# Ultracold Feshbach molecules in an orbital optical lattice

Yann Kiefer  Institut für Laser-Physik, Universität Hamburg, Luruper Chaussee
149, 22761 Hamburg, Germany  Zentrum für optische Quantentechnologien,
Universität Hamburg, 22761 Hamburg, Germany  Max Hachmann  Institut für Laser-
Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
Zentrum für optische Quantentechnologien, Universität Hamburg, 22761 Hamburg,
Germany  Andreas Hemmerich  Institut für Laser-Physik, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany  Zentrum für optische
Quantentechnologien, Universität Hamburg, 22761 Hamburg, Germany
<EMAIL_ADDRESS>
Quantum gas systems provide a unique experimental platform to study a
fundamental paradigm of quantum many-body physics: the crossover between Bose-
Einstein condensed (BEC) molecular pairs and Bardeen Cooper Schrieffer (BCS)
superfluidity. Some studies have considered quantum gas samples confined in
optical lattices, however, focusing on the case, when only the lowest Bloch
band is populated, such that orbital degrees of freedom are excluded. In this
work, for the first time, ultracold Feshbach molecules of fermionic ${}^{40}K$
atoms are selectively prepared in the second Bloch band of an optical square
lattice, covering a wide range of interaction strengths including the regime
of unitarity. Binding energies and band relaxation dynamics are measured by
means of a method resembling mass spectrometry. The longest lifetimes arise
for strongly interacting Feshbach molecules at the onset of unitarity with
values around $300\,$ms for the lowest band and $100\,$ms for the second band.
In the case of strong confinement in a deep lattice potential, we observe
bound dimers also for negative values of the $s$-wave scattering length,
extending previous findings for molecules in the lowest band. Our work
prepares the stage for orbital BEC-BCS crossover physics.

The crossover between the regimes of BEC and BCS superfluidity is a hallmark
of quantum gas physics Reg1:04 ; Reg2:04 ; Zwi:04 ; Bou:04 ; Bar:04 ; Ket:08 ;
Zwe:12 ; Toe:15 . In most studies, the quantum gas sample is held in a nearly
harmonic optical trapping potential. BEC-BCS crossover physics in optical
lattices Lew:07 ; Gro:17  has been much less explored, and this research is
limited to the lowest Bloch band, which exclusively provides local
$s$-orbitals. For example, in the ground state of a three-dimensional (3D)
optical lattice, binding energies of strongly interacting fermionic potassium
pairs have been studied Sto:06 . Signatures of coherence and superfluidity
have been reported Chi:06  for strongly interacting fermionic lithium pairs.
In an earlier work with fermionic potassium, prepared in the transverse ground
state of an array of effectively one-dimensional wave guides Mor:05 , Feshbach
dimers have been shown to exist even at negative scattering lengths owing to a
confinement induced scattering resonance Ols:98 ; Ber:03 . Very recently,
$p$-wave interacting atomic pairs tightly confined in excited motional states
of isolated microscopic traps have been investigated Ven:22 . Similarly, as
orbital degrees of freedom in electronic condensed matter, e.g. in transition
metal oxides Tok:00 ; Mae:04 , can give rise to unconventional order, the
combination of the conventional BEC-BCS scenario with orbital physics in
higher Bloch bands holds the intriguing perspective to discover unexplored
fundamental many-body phases such as exotic forms of superfluidity Li:16 .
Examples of chiral, nematic, or topological superfluids have been
experimentally demonstrated for bosonic atoms in the second Bloch bands of
square Wir:11 ; Koc:16 , triangular Wang:22 , or hexagonal Wang:21  optical
lattices, respectively. Extending these scenarios to composite bosons composed
of pairs of fermionic atoms would open up a new regime of unconventional BEC-
BCS physics.

Figure 1: Production and detection of Feshbach molecules. (a) A single mass
spectrometry image of a mixture of 40K atoms and 40K Feshbach dimers excited
to the second Bloch band. (b) Composition of the image in (a) by the small
second BZ of the molecules and the double-sized second BZ of the atoms. The
velocity scale is $v_{M}=\sqrt{2}\,2\hbar k/M$ with $k=2\pi/\lambda$ and the
mass $M$ of the Feshbach dimers. (c) Feshbach resonance and protocol for
molecule preparation. See text. (d) Sketch of the lattice geometry in the xy-
plane for $\Delta V=0$ (left) and $\Delta V<0$ (right). The grey shaded
squares denote the corresponding Wigner-Seitz unit cells. At the lower edge,
sections along the dashed lines in the upper panels of (d) are shown.

In a recent work Hac:21 , we have demonstrated spin-polarized non-interacting
fermionic potassium atoms (40K) and weakly interacting spin-mixtures in higher
Bloch bands of an optical square lattice. The present work investigates the
vastly different regime of strong interactions between spin-up and spin-down
fermions accessed by tuning a Feshbach resonance. This allows us to
demonstrate and investigate bosonic Feshbach dimers Chi:10 , composed of
fermionic 40K atoms, in higher Bloch bands of an optical square lattice. We
study shallow and deep lattices, where the dimers can tunnel or are confined
in one-dimensional channels. A method inspired by mass spectrometry is applied
to discriminate atoms and dimers by separating them in a ballistic time-of-
flight protocol (see Methods). This leads to images of the Brillouin zone (BZ)
structures of atoms and molecules in velocity space, which differ in size by
their mass ratio. This is seen in Fig. 1(a) for a mixture of molecules and
atoms populating the second Bloch band. The molecules appear in the small -
and the atoms in the large second BZ, forming a nested structure according to
the sketch in Fig. 1(b). Hence, atoms and molecules can be separately counted.
Using this method, we present measurements of binding energies, dissociation
dynamics, and exceptionally long molecule lifetimes for a wide range of
interaction strengths including the strongly correlated regime. Such long
lifetimes exceeding all other relevant time scales are crucial, to study long-
lived equilibrium states of the system.

Production of ultracold Feshbach molecules. In short, the production of
Feshbach dimers in the second Bloch band of a square lattice proceeds as
follows: A balanced fermionic spin mixture is produced in the lowest Bloch
band of the lattice. Subsequently, spin-up and spin-down atoms are associated
to form Feshbach dimers by rapid adiabatic tuning of the magnetic field across
an $s$-wave Feshbach resonance. Finally, a quench of the lattice potential
selectively excites the dimers to the second Bloch band.

The following detailed protocol is applied. A non-interacting spin-polarized
degenerate Fermi gas of $6\times 10^{5}$ 40K atoms in the state
$\ket{F=9/2,m_{F}=9/2}$ at a temperature of $T=0.17\,T_{F}$ is prepared in an
optical dipole trap (ODT), formed by two orthogonally intersecting laser beams
with a wavelength of $\lambda=1064\,$nm, where $T_{F}$ denotes the Fermi
temperature. Note that for spin mixtures prepared at low magnetic fields, the
lowest temperature reached in the ODT is $T/T_{F}=0.09$. Next, a radio-
frequency with a constant value of $46\,$MHz is applied, while a homogeneous
magnetic field $B$ (pointing along the $z$-axis) is ramped up from zero to
approximately $B_{0}\approx 209.9\,$G, such that a rapid adiabatic passage is
obtained that inverts the sample to the $\ket{\downarrow}\equiv$
$\ket{F=9/2,m_{F}=-9/2}$ state. Due to a Feshbach resonance located at
$B_{\textrm{res}}=202.1\,$G Reg1:04 , the $s$-wave scattering length for
contact interaction between the states $\ket{\downarrow}$ and
$\ket{\uparrow}\equiv$ $\ket{F=9/2,m_{F}=-7/2}$, approximated as

$\displaystyle a_{S}(B)=a_{bg}\,[1-\Delta B/(B-B_{\textrm{res}})]\,,$ (1)

takes the value $a_{S}(B_{0})\approx 0$, which corresponds to position (1) in
Fig. 1 (c). Here, $a_{bg}=174\,a_{0}$ is the background scattering length and
$\Delta B=7.8\,$G is the width of the Feshbach resonance. Subsequently, the
sample is adiabatically loaded into the lowest Bloch band of a bipartite
optical square lattice, formed by two mutually orthogonal standing waves with
wavelengths $\lambda=1064\,$nm, oriented perpendicularly to the $z$-axis. The
optical lattice is shaped in a Michelson-Sagnac interferometer that allows for
precise control of the associated band structure (for details see Ref. Hac:21
). The resulting optical potential is composed of two classes of independently
tunable potential wells $\mathcal{A}$ and $\mathcal{B}$ arranged as the black
and white squares of a chequerboard (see Fig. 1 (d)). In the $xy$-plane, the
lattice potential can be approximated by

$\displaystyle V(x,y)=$ $\displaystyle-
V_{0}\left[\cos^{2}(kx)+\cos^{2}(ky)\right]$ (2)
$\displaystyle-\frac{1}{2}\Delta V\cos(kx)\cos(ky)$

with the wave number $k=2\pi/\lambda$, the lattice depth parameter $V_{0}$,
and $\Delta V\equiv-4V_{0}\cos(\theta)$ denoting the potential difference
between the $\mathcal{A}$ and $\mathcal{B}$ wells. The experimental parameter
$\theta$ can be tuned within the interval $[0,\pi]$, i.e., $\Delta V\in
4V_{0}\times[-1,1]$. Note that tightly bound dimers with mass $M\equiv 2m$
possess twice the polarizability and hence twice the value of $V_{0}$ as
compared to atoms with mass $m$. Henceforth, we indicate lattice depth
parameter values for atoms and molecules as $V_{0}^{(m)}$ and
$V_{0}^{(M)}=2\,V_{0}^{(m)}$, respectively. Along the $z$ direction, the atoms
are held by the weak approximately harmonic confinement of the optical dipole
trap, such that the lattice wells acquire a tubular shape. After lattice
loading, we apply a radio frequency pulse during $13\,\mu$s to create a Fermi
gas with equal populations in the states $\ket{\uparrow}$ and
$\ket{\downarrow}$ at position (1) in Fig. 1(c). Next, by tuning $B$ to
$202.3\,$G (cf. position (2) in Fig. 1 (c)), the $s$-wave scattering length
$a_{S}$ for collisions between $\ket{\uparrow}$ and $\ket{\downarrow}$ is
adjusted to a large negative value, in order to obtain efficient evaporative
cooling of the atomic sample in presence of the lattice. Finally, the
previously unpaired mixture of Fermions is converted into bosonic molecules by
adiabatically sweeping the magnetic field across the Feshbach resonance from
$202.3\,$G to $200.46\,$G (indicated as step (3) in Fig. 1(c)). For the lowest
temperatures, upon arrival at position (4) in Fig. 1(c), we observe close to
hundred percent conversion efficiencies with no discernible atomic fraction.
Next, a quench from an initial value $\theta\approx 0.4\,\pi$, used for
loading the lowest band, to $\theta\approx 0.53\,\pi$ efficiently prepares a
large fraction of the atoms or molecules in the second band (for details see
Ref. Hac:21 ). By means of a final adiabatic change of $B$ (position (5) in
Fig. 1 (c)), we may subsequently tune to a desired target value of $a_{S}$,
which lets us adjust the molecular binding energy.

Figure 2: Binding energies in a deep lattice. (a) A typical dissociation
spectrum for a magnetic field $B=201.95\,$G, i.e., corresponding to the data
point in (d) highlighted by an arrow, with molecules initially prepared in the
lowest Bloch band ($V_{0}^{(M)}=40\,E_{\textrm{rec}}^{(M)}$,
$\theta=0.4\,\pi$). The orange diamonds and dark purple disks show the
molecule and atom populations in the first band, respectively, plotted against
$\Delta f=f-f_{0}$, with the irradiated radio frequency $f$, and $f_{0}$
according to (b). The leftmost vertical red dashed line indicates the
frequency, where the first significant drop of the molecule population and a
corresponding peak of the atom population in the first band is observed. This
frequency is identified with the molecular binding energy $E_{B}$. Further
dashed vertical lines denote the positions of higher Bloch bands
$\ket{\textrm{Aux},\downarrow\,:\nu}$ with $\nu\in\\{2,3,4,5,6,7\\}$, leading
to further resonant molecule dissociation (cf. (b)). The error bars show the
standard deviation of the mean for $15$ experimental runs. (c) Mass
spectrometry images for $\Delta f=-5.3\,$kHz (c1) and $\Delta f=22.7\,$kHz
(c2). The white dashed rectangles show the first BZ for molecules (c1) and
atoms (c2). (d) Measured binding energies for molecules prepared in the first
(dark purple disks) and second (orange diamonds) Bloch band with
($V_{0}^{(M)}=40\,E_{\textrm{rec}}^{(M)}$, $\theta=0.4\,\pi$) and
($V_{0}^{(M)}=60\,E_{\textrm{rec}}^{(M)}$, $\theta=0.54\,\pi$), respectively.
The error bars are estimated from the spectral width of the underlying
dissociation resonance. The dashed vertical line indicates the position of
Feshbach resonance $B_{\textrm{res}}$. The grey line shows a calculation using
Eq. 3.

Binding energies Binding energy of Feshbach molecules are measured by
dissociating them with a $5\,$ms long radio frequency pulse, converting
$\ket{\uparrow}$ atoms into atoms in the auxiliary state
$\ket{\textrm{Aux}}\equiv\ket{F=9/2,m_{F}=-5/2}$. The unbound
$\ket{\downarrow}$ atoms, which remain trapped in the optical lattice, are
readily discriminated from the molecules as explained in the context of Fig. 1
(a,b) and in Methods.

Let us first discuss the case of a deep optical lattice, such that tunnelling
is practically suppressed for molecules and their motion is restricted to
quasi one-dimensional (1D) tubes along the $z$-direction. An example for this
regime is realized by adjusting $V_{0}^{(M)}=40\,E_{\textrm{rec}}^{(M)}$,
$\theta=0.4\,\pi$ and hence $\Delta V^{(M)}=-49.4\,E_{\textrm{rec}}^{(M)}$
with $E_{\textrm{rec}}^{(M)}\equiv\frac{\hbar^{2}k^{2}}{2M}$ denoting the
single photon recoil energy for molecules. A typical example of a dissociation
spectrum for molecules $\ket{\textrm{Mol}:1}$ prepared in the lowest band is
shown in Fig. 2(a) for $B=201.95\,$G corresponding to the data point in (d)
indicated by an arrow. We plot the numbers of molecules (diamonds) and atoms
(disks) in the lowest band, the atoms in all excited bands (squares), and the
total number of particles, i.e., atoms plus molecules (triangles), against
$\Delta f=f-f_{0}$, i.e., the applied radio frequency $f$ minus the
$\ket{\uparrow}\rightarrow\ket{\textrm{Aux}}$ transition frequency $f_{0}$
(see Fig. 2(b)). The latter depends only on $B$ and is readily measured after
preparing a pure $\ket{\uparrow}$ sample in the dipole trap. Hence, $\Delta
f=0$ corresponds to zero molecular binding energy.

To understand the spectral features in Fig. 2(a), it is helpful to first look
at the atomic and molecular levels sketched in Fig. 2(b). The black horizontal
bars show the energies of the bare two-atom states $\ket{\uparrow,\downarrow}$
and $\ket{\textrm{Aux},\downarrow}$ separated by the frequency $f_{0}$. The
blue horizontal bar shows the energy of the Feshbach molecules
$\ket{\textrm{Mol}}$ made of atom pairs $\ket{\uparrow,\downarrow}$, shifted
by the binding energy $E_{B}$. The red horizontal bars show the additional
energy shifts for the Bloch bands due to the presence of the optical lattice.
The molecules $\ket{\textrm{Mol}}$ are prepared in the lowest molecular band,
denoted $\ket{\textrm{Mol}:1}$ in Fig. 2(b). As the radio frequency $f$ is
increased, we expect the first drop of the molecular population in
$\ket{\textrm{Mol}:1}$, when $f$ reaches the resonance frequency $f_{1}$ for
the transition
$\ket{\textrm{Mol}:1}\rightarrow\ket{\textrm{Aux},\downarrow\,:1}$ such that
unbound atoms in the first Bloch band are produced. According to Fig. 2(a),
this occurs at $\Delta f=\Delta f_{1}\equiv 22.7\,$kHz, which is identified
with the value of the binding energy $E_{B}$. To further illustrate the
conversion of molecules into atoms, mass spectrometry images are shown in Fig.
2(c) for $\Delta f=-5.3\,$kHz, well below $\Delta f_{1}$ (c1), and at $\Delta
f=\Delta f_{1}$ (c2). In fact, in (c1), predominantly molecules in the first
BZ are seen, while in (c2) most molecules are dissociated into atoms, giving
rise to a first BZ expanded by a factor two.

At larger values of $\Delta f>\Delta f_{1}$, further resonances occur, leading
to reduced molecule numbers due to dissociation into higher bands
$\ket{\textrm{Aux},\downarrow\,:\nu}$ with $\nu\in\\{2,3,5,6,7\\}$. The
respective resonance frequencies are readily calculated by a band structure
calculation and are plotted into Fig. 2(a) as vertical dashed red lines. Note
that, for the fourth band $\nu=4$, no resonance arises, which can be explained
by the small Franck-Condon overlap between the wave function of
$\ket{\textrm{Mol}:1}$ and $\ket{\textrm{Aux},\downarrow\,:4}$. This has been
checked by calculating the respective Bloch functions, with the result that
$\ket{\textrm{Mol}:1}$ predominantly resides in the deep wells and
$\ket{\textrm{Aux},\downarrow\,:4}$ in the shallow wells.

Dissociation spectra as in Fig. 2(a) let us determine the binding energies for
molecules prepared in the first (dark purple disks) and second (orange
diamonds) bands, shown in Fig. 2(d). The gray solid line shows a calculation
using the implicit equation

$\frac{a_{S}-r_{0}}{(1+\delta)a_{r}}=-\frac{1}{\zeta\left(1/2,-E_{B}/2\hbar\omega_{r}\right)}.$
(3)

adapted from Ref. Ber:03  for a single 1D tubular potential, where $\zeta$
denotes the Hurwitz zeta function. We insert the radial fundamental frequency
$\omega_{r}$, measured in the radially symmetric quasi 1D tubes of our
lattice, and the associated harmonic oscillator length
$a_{r}\equiv\sqrt{\hbar/\mu\omega_{r}}$ for the relative atomic motion with
$\mu$ denoting the reduced atomic mass. We multiply $a_{r}$ by $1+\delta$ with
a small positive $0<\delta\ll 1$, to account for the anharmonicity of the tube
potential. The free space $s$-wave scattering length $a_{S}$ is expressed as a
function of the magnetic field $B$ according to Eq. 1. Finally, following Ref.
Gri:93 , to account for a realistic van der Waals scattering potential
$-C_{6}\,r^{-6}$, we introduce the finite range parameter
$r_{0}=(mC_{6}/32\hbar^{2})^{1/4}\,\Gamma(3/4)/\Gamma(5/4)$, and replace
$a_{S}$ by $a_{S}-r_{0}$. From Refs. Tic:04 ; Fal:08  we take
$C_{6}=3926\,a_{0}$ and $r_{0}=65\,a_{0}$. Note that Eq. 3 is configured such
that for zero interaction, zero binding energy is obtained. This theoretical
model, based on Refs. Gri:93 ; Ols:98 ; Ber:03 , reproduces the binding
energies in Fig. 2(d) remarkably well if one sets $\delta=0.142$, which
reasonably well agrees with the expected anharmonicity. The model accounts for
effects of reduced dimensionality arising if the degree of radial confinement
becomes comparable with $a_{S}$, giving rise to a confinement induced
resonance of the effective 1D scattering cross section. As a consequence,
bound states become possible for negative values of $a_{S}$ as is seen in Fig.
2(d). This has been previously reported for potassium Feshbach molecules in
the lowest transverse state of an array of isolated 1D optical traps in Ref.
Mor:05 . Note that Fig. 2(d) shows equal binding energies for molecules
prepared in the first and second bands. This results from engineering the
lattice potentials via adjustment of $V_{0}$ and $\Delta V$, such that the
lattice wells with predominant population, i.e., the deep wells, if the
molecules are prepared in the first and the shallow wells if prepared in the
second band, respectively, exhibit equal values of $\omega_{r}$ and $a_{r}$.
For example, with $V_{0}^{(M)}=40\,E_{\textrm{rec}}^{(M)}$ and
$\theta=0.4\,\pi$ for molecules in the first band, and
$V_{0}^{(M)}=60\,E_{\textrm{rec}}^{(M)}$ and $\theta=0.54\,\pi$ for molecules
in the second band, as used in Fig. 2(d), $\omega_{r}=2\pi\,\times 30\,$kHz is
obtained.

Next, we discuss molecules prepared in the second band
($\ket{\textrm{Mol}:2}$) for a shallow lattice with
$V_{0}^{(M)}=10\,E_{\textrm{rec}}^{(M)}$, $\theta=0.54\,\pi$ and hence $\Delta
V^{(M)}=5\,E_{\textrm{rec}}^{(M)}$ for an extended range of the magnetic field
below the Feshbach resonance, $B<B_{\textrm{res}}$. In Fig. 3(a), we show a
dissociation spectrum for molecules at $B=200.6\,$G corresponding to the dark
purple diamond-shaped data point in (c). The plot in (a) shows the second band
molecules (grey triangles), whose number is initially maximized, second band
atoms (magenta squares), and molecules (orange diamonds) and atoms (dark
purple disks) in the first band. The red dashed lines emphasize the
frequencies $\Delta f_{1}$ and $\Delta f_{2}$, where local minima in the
number of molecules in the second band are found, due to maximal efficiency of
the dissociation process. At frequencies $\Delta f\ll\Delta f_{1}$
dissociation is not resonant such that in mass spectrometry images one
observes the second BZ filled with molecules, as exemplified in the panel (I)
of Fig. 3(b), recorded at position (I) in (a). At the left red dashed line
($\Delta f_{1}$), dissociation arises due to a transition, mainly exciting
second band molecules $\ket{\textrm{Mol}:2}$ to atomic pairs
$\ket{\textrm{Aux},\downarrow\,:1}$ in the first band. This is confirmed by
the mass spectrometry image in panel (II) in (b), recorded at position (II) in
(a), close to the left red dashed line. Here, a partial filling of the first
BZ with atoms is observed. The dissociation process shows limited efficiency
due to the small Franck Condon overlap between the wave functions involved,
similarly as discussed in the context of Fig. 2(a) for the transition
$\ket{\textrm{Mol}:1}\rightarrow\ket{\textrm{Aux},\downarrow\,:4}$. Around the
second red dashed line ($\Delta f_{2}$), the dissociation transition couples
$\ket{\textrm{Mol}:2}$ to $\ket{\textrm{Aux},\downarrow\,:2}$. Both states
belong to second bands with a sizable Franck Condon overlap, such that the
dissociation around $\Delta f_{2}$ is notably more efficient than for $\Delta
f_{1}$, as seen by the nearly complete depletion of the molecule population in
(a) around $127\,$kHz. Panel (III) in Fig. 3(b) confirms that the dissociated
atoms in fact arise in the second band, giving rise to a filled second BZ for
atoms, twofold increased as compared to the second BZ for molecules in panel
(I). Note that $\Delta f_{2}-\Delta f_{1}$ is approximately given by the
separation of the second and first bands for atoms. A determination of $\Delta
f_{1},\Delta f_{2}$ below the scale of a few kHz is not supported by the width
of the spectral features observed near the red dashed lines in Fig. 3(a). In
Fig. 3(c), binding energies for molecules in the second band are shown,
measured by analyzing dissociation spectra as in (a) for different magnetic
fields. The grey solid line represents the theoretical prediction according to
Eq. 3, showing remarkable agreement.

Figure 3: Binding energies in a shallow lattice. (a) Dissociation spectrum for
molecules initially prepared in the second band with a lattice depth parameter
$V_{0}^{(M)}=10\,E_{\textrm{rec}}^{(M)}$ and $\theta=0.54\,\pi$. The magnetic
field is $B=200.6\,$G corresponding to the dark purple diamond in (c). Orange
diamonds (dark purple disks) show populations of molecules (atoms) in the
first band. Grey triangles (magenta squares) show populations of molecules
(atoms) in the second band. The red dashed lines indicate the two frequencies
$\Delta f_{1}$ and $\Delta f_{2}$, where dissociation maximally depletes the
population of molecules in the second band. The error bars show the standard
deviation of the mean for $15$ experimental runs. (b) Mass spectrometry images
for the values of $\Delta f$ indicated by I, II, III in (a). The orange
diamonds in (c) show binding energies obtained from spectra as plotted in (a)
for varying magnetic fields. The errors are estimated from the spectral width
of the underlying dissociation resonance to be less than $5\,$kHz, i.e. an
order of magnitude smaller than the data symbols. The grey solid line presents
the theoretical prediction of Eq. 3.

Molecular decay dynamics. In this section we discuss the observation of two
relaxation channels for Feshbach dimers Shi:11 . The first process dominates
for $\xi\equiv(k_{F}a_{S})^{-1}\gg 1$ ($k_{F}\equiv$ Fermi momentum), i.e.,
for relatively weak scattering lengths, where Feshbach dimers become deeply
bound. In this regime, the primary relaxation process is based on inelastic
dimer-dimer collisions, where one molecule gains binding energy, while the
other is dissociated, such that both molecules are lost. Dimer-atom collisions
are less relevant since our experiments start with a nearly pure molecule
sample. Larger binding energies provide larger Franck-Condon overlap between
the involved molecular wave functions, so that the molecular lifetime $\tau$
should decrease with $\xi$, in accordance with the scaling
$\tau\propto(a_{S})^{2.55}$ predicted in absence of a lattice Pet:05 . In the
other extreme for $\xi\ll 1$, according to Ref. Pet:05 , the elastic dimer-
dimer scattering length $a_{dd}$ scales as $a_{dd}\approx 0.6\,a_{S}$ and
hence due to the large size of $a_{S}$, molecular three-body collisions are
expected to introduce molecule loss and to give rise to a lifetime that
decreases, when $\xi$ approaches zero.

Figure 4: Relaxation dynamics. (a) Half-lives $\tau$ of Feshbach dimers in the
dipole trap plotted versus $\xi$ (upper axis) and $a_{S}$ (lower axis). The
inset shows an exemplary measurement of $\tau$ for the light green data point
indicated by the black arrow. The black solid line is a fit for the domain
$\xi>2$ with a straight line, extrapolated to the domain $\xi<2$, indicating a
power law $\propto\xi^{-\kappa}$ with $\kappa=0.84\pm 0.05$. The orange disks
and magenta squares in (b) show half-lives of Feshbach dimers prepared in the
first and second Bloch bands of an optical lattice, respectively. The lattice
depth is $V_{0}^{(M)}=8\,E_{\textrm{rec}}^{(M)}$ and $\theta=0.4\,\pi$ for the
first Bloch band and $0.535\,\pi$ for the second Bloch band. The black solid
line is a common fit for data of both bands within the domain $\xi>2$ with a
straight line, extrapolated to the domain $\xi<2$ (dashed continuation),
indicating a power law $\propto\xi^{-\kappa}$ with $\kappa=1.28\pm 0.06$. The
error bars for $\tau=(n_{0}\,\beta)^{-1}$ in (a) and (b) are determined by
propagating the errors for $n_{0}$ and $\beta$ obtained in the fit procedure.
The insets at the top boundary of (b) illustrate the different relaxation
paths found in the second Bloch band for small and large $\xi$ (cf. text).

In Fig. 4(a), we benchmark the lifetimes of the Feshbach molecules trapped in
the approximately harmonic optical dipole trap. After the dimers are formed
(Fig. 1(c)), the scattering length $a_{S}$ is adjusted to a desired value via
the magnetic field according to Eq. 1 and the dimers are held for a variable
time in the trap. In order to count the remaining number of molecules $n_{M}$,
the magnetic field is rapidly tuned to $B_{\textrm{img}}=200.46\,$G,
associated with a moderate value of $a_{S}$ and the molecules are allowed to
ballistically expand (nearly unimpaired by interaction) during $22\,$ms,
before an absorption image of the $\ket{\uparrow}$ atoms is recorded. The
dimer population plotted against the hold time is fitted with the two-body
decay model $\dot{n}_{M}=-\beta\,n_{M}^{2}$, with the solution
$n(t)=n_{0}\cdot(1+t/\tau)^{-1}$, the initial number of molecules $n_{0}\equiv
n(0)$, and the half-time of the molecule sample $\tau\equiv(n_{0}\beta)^{-1}$.
An exponential fit, assuming density independent loss, or a three-body decay
model fail to describe the data. The obtained half-time $\tau$ is plotted in
Fig. 4(a) versus $\xi$ and $a_{S}$. The inset shows an exemplary measurement
of $\beta$ and $n_{0}$ leading to the light green data point indicated by a
black arrow. The main panel shows strikingly different behaviour in the
unitarity regime $\xi<1$, where a nearly constant half-time $\tau$ around
$100\,$ms is observed, and in the regime $\xi>1$, where the data are well
fitted with a straight line, indicating a power law $\tau\propto\xi^{-\kappa}$
with an exponent $\kappa=0.84\pm 0.05$. The largest values of $\tau$ are found
at $\xi\approx 1$. Note that, $\kappa$ does not agree with the prediction
$2.55$ for dimer-dimer collisions in Ref. Pet:05  or the experimental value
$\approx 2.3$ for mixed dimer-dimer and dimer-atom collisions, reported in
Ref. Reg2:04 .

In Fig. 4(b), an analogous analysis is carried out in presence of a shallow
optical lattice with $V_{0}^{(M)}=8\,E_{\textrm{rec}}^{(M)}$, which permits
nearest-neighbour tunneling on a sub-millisecond time scale. The orange disks
show the observed half-lives for molecules prepared in the first Bloch band
with $\theta=0.4\,\pi$ and hence $\Delta
V_{0}^{(M)}=-9.88\,E_{\textrm{rec}}^{(M)}$. The magenta squares correspond to
$\theta=0.535\,\pi$ (i.e., $\Delta V_{0}^{(M)}=3.51\,E_{\textrm{rec}}^{(M)}$)
for molecules prepared in the second Bloch band. In the regime $\xi>2$, both
data sets show the same dependence on $\xi$ and are fitted with the same
straight line (black solid line in (b)), indicating a power law model
$\propto\xi^{-\kappa}$ with $\kappa=1.28\pm 0.06$. The dashed black line is a
continuation into the $\xi<2$ domain. Note that also in the presence of a
lattice, $\kappa$ does not agree with the prediction $2.55$ in Ref. Pet:05  or
the experimental value $\approx 2.3$ from Ref. Reg2:04 , both obtained for a
scenario without a lattice. For $\xi<1$, we find that $\tau$ grows with
increasing $\xi$. While the data for the second band very well agree with the
results for the first band in the entire range $\xi>2$, for $\xi<2$ a dramatic
decrease of $\tau$ is observed for the second band, which only rises up to a
threefold shorter lifetime than observed for the first band. This indicates
that an additional relaxation channel opens for the second band if $\xi<2$. In
fact, as illustrated in the two insets at the upper edge of Fig. 4(b), for
$\xi$ close to zero, e.g. for $\xi\approx 0.1$, we see an initial pronounced
decay of molecules into the first band, followed by a subsequent decay of the
first band population, while in the domain $\xi>2$, e.g., close to $\xi\approx
6$, the molecules do not initially transit to the first band during
relaxation. This may be explained as follows: For small $\xi$, i.e., large
$a_{S}$, in the second band, the increasing elastic binary collision cross
section can lead to a redistribution of energy from the lattice plane into the
tube direction, which gives rise to an additional escape channel from the
lattice. A similar relaxation channel for weakly interacting bosonic atoms has
been recently reported in Ref. Nus:20 . Also effects of reduced dimensionality
in the tubular lattice sites are expected to be larger in the first than in
the second band, which may also contribute to explain the longer lifetimes of
first band dimers Ven:22 .

In summary, our work demonstrates strongly correlated ultracold Feshbach
dimers prepared in higher Bloch bands of an optical lattice, which give rise
to orbital physics. Using a method reminiscent of mass spectrometry, we find
surprisingly long molecular lifetimes in the unitarity regime on the order of
hundred milliseconds and study binding energies and dissociation and
relaxation dynamics. Our work prepares the stage for future studies of BEC-BCS
cross over physics with orbital degrees of freedom.

Methods

Mass spectrometry protocol In order to uniquely distinguish and selectively
count the populations of atoms and Feshbach dimers in presence of an optical
lattice, they are separated in a ballistic time-of-flight protocol, which maps
the population of the $n$-th band to the $n$-th Brillouin zone (BZ). This
protocol consists of a rapid adiabatic termination of the lattice potential
(in $5\,$ms) followed by a ballistic expansion (in $22\,$ms). During the
second half of the expansion the magnetic field is tuned to
$B_{\textrm{img}}=200.6\,$G in order to enable absorption imaging at the same
resonant frequency in each experimental run. For the case that the dimer mass
$M$ is twice that of an atom $m$, atoms travel twice as fast compared to
homonuclear dimers with the same initial momentum. Hence, this protocol
arranges the atoms and molecules in velocity space with a BZ structure scaled
up by a factor two for atoms as compared to dimers. This is sketched in Fig.
5. The figure shows the first and second BZs for atoms in the background
(black and orange areas) and for molecules in the foreground (grey and red
areas), giving rise to a nested structure of four white squares denoted 1-4.
The optical densities integrated across the areas enclosed by these squares
are denoted as $\Box_{\nu},\nu\in\\{1-4\\}$. We denote the total populations
as $A_{\nu}$ and $D_{\nu}$ for atoms and molecules in the band with band index
$\nu\in\\{1,2\\}$, respectively. If sectors in Fig. 5 comprise atoms and
dimers, the counting protocol requires the assumption that the first band for
atoms is uniformly filled. This assumption is reasonable since, in the
experiments described in the main text, we typically start with the lowest
band filled with fermionic atoms, form Feshbach dimers, and subsequently
excite a fraction of them to the second band without changing their quasi-
momenta Hac:21 . Note that due to the bosonic nature of the Feshbach dimers,
the respective Brillouin zones (red and grey areas in Fig. 5) are not
necessarily filled homogeneously, since Bose-enhancement and hence multiple
occupancy of available energy states are possible. Under the condition of a
homogeneous filling of the first atomic band, the populations
$A_{1},A_{2},D_{1},D_{2}$ in different sectors of Fig. 5 are indicated in the
figure accounting for the four-fold rotation symmetry of the BZs and the four
times larger BZ areas for atoms. This leads to the following relations:
$A_{2}=\Box_{1}-\Box_{2}$, $A_{1}=2(\Box_{2}-\Box_{3})$,
$D_{2}=\frac{3}{2}\Box_{3}-\Box_{4}-\frac{1}{2}\Box_{2}$, and
$D_{1}=\frac{1}{2}\Box_{3}-\frac{1}{2}\Box_{2}+\Box_{4}$. If all atoms and
molecules reside in the second Band, i.e. $A_{1}=D_{1}=0$, the simple case
$A_{2}=\Box_{1}-\Box_{2}$ and $D_{2}=\Box_{3}-\Box_{4}$ arises, in which case
the condition of uniform filling is not required.

Figure 5: Evaluation of atom and dimer populations. $xy$-plane of velocity
space with $v_{m}=\sqrt{2}\,2\hbar k/m$. $A_{\nu}$ and $D_{\mu}$ denote the
total atom and dimer populations, respectively, within the Bloch bands with
band index $\nu,\mu\in\\{1,2\\}$. The terms $a_{\nu}A_{\nu}+d_{\mu}D_{\mu}$,
plotted inside different sectors bordered by white lines, specify the
respective total particle numbers. See text.

Acknowledgments We thank Raphael Eichberger for help in the early stage of the
experiment. We acknowledge support from the Deutsche Forschungsgemeinschaft
(DFG) through the collaborative research center SFB 925 (Project No.
170620586, C1). M.H. was partially supported by the Cluster of Excellence CUI:
Advanced Imaging of Matter of the Deutsche Forschungsgemeinschaft (DFG) - EXC
2056 - project ID 390715994.

References

## References

  * (1) Regal, C. A., Greiner, M. & Jin, D. S. Observation of resonance condensation of fermionic atom pairs, Phys. Rev. Lett. 92, 040403 (2004).
  * (2) Regal, C. A., Greiner, M. & Jin, D. S. Lifetime of molecule-atom mixtures near a Feshbach resonance in 40K, Phys. Rev. Lett. 92, 083201 (2004).
  * (3) Zwierlein, M. W., Stan, C. A., Schunck, C. H., Raupach, S. M. F., Kerman, A. J. & Ketterle, W. Condensation of pairs of fermionic atoms near a Feshbach Resonance, Phys. Rev. Lett. 92, 120403 (2004).
  * (4) Bourdel, T., Khaykovich, L., Cubizolles, J., Zhang, J., Chevy, F., Teichmann, M., Tarruell, L., Kokkelmans, S. J. J. M. F. & Salomon, C. Experimental study of the BEC-BCS crossover region in 6Lithium, Phys. Rev. Lett. 93, 050401 (2004).
  * (5) Bartenstein, M., Altmeyer, A., Riedl, S., Jochim, S., Chin, C., Hecker Denschlag, J. & Grimm, R. Crossover from a molecular Bose-Einstein condensate to a degenerate Fermi gas, Phys. Rev. Lett. 92, 120401 (2004).
  * (6) Ketterle, W. & Zwierlein, M. W. Making, probing and understanding ultracold Fermi gases , Rivista del Nuovo Cimento 31, p.247-422 (2008).
  * (7) Zwerger, W., ed., The BCS-BEC crossover and the unitary Fermi gas, Springer-Verlag Berlin, Heidelberg (2012).
  * (8) Törmä, P. & Sengstock, K., eds., Quantum gas experiments, exploring many-body states , Imperial College Press London (2015).
  * (9) Lewenstein, M., Sanpera, A., Ahufinger, V., Damski, B., Sen(De), A. & Sen, U. Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond, Adv. Phys. 56, p.243-379 (2007).
  * (10) Gross, C. & Bloch, I. Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
  * (11) Stöferle, T., Moritz, H., Günter, K., Köhl, M. & Esslinger, T. Molecules of fermionic atoms in an optical lattice, Phys. Rev. Lett. 96, 030401 (2006).
  * (12) Chin, J. K., Miller, D. E., Liu, Y., Stan, C., Setiawan, W., Sanner, C., Xu, K. & Ketterle, W. Evidence for superfluidity of ultracold Fermions in an optical lattice, Nature 443, 961 (2006).
  * (13) Moritz, H., Stöferle, T., Günter, K., Köhl, M. & Esslinger, T. Confinement-induced molecules in a 1D Fermi gas, Phys. Rev. Lett. 94, 210401 (2005).
  * (14) Olshanii, M. Atomic scattering in the presence of an external confinement and a gas of impenetrable Bosons, Phys. Rev. Lett. 81, 938 (1998).
  * (15) Bergeman, T., Moore, M. G. & Olshanii, M. Atom-atom scattering under cylindrical harmonic confinement: numerical and analytic studies of the confinement-induced resonance, Phys. Rev. Lett. 91, 163201 (2003).
  * (16) Venu, V., Xu, P., Mamaev, M., Corapi, F., Bilitewski, T., D’Incao, J. P., Fujiwara, C. J., Rey, A. M. & Thywissen, J. H. Observation of unitary $p$-wave interactions between fermions in an optical lattice, Nature 613, 262-267 (2023).
  * (17) Tokura, Y. & Nagaosa, N. Orbital physics in transition-metal oxides, Science 288, pp.462-468 (2000).
  * (18) Maekawa, S., Tohyama, T., Barnes, S. E., Ishihara, S., Koshibae, W. & Khaliullin, G. Physics of transition metal oxides, Springer Series in Solid-State Sciences Vol. 144, Springer (2004).
  * (19) Li, X. & Liu, W. V. Physics of higher orbital bands in optical lattices: a Review, Rep. Prog. Phys. 79, 116401 (2016).
  * (20) Wirth, G., Ölschläger, M. & Hemmerich, A. Evidence for orbital superfluidity in the P-band of a bipartite optical square lattice, Nat. Phys. 7, 147-153 (2011).
  * (21) Kock, T., Hippler, C., Ewerbeck, A. & Hemmerich, A. Orbital optical lattices with bosons, J. Phys. B: At. Mol. Opt. Phys. 49, 042001 (2016).
  * (22) Wang, X.-Q., Luo, G.-Q., Liu, J.-Y., Huang, G.-H., Li, Z.-X., Wu, C., Hemmerich, A. & Xu, Z.-F. Observation of nematic orbital superfluidity in a triangular optical lattice, arXiv:2211.05578 (2022).
  * (23) Wang, X.-Q., Luo, G.-Q., Liu, J.-Y., Liu, W. V., Hemmerich, A. & Xu, Z.-F. Evidence for an atomic chiral superfluid with topological excitations, Nature. 596, 227-231 (2021).
  * (24) Hachmann, M., Kiefer, Y. H., Riebesehl, J., Eichberger, R. & Hemmerich, A. Quantum degenerate Fermi gas in an orbital optical lattice, Phys. Rev. Lett. 127, 21, 033201 (2021).
  * (25) Chin, C., Grimm, R., Julienne, P. & Tiesinga, E. Feshbach resonances in ultracold gases, Rev. of Mod. Phys. 82, 1225 (2010).
  * (26) Gribakin, G. F. & Flambaum, V. V. Calculation of the scattering length in atomic collisions using the semiclassical approximation, Phys. Rev. A 48, 546 (1993).
  * (27) Ticknor, C., Regal, C. A., Jin, D.S. & Bohn, J. L. Multiplet structure of Feshbach resonances in nonzero partial waves, Phys. Rev. A 69, 042712 (2004).
  * (28) Falke, S., Knöckel, H., Friebe, J., Riedmann, M., Tiemann, E. & Lisdat, Ch. Potassium ground-state scattering parameters and Born-Oppenheimer potentials from molecular spectroscopy, Phys. Rev. A 78, 012503 (12008).
  * (29) Zhang, S. & Ho, T.-L. Atom loss maximum in ultracold Fermi gases, New J. of Phys. 13 055003 (2011).
  * (30) Petrov, D. S., Salomon, C. & Shlyapnikov, G. V. Diatomic molecules in ultracold Fermi gases - novel composite bosons, J. Phys. B: At. Mol. Opt. Phys. 38, pp.645-660 (2005).
  * (31) Nuske, M., Vargas, J., Hachmann, M., Eichberger, R., Mathey, L. & Hemmerich, A. Metastable order protected by destructive many-body interference, Phys. Rev. Res. 2, 043210 (2020).