# Emergent s-wave interactions between identical fermions
in quasi-one-dimensional geometries

Kenneth G. Jackson  Department of Physics and CQIQC, University of Toronto,
Toronto, Ontario M5S 1A7, Canada  Colin J. Dale  Department of Physics and
CQIQC, University of Toronto, Toronto, Ontario M5S 1A7, Canada  Jeff Maki
Department of Physics and HKU-UCAS Joint Institute for Theoretical and
Computational Physics at Hong Kong, The University of Hong Kong, Hong Kong,
China  Kevin G. S. Xie  Department of Physics and CQIQC, University of
Toronto, Toronto, Ontario M5S 1A7, Canada  Ben A. Olsen  Yale-NUS College,
Singapore 138527, Singapore  Denise J. M. Ahmed-Braun  Department of Physics,
Eindhoven University of Technology, The Netherlands  Shizhong Zhang
Department of Physics and HKU-UCAS Joint Institute for Theoretical and
Computational Physics at Hong Kong, The University of Hong Kong, Hong Kong,
China  Joseph H. Thywissen  Department of Physics and CQIQC, University of
Toronto, Toronto, Ontario M5S 1A7, Canada

###### Abstract

Orbital degrees of freedom play an essential role in metals, semiconductors,
and strongly confined electronic systems. Experiments with ultracold atoms
have used highly anisotropic confinement to explore low-dimensional physics,
but typically eliminate orbital degrees of freedom by preparing motional
ground states in strongly confined directions. Here we prepare multi-band
systems of spin-polarized fermionic potassium (40K) in the quasi-one-
dimensional (q1D) regime and quantify the strength of atom-atom correlations
using radio-frequency spectroscopy. The activation of orbital degrees of
freedom leads to a new phenomenon: a low-energy scattering channel that has
even particle-exchange parity along the q1D axis, as if the underlying
interactions were s-wave. This emergent exchange symmetry is enabled by
orbital singlet wave functions in the strongly confined directions, which also
confer high-momentum components to low-energy q1D collisions. We measure both
the q1D odd-wave and even-wave “contact” parameters for the first time, and
compare them to theoretical predictions of one-dimensional many-body models.
The strength and spatial symmetry of interactions are tuned by a p-wave
Feshbach resonance and by transverse confinement strength. Near resonance, the
even-wave contact approaches its theoretical unitary value, whereas the
maximum observed odd-wave contact remains several orders of magnitude below
its unitary limit. Low-energy scattering channels of multi-orbital systems,
such as those found here, may provide new routes for the exploration of
universal many-body phenomena.

##  I Introduction

Strong confinement of many-body systems has proven to be a fruitful approach
for the creation of novel and emergent quantum states. For instance, the
confinement of an electron gas in a two-dimensional quantum well with a strong
perpendicular magnetic field leads to the manifestation of the integer and
fractional quantum Hall effects, which exhibit exceptional transport
properties [1]. Similarly, relaxing the electron spin degrees of freedom can
result in the emergence of novel excitations such as skyrmions [2, 3]. In
ultracold atomic gases, recent advances in the manipulation of quantum gases
in low dimensions have led to similar progress. For instance, fractional spin
and charge degrees of freedom have been observed in atomic gases under strong
transverse confinement [4, 5]. Moreover, confinement-induced resonances [6]
can be used to change the interaction between atoms, providing a platform to
investigate many-body physics in a disorder-free environment [7].

In studies with cold atoms, research has typically focused on regimes where
transverse excitations along the strong confinement directions are irrelevant,
with some exceptions such as Ref. [8]. For instance, in a highly elongated
trap, it is commonly assumed that the atoms occupy the lowest transverse
oscillator state, such that the only active degrees of freedom are those along
the weakly confined (axial) direction. As a result, three-dimensional (3D)
systems of spin-polarized bosons and spin-half fermions under such confinement
have been successfully modeled as quasi-one-dimensional (q1D) systems whose
interactions are even-wave, i.e., described by one-dimensional (1D) pair wave
functions that have exchange-even symmetry [6, 9, 10, 11, 12, 13, 14, 15].

Here we investigate the role of transverse orbital degrees of freedom in a
simplified setting: a spin-polarized Fermi gas of 40K with p-wave interactions
confined in a q1D geometry. By eliminating the spin degree of freedom, the
possibility of spin-singlet s-wave scattering is excluded, so that the role of
orbital degrees of freedom is clearly exhibited. Our work builds on prior
studies of odd-wave interactions, i.e., described by one-dimensional (1D) pair
wave functions that have exchange-odd symmetry. Experimental work with low-
dimensional p-wave systems have suppressed collisions along the strongly
confined directions [16, 17, 18, 19]. Q1D p-wave systems have been the subject
of numerous theoretical studies [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32] including a proposal to enable a p-wave halo dimer state [33] and to
use strong confinement to reduce the three-body recombination loss [34, 35]
that plagues 3D p-wave gases [36, 37, 38, 39, 40].

Figure 1: Experimental overview. (a) An array of q1D traps aligned in the
vertical ($y$) direction is created by two orthogonal standing waves of light
at wavelength $\lambda_{L}$. A static magnetic field $B$ is oriented along
$z$. Radio-frequency fields are generated by a pair of proximate
microfabricated conductors. (b) Experimental sequence, depicting the potential
depth of the confining two-dimensional optical lattice versus time. (i)
Following the creation of a degenerate Fermi gas, the optical standing waves
are ramped on to generate the array of q1D traps. The two-part ramp includes a
slow increase (in $t_{\mathrm{ramp}}$) to minimize heating followed by a fast
jump to to isolate q1D ensembles. (ii) At this stage, time-of-flight band
mapping typically indicates that the majority of atoms are in the ground band
(central dashed box) but some atoms are also loaded into the first excited
band (lateral boxes). (iii) The lattice depth is optionally modulated for
$t_{\mathrm{AM}}$ before being ramped to its full depth. The frequency is
chosen to drive excited-band atoms resonantly into the third excited band,
from which they leave the trap. (iv) Time-of-flight images confirm that AM
reduces population in the excited band. (v) Feshbach-enhanced p-wave
interactions are activated by a spin flip from $\ket{a}$ to $\ket{b}$. After a
time $t_{\mathrm{hold}}$ the system is probed with a spectroscopic pulse at
detuning $\omega$. (vi) The fraction of atoms transferred from $\ket{b}$ to
$\ket{c}$ by the spectroscopic rf pulse is measured with state-selective
imaging.

In our experiment, we activate the orbital degree of freedom through
controlled population of the first excited transverse band. We discover
conditions under which fermions interact through wave functions with even
particle-exchange symmetry for the q1D degree of freedom along the axial
direction. We explain how fermionic symmetry is preserved by exchange-
antisymmetric transverse orbital states. The magnetic Feshbach resonance used
to tune p-wave interactions also adds an anisotropic nature to scattering.
This vector character determines the magnetic field values that can host
emergent s-wave resonances and can be used to unambiguously identify the
nature of the observed interactions.

We find clear signatures of interaction-induced correlations between atoms
through rf spectroscopy for both odd- and even-wave scattering along the axial
direction. Experimental investigations are supplemented by a theoretical
analysis that re-examines the two-body problem and presents a unified
treatment for arbitrary energies when multiple transverse bands are relevant.
This allows us to describe emergent even-wave scattering on equal footing with
odd-wave scattering and to relate this to novel correlations observed in rf
spectroscopy. Specifically, we relate the high-frequency limit of the rf
transfer rate between two internal spin states to the short-range correlations
between atoms through universal contact relations [41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52, 53, 54, 55, 56]. Our theoretical treatment delineates the
conditions under which the rf spin-flip rate is a measure of the 1D contact
parameter. Notably, the rf spectra match true 1D scaling up to
$2\omega_{\perp}$, where $\hbar\omega_{\perp}$ is the single-particle band
gap. We extract the odd-wave contact parameter anticipated by Refs. [57, 58,
59, 28, 60, 29] for the first time, and also measure the even-wave contact
parameter. A distinguishing feature of the latter is a novel singular
structure at $2\omega_{\perp}$ that is anticipated by our analysis.

Our work suggests that introducing energy-gapped orbital degrees of freedom
can be used more broadly to engineer the low-energy exchange symmetry of few-
body and many-body systems. Our theoretical analysis provides a framework for
understanding how these novel orbit-dressed collision channels emerge from the
one- to three-dimensional crossover. Both the conceptual paradigm and the
measurement protocols introduced here could be expanded to bosonic systems,
spin mixtures, quasi-two-dimensional systems, and alternative partial waves.

The manuscript is structured as follows. In Sec. II, we provide an overview of
the experimental protocol. In Sec. III, we describe p-wave scattering in the
presence of strong q1D confinement and orbital degrees of freedom. Adapted to
the experimental scenario, our treatment identifies three scattering continua
with distinct pair wave functions. In Sec. IV, we measure the q1D odd-wave
correlation strength and delineate the range of confinement strengths and
magnetic fields in which the contact parameter is governed by the q1D odd-wave
scattering length. We also demonstrate that the maximum contact parameter
remains two orders of magnitude below its expected unitary value. In Sec. V,
we present three experimental investigations of emergent s-wave interactions.
The even-wave contact parameter is seen to approach its expected unitary
value, and we estimate that the inferred interaction strength is at least an
order of magnitude stronger than that of odd-wave interactions. We conclude in
Sec. VI with a summary of key findings and a discussion of future prospects
enabled by multi-orbital scattering in low-dimensional systems.

##  II Experimental protocol and summary of observations

Our ensemble of q1D systems is prepared as follows. A bulk sample of 40K,
spin-polarized in the lowest hyperfine-Zeeman state $\ket{a}$, is confined in
an optical trap and cooled sympathetically with bosonic 87Rb. After the final
cooling stage, $N_{\mathrm{tot}}=7(1)\times 10^{4}$ spin-polarized 40K atoms
at $T\sim 300$ nK are loaded into a lattice of tube-like anisotropic traps
[see Fig. 1(a)]. The confinement along the $x$ and $z$ directions is created
by two orthogonal standing waves, produced with retro-reflected beams of light
with wavelength $\lambda_{L}=760.6$ nm.

In the deep-lattice limit where tunneling between adjacent tubes can be
neglected, each lattice site is harmonic near its minimum, with longitudinal
oscillation frequency $\omega_{y}\approx 2\pi\times 400$ Hz. The transverse
confinement in turn is set by the lattice depth $V_{L}$, measured in units of
the recoil energy, $E_{R}=h^{2}/(2m\lambda_{L}^{2})\approx h\times 8.6$ kHz,
where $m$ is the atomic mass of 40K. With confinement in both transverse
directions set to $V_{L}=40\,E_{R}$, for instance, the transverse oscillation
frequency is $\omega_{\perp}\approx 2\pi\times 100$ kHz. The transformation of
the 3D cloud into an array of q1D clouds begins with a ramp of the lattice
intensity, across a duration $t_{\mathrm{ramp}}$ (typically $300$ ms), to a
modest depth, less than $10E_{R}$ [Fig. 1(b)(i)]. Tubes are then isolated with
a more rapid ramp to $40(2)\,E_{R}$. We use the loading model described in
App. F to estimate the final number of atoms and temperature distribution in
the tubes. For typical experimental parameters, the peak number per tube is
$N\approx 30$ and $T/T_{F}\approx 0.8$, where $T_{F}$ is the 1D Fermi
temperature.

Band populations are measured using band-mapped time-of-flight imaging. We
adjust the chemical potential of the gas by tuning the optical trap depth and
thereby optimize for either ground-band loading or multi-band loading. For the
latter, when the chemical potential of the bulk gas is comparable to $E_{R}$,
some atoms are loaded into the first excited state of each transverse
direction since loading at the band edge cannot be adiabatic [Fig. 1(b)(ii)].
The fraction of atoms loaded into the first excited state is typically $15\%$
in each direction.

To prepare for dedicated experiments measuring odd-wave correlations without
orbital excitations (Sec. IV), we deplete the excited-band population using
amplitude modulation (AM). The power of both optical standing waves is
modulated for a time $t_{\mathrm{AM}}$ at a frequency resonant with the
transition from the single-atom first-to-third excited bands [see Fig.
1(b)(iii)]. Due to the anharmonicity of the standing-wave confinement, this
modulation is off-resonant from the zero-to-two excitation, leaving the number
of atoms in the lowest band relatively unaffected. When optimizing for ground-
band loading, band mapping after AM [Fig. 1(b)(iv)] indicates that the
excited-band population is reduced to $\lesssim 5\%$ in each lattice
direction. Lattice depth calibration is performed via AM of the single-
particle zero-to-two excitation, with $5\%$ uncertainty. Ensembles for even-
wave correlation measurements (Sec. V) are prepared without AM, or with phase
modulation of the lattice to enhance population in the first excited bands
(Sec. E).

With either preparation, the q1D ensembles of atoms in the lowest hyperfine-
Zeeman state $\ket{a}$ are long-lived and weakly interacting. A p-wave
Feshbach resonance [36, 37, 38] exists in the $bb$ collision channel, where
$\ket{b}$ is the second-lowest hyperfine Zeeman state, adiabatically connected
to the $\ket{F=9/2,m_{F}=-7/2}$ state at low magnetic field. After setting the
field $B$ and lattice depth $V_{L}$ for a particular measurement, interactions
are snapped on by transferring the atoms to the state $\ket{b}$ using a
$40$-$\mu$s-long rf pulse [see Fig. 1(b)(v)].

After a hold time $t_{\mathrm{hold}}$ of typically $160\,\mu$s, a
spectroscopic pulse of duration $\tau=120\,\mu$s is applied. The frequency is
offset $\omega$ above the single-particle $\ket{b}$-to-$\ket{c}$ resonant
frequency, where $\ket{c}$ is the third-lowest hyperfine-Zeeman state (and
assumed to be weakly interacting: see App. C). The number of atoms
transferred, $N_{c}$, is assessed by state-sensitive imaging in time-of-
flight, shown in Fig. 1(b)(vi). The spectroscopic pulse has a Blackman
envelope to minimize Fourier broadening, such that the measured transfer rate
$\Gamma$ is given by $N_{c}/\tau$ times a corrective envelope factor.

Figure 2: Overview of observed correlations. The normalized rf transfer rate
$\widetilde{\Gamma}$, which is a measure of atom-atom correlation strength, is
indicated by color across a range of magnetic fields and optical lattice
depths. The depth of the confining optical lattice, $V_{L}$, is given in units
of the recoil energy $E_{R}=h^{2}/2m\lambda_{L}^{2}$. The first peak in
$\widetilde{\Gamma}$ lies between the odd-wave resonance (indicated by a solid
black line at $B_{0}^{y}$, where $a_{o}^{-1}=0$ as discussed in Sec. III) and
$(k_{F}a_{o})^{-1}=-10$ (indicated by a black dashed line). (a) Without active
orbital degrees of freedom, atom-atom correlations are found only within this
range. (b) With multi-band loading, correlations beyond the odd-wave regime
are found. We show that these additional correlations are due to even-wave
interactions along the q1D direction. The structure of
$\widetilde{\Gamma}(B,V_{L})$ is further explored in Figs. 5, 6, and 8.

Figure 2 provides a side-by-side comparison of correlations found with and
without activated orbital degrees of freedom. The normalized rf transfer rate
$\widetilde{\Gamma}=E_{F}\Gamma/(\hbar\pi\Omega^{2}N_{\mathrm{tot}})$ is shown
versus lattice depth and magnetic field, where $\Omega$ is the Rabi frequency
and $E_{F}$ is the calculated Fermi energy of the central tube. As will be
discussed in Section III.1, $\Gamma$ is proportional to short-range
correlations, which are quantified by so-called contact parameters [41, 42,
43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56] and have been the
object of prior experimental studies in 3D for both s-wave and p-wave systems
[61, 62, 63, 64, 65, 54] and in quasi-two-dimensional systems with s-wave
interactions [66, 67].

Without active orbital degrees of freedom, atom-atom correlations are found
just above the odd-wave resonance ($B_{0}^{y}$ as defined later) indicated by
a solid line in Figure 2(a). At $B_{0}^{y}$, the q1D odd-wave scattering
length $a_{o}$ is predicted to diverge, due to interactions along the
longitudinal direction. The observed normalized transfer rate,
$\widetilde{\Gamma}$, is largest at magnetic fields between $B_{0}^{y}$ and
where $a_{o}=-(10k_{F})^{-1}$ (dashed line), where $k_{F}$ is the Fermi
momentum in the central tube. In Sec. IV, we show that the observed
correlations agree well with those expected from weakly attractive odd-wave
interactions.

Figure 2(b) displays the same quantities, but with orbital degrees of freedom
activated. The normalized rf transfer rate, $\widetilde{\Gamma}$, now reveals
that atom-atom correlations appear well beyond the odd-wave parameter range
(still bracketed by the solid and dashed lines). In Sec. V, we provide clear
assignment of these correlations to even-wave interactions and compare their
strength to that predicted by theory.

##  III p-wave scattering in a quasi-one-dimensional trap

Low-energy scattering in the q1D trap has been considered previously for even-
wave scattering [6, 9, 10, 11, 12, 13, 14, 15] of both identical bosons and
spin mixtures of fermions, as well as for for odd-wave scattering [20, 21, 22,
23, 24, 25, 26, 27] of identical fermions. These treatments consider atoms to
reside in the ground state of the transverse confinement, such that higher-
energy orbital states are only relevant as closed channels [25]. Our
experimental scenario requires a generalization that includes population of
asymptotic states with transverse orbital excitation. When these additional
orbital states are hybridized with 3D p-wave dimer states, whose inter-nuclear
separation is typically much smaller than the transverse oscillator length,
the new scattering resonances which emerge include both even- and odd-wave q1D
symmetries.

Table 1: Summary of notation used to parameterize scattering in this work.  resonance type | scattering | scattering
---|---|---
amplitude | parameters
3D p-wave | $f_{\textrm{3D}}$ |  $V_{\textrm{3D}}$, $R_{\textrm{3D}}$
1D odd-wave | $f_{o}$ |  $a_{o}$, $r_{o}$
1D even-wave | $f_{e}$ |  $a_{e}$, $r_{e}$

To obtain the effective scattering parameters in q1D, let us first recall that
the p-wave scattering amplitude in 3D, $f_{\mathrm{3D}}$, is given by

$-(kf_{\mathrm{3D}})^{-1}=i+(V_{\mathrm{3D}}k^{3})^{-1}+(kR_{\mathrm{3D}})^{-1}+\mathcal{O}(k)\,,$
(1)

where $V_{\mathrm{3D}}$ and $R_{\mathrm{3D}}>0$ are the p-wave scattering
volume and effective range, respectively, and $\hbar k$ is the relative
momentum. Near a p-wave Feshbach resonance, $V_{\mathrm{3D}}$ is widely
tunable [68, 69, 70], but the Wigner bound [71, 72, 73] constrains the maximum
value of $R_{\mathrm{3D}}$ to be on the order of the van der Waals length,
which is small compared to typical $k^{-1}$ in the ultracold regime. Thus
$(kR_{\mathrm{3D}})^{-1}\gg 1$ and, according to Eq. (1), remains numerically
important for 3D p-wave resonant scattering. In fact, it is crucial to include
the effective-range term in Eq. (1) to describe many physical properties of
p-wave Fermi gases appropriately [74, 75, 76, 73, 54, 53, 77, 78].

Now, let us consider a Fermi gas of identical atoms, such as 40K in a single
hyperfine-Zeeman state, confined in a highly anisotropic potential. The
single-particle Hamiltonian within a single tube is, neglecting anharmonic
corrections,

$H_{0}=\frac{1}{2m}{\bm{p}}^{2}+\frac{1}{2}m(\omega_{x}^{2}x^{2}+\omega_{y}^{2}y^{2}+\omega_{z}^{2}z^{2})\,,$
(2)

where ${\bm{p}}=(p_{x},p_{y},p_{z})$ is the 3D momentum of the atom. Quasi-
one-dimensional confinement is achieved when the transverse harmonic
frequencies in the $x$ and $z$ direction are much larger than in the $y$
direction: $\\{\omega_{x},\omega_{z}\\}\gg\omega_{y}$. This allows a
separation of energy scales, in which the collision energy (determined by the
1D Fermi energy $E_{F}$ at low temperature, $T\lesssim E_{F}/k_{B}$) is much
smaller than the energy quanta of the transverse motion, namely
$E_{F}\ll\\{\hbar\omega_{x},\hbar\omega_{z}\\}$. For a true 1D system of $N\gg
1$ spinless fermions, the Fermi energy is $E_{F}=N\hbar\omega_{y}$, so that
the condition for q1D collisions is

$N\lambda\ll 1\,,$ (3)

where $\lambda\equiv\omega_{y}/\omega_{\perp}$ is the trap aspect ratio,
written in the case of cylindrically symmetric confinement,
$\omega_{x}=\omega_{z}\equiv\omega_{\perp}$. For our typical confinement
strength and atom numbers (see Sec. II), $N\lambda\sim 0.1$, i.e., well within
the q1D regime.

###  III.1 Scattering in q1D regime

For non-interacting atoms of equal mass, the two-body Schrödinger equation can
be separated into center-of-mass (CM) and relative (rel) motion, each
described by a Hamiltonian with the same form as Eq. (2) except for the
replacement of the mass $m$ with the total mass $2m$ or the reduced mass
$\mu=m/2$, respectively [79, 80, 81]111In q1D s-wave systems, the
anharmonicity of the transverse confinement potential was observed [79, 81] to
induce couplings between the center-of-mass motion and the relative motion.
This created new scattering resonances to bound states with CM excitation in
the strongly confined direction [80]. However, we did not find any evidence of
this effect in our data. In the limit $N\gg 1$, we can treat the relative
momentum $k$ along the q1D axis ($y$) as continuous, but retain the
quantization of the transverse motion. Note that $\hbar$ has been set to unity
here and for the remainder of this subsection.

In the CM frame, setting $\omega_{x}=\omega_{z}=\omega_{\perp}$ and
$\omega_{y}=0$, we can write the two-body Hamiltonian as

$H_{\mathrm{0,2b}}=\frac{1}{2\mu}{\bm{p}_{\textrm{rel}}^{2}}+\frac{1}{2}\mu\omega_{\perp}^{2}\rho^{2}\,,$
(4)

where $\bm{p}_{\textrm{rel}}$ is the relative momentum and
$\rho=\sqrt{x^{2}+z^{2}}$ (and $\phi$ below) are the standard cylindrical
coordinates. The non-interacting relative wave function can be characterized
conveniently by the principal quantum number in the radial direction,
$\mathcal{N}$, and the azimuthal quantum number, $m$. Up to a normalization
factor, the transverse spatial wave functions of the non-interacting
eigenstates are

$\Psi_{\mathcal{N},m}(\rho,\phi)=\frac{e^{im\phi}\rho^{|m|}}{a_{\perp}^{|m|+1}}e^{-\frac{\rho^{2}}{2a_{\perp}^{2}}}\mathcal{L}^{|m|}_{\mathcal{N}}\\!({\rho^{2}}/{a_{\perp}^{2}})\,,$
(5)

where $\mathcal{L}_{\mathcal{N}}^{|m|}(x)$ is the generalized Laguerre
polynomial and $a_{\perp}=\sqrt{1/\mu\omega_{\perp}}$ is the radial oscillator
length. Including the kinetic energy in the 1D direction, $k^{2}/(2\mu)$, the
energies of the full $\ket{\mathcal{N},m,k}$ states are
$E_{\mathcal{N},m,k}=\frac{k^{2}}{2\mu}+(2\mathcal{N}+|m|+1)\omega_{\perp}$,
which we will also write in dimensionless form as

$\mathcal{E}_{m}\equiv\frac{E_{\mathrm{rel}}-(|m|+1)\omega_{\perp}}{2\omega_{\perp}}=\left(\frac{ka_{\perp}}{2}\right)^{2}\,,$
(6)

where $E_{\mathrm{rel}}$ is the total energy that does not include the CM
motion or CM confinement.

Now, let us include the inter-atomic interaction described in detail in App.
A. We assume for now that the interaction is cylindrically symmetric, such
that $m$ remains a good quantum number; the extension to anisotropic
interactions is discussed in Sec. III.3. For an underlying p-wave interaction,
interacting eigenstates can be sorted into $m=-1$, $m=0$, and $m=+1$, and will
in general be a superposition of $\ket{\mathcal{N},m,k}$ states with different
$\mathcal{N}$.

The symmetry of the asymptotic wave function along the axial direction can be
seen in the emergent 1D wave functions, which are solely a function of $y$.
For an incident wave with momentum $k$ in continuum $\mathcal{N}$ and with
symmetry $m$, the scattered wave component with quantum numbers
$\mathcal{N}^{\prime}$ and $m^{\prime}$ is given by

$\displaystyle\psi_{\mathcal{N}^{\prime},m^{\prime},k}(y)=\delta_{m^{\prime},m}\left[\delta_{\mathcal{N}^{\prime},\mathcal{N}}\,e^{iky}\phantom{\frac{1}{1}^{1}}\right.$
$\displaystyle\left.\qquad+f_{\mathcal{N}^{\prime}}^{(m)}e^{-\frac{2}{a_{\perp}}\sqrt{\mathcal{N}^{\prime}-\mathcal{E}_{m}-i\varepsilon}\,|y|}\left(\frac{y}{|y|}\right)^{\\!|m|+1}\right]\,,$
(7)

where $\varepsilon$ is an infinitesimal positive quantity. Here
$f_{\mathcal{N}^{\prime}}^{(m)}$ are the scattering amplitudes into the
$\mathcal{N}^{\prime}$ radial state,

$\displaystyle f_{\mathcal{N}^{\prime}}^{(0)}=-i\mu k\,T_{o}$ (8) and
$\displaystyle f_{\mathcal{N}^{\prime}}^{(\pm
1)}=-i\mu\frac{a_{\perp}}{2}\frac{\sqrt{\mathcal{N}^{\prime}+1}\sqrt{\mathcal{N}+1}}{\sqrt{\mathcal{N}^{\prime}-\mathcal{E}_{1}-i\varepsilon}}\,T_{e}\,,$

parameterized by odd-wave and even-wave scattering transition matrices
($T$-matrices) $T_{o}$ and $T_{e}$, which are defined below.

For the $m=0$ sector, the two-body wave function is exchange-symmetric along
the transverse direction, as is evidenced by Eq. (5) under parity
transformation $\phi\to\phi+\pi$. The overall exchange antisymmetry of the
two-body wave function then requires the axial wave function to be
antisymmetric under exchange, as is observed in the scattered wave of Eq. (7).
Thus two particles undergo odd-wave scattering along the q1D direction (also
called 1D p-wave scattering).

Consider the incident state $\ket{\mathcal{N}=0,m=0,k}$ just above the lowest
non-interacting band energy, such that $\mathcal{E}_{0}\ll 1$. The interacting
wave function consists of all $\ket{\mathcal{N},m=0,k}$ states, starting with
$\mathcal{N}=0$, for which both atoms are in the single-particle ground-state
orbital of the transverse confinement potential. From Eq. (7), we see that for
higher-energy states with $\mathcal{N}^{\prime}>0$, the associated evanescent
scattering wave vanishes for $|y|\gg a_{\perp}$. Therefore, only the component
with $\mathcal{N}^{\prime}=0$ and $m=0$ survives asymptotically, a state that
describes two particles in the single-particle ground state of the transverse
oscillator.

For the $m=1$ and $m=-1$ sectors, the transverse wave function is odd [see
again Eq. (5) for $\phi\to\phi+\pi$] such that the scattering wave function is
exchange-symmetric along $y$, the longitudinal direction [83]222A
complementary phenomenon can occur for scattering in mixed dimensions [83]. In
that case, the odd parity of the CM motion can induce p-wave scattering in the
relative coordinate, even though the underlying scattering is s-wave, because
two atoms see different trapping potentials. In our case, the CM motion is
irrelevant; instead, the odd parity of the relative motion in the strongly
confined direction allows for the emergence of the even-wave scattering of
identical fermions along the q1D direction.. Consider an incoming state
$\ket{\mathcal{N}=0,m=1,k}$, with collision energies just above the lowest-
energy $m=1$ band, such that $\mathcal{E}_{1}\ll 1$. Similarly to the $m=0$
sector, we see from Eq. (7) that all the $\mathcal{N}^{\prime}>0$ components
of the scattered wave function decay at $|y|\gg a_{\perp}$. Therefore, the
asymptotic q1D wave function features a transverse component that is
characterized by $\mathcal{N}=0$ and $m=1$. Similar arguments hold for the
$m=-1$ sector.

The separation between the odd- and even-wave channels in the presence of
interactions leads to two low-energy scattering matrices. For $m=0$, and in
the low-energy limit ($0<\mathcal{E}_{0}=(E_{\rm
rel}-\omega_{\perp})/2\omega_{\perp}\ll 1$), we find that the effective 1D
scattering $T$-matrix $T_{o}$ is given by

$\displaystyle T^{-1}_{o}$
$\displaystyle=\frac{2\mu}{2}\left[\frac{1}{a_{o}}+k^{2}r_{o}+ik\right]\,,\quad\mbox{with}$
$\displaystyle\frac{1}{a_{o}}$
$\displaystyle=\frac{a_{\perp}^{2}}{6}\left(\frac{1}{V_{\mathrm{3D}}}+\frac{2}{R_{\mathrm{3D}}a_{\perp}^{2}}\right)-\frac{2}{a_{\perp}}\zeta(-1/2)$
$\displaystyle\mbox{and}\quad r_{o}$
$\displaystyle=\frac{a_{\perp}^{2}}{6R_{\mathrm{3D}}}+\frac{a_{\perp}}{4}\zeta(1/2)\,,$
(9)

where $\zeta(s)\equiv\zeta(s,1)$ is the Riemann zeta function, $a_{o}$ is the
1D odd-wave scattering length, and $r_{o}>0$ is the 1D odd-wave effective
range. The key steps in our derivation are described in App. A. These
equations define the effective 1D parameters for the odd-wave scattering
amplitude $f_{o}$ as

$-f_{o}^{-1}=1-i(a_{o}k)^{-1}-ir_{o}k+\mathcal{O}(k^{3})\,.$ (10)

This result connects the strongly confined laboratory scenario to a scattering
theory with a true 1D form and is consistent with previous calculations of
p-wave scattering in strongly confined q1D systems [20, 21, 22, 23, 24, 25,
26, 27]. Effective scattering in the $m=0$ sector asymptotically resembles
that of a spinless Fermi gas in 1D.

Unitary odd-wave scattering, for which $f_{o}\to-1$, extends across the range
of collision momenta $|a_{o}|^{-1}\ll k\ll r_{o}^{-1}$. The effective range
$r_{o}$ decreases with stronger transverse confinement, because it scales as
$a_{\perp}^{2}/R_{\mathrm{3D}}$ and thus as $\omega_{\perp}^{-1}$. Experiments
in this universal regime have not yet been realized, because a typical
$k_{F}r_{o}$ is of order unity; thus the effective-range contribution must be
kept. The persistent relevance of $r_{0}$ is due to the large ratio between
the (optical) confinement length scale and the 3D effective range:
$\lambda_{L}/R_{\mathrm{3D}}$ is approximately $300$. We also note that since
the q1D approach uses a 3D interaction model at short range, our treatment
would break down for $a_{\perp}\lesssim R_{\mathrm{3D}}$. The question of
universality is discussed further in Sec. III.4.

For atoms in the lowest even-wave scattering continuum, i.e. $\mathcal{N}=0$
and $m=\pm 1$, or equivalently $0<\mathcal{E}_{1}=(E_{\rm
rel}-2\omega_{\perp})/2\omega_{\perp}\ll 1$, the even-wave $T$-matrix,
$T_{e}$, in the low-energy limit, reduces to

$\displaystyle
T_{e}^{-1}=\frac{2\mu}{2}\left[-a_{e}+k^{2}r_{e}+\frac{i}{k}\right]\,,\quad\mbox{with}$
$\displaystyle a_{e}$
$\displaystyle=-\frac{a_{\perp}}{2}\left[\frac{a_{\perp}^{3}}{6V_{\mathrm{3D}}}+\frac{2a_{\perp}}{3R_{\mathrm{3D}}}+\zeta(-1/2)+\zeta(1/2)\right]$
$\displaystyle\mbox{and}\quad r_{e}$
$\displaystyle=\frac{a_{\perp}^{3}}{16}\left[\frac{4a_{\perp}}{3R_{\mathrm{3D}}}+\zeta(1/2)+\zeta(3/2)\right]\,,$
(11)

where $a_{e}$ is the 1D even-wave scattering length and $r_{e}>0$ is the 1D
even-wave effective range. (See Tab. 1 for a summary of scattering
parameters.) This result is identical to a 1D calculation with true s-wave
interactions [6, 9, 10, 11, 12, 13, 14, 15] with the effective range $r_{e}$
included. The corresponding 1D even-wave scattering amplitude is

$-f_{e}^{-1}=1+ia_{e}k-ir_{e}k^{3}+\mathcal{O}(k^{5})\,.$ (12)

Compared to the odd-wave effective range term in Eq. (9), we see that the
effective-range term, $k^{3}r_{e}$ here, is smaller than $kr_{o}$ by a factor
of $k^{2}a_{\perp}^{2}$, which must be small in the q1D limit. As a result,
both the resonant $k|a_{e}|\ll 1$ and zero-range $k^{3}r_{e}\ll 1$ limits are
achieved at low energy, with no necessary hierarchy between $a_{e}$ and
$r_{e}$. At the largest lattice depth in our experiments, $k_{F}^{3}r_{e}\sim
0.1$, which enables q1D even-wave scattering to enter a universal regime in
which the effective range can be neglected.

###  III.2 RF spectrum

The $T$-matrices (9) and (11) allow us to calculate the rf lineshape, because
at high frequency, the rf spectrum is related to the short-range correlations
in the many-body system. Our starting point is the relation between $\Gamma$
and the 3D correlation strength [44, 85]:

$\displaystyle\Gamma(\omega)$ $\displaystyle=\frac{\Omega^{2}}{4}{\rm Im}\
i\\!\int dt\,e^{i(\omega+i\varepsilon)t}\int d^{3}\\!{\bm{R}}\int
d^{3}{\bm{r}}$ $\displaystyle\left\langle T_{t}\
\hat{\psi}_{b}^{\dagger}\hat{\psi}_{c}\left({\bm{R}}+\frac{1}{2}{\bm{r}},t\right)\hat{\psi}_{c}^{\dagger}\hat{\psi}_{b}\left({\bm{R}}-\frac{1}{2}{\bm{r}},0\right)\right\rangle\,,$
(13)

where $\Omega$ is the Rabi frequency, ${\bm{r}}$ and ${\bm{R}}$ are the
relative and center-of-mass positions, $T_{t}$ is the time-ordering operator,
and $\hat{\psi}_{b}$ and $\hat{\psi}_{c}$ are the fermionic annihilation
operators for a fermion in hyperfine-Zeeman state $\ket{b}$ and $\ket{c}$,
respectively. Full expressions for the transfer rate are reported in App. B.
In the experimentally relevant regime, where $\omega\gg E_{F}$ and
$\omega<4\omega_{\perp}$,

$\Gamma^{(0)}(\omega)=\frac{\Omega^{2}C_{o}}{2}\left[\frac{\theta(\omega)}{\omega^{3/2}}+\frac{\theta(\omega-2\omega_{\perp})\sqrt{\omega-2\omega_{\perp}}}{\omega^{2}}\right]$
(14)

and

$\Gamma^{(\pm 1)}(\omega)=\frac{\Omega^{2}C^{(\pm
1)}_{e}}{2}\left[\frac{\theta(\omega)}{\omega^{5/2}}+\frac{2\theta(\omega-2\omega_{\perp})}{(\omega-2\omega_{\perp})^{1/2}\omega^{2}}\right]\,.$
(15)

where $C_{o}$ is the 1D odd-wave contact and $C^{(\pm 1)}_{e}$ are the 1D
even-wave contacts and $\theta(x)$ is the Heaviside step function. In both
expressions, the first term represents the contribution from those continuum
states that are below the first orbital gap. The step function
$\theta(\omega-2\omega_{\perp})$ in the second terms indicate the opening of a
new transverse band, when the rf photon energy exceeds the energy difference
to the $\mathcal{N}=1$ scattering continuum, thus making the final scattering
states within the new transverse band available. Each continuum has a density
of states that scales as $1/\sqrt{\mathcal{E}_{m}}$, as is characteristic of a
1D system. This gives rise to a cusp-like $(\omega-2\omega_{\perp})^{1/2}$
dependence for $\Gamma^{(0)}$, and a singular dependence
$(\omega-2\omega_{\perp})^{-1/2}$ for $\Gamma^{(\pm 1)}(\omega)$. The latter
is observed experimentally: see Sec. V.2.

As discussed in App. B, both Eq. (14) and Eq. (15) are constrained to match
the odd-wave contact relation with a characteristic $\omega^{-3/2}$ spin-flip
rate for $\omega<2\omega_{\perp}$ [86, 57, 58, 59, 28, 60, 29],

$\lim_{E_{F}/\hbar\ll\omega<2\omega_{\perp}}\Gamma^{(0)}(\omega)=\frac{\Omega^{2}}{2}\sqrt{\frac{2\mu}{\hbar}}\frac{C_{o}}{\omega^{3/2}}$
(16)

and the even-wave contact relation with a leading $\omega^{-5/2}$ rate,

$\lim_{E_{F}/\hbar\ll\omega<2\omega_{\perp}}\Gamma^{(\pm
1)}(\omega)=\frac{\Omega^{2}}{2}\frac{1}{\sqrt{2\hbar\mu}}\frac{C^{(\pm
1)}_{e}}{\omega^{5/2}}\,,$ (17)

where factors of $\hbar$ have now been restored. These relations will be used
to find the contact parameters in the measured spectra.

Figure 3:  Scattering resonances in q1D. (a) The asymptotic states
$\ket{n_{x},n_{z},k}_{\mathrm{rel}}$ have confinement energies of
$(n_{x}+1/2)\hbar\omega_{x}+(n_{z}+1/2)\hbar\omega_{z}$ plus a continuum of
$\hbar^{2}k^{2}/2\mu$ from motion along $y$. The three lowest continua, shown
here, become strongly interacting when magnetically tuned to resonance with an
appropriate 3D p-wave dimer state (labeled $x$, $y$, and $z$). The resonant
location for $\ket{1_{x},0_{z},k}_{\mathrm{rel}}$ states (blue square at
$B_{0}^{x}$) is at an energy shifted above the ground-band by the transverse
excitation energy $\hbar\omega_{x}$. The resonance for
$\ket{0_{x},1_{z},k}_{\mathrm{rel}}$ states (red square at $B_{0}^{z}$), is
further displaced by the dipole-dipole splitting $\Delta E_{\mathrm{dd}}$ of
the $xy$ and $z$ dimer states, and any cylindrical asymmetry for which the
case $\omega_{z}>\omega_{x}$ is shown. Corrections to this approximate picture
are given by Eqs. (18) and (III.3). (b) The effective 1D scattering cross
section $|f|^{2}$ at resonance versus q1D collision energy. The broader
resonance for asymptotic states with $n_{x}=1$ or $n_{z}=1$ states can be
attributed to their even-wave character, as discussed in the text. (c)
$|f|^{2}$ versus magnetic field, for a typical in-tube scattering energy of
$10\,$kHz. (d) The q1D odd-wave continuum is characterized by a transverse
wave function $\Psi_{\mathrm{rel}}(x,z)$ with even spatial parity. A contour
plot shows the $|f_{o}|^{2}$ versus magnetic field and collision energy. (e,
f) The q1D even-wave continua are each characterized by an asymptotic
transverse wave function $\Psi_{\mathrm{rel}}(x,z)$ with odd spatial parity.
The contour plots of $|f_{e}|^{2}$ versus magnetic field and collision energy
combines the trends of panels (b) and (c), and emphasize the growing magnetic
width in the $k\to 0$ limit that is characteristic of s-wave collisions.
Scattering amplitudes in panels (b) to (f) are based on Eq. (10) with
parameters in Eq. (9), or Eq. (12) with parameters in Eq. (11), with equal
$V_{L}^{x}=V_{L}^{z}=40\,E_{R}$.

###  III.3 Anisotropic interactions

Table 2: Exchange parity and spatial orientation of the two-atom states in the lowest three scattering continua.  continuum | asymptotic | transverse | q1D | resonant
---|---|---|---|---
state | parity | parity | dimer
zeroth | $\ket{0_{x},0_{z},k}_{\mathrm{rel}}$ | even | odd |  $y$ dimer
first | $\ket{1_{x},0_{z},k}_{\mathrm{rel}}$ | odd | even |  $x$ dimer
second | $\ket{0_{x},1_{z},k}_{\mathrm{rel}}$ | odd | even |  $z$ dimer

Our treatment so far has assumed isotropic 3D interactions and radially
symmetric trapping potentials. For the experimental system, however,
interactions are anisotropic: the 3D $xy$ dimer branch is split by $\Delta
E_{\mathrm{dd}}/h\sim 100$ kHz from the 3D $z$ dimer branch due to dipole-
dipole interactions in 40K [87], leading to direction-dependent scattering
volumes $V_{\mathrm{3D}}^{i}$ and effective ranges $R_{\textrm{3D}}^{i}$,
where $i$ = $x$, $y$, or $z$ [see Fig. 1(a) for axis orientation]. Furthermore
the cylindrical anisotropy can be tuned by adjusting confinement strength
independently along the $x$ and $z$ axes, as depicted by Fig. 3(a) in the case
of $\omega_{z}>\omega_{x}$.

For broken cylindrical symmetry, $\mathcal{N}$ and $m$ are no longer good
quantum numbers. Instead, we use $n_{x}$ and $n_{z}$ to index quanta of
relative motion in the Cartesian axes of confinement 333The cylindrically
symmetric states are superpositions of Cartesian states with
$n_{x}+n_{z}=2\mathcal{N}+|m|$ and write the states as
$\ket{n_{x},n_{z},k}_{\mathrm{rel}}$. Even parity in $y$ occurs for odd
$n_{x}+n_{z}$, and vice versa. The exchange antisymmetry of the
$\ket{1_{x},0_{z}}_{\mathrm{rel}}$ and $\ket{0_{x},1_{z}}_{\mathrm{rel}}$
states can be seen by decomposing them back into a single-particle basis.
Labeling the colliding particles A and B, a $\ket{1}_{\mathrm{rel}}$ state
with no CM excitation is
$(\ket{1}_{\mathrm{A}}\ket{0}_{\mathrm{B}}-\ket{0}_{\mathrm{A}}\ket{1}_{\mathrm{B}})/\sqrt{2}$,
or in other words, a singlet-like state in which two particles share a single
quantum of orbital excitation.

The emergence of q1D resonances is due to the hybridization of the 1D bands of
scattering states with p-wave dimers that are now labeled as $x$-, $y$\- and
$z$-dimers due to dipole splitting [see Tab. 2 and Fig. 3(a)]. Due to the
symmetry of the 3D dimers, a selection rule arises regarding the coupling to
the scattering states. For example, the $x$-dimer has the short-range symmetry
corresponding to the spherical harmonic $Y_{1x}\sim x/r$. As a result, the $x$
dimer can only be hybridized with continuum states in the
$\ket{1_{x},0_{z},k}_{\mathrm{rel}}$ band above the threshold energy
$E_{\mathrm{zp}}+\hbar\omega_{x}$, where $E_{\mathrm{zp}}$ is the zero-point
energy $\frac{1}{2}\hbar(\omega_{x}+\omega_{z})$. The magnetic field at which
this crossing occurs is labeled $B_{0}^{x}$ in Fig. 3(a). Similar analyses
also apply to the $y$ and $z$ scattering: the $y$-dimer enhances odd-wave
scattering within the $\ket{0_{x},0_{z},k}_{\mathrm{rel}}$ band, and the
$z$-dimer enhances even-wave scattering in the
$\ket{0_{x},1_{z},k}_{\mathrm{rel}}$ band (see Tab. 2). The three distinct
resonances are shown in Fig. 3(a) and are labeled in order of increasing
magnetic field: $B_{0}^{y}$ (“zeroth”), $B_{0}^{x}$ (“first”), and $B_{0}^{z}$
(“second”). The nature of the scattering state is illustrated in Figs. 3(d),
3(e), and 3(f).

In the presence of anisotropic interactions, we still calculate the q1D
scattering parameters using Eqs. (9) and (11), but now replace
$V_{\mathrm{3D}}$ and $R_{\mathrm{3D}}$ with the values of
$V_{\mathrm{3D}}^{i}$ and $R_{\mathrm{3D}}^{i}$ specific to the underlying 3D
resonance [70]. The low-energy scattering parameters in Eq. (9) and Eq. (11)
can then be tuned by changing the 3D scattering parameters in order to
engineer arbitrarily strong odd- or even-wave interactions.

In terms of the 3D quasi-bound-state energy [87, 70], approximately
$-\hbar^{2}R_{\mathrm{3D}}^{i}/(2\mu V_{\mathrm{3D}}^{i})$, the resonance
condition $a_{o}^{-1}=0$ for odd-wave scattering is, from Eq. (10)

$-\frac{\hbar^{2}R_{\mathrm{3D}}^{y}}{2\mu
V_{\mathrm{3D}}^{y}}=E_{\textrm{zp}}\left[1-6\frac{R_{\mathrm{3D}}^{y}}{a_{\perp}}\zeta(-1/2)\right]\,,$
(18)

while for even-wave scattering, the resonant conditions ($a_{e}^{x}=0$ or
$a_{e}^{z}=0$) are, from Eq. (12)

$\displaystyle-\frac{\hbar^{2}R_{\mathrm{3D}}^{x,z}}{2\mu
V_{\mathrm{3D}}^{x,z}}=$ $\displaystyle(E_{\textrm{zp}}+\hbar\omega_{x,z})$
$\displaystyle\times\left\\{1+\frac{3}{2}\frac{R_{\mathrm{3D}}^{x,z}}{a_{\perp}}\left[\zeta(1/2)+\zeta(-1/2)\right]\right\\}\,.$
(19)

Since $R^{i}_{\mathrm{3D}}/a_{\perp}$ is typically small in our experiments,
these relations show that resonant odd- and even-wave scattering occurs
approximately when the 3D quasi-bound-state reaches the relevant scattering
continuum, as depicted in Fig. 3(a).

The interaction anisotropy is observed in radio-frequency spectroscopy and
thus the contact parameters. We note that the odd-wave contact relation is
only governed by the interactions along the $y$-direction, so Eq. (14) is not
altered. On the other hand, even-wave interactions now appear as $x$ and $z$
correlation peaks at distinct magnetic-field values, instead of the degenerate
$m=\pm 1$ correlation peaks in Eq. (15). Asymptotic rf transfer rates are now
proportional to $C_{e}^{x}$ and $C_{e}^{z}$, respectively.

###  III.4 Width of the universal regime

The qualitative departure of even-wave scattering from odd-wave is illustrated
from its energetic width, i.e., the scaling of elastic scattering versus
momentum $k$ of the q1D collision. Figures 3(b)–(f) show the scattering
probabilities $|f_{o}|^{2}$ and $|f_{e}|^{2}$, as a function of both energy
and magnetic field. A key feature is the enhanced width, in either parameter,
of $|f_{e}|^{2}$. Near resonance, the energy range of unitary scattering is
determined by the effective-range term of Eqs. (10) and (12). For odd-wave
interactions, this energetic width can be characterized by
$\Delta_{o}\equiv(\hbar^{2}/2\mu)r_{o}^{-2}$, while for even-wave interactions
the energetic width is $\Delta_{e}\equiv(\hbar^{2}/2\mu)r_{e}^{-2/3}$. From
Eqs. (9) and (11), we find that
$\Delta_{o}/\Delta_{e}=18^{2/3}(R_{\mathrm{3D}}/a_{\perp})^{4/3}$, which is
small in experiments. This behavior is illustrated in Fig. 3(b) on resonance,
which is a vertical cut at the resonant magnetic field in the contour plots of
$|f_{o,e}|^{2}$ seen in panels (d), (e), and (f).

If $a_{e}$ is tuned to zero [for instance at $B=B_{0}^{z}$ defined by Eq.
(III.3)], then the unitary limit $f_{e}\to-1$ is achieved for $r_{e}k^{3}\ll
1$. The range of scattering energies in which this occurs is given by
$\hbar^{2}k^{2}/2\mu\lesssim\Delta_{e}$. In terms of transverse confinement
strength, the leading-order width is 444The sub-leading correction modifies
$\Delta_{e}$ by $1-2R_{\mathrm{3D}}a_{\perp}^{-1}[\zeta(1/2)+\zeta(3/2)]$,
such that the leading order gives roughly a 10% underestimate for our typical
parameters.

$\Delta_{e}\approx\hbar\omega_{\perp}(3\sqrt{2}R_{\mathrm{3}D}/a_{\perp})^{2/3}\,.$
(20)

Within a degenerate Fermi gas, the range of two-body collision energies is
$2E_{F}$. Scattering is unitary when the energetic width $\Delta_{e}\gtrsim
2\,E_{F}$, or

$N\lambda\lesssim(3R_{\mathrm{3D}}/2a_{\perp})^{2/3}\,.$ (21)

The r.h.s. of Eq. (21) is $\sim 0.2$ for the confinement strength typically
used in low-dimensional quantum gas experiments, and thus poses a comparable
constraint to the q1D criterion $N\lambda\ll 1$. For the $N\lambda\sim 0.1$
conditions in our experiments, resonant even-wave scattering is already
energetically broad [see Fig. 3(b)].

Elastic odd-wave collisions also have a zero-range limit. A parallel argument
finds that when $|a_{o}|^{-1}\to 0$, the energetic width of the unitary regime
($\Delta_{o}=\hbar^{2}/2\mu r_{o}^{2}$) is
$\Delta_{o}\approx\hbar\omega_{\perp}(18R_{\mathrm{3D}}^{2}/a_{\perp}^{2})$ to
leading order 555The sub-leading correction modifies $\Delta_{o}$ by
$1-3R_{\mathrm{3D}}\zeta(1/2)/a_{\perp}$, such that the leading order gives
roughly a 20% underestimate for our typical parameters.. Comparing this width
to $2\,E_{F}$, one finds that the confinement condition for
${\Delta_{o}}\gtrsim 2\,E_{F}$ is

$N\lambda\lesssim(3R_{\mathrm{3D}}/a_{\perp})^{2}\,.$ (22)

The r.h.s. of Eq. (22) is $\sim 0.02$ for our typical conditions, so is much
more restrictive than the q1D criterion, and not satisfied for our
experiments.

The magnetic width of low-energy resonant scattering reveals qualitatively
different scaling in the odd- and even-wave cases. From Eq. (10), at constant
$a_{o}$, as $k\to 0$, we see that $|f_{o}|^{2}\to 0$ and scattering is thus
progressively narrower in $B$ at lower energy [Fig. 3(d)]. A similar analysis
for Eq. (12), however, shows that $f_{e}\to-1$ for a progressively broader
range in $B$ at lower energy [Figs. 3(e) and 3(f)]. At a constant scattering
energy, both odd- and even-wave resonances become broader in field with
increasing confinement. However, the odd-wave field width scales with the
square root of the lattice depth, while the even-wave field width scales
linearly with lattice depth. Therefore, confinement can lead to a
significantly broadened resonance for even-wave scattering, even though the
energetic width of the underlying 3D resonance is still narrow 666This can be
seen from Eqs. (9) and (11) since the 3D scattering volume scales as
$V_{\mathrm{3D}}\sim 1/B$ near the p-wave Feshbach resonance. The widening
field range of correlations at higher lattice depths can be seen in Fig. 2,
with the effect strongest for even-wave correlations in Fig. 2(b). There,
even-wave correlations extend across a 3 G range at $V_{L}=120\,E_{R}$,
compared to a $\lesssim 0.5\,$G range of odd-wave correlations at similar
lattice depth. Both of these are larger than the $\sim 0.2$ G range of
significant correlations observed in 3D..

In the zero-range limit identified here, the emergent even-wave scattering
channel has a solitary relevant parameter that is tunable. This provides a
regime in which a spin-polarized Fermi gas can be considered universal.
Without the need for effective-range terms, its behavior can be understood
with models independent of the particular element used in experiments, similar
to 3D fermionic spin mixtures near an s-wave scattering resonance or two-
dimensional bosons near their non-interacting fixed point [92, 93].

##  IV Observation of odd-wave correlations

Figure 4: RF spectrum of odd-wave interactions. The normalized rf transfer
rate $\widetilde{\Gamma}$ versus dimensionless detuning $\widetilde{\omega}$
from the single-particle resonance [see Fig. 1(b,v)]. Two spectra are shown,
both for $V_{L}=40\,E_{R}$: at $B=203.0\,$G (squares), the gas is weakly
interacting; at $B=199.2$ G (circles), odd-wave interactions are Feshbach-
enhanced [$-1/k_{F}a_{o}\sim 5$; see Fig. 2(a)]. The interacting gas shows an
increased $\widetilde{\Gamma}$ for positive $\widetilde{\omega}$. The inset
shows $\widetilde{\Gamma}(\widetilde{\omega})$ on the same axes, with a
$\widetilde{\omega}^{-3/2}$ fit to the high-frequency tail of the interacting
spectrum, allowing the determination of $C_{o}$ from Eq. (14).

We probe odd-wave interactions by performing rf spectroscopy at magnetic
fields near $B^{y}_{0}$ using the procedure outlined in Sec. II. Figure 4
shows the dimensionless transfer fraction
$\widetilde{\Gamma}(\widetilde{\omega})$ versus dimensionless rf detuning
$\widetilde{\omega}\equiv\hbar\omega/E_{F}$, both rescaled by the calculated
Fermi energy of the central tube (see App. F). Correlations are revealed by an
asymmetric spectrum, i.e., spectral weight that appears at a large positive
$\omega$ but not at a large negative $\omega$ of equal magnitude. The strength
of correlations can be quantified through Eq. (16), in the range
$4\lesssim\widetilde{\omega}\lesssim 15$ (for which $\omega<2\omega_{\perp}$),
where we observed the characteristic $\widetilde{\omega}^{-3/2}$ power-law
decrease. In the strongly interacting regime, we fit the experimental data to
extract $C_{o}$, thereby measuring the odd-wave correlations in a q1D
ultracold gas for the first time to our knowledge. At magnetic fields far from
resonance, correlations are weak and there is no observable asymmetry, as can
be seen in the inset of Fig. 4.

Figure 5: Odd-wave contact and loss. (a) Contact per atom versus odd-wave
scattering length, $-1/k_{F}a_{o}$. Per Eq. (23), the dimensionless odd-wave
contact per atom is related to the scaled transfer from spectroscopy. The
zero-temperature Bethe ansatz prediction for $k_{F}r_{o}=1$ is shown in
dashed-black, and the high-temperature prediction for $T/T_{F}=0.8$ is shown
as a band covering the range of experimental lattice depths. (b) The inset
displays the same $\widetilde{C}_{o}/N$ data versus magnetic field,
emphasizing the data collapse in (a). (c) The total atom number after the
spectroscopy sequence, as a fraction of initial atom number, is shown as a
function of $-1/k_{F}a_{o}$. In the grey-shaded parameter range, 30% or more
atoms are lost. Peak loss occurs at the calculated $a_{o}^{-1}=0$. (d) The
inset displays this atom-number data as a function of magnetic field.

Figure 5 shows the contact strength and atom number measured across a range of
magnetic fields and lattice depths. The contact was measured using transfer
fraction at a single frequency, typically $\widetilde{\omega}=5$, which
fulfills the condition $E_{F}/\hbar\ll\omega<2\omega_{\perp}$ and selects the
first term of Eq. (14). Since we are optimizing for ground-band loading, we
can assume that $\Gamma^{(\pm 1)}\ll\Gamma^{(0)}$, which allows for a single-
shot determination of contact:

$\frac{\widetilde{C}_{o}}{N}=4\sqrt{2}\pi\,\widetilde{\omega}^{3/2}\,\widetilde{\Gamma}\,,$
(23)

where

$\widetilde{C}_{o}\equiv\frac{4m}{\hbar^{2}k_{F}}C_{o}$ (24)

is the dimensionless odd-wave contact 777The numerical factors in $\tilde{C}$
are chosen such that the asymptotic momentum density
$n_{k}\to\tilde{C}_{o}k_{F}/k^{2}$, when $\int(dk/2\pi)n_{k}=N$..

The observed magnitude of $\widetilde{C}_{o}/N$ is $\lesssim 2\times 10^{-2}$,
with the maximum near the q1D scattering resonance. In Figs. 5(b,d), both atom
loss and correlations shift to higher magnetic field with increasing lattice
depth, primarily due to the increasing zero-point energy, $E_{\textrm{zp}}$,
associated with the transverse harmonic confinement. When comparing Fig. 5(a)
to Fig. 5(b), or comparing Fig. 5(c) to Fig. 5(d), we see that field-dependent
data collapses to a single curve when plotted as a function of
$-1/k_{F}a_{o}$, indicating that we are probing the physics in the q1D regime.

Consider first the weakly repulsive regime. No significant correlations are
seen in the range $-20<-1/k_{F}a_{o}<-2$, which can be understood from Fig.
3(a): the 3D quasi-bound p-wave dimer has yet to emerge above the odd-wave
scattering threshold. Similar to the 3D p-wave case [54], we conclude that the
dimer state is not significantly populated. For stronger repulsive
interactions 888Note that inhomogeneous transverse confinement broadens the
range of $a_{o}^{-1}$ for a given magnetic field, which makes the assignment
of small $-1/k_{F}a_{o}$ ambiguous., $-5\lesssim-1/k_{F}a_{o}<0$, a finite
$\tilde{C}_{o}$ is observed, however concurrently with strong losses, which
are perhaps due to the instability of the repulsive odd-wave gas [96], its
value remains small.

Next, consider the regime with relatively weak attractive interactions,
$4\lesssim-1/k_{F}a_{o}\lesssim 15$. In the absence of strong loss, we can
compare the measured values of $C_{o}$ to equilibrium calculations: zero-
temperature Bethe ansatz theory [60] and a high-temperature expansion theory
[29] (App. D). These are shown in Fig. 5(a). The theory band for the high-
temperature expansion reflects the weak 1D effective-range dependence on
lattice depth and includes a correction for the inhomogeneous density and tube
number discussed in App. F. In this weakly interacting regime, the
measurements overlap with both the zero-temperature and the high-temperature
predictions, with no adjustable parameters, and follow the anticipated
monotonic decrease with $a_{o}^{-1}$.

At higher magnetic fields, where $-1/k_{F}a_{o}\gtrsim 15$, the apparent
contact deviates from these predictions. We attribute this departure to
overlapping even-wave correlations (see Sec. V), due to a remaining population
in the excited bands, such that $\Gamma$ can no longer be associated purely
with $\Gamma^{(0)}$.

Finally, consider magnetic fields close to resonance ($B\sim B^{y}_{0}$),
where we observe a smaller contact than is anticipated by theory. The unitary
high-temperature value of the contact is $\widetilde{C}_{o}/N=8/\pi$ in a
uniform system [29] while at zero temperature $\widetilde{C}_{o}/N=4/\pi$ [86,
58, 28]. This far exceeds the highest $\widetilde{C}_{o}/N$ observed. Instead,
within the range $-2\lesssim-1/k_{F}a_{o}\lesssim 4$, indicated in Fig. 5(c)
with a gray band, the contact per atom is decreased. This is attributed to the
onset of strong loss: we find that more than 30% of the initial population is
lost by the end of $t_{\mathrm{hold}}$, from which we infer a three-body loss
coefficient $L_{3}\geq 10^{-5}$ cm2/s. As in prior studies of atom loss in q1D
Fermi gases [19, 18], we find that the loss rate is independent of lattice
depth as indicated by data collapse in Fig. 5(c).

Limited by losses, the system does not develop a contact parameter larger than
$\widetilde{C}_{o}/N\sim 2\times 10^{-2}$. We can interpret this magnitude by
estimating the odd-wave interaction energy that would be found in an
equilibrium cloud with the same correlation strength and negligible effective
range. The adiabatic relation for 1D odd waves is [58]

$\frac{\partial F}{\partial(-a_{o}^{-1})}=C_{o}\,,$ (25)

where $F$ is the free energy of the gas. Integrating under the observed
contact outside of the loss dominated regime, we find that between
$(-k_{F}a_{o})^{-1}=10$ and $(-k_{F}a_{o})^{-1}=4$, the interaction energy is
$\Delta F\sim-0.03\,NE_{F}$. Therefore, at least in the conditions realized
here, an odd-wave Fermi gas can be stable only in the weakly interacting
regime.

##  V Observation of even-wave correlations

We now turn our attention to the features that appear when multi-orbital
loading is allowed, i.e., those features appearing in Fig. 2(b) but not in
Fig. 2(a). In order to test the hypothesis that these spectral features are
due to q1D even-wave correlations and to elucidate their nature, we undertake
a sequence of experiments: spectroscopy with variable asymmetric confinement
(Sec. V.1); spectroscopy at variable frequency (Sec. V.2); and spectroscopy
with variable coupling strength (Sec. V.3). The sample preparation common to
all these tests does not use AM (see Sec. II and App. E), such that $\sim
15\%$ of the atoms are in each of the transverse excited bands as measured by
band mapping.

Figure 6: Vector assignment of the second even-wave correlation peak. (a)
Normalized rf transfer rate $\widetilde{\Gamma}$ at constant
$\widetilde{\omega}\approx 5$ versus magnetic field $B$ applied in the $z$
direction. Three peaks in $\widetilde{\Gamma}(B)$ are resolved for multi-band
loading (circles), while a single clear peak is observed for ground-band
loading (squares). Here, confinement is cylindrically symmetric, with
$V_{L}^{x}=V_{L}^{z}=40\,E_{R}$. The magnetic-field displacements from the
resonance position $B_{0}^{y}$ are indicated for maximum $\widetilde{\Gamma}$
by colored arrows, and reported as $\Delta B$ in the subsequent panel. (b) The
magnetic-field displacements for the three peaks in $\widetilde{\Gamma}(B)$
are shown for variable $V_{L}^{z}$, with fixed $V_{L}^{x}=40\,E_{R}$. While
$B_{\mathrm{0th}}$ and $B_{\mathrm{1st}}$ shift with the zero-point energy
included in $B_{0}^{y}$, $B_{\mathrm{2nd}}$ has a stronger dependence on
$V_{L}^{z}$. This trend supports the hypothesis that the third resolved peak
in $\widetilde{\Gamma}$ versus $B$ corresponds to scattering in the $z$
continuum, whose energy is $\hbar\omega_{z}$ above the odd-wave continuum.
This coincides with the resonance condition for even-wave scattering, Eq.
(26); more direct evidence of even-wave character is given in Sec. V.2.

###  V.1 Vector character of the even-wave resonances

The resonance condition for even-wave interactions is given by Eq. (III.3),
which for our small value of $R_{\mathrm{3D}}/a_{\perp}$ can be approximated
for the $i=x$ or $i=z$ continuua as

$-\left.\frac{\hbar^{2}R_{\mathrm{3D}}^{i}}{2\mu
V_{\mathrm{3D}}^{i}}\right|_{B=B_{0}^{i}}\approx
E_{\mathrm{zp}}+\hbar\omega_{i}\,,$ (26)

where the l.h.s. is the 3D p-wave dimer energy, linear in $B$, as shown in
Fig. 3(a). The incorporation of an extra vibrational excitation quantum
(shared between the atoms in a $\ket{1_{x}}_{\mathrm{rel}}$ or
$\ket{1_{z}}_{\mathrm{rel}}$ motional state) shifts the even-wave resonances
from the odd-wave resonance condition, which is

$-\left.\frac{\hbar^{2}R_{\mathrm{3D}}^{y}}{2\mu
V_{\mathrm{3D}}^{y}}\right|_{B=B_{0}^{y}}\approx E_{\mathrm{zp}}\,,$ (27)

from Eq. (18). In the following, we investigate the dependence of $B_{0}^{i}$
on $\omega_{i}$.

Figure 6(a) shows $\widetilde{\Gamma}(B)$ at $V_{L}=40E_{R}$, a depth at which
three distinct peaks can be resolved in multi-band loading. To subtract off
the common shift from the zero-point energy, we show the displacements of each
peak ($\Delta B_{\mathrm{0th}}$, $\Delta B_{\mathrm{1st}}$, and $\Delta
B_{\mathrm{2nd}}$) from $B_{0}^{y}$ 999The sub-leading term of the isotropic
Eq. 18 provides only a $\sim 50\,$mG correction to the leading order. Still,
we determine $B_{0}^{y}$ through a generalized form of Eq. (18) that includes
anisotropic confinement [25].. Since $\Gamma=\sum\Gamma^{i}$, additional tests
are necessary to confirm the origin of these peaks.

The first peak also appears with a ground-band loading procedure [as shown in
Fig. 6(a) and Fig. 2(a)], so it can be unambiguously associated with odd-wave
interactions. As shown in Fig. 5, the correlation peak is not at the resonance
location, due to strong loss near resonance. For the parameters of Fig. 6, the
magnetic field location of peak correlation $B_{\mathrm{0th}}$ is shifted
$\sim 0.2$ G above $B_{0}^{y}$.

A vector assignment of these two higher peaks can be made using unequal
confinement, $V_{L}^{x}\neq V_{L}^{z}$, whose anticipated effect on
$B_{0}^{x}$ and $B_{0}^{z}$ is illustrated in Fig. 3(a) for
$\omega_{z}>\omega_{x}$. Figure 6(b) shows the best-fit peak displacements
$\Delta B_{\mathrm{0th}}$, $\Delta B_{\mathrm{1st}}$, and $\Delta
B_{\mathrm{2nd}}$ from repeated $\widetilde{\Gamma}(B)$ scans at variable
$V_{L}^{z}$, with fixed $V_{L}^{x}=40\,E_{R}$. We see how $\Delta
B_{\mathrm{1st}}$ is insensitive to $V_{L}^{z}$ (similarly to $\Delta
B_{\mathrm{0th}}$), while $\Delta B_{\mathrm{2nd}}$ has a stronger dependence
on $V_{L}^{z}$. Based on this trend, we can conclude that the peak at
$B_{\mathrm{2nd}}$ must be associated with confinement along the $z$
direction, which would be expected for the even-wave threshold $B_{0}^{z}$,
where scattering states hybridize with the 3D $z$-dimer.

Cylindrical symmetry is broken in our experiments by the applied magnetic
field direction ($z$) orthogonal to the q1D direction ($y$). The field aligns
the magnetic dipole moments of 40K, and thus splits the Feshbach resonances
for the two even-wave features (see Sec. III.3). This $0.5$ G splitting is
evident upon the comparison of $B_{\mathrm{1st}}$ and $B_{\mathrm{2nd}}$ at
equal lattice depth, which is $V_{L}^{z}=40\,E_{R}$ for Fig. 6.

A similar experimental study with variable $V_{L}^{x}$ and fixed $V_{L}^{z}$
indicates that the peak at $B_{\mathrm{1st}}$ is associated with confinement
along $x$. In sum, the vector character and dependence on the lateral
confinement is consistent with the assignment of the correlation peak at
$B_{\mathrm{1st}}$ in Fig. 6 to the $x$ even-wave continuum, and at
$B_{\mathrm{2nd}}$ to the $z$ even-wave continuum.

###  V.2 Spectroscopic signatures

Figure 7: Even-wave rf spectrum. Dimensionless rf transfer rate
$\widetilde{\Gamma}$ versus detuning $\omega$, for a multi-band-loaded
ensemble at 200.5 G, with $V_{L}=80\,E_{R}$. Between $0.7\,\omega_{\perp}$ and
$1.7\,\omega_{\perp}$, the signature even-wave scaling
$\widetilde{\Gamma}\propto\omega^{-5/2}$ is observed [see Eq. (43)]. The solid
lines show Eq. (15), with a free vertical scale factor. The singularity in the
transfer rate (a unique signature of 1D even waves) manifests as a peak near
$\omega\approx 2\omega_{\perp}$, as described in the main text. The inset
shows $\widetilde{\Gamma}(\omega)$ on a linear vertical scale in the
$\omega^{-5/2}$ regime; a single-parameter fit determines the sum of the even-
wave contacts. For these conditions, $k_{F}a_{e}^{z}\approx 0.4$, compared to
$k_{F}a_{e}^{x}\approx 1.6$, so that scattering is dominated by the
$\ket{0_{x},1_{z},k}_{\mathrm{rel}}$ states.

The strongest evidence for the observation of even-wave correlations in our
system comes from the functional form of the rf spectra. As discussed in Sec.
III.2, within the range of frequencies $\omega<2\omega_{\perp}$, in which a
q1D system resembles a true 1D system, the rf transfer rate should scale as
$\omega^{-5/2}$ (as opposed to $\omega^{-3/2}$ found for odd-wave correlations
in Sec. IV). Furthermore, for larger rf frequencies,
$\widetilde{\Gamma}(\omega)$ should exhibit a singularity at
$\omega=2\omega_{\perp}$: see Eq. (15). The experimentally measured
$\Gamma(\omega)$ is a sum of all $\Gamma^{i}$, but will be dominated by only
one Cartesian component near its resonant magnetic field.

Figure 7 shows the rf spectrum observed at $B\approx B_{\mathrm{1st}}$. For
frequencies between $0.7\,\omega_{\perp}$ and $1.7\,\omega_{\perp}$, we
observe a clear $-5/2$ power law consistent with 1D s-wave correlations (see
Fig. 7 inset). In this region, it is possible to extract the 1D even wave
contact parameters $C_{e}^{x,z}$. We note that once $C_{e}^{x}$ and
$C_{e}^{z}$ are fixed, the general form of $\widetilde{\Gamma}(\omega)$ [Eq.
(40) and in particular Eq. (15)] is then determined for higher frequencies, no
matter what the relative strengths of $C_{e}^{x}$ and $C_{e}^{z}$. We do not
understand why the $-5/2$ power law does not extend to lower frequencies,
below $0.7\,\omega_{\perp}/2\pi\sim 90$ kHz. This may be due to incomplete
equilibration of the long-wavelength degrees of freedom [98, 99, 100].

A sharp peak in $\widetilde{\Gamma}(\omega)$ appears, as anticipated, at twice
the band gap 101010Due to anharmonicity of the optical lattice potential,
there is some ambiguity in the experimental value of the transverse trap
frequency. Since the lowest allowed excitation for even-waves is the
excitation of an atom from the first to third single-particle level of the
optical lattice, the best choice for $\hbar\omega_{\perp}$ in this case is
half that energy.. Here, atom pairs are promoted from incident states with
principal radial quantum number $\mathcal{N}=0$ to final states with
$\mathcal{N}=1$ and hyperfine-Zeeman state $\ket{c}$. Since rf transitions
cannot modify the spatial wave functions, the presence of the feature
demonstrates that the scattering states are dressed by higher-$\mathcal{N}$
orbital states at short range, i.e., $y\lesssim a_{\perp}$ in Eq. (7). The
singular feature in the theory results from a combination of two factors: the
divergence of the 1D density of final states (see Sec. III.2) and the
$k$-independent even-wave matrix element [see the $|m|=1$ term of Eq. (31)].
As observed, the $2\omega_{\perp}$ singularity is softened by the
distributions of a finite range of collision energies, the average over
variable lattice depths of 1D tubes, and the Fourier broadening of the rf
pulse.

The observation of these two even-wave signatures in $\Gamma(\omega)$ gives
strong evidence for the emergence of effective s-wave correlations in the q1D
gas.

###  V.3 Even-wave contact parameter

The correlations appearing in Fig. 2(b) across a broad range in magnetic field
can now be quantified through the even-wave contact. At each magnetic field,
the even-wave contacts $C_{e}^{x,z}$ can be probed with a single-shot
measurement of $\Gamma$ at a fixed detuning with the range
$0.7\,\omega_{\perp}\lesssim\omega\lesssim 1.7\,\omega_{\perp}$, as discussed
in Sec. V.2. In the regime where $\Gamma^{(0)}\ll\Gamma^{(x,z)}$, we find

$\widetilde{C}_{e}^{x}+\widetilde{C}_{e}^{z}=2\sqrt{2}\pi
N\widetilde{\omega}^{5/2}\widetilde{\Gamma}\,,$ (28)

where

$\widetilde{C}_{e}^{x,z}\equiv\frac{4m}{\hbar^{2}k_{F}^{3}}C_{e}^{x,z}$ (29)

is the dimensionless even-wave contact. Since only $\sim 15\%$ of atoms
populate each transverse excited band, we base $k_{F}$ on the total atom
number, just as in the odd-wave case. We show in App. E that rf transfer is
proportional to the excited-band population, motivating us to report the
contact divided by the number of atoms in one single-particle excited motional
excited band $N_{1}$, which we assume to be equal for both the $x$ and $z$
directions. In Fig. 8 we report $\widetilde{C}_{e}/N_{1}$ for various
interaction strengths, tuning the scattering length $a_{e}^{z}$ across a range
of $200\,$G to $205\,$G, away from the odd-wave feature. We observe data
collapse between different lattice depths and single-shot detunings
$\omega\sim\omega_{\perp}$ chosen to be within the $-5/2$ power law regime
described in Sec. V.2.

For $k_{F}a_{e}^{z}\gtrsim 2.5$, the observed trend matches that of a high-
temperature model (App. D) that maps the imbalanced band mixture in our system
onto a spin mixture with true 1D s-wave interactions. For the large lattice
depths in Fig. 8, the distinct even-wave resonances overlap [see Fig. 2(b)]
and the slightly larger $a_{e}^{x}$ is shown on the upper axis of Fig. 8. Both
experiment and theory include both $x$ and $z$ even-wave interactions. However
in the region of interest, correlations are dominated by even-wave $z$ pairing
as indicated by the agreement with a solid line that shows
$\widetilde{C}_{e}^{z}/N_{1}$ alone.

For $0\leq k_{F}a_{e}^{z}\lesssim 2$, the observed $\widetilde{C}_{e}/N_{1}$
plateaus to $0.5(1)$. The expected near-unitary contact can be estimated using
a zero-temperature theory [57] developed for 1D s-wave spin mixtures, again
using a mapping of imbalanced band populations to imbalanced spin population
in the model. In the homogeneous case, we find
$\widetilde{C}_{e}^{z}/N_{1}\approx 1.0$, on the same order of magnitude as
the observed value. A full description of the unitary regime would need to
include loss [see inset of Fig. 8], inhomogeneity, and the contribution of
$C_{e}^{x}$. The contrast with the q1D odd-wave case, where the observed
contact was $\lesssim 10^{-2}$ of the unitary value, is already striking.

Figure 8: Even-wave correlations. Even-wave contact
$\widetilde{C}_{e}=\widetilde{C}_{e}^{x}+\widetilde{C}_{e}^{z}$ normalized by
$N_{1}$, plotted versus even-wave scattering lengths $a_{e}^{z}$ (bottom) and
$a_{e}^{x}$ (top). The scattering lengths $a_{e}$ are tuned by the magnetic
field across a range 200 G to 205 G, and scaled by the calculated $k_{F}$ of
the central tube. Single-shot measurements of $C_{e}$ use Eq. (28), repeated
here for two different single-shot detunings: $\omega/2\pi=100$ kHz (squares)
and $\omega/2\pi=125$ kHz (diamonds). Two lattice depths are shown: red and
blue points are $80\,E_{R}$ and $120\,E_{R}$, respectively. The shaded purple
band gives a high-temperature expansion prediction for
$\widetilde{C}_{e}^{x}/N_{1}+\widetilde{C}_{e}^{z}/N_{1}$, from Eq. (D.2),
with no fit parameters. The width of the band reflects one standard deviation
of systematic uncertainties (atom number and band-population fractions) and
the variation of effective-range contributions. Both experiment and theory
include both $x$ and $z$ even-wave interactions. However in the region of
interest, correlations are dominated by even-wave $z$ pairing: a solid line
shows $\widetilde{C}_{e}^{z}/N_{1}$ alone. The inset shows the relative atom
number after spectroscopy, for the same ranges of $k_{F}a_{e}^{z}$ (bottom)
and $k_{F}a_{e}^{x}$ (top).

We can interpret the strength of the observed contact using the adiabatic
relation from 1D s-waves [64]:

$\frac{\partial F}{\partial a_{e}^{z}}=C_{e}^{z}\,.$ (30)

Assuming thermal equilibrium, the interaction energy is then found by
integrating under the observed contact, which contains contributions from both
$x$ and $z$ resonances but is dominated by $\widetilde{C}_{e}^{z}$. As a lower
estimate for the total interaction energy, $\Delta F$, we find that between
$k_{F}a_{e}^{z}=5$ and $k_{F}a_{e}^{z}=2$, $\sim-0.2\,N_{1}E_{F}$, whereas
integrating between $k_{F}a_{e}^{z}=5$ and $k_{F}a_{e}^{z}=0$ we find $\Delta
F\sim-0.7\,N_{1}E_{F}$. The q1D even-wave gas is thus strongly interacting,
with an attractive energy that is at least an order of magnitude larger than
is observed in the low-loss range of the odd-wave interactions.

We can understand the enhanced strength of q1D even-wave interactions through
the proportionality of p-wave coupling to momentum, as indicated by Eq. (31).
For odd waves, the relevant momentum that leads to a finite interaction is in
the $y$ direction, along the tube. For even waves, p-wave coupling is achieved
through the transverse gradient in the spatial wave function, which is
proportional to $a_{\perp}^{-1}$ [see Eq. (32)]. For a typical momentum
$k_{F}$, the ratio of even- to odd-wave coupling is thus
2/$a_{\perp}^{2}k_{F}^{2}$, which is of order $1/N\lambda$, and thus
necessarily large in the q1D regime.

##  VI Discussion and Conclusion

The extension of low-dimensional systems to include multi-orbital scattering
opens the door to several new prospects. We have shown that one result is the
emergence of q1D even-wave interactions in a spin-polarized Fermi gas. The
range of many-body physics accessible with this new interaction channel will
be constrained by atom loss, including three-body recombination. Odd-wave atom
loss processes have been studied both theoretically [34, 35, 28]
experimentally [18, 19]; similar studies in the q1D even-wave scenario are
called for, which also address band relaxation.

Future work could also include a more detailed understanding of rf spectra
such as shown in Fig. 4 and Fig. 7, especially the low-frequency deviations
from the high-frequency power law. In this regime, the sub-leading contacts,
conjugate to $r_{o}$ for odd waves [29] and $r_{e}$ for even waves, are
expected to appear as additional higher-order power-law contributions to the
rf spectral function $\Gamma$.

A shallow q1D dimer state lies below threshold both in the repulsive $a_{o}>0$
gas and in the attractive $a_{e}>0$ gas probed in our experiment. Searching
for these dimer states and characterizing their stability would further
examine the paradigm of an emergent q1D symmetry. In the even-wave case, the
binding energy is expected to be large even near resonance.

Another prospect is to search for emergent s-waves in quasi-two-dimensional
(q2D) ensembles of identical fermions. A similar mechanism would apply: that
antisymmetric orbital wave functions in a strong confinement direction would
enable exchange-even collisional wave functions in the low-energy q2D degrees
of freedom. In either q1D or q2D, a study of the universal dynamics and
thermodynamics would be an interesting comparison to true s-wave systems.

Using a multi-channel framework, new universal contact relations should be
found to clarify, for instance, the correspondence to the closed-channel
fraction. Qualitatively, a q1D (or q2D) polarized gas with emergent s-wave
interactions is expected to be similar to a spin mixture with true s-wave
interactions. However for systems such as metastable helium [102, 103] or
dipolar gases with strong spin-orbit relaxation [104], spin polarization is
necessary for collisional stability. For fermionic isotopes in these systems,
emergent s-wave interactions in the quasi-low-dimensional regime could provide
a new pathway to low-energy universality.

The paradigm introduced here can be extended to higher partial waves. For
example, it could be used to study bosons with d-wave interactions [105, 106,
107]. The d-wave generalization of Eq. (32) would have $m=0$, $|m|=1$, and
$|m|=2$ components. The $|m|=1$ matrix element would be proportional to
$kk^{\prime}/a_{\perp}^{4}$, and would enable odd-wave scattering, i.e.,
emergent q1D p-wave interactions between identical bosons, which is forbidden
by exchange symmetry in true 1D systems. The even-$m$ matrix elements would be
proportional to $1/a_{\perp}^{6}$, which is in contrast to free space, where
low-energy d-wave scattering is suppressed at low $k$. Analogous to the
emergent q1D s-wave observed here, strong confinement enhances coupling by the
high momentum of the confined wave function.

In summary, our study provides a comprehensive understanding of the behavior
of quasi-one-dimensional fermions interacting with short-range p-wave
interactions. Our observations and analyses reveal that multi-orbital
scattering in strongly confined systems opens the door to the exploration of
alternate exchange symmetries in low-energy scattering channels. The energy-
dependence of higher-partial-wave scattering is dominated by the inter-orbital
gap, which offers a new approach to suppressing finite-range corrections and
studying universal many-body phenomena. Our work indicates the potential of
cold-atom experiments with fully activated orbital, spin, and mass degrees of
freedom, whose interplay gives rise to a wealth of phenomena in electronic
materials.

###### Acknowledgements.

The authors would like to thank S. Smale for his early contributions to the
experiment, Xiwen Guan and Xiangguo Yin for discussion of Bethe ansatz
calculations, and D. Clément for discussions. This research is supported by
NSERC, AFOSR FA9550-19-1-7044 and FA9550-19-1-0365, ARO W911NF-15-1-0603, HK
GRF 17304820, 17304719 and CRF C6009-20G and C7012-21G.

## References

  * Prange  _et al._ [2012] R. Prange, M. Cage, K. Klitzing, S. Girvin, A. Chang, F. Duncan, M. Haldane, R. Laughlin, A. Pruisken, and D. Thouless,  _The Quantum Hall Effect_, Graduate Texts in Contemporary Physics (Springer New York, 2012).
  * Barrett  _et al._ [1995] S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West, and R. Tycko, Optically Pumped NMR Evidence for Finite-Size Skyrmions in GaAs Quantum Wells near Landau Level Filling $\nu=1$, Phys. Rev. Lett. 74, 5112 (1995).
  * Aifer  _et al._ [1996] E. H. Aifer, B. B. Goldberg, and D. A. Broido, Evidence of skyrmion excitations about $\nu=1$ in $n$-modulation-doped single quantum wells by interband optical transmission, Phys. Rev. Lett. 76, 680 (1996).
  * Vijayan  _et al._ [2020] J. Vijayan, P. Sompet, G. Salomon, J. Koepsell, S. Hirthe, A. Bohrdt, F. Grusdt, I. Bloch, and C. Gross, Time-resolved observation of spin-charge deconfinement in fermionic hubbard chains, Science 367, 186 (2020).
  * Senaratne  _et al._ [2022] R. Senaratne, D. Cavazos-Cavazos, S. Wang, F. He, Y.-T. Chang, A. Kafle, H. Pu, X.-W. Guan, and R. Hulet, Spin-charge separation in a one-dimensional Fermi gas with tunable interactions, Science 376, 1305 (2022).
  * Olshanii [1998] M. Olshanii, Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons, Phys. Rev. Lett. 81, 938 (1998).
  * Giamarchi [2017] T. Giamarchi, Theory for 1D Quantum Materials Tested with Cold Atoms and Superconductors, Physics 10, 115 (2017).
  * Møller  _et al._ [2021] F. Møller, C. Li, I. Mazets, H.-P. Stimming, T. Zhou, Z. Zhu, X. Chen, and J. Schmiedmayer, Extension of the generalized hydrodynamics to the dimensional crossover regime, Phys. Rev. Lett. 126, 090602 (2021).
  * Bergeman  _et al._ [2003] T. Bergeman, M. G. Moore, and M. Olshanii, Atom-atom scattering under cylindrical harmonic confinement: Numerical and analytic studies of the confinement induced resonance, Phys. Rev. Lett. 91, 163201 (2003).
  * Moore  _et al._ [2004] G. Moore, T. Bergeman, and M. Olshanii, Scattering in tight atom waveguides, J. Phys. IV 116, 69 (2004).
  * Kim  _et al._ [2005] J. I. Kim, J. Schmiedmayer, and P. Schmelcher, Quantum scattering in quasi-one-dimensional cylindrical confinement, Phys. Rev. A 72, 042711 (2005).
  * Yurovsky [2005] V. A. Yurovsky, Feshbach resonance scattering under cylindrical harmonic confinement, Phys. Rev. A 71, 012709 (2005).
  * Naidon  _et al._ [2007] P. Naidon, E. Tiesinga, W. F. Mitchell, and P. S. Julienne, Effective-range description of a Bose gas under strong one- or two-dimensional confinement, New J. Phys. 9, 19 (2007).
  * Dunjko  _et al._ [2011] V. Dunjko, M. G. Moore, T. Bergeman, and M. Olshanii, Confinement-induced resonances, in  _Advances in Atomic, Molecular, and Optical Physics_, Vol. 60, edited by E. Arimondo, P. Berman, and C. Lin (Academic Press, 2011) pp. 461–510.
  * Kristensen and Pricoupenko [2015] T. Kristensen and L. Pricoupenko, Ultracold-atom collisions in atomic waveguides: A two-channel analysis, Phys. Rev. A 91, 042703 (2015).
  * Günter  _et al._ [2005] K. Günter, T. Stöferle, H. Moritz, M. Köhl, and T. Esslinger, $p$-wave interactions in low-dimensional fermionic gases, Phys. Rev. Lett. 95, 230401 (2005).
  * Waseem  _et al._ [2016] M. Waseem, Z. Zhang, J. Yoshida, K. Hattori, T. Saito, and T. Mukaiyama, Creation of $p$-wave Feshbach molecules in selected angular momentum states using an optical lattice, J. Phys. B 49, 204001 (2016).
  * Chang  _et al._ [2020] Y.-T. Chang, R. Senaratne, D. Cavazos-Cavazos, and R. G. Hulet, Collisional loss of one-dimensional fermions near a $p$-wave Feshbach resonance, Phys. Rev. Lett. 125, 263402 (2020).
  * Marcum  _et al._ [2020] A. S. Marcum, F. R. Fonta, A. M. Ismail, and K. M. O’Hara, Suppression of three-body loss near a $p$-wave resonance due to quasi-1D confinement, arXiv:2007.15783 (2020).
  * Granger and Blume [2004] B. E. Granger and D. Blume, Tuning the interactions of spin-polarized fermions using quasi-one-dimensional confinement, Phys. Rev. Lett. 92, 133202 (2004).
  * Kanjilal and Blume [2004] K. Kanjilal and D. Blume, Nondivergent pseudopotential treatment of spin-polarized fermions under one- and three-dimensional harmonic confinement, Phys. Rev. A 70, 042709 (2004).
  * Girardeau  _et al._ [2004] M. Girardeau, H. Nguyen, and M. Olshanii, Effective interactions, Fermi-Bose duality, and ground states of ultracold atomic vapors in tight de Broglie waveguides, Optics Communications 243, 3 (2004).
  * Pricoupenko [2008] L. Pricoupenko, Resonant scattering of ultracold atoms in low dimensions, Phys. Rev. Lett. 100, 170404 (2008).
  * Heß  _et al._ [2014] B. Heß, P. Giannakeas, and P. Schmelcher, Energy-dependent $\ell$-wave confinement-induced resonances, Phys. Rev. A 89, 052716 (2014).
  * Gao  _et al._ [2015] T.-Y. Gao, S.-G. Peng, and K. Jiang, Two-body state with p-wave interaction in a one-dimensional waveguide under transversely anisotropic confinement, Phys. Rev. A 91, 043622 (2015).
  * Saeidian  _et al._ [2015] S. Saeidian, V. S. Melezhik, and P. Schmelcher, Shifts and widths of p-wave confinement induced resonances in atomic waveguides, J. Phys. B 48, 155301 (2015).
  * Zhou and Cui [2017] L. Zhou and X. Cui, Stretching $p$-wave molecules by transverse confinements, Phys. Rev. A 96, 030701(R) (2017).
  * Sekino  _et al._ [2018] Y. Sekino, S. Tan, and Y. Nishida, Comparative study of one-dimensional Bose and Fermi gases with contact interactions from the viewpoint of universal relations for correlation functions, Phys. Rev. A 97, 013621 (2018).
  * Maki [2021] J. Maki, Thermodynamic contacts and breathing-mode physics of one-dimensional $p$-wave Fermi gases in the high-temperature limit, Phys. Rev. A 104, 063314 (2021).
  * Tanaka and Nishida [2022] T. Tanaka and Y. Nishida, Bulk viscosity of dual Bose and Fermi gases in one dimension, Phys. Rev. Lett. 129, 200402 (2022).
  * Nishida [2022] Y. Nishida, Viscous drude weight of dual Bose and Fermi gases in one dimension, Phys. Rev. A 106, 063317 (2022).
  * Maki and Zhang [2023] J. Maki and S. Zhang, Viscous flow in a one-dimensional spin-polarized Fermi gas: The role of integrability on viscosity, Phys. Rev. A 107, 013310 (2023).
  * Fonta and O’Hara [2020] F. Fonta and K. M. O’Hara, Experimental conditions for obtaining halo $p$-wave dimers in quasi-one-dimension, Phys. Rev. A 102, 043319 (2020).
  * Levinsen  _et al._ [2008] J. Levinsen, N. R. Cooper, and V. Gurarie, Stability of fermionic gases close to a $p$-wave Feshbach resonance, Phys. Rev. A 78, 063616 (2008).
  * Kurlov and Shlyapnikov [2017] D. V. Kurlov and G. V. Shlyapnikov, Two-body relaxation of spin-polarized fermions in reduced dimensionalities near a $p$-wave Feshbach resonance, Phys. Rev. A 95, 032710 (2017).
  * Regal  _et al._ [2003] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Tuning p-wave interactions in an ultracold Fermi gas of atoms, Phys. Rev. Lett. 90, 053201 (2003).
  * Zhang  _et al._ [2004] J. Zhang, E. G. M. van Kempen, T. Bourdel, L. Khaykovich, J. Cubizolles, F. Chevy, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon, $p$-wave Feshbach resonances of ultracold ${}^{6}\mathrm{Li}$, Phys. Rev. A 70, 030702(R) (2004).
  * Schunck  _et al._ [2005] C. H. Schunck, M. W. Zwierlein, C. A. Stan, S. M. F. Raupach, W. Ketterle, A. Simoni, E. Tiesinga, C. J. Williams, and P. S. Julienne, Feshbach resonances in fermionic $\sideset{{}^{6}}{}{\mathop{\rm Li}}$, Phys. Rev. A 71, 045601 (2005).
  * Chevy  _et al._ [2005] F. Chevy, E. G. M. van Kempen, T. Bourdel, J. Zhang, L. Khaykovich, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon, Resonant scattering properties close to a $p$-wave Feshbach resonance, Phys. Rev. A 71, 062710 (2005).
  * Inada  _et al._ [2008] Y. Inada, M. Horikoshi, S. Nakajima, M. Kuwata-Gonokami, M. Ueda, and T. Mukaiyama, Collisional properties of p-wave Feshbach molecules, Phys. Rev. Lett. 101, 100401 (2008).
  * Tan [2008a] S. Tan, Energetics of a strongly correlated Fermi gas, Ann. Phys. 323, 2952 (2008a).
  * Tan [2008b] S. Tan, Large momentum part of a strongly correlated Fermi gas, Ann. Phys. 323, 2971 (2008b).
  * Tan [2008c] S. Tan, Generalized virial theorem and pressure relation for a strongly correlated Fermi gas, Ann. Phys. 323, 2987 (2008c).
  * Braaten and Platter [2008] E. Braaten and L. Platter, Exact relations for a strongly interacting Fermi gas from the operator product expansion, Phys. Rev. Lett. 100, 205301 (2008).
  * Braaten  _et al._ [2008] E. Braaten, D. Kang, and L. Platter, Universal relations for a strongly interacting Fermi gas near a Feshbach resonance, Phys. Rev. A 78, 053606 (2008).
  * Werner  _et al._ [2009] F. Werner, L. Tarruell, and Y. Castin, Number of closed-channel molecules in the BEC-BCS crossover, Euro. Phys. J. B 68, 401 (2009).
  * Zhang and Leggett [2009] S. Zhang and A. J. Leggett, Universal properties of the ultracold Fermi gas, Phys. Rev. A 79, 023601 (2009).
  * Braaten  _et al._ [2010] E. Braaten, D. Kang, and L. Platter, Short-time operator product expansion for rf spectroscopy of a strongly interacting Fermi gas, Phys. Rev. Lett. 104, 223004 (2010).
  * Braaten [2012] E. Braaten, Universal Relations for Fermions with Large Scattering Length, in  _The BCS-BEC Crossover and the Unitary Fermi Gas_ , edited by W. Zwerger (Springer, Berlin, 2012) pp. 193–231.
  * Werner and Castin [2012a] F. Werner and Y. Castin, General relations for quantum gases in two and three dimensions: I. Two-component fermions, Phys. Rev. A 86, 013626 (2012a).
  * Werner and Castin [2012b] F. Werner and Y. Castin, General relations for quantum gases in two and three dimensions. II. Bosons and mixtures, Phys. Rev. A 86, 053633 (2012b).
  * Yoshida and Ueda [2015] S. M. Yoshida and M. Ueda, Universal high-momentum asymptote and thermodynamic relations in a spinless Fermi gas with a resonant $p$-wave interaction, Phys. Rev. Lett. 115, 135303 (2015).
  * Yu  _et al._ [2015] Z. Yu, J. H. Thywissen, and S. Zhang, Universal Relations for a Fermi Gas Close to a $p$-Wave Interaction Resonance, Phys. Rev. Lett. 115, 135304 (2015).
  * Luciuk  _et al._ [2016] C. Luciuk, S. Trotzky, S. Smale, Z. Yu, S. Zhang, and J. H. Thywissen, Evidence for universal relations describing a gas with p-wave interactions, Nat. Phys. 12, 599 (2016).
  * He  _et al._ [2016a] M. He, S. Zhang, H. M. Chan, and Q. Zhou, Concept of a contact spectrum and its applications in atomic quantum Hall states, Phys. Rev. Lett. 116, 045301 (2016a).
  * Peng  _et al._ [2016] S.-G. Peng, X.-J. Liu, and H. Hu, Large-momentum distribution of a polarized Fermi gas and $p$-wave contacts, Phys. Rev. A 94, 063651 (2016).
  * He  _et al._ [2016b] W.-B. He, Y.-Y. Chen, S. Zhang, and X.-W. Guan, Universal properties of Fermi gases in one dimension, Phys. Rev. A 94, 031604(R) (2016b).
  * Cui [2016] X. Cui, Universal one-dimensional atomic gases near odd-wave resonance, Phys. Rev. A 94, 043636 (2016).
  * Cui and Dong [2016] X. Cui and H. Dong, High-momentum distribution with a sub-leading $k^{-3}$ tail in odd-wave interacting one-dimensional Fermi gases, Phys. Rev. A 94, 063650 (2016).
  * Yin  _et al._ [2018] X. Yin, X.-W. Guan, Y. Zhang, H. Su, and S. Zhang, Momentum distribution and contacts of one-dimensional spinless Fermi gases with an attractive p-wave interaction, Phys. Rev. A 98, 023605 (2018).
  * Schneider and Randeria [2010] W. Schneider and M. Randeria, Universal short-distance structure of the single-particle spectral function of dilute Fermi gases, Phys. Rev. A 81, 021601(R) (2010).
  * Stewart  _et al._ [2010] J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin, Verification of universal relations in a strongly interacting Fermi gas, Phys. Rev. Lett. 104, 235301 (2010).
  * Kuhnle  _et al._ [2010] E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D. Drummond, P. Hannaford, and C. J. Vale, Universal behavior of pair correlations in a strongly interacting Fermi gas, Phys. Rev. Lett. 105, 070402 (2010).
  * Barth and Zwerger [2011] M. Barth and W. Zwerger, Tan relations in one dimension, Ann. Phys. 326, 2544 (2011).
  * Hoinka  _et al._ [2013] S. Hoinka, M. Lingham, K. Fenech, H. Hu, C. J. Vale, J. E. Drut, and S. Gandolfi, Precise determination of the structure factor and contact in a unitary Fermi gas, Phys. Rev. Lett. 110, 055305 (2013).
  * Fröhlich  _et al._ [2012] B. Fröhlich, M. Feld, E. Vogt, M. Koschorreck, M. Köhl, C. Berthod, and T. Giamarchi, Two-dimensional Fermi liquid with attractive interactions, Phys. Rev. Lett. 109, 130403 (2012).
  * Luciuk  _et al._ [2017] C. Luciuk, S. Smale, F. Böttcher, H. Sharum, B. A. Olsen, S. Trotzky, T. Enss, and J. H. Thywissen, Observation of quantum-limited spin transport in strongly interacting two-dimensional Fermi gases, Phys. Rev. Lett. 118, 130405 (2017).
  * Gaebler  _et al._ [2007] J. P. Gaebler, J. T. Stewart, J. L. Bohn, and D. S. Jin, p-wave Feshbach molecules, Phys. Rev. Lett. 98, 200403 (2007).
  * Fuchs  _et al._ [2008] J. Fuchs, C. Ticknor, P. Dyke, G. Veeravalli, E. Kuhnle, W. Rowlands, P. Hannaford, and C. J. Vale, Binding energies of $\sideset{{}^{6}}{}{\mathop{\rm Li}}$ p-wave Feshbach molecules, Phys. Rev. A 77, 053616 (2008).
  * Ahmed-Braun  _et al._ [2021] D. J. M. Ahmed-Braun, K. G. Jackson, S. Smale, C. J. Dale, B. A. Olsen, S. J. J. M. F. Kokkelmans, P. S. Julienne, and J. H. Thywissen, Probing open- and closed-channel $p$-wave resonances, Phys. Rev. Research 3, 033269 (2021).
  * Hammer and Lee [2009] H. W. Hammer and D. Lee, Causality and universality in low-energy quantum scattering, Phys. Lett. B 681, 500 (2009).
  * Hammer and Lee [2010] H. W. Hammer and D. Lee, Causality and the effective range expansion, Ann. Phys. 325, 2212 (2010).
  * Braaten  _et al._ [2012] E. Braaten, P. Hagen, H. W. Hammer, and L. Platter, Renormalization in the three-body problem with resonant p-wave interactions, Phys. Rev. A 86, 012711 (2012).
  * Bertulani  _et al._ [2002] C. A. Bertulani, H.-W. Hammer, and U. van Kolck, Effective field theory for halo nuclei: shallow p-wave states, Nuc. Phys. A 712, 37 (2002).
  * Bedaque  _et al._ [2003] P. Bedaque, H.-W. Hammer, and U. van Kolck, Narrow resonances in effective field theory, Phys. Lett. B 569, 159 (2003).
  * Gubbels and Stoof [2007] K. B. Gubbels and H. T. C. Stoof, Theory for p-wave Feshbach molecules, Phys. Rev. Lett. 99, 190406 (2007).
  * Maki and Zhang [2020] J. Maki and S. Zhang, Role of effective range in the bulk viscosity of resonantly interacting $s$\- and $p$-wave fermi gases, Phys. Rev. Lett. 125, 240402 (2020).
  * Ding and Zhang [2019] S. Ding and S. Zhang, Fermi-liquid description of a single-component Fermi gas with $p$-wave interactions, Phys. Rev. Lett. 123, 070404 (2019).
  * Haller  _et al._ [2010] E. Haller, M. J. Mark, R. Hart, J. G. Danzl, L. Reichsöllner, V. Melezhik, P. Schmelcher, and H.-C. Nägerl, Confinement-induced resonances in low-dimensional quantum systems, Phys. Rev. Lett. 104, 153203 (2010).
  * Sala  _et al._ [2012] S. Sala, P.-I. Schneider, and A. Saenz, Inelastic confinement-induced resonances in low-dimensional quantum systems, Phys. Rev. Lett. 109, 073201 (2012).
  * Sala  _et al._ [2013] S. Sala, G. Zürn, T. Lompe, A. N. Wenz, S. Murmann, F. Serwane, S. Jochim, and A. Saenz, Coherent molecule formation in anharmonic potentials near confinement-induced resonances, Phys. Rev. Lett. 110, 203202 (2013).
  * Note [1] In q1D s-wave systems, the anharmonicity of the transverse confinement potential was observed [79, 81] to induce couplings between the center-of-mass motion and the relative motion. This created new scattering resonances to bound states with CM excitation in the strongly confined direction [80]. However, we did not find any evidence of this effect in our data.
  * Nishida and Tan [2010] Y. Nishida and S. Tan, Confinement-induced $p$-wave resonances from $s$-wave interactions, Phys. Rev. A 82, 062713 (2010).
  * Note [2] A complementary phenomenon can occur for scattering in mixed dimensions [83]. In that case, the odd parity of the CM motion can induce p-wave scattering in the relative coordinate, even though the underlying scattering is s-wave, because two atoms see different trapping potentials. In our case, the CM motion is irrelevant; instead, the odd parity of the relative motion in the strongly confined direction allows for the emergence of the even-wave scattering of identical fermions along the q1D direction.
  * Son and Thompson [2010] D. T. Son and E. G. Thompson, Short-distance and short-time structure of a unitary Fermi gas, Phys. Rev. A 81, 063634 (2010).
  * Bender  _et al._ [2005] S. A. Bender, K. D. Erker, and B. E. Granger, Exponentially decaying correlations in a gas of strongly interacting spin-polarized 1D fermions with zero-range interactions, Phys. Rev. Lett. 95, 230404 (2005).
  * Ticknor  _et al._ [2004] C. Ticknor, C. A. Regal, D. S. Jin, and J. L. Bohn, Multiplet structure of Feshbach resonances in nonzero partial waves, Phys. Rev. A 69, 042712 (2004).
  * Note [3] The cylindrically symmetric states are superpositions of Cartesian states with $n_{x}+n_{z}=2\mathcal{N}+|m|$.
  * Note [4] The sub-leading correction modifies $\Delta_{e}$ by $1-2R_{\mathrm{3D}}a_{\perp}^{-1}[\zeta(1/2)+\zeta(3/2)]$, such that the leading order gives roughly a 10% underestimate for our typical parameters.
  * Note [5] The sub-leading correction modifies $\Delta_{o}$ by $1-3R_{\mathrm{3D}}\zeta(1/2)/a_{\perp}$, such that the leading order gives roughly a 20% underestimate for our typical parameters.
  * Note [6] This can be seen from Eqs. (9) and (11) since the 3D scattering volume scales as $V_{\mathrm{3D}}\sim 1/B$ near the p-wave Feshbach resonance. The widening field range of correlations at higher lattice depths can be seen in Fig. 2, with the effect strongest for even-wave correlations in Fig. 2(b). There, even-wave correlations extend across a 3 G range at $V_{L}=120\,E_{R}$, compared to a $\lesssim 0.5\,$G range of odd-wave correlations at similar lattice depth. Both of these are larger than the $\sim 0.2$ G range of significant correlations observed in 3D.
  * Zwerger [2012] W. Zwerger,  _The BCS-BEC Crossover and the Unitary Fermi Gas_ (Springer, Berlin, 2012).
  * Bloch  _et al._ [2008] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Reviews of Modern Physics 80, 885 (2008).
  * Note [7] The numerical factors in $\tilde{C}$ are chosen such that the asymptotic momentum density $n_{k}\to\tilde{C}_{o}k_{F}/k^{2}$, when $\intop\ilimits@(dk/2\pi)n_{k}=N$.
  * Note [8] Note that inhomogeneous transverse confinement broadens the range of $a_{o}^{-1}$ for a given magnetic field, which makes the assignment of small $-1/k_{F}a_{o}$ ambiguous.
  * Pan  _et al._ [2018] L. Pan, S. Chen, and X. Cui, Many-body stabilization of a resonant $p$-wave Fermi gas in one dimension, Phys. Rev. A 98, 011603(R) (2018).
  * Note [9] The sub-leading term of the isotropic Eq. 18 provides only a $\sim 50\,$mG correction to the leading order. Still, we determine $B_{0}^{y}$ through a generalized form of Eq. (18) that includes anisotropic confinement [25].
  * Makotyn  _et al._ [2014] P. Makotyn, C. E. Klauss, D. L. Goldberger, E. A. Cornell, and D. S. Jin, Universal dynamics of a degenerate unitary Bose gas, Nature Physics 10, 116 (2014).
  * Yin and Radzihovsky [2016] X. Yin and L. Radzihovsky, Postquench dynamics and prethermalization in a resonant Bose gas, Phys. Rev. A 93, 033653 (2016).
  * Eigen  _et al._ [2018] C. Eigen, J. A. P. Glidden, R. Lopes, E. A. Cornell, R. P. Smith, and Z. Hadzibabic, Universal prethermal dynamics of Bose gases quenched to unitarity, Nature 563, 221 (2018).
  * Note [10] Due to anharmonicity of the optical lattice potential, there is some ambiguity in the experimental value of the transverse trap frequency. Since the lowest allowed excitation for even-waves is the excitation of an atom from the first to third single-particle level of the optical lattice, the best choice for $\hbar\omega_{\perp}$ in this case is half that energy.
  * Jeltes  _et al._ [2007] T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. Westbrook, Comparison of the Hanbury Brown–Twiss effect for bosons and fermions, Nature 445, 402 (2007).
  * Thomas  _et al._ [2022] K. F. Thomas, Z. Ou, B. M. Henson, A. A. Baiju, S. S. Hodgman, and A. G. Truscott, Production of a highly degenerate Fermi gas of metastable helium-3 atoms, arXiv:2211.07111 (2022).
  * Chomaz  _et al._ [2023] L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B. L. Lev, and T. Pfau, Dipolar physics: a review of experiments with magnetic quantum gases, Rep. Prog. Phys. 86, 026401 (2023).
  * Cui  _et al._ [2017] Y. Cui, C. Shen, M. Deng, S. Dong, C. Chen, R. Lü, B. Gao, M. K. Tey, and L. You, Observation of broad $d$-wave Feshbach resonances with a triplet structure, Phys. Rev. Lett. 119, 203402 (2017).
  * Yao  _et al._ [2019] X.-C. Yao, R. Qi, X.-P. Liu, X.-Q. Wang, Y.-X. Wang, Y.-P. Wu, H.-Z. Chen, P. Zhang, H. Zhai, Y.-A. Chen, and J.-W. Pan, Strongly interacting Bose gases near a $d$-wave shape resonance, Nat. Phys. 15, 570 (2019).
  * Fouché  _et al._ [2019] L. Fouché, A. Boissé, G. Berthet, S. Lepoutre, A. Simoni, and T. Bourdel, Quantitative analysis of losses close to a $d$-wave open-channel Feshbach resonance in ${}^{39}\mathrm{K}$, Phys. Rev. A 99, 022701 (2019).
  * Pricoupenko [2006] L. Pricoupenko, Modeling interactions for resonant p-wave scattering, Phys. Rev. Lett. 96, 050401 (2006).
  * Idziaszek [2009] Z. Idziaszek, Analytical solutions for two atoms in a harmonic trap: $p$-wave interactions, Phys. Rev. A 79, 062701 (2009).
  * Venu  _et al._ [2023] V. Venu, P. Xu, M. Mamaev, F. Corapi, T. Bilitewski, J. P. D’Incao, C. J. Fujiwara, A. M. Rey, and J. H. Thywissen, Unitary p-wave interactions between fermions in an optical lattice, Nature 613, 262 (2023).
  * He and Zhou [2021] M. He and Q. Zhou, $p$-wave contacts of quantum gases in quasi-one-dimensional and quasi-two-dimensional traps, Phys. Rev. A 104, 043303 (2021).
  * Gao and Yu [2012] C. Gao and Z. Yu, Breathing mode of two-dimensional atomic Fermi gases in harmonic traps, Phys. Rev. A 86, 043609 (2012).
  * Hofmann [2012] J. Hofmann, Quantum anomaly, universal relations, and breathing mode of a two-dimensional Fermi gas, Phys. Rev. Lett. 108, 185303 (2012).
  * Zhang and Zhang [2017] Y.-C. Zhang and S. Zhang, Strongly interacting $p$-wave Fermi gas in two dimensions: Universal relations and breathing mode, Phys. Rev. A 95, 023603 (2017).
  * Hu and Liu [2019] H. Hu and X.-J. Liu, Resonantly interacting $p$-wave Fermi superfluid in two dimensions: Tan’s contact and the breathing mode, Phys. Rev. A 100, 023611 (2019).
  * Rey  _et al._ [2005] A. M. Rey, G. Pupillo, C. W. Clark, and C. J. Williams, Ultracold atoms confined in an optical lattice plus parabolic potential: A closed-form approach, Phys. Rev. A 72, 033616 (2005).
  * Blakie  _et al._ [2007] P. B. Blakie, A. Bezett, and P. Buonsante, Degenerate Fermi gas in a combined harmonic-lattice potential, Phys. Rev. A 75, 063609 (2007).

##  Appendix A Q1D limit of p-wave scattering

To incorporate the correct momentum dependence of p-wave scattering, a model
inter-atomic pseudo-potential $U(y,\rho)$ can be written in the following way.
Given two arbitrary two-body states $\phi$ and $\chi$, the matrix element of
$U$ is

$\langle\phi|U|\chi\rangle=g\bm{\nabla}\phi^{*}({\bm{0}})\cdot\bm{\nabla}\chi({\bm{0}})\,,$
(31)

evaluated at zero separation between the two fermions, where $g$ is a coupling
constant [21, 108, 109, 25, 110]. We note that the potential, as written, is
isotropic. In the q1D regime, the relevant two-particle states are denoted as
$\ket{\mathcal{N},m,k}$, and the matrix elements of $U$ are

$\displaystyle\langle\mathcal{N}^{\prime},m^{\prime},k^{\prime}|U$
$\displaystyle|\mathcal{N},m,k\rangle=\frac{g}{\pi
a_{\perp}^{2}}kk^{\prime}\delta_{m,0}\delta_{m,m^{\prime}}$
$\displaystyle+\frac{2g}{\pi
a_{\perp}^{4}}\sqrt{\mathcal{N}+1}\sqrt{\mathcal{N}^{\prime}+1}\delta_{|m|,1}\delta_{m,m^{\prime}}\,.$
(32)

where $\hbar$ and atomic mass factors have been suppressed. Since the relative
wave function vanishes as $\Psi_{\mathcal{N},m}\propto\rho^{|m|}$, this
pseudopoential only has non-zero matrix elements for $|m|\leq 1$. The
structure of Eq. (32) is as follows. The first term on the r.h.s. corresponds
to $m=0$, where the gradient operator acts along the $y$ direction (parallel
to $k$ and $k^{\prime}$). This produces the correct form factor for odd-wave
scattering in the q1D direction, $kk^{\prime}$. As a result, it inherits the
low-momentum behavior of the 3D p-wave scattering and vanishes at low energy.
On the other hand, the second term corresponds to the gradient operator acting
along the transverse directions, and as such, requires antisymmetric radial
wave functions ($m=\pm 1$) to give rise to a non-vanishing matrix element.
This term is proportional to a constant at low $k$ and $k^{\prime}$, which is
a hallmark of s-wave scattering.

The matrix elements in Eq. (32) carry forward to a similar structure for the
low-energy $T$-matrix in the q1D geometry which we take to have the following
form

$\displaystyle\langle$
$\displaystyle\mathcal{N}^{\prime},m^{\prime},k^{\prime}|T_{\mathrm{q1D}}|\mathcal{N},m,k\rangle=\frac{kk^{\prime}}{\pi
a_{\perp}^{2}}T_{\mathrm{q1D}}^{(0)}(E_{\mathrm{rel}})\delta_{m,0}\delta_{m,m^{\prime}}$
$\displaystyle+\frac{2}{\pi
a_{\perp}^{4}}\sqrt{\mathcal{N}+1}\sqrt{\mathcal{N}^{\prime}+1}\;T_{\mathrm{q1D}}^{(m)}(E_{\mathrm{rel}})\delta_{|m|,1}\delta_{m,m^{\prime}}\,,$
(33)

where $T_{\mathrm{q1D}}^{(m)}$ is the $T$-matrix in the q1D geometry with
relative angular momentum $m$. To evaluate $T_{\mathrm{q1D}}^{(m)}$, we make
use of its relation to the free-space $T$-matrix, $T_{\mathrm{3D}}$ [10]:

$\displaystyle
T_{\mathrm{q1D}}^{(m)}(E_{\mathrm{rel}})=T_{\mathrm{3D}}^{(m)}(E_{\mathrm{rel}})$
(34)
$\displaystyle+T_{\mathrm{3D}}^{(m)}(E_{\mathrm{rel}})\left[G_{\mathrm{q1D}}^{(m)}(E_{\mathrm{rel}})-G_{\mathrm{3D}}^{(m)}(E_{\mathrm{rel}})\right]T_{\mathrm{q1D}}^{(m)}(E_{\mathrm{rel}})\,,$

where $G_{\mathrm{q1D}}^{(m)}$ and $G_{\mathrm{3D}}^{(m)}$ are the propagators
in the q1D geometry and 3D free space, respectively, with definite angular
momentum $m$, and $T_{\mathrm{3D}}=(2\pi/\mu)f_{\mathrm{3D}}$.

Equation (34) can be evaluated analytically for both the odd-wave ($m=0$) and
even-wave ($m=\pm 1$) sectors with respective solutions given by

$\displaystyle T_{o}^{-1}$ $\displaystyle\equiv\left(\frac{1}{\pi
a_{\perp}^{2}}T_{\mathrm{q1D}}^{(0)}(E_{\mathrm{rel}})\right)^{-1}\\!\\!=\frac{2\mu}{2}\left[\frac{a_{\perp}^{2}}{6}\left(\frac{1}{V_{\mathrm{3D}}}+\frac{2\mu
E_{\mathrm{rel}}}{R_{\mathrm{3D}}}\right)-\frac{2}{a_{\perp}}\zeta(-\frac{1}{2},-\mathcal{E}_{0}-i\varepsilon)\right]\quad\mbox{and}$
(35) $\displaystyle T_{e}^{-1}$ $\displaystyle\equiv\left(\frac{2}{\pi
a_{\perp}^{4}}T_{\mathrm{q1D}}^{(\pm
1)}(E_{\mathrm{rel}})\right)^{-1}\\!\\!=\frac{2\mu}{2}\left[\frac{a_{\perp}^{4}}{12}\left(\frac{1}{V_{\mathrm{3D}}}+\frac{2\mu
E_{\mathrm{rel}}}{R_{\mathrm{3D}}}\right)+\frac{a_{\perp}}{2}\zeta(-\frac{1}{2},-\mathcal{E}_{1}-i\varepsilon)+\frac{a_{\perp}}{2}\frac{E_{\mathrm{rel}}}{2\omega_{\perp}}\zeta(\frac{1}{2},-\mathcal{E}_{1}-i\varepsilon)\right]\,.$
(36)

Here $\zeta(s,x)$ is the Hurwitz zeta function, $\varepsilon$ is an
infinitesimally small positive quantity, and
$\mathcal{E}_{m}=\left[E_{\mathrm{rel}}-(|m|+1)\omega_{\perp}\right]/2\omega_{\perp}=(ka_{\perp}/2)^{2}$.
We have also defined the appropriate renormalized q1D scattering $T$-matrices
$T_{o}$ (odd wave) and $T_{e}$ (even wave) that have a direct correspondence
with scattering $T$-matrices in true 1D systems.

The resonance condition for q1D scattering is $a_{o}^{-1}=0$ for the odd-wave
case or $a_{e}=0$ the for even-wave case. In the low-energy regime
$\mathcal{E}_{m}\ll 1$, the bound-state energies for the odd-wave and even-
wave sectors are given by

$E_{B,o}=\omega_{\perp}-\frac{\kappa_{o}^{2}}{2\mu}\quad\mbox{with}\quad\kappa_{o}=\frac{1}{a_{o}}-\kappa_{o}^{2}r_{o}$
(37)

and

$E_{B,e}=2\omega_{\perp}-\frac{\kappa_{e}^{2}}{2\mu}\quad\mbox{with}\quad\frac{1}{\kappa_{e}}=a_{e}+\kappa_{e}^{2}r_{e}\,,$
(38)

where $\kappa_{o}>0$ and $\kappa_{e}>0$ are the odd-wave and even-wave bound-
state wave numbers, respectively. The binding energies are consistent with
true 1D calculations in the shallow-dimer limit ($\kappa a_{\perp}\ll 1$): for
repulsive odd waves $a_{o}>0$, $\omega_{\perp}-E_{B,o}\approx 1/(2\mu
a_{o}^{2})$, and for attractive even waves $a_{e}>0$,
$2\omega_{\perp}-E_{B,e}\approx 1/(2\mu a_{e}^{2})$.

##  Appendix B RF spectral form in the q1D regime

Here we derive the analytic expressions for the rf transition rate
$\Gamma(\omega)$ at high frequency, taking into account the multiple
transverse excited bands. From standard arguments [61], the dominant
contribution to the rf transfer rate at high frequencies is determined by the
short-range part of the many-body wave function, which is essentially the same
as the two-body case at zero energy, apart from a normalization constant
(which we will introduce as $\gamma_{0}$ or $\gamma_{1}$ momentarily) that
depends on the many-body physics. Since we have evaluated the two-body
$T$-matrix exactly for cylindrically symmetric q1D scattering in Sec. III.1,
we can evaluate Eq. (13) in the high-frequency limit [44, 85], which gives the
exact functional dependencies of $\Gamma(\omega)$ apart from overall constants
that characterize the many-body states. We find that the rf transfer rate can
be written as $\Gamma(\omega)=\sum_{m=-1}^{+1}\Gamma^{(m)}(\omega)$ with

$\displaystyle\Gamma^{(0)}(\omega)$
$\displaystyle=\gamma_{0}\frac{\Omega^{2}}{8\pi}\frac{(2\omega_{\perp})^{3/2}}{\omega^{2}}$
(39)
$\displaystyle\times\sum_{\mathcal{N}=0}^{\infty}\bigg{(}\frac{\omega}{2\omega_{\perp}}-\mathcal{N}\bigg{)}^{1/2}\theta\bigg{(}\frac{\omega}{2\omega_{\perp}}-\mathcal{N}\bigg{)}$

and

$\displaystyle\Gamma^{(\pm 1)}(\omega)$
$\displaystyle=\gamma_{1}\frac{\Omega^{2}}{8\pi}\frac{(2\omega_{\perp})^{3/2}}{\omega^{2}}$
(40)
$\displaystyle\times\sum_{\mathcal{N}=0}^{\infty}(\mathcal{N}+1)\bigg{(}\frac{\omega}{2\omega_{\perp}}-\mathcal{N}\bigg{)}^{-1/2}\\!\theta\bigg{(}\frac{\omega}{2\omega_{\perp}}-\mathcal{N}\bigg{)}\,.$

Equations (39) and (40) are valid for large positive detuning $\omega\gg
E_{F}$. For both $\Gamma^{(0)}(\omega)$ and $\Gamma^{(\pm 1)}(\omega)$, the
sum over $\mathcal{N}$ represents the contributions from all radially excited
states. The physical interpretation of this expression is similar to Eqs. (14)
and (15) given in the main text, but now generalized: the step function
$\theta(\omega/2\omega_{\perp}-\mathcal{N})$ indicates the opening of the
transverse band for which the rf photon energy exceeds the energy difference
to the $\mathcal{N}$-th scattering continuum.

This calculation clarifies the condition in which true-1D scaling in the rf
spectrum is observed. Specifically, the 1D odd- and even-wave contact
relations apply when $|\omega|<2\omega_{\perp}$, a regime in which only the
continuum states with $\mathcal{N}=0$ are relevant. In this case,

$\displaystyle\Gamma^{(0)}(\omega)$
$\displaystyle=\gamma_{0}\Omega^{2}\frac{2\omega_{\perp}}{8\pi}\frac{1}{\omega^{3/2}}\quad\mbox{and}$
(41) $\displaystyle\Gamma^{(\pm 1)}(\omega)$
$\displaystyle=\gamma_{1}\Omega^{2}\frac{(2\omega_{\perp})^{2}}{8\pi}\frac{1}{\omega^{5/2}}\,.$
(42)

These two relations should be identified with the expressions for
$\Gamma(\omega)$ from the exact 1D odd-wave [58] and even-wave scattering
calculations:

$\Gamma^{(0)}(\omega)=\frac{\Omega^{2}}{2\omega^{3/2}}C_{o}\quad\mbox{and}\quad\Gamma^{(\pm
1)}(\omega)=\frac{\Omega^{2}}{2\omega^{5/2}}C_{e}\,,$ (43)

where $C_{o}$ and $C_{e}$ are the 1D odd-wave and even-wave contacts,
respectively. We thus identify $\gamma_{0}$ and $\gamma_{1}$ as

$\gamma_{0}=\frac{4\pi}{2\omega_{\perp}}C_{o}\quad\mbox{and}\quad\gamma_{1}=\frac{4\pi}{(2\omega_{\perp})^{2}}C_{a}\,.$
(44)

Using these relations in Eqs. (39) and (40), we obtain the limiting forms Eqs.
(14) to (17) in the main text, which are relevant to q1D physics. However, in
the limit $\omega\to\infty$, increasingly shorter length scales are probed as
the continuum final states of higher transverse bands are made available,
eventually approaching the 3D limit; this behavior is also present in Eqs.
(39),(40). Replacing the sum over transverse radial states with an integral,
one reproduces the correct leading order $\omega^{-1/2}$ rf spectral law for a
p-wave Fermi gas in 3D. A related discussion of the q1D momentum distribution
is presented in Ref. [111].

##  Appendix C Final-state effects in even-wave spectroscopy

The ideal scenario of rf spin-flip spectroscopy is that an atom is transferred
from an interacting state to a non-interacting state. This is a good
approximation when interactions in the initial spin state are near-resonant,
and the final states at the same magnetic field are far from resonance. In the
situation probed in our experiments, atoms are spin-flipped from $\ket{b}$ to
$\ket{c}$, so that $bb$ p-wave interactions should be large compared to $bc$
p-wave interactions. Coupled-channels calculations indicate that the $z$-dimer
$bc$ p-wave Feshbach resonance is located at $B_{0}=217.12$ G, with a
scattering volume that can be parameterized as
$V_{bc}(B)/V_{\mathrm{bg}}=1-\Delta/(B-B_{0})$, with
$V_{\mathrm{bg}}=(-107.95a_{0})^{3}$ and $\Delta=-8.98$ G where $a_{0}$ is the
Bohr radius. This yields a zero-crossing near $208$ G, such that $V_{bc}$ is
below its background value in the range of fields across which we measure q1D
contacts, typically $200$ G – $205$ G, and can hence be considered as weakly
interacting in this regime.

For even-wave spectroscopy, adding final states effects to Eq. (17), the
relation between $\Gamma$ and $C_{e}$ is

$\lim_{\omega\rightarrow\infty}\Gamma(\omega)=\frac{\Omega^{2}}{2\sqrt{\hbar
m}}\frac{C_{e}}{\omega^{5/2}}\frac{\left(a_{e}-a_{e}^{\prime}-m\omega(r_{e}-r_{e}^{\prime})/\hbar\right)^{2}}{(m\omega
r_{e}^{\prime}/\hbar-a_{e}^{\prime})^{2}+(m\omega/\hbar)^{-1}}\,,$ (45)

where $a_{e}^{\prime}$ and $r_{e}^{\prime}$ are the 1D scattering parameters
for the $bc$ channel. Equation (45) predicts a deviation of $\Gamma(\omega)$
from its asymptotic form, however within the range of frequency and scattering
parameters relevant to the experiment, it does not explain the deviation of
the even-wave rf spectrum from the $-5/2$ scaling shown in Fig. 7.

In our regime of interest, Eq. (45) can be approximated as

$\lim_{\omega\rightarrow\infty}\Gamma(\omega)=\frac{\Omega^{2}}{2\sqrt{\hbar
m}}\frac{C_{e}}{\omega^{5/2}}\frac{\left(a_{e}-a_{e}^{\prime}\right)^{2}}{a_{e}^{\prime
2}}\,.$ (46)

This allows us to bound the final-state effects to corrections between $0.90$
and $1.05$ using $V_{bc}$ and a typical value of $R_{\textrm{3D}}$ in the
field and depth range used in Fig. 8. Since this is less than the scatter of
the experimental data, we can safely neglect final-state effects.

##  Appendix D High-temperature expansion for the odd- and even-wave contacts

In the high-temperature regime of small fugacity, $z\ll 1$, analytic forms of
the many-body contact can be found from a virial expansion [112, 113, 114,
115, 29]. Although we explore systems with $T$ comparable to $T_{F}$, the
high-temperature approximation provides a qualitative comparison between
theory and experiment with no adjustable parameters. Here we consider a
homogeneous 1D system with length $L$, uniform density $n$, and total number
$N=Ln$. Comparison to experimental data uses a correction for inhomogeneity as
described in App. F.

###  D.1 Odd-wave contact

The high-temperature calculation of the odd-wave contact comes from a two-
channel model of interacting atoms and closed-channel molecules [29]. For weak
attractive interactions $|a_{o}|/\Lambda_{T}\ll 1$ and $a_{o}<0$,

$C_{o}=\frac{2\pi\hbar^{2}Nna_{o}^{2}}{m\Lambda_{T}^{2}}\left[1-6\pi\frac{a_{o}r_{o}}{\Lambda_{T}^{2}}\right]\,,$
(47)

where $\Lambda_{T}=(2\pi\hbar^{2}\beta/m)^{1/2}$ is the single-particle
thermal de Broglie wavelength. As assumed in App. F, the contact is $Nn$ times
a function of $T$, $a_{o}$, and $r_{o}$. A convenient form uses $k_{F}$ and
$E_{F}$ of the central tube as reference values:

$\frac{C_{o}}{N}=\frac{\hbar^{2}k_{F}^{3}\widetilde{T}a_{o}^{2}}{2\pi
m}\left(1+\frac{3}{2}k_{F}^{2}r_{o}a_{o}\widetilde{T}\right)\,,$ (48)

where we have used $\Lambda_{T}^{-2}=\widetilde{T}k_{F}^{2}/4\pi$ and
$k_{F}=\pi n$.

###  D.2 Even-wave contact

The calculation technique for the even-wave contact follows the high-
temperature treatment for odd waves [29], i.e., using a two-channel model of
spin-half fermions coupled to closed-channel molecules to evaluate the
contacts for arbitrary interaction strengths. To model the anisotropic nature
of the interactions and the trapping potential, we assume that the $x$ and $z$
resonances are independent 1D systems. The expression for the contact
$C_{e}^{x,z}$ has a contribution from the scattering continuum
($C_{e}|_{\mathrm{at}}$) and a contribution from the two-body bound state
($C_{e}|_{\mathrm{mol}}$). Omitting the direction label $(x,z)$, these two
terms are

$C_{e}\big{|}_{\mathrm{at}}=N_{1}np_{0}\int^{\infty}_{0}\\!\\!du\frac{2\pi^{-1/2}u^{1/2}e^{-u}}{\frac{mu}{\hbar^{2}}\big{(}a_{e}-\frac{mu}{\hbar^{2}\beta}r_{e}\big{)}^{2}+\beta}\,,$
(49)

where the integration variable is dimensionless, $p_{0}\equiv
N_{0}/(N_{0}+N_{1})$ is the population fraction of ground-band atoms, and

$C_{e}\big{|}_{\mathrm{mol}}\approx
N_{1}np_{0}\frac{\hbar^{2}}{m}\frac{\sqrt{2}\Lambda_{T}\,\theta(a_{e})}{r_{e}+|a_{e}|^{3}/2}e^{\beta\hbar^{2}/(ma_{e}^{2})}\,,$
(50)

where $\theta()$ is the Heaviside step function. Note that in Eq. (50), we
have used the shallow-dimer limit instead of the full Eq. (38), which is a
good approximation in the $k_{F}a_{e}>2$ range for which the high-temperature
$C_{e}$ is considered in Fig. 8. Just like the odd-wave case, the even-wave
contact is a function of $Nn$, but now multiplied by the fraction in each
band.

Scaling by $k_{F}$ and $E_{F}$ of the central tube, the sum of these
contributions is

$\displaystyle\frac{C_{e}}{N_{1}}=$
$\displaystyle\frac{2p_{0}\hbar^{2}k_{F}^{3}}{\pi^{3/2}m}\bigg{\\{}\frac{\sqrt{2}\pi
e^{2/(k_{F}^{2}a_{e}^{2}\widetilde{T})}}{k_{F}^{3}r_{e}+|k_{F}a_{e}|^{3}/2}\widetilde{T}^{-1/2}$
$\displaystyle+\int^{\infty}_{0}\\!\\!du\frac{u^{1/2}e^{-u}}{u(k_{F}a_{e}-uk_{F}^{3}r_{e}\widetilde{T}/2)^{2}+2\widetilde{T}^{-1}}\bigg{\\}}\,,$
(51)

which is plotted in Fig. 8. In the weakly interacting limit the even-wave
contacts are given by

$\frac{C_{e}}{N_{1}}\approx\frac{2p_{0}\hbar^{2}k_{F}}{\pi m}\,a_{e}^{-2}$
(52)

In the resonantly interacting limit, the high temperature theory breaks down,
as the binding energy $\propto 1/a_{e}^{2}$ quickly exceeds the temperature,
invalidating the high-temperature assumption.

Figure 9: Correlation of transfer with modulation type and strength. (a)
Amplitude modulation of the lattice potential causes a single-particle first-
to-third transition that removes atoms from the first excited band. (b)
Frequency modulation causes a single-particle zero-to-first transition to
populate the excited band while also depleting the ground band. (c) Number of
transferred atoms to probe state from $\widetilde{\omega}/2\pi=50$ kHz for
$40\,E_{R}$ confinement with varying AM strength. No significant change in
transfer at the odd-wave peak ($199.25$ G). Less transfer is noticed at the
even-wave ($x$) peak ($199.75$ G). Transfer is plotted relative to the
transfer with no modulation. (d) Transferred atoms for various frequency
modulation. A slight decrease in transfer for the odd-wave peak is seen as
atoms are transferred out of the ground band. Transfer is increased for the
even-wave peak. (e) Transfer at even-wave peak is plotted versus the
population product excited- and ground-state populations (see text) divided by
$N_{\mathrm{tot}}$, and compared to a linear fit.

##  Appendix E RF spectroscopy with variable band populations

The odd- and even-wave contacts are extensive quantities that are proportional
to population in the relevant bands at high temperature. We are able to
manipulate band populations with two types of lattice modulations, amplitude
modulation (AM) and frequency modulation (FM). In this Appendix, we show their
effects on the measured contacts.

Figure 9(a) illustrates AM and its primary effect on band populations.
Anharmonicity of the lattice allows for the tuning of the frequency of
modulation to be such that only one transition is resonant: AM frequency
resonant first-to-third excited states transfer is off-resonant for excitation
from the ground band to the second excited band. Since atoms in the third band
leave the trap, this process distills the ground band.

Phase modulation of the optical standing wave is implemented experimentally by
FM of the light, which creates a time-dependent displacement of local
potential minima where atoms are trapped, due to the optical propagation to
the retro-reflecting mirror and back. Figure 9(b) illustrates this local
position displacement of the lattice potential. Unlike AM, phase modulation
provides the necessary parity-breaking time-dependent gradient to transfer
atoms from the ground band to the first excited band.

The effect of AM on the odd- and even-wave contacts are shown in Fig. 9(c).
Here circles and squares report $\Gamma$ at magnetic fields near the odd-wave
and even-wave peaks, respectively, which is proportional to the (extensive)
contact. We see that increasing AM strength has no noticeable trend in
$\Gamma$ at the odd-wave peak, which is consistent with odd-wave correlations
being due to scattering between ground-band atoms. At the even-wave peak,
decrease of $\Gamma$ with AM is also consistent with even-wave correlations
being proportional to the exited-band population.

Similarly, the effect of FM is shown in Fig. 9(d). Since FM modulates atoms
from the ground band to the excited band, one might expect an equal change in
the strength of correlations for odd and even-waves. Instead, we observe that
increasing modulation strength only slightly decreases transfer at the odd-
wave peak and causes comparatively large increases in transfer for the even-
wave peak. From this we remark that the relatively equal weights for the peaks
seen in Fig. 2(b) is coincidental for the $\sim 15\%$ of excited-band atoms
populated by our multi-band loading procedure.

Figure 9(e) demonstrates how transfer at the even-wave peak is correlated with
the product of ground- and excited-state atom populations, counted in band-
mapped time-of-flight imaging. A linear fit of the measurements has a zero
crossing within uncertainty of zero and a reduced $\chi^{2}$ of 1.07. This
measurement provides us with the confidence to attribute the transfer seen at
the even-wave peaks as a measure of correlations between ground and excited
band atoms.

##  Appendix F Inhomogeneous averaging

The theoretical description of atom-atom correlations throughout this
manuscript assume a single q1D system with a homogeneous atomic density. Our
experimental measurements average across two types of inhomogeneity: the
density profile within a single tube, and tube-to-tube variation in $N$. In
the high-temperature limit, $C_{\mathrm{hom}}/Nn_{\textrm{1D}}$ for a uniform-
density gas is solely a function of temperature, scattering parameters, and
confinement strength (see Sec. D), i.e., variables that are uniform across our
ensemble. We can thus account for inhomogeneities by separating out the
dependence of contact on atom number and density, as

$\langle C\rangle_{\mathrm{obs}}=\langle
Nn_{\textrm{1D}}\rangle\frac{C_{\mathrm{hom}}}{Nn_{\textrm{1D}}}$ (53)

where $\langle Nn\rangle$ is an average across the ensemble.

The inhomogeneity within each tube is treated with a local density
approximation: that each differential length $dy$ contributes
$n_{\textrm{1D}}^{2}(y)dy$ to $Nn_{\textrm{1D}}$. In the high-temperature
limit, the density distribution is given by a Maxwell-Boltzmann distribution,
so

$\langle Nn_{\textrm{1D}}\rangle_{\mathrm{tube}}=\int
n_{\textrm{1D}}^{2}(y)dy=(4\pi)^{-1/2}{N^{2}}/{R_{T}}$ (54)

where $R_{T}=(k_{B}T/m\omega_{y}^{2})^{1/2}$ is the thermal length of the
cloud. By comparison, the peak 1D density is $(2\pi)^{-1/2}N/R_{T}$, such that
$\langle Nn_{\textrm{1D}}\rangle_{\mathrm{tube}}=Nn_{\mathrm{pk}}/\sqrt{2}$.

We model the distribution of q1D ensembles by assuming the transfer between a
3D Fermi gas and an array of tubes is isentropic. This approach neglects the
fraction of atoms that appear in higher bands, since edges of the Brillouin
zone cannot be adiabatic. This transfer occurs during the first
$t_{\mathrm{ramp}}$ phase of Fig. 1(b)(i), up to a modest depth
($V_{L}<10E_{R}$). The density of states of the low-depth array is a
convolution of a harmonic-lattice density of states
$\rho(\epsilon)=2\sqrt{E_{R}}/\pi\hbar\omega_{\perp}\sqrt{\epsilon}$ [116,
117] in the $x$ and $z$ directions with a 1D harmonic oscillator
$\rho(\epsilon)=1/\hbar\omega_{y}$ along the tube where $\epsilon$ is energy.
We find the chemical potential and temperature of the transformed tube array
by imposing entropy and number conservation between the pre-loaded gas to the
array of tubes. After this loading phase, energy and particle transport
between the tubes are suppressed as the lattice depths increase to their final
values. Including a $-20\%$ corrective factor for atom loss observed in round-
trip loading calibrations, we find the number and temperature of the central
tube to be $N^{*}\approx 30$ and $\widetilde{T}\approx 0.8$ for typical
conditions, where $\widetilde{T}=k_{B}T/E_{F}$; here and below, we will use
$N^{*}$ to define a reference value of the 1D Fermi energy $E_{F}$. The atom-
number distribution of this adiabatic loading model is compared to a diabatic
model in which the initial 3D density distribution is partitioned into an
square array of parallelepipeds. In both models, the distribution of $N$ is
found to be roughly uniform: i.e., as many tubes have near-peak atom number as
have only one atom.

We define a distribution function $w_{N}$ of atom number that satisfies
$\sum_{N}w_{N}=M$, where $M$ is the total number of tubes, and
$\sum_{N}w_{N}N=N_{\mathrm{tot}}$. Without a dispersion in atom number, each
of the $M$ tubes would have $N^{*}$ atoms, so that $N_{\mathrm{tot}}=N^{*}M$,
and the distribution function would be
$w_{N}=N_{\mathrm{tot}}\delta_{N,N^{*}}/N^{*}$. Instead, in the distribution
found by these loading models, $w^{\mathrm{exp}}_{N}$ is approximately
constant between $N=1$ and $N=N^{*}$, and zero otherwise. This gives

$N_{\mathrm{tot}}=w^{\mathrm{exp}}_{N}\sum_{N=1}^{N^{*}}N=\frac{1}{2}(N^{*}+1)N^{*}w^{\mathrm{exp}}_{N}\,.$
(55)

We can rearrange this expression to find
$w^{\mathrm{exp}}_{N}=2N_{\mathrm{tot}}/[(N^{*}+1)N^{*}]$ for $1\leq N\leq
N^{*}$. Since the contact is proportional to $N^{2}$ in the high-temperature
limit [see Eq. (54)], the ratio of the contacts in these two cases is given by

$\frac{\langle
N^{2}\rangle^{\mathrm{exp}}}{(N^{*})^{2}}=\frac{1}{N_{\mathrm{tot}}N^{*}}\sum_{N=1}^{N^{*}}w^{\mathrm{exp}}_{N}N^{2}=\frac{1+2N^{*}}{3N^{*}}\,.$
(56)

which approaches $2/3$ for large $N^{*}$.

Combining the intra-tube and inter-tube inhomogeneity effects yields a
correction in the estimated value of the contact

$\frac{\langle
C_{\mathrm{tube}}\rangle^{\mathrm{exp}}}{C_{\mathrm{hom}}}=\frac{2}{3}\sqrt{\frac{\pi}{8}}\widetilde{T}^{-1/2}\,.$
(57)

The theoretical curves shown in Figs. 5 and 8 use the uniform-gas prediction
multiplied by this correction, which is $\sim 0.4$ for typical loading
parameters.