Creation of Ultracold Sr-2 Molecules in the Electronic Ground State

Abstract

We report on the creation of ultracold Sr-84(2) molecules in the electronic ground state. The molecules are formed from atom pairs on sites of an optical lattice using stimulated Raman adiabatic passage (STIRAP). We achieve a transfer efficiency of 30% and obtain 4 X 10(4) molecules with full control over the external and internal quantum state. STIRAP is performed near the narrow S-1(0)-P-3(1) intercombination transition, using a vibrational level of the 1(0(u)(+)) potential as an intermediate state. In preparation of our molecule association scheme, we have determined the binding energies of the last vibrational levels of the 1(0(u)(+)), 1(1(u)) excited-state and the X 1 Sigma(+)(g) ground-state potentials. Our work overcomes the previous limitation of STIRAP schemes to systems with magnetic Feshbach resonances, thereby establishing a route that is applicable to many systems beyond alkali-metal dimers.

Full text

Page 1

Creation of ultracold Sr2molecules in the electronic ground state
Simon Stellmer,1Benjamin Pasquiou,1Rudolf Grimm,1,2and Florian Schreck1
1Institut f¨ ur Quantenoptik und Quanteninformation (IQOQI),
¨Osterreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
2Institut f¨ ur Experimentalphysik und Zentrum f¨ ur Quantenphysik, Universit¨ at Innsbruck, 6020 Innsbruck, Austria
(Dated: May 22, 2012)
We report on the creation of ultracold
molecules are formed from atom pairs on sites of an optical lattice using stimulated Raman adia-
batic passage (STIRAP). We achieve a transfer efficiency of 30% and obtain 4×104molecules with
full control over the external and internal quantum state. STIRAP is performed near the narrow
1S0-3P1 intercombination transition, using a vibrational level of the 0u potential as intermediate
state. In preparation of our molecule association scheme, we have determined the binding energies
of the last vibrational levels of the 0u, 1u excited-state, and the1Σ+
work overcomes the previous limitation of STIRAP schemes to systems with Feshbach resonances,
thereby establishing a route that is applicable to many systems beyond bi-alkalis.
84Sr2 molecules in the electronic ground state. The
g ground-state potentials. Our
PACS numbers: 03.75.Nt, 33.20.-t, 34.50.Rk, 37.10.Pq
The creation of ultracold molecular gases has made
rapid progress over the last years.
structure of molecules combined with low translational
energy enables precision measurements of fundamental
constants, realizations of novel quantum phases, and ap-
plications for quantum computation [1]. A very success-
ful route to large samples of ultracold molecules with
complete control over the internal and external quantum
state is association of molecules from ultracold atoms.
Early experiments used magnetic Feshbach resonances
to form weakly bound bi-alkali molecules, some of which
have even been cooled to quantum degeneracy [2]. Stim-
ulated Raman adiabatic passage (STIRAP) [3] has en-
abled the coherent transfer of these Feshbach molecules
into the vibrational ground state [4–6]. In particular,
heteronuclear molecules in the vibrational ground state
have received a lot of attention, because they possess
a strong electric dipole moment, leading to anisotropic,
long-range dipole-dipole interactions, which will enable
studies of fascinating many-body physics [7].
are underway to create samples of completely state-
controlled molecules beyond bi-alkalis [8–10], which will
widen the range of applications that can be reached ex-
perimentally.
The rich internal
Efforts
So far, the key step in the efficient creation of ul-
tracold molecules has been molecule association using
magnetic Feshbach resonances.
tion technique cannot be used to form dimers of alkaline-
earth atoms or ytterbium, because of the lack of mag-
netic Feshbach resonances in these nonmagnetic species.
An example is Sr2, which has been proposed as a sen-
sitive and model-independent probe for time variations
of the proton-to-electron mass ratio [11–13]. Another
class of molecules for which magnetoassociation is dif-
ficult, are dimers containing an alkali atom and a non-
magnetic atom, since in these cases magnetic Feshbach
resonances are extremely narrow [14, 15]. This difficulty
occurs in current experimental efforts to create LiYb,
This magnetoassocia-
RbYb, or RbSr molecules [8–10]. Other molecule cre-
ation techniques that are suitable for dimers contain-
ing nonmagnetic atoms have been proposed, for exam-
ple molecule formation by STIRAP from a Bose-Einstein
condensate (BEC) [16–19] or two-color photoassociation
(PA) of atom pairs in a Mott insulator with two atoms
per site [20].
In this Letter, we show that ultracold Sr2molecules in
the electronic ground state can be efficiently formed, de-
spite the lack of a magnetic Feshbach resonance. Instead
of magnetoassociation, we combine ideas from [16–20]
and use optical transitions to transform pairs of atoms
into molecules by STIRAP. The molecule conversion effi-
ciency is enhanced by preparing pairs of atoms in a Mott
insulator on the sites of an optical lattice [21, 22]. We
achieve an efficiency of 30% and create samples of 4×104
84Sr2molecules. We perform PA spectroscopy to iden-
tify the states and optical transitions used for molecule
creation.
STIRAP coherently transfers an initial two-atom state
|a? into a molecule |m? by optical transitions; see Fig. 1.
In our case, the initial state |a? consists of two84Sr atoms
occupying the ground state of an optical lattice well. The
final state |m? is a Sr2 molecule in the second to last
bound state of the1Σ+
gground-state molecular potential
without rotational angular momentum. The molecules
have a binding energy of 645MHz and are also confined
to the ground state of the lattice well. States |a? and |m?
are coupled by lasers fields L1 and L2, respectively, to
state |e?, the third-to-last bound state of the metastable
0upotential, dissociating to1S0-3P1.
We use the isotope84Sr for molecule creation, since
it is ideally suited for the creation of a BEC [25, 26],
and formation of a Mott insulator. The binding ener-
gies of the states involved in our STIRAP scheme are
only known for the isotope88Sr [27, 28] and can be esti-
mated for84Sr by mass-scaling [29, 30]. An essential task
in preparation of molecule creation is therefore to spec-
arXiv:1205.4505v1 [cond-mat.quant-gas] 21 May 2012

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2
energy (cm-1)
internuclear separation (a0)
ν = -2
binding energy/h (GHz)
internuclear separation (a0)
binding energy/h (GHz)
0
2000
4000
6000
8000
10000
12000
14000
16000
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0 10 20
40
1S0 + 1S0
1Σg
50150
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
|e 
|m 
|a 
30
|m 
|e 
100
ν = -1
internuclear separation (a0)
50150100
-1.4
ν = -1
ν = -2
ν = -3
ν = -4
h∆
Γ
1S0 + 3P1 0u
0u
L1, Ω1
L2, Ω2
+
1Σg
+
Figure 1:
of84Sr2 involved in STIRAP. The initial state |a?, an atom
pair in the ground state of an optical lattice well, and the
final molecular state |m?, are coupled by laser fields L1 and
L2 to the excited state |e? with Rabi frequencies Ω1 and Ω2,
respectively. The parameter ∆ is the detuning of L1 from
the
insets show the last bound states of the molecular potentials
and the wavefunctions of states |m? and |e?. For compari-
son, the wavefunction of atomic state |a? (not shown) has its
classical turning point at a radius of 800a0, where a0 is the
Bohr radius. The potentials are taken from [23, 24] and the
wavefunctions are calculated using the WKB approximation.
The energies of states |m? and |e? are not to scale in the main
figure.
(Color online) Molecular potentials and states
1S0-3P1 transition and Γ is the decay rate of |e?. The
troscopically determine the binding energies of relevant
84Sr2states.
We perform PA spectroscopy [31] on a84Sr BEC, pro-
duced similarly to our previous work [25]. The BEC is
confined in an oblate crossed-beam optical dipole trap
with oscillation frequencies of 55Hz in the horizontal
plane and 180Hz in the vertical direction, based on two
5-W laser sources operating at 1064nm with a linewidth
of 3nm. Laser fields L1and L2, which are used for spec-
troscopy and STIRAP, have linewidths of ∼ 2kHz and
are referenced with an accuracy better than 1kHz to the
1S0-3P1intercombination line, which has a natural width
of Γ/2π = 7.4kHz. To achieve the coherence of the laser
fields required for STIRAP, L1and L2are derived from
Table I: Energy of the last vibrational levels of the 0u, 1u,
and1Σ+
bound states of the potentials, starting with ν = −1 for the
first level below the free atom threshold. l is the angular
momentum quantum number.
g potentials. The quantum number ν labels the last
ν0u
1u
1Σ+
g (l = 0)
(MHz)
-13.7162(2)
-644.7372(2)
1Σ+
g (l = 2)
(MHz)
-519.6177(5)
(MHz)
-0.32(1)
-23.01(1)
-228.38(1)
-1288.29(1)
(MHz)
-351.45(2) -1
-2
-3
-4
the same master laser by means of acousto-optical modu-
lators. These laser beams are copropagating in the same
spatial mode with a waist of 100(25)µm on the atoms
and are linearly polarized parallel to a guiding magnetic
field of 120mG.
One-color PA spectroscopy is used to determine the
binding energies of the last four vibrational levels of the
0upotential. To record the loss spectrum, we illuminate
the BEC for 100ms with L1 for different detunings ∆
with respect to the1S0-3P1transition. The binding en-
ergies derived from these measurements are presented in
Tab. I.
We then use two-color PA spectroscopy to determine
the binding energies of the last vibrational levels of
the1Σ+
gground-state potential. The loss spectra are
recorded in the same manner as for one-color PA spec-
troscopy, just with the additional presence of L2. Fig-
ure 2 shows two spectra, for which L2 is on resonance
with the transition from state |m? to state |e?. The dif-
ference between the spectra is the intensity of L2. The
spectrum shown in Fig. 2(a) was recorded at high in-
tensity and displays an Autler-Townes splitting [32]. In
this situation, the coupling of states |m? and |e? by L2
leads to a doublet of dressed states, which is probed by
L1. The data of Fig. 2(b) was recorded at low intensity
and shows a narrow dark resonance at the center of the
PA line. Here, a superposition of states |a? and |m? is
formed, for which excitation by L1and L2destructively
interfere. The binding energies of the ground-state vi-
brational levels derived from measurements of dark res-
onances are given in Tab. I.
The Rabi frequencies Ω1,2 of our coupling lasers are
determined by fitting a three-mode model to the spec-
tra [33]; see Fig. 2. The free-bound Rabi frequency Ω1
scales with intensity I1 of L1 and atom density ρ as
Ω1 ∝
Ω2∝√I2depends only on the intensity I2of L2. We
obtain Ω1/(√I1
a peak density of ρ0 = 4 × 1014cm−3and Ω2/√I2 =
2π×50(15)kHz/?W/cm2, where the error is dominated
To enhance molecule formation, we create a Mott in-
sulator by loading the BEC into an optical lattice [34].
The local density increase on a lattice site leads to an
√I1√ρ [17]. The bound-bound Rabi frequency
?ρ/ρ0) = 2π×10(4)kHz/?W/cm2at
by uncertainty in the laser beam intensity.

Page 3

3
(a)
(b)
(∆ - 228.37 MHz) (kHz)
-100 -50050100
0.0
0.5
1.0
normalized atom number
-200-1000 100200
0.0
0.5
1.0
Figure 2: (Color online) Two-color PA spectra near state |e?
for two intensities of L2. (a) For high intensity (20W/cm2)
the spectrum shows an Autler-Townes splitting. (b) For low
intensity (80mW/cm2) a narrow dark resonance is visible.
For both spectra, the sample was illuminated by L1for 100ms
with an intensity of 7mW/cm2at varying detuning ∆ from
the1S0-3P1transition. The lines are fits according to a three-
mode model [33].
increased free-bound Rabi frequency Ω1. Furthermore,
molecules are localized on lattice sites and thereby pro-
tected from inelastic collisions with each other. The lat-
tice is formed by three nearly orthogonal retroreflected
laser beams with waists of 100µm on the atoms, derived
from an 18-W single-mode laser operating at a wave-
length of λ = 532nm. Converting the BEC into a Mott
insulator is done by increasing the lattice depth during
100ms to 16.5Erec, where Erec= ?2k2/2m is the recoil
energy with k = 2π/λ and m the mass of a strontium
atom. After lattice ramp-up, the 1064-nm dipole trap is
ramped off in 100ms.
To estimate the number of doubly occupied sites,
which are the sites relevant for molecule formation, we
analyze the decay of the lattice gas under different con-
ditions. After loading a BEC with less than ∼ 3 × 105
atoms into the lattice, the lifetime of the lattice gas is
10(1)s. For higher BEC atom numbers, we observe an
additional, much faster initial decay on a timescale of
50ms, which removes a fraction of the atoms. We at-
tribute this loss to three-body decay of triply occupied
sites, which are formed only if the BEC peak density is
high enough. To obtain a high number of doubly occu-
pied sites, we load a large BEC of 1.5 × 106atoms into
the lattice. After the initial decay, 6×105atoms remain.
By inducing PA loss using L1, we can show that half of
x 1/10
L2
L1
C
(a)
x 1/10
(b)
(c)
time (µs)
atom number (103)
intensity
6080100 120 140 160 180 20040200
0
1
2
3
0
20
40
60
0.0
0.5
1.0
Figure 3: (Color online) Time evolution of STIRAP transfer
from atom pairs to Sr2 molecules and back. (a) Intensities
of L1, L2, and cleaning laser C, normalized to one. (b),(c)
Atom number evolution. For these measurements, L1 and
L2 are abruptly switched off at a given point in time and
the atom number is recorded on an absorption image after
10ms free expansion. Note the scaling applied to data taken
during the first 100µs (triangles). The starting point for the
time evolution shown in (b) is a Mott insulator, whereas the
starting point for (c) is a sample for which 80% of the atoms
occupy lattice sites in pairs.
these atoms occupy sites in pairs.
We are now ready to convert the atom pairs on
doubly occupied sites into molecules by STIRAP. This
method relies on a counterintuitive pulse sequence [3],
during which L2 is pulsed on before L1. During this
sequence, the atoms populate the dark state |Ψ? =
(Ω1|m? + Ω2|a?)/(Ω2
the time-dependent Rabi frequencies of the two cou-
pling laser fields as defined in [3], which can reach up to
Ωmax
1
∼ 2π×150kHz and Ωmax
case [35]. Initially the atoms are in state |a?, which is the
dark state after L2is suddenly switched on, but L1kept
off. During the pulse sequence, which takes T = 100µs,
L1is ramped on and L2off; see Fig. 3(a). This adiabat-
ically evolves the dark state into |m? if Ωmax
condition, which we fulfill. To end the pulse sequence,
L1is suddenly switched off. During the whole process,
state |e? is only weakly populated, which avoids loss of
atoms by spontaneous emission if Ωmax
dition is easily fulfilled with a narrow transition as the
one used here. The STIRAP transfer does not lead to
molecules in excited lattice bands, since T is long enough
for the band structure to be spectrally resolved.
We now characterize the molecule creation process.
To detect molecules, we dissociate them using a time-
mirrored pulse sequence and take absorption images of
1+ Ω2
2)1/2, where Ω1 and Ω2 are
2
= 2π×170(10)kHz in our
1,2 ? 1/T, a
1,2 ? Γ. This con-

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4
x (µm)
(a)
y
linear density (107 m-1)
-200 -1000100 200
0
2
4
6
8
10
x
(b)
Figure 4:
repulsively bound pairs. (a) Average of 20 absorption images
recorded 10ms after release of the atoms from the lattice. (b)
Integral of the distribution along y.
(Color online) Quasi-momentum distribution of
the resulting atoms. The atom number evolution dur-
ing molecule formation and dissociation is shown in
Fig. 3(b). After the molecule formation pulse sequence,
2 × 105atoms remain, which we selectively remove by
a pulse of light resonant to the1S0-1P1 atomic tran-
sition, out of resonance with any molecular transition
[36]; see “cleaning” laser C in Fig. 3(a). The recovery of
2 × 104atoms by the time-mirrored pulse sequence con-
firms that molecules have been formed. Further evidence
that molecules are the origin of recovered atoms is that
80% of these atoms occupy lattice sites in pairs. Quanti-
tatively this is shown by removing atom pairs using PA
and measuring the loss of atoms. Qualitatively we illus-
trate this fact by creating and detecting one-dimensional
repulsively bound pairs along the x-direction [37]. The
pairs were created by ramping the x-direction lattice
beam to a value of 10Erec before ramping all lattice
beams off, which dissociates the pairs into atoms with
opposite momenta along x. Figure 4 shows the charac-
teristic momentum space distribution of these pairs.
To estimate the STIRAP efficiency and subsequently
the number of molecules, we perform another round of
molecule formation and dissociation on such a sample
of atoms with large fraction of doubly occupied sites;
see Fig. 3(c). We recover f = 9% of the atoms, which
corresponds to a single-pass efficiency of√f = 30%. The
largest sample of atoms created by dissociating molecules
contains Na = 2.5 × 104atoms, which corresponds to
Nm= Na/(2√f) = 4 × 104molecules.
We measure the lifetime of molecules in the lattice,
by varying the hold time between molecule creation and
dissociation. We obtain ∼ 60µs, with little variation
in dependence on lattice depth. Executing the cleaning
laser pulse after the hold time instead of before, does not
change the lifetime. This time is surprisingly short and
can neither be explained by scattering of lattice photons
nor by tunneling of atoms or molecules confined to the
lowest band of the lattice and subsequent inelastic colli-
sions. By band mapping, we observe that 3×104of the
initial 6×105atoms are excited to the second band dur-
ing the STIRAP pulse sequence, and more atoms have
possibly been excited to even higher bands. We spec-
ulate that these atoms, which move easily through the
lattice, collide inelastically with the molecules, resulting
in the observed short molecule lifetime. The cleaning
laser pulse is not able to push these atoms out of the re-
gion of the molecules fast enough to avoid the loss. The
short lifetime can explain the 30%-limit of the molecule
conversion efficiency. Without the loss, the high Rabi
frequencies and the good coherence of the coupling lasers
should result in a conversion efficiency close to 100%.
The excitation of atoms to higher bands cannot be ex-
plained by off-resonant excitation of atoms by L1or L2.
Incoherent light of the diode lasers on resonance with
the atomic transition might be the reason for the excita-
tion. Further investigation of the excitation mechanism
is needed in order to circumvent it.
In conclusion, we have demonstrated that it is possi-
ble to use STIRAP to coherently create Sr2 molecules
from atom pairs on the sites of an optical lattice. The
advantage of this technique compared to the traditional
magnetoassociation approach is that it can be used for
systems that do not possess a suitable magnetic Fesh-
bach resonance. This new approach might be essential
for the formation of alkali/alkaline-earth molecules.
We thank Manfred Mark for helpful discussions. We
gratefully acknowledge support from the Austrian Min-
istry of Science and Research (BMWF) and the Aus-
trian Science Fund (FWF) through a START grant un-
der Project No. Y507-N20. As member of the project
iSense, we also acknowledge financial support of the
Future and Emerging Technologies (FET) programme
within the Seventh Framework Programme for Research
of the European Commission, under FET-Open grant
No. 250072.

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5
SUPPLEMENTAL MATERIAL
One-color PA spectroscopy
We use one-color PA spectroscopy to measure the
binding energies of the last vibrational levels of the 0u
and 1upotentials. A pure BEC of 106atoms is illumi-
nated by L1 for 100ms, and the number of remaining
atoms is measured by absorption imaging. The detun-
ing ∆ of L1with respect to the atomic1S0-3P1transition
is changed for consecutive experimental runs, generating
loss spectra as the ones shown in Fig. 5. To compensate
the difference in line strength for the different vibrational
levels, we adjust the intensity of L1 to obtain a large,
but not saturated signal. Intensities used for the last
four levels of the 0u potential and the last level of the
1upotential are 0.005, 1.8, 3.7, 260, and 260 mW/cm2,
respectively.
The lineshapes of the resonances are very symmetric
and can be described by a simple Lorentz profile. This is
in contrast to similar measurements performed in a ther-
mal gas of88Sr, where the inhomogeneous broadening of
the line had to be considered [27]. The uncertainty in
the position of the resonances amounts to 1, 2, 2, 6, and
20 kHz for the five levels mentioned above, and to 1 kHz
for the atomic transition. Systematic errors arise from
light shifts and mean-field shifts; see Tab. II. We measure
these shifts for the ν = −3 state and find that they are
on the kHz scale; see Fig. 7 (a) and (b). The strength
of the magnetic field does not influence the resonance
position, since we use π-polarized light for spectroscopy
and drive a magnetic field insensitive mJ= 0 to m?
transition. We do not measure light shifts and mean-
field shifts for the other vibrational states, and therefore
give a rather conservative total uncertainty of 10kHz in
Tab. I of the main text.
J= 0
∆ (MHz)
remaining fraction
1.0
0.8
0.6
0.4
0.2
0.0
-22.5-23.0-228.0-228.5 0.0 -0.5
Hallo Hanna!
Figure 5:
L1, and the fraction of remaining atoms is recorded (black
circles). Various loss features are observed in dependence of
the frequency of L1, the ones shown here correspond to the
ν = −3, ν = −2, and ν = −1 vibrational levels of the 0u po-
tential, counting from the left. The feature centered around
zero detuning originates from loss on the atomic transition.
One-color PA spectra. A BEC is illuminated by
Table II: Uncertainties and systematic shifts for the binding
energy of the ν = −3 state of the 0u potential.
error sourceuncertainty
kHz
1
2
< 1
2
position free-free
position free-bound
master laser
total
error source shift
Hz/mW
0.022(1)
170(10)
shiftshift
kHz
1.3
0.6
Hz/(mW/cm2)
300
0.003
mW
0.06
4
L1 Stark
vert DT Stark
mHz/atom
2.4(5)
atoms
≤ 106
mean field
≤ 2.5
Two-color PA spectroscopy
We use two-color PA spectroscopy to determine the
binding energies of the last vibrational levels of the
ground-state potential. The two laser fields L1and L2
couple three states in a Λ-scheme. Laser field L1couples
two BEC atoms to a molecular state |e?? in the 0upoten-
tial. Laser field L2couples |e?? to a molecular state |m??
in the ground-state potential. The frequency difference
between L1and L2gives the binding energy of |m??.
As an example, we present the determination of the
binding energy of the last bound state ν = −1 of the
1Σ+
gground-state potential. Here we choose the excited
state ν = −3 as the intermediate state |e??. A pure BEC
of 6 × 105atoms is prepared in an optical dipole trap
(DT) and illuminated by L1and L2for 100ms. After-
wards, the number of remaining atoms is measured using
absorption imaging. The frequency of L1is always set
to be resonant with the free-bound transition, and the
intensity is chosen such that almost all atoms are ejected
from the trap. To search for the molecular state |m??, the
frequency of L2is increased from one experimental run
to the next. If L2is on resonance with the bound-bound
transition, an atom-molecule dark state is created. Light
from L1is no longer absorbed and no photoassociative
loss occurs. We are searching for this dark resonance as
a signature of state |m?? while we change the frequency
of L2.
Without initial knowledge about the rough position
of the resonance, many 100MHz need to be scanned;
see Fig. 6 (a).Once the resonance is found, we can
reduce the intensity of L2, which reduces the width of
the resonance considerably. Consecutive scans with de-
creasing intensity allow for a very precise determination
of the resonance position, shown in Fig. 6 (a) through
(f). The data can be fitted nicely with a Lorentz pro-
file. For the smallest L2intensity used here, we obtain a
linewidth of 1.0 kHz and an uncertainty in the position

Page 6

6
100 MHz
0
1
2
0
2
0
2
3
1
atom number (105)
0
1
1
0.5
1.5
0.4
0.2
0.6
4
6
(a)
ν (L1) - ν (L2) (MHz)
0 -100 -50-150 -14 -13-15-13.5 -14
-13.7-13.75-13.716-13.718 -13.715-13.72-13.71
2 MHz 500 kHz
50 kHz 10 kHz2 kHz
(b) (c)
(d) (e)(f)
Figure 6: Search for a dark resonance. A BEC is illuminated
by laser fields L1 and L2, where the frequency of L1 is kept
fixed as the frequency of L2 is varied. The number of atoms
remaining after a 100 ms pulse is recorded (black circles).
The resonance feature is described by a Lorentz profile (red
line). The intensity of L2 is continuously reduced from (a)
through (f), assuming values of 110, 13, 3.2, 0.084, 0.038, and
0.013mW/cm2, respectively, thereby drastically reducing the
linewidth of the resonance.
of the resonance of 5 Hz. We find the lineshapes to be
very symmetric, which is in contrast to [28], where the
measurements were performed in a thermal gas.
In the search presented, we had no initial knowledge
about the position of the resonance. The search can be
simplified if the rough position is calculated beforehand.
We can estimate the energy of the last bound state from
the known s-wave scattering length of84Sr [28, 38] using
a simple analytical model [29]. Using general properties
of van der Waals potentials [30], this estimation is ex-
tended to more deeply bound states. This estimation
guides our search for two other bound states of the1Σ+
potential: the ν = −2 state for l = 0 and the ν = −1
state for l = 2.
We will now investigate the systematic errors and un-
certainties of our measurement. The dominant system-
atic errors are caused by light shifts induced by the light
fields involved: L1and L2, as well as the horizontal and
vertical dipole trap beams. For a systematic analysis,
we vary each parameter independently, and record the
resonance position. We fit the data with a straight line
and extrapolate to zero; examples are shown in Fig. 7
g
(a)(b)
(d) (c)
L1 power (mW)
shift of resonance (kHz)
012345
0
20
40
60
80
100
120
atom number (106)
012
0
shift of resonance (kHz)
1
2
3
L1 power (µW)
0100 200 300 400 500
shift of resonance (kHz)
0
2
4
6
8
L2 power (µW)
0100200300400 500
-1
-2
-3
-4
-5
0
shift of resonance (kHz)
Figure 7:
(a) and (b) show the shift of the excited-state ν = −3 level
with varying power of L1 and varying atom number, whereas
panels (c) and (d) show shifts of the ground-state ν = −2
level.
Systematic shifts of the binding energies. Panels
(c) and (d).
Uncertainties in the laser frequencies are small, since
L1and L2are generated by injection-locking two slave
lasers with light from the same master laser. The light
is frequency-shifted using acousto-optical modulators,
where the radiofrequency source is referenced to the
global positioning system (GPS). Frequency drifts of
the master laser do not affect the measurement.
scans taken at the maximum resolution, there is an un-
explained jitter of at most 80Hz between scans taken
some time apart. This is the dominant contribution to
the overall uncertainty. In Tab. III, we present a compi-
lation of all systematic and statistical errors for the case
of the ν = −2 state, which is the one used for STIRAP.
Note that our measurements are a factor 1000 more pre-
cise than previously reported data for the88Sr isotope
[28].
For
Table III:
ν = −2(l = 0) state of the1Σ+
effect
Error budget for the binding energy of the
g potential.
shift
Hz
µW
16.4(4)
-10.2(4)
400(100)
shift
Hz
mW/cm2
+2.8(1) 4.5 × 10−4
-1.7(1)
50(10)
90(40)
shift uncert.
Hz
3
-6
120
300
mWHz
<1
<1
30
100
5
1
80
130
L1 Stark
L2 Stark
hor DT Stark
vert DT Stark 1.1(3) × 104
fitting
RF
shot-to-shot
total
6 × 10−4
320
27
420

Page 7

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and ΩMI
1 . We use
1
from
1 , which we approximate by ΩMI
?ISTIRAP
1
∼
0/ρBEC
01
/IPA
1 ΩBEC
1
. Here ρBEC
0
= 4 ×
= 5 ×
1014cm−3is the peak density of the BEC, ρMI
0
√2 is dis-