# Many-body quantum chaos in stroboscopically-driven cold atoms

Ceren B. Dağ<EMAIL_ADDRESS>ITAMP, Center for Astrophysics $|$
Harvard $\&$ Smithsonian Cambridge, Massachusetts 02138, USA  Department of
Physics, Harvard University, Cambridge, Massachusetts 02138, USA  S. I.
Mistakidis  ITAMP, Center for Astrophysics $|$ Harvard $\&$ Smithsonian
Cambridge, Massachusetts 02138, USA  Department of Physics, Harvard
University, Cambridge, Massachusetts 02138, USA  Amos Chan  Department of
Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom  Princeton
Center for Theoretical Science, Princeton University, Princeton NJ 08544, USA
H. R. Sadeghpour  ITAMP, Center for Astrophysics $|$ Harvard $\&$ Smithsonian
Cambridge, Massachusetts 02138, USA

###### Abstract

In quantum chaotic systems, the spectral form factor (SFF), defined as the
Fourier transform of the two-level spectral correlation function, is known to
follow random matrix theory (RMT), namely a “ramp” followed by a “plateau” in
sufficiently late times. Recently, a generic early-time deviation from the RMT
behavior, which we call the “bump”, was shown to exist in random quantum
circuits and spin chains as toy models for many-body quantum chaotic systems.
Here we demonstrate the existence of the “bump-ramp-plateau” behavior in the
SFF for a number of paradigmatic and stroboscopically-driven 1D cold atom
models: (i) Bose-Hubbard model, (ii) spin$-1/2$ Bose-Hubbard model, and (iii)
nonintegrable spin-$1$ condensate with contact or dipolar interactions. We
find that the scaling of the many-body Thouless time $t_{\mathrm{Th}}$ — the
onset of RMT —, and the bump amplitude are more sensitive to variations in
atom number than the lattice size regardless of the hyperfine structure, the
symmetry classes, or the choice of driving protocol. Moreover,
$t_{\mathrm{Th}}$ scaling and the increase of the bump amplitude in atom
number are significantly slower in spinor gases than interacting bosons in 1D
optical lattices, demonstrating the role of locality. We obtain universal
scaling functions of SFF which suggest power-law behavior for the bump regime
in quantum chaotic cold-atom systems, and propose an interference measurement
protocol.

††preprint: APS/123-QED

##  I Introduction

Quantum chaos is historically diagnosed with level repulsion: A quantum system
is considered chaotic if it exhibits spectral statistics given by the random
matrix theory (RMT), in sufficiently small energy scales Bohigas  _et al._
(1984). Such signatures have been found in a plethora of disciplines including
nuclear resonance spectra Brody  _et al._ (1981); Haq  _et al._ (1982),
mesoscopic physics Beenakker (1997, 2015); Roushan  _et al._ (2017), quantum
chaos D’Alessio  _et al._ (2016); Chan  _et al._ (2018a); Kos  _et al._
(2018), black hole physics García-García and Verbaarschot (2016); Cotler  _et
al._ (2017a, b), and quantum chromodynamics Smilga (1995); Akemann (2018).
With their unprecedented degree of controllability Bloch  _et al._ (2012);
Browaeys and Lahaye (2020), trapped cold atoms Davis  _et al._ (1995);
Anderson  _et al._ (1995); Chu  _et al._ (1985); Pethick and Smith (2008);
Greiner  _et al._ (2001, 2002a); Bakr  _et al._ (2009) are excellent platforms
to study many-body phenomena, including signatures of quantum chaos in
spectral statistics, and for large-scale simulations of quantum systems Jaksch
_et al._ (1998); Weimer  _et al._ (2010); Bloch  _et al._ (2012); Browaeys and
Lahaye (2020); Greiner  _et al._ (2002b); Kinoshita  _et al._ (2006); Rigol
_et al._ (2008); Cheneau  _et al._ (2012); Tarruell  _et al._ (2012); Islam
_et al._ (2015); Schreiber  _et al._ (2015); Kaufman  _et al._ (2016); Bernien
_et al._ (2017); Mazurenko  _et al._ (2017); de Léséleuc  _et al._ (2019);
Rispoli  _et al._ (2019); Semeghini  _et al._ (2021).

A particularly intriguing objective of quantum simulations so far has been to
understand the underlying mechanisms which lead to thermalization, as in
eigenstate thermalization hypothesis (ETH) Deutsch (1991); Srednicki (1994);
Rigol  _et al._ (2008), or its absence via many-body localization (MBL) Gornyi
_et al._ (2005); Basko  _et al._ (2006); Schreiber  _et al._ (2015); Rispoli
_et al._ (2019); Nandkishore and Huse (2015), the existence of quantum many-
body scars Bernien  _et al._ (2017); Turner  _et al._ (2018); Moudgalya  _et
al._ (2018), and Hilbert space fragmentation Sala  _et al._ (2020); Khemani
_et al._ (2020). In this vein, as a probe for quantum chaos and ergodicity
Wang  _et al._ (2004); Kim and Huse (2013); Parra-Murillo  _et al._ (2014);
Hosur  _et al._ (2016); Maldacena  _et al._ (2016); Patel  _et al._ (2017); Li
_et al._ (2017); Chen  _et al._ (2016); Luitz and Bar Lev (2017); Parker  _et
al._ (2019); Dağ and Duan (2019); Murthy and Srednicki (2019); Xu and Swingle
(2020), observables such as entanglement entropy and out-of-time-order
correlators (OTOCs) have been intensively studied, and measured with quantum
simulators Islam  _et al._ (2015); Neill  _et al._ (2016); Kaufman  _et al._
(2016); Gärttner  _et al._ (2017); Li  _et al._ (2017); Wei  _et al._ (2018);
Landsman  _et al._ (2019); Meier  _et al._ (2019); Braumüller  _et al._
(2022).However, the OTOCs could be susceptible to quantum criticality and
order as found by Refs. Shen  _et al._ (2017); Heyl  _et al._ (2018); Dağ  _et
al._ (2019); Sun  _et al._ (2020), even in nonintegrable quantum many-body
models at infinite temperature Dağ  _et al._ (2020). Moreover, it was
demonstrated by Refs. Pilatowsky-Cameo  _et al._ (2020); Xu  _et al._ (2020)
that exponentially fast scrambling detected by OTOCs could be induced at the
fixed points in phase space, and is not necessarily the signature of quantum
chaos. Alternatively, the metrics of spectral statistics can be defined in the
energy domain, e.g. spacing distribution and ratio Bohigas  _et al._ (1984);
Friedrich and Wintgen (1989); Eckhardt (1988); Oganesyan and Huse (2007), as
well as number variance Lombardi and Seligman (1993); D’Alessio  _et al._
(2016). While the spacing distribution can be experimentally obtained for few
particles Haq  _et al._ (1982); Roushan  _et al._ (2017), it is particularly
challenging for many-body systems: One must access an exponentially large
number of energy levels, and resolve exponentially small many-body energy
gaps. Consequently, a direct experimental probe of level repulsion and, more
generally, of spectral correlations in a many-body spectrum remains elusive.

(a)  Figure 1:  (a) Representative behavior of the SFF $K(t)$ for the many-
body driven quantum chaotic spin-$1/2$ Bose-Hubbard system, e.g., with
$u_{\uparrow}=u_{\downarrow}=u_{\uparrow\downarrow}=J$ where
$J_{\uparrow}=J_{\downarrow}\equiv J_{\uparrow\downarrow}=J=1$ is the
characteristic energy scale. Interacting spin-$1/2$ bosons in a lattice with
unit filling factor $\nu=1$, atom number $N=4$, and lattice size $L=4$, are
subjected to a two- (red symbols) or three-step (purple symbols) driving
protocols where the driving periods are $T=1$ and $T=3$, respectively. The
two- and three-step protocols approach the respective RMT behaviors given by
COE (green dashed) and CUE (orange dotted). On the other hand, spin-$1/2$
atoms in a driven single well, $L=1$ (blue line), follows the Poisson
statistics given by the flat line (pink), $K_{\mathrm{Poi}}(t)=D$, where $D$
is the Hilbert space dimension. This is a signature of integrability. The
Thouless times $t_{\mathrm{Th}}$, denoted by stars, and the Heisenberg time
$t_{\mathrm{H}}$ determines the onset of the RMT behavior and the appearance
of the plateaus respectively. (b) $K(t)$ following the three-step driving
protocol in a spin-$1/2$ setting with $\nu=1/4$, $N=3$, $L=12$. The bump
($t<t_{\mathrm{Th}}$) is a signature of many-body quantum chaotic systems. (c)
The nearest-neighbour spacing distribution $P(\delta)$ of a spin$-1/2$ many-
body system with $N=9$ atoms trapped in $M=3$ wells and driven according to
two-step protocol with a period of $T=1$ (blue) coincides with the one for the
random matrix ensemble (red), and displays level repulsion, which is also
captured by the SFF at $t_{\mathrm{H}}$ in (a).

To circumvent this issue, we consider the spectral form factor (SFF), a time-
domain observable defined as the Fourier transform of the two-level spectral
correlation function Mehta (2004); Haake (2010). For a physical system, the
appearance of the RMT-like “ramp” and subsequent “plateau” in SFF, Fig. 1(a),
is a signature of quantum chaos. In particular, the SFF behavior around the
Heisenberg time $t_{\mathrm{H}}$ that separates ramp and plateau, is a
manifestation of the energy level repulsion, which is often probed with
reference to the Wigner-Dyson level spacing distribution Dyson (1962) (Fig.
1(c)). SFF has been instrumental in the discovery of a number of novel and
universal signatures of many-body quantum chaotic systems in both condensed
matter Bertini  _et al._ (2018); Chan  _et al._ (2018b, a); Kos  _et al._
(2018); Friedman  _et al._ (2019); Flack  _et al._ (2020); Šuntajs  _et al._
(2020); Sierant  _et al._ (2020a, b); Liao  _et al._ (2020); Roy and Prosen
(2020); Chan  _et al._ (2021); Moudgalya  _et al._ (2021); Bertini  _et al._
(2021); Garratt and Chalker (2021a); Šuntajs  _et al._ (2021); Garratt and
Chalker (2021b); Li  _et al._ (2021); Prakash  _et al._ (2021); Liao and
Galitski (2022); Chan  _et al._ (2022); Cornelius  _et al._ (2022); Šuntajs
and Vidmar (2022); Roy  _et al._ (2022); Shivam  _et al._ (2023); Bunin  _et
al._ (2022), and in high energy physics Cotler  _et al._ (2017a, b); Gharibyan
_et al._ (2018); Altland and Bagrets (2018); Hunter-Jones and Liu (2018);
Winer  _et al._ (2020); Khramtsov and Lanina (2021); Saad  _et al._ (2019);
Bousso  _et al._ (2022). Black holes have been shown to display RMT behavior
in late times with SFF Cotler  _et al._ (2017a); Saad  _et al._ (2019), and
the SFF has been utilized to investigate the ergodic-to-MBL transition in 1D
spin chains Šuntajs  _et al._ (2020); Garratt and Chalker (2021b); Sierant
_et al._ (2020a); Prakash  _et al._ (2021).

For many-body spin models and local random quantum circuits, it has been shown
that SFF could deviate from RMT. Specifically a “bump” appears at early times
$t\ll t_{\mathrm{H}}$ (see Fig. 1(a) and (b)), and RMT is recovered only after
the many-body Thouless time $t_{\mathrm{Th}}$ Chan  _et al._ (2018a); Kos  _et
al._ (2018); Friedman  _et al._ (2019); Moudgalya  _et al._ (2021); Šuntajs
_et al._ (2020); Gharibyan  _et al._ (2018); Roy  _et al._ (2022); Roy and
Prosen (2020), as illustrated by star markers in Fig. 1. The extension of such
results to cold atom systems is nontrivial, since only an approximate mapping
between them could be established for the Mott insulating regime, where
spinful atoms are confined to lattice sites Duan  _et al._ (2003). A central
result of our work is the existence of the “bump-ramp-plateau” behavior in the
SFF of stroboscopically-driven interacting (spinless and spin-$1/2$) bosons in
optical lattices beyond the Mott insulator regime, as well as spin$-1$
condensates with either short-range contact or long-range dipolar
interactions. We account for the trapped atoms’ motional degree of freedom in
addition to spin degree of freedom. To investigate the dependence of
$t_{\mathrm{Th}}$ on the locality of the underlying Hamiltonian, we employ
spatially-extended Bose-Hubbard (BH) chains Fisher  _et al._ (1989) (Fig.
2a(a)), as opposed to a chaotic spin$-1$ condensate under single-mode
approximation (SMA) Law  _et al._ (1998) (Fig. 2a(b)). The latter refers to an
ensemble of pairwise interacting spin$-1$ atoms, and hence effectively a zero-
dimensional model Sachdev and Ye (1993); Kitaev (2015); Swingle  _et al._
(2016); Gärttner  _et al._ (2017). We find that the Thouless time increases
significantly more slowly as a function of atom number in the case of chaotic
spin$-1$ condensates as compared to the lattice confined interacting bosons.
Furthermore, we reveal that both the bump amplitude and $t_{\mathrm{Th}}$
scaling are more sensitive to the atom number than the lattice size. This
strong dependence on the atom number is not affected by the atomic hyperfine
structure, e.g., single- or two-states; the symmetry class, and the choice of
driving protocol.

Our choice of quantum many-body models is also experimentally motivated. The
BH model describes the physics of cold atoms trapped in optical lattices
Jaksch  _et al._ (1998), as experimentally demonstrated in Greiner  _et al._
(2002b), while different hyperfine states of bosonic atoms can be routinely
used to realize Bose mixtures Mandel  _et al._ (2003); Gadway  _et al._
(2010). The spinor condensates constitute yet another atom-based platform to
simulate many-body physics Stamper-Kurn and Ueda (2013); Anquez  _et al._
(2016); Yang  _et al._ (2019); Evrard  _et al._ (2021). We utilize Floquet
driving, because the stroboscopically-driven setups have the advantage to
produce a uniform density of states, i.e., no unfolding Guhr  _et al._ (1998)
or filtering Gharibyan  _et al._ (2018) procedures are required for the
computation of the spectral statistics, unlike in time-independent Hamiltonian
systems. Floquet driving of cold atom systems have been realized Wintersperger
_et al._ (2020); Lellouch  _et al._ (2017); Goldman and Dalibard (2014).

We empirically obtain a power-law scaling function for the bump regime. This
suggests a power-law correction to the SFF “ramp-plateau” in cold atom
lattices. The deduced scaling function for the bump, in turn, proves to be
practically useful to statistically differentiate the bump from ramp in an
experiment. In addition, we establish a relation between the SFF and the
survival probability thus corroborating the experimental detectability of the
former. The latter has been theoretically studied to probe quantum chaos
Schiulaz  _et al._ (2019); de la Cruz  _et al._ (2020); Wittmann W.  _et al._
(2022) and measured in picosecond spectroscopy of molecular dissociation Pique
_et al._ (1987); Zewail (1993); Alekseyev  _et al._ (2000). We devise a read-
out protocol for the SFF through many-body state interference Daley  _et al._
(2012), which measures survival probability and is a state-of-the-art
measurement scheme for cold atoms in optical lattices Islam  _et al._ (2015).

The focus on SFF in driven atomic systems complements the investigations of
chaos in atomic models utilizing time-independent setups de la Cruz  _et al._
(2020); Wittmann W.  _et al._ (2022); Łydżba and Sowiński (2022), Rydberg
interactions Vasilyev  _et al._ (2020); Joshi  _et al._ (2022), highly long-
range interactions Roy  _et al._ (2022); Roy and Prosen (2020) or tools such
as dynamical fidelity Schlunk  _et al._ (2003); Wimberger and Buchleitner
(2006) and nearest-neighbor energy level statistics Wintgen and Friedrich
(1987); Buchleitner and Kolovsky (2003); Le  _et al._ (2005); Tomadin  _et
al._ (2007); Oganesyan and Huse (2007); Santos and Rigol (2010); Parra-Murillo
_et al._ (2013); Lazarides  _et al._ (2014); Dağ and Duan (2019); Rautenberg
and Gärttner (2020); Anh-Tai  _et al._ (2023). Recent studies Vasilyev  _et
al._ (2020); Joshi  _et al._ (2022) have proposed schemes to measure SFF,
numerically implement spin-1/2 chains and study the SFF in dipole-blockaded
Rydberg atoms, with an emphasis on $t_{\mathrm{H}}$ and the ramp-plateau
behavior. In this work, we argue that $t_{\mathrm{Th}}$ is not only
experimentally more accessible for large systems, because $t_{\mathrm{Th}}\ll
t_{\mathrm{H}}$ and probing $t_{\mathrm{Th}}$ does not require resolving the
many-body level spacing, but it is also physically more appealing —
$t_{\mathrm{Th}}$ and the bump regime emerge in many-body chaotic systems. The
onset of the RMT ramp, as quantified by $t_{\mathrm{Th}}$, provides us with
another physically meaningful timescale in quantum many-body systems.

This work is organized as follows. In Secs. II and III, we define the SFF and
the cold atom models under consideration, respectively. In Sec. IV, we discuss
the bump regime in bosonic quantum gases and obtain a scaling function for it.
Finally in Sec. V, we outline our read-out protocol for the SFF before
concluding in Sec. VI.

##  II Spectral Form Factor

SFF is the Fourier transform of the two-level spectral function, and can be
defined directly in the time domain as

$\displaystyle K(t)$ $\displaystyle=$
$\displaystyle\bigg{\langle}\left|\Tr[\hat{U}(t)]\right|^{2}\bigg{\rangle}=\left\langle\sum_{m,n}e^{i(E_{m}-E_{n})t}\right\rangle\,,$
(1)

where $\hat{U}(t)$ is the time evolution operator of a given system with
spectrum $\\{E_{m}\\}$, and $\langle\dots\rangle$ denotes averaging over
certain ensemble of statistically-similar systems. To understand the basic
features of SFF for quantum chaotic systems, consider the circular unitary
ensemble (CUE) and the circular orthogonal ensemble (COE) in RMT Haake (2010);
Mehta (2004). CUE is the ensemble of $n$-by-$n$ matrices uniformly distributed
in the unitary group $\hat{U}(n)$ according to the Haar measure, and can be
used to model quantum chaotic systems without symmetries. In this case, the
SFF is given by $K_{\text{CUE}}(t)=t$ for $t\leq t_{\mathrm{H}}$ and
$K_{\text{CUE}}(t)=D$ for $t>t_{\mathrm{H}}$, where the Heisenberg time
$t_{\mathrm{H}}=O(\Delta^{-1})=O(D)$ is proportional to the inverse energy
level spacing $\Delta$ and hence to the size of the Hilbert space $D$. COE is
the ensemble of uniformly-distributed unitary symmetric matrices, and can be
utilized to model systems with time-reversal symmetry with the antiunitary
time reversal operator $\mathcal{T}$ satisfying $\mathcal{T}^{2}=1$ Haake
(2010). The corresponding SFF behavior is
$K_{\text{COE}}(t)=2t-t\log(1+2t/t_{\mathrm{H}})$ for $t\leq t_{\mathrm{H}}=D$
and
$K_{\text{COE}}(t)=2t_{\mathrm{H}}-t\log\left[(2t+t_{\mathrm{H}})/(2t-t_{\mathrm{H}})\right]$
for $t>t_{\mathrm{H}}=D$. The RMT-like ramp in a spin$-1/2$ BH model can be
seen in Fig. 1(a) for both symmetry classes.

As mentioned in the Introduction, a deviation from RMT behavior in SFF, the
“bump”, has been demonstrated to exist at times earlier than the many-body
Thouless time $t_{\mathrm{Th}}$ in generic many-body quantum systems (see Fig.
1(a) and (b)). The temporal region $t<t_{\mathrm{Th}}$ of SFF (Fig. 1(b)) will
be referred to as the bump regime throughout the text. Thouless time is an
intrinsic timescale that generically grows with the system size – with the
exception of quantum circuits satisfying the dual unitarity condition Bertini
_et al._ (2018, 2021) – and can be characterized by a set of Lyapunov
exponents Chan  _et al._ (2021). For systems without conserved quantities,
this behavior of $t_{\mathrm{Th}}$ originates from the domain walls in
emergent statistical mechanical models that separates growing RMT-like regions
Chan  _et al._ (2018a); Garratt and Chalker (2021a, b). For systems with
conserved quantities and constraints, the origin of the $t_{\mathrm{Th}}$
scaling can be traced back to the diffusive and sub-diffusive modes of the
charges Friedman  _et al._ (2019); Moudgalya  _et al._ (2021).

(a)  Figure 2:  (a-b) Schematic representation of periodically driven 1D (a)
lattice trapped and interacting spin$-1/2$ bosons and (b) a spin$-1$
condensate in a harmonic potential. The tunnelling ($J_{\uparrow}$,
$J_{\downarrow}$) and the spin-mixing $J_{\uparrow\downarrow}$ (e.g.
facilitated by Raman-assisted tunneling with a certain Rabi frequency, see the
dashed line) are depicted. $u_{\uparrow}$, $u_{\downarrow}$
($u_{\uparrow\downarrow}$) refer to the on-site interactions, while the
induced on-site disorder is illustrated by the irregular wells.

##  III Models

In all of the models, the employed driving protocol is expressed via the
unitary operator $\hat{U}(t^{\prime})\equiv\hat{U}^{t^{\prime}}$. This refers
to the repeated application of the (i) two-step (with period $T=\tau$) and
(ii) three-step (alternating from $\tau_{1}$ to $\tau_{2}$ and having a period
$T=(2\tau_{1}+\tau_{2})/2$) periodic driving scheme defined by the Floquet
operator

$\displaystyle\hat{U}$ $\displaystyle=$
$\displaystyle\begin{cases}e^{-i\hat{H}_{2}\tau}e^{-i\hat{H}_{1}\tau}&\text{two-
step}\\\
e^{-i\hat{H}_{1}\tau_{1}}e^{-i\hat{H}_{2}\tau_{2}}e^{-i\hat{H}_{1}\tau_{1}}\;\;&\text{three-
step}\end{cases}\;,$ (2)

with $\tau_{1}\neq\tau_{2}\neq 0$ which breaks the time-reversal symmetry for
the three-step protocol.

In the following, we introduce the spatially extended spinless and spin$-1/2$
BH models, as well as chaotic spin$-1$ condensates. The system-specific
ingredients of the Floquet protocol are also discussed.

###  III.1 Driven Bose-Hubbard chains

The simplest tight-binding model for interacting bosons, the BH model Fisher
_et al._ (1989), can be effectively realized by loading bosonic cold atoms
into a one-dimensional optical lattice Jaksch  _et al._ (1998). Such a
geometry can be readily implemented in an experiment with tightly confined
transversal directions so that the excitations along the latter are highly
suppressed, and hence the system corresponds to a series of independent 1D
tubes across the $x$ direction Rispoli  _et al._ (2019). Initially, we employ
a setup where the interacting bosonic atoms have two hyperfine states
described by a spin$-1/2$ BH Hamiltonian (Fig. 2a(a)),

$\displaystyle\hat{H}_{\text{BH}}$ $\displaystyle=$
$\displaystyle\sum_{r}\left(\sum_{\sigma}J_{\sigma}\hat{b}_{\sigma,r}^{\dagger}\hat{b}_{\sigma,r+1}+J_{\uparrow\downarrow,r}\hat{b}_{\uparrow,r}^{\dagger}\hat{b}_{\downarrow,r}+\text{h.c.}\right)$
(3) $\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{\sigma,r}u_{\sigma}\hat{n}_{\sigma,r}(\hat{n}_{\sigma,r}-1)+u_{\uparrow\downarrow}\sum_{r}\hat{n}_{\uparrow,r}\hat{n}_{\downarrow,r}$
$\displaystyle+$
$\displaystyle\sum_{\sigma,r}\mu_{\sigma,r}\hat{n}_{\sigma,r}\;.$

Here $\sigma=\,\uparrow,\downarrow$ is the spin index. $J_{\sigma}$,
$u_{\sigma}$ and $u_{\uparrow\downarrow}$ denote the tunneling and on-site
interaction strengths, that can be experimentally tuned through the laser
fields generating the lattice potential Jaksch  _et al._ (1998); Duan  _et
al._ (2003). Additionally, $J_{\uparrow\downarrow}$ refers to the spin-mixing
coupling between two hyperfine states. It can be created and adjusted using
Raman assisted tunneling Weimer  _et al._ (2010) or the radio-frequency fields
Lavoine  _et al._ (2021). The position-dependent random potential
$\mu_{\sigma,r}$ can be realized with the aid of digital micromirror devices
Rispoli  _et al._ (2019); Lukin  _et al._ (2019); Bordia  _et al._ (2017).

The Hamiltonians in the Floquet unitaries above are characterized by the fixed
parameters
$J_{\uparrow}=u_{\uparrow}=u_{\downarrow}=u_{\uparrow\downarrow}=u=J$, while
the remaining ones alternate according to

$\displaystyle\hat{H}_{1}$ $\displaystyle\rightarrow$
$\displaystyle\,\hat{H}_{\mathrm{BH}}\text{ with
}J_{\downarrow}=J,\;J_{\uparrow\downarrow}=0,\;\mu_{\uparrow,r}\in[-J,J]\;,$
$\displaystyle\hat{H}_{2}$ $\displaystyle\rightarrow$
$\displaystyle\,\hat{H}_{\mathrm{BH}}\text{ with
}J_{\downarrow}=0,\;J_{\uparrow\downarrow,r}\in[-J,J],\;\mu=0\;,$

with the constraint $\mu_{\uparrow,r}=-\mu_{\downarrow,r}$. As such, within
the first step, a random potential is turned on across the chain where the
notation $[-J,J]$ means that we randomly choose the site-dependent potential
values from a uniform distribution. However let us note that the strength of
the disorder does not affect the physics so long as it is larger than $J$.
Subsequently, in the second step, we switch off this random potential, while
the tunneling of the atoms confined in $\Ket{\downarrow}$ hyperfine state
vanishes and an onsite random spin-mixing tunneling amplitude between
different hyperfine states is turned on. The driving frequency should not be
on resonance with the spin-mixing coupling $J_{\uparrow\downarrow}$ not to
inadvertently polarize the time-dependent state.

Evolution times for the BH Hamiltonians, e.g., $\tau$, are expressed in units
of inverse tunneling $1/J$, and to compute $\hat{U}^{t^{\prime}}$ we apply
trotterization. When $\tau\gg 0.1$ holds, the application of the two-step
scheme reproduces COE spectral statistics for the spin-$1/2$ bosonic system
sup . In contrast, by following the three-step periodic scheme the system
transitions from exhibiting COE to CUE spectral statistics as $\tau_{1}$ and
$\tau_{2}$ increase and has persistent CUE statistics for
$\tau_{1},\tau_{2}\gg 0.1$. For convenience, throughout, we define a
dimensionless normalized time $t=t^{\prime}/T$ such that our many-body
simulations are directly comparable to the analytic predictions of RMT.

Although our results, to be presented below, are based on the
stroboscopically-driven protocol discussed above, there is flexibility in the
choice of parameters for establishing a chaotic behavior in the spin-$1/2$
setting. Here, we briefly summarize the range of this flexibility and refer to
the SM sup  for numerical evidence. (i) If desired, the number of simultaneous
pulses can be further decreased. (ii) Randomizing either $\hat{H}_{1}$ or
$\hat{H}_{2}$ is sufficient, e.g. $J_{\uparrow\downarrow,r}$ does not have to
be randomized. (iii) The interaction or tunneling strengths do not need to be
equal to each other and one of the three interaction parameters could be set
to zero. We observe that the two-step driven Floquet system moves away from
COE statistics when the atoms with either spin components transition deep into
the superfluid regime, e.g., $J_{\uparrow}=20J$. Overall, we find that the
necessary parameter to simulate RMT statistics in the spin$-1/2$ BH model is
the onsite spin-mixing tunneling $J_{\uparrow\downarrow}$. The underlying
reason can be traced back to the fact that the spin-mixing tunneling term
breaks the SU$(2)$ symmetry of the system Lavoine  _et al._ (2021); Mistakidis
_et al._ (2022a). In the presence of this symmetry and two-step driving, the
chaotic behavior described by COE statistics is not apparent to the SFF of the
entire Hilbert space (i.e., unprojected SFF) sup .

In the absence of spin-mixing coupling $J_{\uparrow\downarrow}=0$, the
projection to a spin-preserving subspace could be achieved through
transferring all atoms to one hyperfine state. This is effectively modeled by
the spinless BH model Fisher  _et al._ (1989). The Hamiltonian of this
spinless model could be written in the form of Eq. (3) with the constraint
that all atoms are, for instance, in the spin-$\downarrow$ state. Namely, we
consider the parameters
$J_{\uparrow\downarrow}=u_{\uparrow\downarrow}=J_{\uparrow}=u_{\uparrow}=\mu_{r,\uparrow}=0$,
and assume that $N=N_{\downarrow}$. The remaining parameters are denoted by
$J_{\downarrow}=J$, $u_{\downarrow}=u$ and $\mu_{r,\downarrow}=\mu_{r}$.
Exploring how to induce chaotic behavior in the spinless BH model and
comparing the predictions with the spin-$1/2$ case enable us to determine how
the bump regime and the Thouless time depend on the hyperfine structure of the
atoms. The periodically driven spinless BH model exhibits chaotic behavior
when the Hamiltonians of the Floquet unitary are given by
$\hat{H}_{1}=\hat{H}_{\text{BH}}$ with $\\{J=1,u=J,\mu_{r}\in[-J,J]\\}$, and
$\hat{H}_{2}=\hat{H}_{\text{BH}}$ with $\\{J=0,u=1,\mu_{r}=0\\}$. Note that
the first two guidelines stated above, (i) and (ii), for the parameter
selection that could lead to RMT behavior, are also valid for the spinless BH
model sup . Moreover, we utilize an alternative protocol based on Refs. Elben
_et al._ (2018); Vermersch  _et al._ (2018) to test whether, and if so how,
$t_{\mathrm{Th}}$ depends on the choice of the driving protocol (see Sec. IV.1
and SM for further details).

###  III.2 Driven short- and long-range interacting spinor condensates

(a)  Figure 3: The bump behavior of spin-$1/2$ BH model when driven according
to (a-b) three-step and (c) two-step protocols. (a) SFF for $N=3$ following
CUE for a driving period $T=3$ with lattice sizes $L\in[4,12]$. Inset:
$t_{\mathrm{Th}}$ increases with $L$, however its function cannot be
unambiguously determined with the available data. Solid and dashed lines are
fit models for power-law and logarithmic, respectively. (b) SFF for a triple
well $L=3$ following CUE for a driving period of $T=1.5$ with atom numbers
$N\in[3,10]$. Inset: $t_{\mathrm{Th}}$ scaling is fitted to a power-law with
$t_{\mathrm{Th}}(N=3,L)\propto N^{2}$. (c) SFF of three atoms $N=3$ following
COE for a driving period $T=1$ with lattice sizes $L\in[4,13]$. The green-
solid denote the COE for largest $D$ in the data set. Inset: $t_{\mathrm{Th}}$
scaling with $L$ is fitted to a power-law with $t_{\mathrm{Th}}(N=3,L)\propto
L^{0.78}$. For all cases, the bumps generally increase with $L$ and $N$. Note
the $t_{\mathrm{Th}}$ is defined as a time at which an error function
$\epsilon(t)=|K(t)-K_{\mathrm{RMT}}(t)|/K_{\mathrm{RMT}}(t)$ is smaller than a
threshold of choice, where $K_{\mathrm{RMT}}$ is the SFF behavior for RMT of
an appropriate symmetry class. See details in sup .

Since we aim to identify the role of spatial extendedness in the bump regime,
we also employ an $s-$wave interacting spin$-1$ Bose gas within the single-
mode approximation (SMA) Law  _et al._ (1998); Ho (1998); Yi  _et al._ (2002);
Mistakidis  _et al._ (2022b) which assumes a separation of the spin and
spatial degrees of freedom. In particular, the wave functions for each
hyperfine state are described by the same spatial mode $\phi_{m=0,\pm
1}(x)=\phi(x)$ and different spin wave functions such that the decomposition
$\hat{\psi}_{m}(\mathbf{r})=\hat{a}_{m}\phi(\mathbf{r})$ holds where
$\hat{a}_{m}$ is the annihilation operator acting on the spin-$m$. SMA is
valid for atom numbers as small as $N=100$ when the condensate is prepared in
a tight laser trap Evrard  _et al._ (2021), and it could break down for long
evolution times either due to the build-up of spatial correlations Stamper-
Kurn and Ueda (2013); Mistakidis  _et al._ (2022b) as well as atom loss
related processes from the condensate Yang  _et al._ (2019); Bookjans  _et
al._ (2011); Stamper-Kurn and Ueda (2013). The SMA Hamiltonian for the
spin$-1$ condensate, with both linear and quadratic Zeeman fields along the
$x-$ and $z-$ spin directions, reads

$\displaystyle\hat{H}_{\text{SC}}=\frac{c_{1}}{2N}\hat{\mathbf{F}}^{2}+\sum_{i=x,z}\left(p_{i}\hat{F}_{i}+q_{i}\hat{F}_{i}^{2}\right).$
(4)

Here $\hat{\mathbf{F}}^{2}=\hat{F}_{x}^{2}+\hat{F}_{y}^{2}+\hat{F}_{z}^{2}$
denotes the spin operator, while $c_{1}>0$ is assumed corresponding to a
ferromagnetic spinor gas Stamper-Kurn and Ueda (2013). A magnetic field across
the $z-$direction, embodied in the quadratic Zeeman term
$q_{z}\hat{F}_{z}^{2}$, breaks the rotational symmetry of the spinor gas
Stamper-Kurn and Ueda (2013).
$c_{1}\hat{\mathbf{F}}^{2}/2N+q_{z}\hat{F}_{z}^{2}$ is still an integrable
model Zhang  _et al._ (2005); Dağ  _et al._ (2018), however, the addition of a
linear Zeeman field in the $x-$direction has been recently shown to break its
integrability Rautenberg and Gärttner (2020); Evrard  _et al._ (2021). This is
because the $p_{x}\hat{F}_{x}$ term breaks the SO$(2)$ symmetry of the spinor
condensate, giving rise to signatures of level repulsion which are captured by
the Brody distribution Brody  _et al._ (1981).

(a)  Figure 4: Bump behavior of spinless BH model in different filling factor
regimes with three-step CUE protocol and a driving period of $T=3$. The
filling factors are (a) $\nu\rightarrow 0$, (b) $\nu=1$ and (c)
$\nu\rightarrow\infty$ as the Hilbert space dimension increases. (a) The atom
number is $N=5$ and lattice size changes, $L\in[5,11]$. Inset:
$t_{\mathrm{Th}}(N=5,L)$ fluctuates with $L$. (b) $N$ and $L$ both change,
$L=N\in[4,8]$. Inset: $t_{\mathrm{Th}}(N=L,L)$ increases either linearly
$1.3L-1$ (dotted) or close to linear $\propto L^{1.12}$ (solid). (c) $L=5$ and
$N\in[5,13]$. The power-law fit to $t_{\mathrm{Th}}$ scaling is
$t_{\mathrm{Th}}(N,L=5)\propto N^{2.16}$.

We find that spin$-1$ condensates with $p_{x}\hat{F}_{x}$ term display
stronger level repulsion and spectral rigidity, i.e., an extended RMT ramp in
SFF, only when a quadratic Zeeman shift in the $x-$direction
$q_{x}\hat{F}_{x}^{2}$ or a linear Zeeman shift in the $z-$direction
$p_{z}\hat{F}_{z}$ are introduced to the system sup .

There is no spatial-extendedness in this model due to the SMA. All atoms,
regardless of the physical distance between them, could interact with one
another equally likely through contact interaction, i.e. such a spinor
Hamiltonian is effectively zero-dimensional. The spinor Hamiltonian described
in Eq. (4) can also be expressed in the Fock basis $\Ket{n_{-1},n_{0},n_{1}}$,
where $n_{i}$ is the population in each hyperfine state. By utilizing the
SU(3) Lie algebra operators Yukawa  _et al._ (2013),

$\displaystyle\hat{H}_{\text{SC}}$ $\displaystyle=$
$\displaystyle\frac{c_{1}}{2N}\bigg{[}\hat{a}_{0}^{\dagger}\hat{a}_{0}^{\dagger}\hat{a}_{1}\hat{a}_{-1}+\text{h.c.}+\hat{N}_{0}(\hat{N}_{1}+\hat{N}_{-1})$
(5) $\displaystyle+$
$\displaystyle\frac{1}{2}\left(\hat{N}_{1}-\hat{N}_{-1}\right)^{2}\bigg{]}+\frac{p_{x}}{\sqrt{2}}\bigg{[}(\hat{a}_{1}^{\dagger}+\hat{a}_{-1}^{\dagger})\hat{a}_{0}+\text{h.c.}\bigg{]}$
$\displaystyle+$ $\displaystyle
p_{z}\left(\hat{N}_{1}-\hat{N}_{-1}\right)+\frac{q_{x}}{2}\bigg{[}\hat{a}_{1}^{\dagger}\hat{a}_{-1}+\text{h.c.}\bigg{]}-q_{z}\hat{N}_{0},$

where $\hat{a}_{\nu}$ ($\hat{a}_{\nu}^{\dagger}$) and $\hat{N}_{\nu}$ refer to
the annihilation (creation) and the number operators of an atom occupying the
spin$-\nu$ hyperfine level, respectively. Note also the presence of spin-
mixing collision interactions Law  _et al._ (1998) in the first two terms of
Eq. (5). Let us emphasize that without $p_{z}\hat{F}_{z}$ term in Eq. (5), the
condensate preserves the inversion symmetry. Therefore, in order to capture
the bump-ramp-plateau behavior in SFF, we either perform projection onto the
symmetry subspace with even parity
$\Ket{N_{0},\mathcal{M}}\rightarrow\left(\Ket{N_{0},\mathcal{M}}+\Ket{N_{0},-\mathcal{M}}\right)/\sqrt{2}$
where $\mathcal{M}=N_{1}-N_{-1}$ is the magnetization, or we add linear Zeeman
field in the $z-$direction, as in Eq. (5), and compute the unprojected SFF.
The latter approach is experimentally feasible, see sup  for details on the
inversion symmetry subspace projection.

Furthermore, we consider the effect of long-range interactions in the spin$-1$
condensate and study the bump regime in the even parity subspace when there
are dipolar magnetic interactions between the spinor atoms Stamper-Kurn and
Ueda (2013); Yi and Pu (2006). For an anisotropic harmonic trap, it is
possible to arrive at the so-called two-axis counter-twisting (TACT)
Hamiltonian Kajtoch and Witkowska (2016), namely

$\displaystyle\hat{H}_{\text{TACT}}$ $\displaystyle=$
$\displaystyle\chi\left(\hat{F}_{x}^{2}-\hat{F}_{y}^{2}-\hat{D}_{xy}\right),$
$\displaystyle=$
$\displaystyle\chi\bigg{[}2\hat{N}_{0}\hat{a}_{-1}^{\dagger}\hat{a}_{1}+\hat{a}_{0}^{\dagger}\hat{a}_{0}^{\dagger}(\hat{a}_{1}^{2}+\hat{a}_{-1}^{2})+\text{h.c.}\bigg{]}.$

The latter term in Eq. (III.2) is one of the eight SU$(3)$ Lie algebra
operators $\hat{D}_{xy}=\hat{a}_{1}^{\dagger}\hat{a}_{-1}+\text{h.c.}$.
Accordingly, we write the Hamiltonian for the dipolar spin$-1$ condensate as

$\displaystyle\hat{H}_{\text{DC}}=\frac{c_{1}}{2N}\hat{\mathbf{F}}^{2}+p_{x}\hat{F}_{x}+q_{z}\hat{F}_{z}^{2}+\hat{H}_{\text{TACT}}.$
(7)

Similar to the BH model protocols described above, the two- and three-step
stroboscopic protocols of Eq. (2) lead to COE and CUE statistics with the
following Hamiltonians in the Floquet unitary,

$\displaystyle\hat{H}_{1}$ $\displaystyle\rightarrow$
$\displaystyle\,\hat{H}_{\text{SC/DC}}\text{ with
}q_{z}\in\left[0,\frac{c_{1}}{N}\right],\;\;p_{x}=\frac{c_{1}}{N}\,,$
$\displaystyle\hat{H}_{2}$ $\displaystyle\rightarrow$
$\displaystyle\,\hat{H}_{\text{SC/DC}}\text{ with
}q_{z}\in\left[-\frac{c_{1}}{N},0\right],\;\;p_{x}=-\frac{c_{1}}{N}\,.$ (8)

For concreteness throughout the text, we will assume a spinor condensate in
the Thomas-Fermi limit and confined in a 1D trap Pu  _et al._ (1999), such
that $c_{1}/N\propto N^{-1/3}$ holds.

##  IV Bump regime and Thouless Time

Now we discuss the behavior of the bump regime and the scaling of
$t_{\mathrm{Th}}$ for the spatially-extended driven BH models, and the spinor
condensates in subsections IV.1 and IV.2, respectively. To summarize, we find
that the bump regime and $t_{\mathrm{Th}}$ scaling are more sensitive to
variations in the atom number than in the chain size. Moreover, we observe
that interacting bosons trapped in a single well display RMT behavior in SFF
much faster than the bosonic system in spatially extended potentials. We
conclude this section by empirically deducing a scaling function for the bump
regime of the SFF for the interacting trapped bosons. Our results discussed
below are based on 10000-50000 realizations of $\hat{U}$.

###  IV.1 Dependence on atom number and lattice size

(a)  Figure 5: (a) Projected SFF of the spinor condensate $H_{\text{SC}}$ on
even parity sector driven according to three-step protocol and with $p_{z}=0$,
$q_{x}=2$. The atom number changes between $N\in[30-100]$ from yellow to red.
Inset: The Thouless timescales with the atom number, either in a slow power-
law ($\propto N^{0.3}N/|c_{1}|\propto N^{0.63}$, solid) or logarithmic
($\propto\log(N)$, dashed). (b) SFF of the entire Hilbert space of the spinor
condensate $H_{\text{SC}}$ driven according to three-step protocol, with
$p_{z}=2$, $q_{x}=0$ and the atom number changing between $N\in[30,80]$.
Inset: The Thouless timescales with the atom number in a slow power-law
$\propto N^{0.63}N/|c_{1}|\propto N^{0.96}$. (c) Projected SFF of the spinor
condensate with dipolar interactions $H_{\text{DC}}$ on even parity sector
driven according to three-step protocol and with $\chi=2$. The dipolar
interaction strength is $\chi=2$ for atom numbers $N\in[30,110]$ from yellow
to red. Inset: The Thouless time fluctuates with the atom number, resulting in
$t_{\mathrm{Th}}(N)\propto N/|c_{1}|\propto N^{1/3}$ scaling in atom number.
In all panels, we set the driving period as $T=3$.

In contrast to spin systems and quantum circuits, the size of the Hilbert
space in BH models is determined by two parameters: the chain size $L$ and the
atom number $N$. These parameters dictate the size of the Hilbert space as
$D=(N+\alpha L-1)!/(N!\times(\alpha L-1)!)$, where $\alpha$ is the number of
hyperfine levels. Therefore, we can explore the dependence of the bump regime
and $t_{\mathrm{Th}}$ scaling not only on $L$ but also on $N$, and hence the
filling factor $\nu$ of the lattice. We confirm that the bump-ramp-plateau
behavior persists in the Mott insulator regime, where $u\gg J$ holds for the
spin$-1/2$ bosonic system sup , and further explore the bump and the emergence
of RMT beyond the Mott insulator regime below.

Figs. 3a and 3b demonstrate that the early time behavior of SFF in a
spin$-1/2$ BH model, converges to the CUE behavior, as a function of $L$ ($N$)
with fixed $N=3$ ($L=3$). Fig. 3(c), on the other hand, shows the presence of
the bump when the system is driven according to the two-step protocol, and
thus exhibits COE statistics as a function of $L$ with three atoms. The
amplitude of the bump increases with $L$ at fixed $N$ for both protocols that
simulate CUE and COE symmetry classes as illustrated in Figs. 3a and 3c.

An increase of $L$ ($N$) while keeping all other parameters fixed results in a
more dilute (denser) lattice trapped gas. Interestingly, the bump is
significantly enhanced for a denser gas as seen by comparing Fig. 3(b) and
Fig. 3(a). In all cases of spin$-1/2$ BH model, $t_{\mathrm{Th}}$ increases
either as a function of $L$ or $N$, but it is difficult to determine the
functional form of $t_{\mathrm{Th}}(N,L)$ based on the accessible finite-size
data. To compare the scaling behavior among different cases, power-law fits
are provided in the insets of Fig. 3. We observe that the scaling exponent of
$t_{\mathrm{Th}}$ is larger in $N$ for filling factors $\nu\rightarrow\infty$
($N\rightarrow\infty$, fixed $L$) e.g., $t_{\mathrm{Th}}(N,L=3)\propto N^{2}$,
than in $L$ for $\nu\rightarrow 0$ ($L\rightarrow\infty$, fixed $N$) e.g.,
$t_{\mathrm{Th}}(N=3,L)\propto L^{0.44}$.

For the two-step protocol simulating COE symmetry class presented in Fig.
3(c), the bump is more prevalent in $L$ compared to the three-step protocol
simulating CUE class shown in Fig. 3(a). Consistently,
$t_{\mathrm{Th}}(N=3,L)\propto L^{0.78}$ scaling is slightly faster. For the
corresponding scaling behavior at $N>3$ or $L>3$, see SM. A similar stark
difference between the scaling in $N$ and $L$ persists also for the two-step
protocol under COE symmetry class with $t_{\mathrm{Th}}(N,L=3)\propto
N^{1.95}$ and $t_{\mathrm{Th}}(N=3,L)\propto L^{0.78}$, respectively sup .

These observations suggest that $t_{\mathrm{Th}}(N,L)$ is more sensitive to
$N$ than to $L$ for spin$-1/2$ interacting bosons in optical lattices. This
dependence on atom number, in turn could be traced back to the interactions
between lattice confined atoms. In fact, increasing the interaction strengths
$u=u_{\uparrow}=u_{\downarrow}=u_{\uparrow\downarrow}$ so that the system
transitions to a Mott insulator, increases the Thouless time and results in a
larger bump amplitude sup .

To further test the generality of the argument given above, we compare the SFF
of the spinless BH model with three different filling factor regimes, namely
$\nu\rightarrow 0$, $\nu=1$ and $\nu\rightarrow\infty$ as $D$ increases. As
depicted in Fig. 4(c), the bump signature is more pronounced for increasing
$\nu$, and $t_{\mathrm{Th}}$ scales faster ($t_{\mathrm{Th}}(N,L=5)\propto
N^{2.16}$) than in Fig. 4(b), where $\nu=1$ holds
($t_{\mathrm{Th}}(N,L=N)\propto N^{1.12}$). We observe that this difference
also depends on the choice of the driving protocol. Indeed, the
$t_{\mathrm{Th}}(N,L)$ scaling turns out to be the same for the filling
factors $\nu=1$ and $\nu\rightarrow\infty$ both leading to a scaling
$t_{\mathrm{Th}}(N,L)\propto N^{4}$ when the stroboscopic driving scheme is
set to be an alternative protocol based on Ref. Vermersch  _et al._ (2018)
which we refer to as the VE-protocol.

The VE protocol is defined by the Floquet unitary,
$\hat{U}_{\text{VE}}=\prod_{m}^{M}e^{-i\hat{H}_{m}\tau}$ where $\hat{H}_{m}$
is a spinless BH Hamiltonian with a different random potential landscape at
each Floquet step $\mu_{r,m}\in[-J,J]$ for $J=1$ and $u=J$. This protocol
reproduces COE symmetry class for a two-step $M=2$ Floquet unitary and
smoothly transitions to simulating CUE symmetry class as the number of steps
in the Floquet unitary increases to $M>5$ sup . The comparison of the SFF with
VE-protocol suggests that the onset of RMT behavior occurs earlier in the
three-step protocol described by Eq. (2), giving rise to CUE in cold atoms
earlier than the VE-protocol does. Nevertheless, for both protocols we observe
a significant difference in the bump regimes and $t_{\mathrm{Th}}$ scaling
between $\nu\rightarrow\infty$ and $\nu\rightarrow 0$. This manifests again
the sensitivity of $t_{\mathrm{Th}}(N,L)$ to $N$.

Differently than the spin-$1/2$ BH model, $t_{\mathrm{Th}}$ fluctuates around
a fixed value instead of exhibiting a monotonic increase with $L$ for
$\nu\rightarrow 0$ as illustrated in the inset of Fig. 4(a). We confirm the
independency of this behavior from the protocol choice sup . In SM, we show a
slowly increasing trend in $L$ with fluctuations for $t_{\mathrm{Th}}(N=4,L)$,
so that we can simulate up to $L=15$. Therefore, observing no scaling for
$t_{\mathrm{Th}}(N=5,L)$ in $L$ might be a finite-size effect.

###  IV.2 Role of locality

For a many-body system that does not extend spatially in a 1D chain, the
Thouless time scaling is significantly slower due to lack of locality
Gharibyan  _et al._ (2018); Chan  _et al._ (2018b). As we show in Fig. 1(a), a
spin$-1/2$ gas trapped in a single well does not exhibit RMT statistics (blue
markers). For this reason below we employ a spin$-1$ condensate within a
harmonic trap, and explore the Thouless time scaling with respect to $N$ as
the dimension of the Hilbert space is $D=(N+2)(N+1)/2$. Fig. 5(a) shows the
presence of the bump regime for $H_{\text{SC}}$ driven according to the three-
step protocol and projected to even parity subspace when $p_{z}=0$ and
$q_{x}=2$ hold. Although this spin$-1$ gas is a zero-dimensional model, we
observe a bump in the SFF whose amplitude increases with $N$, i.e. the bump
feature is not a finite-size effect. However, $t_{\mathrm{Th}}$ scaling with
$N$ is significantly slower compared to 1D BH models. For instance, assuming a
power-law scaling function, we obtain $t_{\mathrm{Th}}(N)\propto
N^{0.3}N/|c_{1}|\propto N^{0.63}$ when we take $N$ dependence of $c_{1}$ into
account (see Sec. III).

An experimental operation to achieve the subspace projection is through
introducing a linear Zeeman term along the spin $z-$direction, i.e.,
$p_{z}\neq 0$ breaking the inversion symmetry. The resulting SFF when taking
into account the entire Hilbert space is presented in Fig. 5(b), where
$p_{z}=2$ and $q_{x}=0$ were set. The bump gets sharper with increasing atom
number, and $t_{\mathrm{Th}}$ scaling can be described with a power-law
increase in $N$ with an exponent that is still smaller than the spatially
extended BH models with $\gamma\sim 1$.

In order to generalize our results beyond the short-range $s-$wave
interactions, we next introduce dipolar interactions into the spin$-1$
condensate, e.g., $\chi=2$ in Eq. (7), and drive it according to the three-
step protocol. The projected SFF on the even parity sector is presented in
Fig. 5(c). A tendency for suppression in the amplitude of the bump takes place
compared to Fig. 5(a), while $t_{\mathrm{Th}}$ does not exhibit any clear
scaling trend with $N$. When $N/|c_{1}|$ rescaling is taken into account, we
find $t_{\mathrm{Th}}(N)\propto N^{1/3}$. Therefore, our findings corroborate
the idea that the locality of the underlying Hamiltonian is indeed reflected
on the bump regime and $t_{\mathrm{Th}}$ scaling.

###  IV.3  Scaling function of the bump regime

(a)  Figure 6: The collapse of data around $t_{\mathrm{Th}}$. (a) Spin$-1/2$
BH model for atom numbers ranging between $N\in[6,10]$ from yellow to red in a
triple well following the CUE statistics and giving $\xi=-1.75$ in Eq. (9).
(b) Spin$-1/2$ BH model for atom numbers ranging between $N\in[4,9]$ from
yellow to red in a triple well following the COE statistics and giving
$\xi=-1.75$. (c) Spinless BH model of five atoms for lattice sizes ranging
between $L\in[9,11]$ from yellow to red in a triple well following the CUE
statistics and giving $\xi=-3.25$. (d) Spinor condensate with $p_{z}=2$ and
$q_{x}=0$ for atom numbers ranging between $N\in[30,80]$ from yellow to red
following the CUE statistics and giving $\xi=-1.6$. The $x-$axes are scaled
with the numerically extracted Thouless times. The $y-$axes are scaled with
the corresponding RMT behavior.

Figs. 3, 4 and 5 suggest the existence of rescaling parameters in both $t-$
and $y-$axes where $K(t)$ for different $D$ could collapse on each other in
the bump region. We rescale time with $t_{\mathrm{Th}}$ and $K(t)$ with the
analytical ramp expression to spotlight the bump region:
$K(t)/K_{\text{rmt}}(t)$ which should saturate at unity for
$t>t_{\mathrm{Th}}$. To determine how the bump approaches unity and whether
there is a universal behavior for all $D$, we plot $K(t)/K_{\text{rmt}}(t)-1$.
This rescaling analysis reveals a collapse of data around $t_{\mathrm{Th}}$.
The data collapses well in the RMT regime $t/t_{\text{th}}\gtrsim 1$ as
expected, and in the bump region $t<t_{\mathrm{Th}}$. We observe no data
collapse for $t/t_{\mathrm{Th}}\rightarrow 0$. Interestingly for all models, a
collapse in the rescaled SFF $K(t)/K_{\text{rmt}}(t)-1$ for sufficiently large
$D$ is present, and $K(t)/K_{\text{rmt}}(t)-1$ approaches the saturation value
$\sim 0$ as a power-law in the rescaled time $t/t_{\text{Th}}$. We notice that
this power-law approach to RMT in fact describes the region where $K(t)$
decreases from certain time that we call $t_{\mathrm{b}}$, reaches a minimum
and increases towards RMT. We can then write the following empirical
expression for the SFF which also characterizes the bump region,

$\displaystyle K(t_{\mathrm{b}}<t<t_{\mathrm{H}})\approx
K_{\textrm{RMT}}(t)\left[\beta\left(\frac{t}{t_{\mathrm{Th}}}\right)^{\xi}+1\right],$
(9)

where $\beta$ and $\xi<0$ are scalar. Based on our analysis, we observe
$\beta\ll 1$ is a small parameter, and the exact value of $\xi$ could depend
on the microscopics, e.g., $-4<\xi<-1.4$. We note that $t_{\mathrm{b}}$
coincides with time where the bump exhibits a local peak before reaching a
local minimum for all cases.

We depict two cases of spin$-1/2$ BH chain in a triple well with different
atom numbers in Figs. 6(a) and 6(b) where (a) data is for CUE and (b) for COE.
The associated bump region and $t_{\mathrm{Th}}$ scaling of (a) are already
given in Fig. 3(b). Hence Fig. 6(b) suggests that the physics discussed
earlier also holds for COE. We plot the rescaled SFF in Fig. 6(c) for the
spinless BH model of five atoms trapped in optical lattices of varying sizes.
The associated bump region is given in Fig. 4(a) whose inset actually shows no
$t_{\mathrm{Th}}$ scaling with lattice size. Nevertheless the bump regime is
present, and consistently with the rest of the observations, it introduces a
power-law correction to the RMT. Finally in Fig. 6(d), we demonstrate the bump
scaling for a chaotic spinor condensate and again find that the expression in
Eq. (9) holds.

The observation that a consistent scaling, Eq. (9), exists for the different
atomic systems and parameter values suggests universality beyond the RMT
regime. This means that the physical signature of quantum chaos in many-body
systems precedes the ramp and the Thouless time, and includes the bump region
$t_{\mathrm{b}}<t<t_{\mathrm{Th}}$. An indirect physical evidence in support
of this argument can be found in the heating times of local observables
Lazarides  _et al._ (2014). For the considered lattice sizes, we find that the
heating times of density populations in the BH models are less than the
corresponding $t_{\mathrm{Th}}$, and in fact sometimes comparable to
$t_{\mathrm{b}}$, i.e. the time when the bump regime collapse starts sup .
Hence, the knowledge of relevant timescales of quantum chaos in an experiment
may have practical consequences for information acquisition.

##  V  Experimental protocol

Next we discuss an experimental protocol applicable to cold atoms trapped in
optical lattices. For concreteness, we will focus on the BH model, however the
arguments are valid for the spin$-1/2$ BH model which requires an additional
hyperfine state. Later, we comment on the case of spin$-1$ condensates.

SFF is defined through a trace operation (see Eq. (1)) meaning that the system
should be initialized in an infinite-temperature state whose controlled
preparation in an experiment is not a simple task. On the other hand, the BH
model in an optical lattice can be routinely initialized in the Mott insulator
regime, i.e., a product state Islam  _et al._ (2015). The VE-protocol could be
utilized to generate a random state $\Ket{\psi}$ from a Mott insulating
product state $\Ket{\mathbf{1}}$, as a variant of this protocol has been shown
to generate unitary$-2$ design Vermersch  _et al._ (2018).

Figure 7: Diagrammatical tensor network representation of (a) the survival
probability, which is a quantum circuit of a pair of replicated systems, a
time evolution operation on system 1 and a swap operator; and (b) the SFF, in
terms of survival probability with initial state chosen to be a certain pure
state, e.g. a Mott insulating product state $\ket{\mathbf{1}}$, and a set of
global rotations $W$ which form a 2-design.

An important ingredient underlying our protocol is the relation between the
survival probability and SFF. The survival probability for an initial state
$\ket{\psi}$ reads,

$\displaystyle P(t,\\{U,\ket{\psi}\\})$ $\displaystyle=$
$\displaystyle|\Bra{\psi}\hat{U}(t)\Ket{\psi}|^{2}\;.$ (10)

Consider replicating the system with the pure initial state $\ket{\psi}$, so
that the density matrix of both systems is
$\rho=(\ket{\psi}\otimes\ket{\psi})(\bra{\psi}\otimes\bra{\psi})$. Defining
the swap operator $\hat{V}_{2}$ by the action
$\hat{V}_{2}\ket{\psi_{1}}\otimes\ket{\psi_{2}}=\ket{\psi_{2}}\otimes\ket{\psi_{1}}$,
it is possible to express the survival probability in the form

$\displaystyle
P(t)=\text{Tr}\left[\left(\hat{U}(t)\otimes\mathds{1}\right)\rho\left(\hat{U}^{\dagger}(t)\otimes\mathds{1}\right)\hat{V}_{2}\right]\;,$
(11)

which has a diagrammatical tensor network representation shown in Fig. 7(a).

SFF can be expressed as the survival probability for certain initial states.
For instance, $P(t,\\{\hat{U},\ket{\psi}\\})=K(t)$ if
$\ket{\psi}=D^{-1}\sum_{a}\ket{E_{a}}$ is chosen to be the equal-amplitude
superposition of the eigenstates $\ket{E_{a}}$ of $\hat{U}$ or $\hat{H}$ with
Hilbert space dimension $D$. Here, instead, we use an experimentally
realizable initial state given by

$\Ket{\psi}_{\mathrm{CUE}}=\hat{W}\Ket{\mathbf{1}},\quad\;\;\hat{W}\in\textrm{CUE}(D)\;.$
(12)

so that

$K^{\text{ex}}(t)=\left\langle\mathbb{E}\left[P(\\{\ket{\psi}_{\mathrm{CUE}},\hat{U}(t)\\})\right]_{W}\right\rangle
D(D+1)-D\;.$ (13)

where we have used $\mathbb{E}\left[\dots\right]_{W}$ to denote averages over
$W$. The circuit to measure $K(t)$ in terms of $P(t)$ is sketched in Fig.
7(b).

Now we show that the survival probability and consequently, the SFF of a many-
body system can be experimentally measured utilizing Eq. (11). A feasible
read-out protocol for probing SFF of cold atoms in optical lattices is the
interference measurement through the measurement of swap operator
$\hat{V}_{2}$ Daley  _et al._ (2012); Islam  _et al._ (2015) which is outlined
below:

(i) Initiate two BH chains in a Mott insulator state and apply VE-protocol to
both copies with global rotations $\hat{W}$ to obtain the same random state
$\Ket{\psi}$. In other words, $\hat{W}=\prod_{m}^{M}e^{-i\hat{H}_{m}\tau}$
where $M$ should be chosen $M\geq 6$ sup .

(ii) Freeze one of the copies, and evolve the other one with the Floquet
unitary of either two-step or three-step protocol, $\hat{U}$ for an evolution
time $t$.

(iii) Switch off the hoping parameters across the 1D chains, and subsequently
lower the potential between the two chains to interfere the respective many-
body states. This corresponds to beam-splitting operation Islam  _et al._
(2015).

(iv) Measure the parity of each site in the first chain, e.g., system 1 in
Fig. 7(a) with a quantum gas microscope Bakr  _et al._ (2009), and compute
$P(t)=\prod_{j\in R}e^{i\pi n_{j}}$ where $R$ is the entire chain.

(v) Repeat steps (i)-(iv) for a number of different realizations of $\hat{U}$
and $\hat{W}$ at each time $t$ to compute
$\left\langle\mathbb{E}\left[P(\\{\ket{\psi}_{\mathrm{CUE}},\hat{U}(t)\\})\right]_{\hat{W}}\right\rangle$.

(vi) Compute SFF via Eq. (13).

An alternative proposal for the experimental detection of SFF has been
recently introduced in Ref. Vasilyev  _et al._ (2020), using interferometry
with a control atom Poulin  _et al._ (2003); Müller  _et al._ (2009) for
Rydberg gases in tweezer arrays which realize spin models. We comment on the
application of interferometry to our systems in the SM, which could be
realized by introducing a cavity to implement the controlled coupling of the
auxiliary atom to the many-body quantum simulator Jiang  _et al._ (2008).
Another alternative protocol is proposed for Rydberg gases by utilizing the
so-called randomized measurement toolbox in Ref. Joshi  _et al._ (2022) where
measurements are performed on the system of interest after a time evolution
and random on-site rotations. Our interference measurement complements this
randomized measurement circuit described in Ref. Joshi  _et al._ (2022): By
introducing a replica of the chain and many-body interfering the copies, we
remove the requirement of applying $\hat{W}^{\dagger}$ before read-out. This
potentially helps with reducing the total time of a single experimental run.

(a)  Figure 8: Eq. (13), interference measurement result, (blue, dotted-
crosses) is compared to Eq. (1), the theoretical prediction (yellow-solid) for
a three-step [CUE] protocol applied to spinless BH model at $N=L=6$. To
prepare $W$ operators, we used the VE-protocol with $M=8$. (a) Focus on the
early-time SFF where the experimental protocol with $50000$ realization number
captures $t_{\mathrm{Th}}=8$ and most of early-time SFF well except $t\leq 3$.
The green line is the CUE. (b) The experimental protocol captures the ramp and
plateau well.

We emphasize that it is important to measure the SFF with its standard
normalization, e.g. Eq. (13), instead of a normalization as
$0<K^{\text{ex}}(t)/D^{2}\leq 1$ as in Vasilyev  _et al._ (2020); Joshi  _et
al._ (2022). This ensures a sufficient resolution for the bump regime and not
to inadvertently suppress the bump amplitude as $D$ increases.

An experimental implication of our results concerns the resolution of
$K(t_{\mathrm{b}}<t\lesssim t_{\mathrm{Th}})$ where we would aim to
differentiate the bump from the ramp. Eq. (9) provides the difference between
bump and ramp amplitudes as $K(t_{\mathrm{b}}<t<t_{\mathrm{H}})-K(t\geq
t_{\mathrm{Th}})\sim\beta t^{1+\xi}t_{\mathrm{Th}}^{|\xi|}$. For the bump to
be measured, this difference should be greater than the standard deviation of
$\mathcal{A}$ measurements. Denoting
$\mathcal{K}\equiv|\text{Tr}[\hat{U}(t)]|^{2}$, and the variance
$\sigma_{\mathcal{K}}(t)\equiv\big{[}\Braket{\mathcal{K}^{2}}-\Braket{\mathcal{K}}^{2}\big{]}^{1/2}$,
an upper bound for $\sigma_{\mathcal{K}}(t)$ must read

$\displaystyle\frac{\sigma_{\mathcal{K}}(t)}{\sqrt{\mathcal{A}}}$
$\displaystyle<$ $\displaystyle\beta t^{1+\xi}t_{\mathrm{Th}}^{|\xi|}=\beta
t^{1+\xi}N^{\gamma|\xi|}.$ (14)

The equality follows from assuming a power-law form for the Thouless time
scaling in atom number, $N^{\gamma}$. For spatially extended models
$\gamma>1$, whereas for spinor condensates $\gamma<1$, and $\xi<-1$ based on
our analysis in the previous subsections resulting in $\gamma|\xi|>1$.
Although setting $D$ sufficiently small is crucial to be able to resolve
$t_{\mathrm{H}}$ and recover the ramp-plateau behavior, we observe that larger
atom numbers increase the upper bound for the error in measuring the bump. For
the data set depicted in Fig. 8 for $\nu=1$, Eq. (14) requires
$\mathcal{A}\sim 5000$ runs of the experiment for a single time point to
differentiate the bump from ramp. On the other hand, $\mathcal{A}\sim 2000$ is
sufficient for the same system with $N=8$ and $L=5$, which has a similar
Hilbert space dimension and $t_{\mathrm{Th}}=10$. In passing, let us mention
that the error bars in all figures on SFF are found by calculating
$\sigma_{\mathcal{K}}(t)/\sqrt{\mathcal{A}}$.

Atomic systems in optical lattices are highly flexible platforms Bloch  _et
al._ (2012). Indeed, their parameters, including the on-site interaction and
hopping terms, can be readily tuned in the experiment over a wide range of
values. For instance, it is possible to adjust the hopping parameter $J$ from
$J/h\propto 2$ Hz preparing a shallow lattice to $J/h\propto 150$ Hz
constructing a deep lattice Sherson  _et al._ (2010) where $h$ is the Planck’s
constant. By considering a hopping amplitude of the order of $100$ Hz for the
spin$-1/2$ BH model with $N=L=4$ as depicted in Fig. 1(a), typical total
evolution times correspond to $\propto 10$ s [$\propto 30$ s] for two [three]
step driving protocol. For this system, Thouless time is around
$t_{\mathrm{Th}}\propto 0.6$ s in both driving protocols which should be
experimentally feasible to monitor. This is in contrast to the respective
Heisenberg times that are attained for $t_{\mathrm{H}}\propto 6.6$ s
[$t_{\mathrm{H}}\propto 19.8$ s] in the case of a two-[three-]step driving
protocol.

Turning to the short-range interacting spin$-1$ condensate and using a spin-
spin interaction coefficient $c_{1}=-2\pi\times 9$ Hz Stamper-Kurn and Ueda
(2013) the Thouless time for a three-step driving protocol corresponds to
$t_{\mathrm{Th}}\approx 3.93$ s and $t_{\mathrm{Th}}\approx 2.17$ s in the
case of $N=100$ and $N=40$ respectively. Along the same lines, the Thouless
time for a dipolar spin$-1$ bosonic gas is $t_{\mathrm{Th}}\approx 4.43$ s
[$t_{\mathrm{Th}}\approx 2.54$ s] for $N=100$ [$N=40$]. For a spinor
condensate trapped in a single well, a single copy randomized measurement
Joshi  _et al._ (2022) might be more feasible than two-copy beam-splitting
operation. The condensate could be initialized in a state where all atoms
occupy the spin-$0$ hyperfine level and a measurement on the number of atoms
in each hyperfine level can be made at the end of the protocol to estimate Eq.
(10).

##  VI Conclusions

We investigate signatures of quantum many-body chaos in various
stroboscopically-driven atomic setups ranging from lattice trapped spinless or
spin$-1/2$ bosons to harmonically confined spin$-1$ condensates. A particular
focus is placed on the behavior of the spectral form factor which was found to
feature a universal bump regime and for longer evolution times, the ramp and
plateau regimes. The latter is a measure of spectral rigidity, and hence
quantum chaos. Our many-body cold atom models, regardless of the locality of
the underlying Hamiltonian, hyperfine structure, driving protocols or the
symmetry classes, exhibit a universal bump, and suggest a power-law correction
to the RMT prediction of the SFF at early experimentally accessible times.
Extension of universality from the RMT to the bump regime and the observation
that heating times of density populations are significantly shorter than the
corresponding $t_{\mathrm{Th}}$ highlight the role of $t_{b}$, the start time
of the bump regime collapse. The practical consequences of such a separation
in the early timescales in many-body quantum chaos, e.g., for information
retrieval from a heating system, is an intriguing endeavor.

We study $t_{\mathrm{Th}}$ scaling with respect to relevant system parameters,
the atom number and chain size. The Thouless time scaling with atom number is
significantly slower in spin$-1$ gases trapped in a single well than the
1D-lattice trapped BH models with $L\geq 3$. This suggests that the spatially-
extended cold atom systems take longer to exhibit RMT. Consequently, the
locality of the underlying Hamiltonian is a key condition for the Thouless
time scaling and the bump amplitude. Importantly, we find that
$t_{\mathrm{Th}}$ of spatially-extended systems is more sensitive to the atom
number than the lattice size. Namely, the scaling in atom number is larger
than in lattice size, and subsequently it takes a denser gas longer to reach
RMT. These conclusions highlight the role of atom number for determining the
onset of RMT in many-body systems, and therefore the impact of interactions.
Consistently, the bump regime is enhanced as the lattice-trapped gas becomes
denser. These observations hold true regardless of the hyperfine structure,
symmetry classes or the choice of driving protocol.

Our results suggest the existence of a universal scaling function for the
Thouless time depending on both the lattice size and the atom number
$t_{\mathrm{Th}}(N,L)$. Determining the exact form of this scaling function is
an important next step to better understand the nature of many-body quantum
chaos in experimentally realizable systems. Constructing dual cold atom
circuits for this purpose could be a fruitful direction. Alternatively, one
could utilize advanced numerical techniques such as time-dependent density
matrix renormalization group Daley  _et al._ (2004) or multiconfiguration
approaches Cao  _et al._ (2017) in order to access larger lattice sizes and
atom numbers while taking into account relevant many-body correlation effects.

As proposed, the bump regime and Thouless time can be measured with cold atoms
in optical lattices with quantum gas microscopy. How the bump regime changes
as one deviates from random initial states is an intriguing and potentially
useful prospect for experiments. Exploring the signatures of the bump regime
and the Thouless time in infinite-temperature correlators and entanglement
entropy is an interesting future direction, and can provide a route to
understand connections between spectral and spatio-temporal correlations.
Finally, devising spectroscopic probes of the bump and the Thouless energy in
many-body systems is also a fruitful direction, as these features do not
require resolving the many-body level spacing.

## Acknowledgements

We thank Soonwon Choi, Bertrand Evrard, Wen-Wei Ho, David Huse, Di Luo, Minh
Tran, Norman Yao, Sina Zeytinoglu and Peter Zoller for stimulating
discussions. C.B.D., S.I.M., and H.R.S. acknowledge support from the NSF
through a grant for ITAMP at Harvard University. A.C. acknowledges funding
support from the PCTS at the Princeton University.

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