Three dimensional cooling and trapping with a narrow line
EPJ manuscript No.
(will be inserted by the editor)
Three dimensional cooling and trapping with a narrow line
T. Chanelière1, L. He2, R. Kaiser3 and D. Wilkowski3
1 + Now at: Laboratoire Aimé Cotton, CNRS, UPR 3321, Université Paris-Sud, Bat. 505, F-91405 Orsay Cedex, France
2 ∗ Now at: State Key Laboratory of Magnetic Resonance and Atomic and Molecular Wuhan Institute of Physics and Mathe-
matics, Chinese Academy of Sciences, Wuhan 430071, P. R. China
3 Institut Non Linéaire de Nice, CNRS, UMR 6618, Université de Nice Sophia-Antipolis, F-06560 Valbonne, France.
October 21, 2018
Abstract. The intercombination line of Strontium at 689nm is successfully used in laser cooling to reach
the photon recoil limit with Doppler cooling in a magneto-optical traps (MOT). In this paper we present
a systematic study of the loading efficiency of such a MOT. Comparing the experimental results to a
simple model allows us to discuss the actual limitation of our apparatus. We also study in detail the final
MOT regime emphasizing the role of gravity on the position, size and temperature along the vertical and
horizontal directions. At large laser detuning, one finds an unusual situation where cooling and trapping
occur in the presence of a high bias magnetic field.
PACS. 3 9.25.+k
1 Introduction
Cooling and trapping alkaline-earth atoms offer interest-
ing alternatives to alkaline atoms. Indeed, the singlet-
triplet forbidden lines can be used for optical frequency
measurement and related subjects [1]. Moreover, the spin-
less ground state of the most abundant bosonic isotopes
can lead to simpler or at least different cold collisions prob-
lems than with alkaline atoms [2]. Considering fermionic
isotopes, the long-living and isolated nuclear spin can be
controlled by optical means [3] and has been proposed to
implement quantum logic gates [4]. It has also been shown
that the ultimate performance of Doppler cooling can be
greatly improved by using narrow transitions whose pho-
ton recoil frequency shifts ωr are larger than their natural
widths Γ [5]. This is the case for the 1S0 →3 P1 spin-
forbidden line of Magnesium (ωr ≈ 1100Γ ) or Calcium
(ωr ≈ 36Γ ). Unfortunately, both atomic species can not
be hold in a standard magneto-optical trap (MOT) be-
cause the radiation pressure force is not strong enough
to overcome gravity. This imposes the use of an extra
quenching laser as demonstrated for Ca [6]. For Stron-
tium, the natural width of the intercombination transition
(Γ = 2π×7.5 kHz) is slightly broader than the recoil shift
(ωr = 2π×4.7 kHz). The radiation pressure force is higher
than the gravity but at the same time the final tempera-
ture is still in the µK range [7,8]. In parallel, the narrow
transition partially prevents multiple scattering processes
and the related atomic repulsive force [10]. Hence impor-
tant improvements on the spatial density have been re-
ported [7]. However, despite experimental efforts, such as
adding an extra confining optical potential, pure optical
methods have not allowed yet to reach the quantum de-
generacy regime with Strontium atoms [9].
In this paper, we will discuss some performances, es-
sentially in terms of temperatures, sizes and loading rates,
of a Strontium 88 MOT using the 689 nm 1S0 →3P1 in-
tercombination line.
Initially the atoms are precooled in a MOT on the
spin-allowed 461 nm 1S0 →1P1 transition (natural width
Γ = 2π × 32MHz) as discussed in [11]. Then the atoms
are transferred into the 689 nm intercombination MOT.
To achieve a high loading rate, Katori et al. [7] have used
laser spectrum, broadened by frequency modulation. Thus
the velocity capture range of the 689 nm MOT matches
the typical velocity in the 461 nm MOT. They report a
transfer efficiency of 30%. The same value of transfer effi-
ciency is also reported in reference [8]. In our set-up, 50%
of the atoms initially in the blue MOT are transferred into
the red one. In section 3 we present a systematic study
of the transfer efficiency as function of the parameters of
the frequency modulation. In order to discuss the intrin-
sic limitations of the loading efficiency, we compare our
experimental results to a simple model. In particular, we
demonstrated that our transfer efficiency is limited by the
size of the red MOT beams. We show that it could be op-
timized up to 90% with realistic laser power (25mW per
beams).
The minimum temperature achieved in the broadband
MOT is about 2.5µK. In order to reduce the tempera-
ture down to the photon recoil limit (0.5µK), we apply
a second cooling stage, using a single frequency laser and
observe similar temperatures, detuning and intensity de-
pendencies as reported in the literature (see references [7],
http://arxiv.org/abs/0704.0855v2
2 T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line
[8], [12] and [13]). In those publications, the role of gravity
on the cooling and trapping dynamics along the z vertical
direction has been discussed. In this paper we compare
the steady state behaviour along vertical (z) direction to
that along the horizontal plane (x−y) where gravity plays
indirectly a crucial role (section 4).
Details about the dynamics are given in references
[8],[12]. In particular the authors establish three regimes.
In regime (I) the laser detuning |δ| is larger than the
power-broadened linewidth ΓE = Γ
(1 + s). Regime (II)
on the contrary corresponds to ΓE > |δ|. In both regimes
(I) and (II) ΓE ≫ Γ, ωr and the semiclassical limit is a
good approximation. In regime (III) the saturation pa-
rameter is small and a full quantum treatment is required.
We will focus here on the semiclassical regime (I). In this
regime, we confirm that the temperature along the z di-
rection is independent of the detuning δ. Following Loftus
et al. [12], we have also found (see section 4.1) that this
behavior is due to the balance of the gravitational force
and the radiation pressure force produced by the upward
pointing laser (the gravity defining the downward direc-
tion). The center of mass of the atomic cloud is shifted
downward from the magnetic field quadrupole center. As a
consequence, cooling and trapping in the horizontal plane
occur at a strong bias magnetic field mostly perpendicu-
lar to the cooling plane. This unusual situation is studied
in detail (section 4.2). Despite different friction and dif-
fusion coefficients along the horizontal and the vertical
directions, the horizontal temperature is found to be the
same as the vertical one (see section 4.3). In reference [12],
the trapping potential is predicted to have a box shape
whose walls are given by the laser detuning. This is in-
deed the case without a bias magnetic field along the z
axis. It is actually different for the regime (I) described in
this paper. Here we have found that the trapping poten-
tial remain harmonic. This leads to a cloud width in the
horizontal direction which is proportional to
|δ| (section
4.2).
2 Experimental set-up
Our blue MOT setup (on the broad 1S0 →1 P1 transi-
tion at 461 nm) is described in references [14,15]. Briefly,
it is composed by six independent laser beams typically
10mW/cm2 each. The magnetic field gradient is about
70G/cm. The blue MOT is loaded from an atomic beam
extracted from an oven at 550 ◦C and longitudinally slowed
down by a Zeeman slower. The loading rate of our blue
MOT is of 109 atoms/s and we trap about 2.106 in a
0.6mm rms radius cloud when no repumping lasers are
used [16]. To optimize the transfer into the red MOT, the
temperature of the blue MOT should be as small as possi-
ble. As previously observed [11], this temperature depends
strongly on the optical field intensity. We therefore de-
crease the intensity by a factor 5 (see figure 1) 4ms before
switching off the blue MOT. The rms velocity right before
the transfer stage is thus reduced down to σb = 0.6m/s
whereas the rms size remains unchanged. Similar two stage
cooling in a blue MOT is also reported in reference [13].
The 689 nm laser source is an anti-reflection coated
laser diode in a 10 cm long extended cavity, closed by a
diffraction grating. It is locked to an ULE cavity using the
Pound-Drever-Hall technique [17]. The unity gain of the
servo loop is obtained at a frequency of 1MHz. From the
noise spectrum of the error signal, we derive a frequency
noise power. It shows, in the range of interest, namely
1Hz − 100 kHz, an upper limit of 160 Hz2/Hz which is
low enough for our purpose. The transmitted light from
the ULE cavity is injected into a 20mW slave laser diode.
Then the noise components at frequencies higher than the
ULE cavity cut-off (300 kHz) are filtered. It is important
to note that the lateral bands used for the lock-in are
also removed. Those lateral bands, at 20MHz from the
carrier, are generated modulating directly the current of
the master laser diode. A saturated spectroscopy set-up on
the 1S0 →3P1 intercombination line is used to compensate
the long term drift of 10−50Hz/s mainly due to the daily
temperature change of the ULE cavity.
The slave beam is sent through an acousto-optical mod-
ulator mounted in a double pass configuration. The laser
detuning can then be tuned within the range of a few
hundreds of linewidth around the resonance. This acousto-
optical modulator is also used for frequency modulation
(FM) of the laser, as required during the loading phase
(see section 3).
The red MOT is made of three retroreflected beams
with a waist of 0.7 cm. The maximum intensity per beam
is about 4mW/cm2 (the saturation intensity being Is =
3µW/cm2). The magnetic gradient used for the red MOT
is varied from 1 to 10G/cm.
To probe the cloud (number of atoms and tempera-
ture) we use a resonant 40µs pulse of blue light (see fig
1). The total emitted fluorescence is collected onto an
avalanche detector. From this measurement, we deduce
the number of atoms and then evaluate the transfer rate
into the red MOT. At the same time, an image of the cloud
is taken with an intensified CCD camera. The typical spa-
tial resolution of the camera is 30µm. Varying the dark
period (time-of-flight) between the red MOT phase and
the probe, we get the ballistic expansion of the cloud. We
then derive the velocity rms value and the corresponding
temperature.
3 Broadband loading of the red MOT
The loading efficiency of a MOT depends strongly on the
width of the transition. With a broad transition, the max-
imum radiation pressure force is typically am =
104 × g, where vr is the recoil velocity [18]. Hence, on
l ≈ 1 cm (usual MOT beam waist) an atom with a veloc-
ity vc =
2aml ≈ 30m/s can be slowed down to zero and
then be captured. During the deceleration, the atom re-
mains always close to resonance because the Doppler shift
is comparable to the linewidth. Thus MOTs can be di-
rectly loaded from a thermal vapor or a slow atomic beam
using single frequency lasers. Moreover typical magnetic
field gradients of few tens of G/cm usually do not dras-
T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line 3
tically change the loading because the Zeeman shift over
the trapping region is also comparable to the linewidth.
An efficient loading is more complex to achieved with
a narrow transition. For Strontium, the maximum radia-
tion pressure force of a single laser is only am ≈ 15 × g.
Assuming the force is maximum during all the capture
process, one gets vc =
2aml ≈ 1.7m/s. Hence, precool-
ing in the blue MOT is almost mandatory. In that case the
initial Doppler shift will be vcλ
−1 ≈ 2.5MHz, 300 times
larger than the linewidth. In order to keep the laser on
resonance during the capture phase, the red MOT lasers
must thus be spectrally broadened. Because of the low
value of the saturation intensity, the spectral power den-
sity can easily be kept large enough to maintain a maxi-
mum force with a reasonable total power (few milliwatts).
The magnetic field gradient of the MOT may also affect
the velocity capture range. To illustrate this point, let us
consider an atom initially in the blue MOT at the center
of the trap with a velocity vc = 1.7m/s. During the decel-
eration, the Doppler shift decreases whereas the Zeeman
shift increases. However, the magnetic field gradient does
not affect the capture velocity as far as the total shift
(Doppler+Zeeman) is still decreasing. This condition is
fulfilled if the magnetic field gradient is lower than [19]:
λgeµbvc
≈ 0.6G/cm (1)
where ge = 1.5 is the Landé factor of the
3P1 level and
µb = 1.4MHz/G is the Bohr magneton. In practice we use
a magnetic field gradient which is larger than bc. In that
case, it is necessary to increase the width of the laser spec-
trum so that the optimum transfer rate is not limited by
the Zeeman shift (see section 3.2). An alternative solution
may consist of ramping the magnetic field gradient during
the loading [7].
3.1 Transfer rate: experimental results
In this section we will present the experimental results re-
garding the loading efficiency of the red MOT from the
blue MOT. To optimize the transfer rate, the laser spec-
trum is broadened using frequency modulation (FM). Thus
the instantaneous laser detuning is ∆(t) = δ+∆ν. sin νmt.
∆ν and νm are the frequency deviation and modulation
frequency respectively, δ is the carrier detuning. Here, the
modulation index ∆ν/νm is always larger than 1, thus the
so-called wideband limit is well fulfilled. Hence one can
assume the FM spectrum to be mainly enclosed in the
interval [δ −∆ν; δ +∆ν].
As shown in figure 2, the transfer rate increases with
νm up to 15 kHz where we observe a plateau at 45% trans-
fer efficiency. On the one hand when νm is larger than
the linewidth, the atoms are in the non-adiabatic regime
where they interact with all the Fourier components of the
laser spectrum. Moreover, the typical intensity per Fourier
component remains always higher than the saturation in-
tensity Is = 3µW/cm
2. As a consequence, the radiation
pressure force should be close to its maximum value for
any atomic velocity. On the other hand when νm < Γ/2π,
the atoms interact with a chirped intense laser where the
mean radiation pressure force (over a period 2π/νm) is
clearly smaller than in the case νm > Γ/2π. As a conse-
quence, the transfer rate is reduced when νm decreases.
In figure 3, the transfer rate is measured as a func-
tion of ∆ν. The carrier detuning is δ = −1MHz and the
modulation frequency is kept larger than the linewidth
(νm = 25 kHz). Starting from no deviation (∆ν = 0), we
observe (fig. 3) an increase of the transfer rate with ∆ν (in
the range 0 < ∆ν < 500 kHz). After reaching its maximum
value, the transfer rate does not depend on ∆ν anymore.
Thus the capturing process is not limited by the laser
spectrum anymore. If we further increase the frequency
deviation ∆ν, the transfer becomes less efficient and fi-
nally decreases again down to zero. This reduction occurs
as soon as ∆ν > |δ|, i.e. some components of the spectrum
are blue detuned. This frequency configuration obviously
should affect the MOT steady regime adding extra heat-
ing at zero velocity (see section 3.3). We can see that it
is also affecting the transfer rate. To confirm that point,
figure 4 shows the same experiment but with a larger de-
tuning δ = −1.5MHz and δ = −2MHz for the figures
4a and 4b respectively. Again the transfer rate decreases
as soon as ∆ν > |δ|. The transfer rate is also very small
on the other side for small values of ∆ν. In that case the
entire spectrum of the laser is too far red detuned. The
radiation pressure forces are significant only for velocities
larger than the capture velocity and no steady state is
expected. Keeping now the deviation fixed and varying
the detuning as shown in figure 5, we observe a maximum
transfer rate when the detuning is close to the deviation
frequency ∆ν ≃ |δ|. Closer to resonance (∆ν < |δ|), the
blue detuned components prevent an efficient loading of
the MOT.
The magnetic field gradient plays also a crucial role for
the loading. We indeed observe (fig. 6) that the transfer
rate decreases when the magnetic field gradient increases.
At very low magnetic field (b < 1G/cm) the reduction
of the transfer rate is most likely due to a lack of stabil-
ity within the trapping region. In that case we actually
observe a strong displacement of the center of mass of
the cloud. This is induced by imperfections of the set-up
such as non-balanced laser intensities which are critical
at low magnetic gradient. Hence, the optimum magnetic
field gradient is found to be the smallest one which ensure
the stability of the cloud in the MOT.
3.2 Theoretical model and comparison with the
experiments
To clearly understand the limiting processes of the transfer
rate, we compare the experimental data to a simple 1D
theoretical model based on the following assumptions:
- An atom undergoes a radiation pressure force and thus a
deceleration if the modulus of its velocity is between vmax
and vmin with
vmax = λ(|δ|+∆ν), vmin = max{λ(|δ|−∆ν);λ(−|δ|+∆ν)}
4 T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line
am = 0 elsewhere. We simply write that the Doppler shift
is contained within the FM spectrum. We add the condi-
tion vmin = λ(−|δ|+∆ν) when some components are blue
detuned ∆ν > |δ|. In this case, we consider the simple
ideal situation where the two counter-propagating lasers
are assumed perfectly balanced and then compensate each
other in the spectral overlapping region.
- Even in the semiclassical model, it is difficult to calculate
the acceleration as a function of the velocity for a FM spec-
trum. However for all the data presented here, the satura-
tion parameter is larger than one. Hence the deceleration
is set to a constant value − 1
am when vmin < |v| < vmax.
The prefactor 1/3 takes into account the saturation by the
3 counter-propagating laser beam pairs.
- The magnetic field gradient is included by giving a spa-
tial dependence of the detuning δ in the expression (2).
- An atom will be trapped if its velocity changes of sign
within a distance shorter that the beam waist.
In figures 3-6 the results of the model are compared to
the experimental data. The agreement between the model
and the experimental data is correct except at large fre-
quency deviation (figures 3 and 4) or at low detuning (fig-
ure 5). In those cases the spectrum has some blue detuned
components. As mentionned before, this is a complex sit-
uation where the assumptions of the simple model do not
hold anymore. Fortunately those cases do not have any
practical interest because they do not correspond to the
optimum transfer efficiency.
At the optimum, the model suggests that the transfer
is limited by the beam waist (see caption of figures (3-6)).
Moreover for all the situation explored in figures 3-5, the
magnetic field gradient is strong enough (b = 1G/cm) to
have an impact on the capture process, as suggested by
the inequality (1). However it is not the transfer limiting
factor because the Zeeman shift is easily compensated by
a larger frequency excursion or by a larger detuning.
Increasing the beam waist would definitely improve the
transfer efficiency as showed in figure 7. If the saturation
parameter would remain large for all values of beam waist,
more than 90% of the atoms would be transferred for a
2 cm beam waist. 25mW of power per beam should be
sufficient to achieve this goal. In our experimental set-up,
the power is limited to 3mW per beam. So the satura-
tion parameter is necessarily reduced once the waist is
increased. To take this into account and get a more realis-
tic estimation of the efficiency for larger beams, we replace
the previous acceleration by the expression ams/(1 + 3s),
with s = I/Is the saturation parameter per beam. In this
case, the transfer efficiency becomes maximum at 70% for
a beam waist of 1.5 cm.
3.3 Temperature
Cooling with a broadband FM spectrum on the intercom-
biaison line decreases the temperature by three orders of
magnitude in comparison with the blue MOT: from 3mK
(σb = 0.6m/s) to 2.5µK (see figure 8). For small detuning,
the temperature is strongly increasing when the spectrum
has some blue detuned components (∆ν > |δ|). Indeed the
cooling force and heating rate are strongly modified at the
vicinity of zero detuning. This effect is illustrated in figure
8. On the other side at large detuning (δ < −1.5MHz), the
temperature becomes constant. This regime corresponds
to a detuning independent steady state, as also observed
in single frequency cooling (see ref. [12] and section 4).
4 Single frequency cooling
About half of the atoms initially in the 461 nm MOT are
recaptured in the red one using a broadband laser. The
final temperature is 2.5µK i.e. 5 times larger than the
photon recoil temperature Tr = 460 nK. To further de-
crease the temperature one has to switch to single fre-
quency cooling (for time sequences: see figure 1). As we
will see in this section, the minimum temperature is now
about 600 nK close to the expected 0.8Tr in an 1D mo-
lasses [5]. Moreover, one has to note that, under proper
conditions described in reference [12], the transfer between
the broadband and the single frequency red MOT can be
almost lossless.
In the steady state regime of the single frequency red
MOT, one has kσv ≈ ωr ≈ Γ . Thus, there is no net sepa-
ration of different time scales as in MOTs operated with a
broad transition where ωr << kσv << Γ . However, here
the saturation parameter s always remains high. It cor-
responds to the so-called regimes (I) and (II) presented
in reference [12]. Thus ωr << Γ
1 + s and the semiclas-
sical Doppler theory describes properly the encountered
experimental situations.
To insure an efficient trapping, the parameter’s val-
ues of the single frequency red MOT are different from
a usual broad transition MOT: the magnetic field gradi-
ent is higher, typically 1000Γ/cm. Moreover the gravity
is not negligible anymore by comparison with the typical
radiation pressure. Those features lead to an unusual be-
havior of the red MOT as we will explain in this section.
We will first independently analyze the MOT properties
along the vertical dimension (section 4.1) then in the hor-
izontal plane (section 4.2), to finally compare those two
situations (section 4.3).
4.1 Vertical direction
In the regime (I) i.e. at large negative detuning and high
saturation (see examples on figure 9a) the temperature is
indeed constant. As explained in reference [12], this be-
havior is due to the balance between the gravity and the
radiation pressure force of the upward laser. At large neg-
ative detuning, the downward laser is too far detuned to
give a significant contribution. In the semiclassical regime,
an atom undergoes a net force of
Fz = h̄k
1 + sT + 4(δ − geµBbz − kvz)2/Γ 2
−mg (3)
Considering the velocity dependence of the force, the first
order term is:
T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line 5
Fz ≈ −γzvz (4)
γz = −4
h̄k2δeff
(1 + sT + 4δ
/Γ 2)2
where the effective detuning δeff = δ − geµBb < z > is
define such as
1 + sT + 4δ
= mg (6)
sT is the total saturation parameter including all the beams.
< z > is the mean vertical position of the cold cloud.
Hence δeff is independent of the laser detuning δ and the
vertical temperature at larger detuning depends only on
the intensity as shown in figures 9a and 9b.
The spatial properties of the cloud are also related to
the effective detuning δeff which is independent of δ. The
mean vertical position depends linearly on the detuning,
so that one has :
d < z >
geµBb
The predicted vertical displacement is compared to the
experimental data in figure 10a. The agreement is excel-
lent (the only adjustable parameter is the unknown origin
of the vertical axe). Because the radiation pressure force
for an atom at rest does not depend on the laser detuning
δ, the vertical rms size should be also δ-independent. This
point is also verified experimentally (see figure 10b).
4.2 x− y horizontal plane
Let us now study the behavior of the cold cloud in the x−y
plane at large laser detuning. As explained in section 4.1,
the position of the cloud is vertically shifted downward
with respect to the center of the magnetic field quadrupole
(see figure 11). The dynamic in the x−y plane occurs thus
in the presence of a high bias magnetic field. To derive
the expression of the semiclassical force in this unusual
situation one has first to project the circular polarizations
states of the horizontal lasers on the eigenstates. We define
the quantification axis along the magnetic field, one gets:
e+x =
1 + sinα
cosα√
1− sinα
e−x =
1− sinα
cosα√
1+ sinα
where e−i , πi and e
i represent respectively the left-handed,
linear and right-handed polarisations along the i axis. The
angle α between the vertical axis and the local magnetic
field is shown on figure 11. For large detuning, α is al-
ways small (α ≪ 1 ) and we write α ≈ −x/ < z >
considering only the dynamics along the x dimension. For
simplicity the magnetic field gradient b is considered as
spatially isotropic with b > 0 as sketched on figure 11b.
The expression of the radiation pressure force is then:
Fx = h̄k
×(10)
s(1− sinα)2/4
1 + sT + 4(δ − geµBb < z > (1− tanα)− kvx)2/Γ 2
s(1 + sinα)2/4
1 + sT + 4(δ − geµBb < z > (1− tanα) + kvx)2/Γ 2
Note that this expression is not restricted to the small α
values. We expect six terms in the expression (11): three
terms for each laser corresponding to the three e−
and e+
polarisation eigenstates. However only two terms,
corresponding to the e+
state, are close to resonance and
thus have a dominant contribution. As for the vertical
dimension, the off resonant terms are removed from the
expression (11). One has also to note that the effective
detuning δeff = δ − geµBb < z > is actually the same as
the one along the vertical dimension.
The first order expansion of (11) in α and kvx/Γ gives
the expression of the horizontal radiation pressure force:
Fx ≈ −καα− γxvx = −κxx− γxvx (11)
κα = − < z > κx = h̄k
1 + sT + 4δ
= mg (12)
= −2 h̄k
2δeff
(1 + sT + 4δ
/Γ 2)2
As for the vertical dimension (equation (6)), the force
depends on δeff but at the position of the MOT does not
depend on the laser detuning δ. Hence, at large detuning,
the horizontal temperature depends only on the intensity
as observed in figures 9a and 9b.
To understand the trapping mechanisms in the x − y
plane, we now consider an atom at rest located at a po-
sition x 6= 0 (corresponding to α 6= 0), i.e. not in the
center of the MOT. The transition rate of two counter-
propagating laser beam is not balanced anymore. This is
due to the opposite sign in the α dependency of the prefac-
tor in expression (11). This mechanism leads to a restoring
force in the x−y plane at the origin of the spatial confine-
ment (equation 11). Applying the equipartition theorem
one gets the horizontal rms size of the cloud:
x2rms =
= −< z > kBT
Without any free adjusting parameter, the agreement with
experimental data is very good as shown in figure 10b.
On the other hand there’s no displacement of the center
of mass in the x − y plane whatever is the detuning δ as
long as the equilibrium of the counter-propagating beams
intensities is preserved (figure 10a).
6 T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line
4.3 Comparing the temperatures along horizontal and
vertical axes
As seen in sections 4.1 and 4.2, gravity has a dominant
impact on cooling in a MOT operated on the intercom-
bination line not only along the vertical axe but also in
the horizontal plane. Even so we expect different behav-
iors along this directions essentially because the gravity
renders the trapping potential anisotropic. This is indeed
the case for the spatial distribution (figures 10a and 10b)
whereas the temperatures are surprisingly the same (fig-
ures 9a and 9b). We will now give few simple arguments
to physically explain this last point.
In the semiclassical approximation, the temperature is
defined as the ratio between the friction and the diffusion
term:
kBTi =
Dabsi +D
with i = x, y, z (15)
Dabs and Dspo correspond to the diffusion coefficients in-
duced by absorption and spontaneous emission events re-
spectively. The friction coefficients has been already de-
rived (equation 13):
γz = 2γx,y (16)
Indeed cooling along an axe in the x − y plane results in
the action of two counter-propagating beams four times
less coupled than the single upward laser beam. The same
argument holds for the absorption term of the diffusion
coefficient:
Dabsz = 2D
x,y (17)
The spontaneous emission contribution in the diffusion co-
efficient can be derived from the differential cross-section
dσ/dΩ of the emitting dipole [20]. With a strong biased
magnetic field along the vertical direction, this calcula-
tion is particularly simple as e+z is the only quasi resonant
state. Hence
dσ/dΩ ∝ (1 + cosφ2) (18)
φ is the angle between the vertical axe and the direction of
observation. After a straightforward integration, one finds
a contribution again two times larger along the vertical
Dspoz = 2D
x,y (19)
From those considerations, the temperature is expected
to be isotropic as observed experimentally (see figures 9a
and 9b).
In the so-called regime (I), the minimum temperature
is given by the semiclassical Doppler theory:
T = NR
s (20)
Where NR is a numerical factor which should be close
to two [12]. This solution is represented in figure 9 by
a dashed line nicely matching the experimental data for
s > 8 but with NR = 1.2. Similar results, i.e. with unex-
pected low NR values, have been found in [12]. For s ≤ 8
we observed a plateau in the final temperature slightly
higher than the low saturation theoretical prediction [5].
We cannot explain why the temperature does not decrease
further down as reported in [12]. For quantitative compar-
ison with the theory, more detailed studies in a horizontal
1D molasses are required.
4.4 Conclusions
Cooling of Strontium atoms using the intercombination
line is an efficient technique to reach the recoil temper-
ature in three dimensions by optical methods. Unfortu-
nately loading from a thermal beam cannot be done di-
rectly with a single frequency laser because of the nar-
row velocity capture range. We have shown experimentally
that more than 50% of the atoms initially in a blue MOT
on the dipole-allowed transition are recaptured in the red
MOT using a frequency-broadened spectrum. Using a sim-
ple model, we conclude that the transfer is limited by the
size of the laser beam. If the total power of the beams
at 689 nm was higher, transfer rates up to 90% could be
expected by tripling our laser beam size. The final tem-
perature in the broadband regime is found to be as low
as 2.5µK, i.e. only 5 times larger than the photon recoil
temperature. The gain in temperature by comparison to
the blue MOT (1−10mK) is appreciable. So in absence of
strong requirements on the temperature, broadband cool-
ing is very efficient and reasonably fast (less than 100ms).
The requirements for the frequency noise of the laser are
also much less stringent than for single frequency cooling.
Using a subsequent single frequency cooling stage, it
is possible to reduce the temperature down to 600 nK,
slightly above the photon recoil temperature. Analyzing
the large detuning regime, we particularly focus our stud-
ies on the comparison between vertical and horizontal di-
rections. We show how gravity indirectly influences the
horizontal parameters of the steady state MOT and find
that the trapping potential remains harmonic along all
directions, but with an anisotropy.
Gravity has a major impact on the MOT as it coun-
terbalances the laser pressure of the upward laser (making
the steady state independent of the detuning). We show
that gravity thus affects the final temperature, which re-
mains isotropic, despite different cooling dynamics along
the vertical and horizontal directions.
5 Acknowledgments
The authors wish to thank J.-C. Bernard and J.-C. Bery
for valuable technical assistances. This research is finan-
cially supported by the CNRS (Centre National de la
Recherche Scientifique) and the former BNM (Bureau Na-
tional de Métrologie) actually LNE (Laboratoire national
de métrologie et d’essais) contract N◦ 03 3 005.
T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line 7
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Blue MOT Laser
Red MOT Laser
Red MOT Laser
Magnetic field gradient
70 G/cm
1-10 G/cm
Du~k vbD
70 ms40 ms
80 ms
Fig. 1. Time sequence and cooling stages of Strontium with
the dipole-allowed transition and with the intercombination
line.
0 5 10 15 20 25 30 35 40 45
Modulation frequency (kHz)
Fig. 2. Transfer rate as a function of the modulation frequency.
The other parameters are fixed: P = 3mW, δ = −1000 kHz,
b = 1G/cm and ∆ν = 1000 kHz
http://arxiv.org/abs/quant-ph/0609111
8 T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line
0 500 1000 1500 2000 2500 3000
Frequency deviation (kHz)
Fig. 3. Transfer rate as a function of the frequency deviation
(squares). The other parameters are fixed: P = 3mW, δ =
−1000 kHz, b = 1G/cm and νm = 25 kHz. The dash and solid
line correspond to a simple model prediction (see text). The
transfer rate is limited by the frequency deviation of the broad
laser spectrum for the dash line and by the waist of the MOT
beam for the solid line.
0 500 1000 1500 2000 2500 3000
Frequency deviation (kHz)
Frequency deviation (kHz)
0 500 1000 1500 2000 2500 3000
(a)                                                     (b)
Fig. 4. Transfer rate as a function of the frequency deviation
(squares). δ = −1500 kHz and δ = −2000 kHz for (a) and (b)
respectively, the other parameters and the definitions are the
same than for figure 3.
0 500 1000 1500 2000 2500
Detuning (kHz)
Fig. 5. Transfer rate as a function of the detuning (squares).
The other parameters are fixed: P = 3mW, ∆ν = 1000 kHz,
b = 1G/cm and νm = 25 kHz. The dashed and solid lines have
the same signification than in figure 3.
0 1 2 3 4 5 6 7 8 9 10
b (G/cm)
Fig. 6. Transfer rate as a function of the magnetic gradient
(squares). The other parameters are fixed: P = 3mW, δ =
−1000 kHz, ∆ν = 1000 kHz and νm = 25 kHz. The transfer
rate is limited by the waist of the MOT beam for all values.
The dotted lines represent the case where the magnetic field
gradient do not affect the deceleration.
0 1 2 3 4 5
Beam waist (cm)
Fig. 7. Transfer rate as a function of the beam waist. The solid
lines correspond to a high saturation parameter where as the
dash line correspond to a constant power of P = 3mW. The
other parameters are fixed: δ = −1000 kHz, ∆ν = 1000 kHz
and b = 0.1G/cm.
-2000 -1500 -1000 -500
δ (kHz)
Fig. 8. Measured temperature as a function of the detuning for
a FM spectrum. The other parameters are fixed: P = 3mW,
b = 1G/cm, ∆ν = 1000 kHz and νm = 25 kHz
T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line 9
10 100 1000
-80 -60 -40 -20 0
Detuning (kHz)
Fig. 9. Measured temperature as a function of the detuning
(a) with I = 4Is or I = 15Is and as a function of the intensity
(b) with δ = −100 kHz of single frequency cooling. The circles
(respectively stars) correspond to temperature along one of
the horizontal (respectively vertical) axis. The magnetic field
gradient is b = 2.5G/cm
-700 -600 -500 -400 -300 -200 -100 0
Detuning (kHz)
-700 -600 -500 -400 -300 -200 -100 0
Detuning (kHz)
Fig. 10. Displacement (a) and rms radius (b) of the cold cloud
in single frequency cooling along the z axis (star) and in the
x−y plane (circle). The intensity per beam is I = 20Is and the
magnetic gradient b = 2.5G/cm along the strong axis in the x−
y plane. The linear displacement prediction correspond to the
plain line (graph a). In graph b, the plain curve correspond to
the rms radius prediction based on the equipartition theorem.
10 T. Chanelière, L. He, R. Kaiser and D. Wilkowski: Three dimensional cooling and trapping with a narrow line
d=-1000kHzd=-100kHz
Cloud
Quantization axe
Fig. 11. (a) Images of the cold cloud in the red MOT. The
cloud position for δ = −100 kHz coincides roughly with the
center of the MOT whereas it is shifted downward for δ =
−1000 kHz. The spatial position of the resonance correspond
dot circle. (b) Sketch representing the large detuning case. The
coupling efficiency of the MOT lasers is encoded in the size of
the empty arrow. The laser form below has maximum efficiency
whereas the one pointing downward is absent because is too
detuned. Along a horizontal axe, the lasers are less coupled
because they do not have the correct polarization. The α angle
is the angular position of an atom M with respect to O, the
center of the MOT.
	Introduction
	Experimental set-up
	Broadband loading of the red MOT
	Single frequency cooling
	Acknowledgments
