# Observation of confinement-induced resonances in a 3D lattice

Deborah Capecchi  Institut für Experimentalphysik und Zentrum für
Quantenphysik, Universität Innsbruck, 6020 Innsbruck, Austria.  Camilo
Cantillano  Institut für Experimentalphysik und Zentrum für Quantenphysik,
Universität Innsbruck, 6020 Innsbruck, Austria.  Manfred J. Mark  Institut für
Experimentalphysik und Zentrum für Quantenphysik, Universität Innsbruck, 6020
Innsbruck, Austria.  Institut für Quantenoptik und Quanteninformation,
Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria.  Florian
Meinert  5.Physikalisches Institut and Center for Integrated Quantum Science
and Technology, Universität Stuttgart, 70569 Stuttgart, Germany.  Andreas
Schindewolf  Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany.
Munich Center for Quantum Science and Technology, 80799 München, Germany.
Manuele Landini  Institut für Experimentalphysik und Zentrum für
Quantenphysik, Universität Innsbruck, 6020 Innsbruck, Austria.  Alejandro
Saenz  Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin,
Germany.  Fabio Revuelta  Grupo de Sistemas Complejos, Escuela Técnica
Superior de Ingeniería Alimentaria y de Biosistemas, Universidad Politécnica
de Madrid, 28040 Madrid, Spain.  Hanns-Christoph Nägerl  Institut für
Experimentalphysik und Zentrum für Quantenphysik, Universität Innsbruck, 6020
Innsbruck, Austria.

###### Abstract

We report on the observation of confinement-induced resonances for strong
three-dimensional (3D) confinement in a lattice potential. Starting from a
Mott-insulator state with predominantly single-site occupancy, we detect loss
and heating features at specific values for the confinement length and the 3D
scattering length. Two independent models, based on the coupling between the
center-of-mass and the relative motion of the particles as mediated by the
lattice, predict the resonance positions to a good approximation, suggesting a
universal behavior. Our results extend confinement-induced resonances to any
dimensionality and open up an alternative method for interaction tuning and
controlled molecule formation under strong 3D confinement.

Cold atoms are a well-established platform for quantum simulation [1] and
quantum computation [2]. They allow for the realization of strongly-correlated
quantum many-body phases [3] and promise the investigation of dynamical
processes in correlated quantum matter [4] with exquisite parameter control.
The cold-atom setting comes with two advantageous and enabling features:
External potentials that can be flexibly created by optical means, e.g.
optical lattices [5, 6], and interatomic interactions that can be tuned almost
at will, via magnetic Feshbach resonances (FRs) [7]. These features have been
instrumental to a multitude of spectacular results, e.g. the observation of
the superfluid-to-Mott-insulator quantum phase transition [8, 9] and the Bose-
Einstein-condensation-to-Bardeen-Cooper-Schrieffer (BEC-to-BCS) crossover
[10]. Besides interaction tuning, FRs serve as an entrance door to the world
of ultracold molecules [11, 12], with great promises for ultracold chemistry
[13], precision measurement [14], and dipolar many-body physics [15]. For a FR
to occur, the free scattering state of two atoms needs to be brought to
degeneracy with a molecular bound state. In most applications, a variable
external magnetic field, as a result of the Zeeman effect, tunes the energy
difference, inducing the resonance.

Tight external confinement as provided by, e. g., a lattice can also lead to a
dramatic modification of the atoms’ scattering properties [16, 17, 18, 19, 20,
21, 22, 23]. Confinement-induced resonances (CIRs) occur if the typical length
scale of the confining potential, e. g., the harmonic-oscillator length
$a_{\mathrm{h}}$, and the 3D s-wave scattering length $a_{\mathrm{s}}$ assume
similar values. CIRs come in two flavors: Elastic CIRs (ECIRs) emerge if the
effective 1D or 2D interaction strength diverges at a particular ratio of
these two length scales [16, 17, 18]. They are especially suited for tuning
the interaction strength. Inelastic CIRs (ICIRs) [23, 24, 25] emerge due to
the coupling between the relative motion (RM) and the center-of-mass motion
(CM) [26, 20, 27, 28]. At ICIRs, which form an infinite series, atoms can be
transferred to the RM bound state after the scattering process, making such
resonances more prone to losses, but also enabling molecule formation [29],
similar to magnetic FRs. For ECIRs and ICIRs to occur, scattering properties
can be tuned by variation of either $a_{\mathrm{s}}$ or $a_{\mathrm{h}}$.

Figure 1: (a) Scattering length $a_{\mathrm{s}}$ for Cs
$(F,m_{F})\\!=\\!(3,3)$ as a function of $B$ (solid black line) [30], with two
narrow FRs at $47.8\,\mathrm{G}$ and $53.8\,\mathrm{G}$. The purple (yellow)
shaded area corresponds to the interval used in experiment E1 (E2). The dashed
lines labeled $P_{0}$, $P_{1}$, $P_{2}$, and $P_{3}$ mark the positions of the
resonant features observed in E1. (b) Schematic representation of the states
involved in the ICIRs: trap (upper) and bound (lower) state according to each
model of the atom pair [23]. For model M1 (M2) the trapping potential is given
by the lattice (sextic) potential. Initially, both atoms are in the same
lattice site in a trap state (separated orange circles), while the CM (blue
area) can be spread (M1) or localized (M2). ICIRs occur if the anharmonicity
of the trapping potential couples the trap state to the least-bound state
(connected orange circles) with some center-of-mass excitations. (c) Energy
diagram for varying $a_{\mathrm{s}}$ in units of $a_{\mathrm{h}}$ for
isotropic trapping. The thick and thin solid lines correspond to
$\psi^{\mathrm{RM}}_{\mathrm{t}}\phi^{\mathrm{CM}}_{\mathrm{iso}}(0,0,0)$ and
$\psi^{\mathrm{RM}}_{\mathrm{t}}\phi^{\mathrm{CM}}_{\mathrm{iso}}(2,0,0)$,
respectively, whereas the dashed, dotted, and dashed-dotted curves represent
$\psi_{b}^{\mathrm{RM}}\phi^{\mathrm{CM}}_{\mathrm{iso}}(4,0,0)$,
$\psi_{b}^{\mathrm{RM}}\phi^{\mathrm{CM}}_{\mathrm{iso}}(2,2,0)$, and
$\psi_{b}^{\mathrm{RM}}\phi^{\mathrm{CM}}_{\mathrm{iso}}(6,0,0)$,
respectively. Intersections causing the ICIRs of the present work are
indicated by crosses.

Experimentally, so far, an ECIR in 1D has been used to put samples of
ultracold bosons deeply into the Tonks-Girardeau regime and to access the
super-Tonks-Girardeau (sTG) state [31]. For spin-mixtures of fermions, the sTG
state has been realized in the limit of two particles by means of an ECIR
[32]. Tuning on the repulsive side of an ECIR has been used to map out three-
body correlations in the strongly interacting 1D regime [33], while in 2D the
existence of an ECIR for attractive interactions has been confirmed [34].
Recently, the interplay of contact and dipolar interactions in conjunction
with ECIRs has allowed the creation of scar states [35]. The observation of
loss features in the 1D-to-2D crossover regime [36] was originally interpreted
as stemming from the ECIR and later recognized to originate from ICIRs [37,
23]. ICIRs have been observed in mixed dimensionality [38] and molecules have
been coherently produced on ICIRs in 1D [29].

While the work discussed so far was limited to 1D or 2D systems, in this
Letter we experimentally detect ICIRs in 0D as a result of strong 3D
confinement and compare our data to theory. Interestingly, we detect the
resonances starting from a Mott-insulator state with predominantly single-site
occupancy, a situation for which one would expect at first glance a strong
suppression of atom-atom scattering. We employ Cs atoms in the hyperfine
$(F,m_{F})\\!=\\!(3,3)$ ground state, which features wide tunability for
$a_{s}$ due to a combination of broad and narrow magnetic FRs [30], as shown
in Fig. 1(a). We take data with two different experimental setups, one (E1)
[39] for tuning $a_{s}$ on a broad background, and the other (E2) [40] for
tuning near a comparatively narrow FR.

Our theoretical methods build on previous models employed for the prediction
of ICIR positions [23, 24]. We study our data by two complementary approaches,
named in the following M1 and M2 [41]. Both descriptions provide an
approximate solution of the two-body problem in the presence of the 3D-lattice
potential.

The potential energy of two atoms is given by
$V(\textbf{r}_{1},\textbf{r}_{2})=V_{\mathrm{RM}}(\textbf{r})+V_{\mathrm{CM}}(\textbf{R})+W(\textbf{r},\textbf{R})$,
where $\textbf{r}_{1}$ and $\textbf{r}_{2}$ are the positions of atoms 1 and
2, while r and R are the relative and center-of-mass coordinates,
respectively. For M1, $V_{\mathrm{RM}}$ is Taylor expanded to order $r^{6}$
(sextic potential), and the eigenenergies of the RM are found perturbatively
employing the solution for two particles in a harmonic trap interacting via a
pseudo-potential [42]. The CM is solved exactly, considering the form of the
lattice potential. In particular, M1 accounts for the full lattice bandwidth,
without imposing any restriction on the quasimomentum, resulting in an allowed
range for each crossing between the levels associated with an ICIR rather than
in a fixed condition [41]. Finally, the coupling term $W$, which originates
from the anharmonicity, is treated perturbatively, leading to avoided
crossings. For M2, the trap-potential terms are expanded in the form of sextic
potentials and the solution is found numerically, with the interaction between
the atoms being described by an ab-initio potential [41]. The sextic Taylor
expansion employed in both models for the RM implies that the two particles
occupy the same well. M1 (as opposed to M2) can describe the motion of the
atoms moving as pairs through the lattice, as illustrated in Fig. 1(b) 111The
numerical simulations for M1 have been carried out using Cesium atoms, while
those for M2 have been conducted with Lithium, whose validity has been
previously assessed by comparison with a Cesium experiment [23]..

In both models the RM eigenfunction can be classified by two families of
states of different character. On the one hand, there are molecular-bound
states, the most loosely (least) one being denoted as $\psi^{\mathrm{RM}}_{\rm
b}$. On the other hand, there are higher-lying unbound states strongly
influenced by the external trapping potential. These states energetically lie
in the dissociation continuum in the absence of the external potential, and
become discrete trap states in a few-well potential or bands in a periodic
lattice. The lowest-lying one is denoted as $\psi^{\mathrm{RM}}_{\mathrm{t}}$.
The function $\phi^{\mathrm{CM}}(n_{x},n_{y},n_{z})$ describes the solutions
for the CM of the atom pair in the three directions with $n_{i}=0,1,2,...$
[41]. In the isotropic lattice the excited bound states are typically three-
fold degenerate because of the factorization of the potential along the three
directions. In the following, we will denote the degenerate states with the
compact notation $\phi^{\mathrm{CM}}_{\mathrm{iso}}(n_{j},n_{k},n_{l})$
implying any permutation of $\\{n_{x},n_{y},n_{z}\\}$. A schematic drawing of
the relevant states for the reported measurements can be found in Fig. 1(c),
where we plot the energy as a function of $a_{\mathrm{s}}$ in units of
$a_{\mathrm{h}}=\sqrt{\hbar/(m\omega)}$, with $\omega$ the harmonic trapping
frequency of the lattice in the isotropic case. The results are similar for
both models. Additional crossings that violate parity conservation and cannot
give rise to RM-CM coupling are not included in Fig. 1(c).

The experimental procedures are similar for both setups, and we give details
here only for E1. We start from a BEC of $\sim$3$\times 10^{4}$ 133Cs atoms
[44, 45] with a condensate fraction of 70 to 80%, levitated against gravity by
a magnetic field gradient $\left|\nabla{B}\right|\approx 31$ G/cm and held in
a crossed optical dipole trap (XODT) at $\lambda=1064.5$ nm by trapping beams
along the horizontal $x$ and $y$ directions. The trap depth of the XODT is
$V_{\textrm{trap}}\\!=\\!k_{\text{B}}\times 0.29$ $\mathrm{\mu K}$ with
trapping frequencies $\nu_{x}=10.6(1.2)$ Hz, $\nu_{y}=16.0(1.7)$ Hz, and
$\nu_{z}=20.8(1.5)$ Hz, where $z$ denotes the vertical direction along
gravity.

Figure 2: Observation of ICIRs in a 3D lattice. (a) Cloud radius $\sigma$ as a
function of $B$ for E1 at $V_{x,y,z}=20.0(3)$ $E_{\mathrm{R}}$. The data are
fit by multi-peak Gaussian (solid line) to guide the eye. (b) Atom number $N$
as a function of $B$ for E2, showing four loss features $P^{\prime}_{j}$, $j=$
1A, 1B, 2, 3. The solid line is also a multi-peak Gaussian fit. The errors
bars in both plots reflect the standard deviation from typically 3
experimental runs.

We load the BEC into a 3D cubic lattice with a depth of up to
$V_{x,y,z}\\!=\\!20.0(3)$ $E_{\mathrm{R}}$ along all three directions, where
$E_{\mathrm{R}}$ is the Cs recoil energy, by ramping up the power in the three
retro-reflected lattice beams with wavelength $\lambda$ within
$500\,\mathrm{ms}$. Together with a fine adjustment of the XODT to control the
chemical potential this creates a Mott insulator with predominant single-site
occupancy [39, 41]. The offset magnetic field $B$ is subsequently increased
from $B_{\textrm{i}}\\!=\\!21.0\,$G to a value $B_{\textrm{0}}$ between
$40\,\mathrm{G}$ and $100\,\mathrm{G}$, while adjusting
$\left|\nabla{B}\right|$ to keep the atoms magnetically levitated. In this
interval, $a_{s}$ varies in the range from $\sim$1.0$\times 10^{3}\,a_{0}$ to
$\sim$1.5$\times 10^{3}\,a_{0}$, where $a_{0}$ is the Bohr radius, skipping
the narrow FRs. We hold the atoms in the lattice for a time
$\tau_{\textrm{h}}=15$ to $20$ ms, after which we reverse the procedure above.
We image the atoms after $52\,\mathrm{ms}$ time of flight and we record the
atom number $N$ and the rms radius $\sigma$ of the sample.

With E1 we observe 4 characteristic heating features. The data are shown in
Fig. 2(a). We attribute the feature $P_{0}$ to the FR at 47.8 G. The
resolution is not sufficient to allow for the observation of substructures
near this FR. The much narrower FR at 53.8 G does not show up in the data. The
features $P_{1}$, $P_{2}$, and $P_{3}$ at $B\\!=\\!56.5(2)$ G, $65.3(3)$ G,
and $93.0(2)$ G, respectively, do not correspond to known FRs. In fact, they
are ICIRs, as is confirmed below. In E2, we tune $a_{\mathrm{s}}$ on the
repulsive side of the FR at 47.8 G. This time we observe 4 atom-loss features
$P^{\prime}_{\mathrm{1A}}$, $P^{\prime}_{\mathrm{1B}}$,
$P^{\prime}_{\mathrm{2}}$, and $P^{\prime}_{3}$ as $B$ is scanned from $46$ G
to $47.6$ G as shown in Fig. 2(b). We convert $B$ into $a_{\mathrm{s}}$
according to Fig. 1(a) and plot both data sets together in Fig. 3. The
positions of the heating features $P_{1}$, $P_{2}$, and $P_{3}$ agree
reasonably well with the positions of the loss features
$P^{\prime}_{\mathrm{1A}}$ and $P^{\prime}_{\mathrm{1B}}$,
$P^{\prime}_{\mathrm{2}}$, and $P^{\prime}_{3}$, respectively. As discussed
below, in E2 the feature $P_{1}$ from E1 appears to be split into two
components $P^{\prime}_{\mathrm{1A}}$ and $P^{\prime}_{\mathrm{1B}}$. We
attribute the splitting to some slight anisotropy of the lattice in E2 [41].
Features $P^{\prime}_{\mathrm{2}}$ and $P^{\prime}_{3}$ are significantly
broader than $P_{2}$ and $P_{3}$, respectively. We attribute the broadening to
the magnetic field gradient, which causes a spread of $a_{s}$ across the
sample [41]. This becomes increasingly larger as one climbs up the narrow FR
at 47.8 G.

Figure 3: Comparison of the data from the two experiments. (a) Cloud radius
$\sigma$ from E1 and (b) atom number $N$ from E2 as a function of
$a_{\mathrm{s}}$. The resonance positions $a_{r}$ are found at $1112(22)$
$a_{0}$, $1236(11)$ $a_{0}$, and $1488(16)$ $a_{0}$ for $P_{1}$, $P_{2}$, and
$P_{3}$, respectively, and at $1094(10)$ $a_{0}$, $1124(26)$ $a_{0}$,
$1233(40)$ $a_{0}$, and $1484(92)$ $a_{0}$ for $P^{\prime}_{\mathrm{1A}}$,
$P^{\prime}_{\mathrm{1B}}$, $P^{\prime}_{\mathrm{2}}$, and
$P^{\prime}_{\mathrm{3}}$, respectively. For each peak, the resonance position
is obtained as the center of a Gaussian fit, and the Gaussian sigma is
indicated in the parenthesis. The colored areas indicate the intervals
obtained from M1, and the dotted lines represent the positions of the
crossings as given by M2:
$\psi^{\mathrm{RM}}_{\mathrm{t}}\phi^{\mathrm{CM}}(0,0,0)$ with
$\psi_{b}^{\mathrm{RM}}\phi^{\mathrm{CM}}_{\mathrm{iso}}(2,2,0)$ (blue) and
with $\psi_{b}^{\mathrm{RM}}\phi^{\mathrm{CM}}_{\mathrm{iso}}(4,0,0)$ (red),
and $\psi^{\mathrm{RM}}_{\mathrm{t}}\phi^{\mathrm{CM}}_{\mathrm{iso}}(2,0,0)$
with $\psi_{b}^{\mathrm{RM}}\phi^{\mathrm{CM}}_{\mathrm{iso}}(6,0,0)$ (green).
The width of the blue area has been increased by a factor of 20 to improve
visibility.

We now compare the experimental data with the predictions of the two models,
without any fitting parameters. Even though the models are quite different,
they both reflect the data reasonably well. The positions of $P_{1}$ and
$P_{2}$ are in good agreement with the avoided crossings predicted by M2 of
the state $\psi^{\mathrm{RM}}_{\mathrm{t}}\phi^{\mathrm{CM}}(0,0,0)$ with the
three-fold degenerate states
$\psi^{\mathrm{RM}}_{b}\phi^{\mathrm{CM}}_{\mathrm{iso}}(2,2,0)$ and
$\psi^{\mathrm{RM}}_{b}\phi^{\mathrm{CM}}_{\mathrm{iso}}(4,0,0)$,
respectively. The difference in energy between the two states (2,2,0) and
(4,0,0) stems from anharmonic corrections and hence leads to two different
ICIRs. In Fig. 3 we indicate the calculated positions of the resonances from
M2 and the intervals from M1. For $P_{1}$ and $P_{2}$ we again find reasonable
agreement.

It is interesting to note that, on the basis of our modelling, $P_{3}$ cannot
arise from a crossing in which the state
$\psi^{\mathrm{RM}}_{\mathrm{t}}\phi^{\mathrm{CM}}_{\mathrm{iso}}(0,0,0)$ is
involved. However, this resonance could be the result of excitations in the
Mott-insulator state. The avoided crossings of
$\psi^{\mathrm{RM}}_{\mathrm{t}}\phi^{\mathrm{CM}}_{\mathrm{iso}}(2,0,0)$ with
$\psi^{\mathrm{RM}}_{b}\phi^{\mathrm{CM}}_{\mathrm{iso}}(6,0,0)$, and
$\psi^{\mathrm{RM}}_{\mathrm{t}}\phi^{\mathrm{CM}}_{\mathrm{iso}}(1,0,0)$ with
$\psi^{\mathrm{RM}}_{b}\phi^{\mathrm{CM}}_{\mathrm{iso}}(5,0,0)$ are in fact
very close to the position of $P_{3}$. We hypothesize that such excitations
can be brought about by mixing of the lowest band with excited bands due to
the strong interparticle interaction. In our case, the on-site interaction
energy is $\sim$78% of the bandgap to the first-excited lattice band [41],
which may open the route for a non-negligible population of the excited
states, i. e., bands, in a fully correlated many-body description.

Figure 4: Tuning of the ICIRs. Resonance position $a_{r}$ for $P_{1}$ (blue),
$P_{2}$ (red), and $P_{3}$ (green) as a function of the oscillator length
$a^{z}_{h}$ along the $z$ direction for the isotropic (circles) and the
anisotropic (triangles) case. For the isotropic case,
$V_{x}\\!=\\!V_{y}\\!=\\!V_{z}$ was set to $20.0(3),18.5(3),17.5(3),16.5(3)$,
and $15.5(3)$ $E_{\mathrm{R}}$ for the data from left to right. For the
anisotropic case, $V_{x}\\!=\\!V_{y}\\!=\\!18.5(3)$ $E_{\mathrm{R}}$ was
chosen for $V_{z}\\!=\\!17.5(3)$ $E_{\mathrm{R}}$ (left triangles) and
$16.5(3)$ $E_{\mathrm{R}}$ (right triangles). The error bars reflect the
statistical uncertainties [41]. The results from M1, applicable to the
isotropic case, are shown as the blue, red, and green areas. The blue area has
been widened by a factor of 3 to improve visibility. The predictions of M2 are
plotted as dotted (dashed) lines for the isotropic (anisotropic) case.

Next, we test how the stiffness of the lattice confinement and the
introduction of some controlled anisotropy affect the resonances, i. e., how
their positions are tuned and to what extent they split. Within E1 we first
vary $V_{x,y,z}$ in the range between $20.0(3)$ and $15.5(3)$ $E_{\mathrm{R}}$
while maintaining $V_{x}\\!=\\!V_{y}\\!=\\!V_{z}$. Model M1, which takes into
account the full lattice in the treatment of the CM [41], allows us to provide
a range 222The actual range is slightly larger due to the experimental
uncertainties in, e. g., the laser intensities. within which the resonances
are more likely to be found. This turns out to be very narrow for $P_{1}$ and
increasingly larger for $P_{2}$ and $P_{3}$ due to the increasing bandwidth of
the lattice bands. In Fig. 4 we plot the resulting resonance position
$a_{\mathrm{r}}$, obtained in the same way as for the resonances shown in Fig.
3, as a function of the harmonic oscillator length
$a^{z}_{\mathrm{h}}=\sqrt{\hbar/(m\omega_{z})}$ along the $z$ direction. As
can clearly be seen for $P_{1}$ and $P_{2}$, the position shifts to higher
values for $a_{\mathrm{r}}$ with increasing $a^{z}_{\mathrm{h}}$. For $P_{3}$
only two data points are available, but also here a trend to higher values is
present. For two specific values $V_{z}\\!=\\!17.5(3)$ $E_{\mathrm{R}}$ and
$16.5(3)$ $E_{\mathrm{R}}$ we then set $V_{x}\\!=\\!V_{y}\\!=\\!18.5(3)$
$E_{\mathrm{R}}$. In such an anisotropic case, $P_{1}$ and $P_{2}$ split into
two components, similarly to what we had observed earlier with E2. The
degeneracy is removed [41] by increasing $a_{\mathrm{h}}^{\mathrm{z}}$ with
respect to $a_{\mathrm{h}}^{\mathrm{x,y}}$. Here, the crossings of e.g.
$\psi^{\mathrm{RM}}_{\mathrm{t}}\phi^{\mathrm{CM}}(0,0,0)$ with
$\psi^{\mathrm{RM}}_{b}\phi^{\mathrm{CM}}(2,0,2)$ and
$\psi^{\mathrm{RM}}_{b}\phi^{\mathrm{CM}}(0,2,2)$ are found at the same
position, but they are at a different position with respect to the crossing of
$\psi^{\mathrm{RM}}_{\mathrm{t}}\phi^{\mathrm{CM}}(0,0,0)$ with
$\psi^{\mathrm{RM}}_{b}\phi^{\mathrm{CM}}(2,2,0)$. For this reason, in the
anisotropic lattice, each resonance splits into two separate resonances, with
a separation that depends on the difference in the trapping potential along
the three directions.

While in this work the positions and physical origin of the ICIRs, found
experimentally to occur in strong 3D confinement, are well explained by the
theoretical models, the details of how the resonance leads to loss and heating
need further theoretical and experimental investigation. In particular, the
degree of suppression of the loss features in the Mott-insulator state and the
possible role of inhomogeneities, allowing for superfluid regions in the
system, is unclear. Studies of the loss dynamics and in particular of its
density dependence would help elucidate the nature of the observed resonances,
with connections to recent observations of tunnelling dynamics of doublons in
the regime of strong interactions and under strong three-body losses [47],
something that is not considered in any of our theoretical models, which
relies on previous observations of ICIRs in 1D geometry, where the loss was
attributed to a two-step process via the creation of a mobile molecule [29].
In this context, detecting the ICIRs out of a two-atom Mott-insulator shell
could greatly elucidate the relevant processes.

Our results contribute to the understanding and characterization of CIRs, with
prospects for the association of dimers in optical lattices [48, 49, 39] and
optical tweezers [50, 51] in the absence of, e. g., magnetic FRs, as a
detection tool or as a step for creating ultracold molecules. In addition,
these results add constraints for the stability of atoms in optical lattices
and become very relevant in the presence of systems with high number of FRs
like lanthanide atoms [52, 53], or large lattice depths as it is the case in
quantum-gas microscopy [54, 55].

We thank Erich Dobler for help with the experiment and discussions. The
Innsbruck team gratefully acknowledges funding by the DFG-FWF Forschergruppe
FOR2247 under the FWF project number I4343-N36, via a Wittgenstein prize grant
under project number Z336-N36, and by the European Research Council (ERC)
under project number 789017. Alejandro Saenz thanks that part of the research
was supported by the National Science Foundation under Grant No. NSF
PHY-1748958. Fabio Revuelta acknowledges support by PGC2018-093854-BI00 and
PID2021-122711NB-C21 funded by MCIN/AEI/10.13039/501100011033, and by the
Comunidad de Madrid under the Grant APOYO-JOVENES-4L2UB6-53-29443N (GeoCoSiM)
financed within the Plurianual Agreement with the Universidad Politécnica de
Madrid in the line to improve the research of young doctors.

The data that support the findings of this study are made publicly available
by the authors at 333Dataset are available via Zenodo with
doi:10.5281/zenodo.7108194

Note added.—We recently became aware of an independent work by Lee et al.
[57].

##  I Supplemental Material

This supplemental material provides further details for the results presented
in the main text. For this purpose, first the adopted theoretical models are
discussed. Second, a description of the two experimental settings is
presented. Finally, we conclude with a brief discussion on the question of
possible excitations to higher Bloch bands, which may explain the otherwise
not assigned resonance P3.

###  I.1 Theoretical description

The Hamiltonian for two atoms in the lattice potential is given by [23]

$\displaystyle H(\textbf{r},\textbf{R})=$ $\displaystyle
T_{\mathrm{RM}}(\textbf{r})+T_{\mathrm{CM}}(\textbf{R})+V_{\mathrm{RM}}(\textbf{r})$
(1)
$\displaystyle+V_{\mathrm{CM}}(\textbf{R})+U_{\mathrm{int}}(\textbf{r})+W(\textbf{r},\textbf{R}),$

with $\textbf{r}=(\textbf{r}_{1}-\textbf{r}_{2})/\sqrt{2}$ and
$\textbf{{R}}=(\textbf{r}_{1}+\textbf{r}_{2})/\sqrt{2}$ the coordinates for
the relative motion (RM) and the center-of-mass motion (CM), respectively, and
the interaction potential $U_{\mathrm{int}}$. The potentials for the RM and
the CM take the form
$V_{\mathrm{RM}}(\textbf{r})=2sE_{\mathrm{R}}\sum_{i}\sin^{2}\left(kr_{i}/\sqrt{2}\right)$
and
$V_{\mathrm{CM}}(\textbf{R})=2sE_{\mathrm{R}}\sum_{i}\sin^{2}(kR_{i}/\sqrt{2})$,
while the coupling between the two motions is given by
$W(\textbf{r},\textbf{R})=-4sE_{\mathrm{R}}\sum_{i}\sin^{2}\left(kr_{i}/\sqrt{2}\right)\sin^{2}(kR_{i}/\sqrt{2})$.
Here, $s$ is the lattice depth in units of the recoil energy
$E_{\mathrm{R}}$=$\frac{\hbar^{2}k^{2}}{2m}$, $k$ is the wave number of the
lattice light, and $i$ denotes $x$, $y$, or $z$.

To describe the system theoretically, we employ two complementary approaches,
differing in the way the interatomic potential as well as the external
potential are treated. A summary of the model differences can be found in
Table 1.

####  I.1.1 Interaction-potential treatment

In the first approach (M1), we employ a pseudopotential approximation for the
interatomic interaction potential, modeled by a regularized delta function
$U_{\mathrm{int}}(r)=\frac{4\pi\hbar^{2}a_{s}}{m}\delta(r)\frac{\partial}{\partial
r}r$, where $a_{s}$ is the scattering length. The advantage of this approach
is that most of the calculations can be carried out analytically. However, for
the case of an anisotropic trapping potential, M1 cannot be applied due to a
substantial complication of the computation.

The second approach (M2) deals with a more realistic interaction potential and
is based on a full ab-initio calculation. The interaction potential with Cs
atoms is, nevertheless, computationally challenging due to the large number of
molecular bound states, which have, as a consequence, a more complex nodal
pattern. In view of this, and assuming a universal regime for which the
properties of the system can be entirely described by $a_{\mathrm{s}}$ and by
the characteristic trap size $a_{\mathrm{h}}$, regime which has been assessed
by comparison with the experiment reported in Ref. [23], we perform the
computation for two Li atoms, which have only 11 bound states and thus
wavefunctions with much less nodes than the Cs2 dimer.

####  I.1.2 Trapping-potential treatment

For the treatment of the external trapping potential, as an introductory step,
we consider the case of an individual potential well in the harmonic
approximation, for which relative motion (RM) and center of mass motion (CM)
can be separated. For the RM, the eigenstates of the system feature one bound
state $\psi^{\mathrm{RM}}_{\textrm{b}}$444for M1, this is the only bound state
supported by the delta function. For M2, this is the bound state closest to
threshold, other bound states are separated by a large energy gap and an
infinite ladder of trap-induced states $\psi^{\mathrm{RM}}_{n}$, whose
energies vary as a function of $a_{\mathrm{s}}$ [42]. To these RM states, the
CM adds a ladder of states $\phi^{\mathrm{CM}}(n_{x},n_{y},n_{z})$ separated
by an integer number of excitations along the three trap directions $x,y$ and
$z$. The term $W$, in our case caused by an anharmonicity of the trapping
potential, can now provide the necessary coupling to create avoided crossings
and hence to give rise to inelastic confinement induced resonances (ICIRs)
[23, 24].

The procedure followed in both models is based on the identification of the
relevant crossings, as just explained for the case of the harmonic oscillator.

For both models, we initially neglect the coupling term $W$ and isolate the RM
Hamiltonian and the CM Hamiltonian and calculate their eigenstates
$\psi^{\mathrm{RM}}$ and $\phi^{\mathrm{CM}}(n_{x},n_{y},n_{z})$,
respectively.

For M1, the potential $V_{\mathrm{RM}}$ is initially approximated by a
quadratic function, with $\hbar\omega_{\mathrm{RM}}=2\sqrt{s}E_{R}$. For this
problem, an exact solution is known [42]. The energies are given implicitly by

$\frac{a_{h}}{a_{s}}=\frac{\sqrt{2}\Gamma\left(\frac{3}{4}-\frac{E_{\mathrm{RM}}}{2\hbar\omega_{\mathrm{RM}}}\right)}{\Gamma\left(\frac{1}{4}-\frac{E_{\mathrm{RM}}}{2\hbar\omega_{\mathrm{RM}}}\right)}$
(2)

with $a_{h}=\sqrt{\frac{\hbar}{m\omega_{\mathrm{RM}}}}=\frac{1}{ks^{1/4}}$.
The corresponding wavefunctions are given by

$\displaystyle\psi^{\mathrm{RM}}\left(\textbf{r}\right)=$
$\displaystyle\frac{1}{2\pi^{3/2}}Ae^{-\frac{r^{2}}{2a_{h}^{2}}}\Gamma\left(\frac{3}{4}-\frac{E_{\mathrm{RM}}}{2\hbar\omega_{\mathrm{RM}}}\right)\times$
(3)
$\displaystyle\mathcal{U}\left(\frac{3}{4}-\frac{E_{\mathrm{RM}}}{2\hbar\omega_{\mathrm{RM}}},\frac{3}{2},\frac{r^{2}}{a_{h}^{2}}\right),$

where $A$ is a normalization constant, $\mathcal{U}$ is the confluent
hypergeometric function, and $\Gamma$ is the Gamma-function. We further
calculate, by first order perturbation theory, corrections to the energies
arising from anharmonicities of the form $r_{i}^{4}$ and $r_{i}^{6}$, as
obtained by a Taylor expansion of $V_{\mathrm{RM}}$. In M1, the CM is solved
exactly on the separable lattice $V_{\mathrm{CM}}$, with periodic boundary
conditions [26]. In this case the wavefunctions
$\phi^{\mathrm{CM}}(n_{x},n_{y},n_{z})(\textbf{q})$ are Bloch functions with
eigenenergies $E^{\mathrm{CM}}_{n_{x},n_{y},n_{z}}(\textbf{q})$, where the
indices $n_{i}$ take the meaning of band indices and q is the quasi-momentum.
The CM-RM coupling energy contribution is considered also as self-coupling on
the CM and RM subspaces:
$\left\langle\phi^{\mathrm{CM}}\right|$W$\left|\phi^{\mathrm{CM}}\right\rangle$
and similarly for $\psi^{\mathrm{RM}}$. In this way, we build a diagram
similar to the one shown in Fig. 1(c) in the main text. The crossings are
calculated for the two edges of the lattice band, $q=0$ and $q=k$. In the
isotropic lattice, each resonance is three-fold degenerate due to the
factorization of the CM in the three directions.

For M2, the potential terms $V_{\mathrm{RM}}$, $V_{\mathrm{CM}}$ and $W$ are
Taylor expanded up to powers of $r_{i}^{6}$ and the wavefunctions
$\psi^{\mathrm{RM}}$ and $\phi^{\mathrm{CM}}(n_{x},n_{y},n_{z})$ are computed
on a basis of $B$ splines for the radial part and spherical harmonics for the
angular part [21, 27]. In order to adequately reproduce the highly oscillatory
inner part of the RM eigenfunctions (usually known as the molecular region),
different knot sequences for the $B$ splines are used. For the region close to
the origin (up to a distance of 15 $a_{0}$) it is 50 $B$ splines following a
linear distribution; in order to cover the smooth long-range part, 100 $B$
splines. with a geometric distribution for larger distances are used (up to
30,000$\sqrt{12}$ $a_{0}$). As the center-of-mass potential is a smooth
function with no singularity at the origin, 100 $B$ splines uniformly
distributed over $30,000\sqrt{3}a_{0}$ were found to be sufficient in the
computation of the CM eigenfunctions. In order to calculate only those states
that are totally symmetric, i. e., those states that belong to the $A_{g}$
irreducible representation of the $D_{2h}$ symmetry group [21, 27], the
angular part of both the RM and the CM is described using the spherical
harmonics with $\ell=0,2,\ldots,34$, and $m=0,2,\ldots,14$.

The eigenstates of the Hamiltonian in Eq. (1) are then described in terms of
configurations
$\Phi_{i,j}$=$\psi^{\mathrm{RM}}_{j}\phi^{\mathrm{CM}}_{j}(n_{x},n_{y},n_{z})$,
which are combinations of the RM and CM eigenfunctions, as detailed in Refs.
[21, 27]. Then, the contribution from the coupling term can be calculated and
the total Hamiltonian can be diagonalized. In M2, 200 eigenfunctions for the
RM and 75 for the CM have been combined, rendering a basis set formed by 15000
elements. The simulation is performed in a lattice of $1064.5\,\mathrm{nm}$,
where the power of the beams is chosen such that the potential depth for Li in
units of recoil is $s^{\mathrm{Li}}=s^{\mathrm{Cs}}$. The validity of the use
of the interaction potential of Li atoms has been previously reported in the
Ref. [23] by explicit comparison with the Cs experiment described in the Ref.
[31]. The results are shown in both plots of Fig. 3 and Fig. 4 in the main
text and are in fair agreement with the experimental data.

Table 1: Main characteristics of the models M1 and M2 that are used in the simulations. Characteristic |  | M1 |  | M2
---|---|---|---|---
Optical trap |  |  Isotropic (Anisotropic demanding) |  |  Isotropic & anisotropic
RM |  |  |  |
a. Optical trap potential |  | Taylor expanded (order 6) |  | Taylor expanded (order 6)
b. Interaction potential |  |  $\delta$ pseudopotential |  |  ab initio (Li2)
c. No. of bound (molecular) states |  | 1 |  | $11$
d. Energies |  | Discrete levels / Bands |  | Discrete levels / Bands
CM |  |  |  |
a. Optical trap potential |  | Full ($\sin^{2}$) |  | Taylor expanded (order 6)
b. Energies |  | Continuous bands |  | Discrete levels

Let us conclude by remarking that M2 allows also for Taylor expansions of
higher order which reproduce multiwell setups. For example, a Taylor expansion
of order 22 reproduces a triple-well system [54]. M2 is also able to describe
a $\cos^{2}$ optical lattice, something that enables the study of double-well
systems when Taylor expanded up to order 12.

Figure 5: Correlation diagram for two Li atoms confined in an optical trap
with $V_{x,y,z}=20E_{R}$ formed by an isotropic single well (blue filled
circles), and in a trap formed by a single well in two directions and a double
well in the remaining one (orange empty circles). The vertical solid lines
show the position of the experimentally observed confinement-induced resonance
$P_{1}$, $P_{2}$, $P_{3}$, respectively in blue, red, and green.

Fig. 5 shows the correlation diagram for two Li atoms interacting with an ab-
initio potential. The blue filled circles show the energy of the atoms when
confined in a trap with a single well in all directions, while the orange
empty circles show the energies for a trap with a single well in two of the
spatial directions and a double well in the remaining one. As can be seen, the
double-well system presents more energy levels than the single-well one. Some
of them emerge due to the symmetry breaking in one direction (like those for
the two molecular states on the right), while others are totally absent in the
single well. That is the case for the first-trap state for the double well
(bottom almost horizontal aligned points) whose energy is almost unaffected by
the interaction strength between the two particles; this state corresponds to
a situation where each of the atoms is found in a different well and then
hardly interact due the short-range interparticle potential. Contrarily, the
first-trap state for the single well strongly increases with the interaction
strength as the atoms lie close to each other and then can repel much more
strongly. Note, similarly, that this trap state is also present in the double
well (but with a slightly larger energy). For vanishing interactions, like
those used in the beginning of the experiment, the previous two trap states
are degenerate, a fact that could explain why the observed resonances, whose
positions are marked as vertical solid lines, involve the upper trap state
instead of the bottom one, which is the one associated with the Mott-
insulating state. The comparison between the M2 single-well results has been
assessed and the double-well calculations, renders differences of only a few
atomic units in the ICIRs positions.

###  I.2 Experimental procedures

####  I.2.1 Details of E2

We start with an essentially pure Bose-Einstein condensate (BEC) of
1$\times$$10^{5}$ Cs atoms in a crossed optical dipole trap (XODT)
adiabatically loaded into a cubic optical lattice with a lattice depth of
$V_{x}=V_{y}=V_{z}=20$ $E_{\mathrm{R}}$, with $\lambda=1064.5\,\mathrm{nm}$
the wavelength of the lattice light. We take care to prepare a pure singly
occupied Mott-insulator state by adapting the XODT during loading. During BEC
preparation and lattice loading we apply a magnetic field gradient of 31 G/cm
to compensate the gravitational force and we set the magnetic offset field $B$
to values around 21 G, giving a s-wave scattering length $a_{s}=210$ $a_{0}$.
The field $B$ is then ramped to its final value with a resolution of $\sim 8$
mG and a ramp speed of $~{}25\,\mathrm{G/ms}$. After a hold time of
$50\,\mathrm{ms}$ we ramp $B$ to its final value of 18.5 G at maximum speed,
switch off the lattice and XODT beams within 2 ms, and subsequently take an
absorption picture of the atom cloud after an expansion time of
$20\,\mathrm{ms}$ to determine the number of atoms. The lattice depth $V_{q}$
is calibrated via Kapitza-Dirac diffraction. The statistical error for $V_{q}$
is 1%, though the systematic error can reach up to 5%. The splitting of
$P_{1}$ into $P^{{}^{\prime}}_{1A}$ and $P^{{}^{\prime}}_{1B}$ is
$30\,\mathrm{a_{0}}$. Assuming the theory from M2, such a splitting
corresponds to an anisotropy of $\sim$1 $E_{\mathrm{R}}$, which is compatible
with the systematic error for the lattice calibration. The scattering length
$a_{\mathrm{s}}$ is calculated via its dependence on $B$ with an estimated
uncertainty of $\pm 5\,\mathrm{mG}$ arising from magnetic field noise and
shot-to-shot variations. The resonances are further broadened by the magnetic
field gradient. An estimate of the broadening can be obtained from the size of
the cloud in the lattice. For simplicity, we assume that all the atoms are
distributed on a sphere with perfect single-occupancy at the sites of the
lattice. The levitation gradient of $31\,\mathrm{G/cm}$ gives then the
variation of the field across the cloud, which can be converted to a variation
$\Delta a_{\mathrm{s}}$ of $a_{s}$ according to Fig. 1(a) in the main text.
For $P^{{}^{\prime}}_{\mathrm{1A}}$, $P^{{}^{\prime}}_{\mathrm{1B}}$,
$P^{{}^{\prime}}_{\mathrm{2}}$, and $P^{{}^{\prime}}_{\mathrm{3}}$ we obtain
8, 13, 37, and $150\,a_{0}$, respectively. These variations are in fair
agreement with the width of the resonances shown in Fig. 3(b) in the main
text.

####  I.2.2 Error bars for Fig. 4 in the main text

The data in Fig. 4 in the main text are obtained by fitting data that is taken
in the same way as the data shown in Fig. 3(a) in the main text. In view of
limited sampling of the Gaussian fitting curves, the error bars are calculated
from the sigma of the Gaussian fit. On top of that, we include the uncertainty
from the lattice and magnetic-field calibration. The magnetic field is
calibrated by microwave spectroscopy and the use of the Breit-Rabi formula,
the uncertainty for E2 is $\Delta B=0.8\,G$. The lattice depth is calibrated
via parametric heating measurements by driving the transition between the
lowest and second excited band of the lattice near 20 $E_{\mathrm{R}}$, the
uncertainty is $\Delta V=0.3\,E_{\mathrm{R}}$. Other systematic errors such as
the ones deriving from drifts of the wavelength of the lattice laser and
slight misalignment of the lattice beams are found to be negligible.

###  I.3 P4 discussion

####  I.3.1 Beyond the Bose-Hubbard model

The standard Bose-Hubbard model is commonly derived under the assumption that
the bandgap is the dominating energy scale of the system and that no mixing
between the different bands takes place. When driving the superfluid-to-Mott-
insulator transition, this leads to atoms confined in the
$\psi^{\mathrm{RM}}_{\mathrm{t}}\phi^{\mathrm{CM}}(0,0,0)$ state. To assess
the validity of the hypothesis the bandgap for different bands for non-
interacting atoms in the lattice is compared to the on-site interaction energy
calculated on the basis of the non-interacting Wannier functions. For a
lattice with $V_{0}=20\,E_{\mathrm{R}}$, we find that difference in energy
between the maximum of the lower band and the minimum of the first, second,
and third excited band is $\Delta_{1}=7.6~{}E_{\mathrm{R}}$,
$\Delta_{2}=13.2E_{\mathrm{R}}$, and $\Delta_{3}=17.3~{}E_{\mathrm{R}}$,
respectively, while for $a_{\mathrm{h}}=1500\,a_{0}$ the onsite interaction is
$U=5.9~{}E_{\mathrm{R}}$. Thus, in the regime in which we are observing the
resonances, the onsite interaction energy is non-negligible with respect to
the bandgap. Noteworthy, a not fully adiabatic variation of the scattering
length may cause an excitation to higher bands via avoided crossings [27].

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