††thanks: Authors P.H. and M.A.P. contributed equally to this work.††thanks:
Authors P.H. and M.A.P. contributed equally to this work.

# Engineering spin squeezing in a 3D optical lattice with interacting spin-
orbit-coupled fermions

P. He  M. A. Perlin  S. R. Muleady  R. J. Lewis-Swan  JILA, National Institute
of Standards and Technology and Department of Physics, University of Colorado,
Boulder, CO, 80309, USA  Center for Theory of Quantum Matter, University of
Colorado, Boulder, CO, 80309, USA  R. B. Hutson  J. Ye  JILA, National
Institute of Standards and Technology and Department of Physics, University of
Colorado, Boulder, CO, 80309, USA  A. M. Rey  JILA, National Institute of
Standards and Technology and Department of Physics, University of Colorado,
Boulder, CO, 80309, USA  Center for Theory of Quantum Matter, University of
Colorado, Boulder, CO, 80309, USA

###### Abstract

One of the most important tasks in modern quantum science is to coherently
control and entangle many-body systems, and to subsequently use these systems
to realize powerful quantum technologies such as quantum-enhanced sensors.
However, many-body entangled states are difficult to prepare and preserve
since internal dynamics and external noise rapidly degrade any useful
entanglement. Here, we introduce a protocol that counterintuitively exploits
inhomogeneities, a typical source of dephasing in a many-body system, in
combination with interactions to generate metrologically useful and robust
many-body entangled states. Motivated by current limitations in state-of-the-
art three-dimensional (3D) optical lattice clocks (OLCs) operating at quantum
degeneracy, we use local interactions in a Hubbard model with spin-orbit
coupling to achieve a spin-locking effect. In addition to prolonging inter-
particle spin coherence, spin-locking transforms the dephasing effect of spin-
orbit coupling into a collective spin-squeezing process that can be further
enhanced by applying a modulated drive. Our protocol is fully compatible with
state-of-the-art 3D OLC interrogation schemes and may be used to improve their
sensitivity, which is currently limited by the intrinsic quantum noise of
independent atoms. We demonstrate that even with realistic experimental
imperfections, our protocol may generate $\sim 10$–$14$ dB of spin squeezing
in $\sim 1$ second with $\sim 10^{2}$–$10^{4}$ atoms. This capability allows
OLCs to enter a new era of quantum enhanced sensing using correlated quantum
states of driven non-equilibrium systems.

##  I Introduction

A major frontier of contemporary physics is the understanding of non-
equilibrium behaviors of many-body quantum systems, and the application of
these behaviors toward the development of novel quantum technologies with
untapped capabilitiesEisert  _et al._ (2015). To this end, ultracold atomic,
molecular, and optical systems are ideal platforms for studying unexplored
regimes of many-body physics due to their clean preparation and readout, high
controllability, and long coherence timesBloch  _et al._ (2008); Gross and
Bloch (2017). The exquisite capabilities of these systems have pushed the
frontiers of metrology, quantum simulation, and quantum information science.

Optical lattice clocks in particular have seen some of the most impressive
developments in recent years, reaching record levels of precision ($\sim
3\times 10^{-19}$)Campbell  _et al._ (2017); Marti  _et al._ (2018) and
accuracy ($\sim 1\times 10^{-18}$)Bloom  _et al._ (2014); McGrew  _et al._
(2018). These advancements required important breakthroughs, including the
capability to cool and trap fermionic alkaline-earth atoms in spin-insensitive
potentialsTakamoto and Katori (2003); Barber  _et al._ (2006); Ye  _et al._
(2008); the development of ultracoherent lasersKessler  _et al._ (2012); Cole
_et al._ (2013); Matei  _et al._ (2017) to fully exploit an ultranarrow clock
transitionLudlow  _et al._ (2015); the detailed characterization of inter-
atomic interactionsScazza  _et al._ (2014); Cappellini  _et al._ (2014); Zhang
_et al._ (2014); and, more recently, the preparation of a quantum degenerate
gas in a three-dimensional (3D) optical latticeCampbell  _et al._ (2017);
Marti  _et al._ (2018); Goban  _et al._ (2018). Nonetheless, all improvements
in sensing capabilities to date have been based on single-particle control of
internal atomic degrees of freedom. Such strategies will eventually have
diminishing returns due to practical difficulties in (i) suppressing
decoherence from external (motional) degrees of freedom, and (ii)
interrogating more particles without additional systematic errors from
interactionsMartin  _et al._ (2013); Ludlow  _et al._ (2015); Marti  _et al._
(2018).

Pushing beyond the current independent-particle paradigm requires leveraging
many-body quantum correlations. Entangled states such as spin-squeezed
statesKitagawa and Ueda (1993); Wineland  _et al._ (1992); Ma  _et al._ (2011)
can enhance measurement sensitivity, i.e. the uncertainty $\Delta\theta$ in
the estimation of a parameter $\theta$, below the standard quantum limit
$\Delta\theta\sim 1/\sqrt{N}$ for $N$ uncorrelated particlesItano  _et al._
(1993); Degen  _et al._ (2017). The major challenge for progress in this
direction is that generating entanglement requires interactions, which are
generally undesirable because they degrade atomic coherence, thereby limiting
clock performanceSwallows  _et al._ (2011); Martin  _et al._ (2013); Rey  _et
al._ (2014); Ludlow  _et al._ (2011); Lemke  _et al._ (2011); Ludlow  _et al._
(2015). In fact, the most precise and accurate optical lattice clocks were
designed to operate with fermionic atoms in identical nuclear and electronic
states to suppress collisional decoherenceCampbell  _et al._ (2009); Swallows
_et al._ (2011); Campbell  _et al._ (2017), as identical fermions cannot
interact via the otherwise dominant ($s$-wave) collisions at ultracold
temperatures. However, an initially spin-polarized Fermi gas still exhibits
interactions at later times due to spin-orbit coupling (SOC) that is induced
by the laser that drives the clock transition (i.e. the “clock laser”)Wall
_et al._ (2016); Kolkowitz  _et al._ (2016); Livi  _et al._ (2016); Bromley
_et al._ (2018). Specifically, the momentum kick imparted by this laser
imprints a position-dependent phase that induces inhomogeneous spin precession
and generates spin dephasing, thereby making atoms distinguishable and
vulnerable to collisions. While a deep lattice can suppress SOC, it also
intensifies the light scattering which currently limits the coherence time of
the clockDörscher  _et al._ (2018); Goban  _et al._ (2018); Hutson  _et al._
(2019).

In this work, we describe a scheme that can lead to metrological advances in
state-of-the-art optical lattice clocks through direct use of quantum
entanglement by harnessing the interplay between nominally undesirable
collisions and SOC. This scheme is made possible in the weak SOC regime by the
formation of an interaction-energy gap that suppresses the SOC-induced
population transfer from the exchange-symmetric Dicke manifold (spanned by
spin-polarized, and thus non-interacting states) to the remainder of Hilbert
space. Interactions thereby prolong inter-particle spin coherence through a
spin-locking effect, which additionally transforms the dephasing effect of SOC
into a collective spin squeezing process. To generate spin squeezing, our
protocol only requires the capability to fix (i) the orientation of the clock
laser and (ii) the optical lattice depth. These controls are straightforward
to incorporate into current 3D clock interrogation sequences without
sacrificing atom numbers or coherence times. Additionally, we show that by
applying a modulated drive from the clock laser, one can further prepare
states that saturate the Heisenberg limit $\Delta\theta\sim 1/N$ for phase
sensitivityKitagawa and Ueda (1993); Ma  _et al._ (2011); Degen  _et al._
(2017). This capability mirrors efforts in other settings, such as nitrogen-
vacancy centers in diamondBauch  _et al._ (2018); Aiello  _et al._ (2013) and
trapped ionsBurd  _et al._ (2019), to enhance quantum metrology through the
use of driven non-equilibrium phenomena.

Despite an abundance of proof-of-principle experiments with entangled
statesDegen  _et al._ (2017); Pezzè  _et al._ (2018), so far only the
remarkable example of LIGOAasi  _et al._ (2013); Abbott  _et al._ (2016) has
demonstrated a quantum advantage in a state-of-the-art quantum sensing or
measurement system. The new generation of 3D optical lattice systems have
fully quantized motional degrees of freedomCampbell  _et al._ (2017), allowing
for precise control of collisional interactions. We demonstrate how these
interactions can naturally give rise to metrologically useful correlated many-
body fermionic states, opening a path to not only generate entanglement, but
also harness it to achieve a quantum advantage in a world-class sensor. Such
an advance will ultimately deliver gains to real-world applications including
timekeeping, navigation, telecommunication, and our understanding of the
fundamental laws of natureSafronova  _et al._ (2018).

##  II Spin squeezing with the Fermi-Hubbard model

We consider $N$ fermionic atoms with two spin states (labeled $\uparrow$ and
$\downarrow$) trapped in a 3D optical lattice. In this discussion, these spin
states are associated with the two electronic states of a nuclear-spin-
polarized gas. At sufficiently low temperatures, atoms occupy the lowest Bloch
band of the lattice and interact only through $s$-wave collisions. A schematic
of this system is provided in Fig. 1(a), where tight confinement prevents
motion along the vertical direction ($z$), effectively forming a stack of
independent 2D lattices. For simplicity and without loss of generality,
however, we first consider the case when tunneling can only occur along one
direction, $x$, and thus model the system as living in one dimension.

Figure 1: Schematic of the setup for spin squeezing. (a) We consider $N$
fermionic atoms with two (pseudo-)spin components, represented by red and blue
spheres, trapped in the ground band of an optical lattice (shown in 2D for the
sake of presentation). Atoms tunnel to neighboring sites at a rate $J$ and
experience on-site interactions with strength $U$. An external laser carrying
a position dependent phase $e^{i\bm{k}_{\text{L}}\cdot\bm{r}}$ couples the
spin states of the atoms. (b) After a gauge transformation, different spin
states exhibit different dispersion relations with a relative phase
$\phi=k_{\text{L}}a$, where $a$ is the lattice spacing. The external laser
couples spin states with identical quasi-momenta $q$ in the gauge-transformed
frame. (c) If interactions are sufficiently weak, all motional degrees of
freedom become frozen in momentum space, with atoms effectively pinned to
fixed quasi-momentum modes $\bm{q}$. The dynamics on the frozen $\bm{q}$-space
lattice can then be mapped to a spin model in which collisional interactions
correspond to a uniform, all-to-all ferromagnetic Heisenberg Hamiltonian with
strength $U/L$, where $L$ is the total number of lattice sites. (d) The spin
dependence of the dispersion relation is captured by a mode-dependent axial
field $B_{q}$ that generates inhomogeneous spin precession. This axial field
couples exchange-symmetric many-body Dicke states with total spin $S=N/2$ to
spin-wave states with $S=N/2-1$. The all-to-all interaction opens an energy
gap $fU$ (with $f=N/L$ the filling fraction of spatial modes) between the
Dicke states and the spin-wave states, which forbids population transfer
between them in the weak-field limit. (e) To generate spin squeezing via one-
axis twisting, we initialize a product state with all spins polarized in $-z$
(i.e. in $\Ket{\downarrow}$), and apply a fast external laser pulse to rotate
all spins into $x$. We then let atoms freely evolve for a variable time $t$
(with a spin-echo pulse), after which the amount of spin squeezing can be
determined experimentally from global spin measurements. The spin-squeezed
state can be used for a follow-up clock interrogation protocol (see Appendix
E).

An external laser with Rabi frequency $\Omega$ and wavenumber $k_{\text{L}}$
along the tunneling axis resonantly couples atoms’ internal states through the
Hamiltonian $\hat{H}_{\text{laser}}/\hbar=\sum_{j}\Omega
e^{ik_{\text{L}}x_{j}}\hat{c}_{j,\uparrow}^{\dagger}\hat{c}_{j,\downarrow}+\text{h.c.}$,
where $\hat{c}_{j\alpha}$ is a fermionic annihilation operator for an atom on
site $j$ with internal state $\alpha\in\left\\{\uparrow,\downarrow\right\\}$
and $x_{j}$ is the position of site $j$. This laser imprints a position-
dependent phase that equates to a momentum kick $k_{\text{L}}$ when an atom
changes internal states by absorbing or emitting a photon, thereby generating
spin-orbit couplingWall  _et al._ (2016); Livi  _et al._ (2016). After
absorbing the position dependence of the laser Hamiltonian into fermionic
operators through the gauge transformation $\hat{c}_{j,\uparrow}\to
e^{ik_{\text{L}}x_{j}}\hat{c}_{j,\uparrow}$, which makes
$\hat{H}_{\text{laser}}$ spatially homogeneous, the atoms are well-described
in the tight-binding limit by the Fermi-Hubbard HamiltonianEsslinger (2010)

$\displaystyle\hat{H}_{\text{FH}}^{(\phi)}/\hbar$
$\displaystyle=-J\sum_{j}\left(e^{i\phi}\hat{c}_{j,\uparrow}^{\dagger}\hat{c}_{j+1,\uparrow}+\hat{c}_{j,\downarrow}^{\dagger}\hat{c}_{j+1,\downarrow}+\text{h.c.}\right)$
$\displaystyle\qquad+U\sum_{j}\hat{n}_{j,\uparrow}\hat{n}_{j,\downarrow},$ (1)

where $J$ is the nearest-neighbor tunneling rate; the SOC angle $\phi\equiv
k_{\text{L}}a$ determines the phase gained by spin-up atoms upon tunneling
from site $j$ to site $j+1$ (in the gauge-transformed frame) with lattice
spacing $a=x_{j+1}-x_{j}$; $U$ is the on-site interaction energy of two atoms;
and $\hat{n}_{j\alpha}\equiv\hat{c}_{j\alpha}^{\dagger}\hat{c}_{j\alpha}$ is a
number operator.

The Fermi-Hubbard Hamiltonian can be re-written in the quasi-momentum basis
with annihiliation operators $\hat{c}_{q\alpha}\equiv
L^{-1/2}\sum_{j}e^{-iqx_{j}}\hat{c}_{j\alpha}$, where $q$ is a quasi-momentum
and $L$ is the total number of lattice sites. In this basis, the single-
particle Hamiltonian exhibits shifted dispersion relations that signify spin-
orbit coupling [see Fig. 1(b)]:

$\displaystyle\hat{H}_{\text{FH,single}}^{(\phi)}/\hbar=-2J\sum_{q}\left[\cos(qa+\phi)\hat{n}_{q,\uparrow}+\cos(qa)\hat{n}_{q,\downarrow}\right].$
(2)

When $U\lesssim J$, interaction energies are too weak for collisions to change
the occupancies of single-particle quasi-momentum modes. Atoms are then pinned
to these modes, which form a lattice in quasi-momentum space [see Fig.
1(c)]Bromley  _et al._ (2018). In this strong-tunneling limit, the Fermi-
Hubbard Hamiltonian [Eqn. (2)] can be mapped to a spin-$1/2$ system with a
collective ferromagnetic Heisenberg interaction and an inhomogeneous axial
field, given byMartin  _et al._ (2013); Rey  _et al._ (2014); Bromley  _et
al._ (2018)

$\displaystyle\hat{H}_{\text{spin}}/\hbar=-\frac{U}{L}\hat{\bm{S}}\cdot\hat{\bm{S}}-\sum_{q}B_{q}\hat{s}_{q}^{\text{z}},$
(3)

where $\hat{\bm{S}}=\sum_{q}\hat{\bm{s}}_{q}$ is a collective spin operator;
$\hat{\bm{s}}_{q}$ is a spin-1/2 operator for mode $q$ with components
$\hat{s}_{q}^{j=\text{x},\text{y},\text{z}}\equiv\frac{1}{2}\sum_{\alpha,\beta}\hat{c}_{q\alpha}^{\dagger}\sigma^{j}_{\alpha\beta}\hat{c}_{q\beta}$
defined in terms of the Pauli matrices
$\sigma^{j=\text{x},\text{y},\text{z}}$; the sums over $q$ run over all
occupied quasi-momentum modes; and $B_{q}\equiv-4J\sin(qa+\phi/2)\sin(\phi/2)$
is the SOC-induced axial field.

On its own, the collective Heisenberg term
($\sim\hat{\bm{S}}\cdot\hat{\bm{S}}$) in Eqn. (3) opens an energy gap $fU$,
with $f\equiv N/L$ the filling fraction of spatial modes, between the
collective Dicke states $\Ket{S=N/2,M_{S}}$ and the remainder of Hilbert
spaceRey  _et al._ (2008); Martin  _et al._ (2013); Norcia  _et al._ (2018);
Smale  _et al._ (2019) with $S<N/2$. Here $S$ and $M_{S}$ respectively label
the eigenvalues of the collective spin operators
$\hat{\bm{S}}\cdot\hat{\bm{S}}$ and $\hat{S}_{\text{z}}$, with eigenvalues
$S(S+1)$ for non-negative $S\in\left\\{N/2,N/2-1,\cdots\right\\}$ and
$M_{S}\in\left\\{-S,-S+1,\cdots,S\right\\}$. The axial field $B_{q}$ generally
couples states within the Dicke manifold to states outside it. In the weak SOC
limit (i.e. $B_{q}\ll fU$), however, the interaction energy gap suppresses
population transfer between states with different total spin $S$ [see Fig.
1(d)]. In this regime, the virtual occupation of states outside the Dicke
manifold can be accounted for perturbatively. The symmetries of SOC as
expressed in Eqn. (3) dictate that this treatment should yield powers of
$\hat{S}_{\text{z}}$ when projected onto the collective Dicke manifold at
higher orders in perturbation theory. At second order in perturbation theory
(see Appendix A), we thus find that SOC effectively yields a one-axis twisting
(OAT) model widely known to generate squeezing dynamicsKitagawa and Ueda
(1993); Ma  _et al._ (2011):

$\displaystyle\hat{H}_{\text{eff}}/\hbar=-\frac{U}{L}\hat{\bm{S}}\cdot\hat{\bm{S}}-\overline{B}\hat{S}_{\text{z}}+\chi\hat{S}_{\text{z}}^{2},$
$\displaystyle\chi\equiv\dfrac{\widetilde{B}^{2}}{(N-1)fU},$ (4)

where $\overline{B}\equiv\sum_{q}B_{q}/N$ is the mean and
$\widetilde{B}^{2}\equiv\sum_{q}\left(B_{q}-\overline{B}\right)^{2}/N$ the
variance of the axial field. The effect of the
$\sim\hat{\bm{S}}\cdot\hat{\bm{S}}$ term is to generate a relative phase
between states with different total spin $S$ and thus has no effect on
dynamics restricted to a fixed $S$. Note also that the collective spin
rotation from $\overline{B}\hat{S}_{\text{z}}$ can be eliminated by going into
a rotating frame or by using a spin echo.

The entire protocol for preparing a squeezed state via OAT, sketched out in
Fig. 1(e), reduces to a standard Ramsey protocol with a spin echo: after
initially preparing a spin-down (i.e. $-\hat{\bm{z}}$) polarized sample of
ultracold atoms populating the lowest Bloch band of a lattice, a fast $\pi/2$
pulse is applied with the clock laser to rotate all spin vectors into
$+\hat{\bm{x}}$. The atoms then freely evolve for a variable time $t$
(possibly with spin-echo $\pi$-pulses), after which the amount of
metrologically useful spin squeezing, measured by the Ramsey squeezing
parameter

$\displaystyle\xi^{2}\equiv\min_{\theta}\braket{\text{var}(\hat{S}^{\perp}_{\theta})}\times
N/\absolutevalue*{\braket{\hat{\bm{S}}}}^{2},$ (5)

can be determined experimentally from global spin measurements. Here
$\braket{\hat{\bm{S}}}$ is the mean collective spin vector and
$\braket{\text{var}(\hat{S}^{\perp}_{\theta})}$ is the variance of spin
measurements along an axis orthogonal to $\braket{\hat{\bm{S}}}$,
parameterized by the angle $\theta\in[0,2\pi)$.

The above protocol concerns only the preparation of a spin-squeezed state,
which would then be used as an input state for a follow-up clock interrogation
protocol without SOC. While increasing the lattice depth to turn off SOC
during clock interrogation is the simplest approach, this will limit the
interrogation time due to light scattering (see discussion below).
Alternatively, it is possible to keep the same lattice depth used for the spin
squeezing generation by adding a specific pulse sequence to suppress SOC. See
details in Appendix E.

###  II.1 Model validity

The validity of the OAT model in Eqn. (4) relies on two key conditions
concerning experimental parameter regimes. First, the on-site interaction
energy $U$ should not be much larger in magnitude than the tunneling rate $J$
(clarified below); otherwise, one cannot assume frozen motional degrees of
freedom (i.e. with atoms pinned to fixed quasi-momentum modes) and map the
Fermi-Hubbard model to a spin model. Second, the SOC-induced fields $B_{q}\sim
J\sin(\phi/2)$ should be considerably smaller in magnitude than the
interaction energy gap $fU$, as otherwise one cannot perturbatively transform
SOC into OAT. These two conditions can be satisfied by appropriate choices of
$U/J$ and the SOC angle $\phi$, which are respectively controlled by tuning
the lattice depth and changing the angle between the clock laser and the
lattice axes [see Fig. 1(a)].

Figure 2: Benchmarking the spin and one-axis twisting models. Comparisons of
maximum squeezing (top panels, a.i and b.i) and optimal squeezing time (lower
panels, a.ii and b.ii) between the Fermi-Hubbard (FH), spin, and one-axis
twisting (OAT) models; obtained numerically via the protocol depicted in Fig.
1(e) in a 1D lattice with $L=12$ sites. Results are shown for half filling
with $N=12,f\equiv N/L=1$ (left panels, a.i and a.ii) and filling $f=5/6$
(right panels, b.i and b.ii) as a function of $U/J$ and the SOC angle $\phi$.
In both cases, the system is initialized in the corresponding ground state.
Insets for both $f=1$ and $f=5/6$ show (in green) regions of the $U$-$\phi$
plane in which both the optimal squeezing (in dB) and the corresponding
squeezing time of all three models agree to within 20%. At half filling (a.i
and a.ii), mode-changing collisions are suppressed by Pauli blocking,
resulting in almost exact agreement between the FH and spin models; both of
these models converge onto the OAT model in the gap-protected, weak SOC regime
of large $U/J$ and small $\phi$. The spin and OAT models show similar behavior
away from half filling (b.i and b.ii), but the presence of mode-changing
collisions results in their disagreement with the FH model as interactions
begin to dominate at larger $U/J$. Even below half filling, however, the FH
exhibits comparable amounts of squeezing to the spin model across a broad
range of $U/J$ and $\phi$, albeit at earlier times when $U/J\gtrsim 2$.

We demonstrate the importance of these conditions in Fig. 2, where we show
numerical results from exact simulations of a 1D system with $L=12$ sites.
Therein, optimal squeezing achievable under unitary dynamics is provided in
dB, i.e. $-10\log_{10}(\xi_{\text{opt}}^{2})$, while the time at which this
squeezing occurs is provided in units of the nearest-neighbor tunneling time
$2\pi/J$. At $f=1$ atom per lattice site, i.e. half filling of all atomic
states in the lowest Bloch band, the spin model [Eqn. (3)] agrees almost
exactly with the Fermi-Hubbard (FH) model [Eqn. (2)] up through (and
exceeding) $U/J=8$. The agreement at half filling ($f=1$) is assisted by Pauli
blocking of mode-changing collisions. Below half filling ($f=5/6$), these two
(FH and spin) models show good agreement at $U/J\lesssim 2$, while at
$U/J\gtrsim 2$ mode-changing collisions start to become relevant and
invalidate the frozen-mode assumption of the spin model. Note that we chose
filling $f=5/6$ to demonstrate that our protocol should work, albeit sub-
optimally, even in this highly hole-doped case; in practice, optimized
experiments are capable of achieving fillings closer to the optimal $f=1$Brown
_et al._ (2017). Interestingly, even with mode-changing collisions the Fermi-
Hubbard model exhibits comparable amounts of squeezing to the spin model, and
achieves this squeezing in less time. The spin and OAT models agree in the
regime of weak SOC with $\widetilde{B}\sim J\sin(\phi/2)\ll fU$, and exhibit
different squeezing behaviors outside this regime as single-particle spin
dephasing can no longer be treated as a weak perturbation to the spin-locking
interactions.

In realistic implementations, the Gaussian profile of the laser beams always
introduces an additional effective harmonic potential that modifies the
translational invariance assumed so far. We present a detailed discussion of
the role of the harmonic trap in Appendix I, where we demonstrate that the
addition of harmonic confinement barely modifies the achievable spin squeezing
with currently accessible trapping frequencies. We find that the existence of
single-particle localized modes in the lattice with harmonic confinementRey
_et al._ (2005); Pupillo  _et al._ (2006) helps to protect spin squeezing and
shifts the optimal parameter window to $U/J\gtrsim 2$.

###  II.2 Two-axis twisting

The above scheme for OAT achieves optimal spin squeezing that scales as
$\xi^{2}_{\text{opt}}\sim N^{-2/3}$ with minimal intervention, i.e. a standard
Ramsey protocol. Further improvements upon this scheme can be made by
introducing a time-dependent driving field that transforms the OAT Hamiltonian
into a two-axis twisting (TAT) one. While the OAT model initially generates
squeezing faster than the TAT model, the squeezing generation rate of OAT
(measured in dB per second) falls off with time, while the squeezing
generation rate for TAT remains approximately constant until reaching
Heisenberg-limited amount of spin squeezing with $\xi^{2}_{\text{opt}}\sim
N^{-1}$Kitagawa and Ueda (1993).

There are two general strategies for converting OAT into TAT: by use of either
a pulsedLiu  _et al._ (2011) or continuousHuang  _et al._ (2015) drive
protocol. For simplicity, we consider the latter in this work, although the
pulsed protocol could provide additional advantages, as explained at the end
of Appendix E. Following the prescription in Ref. [51], we use the clock laser
to apply an amplitude-modulated drive
$\hat{H}_{\text{drive}}(t)/\hbar=\Omega_{0}\cos(\omega t)\hat{S}_{\text{x}}$.
If the modulation frequency $\omega$ satisfies $\omega\gg N\chi$ and
$\mathcal{J}_{0}\left(2\Omega_{0}/\omega\right)=\pm 1/3$, where $\chi$ is the
OAT squeezing strength in Eqn. (4) and $\mathcal{J}_{0}$ is the zero-order
Bessel function of the first kind, then up to (i) an
$\sim\hat{\bm{S}}\cdot\hat{\bm{S}}$ term that contributes only overall phase
factors, and (ii) an $\sim\hat{S}_{\text{z}}$ term that can be eliminated with
a simple dynamical decoupling pulse sequence (see Appendix D), the effective
Hamiltonian becomes
$\hat{H}_{\text{TAT}}^{(+)}/\hbar=(\chi/3)(\hat{S}_{\text{z}}^{2}-\hat{S}_{\text{x}}^{2})$
or
$\hat{H}_{\text{TAT}}^{(-)}/\hbar=(\chi/3)(\hat{S}_{\text{y}}^{2}-\hat{S}_{\text{x}}^{2})$
(see Appendix C), which squeezes an initial state polarized along the $y$ or
$z$ axis, respectively.

##  III Experimental implementation and practical considerations

Thus far, we have largely considered the general preparation of spin-squeezed
states with the Fermi-Hubbard model. Here, we discuss the specific
implementation of the above protocols in the state-of-the-art 3D 87Sr optical
lattice clock (OLC). If successful, such an implementation would (to our
knowledge) for the first time break through the proof-of-principle stage of
spin squeezing efforts, and achieve a genuine metrological enhancement of a
world-class quantum sensor.

As required for our protocol, 3D 87Sr OLC has demonstrated the capability to
load a quantum degenerate gas into a 3D lattice at the “magic wavelength”
($\lambda_{\text{lattice}}=2a\approx 813$ nm) for which both the ground
(${}^{1}S_{0},\downarrow$) and first excited (${}^{3}P_{0},\uparrow$)
electronic states (i.e. the “clock states”) of the atoms experience the same
optical potentialCampbell  _et al._ (2017). Furthermore, the 3D 87Sr OLC
currently operates at sufficiently low temperatures to ensure vanishing
population above the lowest Bloch band, such that its dynamics are governed by
the Fermi-Hubbard Hamiltonian [Eqn. (2)]Esslinger (2010).

An external clock laser with wavelength $\lambda_{\text{L}}\approx 698$ nm
resonantly interrogates the ${}^{1}S_{0}$ and ${}^{3}P_{0}$ states of the
atoms and generates spin-orbit coupling (SOC)Wall  _et al._ (2016). While the
relative wavelengths of the lattice and clock lasers do not allow for weak SOC
along all three lattice axes, weak SOC along two axes can be implemented by,
for example, (i) fixing a large lattice depth along the $z$ axis, effectively
freezing atomic motion along $z$, and then (ii) making the clock laser nearly
collinear with the $z$ axis, with only a small projection of its wavenumber
$\bm{k}_{\text{L}}$ onto the $x$-$y$ plane [see Figure 1(a)]. The entire 3D
OLC then factorizes into an array of independent 2D systems with $N=f\ell^{2}$
atoms each, where $\ell$ is the number of lattice sites along each axis of the
lattice. As in the 1D case, atoms within the 2D system experience all-to-all
interactions, as well as spin-orbit coupling along two directions
characterized by SOC angles
$\phi_{\text{x},\text{y}}=k_{\text{L}}^{\text{x},\text{y}}a$. Generally
speaking, higher-dimensional systems (e.g. 2D vs. 1D) are more desirable
because they allow packing more interacting atoms into a fixed system volume,
thereby increasing the maximally attainable amount of spin squeezing.

Figure 3: Optimal squeezing with one- and two-axis twisting in a 2D section of
the 3D 87Sr optical lattice clock. (a) The maximum amount of squeezing depends
only on the atom number $N=\ell^{2}$, where $\ell$ is the number of lattice
sites along each axis of the lattice. While the time scales for squeezing
generally depend on several experimental parameters, the time at which maximal
squeezing occurs can be minimized at any given lattice depth $V_{0}$ by
choosing SOC angles $\phi$ that saturate $\widetilde{B}/U\approx 0.05$, where
$\widetilde{B}$ is the variance of the SOC-induced axial field and $U$ is the
two-atom on-site interaction energy. Panels (b, c) show these minimal
squeezing times as a function of the depth $V_{0}$ and linear size $\ell$ of
the lattice. Lattice depths $V_{0}$ are normalized to the atomic lattice
recoil energy $E_{\text{R}}$, and the upper axis on panels (b, c) marks values
of $U/J$ at fixed lattice depths. In general, TAT achieves more squeezing than
OAT for any system size, and achieves optimal squeezing faster for $N\gtrsim
400$ atoms, as denoted by a dotted line in panels (b, c).

Figure 3 shows, for both OAT and TAT protocols, the maximally attainable
amount of spin squeezing and the shortest time at which it occurs as a
function of the lattice depth $V_{0}$ and linear lattice size $\ell$ in a
single half-filled 2D layer (i.e. $f=1,N=\ell^{2}$) of the 3D OLC. Atoms are
confined along the direction transverse to the 2D layer by a lattice of depth
60 $E_{\text{R}}$, where $E_{\text{R}}$ is the atomic lattice recoil energy.
The maximally attainable amount of spin squeezing by each protocol in Fig. 3
depends only on the atom number $N$, while the shortest attainable time is
determined by choosing the largest SOC angles
$\phi_{\text{x}}=\phi_{\text{y}}\equiv\phi$ which saturate
$\widetilde{B}/U\approx 0.05$. We impose this constraint on $\widetilde{B}/U$
to ensure validity of the OAT Hamiltonian perturbatively derived in Appendix A
(see also Appendix B).

Figure 4: Optimal squeezing with decoherence via one- and two-axis twisting in
a 2D section of the 3D 87Sr optical lattice clock (OLC). In practice,
decoherence due to light scattering limits the amount of squeezing that is
attainable in the the 3D 87Sr OLC. Due to growing squeezing times with
increasing system size, the maximal squeezing obtainable via OAT saturates
past $\ell\approx 30$ sites along each axis of the lattice, with $N\approx
10^{3}$ atoms total. The more favorable size-dependence of TAT time scales,
however, allow for continued squeezing gains through $\ell=100$ ($N=10^{4}$).
While the OAT results in (a) are exact, the TAT results in (b) reflect only a
lower bound on the maximum squeezing obtainable, albeit one that is likely
close (within a few dB) to the actual value. Optimal squeezing times in the
presence of decoherence are generally smaller than the corresponding times
shown in Fig. 3, as decoherence typically degrades squeezing before it reaches
the decoherence-free maximum. The decoherence considered in this work also
limits maximally achievable squeezing to $\sim 20$ dB less than the
decoherence-free maxima shown in Fig. 3. Sample plots of squeezing over time
for particular choices of lattice size ($\ell$) and depth
($V_{0}/E_{\text{R}}$) are provided in Appendix G.

Currently, light scattering from the lattice beams induces decoherence of the
clock on a time scale of $\sim 10$ secondsGoban  _et al._ (2018); Hutson  _et
al._ (2019), which is much shorter than the natural ${}^{3}P_{0}$ lifetime of
$\sim 160$ seconds (see Appendix F). This limitation imposes significant
constraints on achievable spin squeezing, as shown in Figure 4 where the
maximal squeezing with spin decay in the OAT case was determined using exact
expressions for spin correlators derived in Ref. [52], while in the TAT case
these correlators were determined by solving Heisenberg equations of motion
for collective spin operatorsPerlin and Rey (2019) (see Appendix H). Due to
the fast growth of Heisenberg operators in systems with all-to-all
interactions, the latter method is not always capable of simulating up to the
optimal squeezing time, and thus only provides a lower bound on the maximal
squeezing theoretically obtainable via TAT.

The results in Fig. 4 show that squeezing via OAT saturates with system size
around $N\approx 10^{3}$ ($\ell\approx 30$), while TAT allows for continued
squeezing gains through $N=10^{4}$ ($\ell=100$). Even with decoherence, our
protocol may realistically generate $\sim 10$–$14$ dB of spin squeezing in
$\sim 1$ second with $\sim 10^{2}$–$10^{4}$ atoms in a 2D section of the
lattice, which is compatible with the atom numbers and interrogation times of
state-of-the-art optical lattice clocksCampbell  _et al._ (2017); Marti  _et
al._ (2018). This amount of spin squeezing exceeds those reported in the
ground-state nuclear spin sublevels of a state-of-the-art 171Yb OLC ($\sim
6.5$ dB)Braverman  _et al._ (2019). While the latter protocol might be used to
transfer spin squeezing to the electronic clock state, to date there has been
no demonstration of spin squeezing in an optical clock transition.

In addition to light scattering, $p$-wave losses from inelastic ${}^{3}P_{0}$
collisionsMartin  _et al._ (2013); Zhang  _et al._ (2014); Bishof  _et al._
(2011) can also degrade the maximum achievable spin squeezing, which becomes
more pronounced for shallower lattices. More details on $p$-wave losses are
discussed in Appendix J, where we show that operating at lattice depths
$V_{0}\gtrsim 7E_{\text{R}}$ may be necessary to suppress the impact of
inelastic collisions on spin squeezing, at the cost of slightly increasing
light scattering.

The sources of decoherence considered above are not fundamental, and can be
avoided by, for instance, using two nuclear spin levels as spin-1/2 degrees of
freedom that are interrogated by far-detuned Raman transitions instead of a
direct optical transitionMancini  _et al._ (2015). The strength of SOC for
Raman transitions is tunable and, moreover, the lifetimes of ground nuclear
spin levels are longer than 100 seconds in the latticeGoban  _et al._ (2018).
In this case, our protocol for preparing a squeezed state would additionally
end with a coherent state transfer from nuclear to electronic degrees of
freedom to retain metrological utility for the atomic clock. If, for example,
the $-9/2$ and $-7/2$ nuclear spin states are used for the preparation of a
squeezed state, then the collective-spin entanglement of atoms can be
transferred to electronic states at the end of the spin squeezing protocol
with a $\sigma^{-}$ polarized $\pi$ pulse. Such a pulse can transfer
$\Ket{g,-7/2}$ to $\Ket{e,-9/2}$ without affecting $\Ket{g,-9/2}$, where $g$
and $e$ respectively denote the ground and excited (electronic) clock states.

##  IV Conclusions

We have proposed a new protocol to generate spin squeezing in a fermionic 3D
optical lattice clock by combining nominally undesirable atomic collisions
with spin-orbit coupling. To our knowledge, this is the first proposal to use
quantum correlations in a many-body fermionic system to push state-of-the-art
quantum sensors beyond the independent-particle regime, thereby achieving a
genuine quantum advantage. Such capability could allow for major improvements
in clock sensitivity and bandwidth, enhancing not only traditional timekeeping
applications such as measurement standards, navigation (GPS), and
telecommunications, but also geodesy and gravitational wave detection,
precision tests of fundamental physics, and the search for new physics beyond
the standard modelSafronova  _et al._ (2018).

##  V Acknowledgments

We acknowledge helpful discussions with M. Norcia, C. Sanner, and M. Mamaev.
This work is supported by the Air Force Office of Scientific Research (AFOSR)
grant FA9550-18-1-0319; the AFOSR Multidisciplinary University Research
Initiative (MURI) grant; the Defense Advanced Research Projects Agency (DARPA)
and Army Research Office (ARO) grant W911NF-16-1-0576; the National Science
Foundation (NSF) grant PHY-1820885; JILA-NSF grant PFC-1734006; and the
National Institute of Standards and Technology (NIST).

##  Appendix A Derivation of the effective one-axis-twisting model

Suppose we have a Hamiltonian of the form ($\hbar=1$)

$\displaystyle H=H_{0}+V,$ (6)

with

$\displaystyle H_{0}=-\dfrac{U}{L}\bm{S}\cdot\bm{S},$ $\displaystyle
V=-\sum_{n}B_{n}s_{\text{z}}^{(n)}+\Omega S_{\text{x}},$ (7)

and we consider $N$-particle states initially in the ground-state manifold
$\mathcal{G}_{0}$ of $H_{0}$, which have total spin $S=N/2$. If the largest
eigenvalue of $V$ is smaller in magnitude than half of the collective spin gap
$NU/L=fU$, i.e. the energy gap under $H_{0}$ between $\mathcal{G}_{0}$ and its
orthogonal complement $\mathcal{E}_{0}$, then we can formally develop a
perturbative treatment for the action of $V$ on $\mathcal{G}_{0}$. Such a
treatment yields an effective Hamiltonian on $\mathcal{G}_{0}$ of the form
$H_{\text{eff}}=\sum_{p}H_{\text{eff}}^{(p)}$, where $H_{\text{eff}}^{(p)}$ is
order $p$ in $V$. Letting $\mathcal{P}_{0}$ ($\mathcal{Q}_{0}$) be a projector
onto $\mathcal{G}_{0}$ ($\mathcal{E}_{0}$), we define the super-operators
$\mathcal{O}$ and $\mathcal{L}$ by

$\displaystyle\mathcal{O}V$
$\displaystyle\equiv\mathcal{P}_{0}V\mathcal{Q}_{0}+\mathcal{Q}_{0}V\mathcal{P}_{0},$
(8) $\displaystyle\mathcal{L}V$
$\displaystyle\equiv\sum_{\alpha,\beta}\dfrac{\outerproduct{\alpha}{\alpha}\mathcal{O}V\outerproduct{\beta}{\beta}}{E_{\alpha}-E_{\beta}},$
(9)

where $H_{0}=\sum_{\alpha}E_{\alpha}\outerproduct{\alpha}{\alpha}$. The first
few terms in the expansion of the effective Hamiltonian $H_{\text{eff}}$ are
then, as derived in Ref. [57],

$\displaystyle H_{\text{eff}}^{(0)}=\mathcal{P}_{0}H_{0}\mathcal{P}_{0},$
$\displaystyle H_{\text{eff}}^{(1)}=\mathcal{P}_{0}V\mathcal{P}_{0},$ (10)

and

$\displaystyle
H_{\text{eff}}^{(2)}=-\dfrac{1}{2}\mathcal{P}_{0}\left[\mathcal{O}V,\mathcal{L}V\right]_{-}\mathcal{P}_{0},$
(11)

with $\left[X,Y\right]_{-}\equiv XY-YX$. The zero-order effective Hamiltonian
$H_{\text{eff}}^{(0)}=H_{0}$ within the ground-state manifold. To calculate
$H_{\text{eff}}^{(1)}$, we note that the ground-state manifold
$\mathcal{G}_{0}$ is spanned by the Dicke states

$\displaystyle\Ket{m}\propto S_{+}^{N/2+m}\Ket{\downarrow}^{\otimes n},$
$\displaystyle S_{+}\equiv\sum_{n}s_{+}^{(n)},$ (12)

in terms of which we can expand the collective spin-z operator as
$S_{\text{z}}=\sum_{m}m\outerproduct{m}{m}$. We can likewise expand the
collective spin-x operator $S_{\text{x}}$ in terms of $x$-oriented Dicke
states $\Ket{m_{\text{x}}}$ as
$S_{\text{x}}=\sum_{m}m\outerproduct{m_{\text{x}}}{m_{\text{x}}}$. The ground-
state projector $\mathcal{P}_{0}$ onto $\mathcal{G}_{0}$ can be expanded in
either basis as
$\mathcal{P}_{0}=\sum_{m}\outerproduct{m}{m}=\sum_{m}\outerproduct{m_{\text{x}}}{m_{\text{x}}}$.
Defining the mean and residual fields

$\displaystyle\overline{B}\equiv\dfrac{1}{N}\sum_{n}B_{n},$ $\displaystyle
b_{n}\equiv B_{n}-\overline{B},$ (13)

we can then write

$\displaystyle V$
$\displaystyle=-\sum_{n}\left(b_{n}+\overline{B}\right)s_{\text{z}}^{(n)}+\Omega
S_{\text{x}}$
$\displaystyle=-\sum_{n}b_{n}s_{\text{z}}^{(n)}-\overline{B}S_{\text{z}}+\Omega
S_{\text{x}},$ (14)

and in turn

$\displaystyle H_{\text{eff}}^{(1)}$
$\displaystyle=\mathcal{P}_{0}\left(-\sum_{n}b_{n}s_{\text{z}}^{(n)}-\overline{B}S_{\text{z}}+\Omega
S_{\text{x}}\right)\mathcal{P}_{0}$
$\displaystyle=-\sum_{n}b_{n}\mathcal{P}_{0}s_{\text{z}}^{(n)}\mathcal{P}_{0}-\overline{B}S_{\text{z}}+\Omega
S_{\text{x}},$ (15)

where we used the fact that
$\mathcal{P}_{0}S_{j=\text{z},\text{x}}\mathcal{P}_{0}=S_{j}$ within the
ground-state manifold. By construction, the residual fields are mean-zero,
i.e. $\sum_{n}b_{n}=0$. Using the particle-exchange symmetry of the Dicke
states, we can therefore expand

$\sum_{n}b_{n}\mathcal{P}_{0}s_{\text{z}}^{(n)}\mathcal{P}_{0}=\sum_{n,m,m^{\prime}}b_{n}\outerproduct{m}{m}s_{\text{z}}^{(n)}\outerproduct{m^{\prime}}{m^{\prime}}\\\
=\sum_{n}b_{n}\sum_{m,m^{\prime}}\outerproduct{m}{m}s_{\text{z}}^{(1)}\outerproduct{m^{\prime}}{m^{\prime}}=0,$
(16)

which implies

$\displaystyle H_{\text{eff}}^{(1)}=-\overline{B}S_{\text{z}}+\Omega
S_{\text{x}}.$ (17)

To calculate the second-order effective Hamiltonian $H_{\text{eff}}^{(2)}$, we
let $\mathcal{B}_{0}\left(\mathcal{E}_{0}\right)$ denote an eigenbasis of
$H_{0}$ for the excited subspace $\mathcal{E}_{0}$, and set the ground-state
energy to 0. We then define the operator

$\displaystyle\mathcal{I}\equiv\sum_{\Ket{\alpha}\in\mathcal{B}_{0}\left(\mathcal{E}_{0}\right)}\dfrac{\outerproduct{\alpha}{\alpha}}{E_{\alpha}},$
(18)

which sums over projections onto excited states with corresponding energetic
suppression factors, in terms of which we can write

$\displaystyle
H_{\text{eff}}^{(2)}=-\mathcal{P}_{0}V\mathcal{I}V\mathcal{P}_{0},$ (19)

which is simply an operator-level version of the textbook expression for
second-order perturbation theory. The only part of $V$ which is off-diagonal
with respect to the ground- and excited-state manifolds $\mathcal{G}_{0}$ and
$\mathcal{E}_{0}$ is $-\sum_{n}b_{n}s_{\text{z}}^{(n)}$, and the individual
spin operators in this remainder can only change the total spin $S$ by at most
1. It is therefore sufficient to expand $\mathcal{I}$ in a basis for states
which span the image of $\mathcal{G}_{0}$ under all $s_{\text{z}}^{(n)}$
within the $S=N/2-1$ manifold. Such a basis is provided by the spin-wave
states

$\displaystyle\Ket{mk}\propto\sum_{n=1}^{N}e^{2\pi
ikn/N}s_{+}^{(n)}\Ket{m-1},$ (20)

for $k=1,2,\cdots,N-1$Swallows  _et al._ (2011). Using the fact that all
spin-$z$ operators preserve the projection of total spin onto the $z$ axis, we
then have that

$H_{\text{eff}}^{(2)}=-\dfrac{1}{fU}\sum_{m,k,n,n^{\prime}}b_{n}b_{n^{\prime}}\outerproduct{m}{m}s_{\text{z}}^{(n)}\Ket{mk}\\\
\times\Bra{mk}s_{\text{z}}^{(n^{\prime})}\outerproduct{m}{m},$ (21)

where the relevant matrix elements between the Dicke states and the spin-wave
states areSwallows  _et al._ (2011)

$\displaystyle\Braket{m}{s_{\text{z}}^{(n)}}{mk}=e^{2\pi
ikn/N}\sqrt{\dfrac{(N/2)^{2}-m^{2}}{N^{2}(N-1)}},$ (22)

which implies

$H_{\text{eff}}^{(2)}=-\dfrac{1}{fU}\sum_{m}\dfrac{(N/2)^{2}-m^{2}}{N^{2}(N-1)}\outerproduct{m}{m}\\\
\times\sum_{k,n,n^{\prime}}b_{n}b_{n^{\prime}}e^{2\pi
ik\left(n-n^{\prime}\right)/N}.$ (23)

Using the fact that $\sum_{n}b_{n}=0$, we can expand

$\displaystyle\sum_{k,n,n^{\prime}}b_{n}b_{n^{\prime}}e^{2\pi
ik\left(n-n^{\prime}\right)/N}$
$\displaystyle=\sum_{n,n^{\prime}}b_{n}b_{n^{\prime}}\sum_{k=1}^{N-1}e^{2\pi
ik\left(n-n^{\prime}\right)/N}$
$\displaystyle=\sum_{n,n^{\prime}}b_{n}b_{n^{\prime}}\sum_{k=0}^{N-1}e^{2\pi
ik\left(n-n^{\prime}\right)/N},$ (24)

where the sum over $k$ vanishes for $n\neq n^{\prime}$ and equals $N$ when
$n=n^{\prime}$, so

$\displaystyle\sum_{k,n,n^{\prime}}b_{n}b_{n^{\prime}}e^{2\pi
ik\left(n-n^{\prime}\right)/N}=N^{2}\widetilde{B}^{2},$ (25)

where

$\displaystyle\widetilde{B}^{2}\equiv\dfrac{1}{N}\sum_{n}b_{n}^{2}=\dfrac{1}{N}\sum_{n}\left(B_{n}-\overline{B}\right)^{2}.$
(26)

We therefore have that

$\displaystyle
H_{\text{eff}}^{(2)}=-\sum_{m}\dfrac{\left(N/2\right)^{2}-m^{2}}{\left(N-1\right)fU}~{}\widetilde{B}^{2}\outerproduct{m}{m},$
(27)

where the $\left(N/2\right)^{2}$ term contributes a global energy shift which
we can neglect, while the $m^{2}$ term is proportional to
$m^{2}\outerproduct{m}{m}=S_{\text{z}}^{2}$. In total, the effective
Hamiltonian through second order in perturbation theory is thus

$\displaystyle
H_{\text{eff}}=-\dfrac{U}{L}\bm{S}\cdot\bm{S}-\overline{B}S_{\text{z}}+\Omega
S_{\text{x}}+\chi S_{\text{z}}^{2},$ (28)

with

$\displaystyle\chi\equiv\dfrac{\widetilde{B}^{2}}{\left(N-1\right)fU}.$ (29)

We benchmark the validity of this effective Hamiltonian via exact simulations
of the spin [Eqn. (3)] and OAT [Eqn. (4)] Hamiltonians in a system of 20
spins, finding that the relative error in maximal squeezing (in dB) of the OAT
model is less than 3% when $\widetilde{B}/U<0.06$ (see Appendix B).

##  Appendix B Numerical benchmarking of the OAT model

Here we provide additional information about our benchmarking of the one-axis
twisting model against the spin model. This benchmarking was performed via
exact simulations of a 20-spin system. Fig. B1 shows the relative error in
maximal squeezing of the OAT model (measured against the spin model) as a
function of the reduced field variance $\widetilde{B}/U$. Here squeezing is
measured in decibels (dB) by $-10\log_{10}\xi^{2}$ for the squeezing parameter
$\xi^{2}$ define in Eqn. (5). The relative error in maximal squeezing (in dB)
by the OAT model is less than 3% when $\widetilde{B}/U<0.06$.

Figure B1: Relative error between maximal squeezing (measured in dB) obtained
by the OAT [Eqn. (4)] and spin [Eqn. (3)] models of the main text in a system
of 20 particles. The OAT model correctly captures the maximal squeezing (in
dB) of the spin model to within 3% (marked by the horizontal reference line)
within the gap-protected regime $\widetilde{B}/U<0.06$.

In principle, spin-changing decoherence compromises the validity of the OAT
model, as its perturbative derivation in Appendix A relies on spin population
remaining primarily within the Dicke manifold. This assumption breaks down in
the presence of, for example, spontaneous emission, which transfers population
outside of the Dicke manifold. Nonetheless, we find decent agreement between
the OAT and spin models when decoherence is sufficiently weak (see Fig. B2).

Figure B2: Comparison between the OAT and the spin model in the presence of
decoherence. (a) The difference between the maximal squeezing (measured in dB)
obtained by the OAT [Eqn. (4)] and spin [Eqn. (3)] models increases with the
particle number $N$ and the single-particle spontaneous emission rate
$\gamma$. This disagreement is attributed in part to the fact that spontaneous
emission transfers population of the collective spin state outside of the
Dicke manifold, violating an assumption of the OAT model; see panel (b). The
rate of population transfer outside of the Dicke manifold increases with both
particle number and spontaneous emission rate. (Parameters for simulations in
this figure: $U=1000$ Hz, $J=200$ Hz, and $\phi=\pi/20$).

##  Appendix C Two-axis twisting, decoherence, and the residual axial field

The protocol we use to transform one-axis twisting (OAT) into two-axis
twisting (TAT) is as previously proposed in Ref. [51]; we provide a summary of
this protocol here, in addition to some brief discussion of its implications
for decoherence and the residual $\sim\bm{S}\bm{\cdot}\bm{S}$ and $\sim
S_{\text{z}}$ terms of our OAT protocol. The TAT protocol begins with the OAT
Hamiltonian with a time-dependent transverse field,

$\displaystyle H=\chi S_{\text{z}}^{2}+\Omega\left(t\right)S_{\text{x}},$
$\displaystyle\Omega\left(t\right)=\beta\omega\cos\left(\omega t\right),$ (30)

where $\beta$ is the modulation index of the driving field and the drive
frequency $\omega\gg N\chi$, with $N$ the total number of spins. Moving into
the rotating frame of $\Omega\left(t\right)S_{\text{x}}$ subtracts this term
from the Hamiltonian, and transforms operators $\mathcal{O}$ as

$\displaystyle\mathcal{O}\to
U\left(t\right)^{\dagger}\mathcal{O}U\left(t\right),$ (31)

where

$\displaystyle
U\left(t\right)\equiv\exp\left[-i\int_{0}^{t}\text{d}\tau~{}\Omega\left(\tau\right)S_{\text{x}}\right]=\exp\left[-i\beta\sin\left(\omega
t\right)S_{\text{x}}\right].$ (32)

In particular, the operators $\tilde{S}_{\pm}\equiv-S_{\text{z}}\pm
iS_{\text{y}}$ (i.e. the raising and lowering operators in the $x$ basis)
transform simply as

$\displaystyle\tilde{S}_{\pm}\to U^{\dagger}\tilde{S}_{\pm}U=e^{\pm
i\beta\sin\left(\omega t\right)}\tilde{S}_{\pm}.$ (33)

For any operator $\mathcal{O}$ and drive frequency
$\omega\gg\norm{\mathcal{O}}$, where
$\norm{\mathcal{O}}\equiv\max_{\psi}\sqrt{\braket{\psi}{\mathcal{O}^{\dagger}\mathcal{O}}{\psi}}$
is the operator norm of $\mathcal{O}$ (i.e. the magnitude of the largest
eigenvalue of $\mathcal{O}$), we can generally make a secular approximation to
say

$\displaystyle e^{\pm im\beta\sin\left(\omega t\right)}\mathcal{O}$
$\displaystyle=\sum_{n=-\infty}^{\infty}\mathcal{J}_{n}\left(\pm
m\beta\right)e^{in\omega t}\mathcal{O}$
$\displaystyle\approx\mathcal{J}_{0}\left(\pm
m\beta\right)\mathcal{O}=\mathcal{J}_{0}\left(m\beta\right)\mathcal{O},$ (34)

where $\mathcal{J}_{n}$ is the $n$-th order Bessel function of the first kind.
Expanding
$S_{\text{z}}^{2}=\dfrac{1}{4}\left(\tilde{S}_{+}+\tilde{S}_{-}\right)^{2}$,
one can thus work out that the effective Hamiltonian in the rotating frame of
the drive is

$\displaystyle
H_{\text{eff}}\approx\dfrac{\chi}{2}\left(\left[1+\mathcal{J}_{0}\left(2\beta\right)\right]S_{\text{z}}^{2}+\left[1-\mathcal{J}_{0}\left(2\beta\right)\right]S_{\text{y}}^{2}\right).$
(35)

Driving with a modulation index $\beta$ for which
$J_{0}\left(2\beta\right)=\pm 1/3$ then gives us the effective two-axis
twisting Hamiltonians

$\displaystyle H_{\text{eff}}^{(+)}$
$\displaystyle=\dfrac{\chi}{3}\left(2S_{\text{z}}^{2}+S_{\text{y}}^{2}\right)\simeq\dfrac{\chi}{3}\left(S_{\text{z}}^{2}-S_{\text{x}}^{2}\right),$
(36) $\displaystyle H_{\text{eff}}^{(-)}$
$\displaystyle=\dfrac{\chi}{3}\left(S_{\text{z}}^{2}+2S_{\text{y}}^{2}\right)\simeq\dfrac{\chi}{3}\left(S_{\text{y}}^{2}-S_{\text{x}}^{2}\right),$
(37)

where $\simeq$ denotes equality up to the addition of a term proportional to
$\bm{S}^{2}=S_{\text{z}}^{2}+S_{\text{x}}^{2}+S_{\text{y}}^{2}$, which is
irrelevant in the absence of coherent coupling between states with different
net spin. In a similar spirit, one can work out that single-spin operators
transverse to the $x$-axis transform as

$\tilde{s}_{\pm}\equiv\dfrac{1}{2}\left(-s_{\text{z}}\pm
is_{\text{y}}\right)\\\ \to U^{\dagger}\tilde{s}_{\pm}U=e^{\pm
i\beta\sin\left(\omega
t\right)}\tilde{s}_{\pm}\approx\mathcal{J}_{0}\left(\beta\right)\tilde{s}_{\pm},$
(38)

which implies that shifting into the rotating frame of the time-dependent
drive takes

$\displaystyle s_{\text{x}}\to s_{\text{x}},$ $\displaystyle
s_{\text{y},\text{z}}\to\mathcal{J}_{0}\left(\beta\right)s_{\text{y},\text{z}},$
(39)

and

$\displaystyle
s_{\pm}\to\dfrac{1}{2}\left[1\pm\mathcal{J}_{0}\left(\beta\right)\right]s_{+}+\dfrac{1}{2}\left[1\mp\mathcal{J}_{0}\left(\beta\right)\right]s_{-}.$
(40)

As the TAT Hamiltonians $H_{\text{eff}}^{(\pm)}$ are realized in a rotating
frame, to properly account for decoherence throughout the TAT protocol one
must transform jump operators according to Eqns. (39)-(40).

In practice, our protocols realize the OAT Hamiltonian in Eqn. (30) with
additional $\sim\bm{S}\cdot\bm{S}$ and $\sim S_{\text{z}}$ terms [see Eqn.
(28)]. The effect of the $\sim\bm{S}\cdot\bm{S}$ term is to generate a
relative phase between states with different total spin $S$ (with e.g. $S=N/2$
within the Dicke manifold). In the absence of coherent coupling between states
with different total spin, therefore, the $\sim\bm{S}\cdot\bm{S}$ term has no
effect on system dynamics. The $\sim S_{z}$ term, meanwhile, is important; the
magnitude of this term (as measured by the operator norm) is generally
comparable to that of the squeezing term $\chi S_{\text{z}}^{2}$. Unlike in
the case of OAT, $S_{\text{z}}$ does not commute with the TAT Hamiltonians, so
its effects cannot be eliminated by a single spin-echo $\pi$-pulse
$\exp\left(-i\pi S_{\text{x}}\right)$ half way through the squeezing protocol.
Nonetheless, we find that for $N=10^{2}$ ($10^{3}$) atoms, $\sim 5$ (10)
$\pi$-pulses in a CPMG (Carr-Purcell-Meiboom-Gill) sequenceCarr and Purcell
(1954); Meiboom and Gill (1958) suffice to mitigate the effects of the
$S_{\text{z}}$ term in the TAT protocol (see Appendix D). Phase control over
these pulses, specifically choices of whether to apply $\exp\left(\pm i\pi
S_{\text{x}}\right)$ or $\exp\left(\pm i\pi S_{\text{y}}\right)$ in any given
$\pi$-pulse, can be used to construct XY-$n$ pulse sequencesMaudsley (1986);
Gullion  _et al._ (1990) that are robust to pulse errors.

##  Appendix D Dynamical decoupling in the TAT protocol

The effective Hamiltonian resulting from a perturbative treatment of SOC is
(see Appendix A)

$\displaystyle
H_{\text{eff}}=-\dfrac{U}{L}\bm{S}\cdot\bm{S}-\overline{B}S_{\text{z}}+\Omega
S_{\text{x}}+\chi S_{\text{z}}^{2},$ (41)

where $U$ is a two-atom on-site interaction strength; $L$ is the number of
lattice sites; $\overline{B}\equiv\sum_{n}B_{n}/N$ is a residual axial field
determined by the occupied quai-momentum modes $\left\\{n\right\\}$ (with
$\absolutevalue{\left\\{n\right\\}}=N$ atoms total); $\Omega$ is the magnitude
of a driving field; and $\chi$ is an effective OAT squeezing strength. The
effect of the $\sim\bm{S}\cdot\bm{S}$ term is to generate a relative phase
between states with different total spin $S$ (where $S=N/2$ within the Dicke
manifold). In the absence of coherent coupling between states with different
total spin, therefore, the $\sim\bm{S}\cdot\bm{S}$ term has no effect on
system dynamics, and we are safe to neglect it entirely.

In the parameter regimes relevant to our discussions in the main text, the
operator norms of $\overline{B}\hat{S}_{\text{z}}$ and
$\chi\hat{S}_{\text{z}}^{2}$ in Eqn. (41) will typically be comparable in
magnitude. The OAT protocol sets $\Omega=0$, and eliminates the effect of
$\overline{B}\hat{S}_{\text{z}}$ with a spin-echo $\pi$-pulse
$\exp\left(-i\pi\hat{S}_{\text{x}}\right)$ applied half way through the
squeezing protocol. The TAT protocol, meanwhile, effectively takes
$\chi\hat{S}_{\text{z}}^{2}+\Omega\hat{S}_{\text{x}}\to\hat{H}_{\text{TAT}}^{(\pm)}$
(as defined in Appendix C) and
$\overline{B}\hat{S}_{\text{z}}\to\mathcal{J}_{0}\left(\beta_{\pm}\right)\overline{B}\hat{S}_{\text{z}}$,
where $\mathcal{J}_{0}$ is the zero-order Bessel function of the first kind
and $\beta_{\pm}$ is the modulation index of the amplitude-modulated driving
field $\Omega$, satisfying $\mathcal{J}_{0}\left(2\beta_{\pm}\right)=\pm 1/3$.
Unlike in the case of OAT, $\hat{S}_{\text{z}}$ does not commute with the TAT
Hamiltonian, so its effect cannot be eliminated with a spin-echo. Nonetheless,
this term can be eliminated with a dynamical decoupling pulse sequence that
periodically inverts the sign of $\hat{S}_{\text{z}}$ while preserving
$\hat{H}_{\text{TAT}}^{(\pm)}$.

Figure D1: Optimal squeezing as a function of $\pi$-pulses applied prior to
the optimal TAT squeezing time in a CPMG sequence with (a) $N=100$ and (b)
$N=1000$ atoms. Results are shown for OAT, TAT, and TAT${}_{\pm,\text{z}}$,
where TAT${}_{\pm,\text{z}}$ denotes squeezing via the Hamiltonian
$\hat{H}_{\text{TAT}}^{(\pm,\text{z})}\equiv\hat{H}_{\text{TAT}}^{(\pm)}-\mathcal{J}_{0}\left(\beta_{\pm}\right)\Braket{\overline{B}}^{\text{rms}}_{f}\hat{S}_{\text{z}}$.
Details about experimental parameters for these simulations are provided in
the text.

Fig. D1 shows the maximal squeezing generated by $N=10^{2}$ and $10^{3}$ atoms
via OAT, TAT, and TAT in the presence of the mean field
$\mathcal{J}_{0}\left(\beta_{\pm}\right)\overline{B}\hat{S}_{\text{z}}$ as a
function of the number of $\pi$-pulses performed prior to the optimal TAT
squeezing time. These pulses are applied in a CPMG sequence
$\left(\tau_{n}/2-\pi_{\text{x}}-\tau_{n}/2\right)^{n}$, where $\tau_{n}/2$
denotes Hamiltonian evolution for a time $\tau_{n}/2$, $\pi_{\text{x}}$
denotes the application of an instantaneous $\pi$-pulse
$\exp\left(-i\pi\hat{S}_{\text{x}}\right)$, and $n$ is the number of pulses,
such that the optimal TAT squeezing time is
$t_{\text{opt}}^{\text{TAT}}=\left(\tau_{n}\right)^{n}$. The label
TAT${}_{\pm,\text{z}}$ in Fig. D1 denotes squeezing through the Hamiltonian
$\hat{H}_{\text{TAT}}^{(\pm,\text{z})}\equiv\hat{H}_{\text{TAT}}^{(\pm)}-\mathcal{J}_{0}\left(\beta_{\pm}\right)\Braket{\overline{B}}^{\text{rms}}_{f}\hat{S}_{\text{z}}$,
where $\Braket{\overline{B}}^{\text{rms}}_{f}$ is the root-mean-square average
of $\overline{B}$ over choices of occupied spacial modes $\left\\{n\right\\}$
at fixed filling $f$ of all spatial modes in the lowest Bloch band of a
periodic 2D lattice. While the modulation index $\beta_{+}$ is uniquely
defined by $\mathcal{J}_{0}\left(2\beta_{+}\right)=1/3$, there are two choices
of $\beta_{-}$ for which $\mathcal{J}_{0}\left(2\beta_{-}\right)=-1/3$; we use
that which minimizes $\absolutevalue{\mathcal{J}_{0}\left(\beta_{-}\right)}$.
Fig. D1 assumes an SOC angle $\phi=\pi/50$ (although results are independent
of $\phi$ for $\phi\ll 1$), a reduced field variance $\widetilde{B}/U=0.05$,
and a filling $f=5/6$. Note that as the filling $f\to 1$, the residual axial
field vanishes ($\overline{B}\to 0$), so TAT${}_{\pm,\text{z}}\to$ TAT.

##  Appendix E Clock interrogation after squeezing

The protocols in our work concern the preparation of spin-squeezed states in
an optical lattice clock. Here, we discuss the use of these states in a
follow-up clock interrogation protocol. For simplicity, we restrict our
discussion to the case of squeezing in 1D, as in Section II of the main text,
with the understanding that a generalization of this discussion to higher
dimensions is straightforward.

A spin-squeezed state is generated by interactions and SOC that are generally
undesirable during the clock interrogation protocol. In the parameter regimes
considered in our work, interactions alone have no effect on clock
interrogation: absent of coherence between states with different net spin $S$,
collective $\bm{S}\cdot\bm{S}$ interactions only generate unobservable global
phases within each fixed-$S$ sector of Hilbert space. Therefore, the remaining
task to allow clock interrogation after spin squeezing is to turn off SOC,
which inhomogeneously detunes atomic transition frequencies by an amount
$B_{q}$ that depends on the quasi-momentum $q$ of an atom. The SOC-induced
axial fields $B_{q}\sim J\sin(\phi/2)$ depend on two tunable parameters: the
tunneling rate $J$ and the SOC angle $\phi$. The simplest way to turn off SOC
is thus to increase the lattice depth prior to clock interrogation, taking
$J\sim B_{q}\to 0$. Increasing the lattice depth to turn off SOC is compatible
with the current clock interrogation sequence, but is incompatible with
ongoing efforts to mitigate light scattering from the lattice beams (currently
the primary source of decoherence in the clock; see Appendix F) by using
shallower latticesHutson  _et al._ (2019). We thus devote the rest of this
section to discussing strategies for turning off SOC that are compatible with
using the same lattice depth for clock interrogation as the spin squeezing
generation.

If we cannot take the tunneling rate $J\to 0$, our remaining control parameter
for turning off SOC is the SOC angle $\phi=k_{\text{L}}a$, where
$k_{\text{L}}$ is the projection of the clock laser wavenumber onto the
lattice axis and $a$ is the lattice spacing. The squeezing protocol needs a
clock laser with a small but nonzero SOC angle $\phi\ll 1$, while the clock
interrogation protocol requires a clock laser with $\phi=0$. Simply using one
clock laser with $\phi\neq 0$ for the squeezing protocol and another clock
laser with $\phi=0$ for the clock interrogation protocol, however, does not
resolve this discrepancy, because a state that is squeezed with respect to
spin operators that are homogeneous (i.e. of the form
$S_{\text{x}},S_{\text{y}},S_{\text{z}}$) in a particular gauge is not
necessarily squeezed with respect to spin operators that are homogeneous in a
different gauge. In this appendix, we will work explicitly in the “lab gauge”
of the clock interrogation protocol, in which the Fermi-Hubbard Hamiltonian is
SOC-free and the $\phi=0$ clock laser is homogeneous. To resolve the fact that
our squeezing protocol prepares a state that is squeezed in the “wrong gauge”,
we will construct a simple pulse sequence that transforms the inhomogeneous
spin operators accessible by the $\phi\neq 0$ laser into a homogeneous form in
our lab gauge.

Starting with a spin-down-polarized initial state

$\displaystyle\Ket{-\text{Z}}\equiv\left(\prod_{j}c_{j,\downarrow}^{\dagger}\right)\Ket{\text{vacuum}},$
(42)

our OAT protocol prepares the state

$\displaystyle\Ket{\xi_{\text{OAT}}^{(\theta)}}=e^{-iH_{\text{FH}}^{(0)}t}e^{-i\left(\pi/2\right)S_{\text{x}}^{(\theta)}}\Ket{-\text{Z}},$
(43)

where $H_{\text{FH}}^{(0)}$ is the Fermi-Hubbard Hamiltonian in Eqn. (2)
without SOC; $t$ is some free evolution time; and $S_{\text{x}}^{(\theta)}$ is
the “rotated” spin-$x$-like generator induced by a clock laser with SOC angle
$\phi=\theta$, namely

$\displaystyle S_{\text{x}}^{(\theta)}=\dfrac{1}{2}\sum_{j}e^{i\theta
j}c_{j,\uparrow}^{\dagger}c_{j,\downarrow}+\text{h.c.}.$ (44)

Defining on-site spin operators (in the lab gauge)

$\displaystyle s_{\text{z}}^{(j)}$
$\displaystyle\equiv\dfrac{1}{2}\left(c_{j,\uparrow}^{\dagger}c_{j,\uparrow}-c_{j,\downarrow}^{\dagger}c_{j,\downarrow}\right)$
(45) $\displaystyle s_{\text{x}}^{(j)}$
$\displaystyle\equiv\dfrac{1}{2}\left(c_{j,\uparrow}^{\dagger}c_{j,\downarrow}+c_{j,\downarrow}^{\dagger}c_{j,\uparrow}\right)$
(46) $\displaystyle s_{\text{y}}^{(j)}$
$\displaystyle\equiv\dfrac{i}{2}\left(c_{j,\downarrow}^{\dagger}c_{j,\uparrow}-c_{j,\uparrow}^{\dagger}c_{j,\downarrow}\right),$
(47)

we can identify the rotated collective spin operators

$\displaystyle S_{\text{x}}^{(\theta)}$
$\displaystyle\equiv\sum_{j}\left(\cos\left(\theta
j\right)s_{\text{x}}^{(j)}+\sin\left(\theta
j\right)s_{\text{y}}^{(j)}\right),$ (48) $\displaystyle
S_{\text{y}}^{(\theta)}$ $\displaystyle\equiv\sum_{j}\left(\cos\left(\theta
j\right)s_{\text{y}}^{(j)}-\sin\left(\theta
j\right)s_{\text{x}}^{(j)}\right).$ (49)

The state $\Ket{\xi_{\text{OAT}}^{(\theta)}}$ is squeezed with respect to
components of the rotated collective spin vector
$\bm{S}_{\theta}\equiv\left(S_{\text{x}}^{(\theta)},S_{\text{y}}^{(\theta)},S_{\text{z}}\right)$.
Therefore, to take advantage of the squeezing in
$\Ket{\xi_{\text{OAT}}^{(\theta)}}$, the clock interrogation protocol
effectively needs to rotate this state by some unitary
$\exp\left(-i\bm{\eta}\cdot\bm{S}_{\theta}\right)$, and then extract
information about the rotation vector $\bm{\eta}$ from collective spin
observables of the form

$\displaystyle\Braket{\mathcal{O}_{\theta}}_{\text{OAT}}^{\bm{\eta}}\equiv\Braket{\xi_{\text{OAT}}^{(\theta)}}{e^{i\bm{\eta}\cdot\bm{S}_{\theta}}\mathcal{O}_{\theta}e^{-i\bm{\eta}\cdot\bm{S}_{\theta}}}{\xi_{\text{OAT}}^{(\theta)}},$
(50)

where $\mathcal{O}_{\theta}$ is some product of the rotated collective spin
operators in $\bm{S}_{\theta}$, e.g. $S_{\text{x}}^{(\theta)}$ or
$S_{\text{x}}^{(\theta)}S_{\text{y}}^{(\theta)}$. In order to turn off SOC
during clock interrogation, however, we are restricted to performing rotations
of the form $\exp\left(-i\bm{\eta}\cdot\bm{S}_{0}\right)$ and measuring
homogeneous operators $\mathcal{O}_{0}$. We thus seek a “gauge-switching”
operation $G_{\theta}$ that maps homogeneous operators $\mathcal{O}_{0}$ onto
rotated operators $\mathcal{O}_{\theta}$ via
$G_{\theta}^{\dagger}\mathcal{O}_{0}G_{\theta}=\mathcal{O}_{\theta}$. Equipped
with $G_{\theta}$, we could decompose

$\displaystyle\Braket{\mathcal{O}_{\theta}}_{\text{OAT}}^{\bm{\eta}}$
$\displaystyle=\Braket{\xi_{\text{OAT}}^{(\theta)}}{G_{\theta}^{\dagger}e^{i\bm{\eta}\cdot\bm{S}_{0}}\mathcal{O}_{0}e^{-i\bm{\eta}\cdot\bm{S}_{0}}G_{\theta}}{\xi_{\text{OAT}}^{(\theta)}}$
$\displaystyle=\Braket{\tilde{\xi}_{\text{OAT}}^{(\theta)}}{e^{i\bm{\eta}\cdot\bm{S}_{0}}\mathcal{O}_{0}e^{-i\bm{\eta}\cdot\bm{S}_{0}}}{\tilde{\xi}_{\text{OAT}}^{(\theta)}}$
(51)

for a transformed state

$\displaystyle\Ket{\tilde{\xi}_{\text{OAT}}^{(\theta)}}\equiv
G_{\theta}\Ket{\xi_{\text{OAT}}^{(\theta)}}$ (52)

that is now squeezed with respect to the homogeneous collective spin operators
in $\bm{S}_{0}$, accessible with the SOC-free ($\phi=0$) clock laser used
during clock interrogation.

Given the definitions of the rotated spin operators
$S_{\text{x}}^{(\theta)},S_{\text{x}}^{(\theta)}$ in Eqns. (48), (49), a
suitable candidate for the gauge-switching operator $G_{\theta}$ is the site-
dependent rotation

$\displaystyle G_{\theta}=\prod_{j}\exp\left(i\theta
js_{\text{z}}^{(j)}\right).$ (53)

To implement $G_{\theta}$ with “global” (i.e. non-site-selective) experimental
controls, we decompose each local rotation into a product of two reflections:

$\displaystyle\exp\left(i2\alpha s_{\text{z}}^{(j)}\right)\simeq\exp\left(i\pi
s_{\text{x}}^{(j)}\right)\exp\left(i\pi s_{\alpha}^{(j)}\right)$ (54)

where $\simeq$ denotes equality up to an overall phase, and
$s_{\alpha}^{(j)}\equiv\cos\alpha s_{\text{x}}^{(j)}+\sin\alpha
s_{\text{y}}^{(j)}$. This decomposition implies

$\displaystyle G_{\theta}\simeq\exp\left(i\pi
S_{\text{x}}^{(0)}\right)\exp\left(i\pi S_{\text{x}}^{(\theta/2)}\right),$
(55)

which can be implemented using one SOC-free ($\phi=0$) clock laser, and one
clock laser with SOC angle $\phi=\theta/2$. Appending the two $\pi$-pulses
given by Eqn. (55) to our squeezing protocol thus prepares a state
$\Ket{\tilde{\xi}_{\text{OAT}}^{(\theta)}}$ that is squeezed with respect to
the homogeneous collective spin operators
$S_{\text{x}},S_{\text{y}},S_{\text{z}}$ accessible to the SOC-free ($\phi=0$)
clock laser used during clock interrogation.

As presented, the combined squeezing and clock interrogation protocols require
three clock lasers in total: one without SOC ($\phi=0$), and one each for SOC
angles $\phi\in\left\\{\theta/2,\theta\right\\}$. We can use $G_{\theta}$,
however, to decompose any pulse
$\exp\left(-i\bm{\beta}\cdot\bm{S}_{\theta}\right)$ into composite pulses that
use only $\bm{S}_{0}$ and $\bm{S}_{\theta/2}$:

$\displaystyle\exp\left(-i\bm{\beta}\cdot\bm{S}_{\theta}\right)=G_{\theta}^{\dagger}\exp\left(-i\bm{\beta}\cdot\bm{S}_{0}\right)G_{\theta}.$
(56)

The state prepared by the OAT squeezing protocol, for example, can be
equivalently prepared via

$\displaystyle\Ket{\xi_{\text{OAT}}^{(\theta)}}$
$\displaystyle=e^{-iH_{\text{FH}}^{(0)}t}G_{\theta}^{\dagger}e^{-i\left(\pi/2\right)S_{\text{x}}^{(0)}}G_{\theta}\Ket{-\text{Z}}$
$\displaystyle\simeq e^{-iH_{\text{FH}}^{(0)}t}e^{-i\pi
S_{\text{x}}^{(\theta/2)}}e^{i\left(\pi/2\right)S_{\text{x}}^{(0)}}\Ket{-\text{Z}}.$
(57)

Spin-echo pulses applied throughout OAT can likewise be decomposed according
to (56), eliminating the need for a clock laser with SOC angle $\phi=\theta$.
Applying a continuous drive during a squeezing protocol, however, still
requires all three clock lasers. If carefully tuning the relative orientations
of three clock lasers proves to be too difficult in practice, converting OAT
into TAT would therefore need to be done with a pulsed drive protocol, as in
Ref. [50].

##  Appendix F Decoherence in the 3D 87Sr optical lattice clock

Currently, light scattering from lattice beams in the 3D 87Sr optical lattice
clock induces decoherence on a time scale of $\sim$10 secondsGoban  _et al._
(2018); Hutson  _et al._ (2019). This decoherence acts identically on all
atoms in an uncorrelated manner, and can be understood by considering the
density operator $\rho$ for a single atom, with effective spin states
$\downarrow$ and $\uparrow$ respectively corresponding to the
${}^{1}\text{S}_{0}$ and ${}^{3}\text{P}_{0}$ electronic states. Empirically,
the effect of decoherence after a time $t$ within the
$\left\\{\downarrow,\uparrow\right\\}$ subspace of a single atom is to take
$\rho\to\rho\left(t\right)$ with $\rho\left(0\right)\equiv\rho$ and

$\displaystyle\rho\left(t\right)\coloneqq\begin{pmatrix}\rho_{\uparrow\uparrow}e^{-\Gamma_{\uparrow\uparrow}t}&&\rho_{\uparrow\downarrow}e^{-\Gamma_{\uparrow\downarrow}t}\\\
\rho_{\uparrow\downarrow}^{*}e^{-\Gamma_{\uparrow\downarrow}t}&&\rho_{\downarrow\downarrow}+\left(1-e^{-\Gamma_{\uparrow\uparrow}t}\right)\rho_{\uparrow\uparrow}\end{pmatrix},$
(58)

where
$\Gamma_{\uparrow\uparrow}\approx\Gamma_{\uparrow\downarrow}\approx\Gamma=0.1~{}\text{sec}^{-1}$
are respectively decay rates for $\rho_{\uparrow\uparrow}$ and
$\rho_{\uparrow\downarrow}$. This form of decoherence can be effectively
modeled by decay and dephasing of individual spins (respectively denoted
$\Gamma_{\text{ud}}$ and $\Gamma_{\text{el}}$ in Ref. [52]) at rates $\Gamma$.
In the language of the section that follows, we would say that this
decoherence is captured by the sets of jump operators
$\mathcal{J}_{-}\equiv\left\\{s_{-}^{(j)}\right\\}$ and
$\mathcal{J}_{\text{z}}\equiv\left\\{s_{\text{z}}^{(j)}\right\\}$ with
corresponding decoherence rates $\gamma_{-}=\gamma_{\text{z}}=\Gamma$.

##  Appendix G Time-series of squeezing via OAT and TAT

Figure G1 shows an example of squeezing over time via OAT and TAT, both with
and without decoherence via decay and dephasing of individual spins. The OAT
model initially generates squeezing faster than the TAT model, but the
squeezing generation rate of OAT (measured in dB per second) falls off with
time. The squeezing generation rate for TAT, meanwhile, remains approximately
constant (in the absence of decoherence) until squeezing via TAT surpasses
that of OAT. In the absence of decoherence, OAT achieves a maximal amount of
squeezing that scales as $\xi^{2}\sim N^{-2/3}$, while TAT achieves
Heisenberg-limited squeezing with $\xi^{2}\sim N^{-1}$. Note that our method
for computing squeezing via TAT in the presence of decoherence (described in
Appendix H) is not capable of computing squeezing for the full range of times
shown in Fig. G1; the corresponding time-series data in this figure is
therefore shown up to the point at which this method breaks down.

Figure G1: Squeezing via OAT and TAT in a 2D section of the 3D 87Sr optical
lattice clock, shown for (a) $\ell=40$ and (b) $\ell=100$ sites per axis (with
$N=\ell^{2}$ atoms total), and a lattice depth of $V_{0}=4~{}E_{\text{R}}$,
where $E_{\text{R}}$ is the atomic lattice recoil energy. Atoms are confined
along the direction transverse to the 2D layer by a lattice of depth 60
$E_{\text{R}}$. Squeezing over time is shown for OAT (blue) and TAT (green),
both with (solid lines) and without (dashed lines) decoherence via
uncorrelated decay and dephasing of individual spins at rates of
$0.1~{}\text{sec}^{-1}$ (see Appendix F).

##  Appendix H Solving Heisenberg equations of motion for collective spin
systems

In order to compute squeezing of a collective spin system, we need to compute
expectation values of (homogeneous) collective spin operators. We compute
these expectation values using a method recently developed in Ref. [53], and
provide a short overview of the method here. Choosing the basis
$\left\\{\mathcal{S}_{\bm{m}}\right\\}$ for all collective spin operators,
where $\mathcal{S}_{\bm{m}}\equiv
S_{+}^{m_{+}}S_{\text{z}}^{m_{\text{z}}}S_{-}^{m_{-}}$ with
$\bm{m}\equiv\left(m_{+},m_{\text{z}},m_{-}\right)\in\mathbb{N}_{0}^{3}$, we
can expand all collective spin Hamiltonians in the form

$\displaystyle H=\sum_{\bm{m}}h_{\bm{m}}\mathcal{S}_{\bm{m}}.$ (59)

The evolution of a general correlator $\Braket{\mathcal{S}_{\bm{n}}}$ under a
Hamiltonian of the form in Eqn. (59) is then given by

$\displaystyle\dfrac{d}{dt}\Braket{\mathcal{S}_{\bm{n}}}$
$\displaystyle=i\sum_{\bm{m}}h_{\bm{m}}\Braket{\left[\mathcal{S}_{\bm{m}},\mathcal{S}_{\bm{n}}\right]_{-}}+\sum_{\mathcal{J}}\gamma_{\mathcal{J}}\Braket{\mathcal{D}\left(\mathcal{J};\mathcal{S}_{\bm{n}}\right)}$
$\displaystyle\equiv\sum_{\bm{m}}\Braket{\mathcal{S}_{\bm{m}}}T_{\bm{m}\bm{n}},$
(60)

where $\left[X,Y\right]_{\pm}\equiv XY\pm YX$; $\mathcal{J}$ is a set of jump
operators with corresponding decoherence rate $\gamma_{\mathcal{J}}$; the
decoherence operator $\mathcal{D}$ is defined by

$\displaystyle\mathcal{D}\left(\mathcal{J};\mathcal{O}\right)\equiv\sum_{J\in\mathcal{J}}\left(J^{\dagger}\mathcal{O}J-\dfrac{1}{2}\left[J^{\dagger}J,\mathcal{O}\right]_{+}\right);$
(61)

and $T_{\bm{m}\bm{n}}$ is a matrix element of the time derivative operator
$T\equiv d/dt$. These matrix elements can be calculated analytically using
product and commutation rules for collective spin operators. We can then
expand correlators in a Taylor series about $t=0$ to write

$\displaystyle\Braket{\mathcal{S}_{\bm{n}}}$ $\displaystyle=\sum_{k\geq
0}\dfrac{t^{k}}{k!}\Braket{\dfrac{d^{k}}{dt^{k}}\mathcal{S}_{\bm{n}}}_{t=0}$
$\displaystyle=\sum_{k\geq
0}\dfrac{t^{k}}{k!}\sum_{\bm{m}}T_{\bm{m}\bm{n};k}\Braket{\mathcal{S}_{\bm{m}}}_{t=0},$
(62)

where $T_{\bm{m}\bm{n};k}\equiv\left[T^{k}\right]_{\bm{m}\bm{n}}$ are matrix
elements of the $k$-th time derivative. Expectation values of collective spin
operators can thus be computed via the expansion in Eqn. (62), which at short
times can be truncated at some finite order beyond which all terms have
negligible contribution to $\Braket{\mathcal{S}_{\bm{n}}}$.

##  Appendix I Effect of a harmonic confining trap

Figure I1: Dynamics of non-interacting spin-orbit coupled fermions in a 1D
lattice with SOC angle $\phi=\pi/50$, plus a harmonic trap with
$\Omega/J=0.01$. Starting with a spin-polarized cloud in $\downarrow$ ground
state, an initial clock laser pulse is applied to rotate spins into $x$, and
the atoms are allowed to evolve during the dark time. We track the dynamics of
the $\uparrow$ particle density for the cases of (a) $N=20$ and (b) $N=60$
atoms. Panel (c) shows the time-averaged fluctuations of the $\uparrow$
particle density for each site index $j$ from its initial value following the
Ramsey pulse; see Eqn. (65). For $N=60$, we have filled all delocalized modes
as well as several localized modes, resulting in a large region of no density
fluctuations at the trap center. Panel (d) contains the eigenspectrum for a
single internal state in the presence of the trap (with the index $n$ labeling
the eigenvalues in order of increasing energy), where the critical mode
$n_{c}$ dividing the spatially delocalized and localized modes is indicated by
a black dash-dotted line. The highest occupied mode in the $\downarrow$ ground
state for $N=20$ and $N=60$ is indicated by the green and red solid lines,
respectively. Figure I2: Dynamics of interacting spin-orbit coupled fermions
in a 1D lattice plus a harmonic trap for $U/J=1$ (a), 2 (b), and 4 (c). For a
1D lattice with 10 sites and an SOC angle $\phi=\pi/50$, we apply a $\pi/2$
clock laser pulse to the $\downarrow$ ground state and let the system evolve
during the dark time. In (a.i)-(c.i) we show the squeezing dynamics of the
system for both $N=10$ (solid lines) and $N=9$ (dashed lines) for a variety of
trapping strengths. In (a.ii)-(c.ii), we plot the time-averaged fluctuations
in total particle density, $\overline{\delta n_{j}}$ (as in Eqn. (65) but with
$\hat{n}_{j,\uparrow}$ replaced by $\sum_{\alpha}\hat{n}_{j,\alpha}$). In
(a.iii)-(c.iii), we plot the growth of the doublon population $N_{d}(t)$ (see
Eqn. (66)) as a function of time, noting the absence of squeezing in the
presence of a large doublon population. For the chosen trap strengths, the
corresponding values of $n_{c}$ are 28 ($\Omega/J=0.01$), 14
($\Omega/J=0.04$), and 6 ($\Omega/J=0.2$). In panels where the results for the
homogeneous case (orange curves) are not visible, they are nearly identical to
the results for $\Omega/J=0.01$ (green curves). Here, we utilize periodic
boundary conditions to minimize finite size effects.

Current 3D optical lattice implementations involve a harmonic confining
potential, which significantly alters the underlying single-particle
eigenstructure and can potentially degrade accessible squeezing within our
protocol. In this appendix, we examine the effect of a harmonic trap on our
protocol and discuss strategies to mitigate undesired effects. We model the
trap by the addition of the term

$\displaystyle\hat{H}_{\Omega}=\hbar\Omega\sum_{j,\alpha}(j-j_{0})^{2}\hat{n}_{j,\alpha}$
(63)

to our Fermi-Hubbard model (Eqn. (2)), where $j_{0}$ denotes the trap center
and $\Omega=m(\omega a)^{2}/2\hbar$ characterizes the trap strength for atom
mass $m$, trap frequency $\omega$, and lattice spacing $a$. In current state-
of-the-art 3D 87Sr OLC implementations, values of $\omega\approx 56\times
2\pi~{}\text{sec}^{-1}$ can be achieved within each 2D layer of weak SOC by
utilizing in-plane lattice depths of $5E_{R}$ and a lattice depth of $60E_{R}$
in the axial direction, resulting in a value of $\Omega/J\approx 0.01$. We
restrict our discussion to 1D, although for a separable 3D lattice our
arguments should extend in a straightforward manner.

We briefly review the structure of the single-particle eigenstates of the
system, before discussing the effects on squeezing. In the quasi-momentum
basis, the eigenfunctions $\psi_{n,\alpha}(q)=\braket{q}{n,\alpha}$ are given
by the $\pi$-periodic Mathieu functions, with the corresponding energies
described by the Mathieu characteristic valuesRey  _et al._ (2005). In the
presence of SOC, using the gauge transformation described in the main text, we
obtain the relation

$\displaystyle\psi_{n,\uparrow}(q)=\psi_{n,\downarrow}(q-\phi/a).$ (64)

In contrast to the case of a pure harmonic potential, which generically has
spatially delocalized single-particle eigenstates, the addition of a tight-
binding lattice causes eigenmodes with quantum number $n$ (index $n$ labels
the eigenvalues in order of increasing energy) larger than $n_{c}\approx
2\sqrt{2J/\Omega}$ to become localized at corresponding lattice sites.
Therefore the sites at a distance $n_{c}/2$ from the trap center with
potential energy $2\hbar J$ define the boundary between the delocalized modes
at the trap center and the high-energy localized trap edges. Tunneling in the
region of localized modes is typically suppressed by large potential energy
differences even in the presence of SOC. These modes are thus largely
decoupled and do not contribute to the trap center dynamics. On the other hand
the delocalized modes may be approximated by those of a quantum harmonic
oscillator with effective mass $m^{*}=\hbar/(2Ja^{2})$ and frequency
$\omega^{*}=\sqrt{4J\Omega}$.

As emphasized in the main text, the key requirements for our protocol are 1)
the validity of the spin model, which depends on the pinning of particles in
their initial single particle modes, and 2) the gap protection against SOC
dephasing, which arises from collective spin interactions. Concerning the
latter point, it is desirable to maintain a weak trap so as to enable a large
number of delocalized modes in the trap center, which are the only type
capable of contributing to the generation of squeezing. Though the
interactions between these modes are not strictly all-to-all, they remain
long-ranged, and can thus still lead to a spin-locking effect and a protective
gapRey  _et al._ (2014); Smale  _et al._ (2019). For $\Omega/J=0.01$ we have
$n_{c}=28$, enabling $\sim 10^{3}$ contributing modes in each 2D layer of our
system. Concerning the validity of the spin model, from a single-particle
perspective the eigenmodes of our $\uparrow$ states will be initially
displaced in quasi-momentum space from equilibrium by $\phi/a$ as per Eqn.
(64), and will generally undergo dipole oscillations and not remain strictly
pinned to their initial modes. However, as long as we ensure the displacement
is small enough to guarantee a constant density distribution across the trap
center, the spin model will remain valid. The localized modes at the trap
edges can actually help to satisfy this condition, since they can serve as a
barrier against motion. This is demonstrated in Fig. I1 where we show that
filling all delocalized modes guarantees that the trap center maintains a
constant density; we characterize this by the time-averaged fluctuations of
the $\uparrow$ density at each site $j$ about its initial value following the
Ramsey pulse,

$\displaystyle\overline{\delta
n_{j,\uparrow}}\equiv\sqrt{\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}\text{d}\tau\bigg{(}\langle\hat{n}_{j,\uparrow}(\tau)\rangle-\langle\hat{n}_{j,\uparrow}(0)\rangle\bigg{)}^{2}},$
(65)

choosing sufficiently large evolution times to ensure convergence.

In the presence of interactions, an additional point of concern is that the
interplay between the trap and interactions may induce resonances that enable
the formation of a significant doublon population,

$\displaystyle
N_{d}(t)=\sum_{j}\langle\hat{n}_{j,\uparrow}(t)\hat{n}_{j,\downarrow}(t)\rangle,$
(66)

which in turn may alter the density distribution and invalidate the spin
model. Since doublon formation in the localized edges will not have
consequences for our squeezing protocol, we must only ensure that doublons are
not formed in the trap center, which requires
$U>\Omega(n_{c}/2)^{2}=2J$Pupillo  _et al._ (2006). In Fig. I2, we perform
exact simulations to assess the effect of the trap on our system. Though
restricted to small system sizes, the results demonstrate that for
$U/J\lesssim 2$, the trap will always lead to a decrease of squeezing due to
the formation of doublons in the trap center, while for $U/J\gtrsim 2$, we are
protected from this process even for trap strengths much stronger than the
experimentally relevant ones.

##  Appendix J Accounting for $p$-wave inelastic collisions

Figure J1:  $p$-wave loss rates. Both the averaged $p$-wave inelastic
collision rate $\gamma$ (orange) and the ratio of this collision rate to the
optimal squeezing rate $\chi_{\text{opt}}$ (blue) are suppressed as the
lattice depth increases. $\chi_{\text{opt}}$ is obtained by choosing SOC
angles $\phi$ that saturate $\widetilde{B}/U\approx 0.05$, where
$\widetilde{B}$ is the variance of the SOC-induced axial field and $U$ is the
two-atom on-site interaction energy.  Figure J2: Squeezing via OAT in the
presence of inelastic collisions. (a) For fixed particle number $N=100$, the
optimal squeezing decreases as the inelastic collision rate increases. Panel
(b) shows squeezing over time for $\gamma/\chi_{\text{opt}}=0.04$ (solid
lines), which corresponds to $U/J=6$, and compares it with $\gamma=0$ (dashed
lines) for different particle numbers. Inelastic collisions prevent the growth
of optimal squeezing with particle number. For $N=1000$, the maximum squeezing
saturates to $\sim 10$ dB.

Inelastic ${}^{3}\text{P}_{0}$ (electronic state $e$ or $\uparrow$) collisions
are detrimental for optical lattice clocks. For the nuclear-spin-polarized gas
discussed in this work, $ee$ losses are only possible via the $p$-wave
scattering channel since $s$-wave collisions are suppressed by Fermi
statistics. The big advantage here compared to prior experiments done in a 1D
lattice at $\mu$K temperatureMartin  _et al._ (2013) is that in a Fermi
degenerate gas loaded in a 3D optical lattice, $p$-wave losses are further
suppressed by the centrifugal barrier and Pauli blocking, and only happen
through a wave-function overlap between atoms at different lattice sites. In
this appendix, we quantify the effect of $p$-wave interactions on squeezing.
To account for $p$-wave losses, we describe the dynamics using a master
equation for the system’s density matrix $\hat{\rho}$:

$\displaystyle\frac{d\hat{\rho}}{dt}=-\frac{i}{\hbar}[\hat{H}_{\text{eff}},\,\hat{\rho}]+\mathcal{L}\hat{\rho},$
(67)

where $\hat{H}_{\text{eff}}=\chi\hat{S}_{z}^{2}$ is the effective one-axis
twisting Hamiltonian obtained from the original Fermi-Hubbard Hamiltonian with
spin-orbit coupling, and $\mathcal{L}$ is the Lindblad superoperator that
accounts for $p$-wave $ee$ inelastic collisions. This latter term can be
written using a pseudo-potential approximation asRey  _et al._ (2014):

$\mathcal{L}\hat{\rho}=\sum_{{\bm{k}}{\bm{k}^{\prime}}}\Gamma_{{\bm{k}}{\bm{k}^{\prime}}}\bigg{[}\hat{A}_{{\bm{k}}{\bm{k}^{\prime}}}\hat{\rho}\hat{A}_{{\bm{k}}{\bm{k}^{\prime}}}^{\dagger}\\\
-\dfrac{1}{2}\left(\hat{A}_{{\bm{k}}{\bm{k}^{\prime}}}^{\dagger}\hat{A}_{{\bm{k}}{\bm{k}^{\prime}}}\hat{\rho}+\hat{\rho}\hat{A}_{{\bm{k}}{\bm{k}^{\prime}}}^{\dagger}\hat{A}_{{\bm{k}}{\bm{k}^{\prime}}}\right)\bigg{]},$
(68)

where the jump operators are
$\hat{A}_{{\bm{k}}{\bm{k}^{\prime}}}=\hat{c}_{{\bm{k}},\uparrow}\hat{c}_{{\bm{k}^{\prime}},\uparrow}$,
and $\bm{k}$, $\bm{k}^{\prime}$ sum over all the populated quasi-momentum
modes. The decay matrix elements $\Gamma_{{\bm{k}}{\bm{k}^{\prime}}}$ are
given by:

$\displaystyle\Gamma_{{\bm{k}}{\bm{k}^{\prime}}}=\frac{3\pi\hbar
b_{\text{im}}^{3}}{m}\left(\int
d{\bm{r}}^{\,3}W[\phi_{{\bm{k}}}({\bm{r}}),\phi_{{\bm{k}^{\prime}}}({\bm{r}})]\right),$
(69)

where $b_{\text{im}}=121a_{0}$Zhang  _et al._ (2014); Goban  _et al._ (2018)
is the $p$-wave inelastic scattering length (with $a_{0}=5.29\times 10^{-11}$
m the Bohr radius), $\phi_{{\bm{k}}}({\bm{r}})$ is the Bloch function with
quasi-momentum ${\bm{k}}$, and

$W\left[\phi_{{\bm{k}}}({\bm{r}}),\phi_{{\bm{k}^{\prime}}}({\bm{r}})\right]\\\
\equiv\left[\left({\bm{\nabla}}\phi^{*}_{{\bm{k}}}(\bm{r})\right)\phi^{*}_{{\bm{k}^{\prime}}}(\bm{r})-\phi^{*}_{{\bm{k}}}(\bm{r})\left({\bm{\nabla}}\phi^{*}_{{\bm{k}^{\prime}}}(\bm{r})\right)\right]\\\
\cdot\left[\left({\bm{\nabla}}\phi_{{\bm{k}}}(\bm{r})\right)\phi_{{\bm{k}^{\prime}}}(\bm{r})-\phi_{{\bm{k}}}(\bm{r})\left({\bm{\nabla}}\phi_{{\bm{k}^{\prime}}}(\bm{r})\right)\right]$
(70)

In Fig. J1, we show the averaged decay rate
$\gamma\equiv\sum_{{\bm{k}}{\bm{k}^{\prime}}}\Gamma_{{\bm{k}}{\bm{k}^{\prime}}}/\ell^{2}$,
where $\ell$ is the number of lattice sites along the $x$ and $y$ axes, as a
function of the lattice depth $V_{0}$ along these axes. Here, we assume the
same lattice depth in the $z$ direction used in the main text,
$V=60E_{\text{R}}$. The decay rate $\gamma$ is suppressed exponentially with
increasing lattice depth $V_{0}$. To quantify the effect of these losses on
the spin squeezing generation process, we follow a similar methodology to the
one described in detail in Ref. Rey  _et al._ (2014). The basic idea is to
take advantage of the so-called Truncated-Wigner Approximation
(TWA)Polkovnikov (2010); Schachenmayer  _et al._ (2015), which allows us to
capture the development of spin squeezing using semi-classical phase-space
methods. In the TWA the quantum dynamics are accounted for by solving mean
field equations of motion supplemented by noise. The mean field equations are
derived by assuming that the many-body density matrix of the system can be
factorized as $\hat{\rho}=\bigotimes_{i}\hat{\rho}({i})$, where
$\hat{\rho}(i)$ is the reduced density matrix of the particle in quasi-
momentum mode ${\bm{q}}_{i}$ [see Eqn. (58)]. Under this assumption, the non-
linear mean field equations are given by

$\displaystyle\frac{d\rho_{\uparrow\uparrow}(j)}{dt}=-\sum_{j^{\prime}}\Gamma_{{\bm{k}}_{j}{\bm{k}}_{j^{\prime}}}\rho_{\uparrow\uparrow}(j)\rho_{\uparrow\uparrow}(j^{\prime}),$
$\displaystyle\frac{d\rho_{\downarrow\downarrow}(j)}{dt}=0$ (71)

and

$\frac{d\rho_{\uparrow\downarrow}(j)}{dt}=\rho_{\uparrow\downarrow}(j)\sum_{j^{\prime}}\bigg{[}i\chi(\rho_{\uparrow\uparrow}(j^{\prime})-\rho_{\downarrow\downarrow}(j^{\prime}))\\\
-\dfrac{1}{2}\Gamma_{{\bm{k}}_{j}{\bm{k}}_{j^{\prime}}}\rho_{\uparrow\uparrow}(j^{\prime})\bigg{]},$
(72)

where
$\rho_{\sigma\sigma^{\prime}}\equiv\langle\hat{\rho}_{\sigma\sigma^{\prime}}\rangle$.
Since we are interested in the collective behavior, one can define
$\rho_{\sigma\sigma^{\prime}}^{T}=\sum_{j}\rho_{\sigma\sigma^{\prime}}(j)$.
For these observables the equations of motion simplify to

$\displaystyle\frac{d\rho_{\uparrow\uparrow}^{T}}{dt}=-f\gamma(\rho_{\uparrow\uparrow}^{T})^{2},$
$\displaystyle\frac{d\rho_{\downarrow\downarrow}^{T}}{dt}=0$ (73)

and

$\displaystyle\frac{d\rho_{\uparrow\downarrow}^{T}}{dt}=\rho_{\uparrow\downarrow}^{T}\left[i\chi(\rho_{\uparrow\uparrow}^{T}-\rho_{\downarrow\downarrow}^{T})-\dfrac{1}{2}f\gamma\rho_{\uparrow\uparrow}^{T}\right],$
(74)

where $f\equiv N/\ell^{2}$ is the filling fraction.

Under the TWA, one accounts for quantum fluctuations during the dynamics by
averaging over different mean field trajectories generated by sampling over
different initial conditions chosen to reconstruct the Wigner function of the
initial coherent spin state at $t=0$Rey  _et al._ (2014). This method has
proven to be successful in simulating quantum spin dynamics. Using this
approach, Fig. J2 shows numerical simulation results of squeezing over time in
the presence of inelastic collisions. For shallow lattices ($V_{0}\lesssim
7E_{\text{R}}$), the effect of inelastic collisions can limit the spin
squeezing to $\sim 10$ dB. Thus, in this regime, losses are as relevant as
light scattering. The role of inelastic interactions could be mitigated by
either operating at deeper lattices as shown in Fig. J2, or by using nuclear
spin states to generate the squeezing instead of the clock states directly.

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