# Verifying a quasi-classical spin model of perturbed quantum rewinding in a
Fermi gas

J. Huang, Camen A. Royse, I. Arakelyan, and J. E. Thomas  1Department of
Physics, North Carolina State University, Raleigh, NC 27695, USA

###### Abstract

We systematically test a quasi-classical spin model of a large spin-lattice in
energy space, with a tunable, reversible Hamiltonian and effective long-range
interactions. The system is simulated by a weakly interacting Fermi gas
undergoing perturbed quantum rewinding using radio-frequency(RF) pulses. The
model reported here is found to be in a quantitative agreement with
measurements of the ensemble-averaged energy-resolved spin density. This work
elucidates the effects of RF detunings on the system and measurements,
pointing the way to new correlation measurement methods.

Measurement of coherence, entanglement, and correlations in time-reversible
many-body spin lattices is of great interest, broadly impacting our
understanding of quantum measurement and information processing Lewis-Swan
_et al._ (2019); Eisert  _et al._ (2015a); Kaufman and et al. (2016). A nearly
ideal platform for simulating large spin-lattices is a weakly interacting
Fermi gas, containing $N\simeq 10^{5}$ atoms with a tunable, reversible
Hamiltonian. The trapped cloud behaves as a spin lattice in energy-space with
effective long-range interactions Du  _et al._ (2008, 2009); Ebling  _et al._
(2011); Pegahan  _et al._ (2019); Piéchon  _et al._ (2009); Natu and Mueller
(2009); Deutsch  _et al._ (2010); Smale  _et al._ (2019); Koller  _et al._
(2016); Gärttner  _et al._ (2017); Eisert  _et al._ (2015b) and enables new
tests of classical versus quantum spin evolution Schubert  _et al._ (2021);
Das  _et al._ (2019); Lakshmanan  _et al._ (1976); Ball (2021).

Spin waves are observed in weakly interacting, nearly collisionless Fermi
gases, which have been explained by several models Ebling  _et al._ (2011); Du
_et al._ (2009); Piéchon  _et al._ (2009); Natu and Mueller (2009); Deutsch
_et al._ (2010); Koller  _et al._ (2016); Pegahan  _et al._ (2019).
Previously, a 1D quasi-classical spin evolution model that uses the exact
energy-dependent couplings was found to yield agreement with spin-density
profiles measured for the evolution of an initially $x$-polarized spin sample
Pegahan  _et al._ (2019). However, it appeared that this model failed to
explain perturbed quantum rewinding experiments, where an RF pulse rotates the
entire spin system by an angle $\phi_{x}$ about the $x$ axis as a perturbation
in between forward and backward evolutions. In a quantum picture, the
$\phi_{x}$ rotation changes the relative phases of the superposed total
angular momentum states that describe the system, i.e.,
$|S,M_{x}\rangle\rightarrow e^{-iM_{x}\phi_{x}}|S,M_{x}\rangle$ for each
state, leading to $\phi_{x}$-dependent coherence amplitude between states
differing in $M_{x}$ Gärttner  _et al._ (2017, 2018); Pegahan  _et al._
(2021). To fit the data, a scattering length of $\approx 2.5$ times the
measured value was needed in the previous work Pegahan  _et al._ (2021),
questioning the adequacy of the quasi-classical model.

In this work, we report precise, systematic tests of a modified quasi-
classical spin model using single-shot measurements of the spin density
profiles from perturbed quantum rewinding experiments. Such experiments are
ideal for testing the model, since unperturbed rewinding experiments can be
implemented in advance to prove that the system is reversed properly without
model-dependent fits. We show the advantages of single-shot data analysis for
studies of ensemble-averaged energy-resolved spin density, and quantitatively
demonstrate the important roles of different RF detunings during the forward
and backward evolution periods. By using two detunings as separate fit
parameters, the data is explained by the model using the measured scattering
length. The new approach reported here validates the modified quasi-classical
treatment of this quantum spin system and suggests detuning-independent
measurement methods for future correlation studies, avoiding probabilistic
methods in data selection Smale  _et al._ (2019).

Our experiments Sup , employ degenerate clouds of 6Li containing a total of
$N=6.5\times 10^{4}$ atoms initially in a single spin state. The cloud is
confined in a harmonic, cigar-shaped optical trap, with oscillation
frequencies $\omega_{x}/2\pi=24.4$ Hz in the axial direction and
$\omega_{r}/2\pi=650$ Hz in the radial direction. The corresponding Fermi
temperature $T_{F}=0.73\,\mu$K and $T/T_{F}=0.31$. RF pulses prepare coherent
superpositions of the two lowest hyperfine-Zeeman states, which are denoted by
$|1\rangle\equiv|\\!\uparrow_{z}\rangle$ and
$|2\rangle\equiv|\\!\downarrow_{z}\rangle$. The experiments are done in the
weakly interacting regime, where the energy-state changing collision rate is
negligible over the time scale of the measurements Pegahan  _et al._ (2019).

As the single particle energies are fixed and the energy distribution is time
independent Pegahan  _et al._ (2019), we approximate the cigar-shaped weakly
interacting Fermi gas as a one-dimensional (1D) spin “lattice” in energy space
Pegahan  _et al._ (2019), with a Hamiltonian

$\frac{H(a)}{\hbar}=a\\!\sum_{i,j\neq
i}\\!g_{ij}\,{\vec{s}}_{i}\cdot{\vec{s}}_{j}+\sum_{i}\Omega^{\prime}E_{i}\,s_{zi}+\Delta(t)S_{z}.$
(1)

Figure 1: Observing quantum rewinding by comparing the spin densities for
forward evolution (red circles) and backward evolution (blue circles). The
scattering length $a=8.0\,a_{0}$ and $\sigma_{TF}\approx 335\,\mu$m. The
Hamiltonian is reversed at $\tau_{f}=280$ ms (top) and $\tau_{f}=400$ ms
(bottom). For forward evolution data, the time $t_{k}$ shown on the top right
corner of each tile is the total evolution time $t_{\rm fk}$: For the backward
evolution data, the total evolution time $t_{\rm bk}=\tau_{f}+\tau_{b}$, where
$\tau_{f}$ is the time at which reversal occurs, and $\tau_{b}=\tau_{f}-t_{k}$
is the backward evolution time, see Fig. 2. To avoid confusion with the
$x$-axis of RF or Bloch frame, here, we clarify that $x$ in all spatial
profiles means axial direction along the longitudinal axis of the cigar-shaped
cloud.

We associate a “site” $i$ with the energy
$E_{i}\\!=\\!(n_{i}\\!+\\!1/2)\,h\nu_{x}$ of the ith harmonic oscillator state
along the cigar axis $x$. For each $E_{i}$, we define a dimensionless
collective spin vector $\vec{s}\,(E_{i})\equiv\vec{s}_{i}$.

The first term in Eq. 1 is the site-to-site interaction, proportional to the
s-wave scattering length $a$ and to the overlap of the harmonic oscillator
probability densities for colliding atoms. In a WKB approximation,
$g_{ij}\propto 1/\sqrt{|E_{i}-E_{j}|}$, which is an effective long-range
interaction in the energy-space lattice Pegahan  _et al._ (2019). For a zero
temperature Fermi gas, the average interaction energy (in rad/s) is
$a\bar{g}=6.8\,n_{0}\hbar a/m$, where $n_{0}$ is the peak density. For our
experimental parameters, with $a=5.2\,a_{0}$, $a\bar{g}/2\pi\simeq 2.0$ Hz.

The second term in Eq. 1 is an effective site-dependent Zeeman energy, arising
from the quadratic spatial variation of the bias magnetic field along $x$,
which produces a spin-dependent harmonic potential. As
$\omega_{r}/\omega_{x}=26.6$, the corresponding effect on the radial motion is
negligible, enabling a 1D approximation, where all atoms at site $i$ have the
same Zeeman energy. In Eq. 1,
$\Omega^{\prime}=-\delta\omega_{x}/(\hbar\omega_{x})$, with
$\delta\omega_{x}/2\pi=14.9$ mHz for our trap Pegahan  _et al._ (2019). For
the mean energy $\bar{E}_{x}\simeq k_{B}T_{F}/4$,
$\Omega^{\prime}\,\bar{E}_{x}/2\pi\simeq 2.0$ Hz.

The last term in Eq. 1 arises from the time-dependent global detuning
$\Delta(t)$, which plays a central role in the analysis of the rewinding data.
Here, $S_{z}=\sum_{i}s_{zi}$. For a typical evolution time $~{}200$ ms,
$\Delta(t)\simeq 0.4$ Hz. Fluctuations in the bias magnetic field and magnetic
tuning of the scattering length cause $\Delta(t)$ to change at 5 kHz/G for
$|1\rangle$-$|2\rangle$ superposition states.

To implement perturbed quantum rewinding, we employ the pulse sequence shown
in Fig. 2 Pegahan  _et al._ (2021). The system is initially prepared in a pure
z-polarized spin state, $|\psi_{0z}\rangle$. The first $(\pi/2)_{y}$ pulse
(0.5 ms), defined to be about the $y$-axis, creates an $x$ polarized state,
$|\psi_{0x}\rangle$. Here, the $y$\- and $x$-axes are defined in the rotating
frame of the RF pulses (RF-frame). Then, the system is allowed to evolve
forward for a time $\tau_{f}$. A voltage-controlled change of the RF phase by
$\pi/2$ permits rotation about the $x$-axis by an angle $\phi_{x}$. Applying a
$(\pi)_{y}$ pulse (1 ms) and magnetically tuning the scattering length from
$a\rightarrow-a$ (10 ms) inverts the sign of Hamiltonian shown in Eq. 1,
causing the system to evolve backward for a time $\tau_{b}$ Sup . As described
below, we perform experiments both with and without the final $(\pi/2)_{y}$
pulse, after which the spatial profiles of the $|\uparrow_{z}\rangle$ and
$|\downarrow_{z}\rangle$ states are measured by two resonant absorption
imaging pulses, separated by $10\,\mu$s, to obtain the single-shot spin
density $S_{z}(x)=[n_{\uparrow_{z}}(x)-n_{\downarrow_{z}}(x)]/2$. For each
shot, $S_{z}(x)$ is normalized to the total central density
$n(0)=n_{\uparrow_{z}}(0)+n_{\downarrow_{z}}(0)$ to minimize errors arising
from shot-to-shot variation in the atom number and cloud width. All spatial
profiles are folded about $x=0$ and displayed for $0\leq x\leq\sigma_{TF}$.

Figure 2: Characterizing perturbed quantum rewinding. The atom cloud is
initially prepared in a pure $z$-polarized spin state. The system runs forward
for a time $\tau_{f}$ and backward for $\tau_{b}$, after which the spatial
profiles of the $|\uparrow_{z}\rangle$ and $|\downarrow_{z}\rangle$ states are
measured. Green and red blocks represent RF pulses and rotation angles about
the $y$\- and $x$-axes respectively, whose durations are
$<<\tau_{f},\,\tau_{b}$.

The reversibility of the system is tested (result shown in Fig. 1) using the
pulse sequence of Fig. 2 with $\phi_{x}=0$ and without the final $(\pi/2)_{y}$
pulse. This sequence measures the component of the collective spin vector
${\vec{s}}_{i}$ that was along the $z$-axis just prior to imaging. The
longitudinal ($z$) component is insensitive to the detuning $\Delta(t)$ that
causes a rotation of ${\vec{s}}_{i}$ about the $z$-axis relative to the RF-
frame, enabling a robust test. In the data analysis, since $S_{z}=0$ for
$\phi_{x}=0$, global spin balance is enforced to minimize the error from small
shot-to-shot changes in the detuning of the RF pulses, which arises from
magnetic field fluctuation.

In these experiments, it is essential to carefully calibrate the bias magnetic
field $B_{0}$ at which the s-wave scattering length vanishes. This is best
done by quantifying the reversal results using different magnetic fields,
which is independent of fitting models and less sensitive to the initial
conditions in contrast to the method adopted in Ref. Pegahan  _et al._ (2019).
$B_{0}$ is found by minimizing the sum of the mean square differences between
the forward and backward spin density profiles at corresponding times Sup .
Unperturbed rewinding experiments done at scattering lengths of $\pm
5.2\,a_{0}$ and $\pm 8.0\,a_{0}$, suggest that $B_{0}=527.150(5)$ G, which is
lower by $30$ mG compared to the $B_{0}$ of Ref. Pegahan  _et al._ (2019).

Fig. 1 shows rewinding data (6-shot average) at corresponding forward(red) and
backward(blue) evolution times for $a=8.0\,a_{0}$ and $-8.0\,a_{0}$
respectively. With the calibrated $B_{0}$, the corresponding forward and
backward spin density profiles demonstrate good agreement for reversal at
$280$ ms (top row), while reversal at 400 ms (bottom row) leads to greater
differences between corresponding forward and backward data profiles.

Figure 3: Relation between the spin vector components in the
radiofrequency(RF) and Bloch(B) frames for nonzero $\varphi_{fb}$.

Having established that the system is reversible for scattering lengths up to
$\pm 8.0\,a_{0}$ and $\tau_{f}=\tau_{b}\leq 280$ ms, data are mainly obtained
with $\tau\equiv\tau_{f}=\tau_{b}=200$ ms at $\pm 5.2\,a_{0}$ using the full
pulse sequence of Fig. 2. This provides stringent tests of quasi-classical
collective spin vector models. Here, the final $(\pi/2)_{y}$ pulse is included
to measure the transverse spin components that were along the $x$-axis in the
RF frame in Fig. 3 just prior to imaging. For $\phi_{x}=0$ and a detuning
$\Delta(t)$ that is constant over the total sequence, the system is expected
to rewind to the initial state, where the density profiles for both spins are
Thomas-Fermi. For $\phi_{x}\neq 0$, however, the rewinding is perturbed,
producing complex spin density profiles after the full sequence. Fig. 4 shows
single-shot spin density profiles for $\phi_{x}=\pi/2,\pi,3\pi/2$. We obtain
the corresponding energy-space profiles, $s_{zi}\equiv s_{z}(E)$, by inverse
Abel-transformation Pretzier (1991) of the spatial profiles, which is valid in
a WKB approximation when energy space coherence is negligible and a quasi-
continuum approximation is valid, as in our experiments Pegahan  _et al._
(2019).

Figure 4: Single-shot spin-density profiles in space and in energy space for
perturbed quantum rewinding. For this set of data, $\sigma_{TF}\approx
340\,\mu$m, (a,e) measured scattering length $a=5.2\,a_{0}$, $\phi_{x}=\pi/2$,
$\tau=200$ ms; (b,f) $a=5.2\,a_{0}$, $\phi_{x}=\pi$, $\tau=200$ ms; (c,g)
$a=8.0\,a_{0}$, $\phi_{x}=3\pi/2$, $\tau=200$ ms; (d,h); $a=5.2\,a_{0}$,
$\phi_{x}=\pi/2$, $\tau=400$ ms. Blue dots are single-shot data. Red curves:
Quasi-classical collective spin vector model using the measured scattering
length, with forward and backward evolution phases $\varphi_{f}$ and
$\varphi_{b}$ as fit parameters. Black-dashed curves show the fits with
$\varphi_{b}=\varphi_{f}$ as one free parameter and scattering length
$a_{fit}$ as the other, which requires $a_{fit}=9.0\,a_{0}$.

To understand the perturbed rewinding data of Fig. 4, we include a time-
dependent global detuning, $\Delta(t)$, in the Hamiltonian of Eq. 1. The
detuning determines the relative angle between the RF-frame and the Bloch
frame $\varphi_{fb}$ in Fig. 3. Here, the RF frame is defined by $x_{RF}$ and
$y_{RF}$ axes that rotate about the $z$-axis at the instantaneous RF
frequency, $\omega_{RF}(t)$, tracking the total phase of the RF field. We
define the rotation axes for all of the RF pulses in Fig. 2 to be in the RF
frame, i.e., $x\equiv x_{RF}$ and $y\equiv y_{RF}$. The Bloch frame is defined
by $x_{B}$ and $y_{B}$ axes that rotate at the instantaneous hyperfine
resonance frequency $\omega_{HF}(t)$ for an atom of axial energy $E=0$.

The detuning, $\Delta(t)=\omega_{HF}(t)-\omega_{RF}(t)$, causes the components
of spin vectors in the Bloch frame to rotate relative to the RF-frame by
generally different angles $\varphi_{f}=\int_{\tau_{f}}\\!\\!dt\,\Delta(t)$
and $\varphi_{b}=\int_{\tau_{b}}\\!\\!dt\,\Delta(t)$, during the forward and
backward evolution times, respectively, even for $\tau_{b}=\tau_{f}$. For
measurements of spin components in the RF frame, the final state of the cloud
can be written as $|\psi_{f}\rangle=e^{-i\pi/2\,S_{y}}|\psi_{f1}\rangle$,
where $|\psi_{f1}\rangle$ is the state just prior to the final $(\pi/2)_{y}$
pulse. Taking $\tau_{f}=\tau_{b}=\tau$, we find Sup

$|\psi_{f1}\rangle=e^{-i\pi
S_{y}}e^{i(\varphi_{b}-\varphi_{f})S_{z}}\,W_{\phi}(\varphi_{f},\tau)|\psi_{0x}\rangle,$
(2)

where $|\psi_{0x}\rangle=e^{-i\pi/2\,S_{y}}|\psi_{0z}\rangle$ is the fully
$x$-polarized state and

$W_{\phi}(\varphi_{f},\tau)=e^{\frac{i}{\hbar}H_{0}(a)\tau}e^{-i\,\phi_{x}S_{x}(\varphi_{f})}e^{-\frac{i}{\hbar}H_{0}(a)\tau}.$
(3)

Here, $S_{x}(\varphi_{f})=S_{x}\cos\varphi_{f}-S_{y}\sin\varphi_{f}$ with
$S_{x}$ and $S_{y}$ the $x$\- and $y$-components of the total spin vector in
the RF frame. $H_{0}(a)$ is defined by Eq. 1 for $\Delta(t)=0$.

For each shot, the operator $s_{zi}$ is measured for an ensemble of atoms in a
selected energy group $E_{i}\in[E,E+\Delta E]$. The energy resolution $\Delta
E$ of the inverse Abel-transform method is small enough that all of the atoms
in the energy group evolve identically over the time scale of the pulse
sequence. A single-shot measurement of the spin density profile then yields
the ensemble-averages,
$\langle\psi_{f}|s_{zi}|\psi_{f}\rangle\equiv\langle\psi_{0x}|\tilde{s}^{\prime}_{xi}|\psi_{0x}\rangle$.
Here,

$\tilde{s}^{\prime}_{xi}=\cos\varphi_{fb}\,\tilde{s}_{xi}-\sin\varphi_{fb}\,\tilde{s}_{yi},$
(4)

with $\tilde{s}^{\prime}_{xi}$ being the $x$-component of the spin vector
operator relative to the RF frame just before the final $(\pi/2)_{y}$ pulse as
shown in Fig. 3. $\tilde{s}^{\prime}_{xi}$ is given in terms of the components
in the Bloch frame, $\tilde{s}_{xi}\equiv
W^{\dagger}_{\phi}(\varphi_{f},\tau)s_{xi}W_{\phi}(\varphi_{f},\tau)$ and
similarly for $\tilde{s}_{yi}$. For each measurement, we see that the
difference between the backward and forward the phase shifts,
$\varphi_{f}-\varphi_{b}\equiv\varphi_{fb}$, determines the relative
contribution of the $\tilde{s}_{xi}$ and $\tilde{s}_{yi}$ spin components in
the Bloch frame to the measured projection in the RF frame,
$\tilde{s}^{\prime}_{xi}$. In addition, Eq. 3 shows that the forward phase
shift $\varphi_{f}$ determines the effective rotation axis for the $\phi_{x}$
pulse.

To predict the measured $\langle\psi_{f}|s_{zi}|\psi_{f}\rangle$, we employ a
mean-field approximation to obtain a quasi-classical model Pegahan  _et al._
(2019), where the Heisenberg equations are solved numerically by treating the
collective spin vectors as classical variables, which ignores quantum
correlations between the spin vectors for different energy groups. The
Heisenberg equations of motion for the collective spin vectors take a simple
form in energy space,
$\dot{\vec{s}}_{i}(t)=\vec{\omega}_{i}(t)\times{\vec{s}}_{i}(t)$, with

$\vec{\omega}_{i}(t)=a\\!\sum_{j\neq
i}g_{ij}\,{\vec{s}}_{j}(t)+\Omega^{\prime}E_{i}\,\hat{e}_{z}+\Delta(t)\hat{e}_{z}.$
(5)

For a given choice of the forward and backward detunings, i.e., the phases
$\varphi_{f}$ and $\varphi_{b}$, $s_{zi}$ is determined by numerical
integration. An Abel transform of $s_{zi}\equiv s_{z}(E)$ then yields the
corresponding spin density $s_{z}(x)$ Pegahan  _et al._ (2019).

Experimentally, 60 shots are taken for each set of parameters. Examples of
single-shot data are shown in Fig. 4 and in the supplement Sup . Due to the
complexity of the spatial profiles for $\phi_{x}\neq 0$, single-shot data
analysis is essential for this experiment. Small variation ($\leq 5\%$) in
cloud parameters results in shifted spatial profiles, even for fixed
$\varphi_{f}$ and $\varphi_{b}$, so averaging over shots with slightly
different initial conditions can wash out the fine structure. Fig. 4 compares
two quasi-classical models with the single-shot data (blue dots). The first
model, adopted from Ref. Pegahan  _et al._ (2021), assumes
$\varphi_{f}\equiv\varphi_{b}$ mod $2\pi$, and the fits (black-dashed curves)
to the data in Fig. 4 (a,e) and (b,f) for $\tau=200$ ms and $a=5.2\,a_{0}$
require a fitted scattering length of $a_{fit}=9.0\,a_{0}$ in disagreement
with the measured value. These results confirm the large discrepancy between
the data and the quasi-classical model ignoring $\varphi_{fb}$ that was
observed in our previous study of information scrambling Pegahan  _et al._
(2021). For the second, modified model, the forward and backward evolution
phases $\varphi_{f}$ and $\varphi_{b}$ are treated as two free parameters. In
this case, the model (red curves) is in good agreement with data taken for
$\phi_{x}=\pi/2$, $\pi$ and $3\pi/2$ with $\tau=200$ ms at both $5.2\,a_{0}$
and $8.0\,a_{0}$ and for $\tau=400$ ms at $5.2\,a_{0}$. Additional data with
$\phi_{x}$ in steps of $\phi_{x}=\pi/4$ were obtained to test the model
further and demonstrate equally good agreement Sup . Section IV B of the
supplement explains the sources of minor defects observed in the data series
for $8.0\,a_{0},~{}\tau=200$ ms and $5.2\,a_{0},~{}\tau=400$ ms.

It is observed that, for the small scattering length, $a=5.2\,a_{0}$, and
short forward evolution time $\tau=200$ ms, the data can be fitted using
$\varphi_{f}$ and $\varphi_{b}$ as two free parameters (red curves) or by
using $\varphi_{f}=\varphi_{b}$ as one parameter and the scattering length $a$
as another free parameter (black-dashed curves). However, for the long forward
evolution time of 400 ms or for the large scattering length of $8.0\,a_{0}$,
the data cannot be fitted for any scattering length with the assumption of
$\varphi_{b}=\varphi_{f}$. In contrast, the modified model reported in this
work, which includes forward and backward evolution phases as separate
parameters, fits the data very well using the measured scattering length.

The modified model explicitly shows the difficulty of multi-shot averaged
measurements of transverse spin components, such as $s_{x}$, where the
averages of $\cos\varphi_{fb}$ and $\sin\varphi_{fb}$ in Eq. 4 tend to vanish.
Previously, the imperfect phase control problem was partially circumvented by
using a maximum likelihood estimation Smale  _et al._ (2019). However, Eq. 4,
which is valid for both quasi-classical and full quantum treatments, suggests
that multi-shot averaged measurements of energy-space spin operator products,
such as
$\langle\psi_{f}|s_{zi}s_{zj}|\psi_{f}\rangle=\langle\psi_{0x}|\tilde{s}^{\prime}_{xi}\tilde{s}^{\prime}_{xj}|\psi_{0x}\rangle$,
are important, since the random-phase averages of $\cos^{2}\varphi_{fb}$ and
$\sin^{2}\varphi_{fb}$ are nonzero. This method enables improved out-of-time-
order correlation measurements in quantum gases, where the $W$ operator is
unchanged and the operator $V=s_{xi}$ is replaced with $V=s_{xi}s_{xj}$, since
the initial x-polarized state is an eigenstate of both operators Gärttner  _et
al._ (2017, 2018); Pegahan  _et al._ (2021).

In summary, this work verifies that a modified quasi-classical spin vector
model of weakly interacting Fermi gases explains perturbed quantum rewinding
experiments, using measurements of single-shot spin density profiles with
sufficient resolution to enable quantitative study. The modified model
reported here elucidates the effects of uncontrolled forward and backward
evolution phases, $\varphi_{f}$ and $\varphi_{b}$, on the system and
measurements, resolving an outstanding conflict with a previous treatment
Pegahan  _et al._ (2021). Our results suggest new correlation analysis methods
based on energy-resolved operator products, which yield signals that are
independent of the uncontrolled RF detuning without assuming phase
distributions Smale  _et al._ (2019). Applying such methods to measure the
time dependence of correlations between transverse components
$\langle\psi_{0x}|\tilde{s}_{\perp i}\cdot\tilde{s}_{\perp
j}|\psi_{0x}\rangle$ allows the study of entanglement development in a large
system Jurcevic  _et al._ (2014) and investigations of many-body dynamics and
information propagationHauke and Tagliacozzo (2013). Such experiments will be
a topic of future work.

Primary support for this research is provided by the Air Force Office of
Scientific Research (FA9550-16-1-0378). Additional support is provided by the
National Science Foundation (PHY-2006234).

∗Corresponding authors<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
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##  Appendix A Supplemental Material

This supplemental material presents the experimental and theoretical details
of the measurements and modeling of quantum rewinding in a weakly interacting
Fermi gas. A new method is introduced for calibration of the magnetic field
where the scattering length vanishes. With this precise measurement,
systematic experimental defects in the perturbed quantum rewinding experiments
are minimized. The critical role of time-dependent detuning is explained by
deriving a new collective spin vector model that properly includes it.
Finally, additional single-shot data from the perturbed quantum rewinding
experiments are presented, illustrating excellent agreement with the quasi-
classical model reported in this work.

###  A.1 Experimental Methods

For the experiments presented in this work, the sample comprises a mixture of
two lowest hyperfine states 6Li atoms, denoted
$|1\rangle\equiv|\uparrow_{z}\rangle$ and
$|2\rangle\equiv|\downarrow_{z}\rangle$, which is evaporatively cooled to
degeneracy near the $1-2$ broad Feshbach resonance at a bias magnetic field of
$832.2$ G. After the sample is prepared, $|1\rangle$ is eliminated by an
imaging pulse applied in the weakly interacting regime near $1200$ G to create
a $z$-polarized sample. The bias magnetic field is then tuned close to the
zero-crossing $527.150$ G, where the scattering length $a$ nearly vanishes. At
this field, a resonant radio-frequency(RF) $(\pi/2)_{y}$ pulse, i.e., a
rotation around the $y$-axis in the RF frame, coherently excites the spin
state $|2\rangle$, creating a $50$-$50$ superposition of states $|1\rangle$
and $|2\rangle$. In addition to the bias magnetic field, a control magnetic
field is applied along the bias field axis by a pair of auxiliary magnet
coils, which are wrapped around the primary bias magnet containers, located on
the top and bottom of the experimental vacuum chamber. The auxiliary coils
enable magnetic field control of the scattering length in the zero crossing
region.

With the coherent superposition state created, the trapped cloud is
$x$-polarized and evolves at the chosen scattering length $a$ for a selected
evolution time $t_{fk}$, after which the spatial profiles of both hyperfine
components are measured. Spin wave formation leads to spin segregation, where
the maxima in the $|1\rangle$ and $|2\rangle$ densities are spatially
separated Du  _et al._ (2008). This initial evolution is defined as “forward”
evolution. To implement ”backward” evolution, a $\pi_{y}$ pulse is applied
after a forward evolution time $\tau_{f}$, interchanging the two spin states,
and the auxiliary magnetic field is swept down over 10 ms to flip the
scattering length from $a$ to $-a$. Ideally, from this point, the sign of the
Hamiltonian is inverted, which is equivalent to letting the system evolve
backward in time. After a backward evolution time $\tau_{b}=\tau_{f}$, the
system is expected to evolve back to its unsegregated $x$-polarized state.

####  A.1.1 Quantum Rewinding Measurements

The system status is observed for a number of different forward evolution
times $t_{fk}$ and backward evolution times $t_{bk}$, by imaging the density
of both spins, using two resonant optical pulses, separated in time by
$10\,\mu$s. Spin density spatial profiles are extracted by subtracting
normalized spatial profiles for the two states and dividing by 2:

$S_{z}(x)\equiv\frac{1}{2}\left(\frac{n_{1}(x)-n_{2}(x)}{n_{1}(0)+n_{2}(0)}\right)=\frac{1}{2}\frac{\Delta
n(x)}{n_{tot}(0)}.$ (S1)

Spin segregation is quantified by measuring the central spin density
$S_{z}(x=0)=\frac{1}{2}(\frac{\Delta n(0)}{n_{tot}(0)})$ at the center of the
cloud. The larger the absolute value of the central spin density is, the more
segregated the system is: For the unsegregated system, i.e., immediately after
the coherent excitation RF $(\pi/2)_{y}$ pulse, the central spin density is
zero in theory.

Perturbed quantum rewinding experiments are done by adding a perturbing
$\phi_{x}$ pulse before the $\pi_{y}$ pulse. This pulse is generated by
connecting a voltage-controlled phase shifter in series with the output of the
RF generator, so that a $\phi_{y}$ pulse can be either unshifted in phase or
phase shifted by $90^{\circ}$ to obtain a $\phi_{x}$ pulse. Immediately after
the perturbing $\phi_{x}$ pulse, the sign of the Hamiltonian is reversed, as
described in the previous paragraph, and the system evolves backward for a
time period of $\tau_{b}=\tau_{f}$. Just before the final $(\pi/2)_{y}$ pulse,
the magnetic field is swept with the auxiliary coils to give the original
scattering length $a$, and the final $(\pi/2)_{y}$ pulse is applied to observe
the transverse components of the spin vectors.

####  A.1.2 Detuning

Ideally, during the experimental cycle, for zero detuning, the Bloch frame
overlaps with the RF frame, which means that the $\phi_{x}$, $\pi_{y}$ and the
last $(\pi/2)_{y}$ rotations in the RF frame are also done about the x-axis or
y-axis in the Bloch frame. However, there are uncontrolled time-dependent
global detunings $\Delta(t)$, producing relative angles between the RF and
Bloch frames during the evolution periods $\tau_{b}$ and $\tau_{f}$.
Experimentally, all rotations are done in the RF frame, which is defined by
the RF generator. At a forward evolution magnetic field $B_{f}$, which is near
resonance, $\Delta\simeq 0$. However, as the magnetic field is swept down to
$B_{b}$ during an experimental cycle, the detuning changes by up to several
kHz. This results in a large phase difference and corresponding angle between
the two frames, which is imperfectly controlled due to field fluctuations. As
described in detail in § A.2, the phase shift accumulated during the forward
evolution period, $\varphi_{f}$, controls the effective rotation axis of the
perturbation relative to the Bloch vector. In addition, the difference between
$\varphi_{f}$ and the phase accumulated during the backward evolution period,
$\varphi_{b}$, determines the measured spin components.

For Hamiltonian reversal experiments, when only the $z$-component of the spin
vectors is measured, there is no sensitivity to the detuning, because the
detuning is equivalent to a rotation about the $z$-axis. In contrast, for
perturbed quantum rewinding experiments, where the transverse components of
the spin vectors are measured, understanding the roles of $\varphi_{f}$ and
$\varphi_{b}$ is critical for correct data analysis and comparison with
predictions. Although the detuning is not controlled, the experimental results
of this work show that complex single-shot data are surprisingly well fitted
by the model described below, where the different accumulated phase shifts for
the two evolution periods, $\varphi_{f}$ and $\varphi_{b}$, are properly
included.

###  A.2 Collective Spin Vector Evolution Model

To understand how the time-dependent global detuning $\Delta(t)$ affects the
perturbed quantum rewinding measurements, we derive the final state of the
system after the pulse sequence of Fig. 2 of the main text, which is
reproduced here for convenience, Fig. S1. Prior to the pulse sequence, the
optically trapped atoms are initially prepared in a $z$-polarized state,

$|\psi_{0z}\rangle=\Pi_{i}\,|\\!\\!\uparrow_{zi}\rangle.$ (S2)

The system runs forward for a time $\tau_{f}$ and backward for $\tau_{b}$,
after which the spatial profiles of the $|\uparrow_{z}\rangle$ and
$|\downarrow_{z}\rangle$ states are measured. Note that the pulse durations
for the $y$\- and $x$\- axis rotations are $<<\tau_{f},\,\tau_{b}$. In between
the forward and backward evolutions, a perturbing pulse is applied, which
rotates the system about the $x$-axis by an angle $\phi_{x}$.

Figure S1: Pulse sequence for perturbed quantum rewinding.

In Fig. S1, the $x$ and $y$ axes are defined in the “RF frame,” where the
$x$-axis of the RF frame is defined to rotate about the $z$-axis at the
instantaneous frequency of the RF generator. The $x$-axis, therefore, tracks
the total phase of the RF field. The detuning $\Delta(t)$ is defined as the
difference between the instantaneous hyperfine resonance frequency for an atom
at rest and the instantaneous radiofrequency. When the hyperfine frequency is
larger than the radiofrequency, spin vectors will appear to rotate
counterclockwise as seen looking down the z-axis from above, i.e., through a
positive angle, relative to the RF frame. In the experiments, changes in the
applied bias magnetic field, as used to reverse the scattering length, and
uncontrolled magnetic field fluctuations, tune the hyperfine frequency at a
rate of $\simeq 5$ kHz/G. The detuning causes the components of spin vectors
in the Bloch frame to rotate relative to the RF-frame by generally different
angles, during the forward and backward evolution times, respectively, even
for $\tau_{b}=\tau_{f}=\tau$ as used in the experiments. We define the forward
and backward phase shifts,

$\varphi_{f}=\int_{\tau_{f}}\\!\\!dt\,\Delta(t)\hskip 18.06749pt{\rm
and}\hskip 18.06749pt\varphi_{b}=\int_{\tau_{b}}\\!\\!dt\,\Delta(t).$ (S3)

To find the final state including the global detuning, we write the
Hamiltonian of Eq. 1 of the main text in the general form

$\frac{H(a)}{\hbar}=\frac{H_{0}(a)}{\hbar}+\Delta(t)S_{z},$ (S4)

where $S_{z}$ is the $z$-component of the dimensionless total spin vector.
Here, the time-independent part of the Hamiltonian, for $\Delta=0$, is defined
by

$\frac{H_{0}(a)}{\hbar}=a\\!\sum_{i,j\neq
i}\\!g_{ij}\,{\vec{s}}_{i}\cdot{\vec{s}}_{j}+\sum_{i}\Omega^{\prime}E_{i}\,s_{zi}$
(S5)

and

$[H_{0}(a),S_{z}]=0.$ (S6)

Referring to Fig. S1, for measurements of spin components in the RF frame, we
see that the final state of the cloud for $\tau_{f}=\tau_{b}=\tau$ is

$|\psi_{f}\rangle=e^{-i\pi/2\,S_{y}}e^{-\frac{i}{\hbar}\,H_{0}(-a)\tau-i\varphi_{b}S_{z}}e^{-i\pi\,S_{y}}e^{-i\,\phi_{x}S_{x}}e^{-\frac{i}{\hbar}\,H_{0}(a)\tau-i\varphi_{f}S_{z}}e^{-i\pi/2\,S_{y}}|\psi_{0z}\rangle.$
(S7)

Eq. S7 is readily simplified using

$e^{-\frac{i}{\hbar}\,H_{0}(-a)\tau-i\varphi_{b}S_{z}}e^{-i\pi\,S_{y}}=e^{-i\pi\,S_{y}}\left[e^{i\pi\,S_{y}}e^{-\frac{i}{\hbar}\,H_{0}(-a)\tau-i\varphi_{b}S_{z}}e^{-i\pi\,S_{y}}\right].$

Using Eq. S5 and noting that the $(\pi)_{y}$ rotation inverts $S_{z}$, we see
that

$e^{i\pi\,S_{y}}\left[\frac{H_{0}(-a)}{\hbar}\tau+\varphi_{b}\,S_{z}\right]e^{-i\pi\,S_{y}}=-\frac{H_{0}(a)}{\hbar}\tau-\varphi_{b}\,S_{z}.$

With Eq. S6, we obtain

$|\psi_{f}\rangle=e^{-i3\pi/2\,S_{y}}e^{+i\varphi_{b}S_{z}}e^{+\frac{i}{\hbar}\,H_{0}(a)\tau}e^{-i\,\phi_{x}S_{x}}e^{-i\varphi_{f}S_{z}}e^{-\frac{i}{\hbar}\,H_{0}(a)\tau}e^{-i\pi/2\,S_{y}}|\psi_{0z}\rangle.$
(S8)

Now,

$e^{-i\,\phi_{x}S_{x}}e^{-i\varphi_{f}S_{z}}=e^{-i\varphi_{f}S_{z}}\left[e^{+i\varphi_{f}S_{z}}e^{-i\,\phi_{x}S_{x}}e^{-i\varphi_{f}S_{z}}\right].$

It is easy to show that

$S_{x}(\varphi_{f})\equiv
e^{+i\varphi_{f}S_{z}}S_{x}\,e^{-i\varphi_{f}S_{z}}=S_{x}\cos\varphi_{f}-S_{y}\sin\varphi_{f},$
(S9)

which follows from $S_{x}^{\prime\prime}(\varphi_{f})=-S_{x}(\varphi_{f})$ and
the initial conditions $S_{x}(0)=S_{x}$ and $S_{x}^{\prime}(0)=-S_{y}$. Then

$e^{-i\,\phi_{x}S_{x}}e^{-i\varphi_{f}S_{z}}=e^{-i\varphi_{f}S_{z}}e^{-i\phi_{x}S_{x}(\varphi_{f})}.$

Again using Eq. S6, we then obtain

$|\psi_{f}\rangle=e^{-i3\pi/2\,S_{y}}e^{+i(\varphi_{b}-\varphi_{f})S_{z}}e^{+\frac{i}{\hbar}\,H_{0}(a)\tau}e^{-i\,\phi_{x}S_{x}(\varphi_{f})}e^{-\frac{i}{\hbar}\,H_{0}(a)\tau}e^{-i\pi/2\,S_{y}}|\psi_{0z}\rangle.$
(S10)

Defining the operator

$W_{\phi}(\varphi_{f},\tau)=e^{\frac{i}{\hbar}H_{0}(a)\tau}e^{-i\,\phi_{x}S_{x}(\varphi_{f})}e^{-\frac{i}{\hbar}H_{0}(a)\tau},$
(S11)

and the $x$-polarized state just after the first $(\pi/2)_{y}$ rotation,

$|\psi_{0x}\rangle=e^{-i\pi/2\,S_{y}}|\psi_{0z}\rangle$ (S12)

we obtain the final state in the simple form,

$|\psi_{f}\rangle=e^{-i3\pi/2\,S_{y}}e^{+i(\varphi_{b}-\varphi_{f})S_{z}}W_{\phi}(\varphi_{f},\tau)|\psi_{0x}\rangle.$
(S13)

As explained in the main text, in a single shot, we can measure the operator
$s_{zi}$ for the ensemble of atoms in the ith energy group. Noting that
$e^{+i3\pi/2\,S_{y}}s_{zi}e^{-i3\pi/2\,S_{y}}=+s_{xi}$, we find

$\langle\psi_{f}|s_{zi}|\psi_{f}\rangle=\langle\psi_{0x}|W^{\dagger}_{\phi}(\varphi_{f},\tau)e^{-i(\varphi_{b}-\varphi_{f})S_{z}}s_{xi}\,e^{i(\varphi_{b}-\varphi_{f})S_{z}}W^{\dagger}_{\phi}(\varphi_{f},\tau)|\psi_{0x}\rangle.$
(S14)

By using Eq. S9 with
$\varphi_{f}\rightarrow\varphi_{fb}\equiv\varphi_{f}-\varphi_{b}$,
$S_{x}\rightarrow s_{xi}$ and $S_{y}\rightarrow s_{yi}$, we see that

$e^{i(\varphi_{f}-\varphi_{b})S_{z}}s_{xi}\,e^{-i(\varphi_{f}-\varphi_{b})S_{z}}=s_{xi}\cos\varphi_{fb}-s_{yi}\sin\varphi_{fb}.$

Then,

$\langle\psi_{f}|s_{zi}|\psi_{f}\rangle=\langle\psi_{0x}|W^{\dagger}_{\phi}(\varphi_{f},\tau)s_{xi}W^{\dagger}_{\phi}(\varphi_{f},\tau)|\psi_{0x}\rangle\cos\varphi_{fb}-\langle\psi_{0x}|W^{\dagger}_{\phi}(\varphi_{f},\tau)s_{yi}W^{\dagger}_{\phi}(\varphi_{f},\tau)|\psi_{0x}\rangle\sin\varphi_{fb}$
(S15)

Eq. S15 shows that a single-shot measurement of the spin density profile then
yields, via inverse-Abel transformation Pegahan  _et al._ (2021), the
ensemble-averages,

$\langle\psi_{f}|s_{zi}|\psi_{f}\rangle\equiv\langle\psi_{0x}|\tilde{s}^{\prime}_{xi}|\psi_{0x}\rangle,$
(S16)

where

$\tilde{s}^{\prime}_{xi}=\cos\varphi_{fb}\,\tilde{s}_{xi}-\sin\varphi_{fb}\,\tilde{s}_{yi}.$
(S17)

Here,

$\tilde{s}_{xi}\equiv
W^{\dagger}_{\phi}(\varphi_{f},\tau)s_{xi}W_{\phi}(\varphi_{f},\tau)$ (S18)

and similarly for $\tilde{s}_{yi}$, reproducing the results given in the main
text.

We see that for each measurement, the difference between the forward and
backward phase shifts, $\varphi_{f}-\varphi_{b}\equiv\varphi_{fb}$, determines
the relative contribution of the $\tilde{s}_{xi}$ and $\tilde{s}_{yi}$ spin
components in the Bloch frame to the measured projection in the RF frame,
$\tilde{s}^{\prime}_{xi}$. In addition, Eq. S11 shows that the forward phase
shift $\varphi_{f}$ determines the effective rotation axis for the $\phi_{x}$
pulse.

To compare the prediction of Eq. S16 to single-shot measurements in a system
containing a large number of spins, we employ a quasi-classical model. In this
case, we treat the Heisenberg equations for the spin-vectors $\vec{s}_{i}$ as
evolution equations for classical vectors, which neglects quantum
correlations. These equations are readily evaluated for any chosen
$\varphi_{f}$ and $\varphi_{b}$ by numerical integration, enabling fits to
single-shot data with $\varphi_{f}$ and $\varphi_{b}$ as fit parameters.

###  A.3 Quantifying Hamiltonian Reversal

This section introduces the method of quantifying the quality of rewinding by
Hamiltonian reversal. Two sets of data are required to examine the result of
the Hamiltonian reversal in the experiments. One set of data represents the
state of the system at different forward evolution times, and the other set
represents the state at corresponding backward evolution times. The first set
is taken at a magnetic field $B_{f}$ for a number of times $0\leq
t_{fk}\leq\tau_{f}$, where $t=0$ is the time of the initial coherent
excitation pulse and $\tau_{f}$ is the maximum forward evolution time. The
second set is taken at a backward evolution magnetic field $B_{b}$ at
corresponding times $t_{bk}=\tau_{f}+\tau_{bk}$, as discussed below.

The $z$-component of the spin density for the forward evolving system,
$S^{f}_{z}(x,t_{k})$, is measured at different evolution times $t_{fk}\equiv
t_{k}$ by imaging the density of both spins at a time $t_{fk}$ relative to the
time $t\equiv 0$ of the initial $(\pi/2)_{y}$ RF pulse. To enable a
determination of the reversal quality, the same quantity,
$S^{b}_{z}(x,t_{k})$, is measured for the Hamiltonian reversed system at
$t_{bk}\equiv\tau_{f}+\tau_{bk}$, where $t_{bk}$ is the total time relative to
the coherent excitation pulse. Here, $\tau_{bk}$ is the amount of time that
the system evolves backward with the reversed Hamiltonian. To match the spin
density spatial profiles for the forward evolution with the corresponding
backward evolution ones, the evolution times need to be matched, i.e.,
$\tau_{bk}=\tau_{f}-t_{fk}=\tau_{f}-t_{k}$ and $t_{bk}=2\tau_{f}-t_{k}$.
$\chi^{2}_{k}$ is defined as the normalized mean square difference between
forward and backward evolution spin density profiles for $t_{k}$,

$\chi^{2}_{k}\equiv\sum_{x}\left(\frac{S^{b}_{z}(x,t_{k})-S^{f}_{z}(x,t_{k})}{S^{f}_{z}(x,t_{k})}\right)^{2}.$
(S19)

To quantify the reversal, we employ
$\chi^{2}\equiv\langle\chi^{2}_{k}\rangle$, the average $\chi_{k}^{2}$ for all
of the $t_{k}$ in the data set. Small $\chi^{2}$ means that the forward and
backward $S_{z}(x)$ profiles overlap very well, which corresponds to a good
reversal.

####  A.3.1 Zero-crossing Measurement

Figure S2: Finding the optimum magnetic fields for reversing the spin
evolution at two initial magnetic fields. (a)(b) are done at the forward
evolution field $B_{f}=528.803$ G ($a\approx 5\,a_{0}$), where the resonant RF
frequency is $RF_{f}=75608.85$ kHz, with the sign of Hamiltonian inverted
after forward evolution time $\tau_{f}=400$ ms; (c)(d) are done at
$B_{f}=529.713$ G ($a\approx 8\,a_{0}$), where the resonant RF frequency is
$RF_{f}=75612.10$ kHz, with the sign of Hamiltonian inverted after
$\tau_{f}=240$ ms. (a)(c) quantify the reversibility with different reversal
magnetic fields $B_{b}$. Blue crosses are calculated $\chi^{2}$ results for
forward and backward evolution spin density spatial profiles $S_{z}(x)$. Red
curves are parabolas fitted to $\chi^{2}$ as a function of reversal magnetic
field $B_{b}$. The parabola fits suggest that the optimum reversal field for
$B_{f}=528.803$ G is $B_{b,opt}=525.488$ G and the optimum reversal field for
$B_{f}=529.713$ G is $B_{b,opt}=524.596$ G. (b)(d) show the central spin
density $\Delta n(0)/n_{tot}(0)$ at different evolution times for both forward
(red circle) and backward (blue triangles) evolution data. Darker blue
corresponds to higher resonant frequency $RF_{b}$ at $B_{b}$, which means a
higher magnetic field.

A critical constant for the implementation of any experiments involving
quantum rewinding is the zero-crossing magnetic field $B_{0}$ where the
scattering length vanishes. Careful calibration of this field ensures that the
Hamiltonian reversal in the experimental sequence is done properly. In this
work, $B_{0}$ is precisely measured by quantifying Hamiltonian reversal for
two different forward evolution magnetic fields $B_{f}=528.803$ G and
$529.713$ G, where the scattering lengths are $\approx 5\,a_{0}$ and $\approx
8\,a_{0}$ respectively. Data is taken for five slightly different backward
evolution magnetic fields $B_{b}$ near the zero crossing for each $B_{f}$.

The magnitudes of all of the magnetic fields are measured precisely using RF
spectroscopy, by applying a $\pi$ pulse (15 ms) with a known RF frequency. The
resonance frequencies of the RF pulse for the atomic transition are in one-to-
one correspondence with the magnetic fields. With this property, the magnetic
field can be calculated with mG precision from the resonance RF frequency for
a $\pi$ pulse that fully transfers atoms from $|2\rangle$ to $|1\rangle$.

By fitting a parabola to $\chi^{2}$ as defined above for five different
$B_{b}$, the optimum reversal magnetic $B_{b,opt}$ is obtained for the
corresponding $B_{f}$. The zero-crossing magnetic field is located at the
midpoint between $B_{f}$ and $B_{b,opt}$. Fig. S2 displays the results of this
measurement. Because the measurement for this experiment is insensitive to the
detuning as described in the main paper, averaging is allowed. Each data point
is the result after averaging 5 shots. The top two figures (a) and (b) are the
result of data series taken for $a=\pm 5.2~{}a_{0}$ with Hamiltonian reversal
done at $\tau_{f}=400$ ms, and the bottom two (c) and (d) are the results for
$a=\pm 8.0~{}a_{0}$ with Hamiltonian reversal done at $\tau_{f}=240$ ms. The
parabolic fit for $\chi^{2}$ with $a=\pm 5.2~{}a_{0}$ series suggests that
$B_{b,opt}=525.488$ G for $B_{f}=528.803$ G, and for $\pm 8.0~{}a_{0}$
suggests that $B_{b,opt}=524.596$ G for $B_{f}=529.713$ G. The two series of
experiments yield the result $B_{0}=527.150(5)$ G, which is $30$ mG lower than
the previous result Pegahan  _et al._ (2019). Note that the scattering lengths
”$5.2\,a_{0}$” and ”$8.0\,a_{0}$” are calculated based on the zero-crossing
magnetic field measured in the above experiment.

####  A.3.2 Reversibility in Different Regimes

This technique of quantifying the reversal quality can also be used to compare
the reversibility for systems in different regimes of scattering length and
forward evolution time. This is achieved by weighting the disagreement between
forward and backward in spatial profiles differently for different $t_{k}$: By
construction, the disagreement is weighted more heavily for small $t_{k}$,
where the system has evolved backward long enough for discrepancies between
backward and forward evolution to appear. For small $t_{k}$,
$S^{f}_{z}(x,t_{k})$ is usually small, i.e., close to a horizontal line around
$x$-axis, because the system has not segregated very much for short forward
evolution times. Hence, the denominator in Eqn. S19 exaggerates the magnitude
of $\chi^{2}_{k}$ for small $t_{fk}$.

For the perturbed quantum rewinding experiments, the evolution times are
always chosen to be $\tau_{b}=\tau_{f}$, which means that the reversal quality
at $t_{k}=0$ ms is extremely critical. Hence, $\chi^{2}_{k}$ at $t_{k}=0$ ms
needs to be weighted highly in the test of reversibility, especially for the
purpose of choosing regimes to do perturbed quantum rewinding experiments.

Note that the $\chi^{2}$ in Fig. S2(a) is one order of magnitude larger than
it is in (c), which means that the system reverses more perfectly with
$a=8.0~{}a_{0}$ and $\tau_{f}=240$ ms than with $a=5.2~{}a_{0}$ and
$\tau_{f}=400$ ms. This matches the comparisons of the central spin density
evolutions shown in Fig. S2 (b)(d): For the $a=5.2~{}a_{0}$ and $\tau_{f}=400$
ms data series, there is a clear sign of segregation ($\Delta
n(0)/n_{tot}(0)>0$) for Hamiltonian reversed data at $t_{k}=0$ ms even if the
optimum reversal magnetic field is adopted. In contrast, for $a=8.0~{}a_{0}$
and $\tau_{f}=240$ ms, an almost perfect reversal is observed at $t_{k}=0$ ms.

With this method of quantifying quantum rewinding, a systematic study of the
reversibility of a system in different regimes can be done by fixing
$\tau_{f}$ and varying the scattering length and by fixing the scattering
length and varying $\tau_{f}$. This study is ongoing and requires a large
amount of data, and so will not be pursued further here.

###  A.4 Testing the Quasi-Classical Spin Model

To test the quasi-classical spin model, as reported in this work, three series
of perturbed quantum rewinding experiments are performed at: $5.2\,a_{0}$ with
$\tau=200$ ms, $8.0\,a_{0}$ with $\tau=200$ ms, and $5.2\,a_{0}$ with
$\tau=400$ ms. The $S_{z}(x)$ profiles are measured as described in § A.1. All
spatial profiles shown in this work are folded over the center of the cloud,
$x=0$, followed by equal-width binning into 50 bins. To extract energy-space
information, $S_{z}(E)$, Abel inversion is applied to the spatial profiles
with 16 expansion terms Pretzier (1991). As energy-space profiles are less
sensitive to experimental defects shown in Fig. S4, the model is fitted to the
$S_{z}(E)$ extracted from single-shot data with $\varphi_{f}$ and
$\varphi_{b}$ as free parameters. Data from the three experimental series are
in good agreement with the model.

####  A.4.1 Primary Data for $a=5.2~{}a_{0}$ and $\tau=200$ ms

Figure S3: Additional single-shot spin density profiles for perturbed quantum
reversal experiment at $5.2~{}a_{0}$, $\tau=200$ ms, with different
$\phi_{x}$. Blue dots in (a)(b)(c)(d) are measured single shot $S_{z}(x)$ and
those in (e)(f)(g)(h) are Abel inversion of (a)(b)(c)(d), which give
$S_{z}(E)$. Red curves are model fitting data with $\varphi_{f}$ and
$\varphi_{b}$ as two free parameters. Perturbation rotation angle about the
x-axis $\phi_{x}$ is: $\pi/4$ in (a)(f), $3\pi/4$ in (b)(g), $5\pi/4$ in
(c)(h), $3\pi/2$ in (d)(i), $7\pi/4$ in (e)(j). $\phi_{x}=\pi/2$ and $\pi$
examples are shown in the main paper, Fig. 4.

Perturbed quantum rewinding measurements are employed to precisely test the
quasi-classical spin model presented in § A.2. Data is mainly taken at
$5.2~{}a_{0}$ with $\tau=200$ ms, as this set of parameters is expected to
provide an almost perfect Hamiltonian reversal. Nine different $\phi_{x}$
values are used as a perturbation to the reversal, ranging from $0$ to $2\pi$
in steps of $\pi/4$. Fig. S3 shows examples of single shot data taken for
different $\phi_{x}$. Note that in these figures $\varphi_{f}$ and
$\varphi_{b}$ are not necessarily the same across the whole data set because
of fluctuations in RF detuning as discussed in § A.2. The systematic
measurements and fits done for the perturbed quantum rewinding experiments
show that the complicated structure observed in the spatial profiles is very
sensitive to the initial conditions (cloud size $\sigma_{TF}$ and atom number
$N$), as well as the RF detunings for two evolution periods $\tau_{f}$ and
$\tau_{b}$. Even small variations in these parameters result in a slightly
shifted/skewed spatial profiles. Hence, for the purpose of quantitative tests
of the quasi-classical spin model using perturbed quantum rewinding
experiments, as presented in this work, single-shot analysis is essential for
processing data with imperfectly controlled experimental parameters, which
tends to wash out the fine structure in the spatial profiles. The measured
single-shot profiles presented in this work have adequate spatial resolution
to capture small details in the profiles. Having minimized experimental
defects by careful calibrations, the measured single-shot data provide
stringent tests of predictions based on the model of § A.2.

####  A.4.2 Additional Data Sets

In this section, we present additional data for increased scattering length
and evolution time. The validity of the modified quasi-classical spin model of
§ A.2 is demonstrated for the primary data set with $a=5.2~{}a_{0}$ and
$\tau=200$ ms in § A.4.1. To test the model further, a series of additional
perturbed quantum rewinding experiments are done with $\tau=200$ ms and
$8.0~{}a_{0}$ and with $\tau=400$ ms and $a=5.2~{}a_{0}$ for three $\phi_{x}$
values: $\pi/2$, $\pi$ and $3\pi/2$. Less quantitative agreement between the
model and data is observed in many of the single shots from these additional
experiments, especially in the spatial profiles.

The disagreement appears from differences between the spin imbalances in the
model and in the data. In Fig. S4, the basic model (red curves) assumes that
all of the applied RF pulses are on resonance, while drifts in the detuning
alter the pulse area. It is clear that the basic model still captures the
shape and oscillations of the data (blue dots), but with an offset. In the
experiments with longer experimental cycles or larger magnetic field sweeps,
there is a larger probability that one or more of the RF pulses are slightly
off resonance with the hyperfine frequency, since the RF frequency is fixed
for all pulses, but the magnetic field can fluctuate because of the limitation
on the stability of the auxiliary coils. Imperfect RF pulses result in a
measurement with the wrong spin imbalance for the given $\phi_{x}$, compared
to ideal measurements. To include the effect of imbalance in the model, the
atom numbers for the two spin states are adjusted from $N_{1}$ and $N_{2}$ to
$N_{1}-\delta N_{tot}$ and $N_{2}+\delta N_{tot}$, with $N_{tot}=N_{1}+N_{2}$
being total atom number and $\delta$ a reasonably small ($\leq 10$%) fraction.
In our system, the total density $n_{tot}(x)=n_{1}(x)+n_{2}(x)$ is invariant
during the experimental cycle. Hence, to include the adjustment of spin
imbalance in the modeled spatial profile, the outcomes $n_{1}(x)$ and
$n_{2}(x)$ are first determined from the model and then scaled to
$n_{1}(x)-\delta n_{tot}(x)$ and $n_{2}(x)+\delta n_{tot}(x)$. In this way,
the total atom number and density profile remain the same in the model output
before and after the adjustment. With this adjustment to the outcome of the
regular model, the improved fits shown as the green curves in Fig. S4 are
obtained.

Another reason for the disagreement between the model and the data set with
$\tau=400$ ms at $a=5.2~{}a_{0}$ is the imperfect Hamiltonian reversal shown
in Section A.3.1. The unperturbed quantum rewinding experiment done with this
set of experimental parameters suggests that the system is not precisely
reversed in this regime. Therefore, it is reasonable that the perturbed
rewinding data can not be predicted by the model as quantitatively as for the
data obtained in the regime where the reversibility of the system is clearly
better.

Figure S4: Adjusting the atom number ratio for the two spin states in the
model results in a much better agreement with data. In (a)(c), blue dots are
measured single-shot spin density $S_{z}(x)$ with $8.0~{}a_{0}$, $\tau=200$
ms, $\phi_{x}=3\pi/2$ and $5.2~{}a_{0}$, $\tau=400$ ms, $\phi_{x}=\pi/2$
respectively. Red curves are generated by the model assuming a perfectly
stable $B$-field, so that all RF pulses are on resonance. Green curves are
obtained by adjusting the spin imbalance of the two spin states as described
in the text: $n_{1}(x)\rightarrow n_{1}(x)-\delta n_{tot}(x)$ and
$n_{2}\rightarrow n_{2}(x)+\delta n_{tot}(x)$, with $n_{1}(x)$ and $n_{2}(x)$
from the red curve model. $\delta=10\%$ for (a) and $\delta=5\%$ for (c).
(b)(d) are Abel inversion of (a)(c) with 16 cosine terms, with color coding
the same as (a)(c).