††thanks: Now at NPL, London UK††thanks: Contributed equally††thanks: Now at
UvA, Amsterdam NL

# Compact and high flux strontium atom source

C.-H. Feng† P. Robert  Université Bordeaux, CNRS, IOGS, LP2N, UMR 5298,
F-33400 Talence, France  P. Bouyer  Université Bordeaux, CNRS, IOGS, LP2N, UMR
5298, F-33400 Talence, France  B. Canuel  Université Bordeaux, CNRS, IOGS,
LP2N, UMR 5298, F-33400 Talence, France  J. Li  MajuLab, International Joint
Research Unit UMI 3654, CNRS, Université Côte d’Azur, Sorbonne Université,
National University of Singapore, Nanyang Technological University, Singapore
Nanyang Quantum Hub, School of Physical and Mathematical Sciences, Nanyang
Technological University, 21 Nanyang Link, Singapore 637371, Singapore  S. Das
MajuLab, International Joint Research Unit UMI 3654, CNRS, Université Côte
d’Azur, Sorbonne Université, National University of Singapore, Nanyang
Technological University, Singapore  Nanyang Quantum Hub, School of Physical
and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link,
Singapore 637371, Singapore  C. C. Kwong  MajuLab, International Joint
Research Unit UMI 3654, CNRS, Université Côte d’Azur, Sorbonne Université,
National University of Singapore, Nanyang Technological University, Singapore
Nanyang Quantum Hub, School of Physical and Mathematical Sciences, Nanyang
Technological University, 21 Nanyang Link, Singapore 637371, Singapore  D.
Wilkowski  MajuLab, International Joint Research Unit UMI 3654, CNRS,
Université Côte d’Azur, Sorbonne Université, National University of Singapore,
Nanyang Technological University, Singapore  Nanyang Quantum Hub, School of
Physical and Mathematical Sciences, Nanyang Technological University, 21
Nanyang Link, Singapore 637371, Singapore  Centre for Quantum Technologies,
National University of Singapore, 117543 Singapore, Singapore  M. Prevedelli
Dipartimento di Fisica e Astronomia, Università di Bologna,
Via Berti-Pichat 6/2, I-40126 Bologna, Italy  A. Bertoldi
<EMAIL_ADDRESS>Université Bordeaux, CNRS, IOGS, LP2N, UMR
5298, F-33400 Talence, France

###### Abstract

We present a novel cold strontium atom source designed for quantum sensors,
addressing key factors such as flux, brightness, temperature, complexity,
footprint, and power efficiency. We optimize the deceleration process to
collect a large velocity class of atoms emitted from an oven to achieve a
compact and low-power setup capable of generating a high atomic flux. Our
approach involves the velocity dependent transverse capture of atoms using a
two-dimensional magneto-optical trap. To enhance the atomic flux, we employ
tailored magnetic fields that minimize radial beam expansion and incorporate a
cascaded Zeeman-slowing configuration utilizing two optical frequencies. The
performance is comparable to that of conventional Zeeman slower sources, and
the scheme is applicable to other atomic species. Our results represent a
significant advancement towards the deployment of portable and, possibly,
space-based cold atom sensors.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

The successful implementation of quantum-based atom sensing in various applied
and fundamental contexts [1, 2, 3] requires a low SWaP (size, weight, and
power) package, while maintaining or improving the performance of each
subsystem within the experimental setup. Challenges arise already at the
atomic source stage when dealing with atomic species possessing low vapor
pressure, such as alkaline-earth and alkaline-earth-like atoms. The
conventional approach involves using an oven to increase the vapor pressure
necessary for cold atom experiments [4]. However, this method necessitates a
bulky Zeeman slower, coupled with a large magnetic field gradient, to
accommodate the broad velocity distribution caused by the high-temperature
oven [5].

Figure 1: Scheme of the Sr atomic source. (above) An atomic thermal beam is
effused by an oven along the $z$ axis; four stacks of permanent magnets (blue
and red rectangles) generate the linear quadrupole magnetic field, which
vanishes along the $y$ axis and is used to implement a 2D MOT. A laser beam
enters the vacuum system along the $z$ axis through a sapphire viewport, and
points against the atomic beam. (below) Horizontal magnetic field along the
$z$ axis, exhibiting a symmetric magnet configuration with respect to $z=0$
(black, solid line) and when the two stacks of magnets with $z>0$ are
displaced from $z=5$ cm to $z=6.23$ cm, and the number of neodymium bars they
adopt increased from 9 to 13 (red, dashed line). The four regions used to slow
down the atoms are indicated with circled numbers.

A potential solution to achieve the miniaturization of the cold atomic jet is
the replacement of the oven with laser-controlled atomic desorption or
ablation from a solid atomic source [6, 7]. Still, the large velocity
dispersion requires to either slow atoms or select a small velocity class of
atoms. While a Zeeman slower (ZS) is the most common configuration for slowing
a thermal atomic beam, it typically exhibits large size and high power
consumption. Additionally, it inherently generated substantial stray magnetic
fields, and its design must mitigate the deleterious effects of a direct line
of sight of the oven on the downstream ultracold atoms used in precision
measurements. These effects include a high flux of photons resulting from
blackbody emission, and a high collision rate with atoms in the thermal beam.
Measures such as deflecting the atomic beam produced by the ZS [5, 8] and
incorporating in-vacuum optical shutters [9] are commonly employed to address
these issues, but they further increase the complexity of the design.

Recent advancements have aimed to replace conventional ZSs with a more compact
alternative, leveraging the cold atomic beam generated by a two-dimensional
magneto-optical trap (2D MOT) that is transversely loaded by the oven.
Notably, this configuration allows for the manipulation of the cold atoms
within an ultra-high-vacuum chamber, shielded from the hot oven surfaces and
the atomic beam it produces, and situated along the axis of the 2D MOT where
the magnetic field is negligible. This scheme has been successfully
implemented for lithium (Li) [10], sodium (Na) [11], and strontium (Sr) [12,
13, 14]. Building upon this basic configuration, further improvements have
been achieved by realizing a compact ZS based on the magnetic field slope
created by the permanent magnets generating the 2D MOT linear quadrupole [11,
12], implementing sideband-enhanced cooling within the 2D MOT [13], and
utilizing a two-color laser beam to enhance 2D MOT loading efficiency [14].
Despite these solutions, the resulting atomic source only surpasses a standard
ZS in terms of compactness and power dissipation, not in atomic flux.

In this work, we present an improved compact atomic source of cold Sr, which
produces a state-of-the-art atomic flux, comparable to that of a standard ZS.
This result is explained by two main features: Firstly, the distance between
the oven and the initial collection region is significantly reduced by up to
one order of magnitude, as in [12, 13, 14], resulting in a 100-fold increase
in the useful solid angle for the outgoing atomic thermal beam. Secondly, we
achieve optimal deceleration of the oven-generated atomic beam across a
velocity range comparable to that of standard Zeeman slowers.

This outcome is facilitated by several factors, including the optimal
utilization of the magnetic field within the 2D MOT to enable effective atomic
deceleration despite the compact nature of the system. Furthermore, we employ
a cascade operation of two laser frequencies within the ZS to increase the
range of velocities being captured by the 2D MOT, and modify the magnetic
field configuration to shorten the trajectories of captured atoms thereby
mitigating losses caused by the radial expansion of the atomic beam.

To evaluate the expected atomic flux, we conduct numerical simulations, which
are subsequently confirmed through experimental characterization.

##  II Atomic source

Our atomic source, depicted schematically in Figure 1, bears a close
resemblance to the setups described in previous works [11, 12, 14]. The setup
comprises an effusive oven, which generates an atomic thermal beam along the
$z$ direction. A series of four stacks, each consisting of 9 neodymium
permanent magnets (Eclipse Magnetics Ltd, model N750-RB), is employed to
create a linear quadrupole vanishing along the $y$ axis at a distance of 15 cm
from the oven, and that enables the operation of a 2D MOT. The magnetic field
produced by these magnets on the propagation axis of the thermal beam ($z$
axis) is transversal, and more precisely is directed along the $x$ axis. The
magnets are spaced 10 cm apart along the $z$ axis, in a symmetric fashion with
respect to the $y$ axis.

To slow down the atomic thermal beam, laser light is applied in the $-z$
direction through a heated sapphire viewport positioned 35 cm away from the
point where the magnetic field vanishes, on the side opposite to the oven.
Additionally, a weak push beam is employed along the $y$ axis to enhance the
transfer of atoms, which have undergone pre-cooling in the 2D MOT, to a second
cell. This transfer occurs through a differential vacuum passage, and the
atoms are subsequently captured in a three-dimensional magneto-optical trap
(3D MOT) operating under ultra-high vacuum (UHV) conditions.

To date, two distinct regions have been considered for implementing a compact
ZS [12]. The first region, denoted as \raisebox{-.9pt}{1}⃝ in Figure 1,
encompasses the portion of the setup with a positive magnetic field gradient
extending from the oven to the position of the first set of magnets. This
region spans the range $-150~{}\rm{mm}<z<-50~{}\rm{mm}$ and has a length of
$\Delta l_{1}=100$ mm. The second region, denoted as \raisebox{-.9pt}{2}⃝,
comprises the portion with a negative magnetic field gradient from the first
set of magnets to the point where the magnetic field becomes zero. This region
covers the range $50~{}\rm{mm}<z<0~{}\rm{mm}$ and has a length of $\Delta
l_{2}=50$ mm. Since the magnetic field is orthogonal to the propagation
direction of the atomic beam, only half of the optical power of the Zeeman
cooling beam is effectively utilized for atom deceleration [15, 11, 12].
Achieving optimal Zeeman slowing with a maximum deceleration
$a_{\rm{max}}=v_{r}/(2\tau)$ across these regions, where $v_{r}$ represents
the recoil velocity of Sr atoms interacting with photons at 461 nm on the
5s${}^{2}\,{}^{1}$S0 – 5s5p$\,{}^{1}$P$1$ transition and $\tau$ denotes the
lifetime of the corresponding excited state, would enable to bring at rest
atoms with velocities at the oven of up to $v_{\rm{max,1}}=434$ m/s in
\raisebox{-.9pt}{1}⃝ and $v_{\rm{max,2}}=306$ m/s in \raisebox{-.9pt}{2}⃝.

In our study, we achieve an optimal use of the 2D MOT magnetic field for the
Zeeman deceleration, with an associated remarkable increase of the captured
velocity interval. In detail, we demonstrate the following findings: i) In the
region \raisebox{-.9pt}{2}⃝+\raisebox{-.9pt}{3}⃝, which encompasses a distance
of $\Delta l_{2+3}=100$ mm, we are able to effectively utilize the entire
negative gradient slope of the magnetic field, i.e. also its portion beyond
the 2D MOT. This allows us to capture atoms that pass through the 2D MOT once
and undergo trajectory inversion near or beyond the second set of magnet
stacks (Sec. II.2). ii) By employing both polarization components of the
cooling beam, we can simultaneously decelerate the adjacent low velocity
classes of atoms in both regions \raisebox{-.9pt}{1}⃝ and
\raisebox{-.9pt}{2}⃝+\raisebox{-.9pt}{3}⃝ (Sec. II.3). iii) Furthermore, we
can increase the upper capture velocity by incorporating a second frequency
component in the cooling beam, to decelerate faster atoms between the oven and
the first set of magnets so that they are then decelerated by the first
frequency component in the region \raisebox{-.9pt}{2}⃝+\raisebox{-.9pt}{3}⃝
(Sec. II.4). This additional component enables the effective atomic
deceleration from the oven to the second set of magnet stacks, covering a
distance of 20 cm. For a strontium atom beam at a temperature of 550∘C and
considering again the optimal deceleration $a_{\rm{max}}$ over all the
distance, this results in an effective capture velocity
$v_{\rm{max,1+2+3}}=613$ m/s, potentially addressing more than 80% of the
atoms emitted from the oven.

###  II.1 Numerical model

To accurately assess the atomic trajectories and estimate the 2D MOT capture
efficiency $\eta_{\mathrm{1D}}$, we employ a numerical model, which takes into
account the magnetic field generated by the permanent magnets as well as the
radiation pressure exerted by the slowing beam directed against the atomic
flux and the 2D MOT beams in close proximity to the $x=0$ position. Our model
is based on certain assumptions, consistent with those presented in [11]: (i)
Each atom is treated as a two-level system; (ii) The atomic motion is
considered as one-dimensional (1D); (iii) Interatomic collisions are
neglected; (iv) We consider classical radiation pressure and neglect the
effects of multiple scattering of photons. However, we do account for the
small gravitational pull on the atoms. Moreover, to achieve estimated atomic
fluxes that are more consistent with experimental measurements, we next
consider two factors which reduce the atomic flux captured by the 2D MOT: the
radial expansion of the thermal beam, and the radiative decay to levels
outside the cooling transition.

To account for the ballistic radial expansion of the atomic beam in the $x-y$
plane and go beyond a purely 1D treatment of the problem, we incorporate the
approach described in [16] into our model; we neglect instead the radial
heating due to spontaneous emission. This allows us to estimate the loss
parameter $\xi_{\mathrm{E}}$ associated with each trajectory, which integrated
over the velocity interval nominally trapped in the 2D MOT provides the
fraction of atoms effectively lost on the surfaces of the vacuum system
$\Xi_{\mathrm{E}}$. The value of $\xi_{\mathrm{E}}$ depends on various
factors, including the atomic parameters, the design of the thermal beam
collimating devices, and most notably the time-of-flight of each trajectory.
In our experimental setup, to reduce thermal beam divergence, we employ a
square array of about 900 Monel 400 micro-sized nozzle tubes mounted at the
aperture of the oven. These micro-tubes have an outer(inner) diameter of
0.4(0.2) mm and a length of 9 mm.

Regarding the radiative losses, we consider every atom decaying from the
$5s5p\,^{1}\textrm{P}_{1}$ to the $5s4d\,^{1}\textrm{D}_{2}$ during the
cooling process (branching ratio $\eta\simeq 1/\left(5\times 10^{4}\right)$)
as lost. This is because once it decays back to the ground level, it will be
out of resonance with the cooling beam. The loss parameter $\xi_{\mathrm{R}}$
for each trajectory ending trapped in the 2D MOT is obtained by evaluating the
probability to quit the cooling transition before reaching for the first time
zero velocity. We remark that in this way we slightly underestimated the
radiative losses for the trajectories reaching a turning point at $z>0$. An
overall radiative loss parameter $\Xi_{\mathrm{R}}$ is computed over the
velocity interval being trapped in the 2D MOT.

To obtain the effective capture efficiency $\eta_{\mathrm{E,R}}$ for the 2D
MOT, we integrate over the range of captured velocities. We take into account
the oven temperature, which is set at 550∘C (corresponding to an approximate
root mean square velocity of $v_{\textrm{rms}}\sim 490$ m/s), to weight the
relative population according to the initial Maxwell-Boltzmann distribution.
Finally, the losses due to radial expansion and radiative decay are
incorporated into the model by considering for each velocity class ending in
the 2D MOT a reduced capture efficiency equal to
$(1-\xi_{\mathrm{E}})\cdot(1-\xi_{\mathrm{R}})$.

Our primary goal is to optimize the flux of atoms captured by the 2D MOT. To
achieve this, we employ a fourth-order Runge-Kutta method to calculate the
trajectories of the atoms starting from the oven output at $z=-15$ cm. The
calculation continues until one of the following three conditions is met: the
atoms hit the sapphire viewport at $z=35$ cm; they return to the oven at
$z=-15$ cm; they are captured by the 2D MOT at $z=0$ cm. We identify the
capture by the 2D MOT when the trajectory becomes stationary within the region
where the 2D MOT cooling beams overlap.

Figure 2: Capture efficiency of the compact ZS. (above) The atomic flux
fraction captured by the 2D MOT is calculated versus the detuning $\Delta_{1}$
of the laser beam used to slow the atoms. The laser intensity is set to
1.5$\,I_{\textrm{sat}}$ on a circular cross section with diameter 9.6 mm. The
contribution due to the action of the $\sigma^{-}$ ($\sigma^{+}$) polarization
of the ZS are highlighted in red (blue). The captured flux is obtained by
considering a purely 1D problem (solid line), and by adding the combined
losses due to the atomic radial expansion and radiative decay (dashed line);
the latter curve alone is depicted more in detail in the central panel. The
atomic flux for $\Delta_{1}=-14.0\,\Gamma/2\pi$ is equal to 27.6% for the 1D
treatment (open black square), reduced to 1.88% (full black circle) when the
contributions of the expansion and radiative losses are taken into account; as
indicated by the vertical arrow, the latter value is increased to 2.92% by the
asymmetric magnet configuration (red full square). (below) Displacement of the
2D MOT along the $z$ axis induced by the radiation pressure imposed on the
trapped atoms by the slowing beam.

As a reference benchmark, in the absence of any Zeeman slowing beam,
approximately 0.41% of the atoms leaving the oven would be captured by the 2D
MOT in the 1D treatment. When accounting for radial losses and considering the
38 mm internal diameter of the vacuum pipe used for the thermal atomic beam,
the capture fraction is reduced to $\eta_{\rm{MOT}}=0.11$%. This reference
value gives us an indication of the capture efficiency achievable without any
additional measures to slow down the atomic flux. The 2D MOT beams have a
$1/e^{2}$ waist of 12 mm, an optical power of 80 mW, and a detuning of
$-1.15\,\Gamma/2\pi$ from the cooling transition ($\Gamma=2\pi\times 30.5$ MHz
is the transition natural linewidth), which yields a capture velocity of
$\approx$70 m/s.

The laser beam directed towards the oven along the $z$ direction can be
polarized in two main different configurations, taking into account that the
magnetic field along the atomic trajectories is primarily directed along $x$.
These configurations are as follows: a linear polarization along $x$, which
induces $\pi$ transitions in the atoms (e.g.,
$\ket{{}^{1}\textrm{S}_{0}}\rightarrow\ket{{}^{1}\textrm{P}_{1},m_{F}=0}$); a
linear polarization along $y$, which induces $\sigma^{\pm}$ transitions in the
atoms (e.g.,
$\ket{{}^{1}\textrm{S}_{0}}\rightarrow\ket{{}^{1}\textrm{P}_{1},m_{F}=\pm
1}$). In our setup, we utilize only the second configuration, where the linear
polarization is along $y$, inducing $\sigma^{\pm}$ transitions. This
arrangement allows for resonance conditions between light and atoms over an
extended spatial region due to the varying Zeeman effect associated with a
magnetic field slope and the decelerating atomic particles. Efficient slowing
of the atoms can be achieved in two ways: they can scatter photons with a
$\sigma^{-}$ polarization along the increasing slope of $B_{x}(z)$ in region
\raisebox{-.9pt}{1}⃝, or photons with a $\sigma^{+}$ polarization along the
decreasing slope of $B_{x}(z)$ in region \raisebox{-.9pt}{2}⃝. Until now,
these two conditions have been considered mutually exclusive in previous works
[11, 12]. As a consequence, in transverse and permanent magnet-based Zeeman
slowers (ZSs) [17], where the magnetic field is orthogonal to the atomic beam,
half of the optical power used for atom deceleration is wasted [15]. However,
we demonstrate here how to fully utilize the available optical power by
addressing different velocity intervals for the atoms effused by the oven with
each polarization component.

The slowing beam is linearly polarized along the $y$ axis, so that its power
is equally split over the $\sigma^{\pm}$ polarization components relative to
the magnetic field. The local acceleration it produces on atoms along the $z$
axis, for a saturation parameter $s$ and red detuning $\Delta_{1}<0$ with
respect to the transition 1S0 – 1P1 at 461 nm, is

$\displaystyle a(z)=-v_{r}\,\frac{\Gamma}{2}\sum_{m_{\rm{F}}=\pm
1}\frac{s}{1+s+4\left(\Delta_{1,\rm{eff}}(z)/\Gamma\right)^{2}},$ (1)

where $\Delta_{1,\rm{eff}}$ depends on $m_{\rm{F}}$, on the atomic velocity
$v_{\rm{z}}$ along $z$, and on the local magnetic field $B_{\rm{x}}(z)$:

$\Delta_{1,\rm{eff}}(z)=\Delta_{1}-kv_{z}+\frac{\mu_{\rm{B}}\,g_{\rm{F}}\,m_{\rm{F}}\,B_{\rm{x}}(z)}{\hbar},$
(2)

where $k$ is the laser wavenumber, $\mu_{B}$ is the Bohr’s magneton and
$g_{F}$ the Landé factor. For the analysis in this section we set the
following simulation parameters, in accordance with what can be implemented in
our laboratory: the saturation parameter for the slowing beam is set to 1.5
for each of the two polarizations, in line with the total optical power
available of 90 mW and the cross section of 0.72 cm2 of the slowing beam (the
saturation intensity of the
$\ket{{}^{1}\textrm{S}_{0}}\rightarrow\ket{{}^{1}\textrm{P}_{1}}$ transition
is $I_{\textrm{sat}}=42$ mW/cm2); the number of magnets in each of the four
stacks is set to 9. Increasing such value will give access to larger magnetic
field slopes, and a potentially more efficient slowing; however, it will be
not possible to exploit them, given the limited photon scattering rate and the
frequency detuning allowed by our optical setup.

###  II.2 One frequency ZS

The atomic flux fraction captured by the 2D MOT is plotted as a function of
$\Delta_{1}$ in the upper panel of Figure 2. The solid line represents the
result of the 1D problem. The dashed line takes into account losses due to the
radial ballistic expansion of the atomic beam and to the radiative decay,
incorporating the loss parameters $\xi_{\mathrm{E}}$ and $\xi_{\mathrm{R}}$
associated with each trajectory. To each trajectory captured by the 2D MOT in
the 1D simulation is associated a reduced capture efficiency equal to
$\left(1-\xi_{\mathrm{E}}\right)\cdot\left(1-\xi_{\mathrm{R}}\right)$; the
result is a more comprehensive view of the captured atomic fraction.

The comprehension of the features detailed in the rest of the section required
the detailed examination of the phase-space plots obtained at various values
of $\Delta_{1}$. This approach allowed us to validate observations made in
prior experimental setups [11, 12, 14], as well as highlight some of the
innovations introduced in our work, as summarized in the configuration in Fig.
3. The graph in Fig. 2 exhibits two main contributions highlighted with shaded
colours: the blue one corresponds to the exploitation of the magnetic field
slope in regions \raisebox{-.9pt}{2}⃝ and \raisebox{-.9pt}{3}⃝ – and in some
cases \raisebox{-.9pt}{4}⃝ – for atomic slowing. It ranges from $\Delta_{1}=0$
to $\Delta_{1}=-14.1\,\Gamma/2\pi$. In the 1D case, the atomic flux in this
region linearly increases from 1.0% to 15.3% with a slope of
1.014%/$\Gamma/2\pi$, except for three intervals. First, in the range
$-2.5\,\Gamma/2\pi\leq\Delta_{1}\leq-0.3\,\Gamma/2\pi$, the operation of the
2D MOT is destabilized. Second, in the range
$-10.0\,\Gamma/2\pi\leq\Delta_{1}\leq-5.8\,\Gamma/2\pi$, approximately 1.9%
more atoms are slowed down in region \raisebox{-.9pt}{4}⃝ by a combination of
Zeeman slowing and radiation pressure from the off-resonance beam, enabling
their capture by the 2D MOT from the positive $y$ direction. Thirdly, for
$\Delta_{1}\leq-12.4\,\Gamma/2\pi$, the blue contribution is combined with the
second main contribution, highlighted in red.

Figure 3: (above) Atomic trajectories from the oven output at $z=-15$ cm, in
the presence of a ZS beam with $s=1.5$ for each of the two circular
polarizations and $\Delta_{1}=-14.0\,\Gamma/2\pi$. The acceleration due to the
ZS, calculated with Eq. 1, and 2D MOT beams is indicated in color code on the
phase space diagram. The targeted trapping axis of the 2D MOT at $z$=0 is
marked with a dashed cross. The trajectories of atoms captured by the 2D MOT
and decelerated by the $\sigma^{-}$ ($\sigma^{+}$) polarization of the ZS are
traced in red (blue). (below) Probability density versus velocity of atoms
emitted by the oven at 550∘C (blue), and when both the ZS and the 2D MOT are
operated (red). The atomic flux captured by the 2D MOT in the 1D case is equal
to 27.6%, indicated by the narrow peak at $v_{\mathrm{z}}=0$ m/s, reduced to
2.79% when radial losses are taken into account, and further to 1.88% because
of radiative losses. Inset: velocity $v_{1}$ at the end of each trajectory of
the ZS as a function of the atomic velocity $v_{0}$ at the oven output.

The red contribution extends from $\Delta_{1}=-12.4\,\Gamma/2\pi$ to nearly
$\Delta_{1}=-15.4\,\Gamma/2\pi$ and corresponds to the utilization of region
\raisebox{-.9pt}{1}⃝ for decelerating the atoms. In the 1D case, the atomic
flux provided by the red contribution linearly increases from 9.6% to 15.6%
with a slope of 2.00%/$\Gamma/2\pi$. This contribution is added to the blue
contribution for $\Delta_{1}\leq 14.1\,\Gamma/2\pi$. The slope of the red area
is approximately twice that of the blue area: this difference arises because
increasing the detuning leads only to a shift upwards of the velocity window
captured by the 2D MOT for the blue contribution, whereas it also extends the
velocity window towards higher velocities for the red contribution.

The abrupt efficiency drop at high negative detunings in each of the two
contributions has different causes. For the blue contribution, it is due to
the atoms’ trajectories reaching the viewport used to shine the cooling beam.
For the red contribution, it is because the atoms’ trajectories are too fast
to be captured when they transit through the 2D MOT region.

The lower panel of Fig. 2 shows the vertical displacement of the 2D MOT
trapping axis, induced by the radiation pressure exerted by the slowing beam.
As anticipated, the displacement approaches zero for larger values of
$\Delta_{1}$. This condition enhances the stability of the experiment since
the alignment of the 2D MOT axis must be precise with respect to the small
aperture that maintains the differential vacuum between the atomic source
chamber and the 3D MOT chamber.

The highest atomic flux is achieved at $\Delta_{1}=-14.1\,\Gamma/2\pi$,
corresponding to the high detuning limit, where both polarization components
of the Zeeman slower contribute to decelerating the atoms. We investigate the
$\Delta_{1}=-14.0\,\Gamma/2\pi$ configuration, which maintains a stability
margin of a few MHz, just before an abrupt drop in the 1D atom capture
efficiency occurs, as illustrated by the empty square in Fig. 2. The resulting
atomic trajectories are depicted in the upper panel of Fig. 3, where all four
previously defined regions are employed to decelerate specific velocity
intervals of the atomic beam emitted by the oven.

In region \raisebox{-.9pt}{1}⃝, the $\sigma^{-}$ light is employed to slow
down atoms with initial velocities $v_{0}\lesssim 230$ m/s until they reach
$z=-5$ cm. Subsequently, these atoms continue at low speed, which gradually
decreases due to off-resonant photon scattering, until they are captured by
the 2D MOT when approaching from the negative $z$ direction.

In regions \raisebox{-.9pt}{2}⃝ and \raisebox{-.9pt}{3}⃝, the $\sigma^{+}$
light decelerates atoms with initial velocities in the range of approximately
240 m/s to 320 m/s. These atoms pass through the 2D MOT region once without
being captured, enter region \raisebox{-.9pt}{4}⃝ where they are further
slowed down through off-resonant photon scattering from the ZS beam.
Consequently, they are reflected and ultimately captured by the 2D MOT when
approaching from the positive $z$ direction.

Figure 4: The trajectories extending far on the positive $z$ axis in the
configuration of Fig. 3 are brought to an inversion close to the position of
the right stacks of magnets, which use 13 instead of 9 elements and are
shifted on the $z$ axis so as to maintain the 2D MOT along the $y$ axis. The
atomic effective flux captured by the 2D MOT when radial and radiative losses
are taken into account raises from 1.88% of the configuration in Fig. 3 to
2.92%; this 55% improvement is depicted in the middle panel of Fig. 2 by the
black arrow going from the full black circle to the red full square.

In the lower panel of Fig. 3, the Maxwell-Boltzmann probability density of the
atomic beam emitted by the oven at 550∘C is depicted by the blue distribution,
whereas the red curve shows how it is modified by the ZS and the 2D MOT. The
inset illustrates the effect on each specific velocity, where the final
velocity $v_{1}$ of each trajectory is plotted against the initial velocity
$v_{0}$ at the oven. The positions considered for determining $v_{1}$ are
$z=z_{\textrm{MOT}}$ for atoms captured by the 2D MOT, $z=-15$ cm for atoms
redirected back towards the oven, and $z=35$ cm for atoms reaching the
sapphire viewport.

Atoms with velocities in the range of 50 m/s $\lesssim v_{0}\lesssim 320$ m/s
are captured by the 2D MOT, which corresponds to 27.6% of the initial
distribution (open square in upper panel of Fig. 2). However, when considering
the radial expansion of the atomic beam, this capture fraction is reduced to
2.79% (open circle in upper panel of Fig. 2), and further down to 1.88% when
radiative losses are also taken into account (full circle in upper and middle
panels of Fig. 2). The expansion losses are particularly pronounced for long
trajectories that are decelerated in regions \raisebox{-.9pt}{2}⃝ and
\raisebox{-.9pt}{3}⃝. These trajectories spend a long interval at their
turning point, located not far from the sapphire viewport at $z=35$ cm, before
falling back to the 2D MOT, for a total duration of $\sim$28 ms. For
comparison, short red trajectories passing only through regions
\raisebox{-.9pt}{1}⃝ and \raisebox{-.9pt}{2}⃝ take only $\sim$5 ms to end up
captured by the 2D MOT.

Figure 5: (above) Atomic trajectories from the oven in the presence of a
cooling beam with two frequency components, one with a detuning of
$-14.0\,\Gamma/2\pi$ and $s$=1.5 for each of the two circular polarizations,
the other at $=-30.0\,\Gamma/2\pi$ and with $s$=2.1. The frequency of the
second laser is chosen to decelerate the faster atoms from the oven and up to
the second stack of magnets all along, as defined in Eq. 3; the relative
trajectories are indicated in orange. (below) Probability density versus
velocity of atoms emitted by the oven at 550∘C (blue), and after the action of
the double frequency ZS and the 2D MOT is taken into account (red). The atomic
flux captured by the 2D MOT is equal to 54.12%, indicated by the narrow peak
at $v_{\mathrm{z}}=0$ m/s, reduced to 8.03% when radial losses are taken into
account and further down to 4.48% because of radiative losses; the performance
of Fig. 4 is improved by 53%. Inset: velocity $v_{1}$ at the end of each
trajectory of the ZS as a function of the atomic velocity $v_{0}$ at the oven
output.

###  II.3 Modified magnetic field

To address this issue and reduce the duration of these long trajectories while
preserving other features, a modification is made to the configuration
depicted in Fig. 3. Specifically, the number of magnets in the two stacks at
positive $z$ is increased from 9 to 13, and their position is shifted from
$z=5$ cm to $z=6.23$ cm. This adjustment increases the magnetic field
“barrier” on the positive side of the 2D MOT, while maintaining the vanishing
position of the magnetic field and its gradient, as shown by the dashed line
in the lower panel of Fig. 1. As a result, the previously long trajectories
captured by the 2D MOT are shortened to a duration of $\sim$7 ms (see Fig. 4).
The atomic flux captured by the 2D MOT, corrected for radial and radiative
losses, increases by 55% and from 1.88% it reaches 2.92%.

###  II.4 Two frequencies ZS

In order to efficiently decelerate higher velocities with respect to the best
configuration defined in the previous paragraph, one possible improvement
consists in increasing the number of magnets in each stack proportionally,
while adjusting their positions to maintain the center of the 2D MOT. This
adjustment will result in larger magnetic field slopes. However, even with
this modification, the effective deceleration distance remains limited to
$\Delta l\approx 10$ cm both in region \raisebox{-.9pt}{1}⃝ and in region
$\raisebox{0.5pt}{\raisebox{-.9pt}{2}⃝}+\raisebox{0.5pt}{\raisebox{-.9pt}{3}⃝}$.
Additionally, due to the limited optical intensity available, it may not be
possible to achieve the required scattering rate to keep the faster incoming
atoms in resonance with the light all along the magnetic field slope.

A more intriguing option is to bring the atoms within the velocity interval
above that which generates the blue trajectories in Fig. 4 into the capture
range of the ZS defined there. To achieve this, a second optical frequency can
be added to the Zeeman cooling beam, with a detuning $\Delta_{2}$ chosen as

$\Delta_{2}\simeq\Delta_{1}-\frac{2\,\mu_{\rm{B}}B_{x,\textrm{max}}}{\hbar}-\Gamma/\pi\>,$
(3)

where $B_{x,\textrm{max}}$ represents the maximum value of the vertical
magnetic field along the $z$ axis near the two stacks of magnets located
between the oven and the 2D MOT. The additional term $-\Gamma/\pi$ is included
to avoid resonantly addressing the same velocities with both optical
frequencies simultaneously at $z=-5$ cm.

For this second frequency component, we consider a saturation parameter
s2=2.1, in line with the optical power available in the experiment. This setup
enables the continuous deceleration of the fastest atoms captured by the 2D
MOT, from the oven all the way to the position of the second stack of magnets:
initially, in region \raisebox{-.9pt}{1}⃝, they scatter the $\sigma^{-}$ light
component of the second optical frequency; then, in regions
\raisebox{-.9pt}{2}⃝ and \raisebox{-.9pt}{3}⃝, the $\sigma^{+}$ light
component of the first optical frequency of the ZS beam further decelerates
these atoms. As depicted in Fig. 5, this extended deceleration distance spans
over more than 20 cm. Consequently, atomic velocities up to 440 m/s can be
efficiently trapped, considering the specific experimental configuration under
consideration.

Using our numerical tool, we compared the 2D MOT capture efficiency with this
cascaded scheme and that obtained by a standard Zeeman slower, with a magnetic
field slope extending over 50 to 100 cm. The result is a comparable
performance, but in a much more compact and power efficient configuration.
Moreover, our design is better suited for the implementation of a radial
optical molasse at the oven’s output, given the almost homogeneous magnetic
field there and the absence of magnetic sources obstructing the optical
access.

##  III Experimental results

Figure 6: The different laser frequencies needed in the experiment are obtained by using 5 AOMs to shift the master laser light, which is set to $-11.7\,\Gamma/2\pi$ from the 1S0–1P1 transition used for the spectroscopy (blue horizontal line). Table 1:  Capture efficiency ($\eta_{\mathrm{1D}}$, $\eta_{\mathrm{E,R}}$), losses ($\Xi_{\mathrm{E}}$, $\Xi_{\mathrm{R}}$), and gain with respect to the 2D MOT only ($\eta_{\mathrm{E,R}}/\eta_{\mathrm{MOT}}$) for different experimental configurations of the Sr atomic source, as listed in the three leftmost columns; the capture efficiency when only the 2D MOT is operative is given in the first line of $\eta_{\mathrm{E,R}}$, and is $\eta_{\mathrm{MOT}}\equiv 0.11\%$ . The different lines relates to: (i) No Zeeman cooling beams active, and hence to the performance of the 2D MOT only; (ii) Single Zeeman cooling beam at the optimal detuning, with symmetric magnetic field configuration (see Fig. 3); (iii) Same as previous, but with asymmetric magnetic field - as in all the following configurations (see Fig. 4); (iv) two frequencies in the cooling beam, with the second one as implemented experimentally; (v) two frequencies in the cooling beam, both at their optimal value. The uncertainty on the gain factor obtained by the simulation is obtained by considering a $\pm 5\%$ variation on the cooling beam diameter, when the other parameters are maintained fixed; for the gain obtain experimentally the uncertainty results form the fluorescence image analysis. All the values refer to the experimental setup considered in the study, and to an atomic source at 550∘C. configuration | efficiency and losses | $\eta_{\mathrm{E,R}}/\eta_{\mathrm{MOT}}$
---|---|---
Fig. | $\Delta_{1}(\Gamma/2\pi)$ | $\Delta_{2}(\Gamma/2\pi)$ |  B field | $\eta_{\mathrm{1D}}(\%)$ | $\Xi_{\mathrm{E}}(\%)$ | $\Xi_{\mathrm{R}}(\%)$ | $\eta_{\mathrm{E,R}}(\%)$ | simulation | experiment
7a | na | na | sym. | 0.41 | 73 | 0 | $0.11$ | 1 | 1
7b, 3 | $-14$ | na | sym. | 27.6 | 89.9 | 32.6 | 1.88 | 17.1(5) | 14.2(16)
7c, 4 | $-14$ | na | asym. | 27.5 | 83.4 | 35.9 | 2.92 | 26.5(7) | 21.3(26)
7d | $-14$ | $-24.5$ | asym. | 39.38 | 84.71 | 39.70 | 3.63 | 33.0(8) | 35.4(28)
5 | $-14$ | $-30.0$ | asym. | 54.12 | 85.16 | 44.20 | 4.48 | 40.7(10) | na
Figure 7: Fluorescence produced by the 2D MOT, observed along the $y$ axis, in
four different experimental conditions: no Zeeman slowing beams along the $-z$
direction (a); a cooling beam with frequency set at $-14\,\Gamma/2\pi$ from
the transition at 461 nm is shined on the atoms along the $-z$ direction, with
a symmetric (b; magnetic field profile in Fig. 1, black solid line) and
asymmetric (c; magnetic field profile in Fig. 1, red dashed line) magnetic
field configuration; a cooling beam containing two frequencies at
$-14\,\Gamma/2\pi$ and $-24.5\,\Gamma/2\pi$, respectively, is shined on the
atoms along the $-z$ direction, in the asymmetric field configuration (d). The
exposure interval for the pictures is 5 ms; the 2D MOT is slightly misaligned
with respect to the 2 mm hole used to maintain the differential vacuum between
the two chambers and to transfer the cold atoms jet generated by the 2D MOT.

To experimentally validate the ZS schemes developed with our numerical tool,
we implemented a laser system generating all the required optical frequencies.
This was achieved using 5 acousto-optic modulators (AOMs) as shown in Fig. 6.
We used a master laser (Toptica, mod. DLC DL pro) locked at
$-11.7\,\Gamma/2\pi$ from the 1S0 – 1P1 cooling transition at 461 nm, and we
optically shifted it to generate the cooling and trapping light components.
The first Zeeman cooling beam (Z1) was set at the optimal detuning
$\Delta_{1}=-14.0\,\Gamma/2\pi$, whereas the second one (Z2) was shifted to
$\Delta_{2}=-24.5\,\Gamma/2\pi$, i.e. which corresponds to a point where the
AOM exhibits good diffraction efficiency. To set the frequency at the optimal
value of $-30.0\,\Gamma/2\pi$ as prescribed by Eq. 3, a different AOM
configuration would be required. The 2D MOT beams, and the two beams used for
Zeeman cooling were further amplified by optically injecting high-power laser
diodes (Nichia, mod. NDB4916). The two Zeeman beams are combined using a 50/50
beam splitter, resulting in a 50% optical power loss. They are then directed
towards the oven along the $z$ axis through the sapphire viewport. The beams
are expanded using a common telescope to achieve the following parameters: Z1
an optical power of 45 mW on each polarization, an effective surface of 0.72
cm2 and an intensity of 1.5 times Isat; Z2 an optical power of 75 mW on each
polarization, an effective surface of 0.84 cm2 and an intensity parameter of
2.1 times Isat. We allocated the laser with the higher output power to the
far-detuned frequency component, because it addresses faster atoms and must
therefore provide a higher on resonance scattering rate.

To assess the efficiency of the ZS in various configurations, we measured the
2D MOT’s fluorescence using a CCD camera oriented along the $y$ direction,
which is typically reserved for the push beam used to increase the atomic
transfer through the differential vacuum pipe.

Panel (a) of Fig 7 shows the 2D MOT fluorescence when the ZS is inactive. The
subsequent two panels display the signal when the Zeeman cooling beam at
$\Delta_{1}=-14\,\Gamma/2\pi$ is directed at the thermal atom beam effused by
the oven, corresponding to the symmetric (b, relative to Fig. 3) and
asymmetric (c, relative to Fig. 4) magnetic field configurations,
respectively. Finally, panel (d) shows the 2D MOT fluorescence when the second
slowing beam Z2, with a detuning $\Delta_{2}=-24.5\,\Gamma/2\pi$ and $s=2.1$,
is incorporated into the setup described in Fig. 4. The brightness of the 2D
MOT, when the Zeeman cooling beam is present, allow us to fit the 10
mm$\times$10 mm central region of the related images with the sum of a 2D
Gaussian functions for the signal originating from the trapped atoms, and a
flat offset for the broad fluorescence produced by the MOT and Zeeman beams.
However, this approach was unsuccessful when only the 2D MOT was operating
(panel (a) of Fig. 7), because the signal from trapped atoms was too faint
with respect to the background. Instead, we used the picture from panel (b),
from which we subtracted the signal from the trapped atoms, to isolate the 2M
MOT fluorescence in panel (a). We could thus estimate that the ZS enhances the
2D MOT by a factor of 14.2(16) in the symmetric configuration, 21.3(26) in the
asymmetric one, and 35.4(28) in the two cooling beam frequency configuration,
as reported in Table 1. The best enhancement factor we measured is at least a
factor 4 better than that reported for similar compact Sr sources [12, 14],
and approximately 3 times better than the result in [11] for Na, where,
however, the lower oven temperature results in higher ZS performance.

The two cooling frequency configuration involves a cascaded deceleration
operation for the high atomic speeds at the oven (orange trajectories), in
combination with deceleration performed on lower initial speeds over 10 cm
using only one frequency, either in region \raisebox{-.9pt}{1}⃝ (red
trajectories) or
$\raisebox{0.5pt}{\raisebox{-.9pt}{2}⃝}+\raisebox{0.5pt}{\raisebox{-.9pt}{3}⃝}$
(blue trajectories). We confirmed that the dual-frequency ZS system indeed
relies on the presence of both ZS beams to function effectively. If Z1 is
removed from the dual-frequency configuration, Z2 operates at a frequency
where the capture efficiency for a single beam is negligible. The modified
setup also retains the advantage of shortening the atomic trajectories through
the implementation of a tailored, asymmetric magnetic field. The measured
values align reasonably with those obtained by the numerical simulation, as
listed in the second-to-last column of Table 1. Several factors could account
for the observed differences: i) Uncertainty in the intensity parameter used
in the experiment for the cooling beam, which arises from the relatively low
quality of the cooling beam’s mode profile; ii) The assumption made in the
numerical simulation that radial losses are determined by the expansion from a
single microtube, and not from an array, which leads to an overestimation of
the effective atomic flux; iii) Heating due to spontaneous emission during the
deceleration process.

We also observed a significant reduction in atomic flux at
$\Delta_{1}=-15\,\Gamma/2\pi$, when shifting the working frequency of the AOM
controlling the cooling beam. However, this measurement was strongly affected
by the AOM operating at the limit of its range. A more precise and
comprehensive study of the atomic flux would require a cooling laser capable
of being frequency shifted over a wide range without a significant impact on
the optical beam’s intensity. Such a laser system would offer greater
flexibility for exploring various detuning values, optimizing the cooling
process, and, most importantly, validating all the results obtained through
our numerical analysis.

Boosting the available optical power could significantly increase the cross
section of the Zeeman cooling beam, enhancing the 2D MOT’s capture efficiency.
In a set of experiments, we directed the two frequency components along the
$z$ axis without overlapping them on the 50/50 beam splitter, instead imposing
a small relative angle estimated to be a few tens of milliradians. However,
this approach was unsuccessful, as precise alignment of the two beams proved
critical to achieve the cascaded operation of Z1 and Z2.

To create an optimal two-frequency ZS, an enticing solution is to generate the
second frequency component superimposed with the first using the serrodyne
technique [18, 19]. Given the lack of wideband EOMs in the blue range, the
solution could be implemented in a laser system using a fundamental laser at
922 nm, where the saw-tooth modulation would be applied, before doubling in a
second harmonic generation (SHG) system to obtain the cooling light at 461 nm.

To finally verify the atomic source efficiency, we put in place the push beam
and observed the formation of a 3D MOT in the second vacuum chamber. To this
aim, we first precisely aligned the 2D MOT free axis to have it passing
through the differential vacuum passage, by acting on precision screws
controlling the magnet stacks’ $x$-$z$ positions. The trapping region is
located at a distance of 35 cm from the differential vacuum pipe, which
results in a recapture efficiency of approximately 10% for the 3D MOT due to
the divergence of the pre-cooled atomic beam generated by the 2D MOT. The
atomic flux we measured for the 3D MOT is exceeding $5\times 10^{9}$ atoms/s
for 88Sr when using the cascade frequency configuration for the ZS. . This
flux is reduced in accordance with the efficiency gains measured in Table 1
when the atomic source configuration is varied.

##  IV Conclusions

We have implemented several critical improvements in an already compact Sr
atomic source, which combines a 2D MOT transversely loaded by a compact ZS.
Experimental results are corroborated by a numerical simulation, first
developed to reproduce the measured data, and later adopted to evaluate the
impact of modified setup configurations and optimize the atomic source
efficiency. We make an optimal use of the magnetic field required to operate
the MOT, by exploiting both light polarizations to slow down the atoms effused
by the oven, by implementing Zeeman slowing all along from the oven to the MOT
position and beyond, and by modifying the magnetic field profile to mitigate
the issue represented by the thermal radial expansion of the thermal beam
produced by the oven. We also operated a novel cascade frequency configuration
for the ZS, to extend its capture velocity.

To further increase the atomic flux captured by the 2D MOT one could adopt
different solutions. One possibility would be to exploit the almost vanishing
magnetic field at the oven’s output to implement collimation beams, thus
increasing the brightness and collimation of the atomic beam and reducing
radial expansion losses. Radiative losses could also be mitigated. Recent
research of rapid pumping schemes to efficiently return Sr atoms to the
cooling transition at 461 nm predicts that atoms falling in the 1D2 level can
be pumped back in a very short time to the 1S0–1P1 transition using a repumper
at 448 nm towards the 5$s$8$p$1P1 level or at 475 nm towards the 4$d$5$p$1P1
level [20].

The atomic source design, which combines high flux, compactness, and low
consumption, could be adapted straightforwardly to other atomic species where
a ZS is required, such as Ca, Cd, and Yb, among others. The atomic source
design can also be integrated in a continuous scheme [21, 22], contributing to
ongoing research in continuous matter wave interferometry [23] and continuous
wave superradiant lasing [24].

## Acknowledgments

We thank P. Teulat for the realization of several mechanical components of the
experimental setup. P.R. acknowledges funding from the “Direction Générale de
l’Armement” (DGA) for his doctoral fellowship. This work was partly supported
by the “Agence Nationale pour la Recherche” (grant EOSBECMR #
ANR-18-CE91-0003-01, grant ALCALINF # ANR-16-CE30-0002-01, and grant MIGA #
ANR-11-EQPX-0028), the European Union (EU) (FET-Open project CRYST3, Horizon
Europe Qu-Test), IdEx Bordeaux - LAPHIA # ANR-10-IDEX-03-02 (grant OE-TWC)
Horizon 2020 QuantERA ERA-NET (grant TAIOL # ANR-18-QUAN-00L5-02), and Conseil
Régional d’Aquitaine (grant USOFF).

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