Agile low phase noise radio-frequency sine wave generator applied to experiments on
ultracold atoms
O. Morizot, J. de Lapeyre de Bellair, F. Wiotte, O. Lopez, P.-E. Pottie, and H. Perrin∗
Laboratoire de physique des lasers, Institut Galilée, Université Paris 13 and CNRS,
Avenue J.-B. Clément, F-93430 Villetaneuse, France
(Dated: October 31, 2018)
We report on the frequency performance of a low cost (∼500 $) radio-frequency sine wave genera-
tor, using direct digital synthesis (DDS) and a field-programmable gate array (FPGA). The output
frequency of the device may be changed dynamically to any arbitrary value ranging from DC to
10 MHz without any phase slip. Sampling effects are substantially reduced by a high sample rate,
up to 1 MHz, and by a large memory length, more than 2 × 105 samples. By using a low noise
external oscillator to clock the DDS, we demonstrate a phase noise as low as that of the master
clock, that is at the level of −113 dB.rad2/Hz at 1 Hz from the carrier for an output frequency of
3.75 MHz. The device is successfully used to confine an ultracold atomic cloud of rubidium 87 in a
RF-based trap, and there is no extra heating from the RF source.
PACS numbers: 39.25.+k, 06.30.Ft, 07.57.Hm
I. INTRODUCTION
Radio-frequency (RF) fields are used in cold atom ex-
periments for different purposes: for instance, evapo-
rative cooling performed in a magnetic trap relies on
RF field coupling between the different atomic magnetic
states [1, 2]. This technique led to the first observa-
tion of Bose-Einstein condensation (BEC) [3, 4]. Also,
RF pulses are used for dissociating ultracold molecules
produced from ultracold gases through Feshbach reso-
nances [5]. More recently, RF fields have been used to-
gether with static magnetic fields for trapping utracold
atoms at a temperature of a few µK in unusual geome-
tries [6, 7]. There is a growing interest for these “RF-
based traps” among atomic physicists, for creating dou-
ble well traps on atom chips [8, 9, 10] or proposing new
kinds of confining potentials [11, 12, 13]. In both cases,
a single frequency RF signal must be frequency swept
over some range, often larger than the initial frequency,
following a precise time function lasting several seconds.
Typically the RF frequency is varied between 1 MHz and
a few tens of MHz in 0.1 s to 10 s in the ramping stage,
and held at the final frequency for seconds in the plateau
stage. For cooling purposes, commercial RF generators
fit physicists’ needs reasonably well, even if a better res-
olution in arbitrary frequency ramps would be appreci-
ated. However, in the case of RF-based trapping, the
requirements are stronger. The main difference between
these two situations is as follows: in evaporative cooling
the cold atomic sample is located away from the region
of efficient coupling, whereas in the RF-based trapping
scheme the atoms sit exactly at the point where the RF
field has the largest effect. The quality of the RF source is
then much more important than for evaporative cooling.
In fact, the cloud position is directly related to the value
∗Electronic address: helene.perrin@galilee.univ-paris13.fr
of the RF field frequency, and the trap restoring force, or
equivalently the oscillation frequency νt in the harmonic
approximation, is linked to the RF amplitude. As a re-
sult, any amplitude noise, frequency noise or phase noise
of the RF signal during the ramp or the plateau leads to
a heating of the cold atomic cloud. This motivated the
construction of a synthesizer fitting our requirements.
This paper is organized as follows. In section II we
give explicit expressions for the heating of the cold atom
sample for frequency and amplitude noise in the case of
RF-based trapping. In section III, we describe our RF
synthesizer. Finally, section IV is devoted to experimen-
tal results on its performance and comparison between
the different RF sources tested on the BEC experiment.
II. REQUIREMENTS ON THE RF SOURCE
FOR RF-BASED TRAPPING
In this section, we will focus on the RF-dressed trap
that we experimentally produce in the laboratory [7].
The extension of the main conclusions to other RF-
dressed trap geometries is straightforward.
The trap confines the atoms in all three space di-
mensions. The trapping force arises from the inter-
action between the linearly polarized RF field B(t) =
BRF cos(2πνRFt) and the atoms in the presence of an in-
homogeneous magnetic field. This interaction results in
a transverse confinement of the atoms to the surface of
an ellipsoid. The atoms are free to move along the con-
fining surface, resulting in a kind of “bubble trap” [6].
Due to gravity, however, the atoms are concentrated at
the bottom of the ellipsoid. Their motion is pendulum-
like in the horizontal directions, and imposed by the RF
interaction along the vertical z axis. This last direction
is thus the most sensitive to the RF field properties (fre-
quency νRF, amplitude BRF) and we will concentrate on
the vertical motion in the following. Along this direc-
tion, heating or atomic losses may arise from frequency
http://arxiv.org/abs/0704.1974v1
mailto:helene.perrin@galilee.univ-paris13.fr
or amplitude noise, phase hops or sudden frequency hops
during the RF ramp.
A. Dipolar excitation heating
Very generally, for atoms in a one dimensional har-
monic trap with a trapping frequency νz, any effect pro-
ducing a jitter in the trap position z results in linear
heating through dipolar excitation. The average energy
of the cold atomic cloud E increases linearly as [14]:
Mω4z Sz(νz) (1)
where ωz = 2πνz, M is the atomic mass and Sz is the
one-sided Power Spectral Density (PSD) of the position
fluctuations δz, defined as the Fourier transform of the
time correlation function [14]
Sz(ν) = 4
dτ cos(2πντ)〈δz(t) δz(t + τ)〉. (2)
The time variations of energy, E, and temperature, T , are
related by Ṫ = Ė/3kB. The factor 3 arises because only
one degree of freedom is responsible for the temperature
increase, as is the case in our atom trap. The vertical
trap position z is linked to the RF frequency νRF by
z = Z(νRF) such that Sz is directly proportional to Sy,
the PSD of relative frequency noise of the RF source,
through:
Sz(ν) =
Sy(ν). (3)
The function Z depends on the geometry of the static
magnetic field. In a quadrupolar field, for instance, Z is
linear with νRF and its derivative is simply a constant.
From Eqs. (1) and (3), we infer that the linear heating
rate is proportional to Sy(νz).
To fix orders of magnitude, within the static mag-
netic field of our Ioffe-Pritchard trap [7], νz may be ad-
justed between 600 and 1500 Hz and the typical tem-
perature of the cold rubidium 87 atoms ranges from 0.5
to 5 µK. To maintain a temperature below condensa-
tion threshold for a few seconds, a linear temperature
increase below 0.1 µK/s is necessary. This rate corre-
sponds to
Sz(νz) = 0.27 nm/
Hz for an intermediate
trap frequency of 1000 Hz and νRF = 3 MHz, which in
turn corresponds to a one-sided PSD of relative frequency
fluctuations of the RF source Sy(νz) = −118 dB/Hz.
B. Parametric heating
Fluctuations of the RF field amplitude BRF are re-
sponsible for parametric heating in the vertical direction.
The trapping frequency νz is inversely proportional to
BRF [6]:
2F h̄
MγBRF
. (4)
Here, γ is the gyromagnetic ratio of the atom and F is
the total atomic spin (F = 2 for rubidium 87 in its upper
hyperfine state). The atoms are assumed to be polarized
in their extreme mF = F substate. The cloud tempera-
ture increases exponentially due to amplitude noise with
a rate Γ, where
Γ = π2ν2zSa(2νz) (5)
and Sa is the PSD of the relative RF amplitude noise [14].
In order to perform experiments with the BEC within a
time scale of a few seconds, Γ should not exceed 10−2 s−1.
Again, for a typical oscillation frequency of 1000 Hz, this
corresponds to Sa < −90 dB/Hz. This requirement is
rather easy to match and does not limit the choice of the
RF source, as -110 dB/Hz is commonly reached. How-
ever, particular care must be taken in the choice and
installation of the RF amplifier usually used after the
source.
C. Phase hops
Controlling the phase of the RF source is not a cru-
cial point for evaporative cooling, but is an issue in the
case of RF-based traps, where it is associated with trap
losses. In the latter situation, the atomic spin follows an
effective magnetic field oscillating at the RF frequency.
A phase hop results in a sudden flip of this effective field,
the atomic spin being then misaligned with the new di-
rection of the field. Some of the atoms end up with a
spin oriented incorrectly and escape the trap.
For this reason, phase hops should be avoided. This
is difficult to achieve with an analog synthesizer over a
wide frequency sweep. By contrast, Direct Digital Syn-
thesis (DDS) technology is well adapted to this require-
ment [16].
D. Frequency steps
The drawback of DDS technology is that, although
the phase is continuous, the frequency is increased by
N successive discrete steps δν. A sudden change in
the RF frequency also results in atomic losses, through
the same mechanism as for phase hops. The effec-
tive magnetic field rotates, at most, by the small angle
δθ = 2π δν/(γBRF/2) [15]. For a linear ramp over a fre-
quency range ∆ν = Nδν, the fraction of atoms remaining
after the full ramp is of order (1− Fδθ2/2)N . Given the
expression for δθ, this reads:
NγBRF
≃ 1− F
. (6)
Thus, for the remaining fraction to be larger than 95%,
the number of frequency steps should be larger than
10F (4π∆ν/γBRF)
2. For example, for a 2 MHz ramp
with a typical RF amplitude of 200 mG, N should be
larger than 16,000.
In addition to this loss effect, a sudden change in the
RF frequency results in a sudden shift of the position
of the RF-dressed trap. This may cause dipolar heating
of the atoms, especially if this frequency change occurs
every trap period. The frequency steps should thus be as
small as possible, a few tens Hz to a hundred Hz typically.
III. DEVICE DESCRIPTION
Our experiment has the following requirements. First,
during the ramp the gap between two successive frequen-
cies must fulfil the criterion discussed in section IID. Sec-
ond, adiabaticity criteria require a controlled, optimized
ramp. Third, the ramp duration should be tunable from
one experiment to the other on a time scale ranging from
50 ms to 10 s. Finally, frequency and amplitude noise
must be small enough, as discussed in previous section.
Given the amplitude of the frequency sweep we need
to perform in our experiment, DDS technology appears
to be an ideal solution. We previously used a com-
mercial DDS-based RF generator, the Stanford DS-345.
Its memory length is limited to 1,500 frequency points
for each waveform with an adjustable step duration of
40 Msample/s/N , with N=1 to 234 − 1. The major in-
convenience of this device is that it is unable to hold
the final frequency at the end of the ramp. Instead, the
frequency sweep is looped indefinitely. It forced us to
sacrifice either frequency resolution during the ramp or
duration of the plateau. To benefit from both a low noise
RF spectrum during the ramp and a very small frequency
step, and to improve the possibilities of the RF source,
we designed a digital RF synthesizer with a > 200, 000
memory length and great agility, fitted to our experimen-
tal requirements.
The main features of the RF synthesizer are as fol-
lows. It is able to generate 262,144 sine waves in a row
in the radio frequency band (DC - 10 MHz), owing to
its 1 M-byte fast asynchronous Static Random Access
Memory (SRAM). Each frequency is an integer chosen
by the user. A key feature of the device is a variable
sample frequency over the sequence, as the duration of
each generated frequency can be tuned from 1 µs to 1
hour.
The general architecture of the device is sketched in
Fig. 1. It is made up of one evaluation kit DDS board,
and a “starter kit” Field Programmable Gate Array
(FPGA) board. The device is managed by a Personal
Computer (PC). The DDS is clocked by an ultra-stable
external reference signal. The output of the DDS is a
sine wave, filtered through a 10 MHz low-pass filter. The
frequency ramp synthesis starts when a TTL signal is
sent to the device.
File *.txt
Memory
Spartan-3
Clock frequency
AD 9851
PC FPGA board DDS board
TTL start DDS device
filter
10MHz
RF output
FIG. 1: Layout of the system.
The DDS board combines digital parameters and
an analog reference clock frequency to generate a sine
wave [16]. The heart of the DDS board is a digitally pro-
grammable device using DDS technology, the AD9851. It
has a 32-bit phase accumulator, a 14-bit digital phase-to-
amplitude converter and a 10-bit Digital-to-Analog Con-
verter (DAC). Its maximum clock frequency is 180 MHz,
and its maximum output frequency is 70 MHz. The
phase, relative to the clock signal, is encoded in 5 bits,
and is adjustable to any value from 0 to 2π. This results
in a rather poor phase resolution of 196 mrad.
The FPGA board manages the 1 Mb memory, the time
settings and the input/output of the device through se-
rial port. The FPGA is a Xilinx spartan-3 XC3S200,
providing 200, 000 logic gates. These logic gates are
designed with VHDL [17]. A Universal Asynchronous
Receiver Transmitter(UART) and a Picoblaze microcon-
troller are loaded into the FPGA, in order to commu-
nicate through serial port to the PC and to load the
on-board memory. We wrote our own VHDL scripts to
manage the DDS board and the FPGA in-board memory.
The FPGA board is clocked internally at 50 MHz. The
output sample rate of the device depends on the num-
ber of clock cycles ncycle between frequency data trans-
ferred to the DDS board. We set ncycle = 50 so that
ncycle/50 MHz = 1 µs, large enough thus ensuring safe
operation of the frequency data.
Software was developed in C with CVI Labwindows in
order to configure the device. The user writes a plain text
file ordering all the frequencies of the desired frequency
ramp. The set of frequencies is separated into 10 groups
of adjustable length, with a given sample rate for each.
The group lengths and the corresponding sample rate are
each translated by the software into 4 bytes. In addition,
each frequency in a given group, an integer written as a
decimal number, is translated into 4 bytes (32 bits). The
software sends these bytes by serial port to the FPGA
board.
The clock frequency fc and its phase noise level are the
key points for setting the frequency performance of the
device. The lower the phase noise of the clock signal, the
lower is theminimum phase noise of the output frequency
of the device (see section IVA).
The clock signal used for the experiment, see next sec-
tion, is the 10 MHz clock signal from an ultra-stable
Oven Controlled Crystal Oscillator (OCXO) BVA-8600.
Its phase noise PSD is −115 dB.rad2/Hz at 1 Hz. As
this clock frequency is very close to the desired maxi-
mum output frequency (≃ 10 MHz, see next section),
and to fulfil Shannon’s theorem [18, 19], we generate a
higher clock frequency by using the internal frequency
clock multiplier, at ×6, of the DDS board.
IV. RESULTS
The device presented in section III was first tested for
its frequency stability performance, as described in sec-
tion IVA and summarized in Table I. It was then in-
tegrated into a Bose-Einstein condensation experiment,
see section IVB.
A. Device frequency performance
1. Quantization error, Phase Accumulator truncation and
“magic” frequencies
By construction, digitization yields to inaccuracies in
frequency synthesis. The output frequency of 32-bit res-
olution DDS is given by
fRF = fc ×
where w is a binary 32-bit tuning word. The output fre-
quency can thus differ slightly from the desired frequency.
As fc = 60 MHz, the maximum digitization error δf is
6× 107/232 = 0.014 Hz.
As our software only takes integer frequencies as in-
put, a given frequency fRF will be synthesized without
sampling error if it may be written exactly as an integer
in the form given at Eq. (7). This condition is written
w = n× 232−p (8)
where n is a positive integer and p is the power of 2 in the
prime factorization of the clock frequency fc. In our case,
fc = 2
8 × 3 × 57 Hz and p = 8 so that every frequency
verifying
fRF = n× 234375 Hz (9)
will yield to no digitization error. n should be less than
2p/3 for the desired frequency to be in the synthesizer
range (fc/3).
In addition, when the AD9851 converts the calcu-
lated phase to an effective output amplitude, only the
first most significant 14 bits are used, even though the
AD9851 is a 32-bit synthesizer, in order to handle prac-
ticable number of entries in a lookup table. Truncating
the phase results in errors in amplitude that are periodic
in the time domain. These errors will be seen as spurs in
the frequency domain. However, for particular frequen-
cies which are exactly encoded by the first 14 bits (The
TABLE I: Performance of the device with fc = 60 MHz. The
relative frequency noise is computed from the phase noise
data, and given for 3.75 MHz (“magic” frequency) and 3 MHz
(larger noise value).
Parameters Min. Max. Units
Dynamic 0 10 MHz
Line-width - 30 mHz
Digitalization error 0 14 mHz
Sample rate adjustable 1 MHz
Memory length 1 262,144 pts
Phase noise@1Hz -113 -78 dB.rad2/Hz
Rel. freq. noise@1Hz -244 -207 dB/Hz
Rel. ampl. noise@1Hz - -120 dB/Hz
last 18 bits are 0.), the phase is not truncated at all,
yielding no spurious effects and the best PSD of phase
noise. This occurs for every frequency satisfying
fRF = n× fc/214 (10)
with n a positive integer. As fc = 60 MHz, we have
fRF = n × 3662.109375 Hz. The most stringent condi-
tion being the first one, we will denote the frequencies
satisfying Eq. (9) as “magic frequencies”. In order to il-
lustrate the difference between a “magic frequency” and
another one, we performed a set of noise PSD measure-
ments for two frequencies: a first set for fRF = 3 MHz
which is not a “magic” frequency, and a second set for
fRF = 3.75 MHz which is a “magic” frequency.
2. Spectral density of noise
We recorded the spectral density of noise of our syn-
thesizer at a given frequency fRF by FFT analysis of
the beat note at 0 Hz with a second synthesized ref-
erence signal. The measurement bench is sketched in
Fig. 2. In order to generate a tunable reference signal in
the RF range we used an analog synthesizer, Rhode &
Schwartz SML-01 (R&S), for synthesizing a signal at a
high frequency, and then divided by 100 to give fRF for
subsequent mixing. The beat note was recorded and an-
alyzed by a digital FFT analyzer HP 3562A sampling at
256 kHz. The R&S, as all the measurement devices, was
clocked at 10 MHz by the ultra-stable BVA-8600 quartz
oscillator. All the plots shown on Fig. 3 are raw spectra
of the beat note. The reference signal (R&S) itself was
characterized by making a beat note with a second iden-
tical R&S based synthesis. This corresponds to the line
labelled as “Reference” in Fig. 3.
By tuning the phase difference φ between the RF sig-
nal and the reference signal to π/2, we recorded phase
noise. At fRF = 3 MHz, the PSD of phase noise is
−78 dB.rad2.Hz−1 at 1 Hz, which corresponds to a PSD
of relative frequency noise of −207 dB.Hz−1. From the
data reported on Fig. 3, and assuming a Lorentzian line-
shape for the beat note, the linewidth δf of the RF signal,
given by
δf = πν2RF
df Sy(f) , (11)
is as small as 30 mHz over a bandwidth ∆f = 100 kHz
for an output frequency of 3 MHz [23]. We found similar
results for output frequencies from 1 to 5 MHz. At a
“magic” frequency, as for example at 3.75 MHz, where
truncation effects cancel out, the results are even better,
with a PSD of phase noise as low as −113 dB.rad2.Hz−1
at 1 Hz, only 2 dB higher than the phase noise of
the BVA 8600. The observed value corresponds to
the ultimate phase noise of the DDS chip specified by
the manufacturer. The relative frequency noise is then
−244 dB.Hz−1.
The frequency noise performance is naturally linked to
the quality of the master oscillator. To illustrate this
fact, the same measurements were repeated with the im-
proved OCXO of a DS-345, the ERC EROS-750-SBR-4,
as master oscillator. No significant change in the fre-
quency performances were noticed at non magic frequen-
cies. At a magic frequency, the PSD of phase noise in-
creased to −100 dB.rad2/Hz, which is consistent with the
phase noise specifications of this quartz.
The PSD of relative amplitude noise Sa was recorded
with the same measurement bench, by tuning the phase φ
to 0. The reference signal was also delivered by the R&S.
The recorded spectrum is very close to the “Reference”
line itself at the level of −120 dB.Hz−1 at 1 Hz, close to
the input noise of the FFT analyzer. Note that for rele-
vant excitation frequencies ν = 2νz (larger than 1.2 kHz),
the PSD of amplitude noise is lower than −130 dB.
In practice, we use in our BEC experiment a pro-
grammable RF attenuator Minicircuit ZAS-3, driven by
an analog output channel of a National Instrument PC
card PCI-6713, in order to vary the RF amplitude sent
to the RF coil. We repeated the measurements described
above with this attenuator and found an increase of Sa,
starting at −115 dB.Hz−1 at 1 Hz, which corresponds to
a noise figure of the programmable attenuator of 5.1 dB.
At the output of the attenuator, the RF signal is am-
plified by a class-A amplifier Kalmus 505 F. Its gain is
¸100SML-01
-8600
10 MHz
ZAS-3
FIG. 2: Noise spectrum measurement bench. The gain in the
mixer was fitted to the output signal, depending on the phase
between the two beating signals.
1
 10
 100
 1k
 10k
 100k
-180
-160
-140
-120
-100
1
 10
 100
 1k
 10k
 100k
-160
-140
-120
-100

2
 .
Frequency (Hz)
 DDS 3.75 MHz
 DDS 3 MHz
 Reference
 BVA 8600
FIG. 3: PSD of relative amplitude noise, upper plot, and
phase noise,lower plot. Averaging = 100. FFT’s sampling
frequency = 256 kHz. The low pass filter cutting frequency of
the mixer was set to 200 kHz. The reference curve corresponds
to the beat note of two identical R&S synthesizers. Plotted
for comparison is the phase noise specifications of the BVA
8600 quartz oscillator, as given by the manufacturer.
37 dB and its noise figure is typically +10 dB according
to the manufacturer specifications.
B. Integration into the experiment
In this section, we present the main experimental re-
sults concerning the heating rate of the atomic sample.
As described in section II, the RF signal is used for
producing the bubble-like trap, where ultracold rubid-
ium atoms are accumulated at the bottom of the surface.
The trap is very anisotropic with stronger confinement
in the vertical direction [7]. For the RF-dressed trap to
be efficiently loaded from the standard Ioffe-Pritchard
static magnetic trap, described in [20], the RF frequency
fRF has to be ramped up from 1 MHz to a final fixed
frequency fendRF ranging from 2 to 10 MHz. The static
magnetic field, necessary both for magnetic trapping and
RF-induced trapping, is always present. A typical ramp
is shown on Fig. 4. The frequency is ramped more slowly
around 1.3 MHz, corresponding to the resonant frequency
at the center of the magnetic trap where adiabaticity of
spin rotation is more difficult to obtain. At the end of
this ramp, which may last between 75 ms and 500 ms, the
RF frequency is held between 0.1 and 10 s for testing the
lifetime and heating rate of the atoms in the RF-based
trap.
The RF signal is amplified by 37 dB with a single stage
amplifier, and the RF field is produced by a small circular
antenna. The RF field is linearly polarized and its ampli-
tude BRF may be adjusted between 70 and 700 mG. We
0
 50
 100
 150
 200
Time (ms)
FIG. 4: Typical shape of a radio-frequency ramp applied to
the ultracold atomic sample. In the present example fRF is
increased from 1 to 3 MHz within 150 ms. At the end of this
ramp, the RF frequency is maintained at its final value for
some holding time in the RF-based trap, dashed line.
record the atomic temperature after the RF ramp as a
function of time while the atoms are confined in the RF-
based trap. The temperature is deduced from the cloud
size along z after a 7 ms of ballistic expansion. The same
measurement was repeated with different RF sources.
First, we used a Agilent 33250A synthesizer with a RF
amplitude larger than 500 mG for both the frequency
ramp and the final holding frequency. Such RF ana-
log synthesizers operated at fixed frequency exhibit very
good relative frequency noise in most cases, typically at
the −180 dB/Hz level or better. However, as mentioned
by Colombe et al. [7] and confirmed by White et al. [21],
the relative frequency noise increases by a few decades if
the output frequency is driven with an external analog
voltage. The frequency was indeed tuned through an ex-
ternal voltage control provided by the PC analog board
(NI 6713), such that the modulation depth was ±1 MHz
on a central frequency of 2 MHz. We obtained both a
short lifetime, typically 400 ms at 1/e, and a strong lin-
ear heating, as shown on Fig. 5 full circles. The heating
rate is measured to be 5.0 µK/s. This rate, given the RF
amplitude, corresponds to a relative frequency noise of
Sy = −100 dB/Hz at the trap frequency of 600 Hz. This
noise is quite high and is related to voltage noise on the
external frequency control of the synthesizer. This effect
is strong in our case as the frequency is varied with a
large modulation depth (∆f/f = 1).
We also tested a two step scheme, with a first ramp
performed by a Stanford DS-345 DDS (1,500 frequency
points), followed by the R&S maintained at fixed fre-
quency for the full holding time. This scheme allows one
to benefit from the excellent frequency stability of the
second device used at fixed frequency. With this setup,
0
 1
 2
 3
 4
 5
 6
 7
 8
 Agilent
 Our DDS device
holding time in the trap (s)
FIG. 5: Comparison of heating of the atomic cloud in the bub-
ble trap: Agilent 33250A synthesizer, full circles, or present
device, open diamonds, is used for creating the RF ramp and
the final radio-frequency fendRF . We observe a heating rate of
5.0 µK/s in the first case and 0.47 µK/s the second one, as
given by a linear fit, full lines.
heating during the plateau stage seemed to be completely
suppressed [7]. However, a large dispersion was observed
in the atom number data after the switching, which pre-
vented us from characterizing the heating very precisely.
This dispersion is due to a random phase hop at the
switching time between the two synthesizers, resulting in
atomic losses. We studied the effect of the random phase
hop on the atom number by recording, for each experi-
ment, the phase difference at the switching time with a
control oscilloscope. The results are presented on Fig. 6,
full circles. For the maximum phase hop, π, 80% of the
atoms are lost. This figure depends on the atomic tem-
perature, the losses being higher at lower temperature,
and is well reproduced by theory, as shown on Fig. 6,
black line. The theoretical curve is calculated for an RF
amplitude of 470 mG by averaging the loss probability
over the positions of the atoms, as deduced from a ther-
mal distribution at a temperature of 4 µK. The fact that
the trap is able to hold two of the five spin components
of the F = 2 hyperfine state is taken into account.
Finally, we performed the heating rate and lifetime
measurements with our DDS device and an amplitude
of 70 mG. We found an increased lifetime, up to 9.6 s.
During the first second, the vertical “kinetic tempera-
ture”, deduced from the vertical cloud size after time of
flight, decreases due to thermalization with the horizon-
tal degrees of freedom which initially have a lower energy
due to the loading procedure: Fig. 5, open diamonds.
After 1 second, a linear heating rate of 0.47 µK.s−1 is
clearly observable. No exponential parametric heating is
measurable. We believe that the remaining linear heat-
ing is due to residual excitation by scattered light, which
was not present in the previous case. The same heating
-180
 -150
 -120
 -90
 -60
 -30
 0
 30
 60
 90
 120
 150
 180

6
 )
phase hop at switching (degrees)
 experiment
 theory
FIG. 6: Number of atoms remaining after switching between
the two synthesizers as a function of the phase hop. Exper-
imental data, full circles, are compared to a calculation, full
line, assuming a RF amplitude of 470 mG and a temperature
of 4 µK.
rate was indeed observed within the static magnetic trap,
when the RF source was off. Heating directly linked to
the RF source is not observable.
V. CONCLUSION
We built a low cost DDS RF generator, based on a
DDS chip and its evaluation kit. It is extremely agile
since we have the ability of using up to 10 different time
steps during the sequence, allowing an efficient use of the
FPGA on-board memory. In contrast to Stanford DS-345
RF synthesizers, the frequency ramp is not repeated at
the end of the sequence. The last frequency is kept until
a new TTL pulse is proceeded to end the sequence. Its
large memory length, up to 262,144 frequency samples,
allows very small frequency steps, thus strongly reduc-
ing the atomic losses in our RF-dressed trap experiment.
The DDS technology ensures that the phase is continu-
ous all over the frequency ramp, which is essential for RF
trapping applications.
Apart from these high resolution features, the device
presents excellent spectral properties. The power spec-
tral density of phase noise is as low as−113 dB.rad2.Hz−1
at well chosen “magic frequencies”, and in any case re-
mains below than −78 dB.rad2.Hz−1 at worst, dominated
by phase truncation. This last value yields a linewidth of
30 mHz over a 100 kHz bandwidth. The corresponding
relative frequency noise level is 90 dB below the require-
ments of our RF-dressed trapping experiment.
The amplitude noise of the DDS synthesizer is mea-
sured to be below −130 dB.Hz−1 at frequencies above
1 kHz. Amplitude noise turns out to be the limiting noise
present in our RF chain at the present time due to the
several dB noise figures of the attenuators and amplifiers
in use in our experiment. They actually increase the RF
amplitude noise to −115 dB.Hz−1 at 1 kHz, which still
remains 25 dB below our experiment requirements.
All these features make our DDS apparatus very well-
suited for cold atom experiments. It may be used for
optimized evaporative cooling implementation, with ar-
bitrary frequency ramp profiles including a final “RF-
knife”, in a standard BEC experiment. It is of particular
importance in the case of atom chip experiments [22],
where the cooling ramp is usually done very quickly (1 s
typically). It is also ideal for developing new RF-based
traps. In particular, the possibility of configuring the
time step, that is the duration of each frequency point, is
very attractive for our application. This feature is all the
more interesting in that the memory can be separated
in several zones, each being allocated a chosen number
of points and a chosen time step, so we can dilate or
compress in time the whole or a part of the ramp, while
keeping the same frequency series. For instance, if we
want to accomplish a ramp of N frequency values with a
small time resolution, say 10 µs, and then keep the final
frequency constant over ten seconds, we have the ability
to address a first memory zone with the first N−1 points
and a 10 µs time step, and a second memory zone with
the last frequency value and a 10 s time step. A sin-
gle time step over the whole sequence would have forced
us to sacrifice either the resolution of the ramp or the
duration of the final plateau.
The device was successfully implemented in our ultra-
cold atom experiment. First, we set boundary specifica-
tions on the RF source performances for trapping exper-
iments regarding heating rate and lifetime. In the same
manner, we set conditions on maximum phase hops and
frequency step magnitude. With our home made device,
we observed no heating due to the RF source during the
plateau step. A remaining heating rate of 0.47µK/s was
identical to the one obtained in the magnetic trap and
was limited by other noise sources, presumably scattered
photons. Despite the very large number of frequency
points, heating along the vertical axis is still present dur-
ing the loading stage of the RF-dressed trap. We at-
tribute this heating to non-adiabatic deformation of the
confining potential during the loading process, which we
could not make slower due to the other noise sources.
While the actual performances of the device are al-
ready very good, a few improvements are still possible.
First, the memory length of the DDS is basically deter-
mined by the FPGA in-board memory size. A larger
sample size may easily be implemented by replacing the
board, provided that the data transfer rate is improved
through the use of an Ethernet port or a USB port. Sec-
ond, the sample rate is 1 MHz, limited by the data trans-
fer rate from the FPGA board. Higher FPGA clock fre-
quency, or smaller clock counts ncycle, would allow higher
sample rates. The frequency limit due to the AD-9851
DDS chip itself is 3 MHz. This modification would im-
prove the temporal resolution by a factor of 3. Finally,
the maximum output frequency of 10 MHz may be very
easily increased to 20 MHz by replacing the low-pass fil-
ter. By clocking the DDS with a 180 MHz clock, one
could even reach output frequencies up to 60 MHz. Note
that with a similar DDS chip, the AD 9858, frequency
clocks as high as 1 GHz are allowed, which makes possi-
ble the synthesis of RF frequencies up to 330 MHz.
Acknowledgments
We are indebted to R. J. Butcher for a critical read-
ing of the manuscript. This work was supported by the
Région Ile-de-France (contract number E1213) and by
the European Community through the Research Train-
ing Network “FASTNet” under contract No. HPRN-CT-
2002-00304 and Marie Curie Training Network “Atom
Chips” under contract No. MRTN-CT-2003-505032.
Laboratoire de physique des lasers is UMR 7538 of CNRS
and Paris 13 University. The LPL is a member of the In-
stitut Francilien de Recherche sur les Atomes Froids.
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  We report on the frequency performance of a low cost (~500$) radio-frequency
sine wave generator, using direct digital synthesis (DDS) and a
field-programmable gate array (FPGA). The output frequency of the device may be
changed dynamically to any arbitrary value ranging from DC to 10 MHz without
any phase slip. Sampling effects are substantially reduced by a high sample
rate, up to 1 MHz, and by a large memory length, more than 2.10^5 samples. By
using a low noise external oscillator to clock the DDS, we demonstrate a phase
noise as low as that of the master clock, that is at the level of -113
dB.rad^2/Hz at 1 Hz from the carrier for an output frequency of 3.75 MHz. The
device is successfully used to confine an ultracold atomic cloud of rubidium 87
in a RF-based trap, and there is no extra heating from the RF source.

Agile low phase noise radio-frequency sine wave generator applied to experiments on
ultracold atoms
O. Morizot, J. de Lapeyre de Bellair, F. Wiotte, O. Lopez, P.-E. Pottie, and H. Perrin∗
Laboratoire de physique des lasers, Institut Galilée, Université Paris 13 and CNRS,
Avenue J.-B. Clément, F-93430 Villetaneuse, France
(Dated: October 31, 2018)
We report on the frequency performance of a low cost (∼500 $) radio-frequency sine wave genera-
tor, using direct digital synthesis (DDS) and a field-programmable gate array (FPGA). The output
frequency of the device may be changed dynamically to any arbitrary value ranging from DC to
10 MHz without any phase slip. Sampling effects are substantially reduced by a high sample rate,
up to 1 MHz, and by a large memory length, more than 2 × 105 samples. By using a low noise
external oscillator to clock the DDS, we demonstrate a phase noise as low as that of the master
clock, that is at the level of −113 dB.rad2/Hz at 1 Hz from the carrier for an output frequency of
3.75 MHz. The device is successfully used to confine an ultracold atomic cloud of rubidium 87 in a
RF-based trap, and there is no extra heating from the RF source.
PACS numbers: 39.25.+k, 06.30.Ft, 07.57.Hm
I. INTRODUCTION
Radio-frequency (RF) fields are used in cold atom ex-
periments for different purposes: for instance, evapo-
rative cooling performed in a magnetic trap relies on
RF field coupling between the different atomic magnetic
states [1, 2]. This technique led to the first observa-
tion of Bose-Einstein condensation (BEC) [3, 4]. Also,
RF pulses are used for dissociating ultracold molecules
produced from ultracold gases through Feshbach reso-
nances [5]. More recently, RF fields have been used to-
gether with static magnetic fields for trapping utracold
atoms at a temperature of a few µK in unusual geome-
tries [6, 7]. There is a growing interest for these “RF-
based traps” among atomic physicists, for creating dou-
ble well traps on atom chips [8, 9, 10] or proposing new
kinds of confining potentials [11, 12, 13]. In both cases,
a single frequency RF signal must be frequency swept
over some range, often larger than the initial frequency,
following a precise time function lasting several seconds.
Typically the RF frequency is varied between 1 MHz and
a few tens of MHz in 0.1 s to 10 s in the ramping stage,
and held at the final frequency for seconds in the plateau
stage. For cooling purposes, commercial RF generators
fit physicists’ needs reasonably well, even if a better res-
olution in arbitrary frequency ramps would be appreci-
ated. However, in the case of RF-based trapping, the
requirements are stronger. The main difference between
these two situations is as follows: in evaporative cooling
the cold atomic sample is located away from the region
of efficient coupling, whereas in the RF-based trapping
scheme the atoms sit exactly at the point where the RF
field has the largest effect. The quality of the RF source is
then much more important than for evaporative cooling.
In fact, the cloud position is directly related to the value
∗Electronic address: helene.perrin@galilee.univ-paris13.fr
of the RF field frequency, and the trap restoring force, or
equivalently the oscillation frequency νt in the harmonic
approximation, is linked to the RF amplitude. As a re-
sult, any amplitude noise, frequency noise or phase noise
of the RF signal during the ramp or the plateau leads to
a heating of the cold atomic cloud. This motivated the
construction of a synthesizer fitting our requirements.
This paper is organized as follows. In section II we
give explicit expressions for the heating of the cold atom
sample for frequency and amplitude noise in the case of
RF-based trapping. In section III, we describe our RF
synthesizer. Finally, section IV is devoted to experimen-
tal results on its performance and comparison between
the different RF sources tested on the BEC experiment.
II. REQUIREMENTS ON THE RF SOURCE
FOR RF-BASED TRAPPING
In this section, we will focus on the RF-dressed trap
that we experimentally produce in the laboratory [7].
The extension of the main conclusions to other RF-
dressed trap geometries is straightforward.
The trap confines the atoms in all three space di-
mensions. The trapping force arises from the inter-
action between the linearly polarized RF field B(t) =
BRF cos(2πνRFt) and the atoms in the presence of an in-
homogeneous magnetic field. This interaction results in
a transverse confinement of the atoms to the surface of
an ellipsoid. The atoms are free to move along the con-
fining surface, resulting in a kind of “bubble trap” [6].
Due to gravity, however, the atoms are concentrated at
the bottom of the ellipsoid. Their motion is pendulum-
like in the horizontal directions, and imposed by the RF
interaction along the vertical z axis. This last direction
is thus the most sensitive to the RF field properties (fre-
quency νRF, amplitude BRF) and we will concentrate on
the vertical motion in the following. Along this direc-
tion, heating or atomic losses may arise from frequency
http://arxiv.org/abs/0704.1974v1
mailto:helene.perrin@galilee.univ-paris13.fr
or amplitude noise, phase hops or sudden frequency hops
during the RF ramp.
A. Dipolar excitation heating
Very generally, for atoms in a one dimensional har-
monic trap with a trapping frequency νz, any effect pro-
ducing a jitter in the trap position z results in linear
heating through dipolar excitation. The average energy
of the cold atomic cloud E increases linearly as [14]:
Mω4z Sz(νz) (1)
where ωz = 2πνz, M is the atomic mass and Sz is the
one-sided Power Spectral Density (PSD) of the position
fluctuations δz, defined as the Fourier transform of the
time correlation function [14]
Sz(ν) = 4
dτ cos(2πντ)〈δz(t) δz(t + τ)〉. (2)
The time variations of energy, E, and temperature, T , are
related by Ṫ = Ė/3kB. The factor 3 arises because only
one degree of freedom is responsible for the temperature
increase, as is the case in our atom trap. The vertical
trap position z is linked to the RF frequency νRF by
z = Z(νRF) such that Sz is directly proportional to Sy,
the PSD of relative frequency noise of the RF source,
through:
Sz(ν) =
Sy(ν). (3)
The function Z depends on the geometry of the static
magnetic field. In a quadrupolar field, for instance, Z is
linear with νRF and its derivative is simply a constant.
From Eqs. (1) and (3), we infer that the linear heating
rate is proportional to Sy(νz).
To fix orders of magnitude, within the static mag-
netic field of our Ioffe-Pritchard trap [7], νz may be ad-
justed between 600 and 1500 Hz and the typical tem-
perature of the cold rubidium 87 atoms ranges from 0.5
to 5 µK. To maintain a temperature below condensa-
tion threshold for a few seconds, a linear temperature
increase below 0.1 µK/s is necessary. This rate corre-
sponds to
Sz(νz) = 0.27 nm/
Hz for an intermediate
trap frequency of 1000 Hz and νRF = 3 MHz, which in
turn corresponds to a one-sided PSD of relative frequency
fluctuations of the RF source Sy(νz) = −118 dB/Hz.
B. Parametric heating
Fluctuations of the RF field amplitude BRF are re-
sponsible for parametric heating in the vertical direction.
The trapping frequency νz is inversely proportional to
BRF [6]:
2F h̄
MγBRF
. (4)
Here, γ is the gyromagnetic ratio of the atom and F is
the total atomic spin (F = 2 for rubidium 87 in its upper
hyperfine state). The atoms are assumed to be polarized
in their extreme mF = F substate. The cloud tempera-
ture increases exponentially due to amplitude noise with
a rate Γ, where
Γ = π2ν2zSa(2νz) (5)
and Sa is the PSD of the relative RF amplitude noise [14].
In order to perform experiments with the BEC within a
time scale of a few seconds, Γ should not exceed 10−2 s−1.
Again, for a typical oscillation frequency of 1000 Hz, this
corresponds to Sa < −90 dB/Hz. This requirement is
rather easy to match and does not limit the choice of the
RF source, as -110 dB/Hz is commonly reached. How-
ever, particular care must be taken in the choice and
installation of the RF amplifier usually used after the
source.
C. Phase hops
Controlling the phase of the RF source is not a cru-
cial point for evaporative cooling, but is an issue in the
case of RF-based traps, where it is associated with trap
losses. In the latter situation, the atomic spin follows an
effective magnetic field oscillating at the RF frequency.
A phase hop results in a sudden flip of this effective field,
the atomic spin being then misaligned with the new di-
rection of the field. Some of the atoms end up with a
spin oriented incorrectly and escape the trap.
For this reason, phase hops should be avoided. This
is difficult to achieve with an analog synthesizer over a
wide frequency sweep. By contrast, Direct Digital Syn-
thesis (DDS) technology is well adapted to this require-
ment [16].
D. Frequency steps
The drawback of DDS technology is that, although
the phase is continuous, the frequency is increased by
N successive discrete steps δν. A sudden change in
the RF frequency also results in atomic losses, through
the same mechanism as for phase hops. The effec-
tive magnetic field rotates, at most, by the small angle
δθ = 2π δν/(γBRF/2) [15]. For a linear ramp over a fre-
quency range ∆ν = Nδν, the fraction of atoms remaining
after the full ramp is of order (1− Fδθ2/2)N . Given the
expression for δθ, this reads:
NγBRF
≃ 1− F
. (6)
Thus, for the remaining fraction to be larger than 95%,
the number of frequency steps should be larger than
10F (4π∆ν/γBRF)
2. For example, for a 2 MHz ramp
with a typical RF amplitude of 200 mG, N should be
larger than 16,000.
In addition to this loss effect, a sudden change in the
RF frequency results in a sudden shift of the position
of the RF-dressed trap. This may cause dipolar heating
of the atoms, especially if this frequency change occurs
every trap period. The frequency steps should thus be as
small as possible, a few tens Hz to a hundred Hz typically.
III. DEVICE DESCRIPTION
Our experiment has the following requirements. First,
during the ramp the gap between two successive frequen-
cies must fulfil the criterion discussed in section IID. Sec-
ond, adiabaticity criteria require a controlled, optimized
ramp. Third, the ramp duration should be tunable from
one experiment to the other on a time scale ranging from
50 ms to 10 s. Finally, frequency and amplitude noise
must be small enough, as discussed in previous section.
Given the amplitude of the frequency sweep we need
to perform in our experiment, DDS technology appears
to be an ideal solution. We previously used a com-
mercial DDS-based RF generator, the Stanford DS-345.
Its memory length is limited to 1,500 frequency points
for each waveform with an adjustable step duration of
40 Msample/s/N , with N=1 to 234 − 1. The major in-
convenience of this device is that it is unable to hold
the final frequency at the end of the ramp. Instead, the
frequency sweep is looped indefinitely. It forced us to
sacrifice either frequency resolution during the ramp or
duration of the plateau. To benefit from both a low noise
RF spectrum during the ramp and a very small frequency
step, and to improve the possibilities of the RF source,
we designed a digital RF synthesizer with a > 200, 000
memory length and great agility, fitted to our experimen-
tal requirements.
The main features of the RF synthesizer are as fol-
lows. It is able to generate 262,144 sine waves in a row
in the radio frequency band (DC - 10 MHz), owing to
its 1 M-byte fast asynchronous Static Random Access
Memory (SRAM). Each frequency is an integer chosen
by the user. A key feature of the device is a variable
sample frequency over the sequence, as the duration of
each generated frequency can be tuned from 1 µs to 1
hour.
The general architecture of the device is sketched in
Fig. 1. It is made up of one evaluation kit DDS board,
and a “starter kit” Field Programmable Gate Array
(FPGA) board. The device is managed by a Personal
Computer (PC). The DDS is clocked by an ultra-stable
external reference signal. The output of the DDS is a
sine wave, filtered through a 10 MHz low-pass filter. The
frequency ramp synthesis starts when a TTL signal is
sent to the device.
File *.txt
Memory
Spartan-3
Clock frequency
AD 9851
PC FPGA board DDS board
TTL start DDS device
filter
10MHz
RF output
FIG. 1: Layout of the system.
The DDS board combines digital parameters and
an analog reference clock frequency to generate a sine
wave [16]. The heart of the DDS board is a digitally pro-
grammable device using DDS technology, the AD9851. It
has a 32-bit phase accumulator, a 14-bit digital phase-to-
amplitude converter and a 10-bit Digital-to-Analog Con-
verter (DAC). Its maximum clock frequency is 180 MHz,
and its maximum output frequency is 70 MHz. The
phase, relative to the clock signal, is encoded in 5 bits,
and is adjustable to any value from 0 to 2π. This results
in a rather poor phase resolution of 196 mrad.
The FPGA board manages the 1 Mb memory, the time
settings and the input/output of the device through se-
rial port. The FPGA is a Xilinx spartan-3 XC3S200,
providing 200, 000 logic gates. These logic gates are
designed with VHDL [17]. A Universal Asynchronous
Receiver Transmitter(UART) and a Picoblaze microcon-
troller are loaded into the FPGA, in order to commu-
nicate through serial port to the PC and to load the
on-board memory. We wrote our own VHDL scripts to
manage the DDS board and the FPGA in-board memory.
The FPGA board is clocked internally at 50 MHz. The
output sample rate of the device depends on the num-
ber of clock cycles ncycle between frequency data trans-
ferred to the DDS board. We set ncycle = 50 so that
ncycle/50 MHz = 1 µs, large enough thus ensuring safe
operation of the frequency data.
Software was developed in C with CVI Labwindows in
order to configure the device. The user writes a plain text
file ordering all the frequencies of the desired frequency
ramp. The set of frequencies is separated into 10 groups
of adjustable length, with a given sample rate for each.
The group lengths and the corresponding sample rate are
each translated by the software into 4 bytes. In addition,
each frequency in a given group, an integer written as a
decimal number, is translated into 4 bytes (32 bits). The
software sends these bytes by serial port to the FPGA
board.
The clock frequency fc and its phase noise level are the
key points for setting the frequency performance of the
device. The lower the phase noise of the clock signal, the
lower is theminimum phase noise of the output frequency
of the device (see section IVA).
The clock signal used for the experiment, see next sec-
tion, is the 10 MHz clock signal from an ultra-stable
Oven Controlled Crystal Oscillator (OCXO) BVA-8600.
Its phase noise PSD is −115 dB.rad2/Hz at 1 Hz. As
this clock frequency is very close to the desired maxi-
mum output frequency (≃ 10 MHz, see next section),
and to fulfil Shannon’s theorem [18, 19], we generate a
higher clock frequency by using the internal frequency
clock multiplier, at ×6, of the DDS board.
IV. RESULTS
The device presented in section III was first tested for
its frequency stability performance, as described in sec-
tion IVA and summarized in Table I. It was then in-
tegrated into a Bose-Einstein condensation experiment,
see section IVB.
A. Device frequency performance
1. Quantization error, Phase Accumulator truncation and
“magic” frequencies
By construction, digitization yields to inaccuracies in
frequency synthesis. The output frequency of 32-bit res-
olution DDS is given by
fRF = fc ×
where w is a binary 32-bit tuning word. The output fre-
quency can thus differ slightly from the desired frequency.
As fc = 60 MHz, the maximum digitization error δf is
6× 107/232 = 0.014 Hz.
As our software only takes integer frequencies as in-
put, a given frequency fRF will be synthesized without
sampling error if it may be written exactly as an integer
in the form given at Eq. (7). This condition is written
w = n× 232−p (8)
where n is a positive integer and p is the power of 2 in the
prime factorization of the clock frequency fc. In our case,
fc = 2
8 × 3 × 57 Hz and p = 8 so that every frequency
verifying
fRF = n× 234375 Hz (9)
will yield to no digitization error. n should be less than
2p/3 for the desired frequency to be in the synthesizer
range (fc/3).
In addition, when the AD9851 converts the calcu-
lated phase to an effective output amplitude, only the
first most significant 14 bits are used, even though the
AD9851 is a 32-bit synthesizer, in order to handle prac-
ticable number of entries in a lookup table. Truncating
the phase results in errors in amplitude that are periodic
in the time domain. These errors will be seen as spurs in
the frequency domain. However, for particular frequen-
cies which are exactly encoded by the first 14 bits (The
TABLE I: Performance of the device with fc = 60 MHz. The
relative frequency noise is computed from the phase noise
data, and given for 3.75 MHz (“magic” frequency) and 3 MHz
(larger noise value).
Parameters Min. Max. Units
Dynamic 0 10 MHz
Line-width - 30 mHz
Digitalization error 0 14 mHz
Sample rate adjustable 1 MHz
Memory length 1 262,144 pts
Phase noise@1Hz -113 -78 dB.rad2/Hz
Rel. freq. noise@1Hz -244 -207 dB/Hz
Rel. ampl. noise@1Hz - -120 dB/Hz
last 18 bits are 0.), the phase is not truncated at all,
yielding no spurious effects and the best PSD of phase
noise. This occurs for every frequency satisfying
fRF = n× fc/214 (10)
with n a positive integer. As fc = 60 MHz, we have
fRF = n × 3662.109375 Hz. The most stringent condi-
tion being the first one, we will denote the frequencies
satisfying Eq. (9) as “magic frequencies”. In order to il-
lustrate the difference between a “magic frequency” and
another one, we performed a set of noise PSD measure-
ments for two frequencies: a first set for fRF = 3 MHz
which is not a “magic” frequency, and a second set for
fRF = 3.75 MHz which is a “magic” frequency.
2. Spectral density of noise
We recorded the spectral density of noise of our syn-
thesizer at a given frequency fRF by FFT analysis of
the beat note at 0 Hz with a second synthesized ref-
erence signal. The measurement bench is sketched in
Fig. 2. In order to generate a tunable reference signal in
the RF range we used an analog synthesizer, Rhode &
Schwartz SML-01 (R&S), for synthesizing a signal at a
high frequency, and then divided by 100 to give fRF for
subsequent mixing. The beat note was recorded and an-
alyzed by a digital FFT analyzer HP 3562A sampling at
256 kHz. The R&S, as all the measurement devices, was
clocked at 10 MHz by the ultra-stable BVA-8600 quartz
oscillator. All the plots shown on Fig. 3 are raw spectra
of the beat note. The reference signal (R&S) itself was
characterized by making a beat note with a second iden-
tical R&S based synthesis. This corresponds to the line
labelled as “Reference” in Fig. 3.
By tuning the phase difference φ between the RF sig-
nal and the reference signal to π/2, we recorded phase
noise. At fRF = 3 MHz, the PSD of phase noise is
−78 dB.rad2.Hz−1 at 1 Hz, which corresponds to a PSD
of relative frequency noise of −207 dB.Hz−1. From the
data reported on Fig. 3, and assuming a Lorentzian line-
shape for the beat note, the linewidth δf of the RF signal,
given by
δf = πν2RF
df Sy(f) , (11)
is as small as 30 mHz over a bandwidth ∆f = 100 kHz
for an output frequency of 3 MHz [23]. We found similar
results for output frequencies from 1 to 5 MHz. At a
“magic” frequency, as for example at 3.75 MHz, where
truncation effects cancel out, the results are even better,
with a PSD of phase noise as low as −113 dB.rad2.Hz−1
at 1 Hz, only 2 dB higher than the phase noise of
the BVA 8600. The observed value corresponds to
the ultimate phase noise of the DDS chip specified by
the manufacturer. The relative frequency noise is then
−244 dB.Hz−1.
The frequency noise performance is naturally linked to
the quality of the master oscillator. To illustrate this
fact, the same measurements were repeated with the im-
proved OCXO of a DS-345, the ERC EROS-750-SBR-4,
as master oscillator. No significant change in the fre-
quency performances were noticed at non magic frequen-
cies. At a magic frequency, the PSD of phase noise in-
creased to −100 dB.rad2/Hz, which is consistent with the
phase noise specifications of this quartz.
The PSD of relative amplitude noise Sa was recorded
with the same measurement bench, by tuning the phase φ
to 0. The reference signal was also delivered by the R&S.
The recorded spectrum is very close to the “Reference”
line itself at the level of −120 dB.Hz−1 at 1 Hz, close to
the input noise of the FFT analyzer. Note that for rele-
vant excitation frequencies ν = 2νz (larger than 1.2 kHz),
the PSD of amplitude noise is lower than −130 dB.
In practice, we use in our BEC experiment a pro-
grammable RF attenuator Minicircuit ZAS-3, driven by
an analog output channel of a National Instrument PC
card PCI-6713, in order to vary the RF amplitude sent
to the RF coil. We repeated the measurements described
above with this attenuator and found an increase of Sa,
starting at −115 dB.Hz−1 at 1 Hz, which corresponds to
a noise figure of the programmable attenuator of 5.1 dB.
At the output of the attenuator, the RF signal is am-
plified by a class-A amplifier Kalmus 505 F. Its gain is
¸100SML-01
-8600
10 MHz
ZAS-3
FIG. 2: Noise spectrum measurement bench. The gain in the
mixer was fitted to the output signal, depending on the phase
between the two beating signals.
1
 10
 100
 1k
 10k
 100k
-180
-160
-140
-120
-100
1
 10
 100
 1k
 10k
 100k
-160
-140
-120
-100

2
 .
Frequency (Hz)
 DDS 3.75 MHz
 DDS 3 MHz
 Reference
 BVA 8600
FIG. 3: PSD of relative amplitude noise, upper plot, and
phase noise,lower plot. Averaging = 100. FFT’s sampling
frequency = 256 kHz. The low pass filter cutting frequency of
the mixer was set to 200 kHz. The reference curve corresponds
to the beat note of two identical R&S synthesizers. Plotted
for comparison is the phase noise specifications of the BVA
8600 quartz oscillator, as given by the manufacturer.
37 dB and its noise figure is typically +10 dB according
to the manufacturer specifications.
B. Integration into the experiment
In this section, we present the main experimental re-
sults concerning the heating rate of the atomic sample.
As described in section II, the RF signal is used for
producing the bubble-like trap, where ultracold rubid-
ium atoms are accumulated at the bottom of the surface.
The trap is very anisotropic with stronger confinement
in the vertical direction [7]. For the RF-dressed trap to
be efficiently loaded from the standard Ioffe-Pritchard
static magnetic trap, described in [20], the RF frequency
fRF has to be ramped up from 1 MHz to a final fixed
frequency fendRF ranging from 2 to 10 MHz. The static
magnetic field, necessary both for magnetic trapping and
RF-induced trapping, is always present. A typical ramp
is shown on Fig. 4. The frequency is ramped more slowly
around 1.3 MHz, corresponding to the resonant frequency
at the center of the magnetic trap where adiabaticity of
spin rotation is more difficult to obtain. At the end of
this ramp, which may last between 75 ms and 500 ms, the
RF frequency is held between 0.1 and 10 s for testing the
lifetime and heating rate of the atoms in the RF-based
trap.
The RF signal is amplified by 37 dB with a single stage
amplifier, and the RF field is produced by a small circular
antenna. The RF field is linearly polarized and its ampli-
tude BRF may be adjusted between 70 and 700 mG. We
0
 50
 100
 150
 200
Time (ms)
FIG. 4: Typical shape of a radio-frequency ramp applied to
the ultracold atomic sample. In the present example fRF is
increased from 1 to 3 MHz within 150 ms. At the end of this
ramp, the RF frequency is maintained at its final value for
some holding time in the RF-based trap, dashed line.
record the atomic temperature after the RF ramp as a
function of time while the atoms are confined in the RF-
based trap. The temperature is deduced from the cloud
size along z after a 7 ms of ballistic expansion. The same
measurement was repeated with different RF sources.
First, we used a Agilent 33250A synthesizer with a RF
amplitude larger than 500 mG for both the frequency
ramp and the final holding frequency. Such RF ana-
log synthesizers operated at fixed frequency exhibit very
good relative frequency noise in most cases, typically at
the −180 dB/Hz level or better. However, as mentioned
by Colombe et al. [7] and confirmed by White et al. [21],
the relative frequency noise increases by a few decades if
the output frequency is driven with an external analog
voltage. The frequency was indeed tuned through an ex-
ternal voltage control provided by the PC analog board
(NI 6713), such that the modulation depth was ±1 MHz
on a central frequency of 2 MHz. We obtained both a
short lifetime, typically 400 ms at 1/e, and a strong lin-
ear heating, as shown on Fig. 5 full circles. The heating
rate is measured to be 5.0 µK/s. This rate, given the RF
amplitude, corresponds to a relative frequency noise of
Sy = −100 dB/Hz at the trap frequency of 600 Hz. This
noise is quite high and is related to voltage noise on the
external frequency control of the synthesizer. This effect
is strong in our case as the frequency is varied with a
large modulation depth (∆f/f = 1).
We also tested a two step scheme, with a first ramp
performed by a Stanford DS-345 DDS (1,500 frequency
points), followed by the R&S maintained at fixed fre-
quency for the full holding time. This scheme allows one
to benefit from the excellent frequency stability of the
second device used at fixed frequency. With this setup,
0
 1
 2
 3
 4
 5
 6
 7
 8
 Agilent
 Our DDS device
holding time in the trap (s)
FIG. 5: Comparison of heating of the atomic cloud in the bub-
ble trap: Agilent 33250A synthesizer, full circles, or present
device, open diamonds, is used for creating the RF ramp and
the final radio-frequency fendRF . We observe a heating rate of
5.0 µK/s in the first case and 0.47 µK/s the second one, as
given by a linear fit, full lines.
heating during the plateau stage seemed to be completely
suppressed [7]. However, a large dispersion was observed
in the atom number data after the switching, which pre-
vented us from characterizing the heating very precisely.
This dispersion is due to a random phase hop at the
switching time between the two synthesizers, resulting in
atomic losses. We studied the effect of the random phase
hop on the atom number by recording, for each experi-
ment, the phase difference at the switching time with a
control oscilloscope. The results are presented on Fig. 6,
full circles. For the maximum phase hop, π, 80% of the
atoms are lost. This figure depends on the atomic tem-
perature, the losses being higher at lower temperature,
and is well reproduced by theory, as shown on Fig. 6,
black line. The theoretical curve is calculated for an RF
amplitude of 470 mG by averaging the loss probability
over the positions of the atoms, as deduced from a ther-
mal distribution at a temperature of 4 µK. The fact that
the trap is able to hold two of the five spin components
of the F = 2 hyperfine state is taken into account.
Finally, we performed the heating rate and lifetime
measurements with our DDS device and an amplitude
of 70 mG. We found an increased lifetime, up to 9.6 s.
During the first second, the vertical “kinetic tempera-
ture”, deduced from the vertical cloud size after time of
flight, decreases due to thermalization with the horizon-
tal degrees of freedom which initially have a lower energy
due to the loading procedure: Fig. 5, open diamonds.
After 1 second, a linear heating rate of 0.47 µK.s−1 is
clearly observable. No exponential parametric heating is
measurable. We believe that the remaining linear heat-
ing is due to residual excitation by scattered light, which
was not present in the previous case. The same heating
-180
 -150
 -120
 -90
 -60
 -30
 0
 30
 60
 90
 120
 150
 180

6
 )
phase hop at switching (degrees)
 experiment
 theory
FIG. 6: Number of atoms remaining after switching between
the two synthesizers as a function of the phase hop. Exper-
imental data, full circles, are compared to a calculation, full
line, assuming a RF amplitude of 470 mG and a temperature
of 4 µK.
rate was indeed observed within the static magnetic trap,
when the RF source was off. Heating directly linked to
the RF source is not observable.
V. CONCLUSION
We built a low cost DDS RF generator, based on a
DDS chip and its evaluation kit. It is extremely agile
since we have the ability of using up to 10 different time
steps during the sequence, allowing an efficient use of the
FPGA on-board memory. In contrast to Stanford DS-345
RF synthesizers, the frequency ramp is not repeated at
the end of the sequence. The last frequency is kept until
a new TTL pulse is proceeded to end the sequence. Its
large memory length, up to 262,144 frequency samples,
allows very small frequency steps, thus strongly reduc-
ing the atomic losses in our RF-dressed trap experiment.
The DDS technology ensures that the phase is continu-
ous all over the frequency ramp, which is essential for RF
trapping applications.
Apart from these high resolution features, the device
presents excellent spectral properties. The power spec-
tral density of phase noise is as low as−113 dB.rad2.Hz−1
at well chosen “magic frequencies”, and in any case re-
mains below than −78 dB.rad2.Hz−1 at worst, dominated
by phase truncation. This last value yields a linewidth of
30 mHz over a 100 kHz bandwidth. The corresponding
relative frequency noise level is 90 dB below the require-
ments of our RF-dressed trapping experiment.
The amplitude noise of the DDS synthesizer is mea-
sured to be below −130 dB.Hz−1 at frequencies above
1 kHz. Amplitude noise turns out to be the limiting noise
present in our RF chain at the present time due to the
several dB noise figures of the attenuators and amplifiers
in use in our experiment. They actually increase the RF
amplitude noise to −115 dB.Hz−1 at 1 kHz, which still
remains 25 dB below our experiment requirements.
All these features make our DDS apparatus very well-
suited for cold atom experiments. It may be used for
optimized evaporative cooling implementation, with ar-
bitrary frequency ramp profiles including a final “RF-
knife”, in a standard BEC experiment. It is of particular
importance in the case of atom chip experiments [22],
where the cooling ramp is usually done very quickly (1 s
typically). It is also ideal for developing new RF-based
traps. In particular, the possibility of configuring the
time step, that is the duration of each frequency point, is
very attractive for our application. This feature is all the
more interesting in that the memory can be separated
in several zones, each being allocated a chosen number
of points and a chosen time step, so we can dilate or
compress in time the whole or a part of the ramp, while
keeping the same frequency series. For instance, if we
want to accomplish a ramp of N frequency values with a
small time resolution, say 10 µs, and then keep the final
frequency constant over ten seconds, we have the ability
to address a first memory zone with the first N−1 points
and a 10 µs time step, and a second memory zone with
the last frequency value and a 10 s time step. A sin-
gle time step over the whole sequence would have forced
us to sacrifice either the resolution of the ramp or the
duration of the final plateau.
The device was successfully implemented in our ultra-
cold atom experiment. First, we set boundary specifica-
tions on the RF source performances for trapping exper-
iments regarding heating rate and lifetime. In the same
manner, we set conditions on maximum phase hops and
frequency step magnitude. With our home made device,
we observed no heating due to the RF source during the
plateau step. A remaining heating rate of 0.47µK/s was
identical to the one obtained in the magnetic trap and
was limited by other noise sources, presumably scattered
photons. Despite the very large number of frequency
points, heating along the vertical axis is still present dur-
ing the loading stage of the RF-dressed trap. We at-
tribute this heating to non-adiabatic deformation of the
confining potential during the loading process, which we
could not make slower due to the other noise sources.
While the actual performances of the device are al-
ready very good, a few improvements are still possible.
First, the memory length of the DDS is basically deter-
mined by the FPGA in-board memory size. A larger
sample size may easily be implemented by replacing the
board, provided that the data transfer rate is improved
through the use of an Ethernet port or a USB port. Sec-
ond, the sample rate is 1 MHz, limited by the data trans-
fer rate from the FPGA board. Higher FPGA clock fre-
quency, or smaller clock counts ncycle, would allow higher
sample rates. The frequency limit due to the AD-9851
DDS chip itself is 3 MHz. This modification would im-
prove the temporal resolution by a factor of 3. Finally,
the maximum output frequency of 10 MHz may be very
easily increased to 20 MHz by replacing the low-pass fil-
ter. By clocking the DDS with a 180 MHz clock, one
could even reach output frequencies up to 60 MHz. Note
that with a similar DDS chip, the AD 9858, frequency
clocks as high as 1 GHz are allowed, which makes possi-
ble the synthesis of RF frequencies up to 330 MHz.
Acknowledgments
We are indebted to R. J. Butcher for a critical read-
ing of the manuscript. This work was supported by the
Région Ile-de-France (contract number E1213) and by
the European Community through the Research Train-
ing Network “FASTNet” under contract No. HPRN-CT-
2002-00304 and Marie Curie Training Network “Atom
Chips” under contract No. MRTN-CT-2003-505032.
Laboratoire de physique des lasers is UMR 7538 of CNRS
and Paris 13 University. The LPL is a member of the In-
stitut Francilien de Recherche sur les Atomes Froids.
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