer compression time. For
a compression of 0.5 s a temperature T = 1.12(3) mK is
obtained from a TOF measurement. For compression times
2 s the compression occurs more adiabatically and a slightly
lower temperature T = 1.05(3) mK is measured, which is
closer to the expected temperature for adiabatic compression,
T = 0.91 mK. Compressing the cloud within 0.2 s yields
T = 1.15(3) mK, corresponding closely to the temperature
expected for instantaneous compression.
In the compressed trap, with B0 = 1 G, the harmonic
radius is small (ρH = 0.15 mm) and the harmonic
approximation is good for temperatures T < 2µB B0 /kB ≈
0.1 mK. At temperatures around 1 mK the density distribution
is radially well approximated by a Laplace distribution. Only
in the centre does the Laplace distribution overestimate the
density by 10%.
An absorption image of the cloud after compression in
the cloverleaf trap is shown in figure 2. After 1 ms of
ballistic expansion a Gaussian density distribution fits well to
the relative absorption data giving the rms radii of the cloud
σρ (1 ms) = 2.4(1) mm and σz (1 ms) = 7.5(4) mm. In
order to infer the initial radii of the cloud we approximate
the initial density distribution of the cloud radially by a
Laplace distribution and axially by a Gaussian and calculate
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rel. transmission
N Herschbach et al
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-1 -0.5 0 0.5 1
horizontal position (cm)
-1.5 -1 -0.5 0 0.5 1 1.5
vertical position (cm)
Figure 2. Absorption image of the cloud captured in the
compressed cloverleaf trap for 2 s after compression. The image is
recorded after 1 ms of ballistic expansion.
the evolution of the distribution during ballistic expansion. For
ballistic expansion times 1 ms and at temperatures around
1 mK a Gaussian profile is found to fit well to the calculated
column density in the radial direction as well. The calculation
shows that the initial radius σρ (0) of the Laplace distribution is
recovered within 5% for a 1 ms expansion. The accuracy of this
method gradually decreases as the contribution of expansion
to measured cloud size increases, as for small initial size or
long expansion times. Thus in the radial direction σρ (0) =
1.8(2) mm
√ is found, which corresponds closely to the expected
radius 2kB T /(2µB α) = 1.7 mm for a Laplace distribution.
In the axial direction the potential is√
closely harmonic and the
cloud radius is expected to be σz = kB T /(µB β) ≈ 5.5 mm.
Experimentally a 30% larger value σz (0) = 7.3(4) mm is
obtained for unknown reasons. With a trap depth of 6.3 mK
the rms radii calculated from a truncated distribution [7] are
σρ (0) = 1.55 mm and σz (0) = 5.3 mm.
Using a semi-Laplace density distribution with the cloud
radii obtained from absorption imaging and the number of
atoms N from the TOF measurement
on the MCP, the central
√
density becomes n 0 = N/( 2π 3/2 σρ2 σz ) = 7.5(2.3) ×
109 cm −3 . The density enhancement in the compressed trap
thus increases the elastic collision rate per atom in the centre of
the atomic cloud to nσ vT = 25(8) s−1 , assuming the elastic
collision coefficient as calculated in [9]. During compression
the phase space density increases by 30% to n 0 3 = 1.3(4) ×
10−7 . In comparison, an increase to n 0 3 = 2.1 × 10−7 is
expected under ideally adiabatic conditions and in the absence
of atom loss.
3.5. Measurements of the central offset field B0
Hall probe and flux-gate probe measurements of the magnetic
field produced by the trap coils agreed within 10% with
the calculated field. From these measurements, however,
the central bias field B0 can only be obtained with a large
uncertainty as it is the resultant of the two large magnetic fields
produced by pinch and compensation coils, which partially
compensate in the centre of the trap.
B0 can be sensitively probed with rf by inducing
transitions to untrapped magnetic substates close to the centre
of the trap. Therefore rf-forced evaporative cooling (described
below in section 4) is performed on the trapped cloud. By
sweeping down the rf frequency ωrf the truncation energy
t = h̄ωrf − 2µB B0 is gradually lowered and in principle the
value of B0 is obtained from ωrf when the truncation energy
becomes zero and no atoms are left in the trap. Evaporative
Magnetic trapping and evaporative cooling of metastable triplet helium
cooling during the rf sweep enhances the sensitivity of the
method. The lower temperatures then also increase the
transition probability to untrapped states, as compared to a
single-frequency cut at small truncation energies applied to the
cloud after compression. To determine the number of trapped
atoms the TOF signal on the MCP is measured at the end of
the rf sweep.
The value of B0 is also determined by the ratio of the
pinch coil resistance to that of the bypass resistors connected
in parallel to them. The bypass resistors have the same
temperature coefficient as the coils, so that B0 is only affected
when coils and bypass resistors heat up differently. When the
trap is on for longer times (∼20 s), as typically required for
rf-forced evaporation, B0 is observed to decrease with trapping
time. The effect is particularly large for the first two cycles of
an evaporative cooling experiment. After the second cycle in a
row coils and bypass resistors heat up in the same way such that
the decrease of B0 is smaller than 0.1 G over the last 10 s and
B0 takes closely the same value at the end of the rf-frequency
sweep. This final B0 value could be lowered to 0.5(1) G, still
ensuring stable operation of the trap. With lower offset fields
differences from one cycle to another become too large and
changes in temperature and flow of the cooling fluid are likely
to have an influence.
4. Evaporative cooling
4.1. Theory of evaporative cooling
Thermalization and evaporative cooling are driven by elastic
collisions. For metastable helium in our trap temperatures
are 1 mK and thus well below the threshold for pure
s-wave scattering, which lies at 10 mK. The collision energies
E ≈ kB T are, however, still in the range of the binding energy
ε0 ≈ kB × 3 mK of the weakest bound state (v = 14) of the
scattering potential 5 g+ [10] and the energy dependence of the
elastic collision cross section σ (k) has to be considered [11].
From scattering theory it follows that
σ (k) =
8πa 2
,
1 + k 2a2
(2)
√
m E/h̄ is the wavevector of the collisional
where k =
system (m denoting the atomic mass) and a is the s-wave
scattering length. For a we choose for the rest of this paper
a value of 9 nm, which gives a conservative estimate of
the elastic collision rate while being consistent with reported
experimental values [1, 2, 12–14] and scattering potential
calculations [10, 15]. The rate coefficient for elastic collisions
σ vT = σ (k)2h̄k/mT is obtained by taking the average of
σ v over the thermal distribution:
3
h̄
σ vT = √
πmkB T
∞
2h̄k
h̄ 2 k 2
4πk 2 dk,
×
σ (k)
(3)
exp −
m
mk
T
B
0
where v = 2h̄k/m using the reduced mass m/2. From the
calculated scattering potential [10] Fedichev et al [9] have
calculated σ vT over a temperature range extending beyond
the threshold for s-wave scattering. For T < 10 mK nearly the
same result is obtained from equation (3) (assuming a = 9 nm).
For low temperatures (ka 1) the cross section tends towards
the constant σ = 8πa 2 . Then the collision
√ rate coefficient
becomes σ vT ≈ 8πa 2 vT = 8πa 2 4 kB T /πm. The
elastic collision rate per atom is given by nσ vT with n
denoting the density of the gas.
In a trap of finite depth t elastic collisions tend to
repopulate states with energies t . When these atoms leave
the trap, they evaporate. With each evaporating atom the
ensemble of trapped atoms loses an energy t which is large
compared to the mean energy per particle in the trap, and so
the system is cooling.
In the kinetic theory of evaporative cooling developed by
Luiten et al [7] the rate at which atoms evaporate from the trap
is given by
Ṅev = −n 20 v̄σ e−η Vev ,
(4)
where v̄ = (8kB T /πm)1/2 , η = t /kB T is the truncation
parameter and the effective volume for elastic collisions
leading to evaporation Vev is defined in [7]. The rate of change
in internal energy of the system due to evaporation is obtained
from
(5)
Ė ev = Ṅev [t + (Wev / Vev )kB T ],
where the volume Wev < Vev is also given in [7]. Both Wev
and Vev can be calculated from the trap parameters, temperature
and truncation parameter. The cooling rate of the gas can then
be calculated from Ṫ = ( Ė ev − µ Ṅev )/C, where C denotes
the heat capacity of the gas [7] and µ = (∂ E/∂ N )T = E/N .
In this theory of evaporative cooling the elastic collision
cross section σ is assumed to be energy-independent. For
metastable helium the energy dependence of the elastic
collision cross section obtained from equation (2) is given
in figure 3. Strictly speaking, the energy independence of σ
is only valid in the limit of small collision energies. Using
equations (4) and (5) the evaporation rate and the rate of
change in internal energy are calculated systematically too
large unless the temperature is very low. As a range of
collision energies extending beyond the truncation energy are
involved in the evaporative cooling process it is difficult to
find a correction factor. Judging from figure 3 in the case
where the s-wave scattering length a takes a value of 9 nm,
for example, the overestimation is probably less than a factor
of 2 for temperatures below 0.1 mK and a truncation parameter
η ∼ 5, as only collision energies 2t can contribute. In the
case of a larger value for a these equations can be expected
to deviate further and lower temperatures are required for a
meaningful prediction of the evaporation and cooling rate.
Nevertheless this theory will be used for comparison with our
experimental results.
4.2. Plain evaporation
Before compression the trap depth is only 3.4 mK which gives
a truncation parameter η = 4.9 and the cloud is observed to
cool by plain evaporation. In a trap lifetime measurement the
TOF signals on the MCP provide the decrease in the number of
atoms shown in figure 4(a) as well as the temperature evolution
of the trapped cloud, plotted in figure 4(b). The temperature
clearly decreases by 14%, and with a larger cooling rate
at shorter trapping times. From figure 4(a) a trap lifetime
τ = 11 s is deduced.
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N Herschbach et al
cross section (10-12 cm2)
4.3. rf-forced evaporative cooling
100
80
σ(k) with a=20nm
60
8 π / k2
40
σ(k) with a=9nm
20
0
0.001
0.01
0.1
1
collision energy (kB mK)
Figure 3. Energy dependence of the elastic collision cross
section σ .
(a)
(b)
9
0.75
temperature (mK)
number of atoms N
10
0.7
0.65
0.6
0.55
8
10
0
5
10 15 20 25
time (s)
0
5 10 15 20 25
time (s)
Figure 4. TOF measurements on the MCP performed after different
trapping times in the uncompressed cloverleaf trap: (a) number of
atoms in the trap and fit of an exponential decay yielding a trap
lifetime of τ = 11 s (line); (b) plain evaporative cooling of the cloud
in the uncompressed trap.
The cooling rate calculated for the measured number
of atoms and temperature using equations (4) and (5) with
σ = 8πa 2 and a = 9 nm is about 150 times larger than
is observed experimentally. This large overestimation of the
cooling rate cannot alone be explained from the fact that
the calculation neglects the energy dependence of the elastic
collision cross section σ . Most probably the evaporation rate is
reduced due to low dimensionality in the evaporation process,
since evaporation mainly occurs in small regions around the
four saddle points of the trapping potential which determine the
trap depth. The evaporation rate corresponding to the observed
cooling rate contributes less than 1% to the total loss rate which
is dominated by losses caused by collisions with background
gas molecules.
Simultaneously to evaporation, heating effects may occur
and reduce the cooling rate. In order to measure the heating rate
separately a measurement similar to the one shown in figure 4
was performed. To neglect plain evaporation the heating
rate was measured in the compressed trap, after rf-forced
evaporation was applied to reach temperatures of ∼0.1 mK.
A heating rate of ∼0.7 µK s−1 was found, which does not
counteract significantly evaporative cooling to temperatures
of ∼1 µK.
S70
Forced evaporative cooling is performed directly after
compression in the magnetic trap with B0 = 0.5 G. An
rf field is switched on and its frequency ωrf is gradually
decreased. The frequency of the rf field determines the
truncation energy t = h̄ωrf − 2µB B0 and the rf field
transfers atoms with an energy t to untrapped magnetic
substates. In fact, the cloud of atoms in the trap experiences a
gravitational sag δz = −g/ωz2 ≈ 0.13 mm, with g denoting the
gravitational acceleration. Taking this sag into account leads
to a√
correction to the truncation energy: t = h̄ωrf − 2µB B0 −
gm (h̄ωrf − 2µB B0 )/(µB β). The correction is small except
for rf frequencies ωrf close to 2µB B0 /h̄.
The lowering of the truncation energy causes the removal
of atoms occupying states with energy t without elastic
collisions being required. The atom loss rate due to this spilling
is
(6)
Ṅspill = n 0 3 e−η ρ(t )˙t ,
where ρ(t ) is the energy density of states and ˙t is the rate of
change of the truncation energy [16].
As rf source we use a signal generator (IFR2023A) which
can be programmed to produce linear as well as exponential
frequency sweeps. Its output is amplified by a broadband rf
amplifier (ENI607L) with 45 dB nominal amplification making
a maximum power of 14 W available in our system. The rf is
coupled in via an isolating coaxial vacuum feedthrough and is
transported to a small coil of four windings with 1 cm diameter,
mounted close to the trap centre. In the trap region maximum
magnetic field amplitudes of 10−2 G can be achieved. In situ
measurements of the frequency dependence with a pick-up coil
revealed a number of maxima and minima, which we ascribe
to effects of the surrounding steel structure of the trap setup. Fortunately, the minimum signal did not become very
small over the frequency range up to 150 MHz. Typically a
constant rf power of 10 W was used in the rf sweeps. When
different values between 1 and 14 W were chosen nearly the
same temperature and number of atoms were found from the
TOF measurement at the end of the rf sweep. Hence the rf
power used can be considered sufficiently large to remove the
atoms efficiently from the trap.
The parameters characterizing the rf sweep were chosen
experimentally.
Different starting frequencies between
ωrf /(2π) = 60 and 180 MHz were tried with an rf sweep
of exponential shape with a 5.6 s time constant and ending
at ωrf /(2π) = 15 MHz. An optimum starting frequency
was found with ωrf /(2π) = 120 MHz corresponding to
a truncation parameter η = 5. Using different starting
frequencies resulted in approximately the same temperature
after the rf sweep, whereas the number of atoms at the end
of the sweep decreased when starting frequencies ωrf /(2π) 100 MHz were used.
When linear rf sweeps were used with a duration of 10 s
ending at ωrf /(2π) = 15 MHz, a 10% higher temperature
and 10% smaller number of atoms were measured at the end
of the frequency sweep. Since the computer control of the
rf generator does not (yet) allow us to generate a sequence
of consecutive partial sweeps in a convenient way, a single
exponentially shaped sweep was used. The exponential time
constant τrf characterizing the frequency sweep was varied in
Magnetic trapping and evaporative cooling of metastable triplet helium
MCP-current (nA)
200
150
100
50
0
2
4
10 0
0.3
0.2 0.25
0.15
0.05 0.1
time of flight (s)
20 0
0.25 0.3
0.15 0.2
0.1
0.05
time of flight (s)
6
8
rf-sweep duration (s)
MCP-current (nA)
140
120
100
80
60
40
20
0
10
12
14
rf-sweep duration (s)
16
18
Figure 5. Evolution of TOF signals recorded by interrupting the rf
sweep at different times.
the range from 1 to 10 s, the final frequency of the sweep being
fixed. The number of atoms remaining in the trap exhibits a
broad maximum as a function of τrf . For a final frequency of
15 or 5 MHz the temperature is low and therefore the density
distribution of the cloud is in the harmonic regime, where the
ratio N/ T scales with the elastic collision rate per atom, when
a constant cross section for elastic collisions is assumed. The
ratio N/ T has a maximum in both series of measurements at
the same value of τrf = 5.7(1) s.
By interrupting the rf sweep at different times a series
of TOF signals was measured. It shows the evolution of the
cloud of atoms in the course of the rf sweep and is presented in
figure 5. A starting frequency of 120 MHz and a time constant
τrf = 5 s characterize the rf sweep used in this measurement.
The central bias field of the trap was B0 = 0.5 G. For the
measurements displayed in the second part of figure 5 the
MCP detector was biased with a larger voltage in order to
increase sensitivity. This enhanced the gain of the detector by
a factor of 24. In the course of the rf sweep the TOF peak
comes at later times as evidence for cooling while the signal
strength decreases due to the loss of atoms. After 18 s, when
ωrf /(2π) 3 MHz, the rf appears to have no effect on the
trapped atoms. Since the peak arrival time as well as the width
of the TOF signal then do not change, it can be concluded
that the temperature remains constant. Also the decrease of
the signal strength after 18 s of rf sweep corresponds to the
exponential decrease of the number of atoms due to the trap
lifetime of 12 s. The reason for this failure of the rf to remove
atoms from the trap in this measurement is probably of a
technical nature and remains unclear.
For low temperatures T 5 µK, calculated
TOF signals
√
tend to peak at the same time tp =
2d/g = 0.192 s,
determined by the free fall of the atoms from a height d to
the active surface of the detector. However, for the lowest
temperatures realized in the measurements shown in figure 5,
the TOF signals are observed to peak at a time about 10 ms
later than tp . Since the position of the MCP detector was
determined with a relative uncertainty 2% using several
methods, the later peak arrival time cannot be explained by
a possible error in the difference in height d between the trap
centre and detector. To explain the 10 ms difference a finite
time required for switching off of the trapping field would have
to be approximately equal to the observed delay of the TOF
peak in order to cause this delay. It was, however, measured
to be only ∼0.1 ms. A more probable explanation is the force
exerted on the falling atoms by gradients of residual magnetic
fields. Assuming that most of the atoms remain in the same
Zeeman substate this force can cause a delayed arrival of atoms
at the MCP detector.
To estimate the order of magnitude of the field gradient
bz that causes a 10 ms delay of the TOF peak, a change in
vertical velocity of the atoms, v(t), required for the delay in
arrival time on the detector and occurring at time t after the
trap is switched off, can be considered. One finds, for instance,
v(0.01 s) = 0.1 m s−1 and v(0.1 s) = 0.2 m s−1 . The
corresponding changes in kinetic energy are E k (0.01 s) =
kB ×2.4 µK ≈ 2µB ×0.02 G and E k (0.1 s) = kB ×85 µK ≈
2µB × 0.6 G. Interpreted as work 2µB bz z of the gradient bz
over the distance z this yields bz ≈ 0.36 and 0.13 G cm−1
respectively assuming for z the distance the atoms fell in
the time t. Although Hall probe measurements around the
centre of the trap revealed no gradients of this magnitude over
times 1 ms after the trap is switched off, the falling atoms
may encounter such gradients away from the trap centre near
the metal coil containers where eddy currents can flow. Also
small remnant or permanent magnetization of weldings and
some components of the set-up may contribute. Whatever the
origin may be, these calculated gradients are so small that they
are difficult to prevent in a set-up with metal components close
to the atoms.
Mainly TOF measurements of colder clouds should be
affected by a systematic uncertainty considering the change in
kinetic energy kB 0.1 mK estimated to be involved with the
disturbance of atomic trajectories by residual field gradients.
Also ballistic expansion of the cloud observed in absorption
imaging was found to be in good agreement with TOF
measurements for temperatures T 0.5 mK.
In order to estimate the uncertainty in the number of
atoms and temperature obtained from the measurements
with the MCP detector the fitting routine is applied using
several sets of differently calculated TOF signals. In the
calculation of the fit functions heuristic attempts are made
to account at least partially for the effect of residual field
gradients. The temperature T and number of atoms N
obtained as fit parameters are plotted in figures 6(c) and 7(a)
respectively using different symbols. The corresponding
truncation parameters η are displayed in figure 6(b) whereas
the truncation energy is shown in 6(a). The cooling rate Ṫ and
evaporation rate Ṅev calculated using equations (4) and (5)
with σ = 8πa 2 and a = 9 nm are plotted for comparison in
figures 6(d) and 7(b) respectively.
S71
N Herschbach et al
(a)
(a)
9
10
1
8
10
0.1
N
ε t / k B (mK)
10
0.01
0
5
10
time (s)
15
20
107
106
(b)
20
5
10
η
15
0
5
10
5
10
time (s)
15
20
(c)
1
T (mK)
20
0.1
0.01
15
20
109
-1
0
evaporation rate (s )
0
8
10
107
106
105
0.001
0
5
10
time (s)
15
20
(d)
cooling rate (mK/s)
15
(b)
5
0
5
10
time (s)
Figure 7. (a) Number of atoms and exponential fit function with
time constant τexp = 2.7(2) s (line); (b) calculated evaporation rate
− Ṅev (data points), experimental evaporation rate − Ṅev,ex (full
curve) and atom loss rate due to spilling − Ṅspill (dashed curve). The
symbols +, and are used in the same way as in figure 6.
1
0.1
•
0.01
0.001
0
5
10
time (s)
15
20
Figure 6. (a) Truncation energy t , with (full curve) and without
(dashed curve) gravitational sag correction; (b) truncation parameter
η; (c) temperature T ; (d) calculated cooling rate. For data points
plotted using symbols the effect of residual field gradients on the
TOF signal is corrected for by assuming in the fitting procedure that
the cloud has an initial mean velocity upwards of 0.2 m s−1 . Using
only 0.12 m s−1 for this mean velocity data points plotted with symbols are obtained, while for data points displayed using +
symbols no correction of the effects of magnetic field gradients is
attempted.
•
The results obtained from the fitting procedure without
attempting to account for the effect of residual field gradients
are displayed using + symbols. The temperature exhibits
strong cooling after 10 s of rf sweep. The corresponding
truncation parameter η increases to large values >10 and
consequently the calculated evaporation rate Ṅev and cooling
rate Ṫ are about an order of magnitude too small to be
consistent with the decrease in number of atoms N and
temperature T .
Attempting to correct for the effect of residual field
gradients, the atomic cloud is given an initial mean velocity
upwards in the simulation of TOF signals used for fitting. With
a mean velocity of 0.12 m s−1 a delay of 10 ms is obtained in the
S72
10
time (s)
calculated signals for the lowest temperatures. For these data
points the symbol is used in the plots. Other measurements,
in which the truncation energy could be lowered to zero,
yielded TOF signals of the coldest clouds still detectable on
the MCP with a maximum delay of the TOF peak of 18 ms. In
order to account for this larger delay an initial mean velocity
of 0.2 m s−1 upwards for the cloud had to be assumed in the
calculation of the fit functions. Also the number of atoms
then obtained from the TOF data is in a better agreement with
absorption imaging results: not accounting for an initial mean
velocity upwards yields a (larger) detected fraction on the MCP
at low temperatures that is not in agreement with absorption
imaging.
Since this interpretation of the measured TOF signals
accounts best for the TOF peak shift and detected fraction,
we will use it in the rest of the paper. The resulting
data points are plotted with the symbol •. The truncation
parameter now increases more moderately in the course of
the sweep, such that calculated evaporation and cooling rates
appear also more consistent with the evolution of the number
of atoms and temperature. To see this, we deduce an
experimental evaporation rate Ṅev,ex from the measurement
in order to compare with the result obtained from equation (4).
The decreasing number of atoms is well described by an
exponential decay function with a time constant τexp =
2.7(2) s, which is plotted as a line in figure 7(a). Hence, the
Magnetic trapping and evaporative cooling of metastable triplet helium
5. Discussion and conclusion
If rf-forced evaporation could have been continued in a
runaway regime as described in section 4.3, quantum
degeneracy of the gas would have been reached eventually, but
probably at temperatures <1 µK and with a small number of
atoms 105 remaining in the trap. Apart from the problem
of detection arising from such a small number of atoms,
the gravitational sag of the cloud would also cause forced
evaporative cooling to become less efficient and even to stop
when the temperature approaches 1 µK. By mounting the
trap set-up in such a way that the weak axis of the trap is in
a horizontal direction, the gravitational sag is smaller and this
10
12
10
11
10
10
-3
n0 (cm )
(a)
0
5
10
time (s)
15
20
15
20
15
20
(b)
250
-1
n0 ⟨σv⟩ T (s )
200
150
100
50
0
0
5
10
time (s)
3
(c)
n0 Λ
total atom loss rate is well approximated by −N/τexp . It is the
sum of the evaporation rate Ṅev,ex , the loss rate due to spilling
Ṅspill (equation (6)) and the loss rate −N/τ due to collisions
with residual gas molecules. Using two- and three-body loss
rate coefficients obtained in recent studies [1, 2, 9, 12, 14], we
find at the densities obtained in our experiment a negligibly
small two-body loss rate contribution 10−2 and a three-body
loss rate contribution <10−3 as compared with the total loss
rate. With the separately measured trap lifetime τ = 12 s the
experimental evaporation rate becomes Ṅev,ex = −N (1/τexp −
1/τ )− Ṅspill . It is plotted as a full curve in figure 7(b) and in the
later part of the rf sweep, where the temperature is 0.05 mK,
it is in good agreement with the calculated evaporation rate
from equation (4) using a = 9 nm. In earlier parts of the
rf sweep the calculated evaporation rate is larger than the
experimental result. This is probably due to the assumption
of an energy-independent elastic collision cross section in
the kinetic theory of evaporative cooling, which is expected
to cause an overestimation of the evaporation rate at higher
temperatures. Using a larger value for a in this analysis leads to
a larger inconsistency between the calculated and experimental
evaporation rate. However, this is a rather indirect way of
determining the value of a. The theory used here neglects
the energy dependence of the scattering cross section and the
analysis of our measurement is based on the assumption that
the disturbance of the TOF signal by residual magnetic field
gradients is corrected for appropriately. It is therefore difficult
to estimate the uncertainty of the value of the scattering length
extracted in this way. Comparing the evaporation rate with
Ṅspill , calculated from equation (6) and plotted in figure 7(b)
as well, evaporative cooling appears relatively inefficient in the
first part of the rf sweep, where the spilling loss is larger. After
10 s of rf sweep the evaporation rate becomes dominant and
evaporative cooling is more efficient.
The increase of the truncation parameter η indicates that
evaporative cooling has a runaway character during the rf
sweep. Also the elastic collision rate per atom in the centre of
the trap n 0 σ vT , plotted in figure 8(b), is more than five times
larger at the end of the rf sweep. It is calculated using the rate
coefficient for elastic collisions σ vT from equation (3). The
evolution of the central density n 0 is displayed in figure 8(a).
The central density becomes 20 times larger during the rf
sweep. The phase space density n 0 3 = N/ζ becomes 104
times larger in the course of forced evaporation as is shown
figure 8(c). An increase by another factor of 700 still has to be
achieved to reach BEC of the gas.
10
-2
10
-3
10
-4
10
-5
10-6
10
-7
0
5
10
time (s)
Figure 8. Evolution during the rf sweep of (a) the central density
n 0 ; (b) the elastic collision rate per atom n 0 σ vT in the centre of the
trap and (c) the phase space density n 0 3 .
problem is avoided. Due to the large detection efficiency for
He∗ , the MCP detector can be used to measure TOF signals
of clouds with less atoms, if it is aligned vertically below the
trap centre. However, residual magnetic field gradients have
to be minimized as they perturb the trajectories of atoms and
thereby affect the TOF measurements on the MCP detector.
Although the number of atoms transferred from the MOT
to the magnetic trap is large, at the end of forced evaporative
cooling a larger number of atoms could be obtained with a
longer trap lifetime, as well as with stronger compression.
The problems of gravitational sag and detection efficiency of
absorption imaging would then be less severe. In a new trap
design with coils outside the vacuum chamber and with tighter
confinement, we hope to reach the improved experimental
conditions necessary to achieve BEC with a large number of
atoms.
S73
N Herschbach et al
Acknowledgments
We gratefully acknowledge support from the Foundation for
Fundamental Research on Matter (FOM), the European Union
(grant HPRN-CT-2000-00125), and the Vrije Universiteit
(USF grant).
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