Abstract
In quantum chaotic systems, the spectral form factor (SFF), defined as the Fourier transform of two-level spectral correlation function, is known to follow random matrix theory (RMT), namely a ‘ramp’ followed by a ‘plateau’ in late times. Recently, a generic early-time deviation from RMT, so-called the ‘bump’, was shown to exist in random quantum circuits as toy models for many-body quantum systems. We demonstrate the existence of ‘bump-ramp-plateau’ behavior in the SFF for a number of paradigmatic and stroboscopically-driven 1D cold-atom models: spinless and spin-1/2 Bose-Hubbard models, and nonintegrable spin-1 condensate with contact or dipolar interactions. We find that the scaling of the many-body Thouless time tTh —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, tTh 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.
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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 scales1. Such signatures have been found in a plethora of disciplines including nuclear resonance spectra2,3, mesoscopic physics4,5,6, quantum chaos7,8,9, black hole physics10,11,12, and quantum chromodynamics13,14. With their unprecedented degree of controllability15,16, trapped cold atoms17,18,19,20,21,22,23 are excellent platforms to study many-body phenomena, including signatures of quantum chaos in spectral statistics, and for large-scale simulations of quantum systems15,16,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38.
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)28,39,40, or its absence via many-body localization (MBL)32,37,41,42,43, the existence of quantum many-body scars34,44,45, and Hilbert space fragmentation46,47. In this vein, as a probe for quantum chaos and ergodicity48,49,50,51,52,53,54,55,56,57,58,59,60, observables such as entanglement entropy and out-of-time-order correlators (OTOCs) have been intensively studied, and measured with quantum simulators31,33,54,61,62,63,64,65,66. However, the OTOCs could be susceptible to quantum criticality and order as found by refs. 67,68,69,70, even in nonintegrable quantum many-body models at infinite temperature71. Moreover, it was demonstrated by refs. 72,73 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 ratio1,74,75,76, as well as number variance7,77. While the spacing distribution can be experimentally obtained for few particles3,6, 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.
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 function78,79. For a physical system, the appearance of the RMT-like “ramp” and subsequent “plateau” in SFF, Fig. 1a, is a signature of quantum chaos. In particular, the SFF behavior around the Heisenberg time tH 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 distribution80 (Fig. 1c). SFF has been instrumental in the discovery of a number of novel and universal signatures of many-body quantum chaotic systems in both condensed matter8,9,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104, and in high energy physics11,12,105,106,107,108,109,110,111. Black holes have been shown to display RMT behavior in late times with SFF11,110, and the SFF has been utilized to investigate the ergodic-to-MBL transition in 1D spin chains85,86,95,97.
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 ≪ tH (see Fig. 1a, b), and RMT is recovered only after the many-body Thouless time tTh8,9,83,85,89,91,102,105, as illustrated by star markers in Fig. 1. Thouless time is an intrinsic timescale that generically grows with the system size – with the exception of quantum circuits satisfying the dual unitarity condition81,92 – and can be characterized by a set of Lyapunov exponents90. For systems without conserved quantities, this behavior of tTh originates from the domain walls in emergent statistical mechanical models that separates growing RMT-like regions8,93,95. For systems with conserved quantities and constraints, the origin of the tTh scaling can be traced back to the diffusive and sub-diffusive modes of the charges83,91. The extension of such results to cold atom systems is nontrivial, since only an approximate mapping between spin and cold atom models could be established for the Mott insulating regime, where spinful atoms are confined to lattice sites112.
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. Our focus on SFF in driven atomic systems complements the investigations of chaos in atomic models utilizing time-independent setups113,114,115, Rydberg interactions116,117, highly long-range interactions89,102 or tools such as dynamical fidelity118,119 and nearest-neighbor energy level statistics58,76,120,121,122,123,124,125,126,127,128. Recent studies116,117 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 tH and the ramp-plateau behavior. In this work, we argue that tTh is not only experimentally more accessible for large systems, because tTh ≪ tH and probing tTh does not require resolving the many-body level spacing, but it is also physically more appealing — tTh and the bump regime emerge in many-body chaotic systems. The onset of the RMT ramp, as quantified by tTh, provides us with another physically meaningful timescale in quantum many-body systems.
To investigate the dependence of tTh on the locality of the underlying Hamiltonian, we employ spatially-extended Bose-Hubbard (BH) chains129 (Fig. 2a), as opposed to a chaotic spin − 1 condensate under single-mode approximation (SMA)130 (Fig. 2b). The latter refers to an ensemble of pairwise interacting spin − 1 atoms, and hence effectively a zero-dimensional model62,131,132,133. 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 tTh 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 experimentally motivated. The BH model describes the physics of cold atoms trapped in optical lattices24, as experimentally demonstrated in26, while different hyperfine states of bosonic atoms can be routinely used to realize Bose mixtures134,135. The spinor condensates constitute yet another atom-based platform to simulate many-body physics136,137,138,139. We utilize Floquet driving, because the stroboscopically-driven setups have the advantage to produce a uniform density of states, i.e., no unfolding140 or filtering105 procedures are required for the computation of the spectral statistics, unlike in time-independent Hamiltonian systems. Floquet driving of cold atom systems have been realized141,142,143. We also 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 chaos113,114,144 and measured in picosecond spectroscopy of molecular dissociation145,146,147. We devise a read-out protocol for the SFF through many-body state interference148, which measures survival probability and is a state-of-the-art measurement scheme for cold atoms in optical lattices31.
Results and discussion
SFF is the Fourier transform of the two-level spectral function, and can be defined directly in the time domain as
where \(\hat{U}(t)\) is the time evolution operator of a given system with spectrum {Em}, and 〈… 〉 denotes averaging over certain ensemble of statistically-similar systems. Hence, we can define error bars around K(t) at each t through \({\sigma }_{{{{{{{{\mathcal{K}}}}}}}}}(t)/\sqrt{{{{{{{{\mathcal{A}}}}}}}}}\) where \({\sigma }_{{{{{{{{\mathcal{K}}}}}}}}}(t)\equiv {[\langle {{{{{{{{\mathcal{K}}}}}}}}}^{2}\rangle -{\langle {{{{{{{\mathcal{K}}}}}}}}\rangle }^{2}]}^{1/2}\) is the variance of \({{{{{{{\mathcal{K}}}}}}}}\equiv | \,{{\mbox{Tr}}}\,[\hat{U}(t)]{| }^{2}\) over an ensemble of \({{{{{{{\mathcal{A}}}}}}}}\) statistically similar systems. The error bars in all figures on SFF in this Manuscript are found by calculating \({\sigma }_{{{{{{{{\mathcal{K}}}}}}}}}(t)/\sqrt{{{{{{{{\mathcal{A}}}}}}}}}\). Our results discussed below are based on \({{{{{{{\mathcal{A}}}}}}}}\in [10000,50000]\) realizations of \(\hat{U}\).
To understand the basic features of SFF for quantum chaotic systems, consider the circular unitary ensemble (CUE) and the circular orthogonal ensemble (COE) in RMT78,79. 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 KCUE(t) = t for t ≤ tH and KCUE(t) = D for t > tH, where the Heisenberg time tH = O(Δ−1) = O(D) is proportional to the inverse energy level spacing Δ 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 \({{{{{{{{\mathcal{T}}}}}}}}}^{2}=1\)79. The corresponding SFF behavior is \({K}_{{{{{{{{\rm{COE}}}}}}}}}(t)=2t-t\log (1+2t/{t}_{{{{{{{{\rm{H}}}}}}}}})\) for t ≤ tH = D and \({K}_{{{{{{{{\rm{COE}}}}}}}}}(t)=2{t}_{{{{{{{{\rm{H}}}}}}}}}-t\log [(2t+{t}_{{{{{{{{\rm{H}}}}}}}}})/(2t-{t}_{{{{{{{{\rm{H}}}}}}}}})]\) for t > tH = D. The RMT-like ramp in a spin − 1/2 BH model can be seen in Fig. 1a for both symmetry classes. A deviation from RMT behavior in SFF, the “bump”, has been demonstrated to exist at times earlier than the many-body Thouless time tTh in generic many-body quantum systems (see Fig. 1a, b). The temporal region t < tTh of SFF (Fig. 1b) will be referred to as the bump regime throughout the text.
The 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 = τ) and (ii) three-step (alternating from τ1 to τ2 and having a period T = (2τ1 + τ2)/2) periodic driving scheme defined by the Floquet operator
with τ1 ≠ τ2 ≠ 0 which breaks the time-reversal symmetry for the three-step protocol.
Dependence of the bump regime and Thouless time on atom number and lattice size
The simplest tight-binding model for interacting bosons, the BH model129, can be effectively realized by loading bosonic cold atoms into a one-dimensional optical lattice24. 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 direction37. We employ a setup where the interacting bosonic atoms have two hyperfine states described by a spin − 1/2 BH Hamiltonian (Fig. 2a),
Here σ = ↑, ↓ is the spin index. Jσ, uσ and u↑↓ denote the tunneling and on-site interaction strengths, that can be experimentally tuned through the laser fields generating the lattice potential24,112. Additionally, J↑↓ refers to the spin-mixing coupling between two hyperfine states. It can be created and adjusted using Raman assisted tunneling25 or the radio-frequency fields149. The position-dependent random potential μσ,r can be realized with the aid of digital micromirror devices37,150,151.
In the absence of spin-mixing coupling J↑↓ = 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 model129. 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-↓ state. Namely, we consider the parameters J↑↓ = u↑↓ = J↑ = u↑ = μr,↑ = 0, and assume that N = N↓. The remaining parameters are denoted by J↓ = J, u↓ = u and μr,↓ = μr. 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 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 + αL − 1)!/(N! × (αL − 1)!), where α is the number of hyperfine levels. Therefore, we can explore the dependence of the bump regime and tTh scaling not only on L but also on N, and hence the filling factor ν of the lattice. We confirm that the bump-ramp-plateau behavior persists in the Mott insulator regime, where u ≫ J holds for the spin − 1/2 bosonic system (Supplementary Note 6), and further explore the bump and the emergence of RMT beyond the Mott insulator regime below.
Figure 3a, b 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). Figure 3c, 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 Fig. 3a, c. The model-specific details of the Floquet protocol are given in Eq. (13).
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. 3b, a. In all cases of spin − 1/2 BH model, tTh increases either as a function of L or N, but it is difficult to determine the functional form of tTh(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 tTh is larger in N for filling factors ν → ∞ (N → ∞, fixed L) e.g., tTh(N, L = 3) ∝ N2, than in L for ν → 0 (L → ∞, fixed N) e.g., tTh(N = 3, L) ∝ L0.44.
For the two-step protocol simulating COE symmetry class presented in Fig. 3c, the bump is more pronounced in L compared to the three-step protocol simulating CUE class shown in Fig. 3a. Consistently, tTh(N = 3, L) ∝ L0.78 scaling is slightly faster. For the corresponding scaling behavior at N > 3 or L > 3, see Supplementary Note 4. A similar stark difference between the scaling in N and L persists also for the two-step protocol under COE symmetry class with tTh(N, L = 3) ∝ N1.95 and tTh(N = 3, L) ∝ L0.78, respectively.
These observations suggest that tTh(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↑ = u↓ = u↑↓ so that the system transitions to a Mott insulator, increases the Thouless time and results in a larger bump amplitude (see Supplementary Note 6).
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 ν → 0, ν = 1 and ν → ∞ as D increases. As depicted in Fig. 4c, the bump signature is more pronounced for increasing ν, and tTh scales faster (tTh(N, L = 5) ∝ N2.16) than in Fig. 4b, where ν = 1 holds (tTh(N, L = N) ∝ N1.12). We observe that this difference also depends on the choice of the driving protocol: The tTh(N, L) scaling turns out to be the same for the filling factors ν = 1 and ν → ∞ both leading to a scaling tTh(N, L) ∝ N4 when the stroboscopic driving scheme is set to be an alternative driving protocol, the VE-protocol (see Sec. Methods). The comparison of the SFF obtained with VE-protocol suggests that the onset of RMT behavior in cold atoms occurs earlier in the three-step protocol described by Eq. (2). Nevertheless, for both protocols we observe a significant difference in the bump regimes and tTh scaling between ν → ∞ and ν → 0. This manifests again the sensitivity of tTh(N, L) to N.
For the spinless BH model, tTh fluctuates around a fixed value instead of exhibiting a monotonic increase with L for ν → 0 as illustrated in the inset of Fig. 4a. We confirm the independency of this behavior from the protocol choice (see Supplementary Note 1). In Supplementary Note 4, we show a slowly increasing trend in L with fluctuations for tTh(N = 4, L), so that we can simulate up to L = 15. Therefore, observing no scaling for tTh(N = 5, L) in L might be a finite-size effect.
To summarize this subsection, we find that the bump regime and tTh scaling are more sensitive to variations in the atom number than in the chain size, regardless of the hyperfine structure, symmetry class and the protocol choice.
Role of locality in the bump regime and Thouless time
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 locality82,105. As we show in Fig. 1a, a spin−1/2 gas trapped in a single well does not exhibit RMT statistics (blue markers). Since we aim to identify the role of spatial extendedness in the bump regime of the cold atom models, we employ an s−wave interacting spin−1 Bose gas within the single-mode approximation (SMA)130,152,153,154 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 ϕm=0,±1(x) = ϕ(x) and different spin wave functions such that the decomposition \({\hat{\psi }}_{m}({{{{{{{\bf{r}}}}}}}})={\hat{a}}_{m}\phi ({{{{{{{\bf{r}}}}}}}})\) 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 trap139, and it could break down for long evolution times either due to the build-up of spatial correlations136,154 as well as atom loss related processes from the condensate136,138,155. 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 SMA Hamiltonian for the spin−1 condensate, with both linear and quadratic Zeeman fields along the x−and z−spin directions, reads
Here \({\hat{{{{{{{{\bf{F}}}}}}}}}}^{2}={\hat{F}}_{x}^{2}+{\hat{F}}_{y}^{2}+{\hat{F}}_{z}^{2}\) denotes the spin operator, while c1 > 0 is assumed corresponding to a ferromagnetic spinor gas136. 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 gas136. \({c}_{1}{\hat{{{{{{{{\bf{F}}}}}}}}}}^{2}/2N+{q}_{z}{\hat{F}}_{z}^{2}\) is still an integrable model156,157, however, the addition of a linear Zeeman field in the x − direction has been recently shown to break its integrability127,139. 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 distribution2. 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 (see Supplementary Note 5).
Figure 5 a shows the presence of the bump regime for HSC driven according to the three-step protocol Eq. (15), and projected to even parity subspace when pz = 0 and qx = 2 hold, (see Sec. Methods for symmetry subspace projection). 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, tTh scaling with N is significantly slower compared to 1D BH models. For instance, assuming a power-law scaling function, we obtain tTh(N) ∝ N0.3N/∣c1∣ ∝ N0.63 when we take N dependence of c1 into account (see Sec. Methods).
An experimental operation to achieve the subspace projection is through introducing a linear Zeeman term along the spin z − direction, i.e., pz ≠ 0 breaking the inversion symmetry. The resulting SFF when taking into account the entire Hilbert space is presented in Fig. 5(b), where pz = 2 and qx = 0 were set. The bump gets sharper with increasing atom number, and tTh scaling can be described with a power-law increase in N with an exponent γ ~ 1 that is still smaller than the spatially extended BH models.
In order to generalize our results beyond the short-range s − wave interactions, 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 atoms136,158. For an anisotropic harmonic trap, it is possible to arrive at the so-called two-axis counter-twisting (TACT) Hamiltonian159, namely
The latter term in Eq. (5) is one of the eight SU(3) Lie algebra operators \({\hat{D}}_{xy}={\hat{a}}_{1}^{{{{\dagger}}} }{\hat{a}}_{-1}+\,{{\mbox{h.c.}}}\,\) where \({\hat{a}}_{\nu }\) (\({\hat{a}}_{\nu }^{{{{\dagger}}} }\)) refers to the annihilation (creation) operators of an atom occupying the spin − ν hyperfine level. Accordingly, we write the Hamiltonian for the dipolar spin − 1 condensate as
Here, we fix χ = 2 in Eq. (6), and drive it according to the three-step protocol. The projected SFF on the even parity sector is presented in Fig. 5c. A tendency for suppression in the amplitude of the bump takes place compared to Fig. 5a, while tTh does not exhibit any clear scaling trend with N. When N/∣c1∣ rescaling is taken into account, we find tTh(N) ∝ N1/3. Therefore, our findings corroborate the idea that the locality of the underlying Hamiltonian is indeed reflected on the bump regime and tTh scaling.
To summarize this subsection, 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.
Universal scaling of the bump regime
Figures 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 tTh and K(t) with the analytical ramp expression to spotlight the bump region: K(t)/Krmt(t) which should saturate at unity for t > tTh. To determine how the bump approaches unity and whether there is a universal behavior for all D, we plot K(t)/Krmt(t) − 1. This rescaling analysis reveals a collapse of data around tTh. The data collapses well in the RMT regime t/tth ≳ 1 as expected, and in the bump region t < tTh. We observe no data collapse for t/tTh → 0. Interestingly for all models, a collapse in the rescaled SFF K(t)/Krmt(t) − 1 for sufficiently large D is present, and K(t)/Krmt(t) − 1 approaches the saturation value ~ 0 as a power-law in the rescaled time t/tTh. We notice that this power-law approach to RMT in fact describes the region where K(t) decreases from certain time that we call tb, reaches a minimum and increases towards RMT. We can then write the following empirical expression for the SFF which also characterizes the bump region,
where β and ξ < 0 are scalar. Based on our analysis, we observe β ≪ 1 is a small parameter, and the exact value of ξ could depend on the microscopics, e.g., − 4 < ξ < − 1.4. We note that tb 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 Fig. 6a, b where (a) data is for CUE and (b) for COE. The associated bump region and tTh scaling of (a) are already given in Fig. 3b. Hence Fig. 6b suggests that the physics discussed earlier also holds for COE. We plot the rescaled SFF in Fig. 6c for the spinless BH model of five atoms trapped in optical lattices of varying sizes. The associated bump region is given in Fig. 4a whose inset actually shows no tTh 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. 6d, we demonstrate the bump scaling for a chaotic spinor condensate and again find that the expression in Eq. (7) holds.
The observation that a consistent scaling, Eq. (7), 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 tb < t < tTh. An indirect physical evidence in support of this argument can be found in the heating times of local observables126. For the considered lattice sizes, we find that the heating times of density populations in the BH models are less than the corresponding tTh, and in fact sometimes comparable to tb, i.e. the time when the bump regime collapse starts (see Supplementary Note 7). Hence, the knowledge of relevant timescales of quantum chaos in an experiment may have practical consequences for information acquisition.
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 state31. The VE-protocol could be utilized to generate a random state \(\left\vert \psi \right\rangle\) from a Mott insulating product state \(\left\vert {{{{{{{\bf{1}}}}}}}}\right\rangle\), as a variant of this protocol has been shown to generate unitary−2 design160.
An important ingredient underlying our protocol is the relation between the survival probability and SFF. The survival probability for an initial state \(\left\vert \psi \right\rangle\) reads,
Consider replicating the system with the pure initial state \(\left\vert \psi \right\rangle\), so that the density matrix of both systems is \(\rho =(\vert \psi \rangle \otimes \vert \psi \rangle )(\langle \psi \vert \otimes \langle \psi \vert )\). Defining the swap operator \({\hat{V}}_{2}\) by the action \({\hat{V}}_{2}\left\vert {\psi }_{1}\right\rangle \otimes \left\vert {\psi }_{2}\right\rangle =\left\vert {\psi }_{2}\right\rangle \otimes \left\vert {\psi }_{1}\right\rangle\), it is possible to express the survival probability in the form
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, \(P(t,\{\hat{U},\left\vert \psi \right\rangle \})=K(t)\) if \(\left\vert \psi \right\rangle ={D}^{-1}{\sum }_{a}\left\vert {E}_{a}\right\rangle\) is chosen to be the equal-amplitude superposition of the eigenstates \(\left\vert {E}_{a}\right\rangle\) of \(\hat{U}\) or \(\hat{H}\) with Hilbert space dimension D. Here, instead, we use an experimentally realizable initial state given by
so that
where we have used \({\mathbb{E}}{[\ldots ]}_{W}\) to denote averages over W. See Methods Section for the derivation of Eq. (11). 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. (9). A feasible read-out protocol for probing SFF of cold atoms in optical lattices is the interference measurement through the measurement of swap operator \({\hat{V}}_{2}\)31,148 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 \(\left\vert \psi \right\rangle\). In other words, \(\hat{W}=\mathop{\prod }\nolimits_{m}^{M}{e}^{-i{\hat{H}}_{m}\tau }\) where M should be chosen M ≥ 6 (see Supplementary Note 1).
-
(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 operation31.
-
(iv)
Measure the parity of each site in the first chain, e.g., system 1 in Fig. 7a with a quantum gas microscope23, 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 \(\big\langle {\mathbb{E}}{[P(\{{\big| \psi \big\rangle }_{{{{{{{{\rm{CUE}}}}}}}}},\hat{U}(t)\})]}_{\hat{W}}\big\rangle\).
-
(vi)
Compute SFF via Eq. (11).
An alternative proposal for the experimental detection of SFF has been recently introduced in ref. 116, using interferometry with a control atom161,162 for Rydberg gases in tweezer arrays which realize spin models. We comment on the application of interferometry to our systems in the Supplementary Note 2, which could be realized by introducing a cavity to implement the controlled coupling of the auxiliary atom to the many-body quantum simulator163. Another alternative protocol is proposed for Rydberg gases by utilizing the so-called randomized measurement toolbox in ref. 117 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. 117: 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.
We emphasize that it is important to measure the SFF with its standard normalization, e.g. Eq. (11), instead of a normalization as 0 < Kex(t)/D2 ≤ 1 as in refs. 116,117. 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(tb < t ≲ tTh) where we would aim to differentiate the bump from the ramp. Eq. (7) provides the difference between bump and ramp amplitudes as \(K({t}_{{{{{{{{\rm{b}}}}}}}}} \, < \, t \, < \, {t}_{{{{{{{{\rm{H}}}}}}}}})-K(t\ge {t}_{{{{{{{{\rm{Th}}}}}}}}}) \sim \beta {t}^{1+\xi }{t}_{{{{{{{{\rm{Th}}}}}}}}}^{| \xi | }\). For the bump to be measured, this difference should be greater than the standard deviation of \({{{{{{{\mathcal{A}}}}}}}}\) measurements. Hence, an upper bound for the variance \({\sigma }_{{{{{{{{\mathcal{K}}}}}}}}}(t)\) (see Methods Section) must read
The equality follows from assuming a power-law form for the Thouless time scaling in atom number, Nγ. For spatially extended models γ > 1, whereas for spinor condensates γ < 1, and ξ < − 1 based on our analysis in the previous subsections resulting in γ∣ξ∣ > 1. Although setting D sufficiently small is crucial to be able to resolve tH 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 ν = 1, Eq. (12) 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 tTh = 10.
Atomic systems in optical lattices are highly flexible platforms15. 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 ∝ 2 Hz preparing a shallow lattice to J/h ∝ 150 Hz constructing a deep lattice164 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. 1a, typical total evolution times correspond to ∝ 10 s [ ∝ 30 s] for two [three] step driving protocol. For this system, Thouless time is around tTh ∝ 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 tH ∝ 6.6 s [tH ∝ 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 c1 = − 2π × 9 Hz136 the Thouless time for a three-step driving protocol corresponds to tTh ≈ 3.93 s and tTh ≈ 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 tTh ≈ 4.43 s [tTh ≈ 2.54 s] for N = 100 [N = 40]. For a spinor condensate trapped in a single well, a single copy randomized measurement117 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. (8).
Conclusions
We investigated 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 tTh highlight the role of tb, 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 tTh 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 ≥ 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 tTh 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 tTh(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 group165 or multiconfiguration approaches166 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.
Methods
Throughout the study tTh is numerically defined as a time at which an error function ϵ(t) = ∣K(t) − KRMT(t)∣/KRMT(t) is smaller than a threshold of choice, where KRMT is the SFF behavior for RMT of an appropriate symmetry class.
Models
Here we discuss the system-specific ingredients of the Floquet protocols, and further details of our models, the spatially extended spinless and spin − 1/2 BH models and chaotic spin − 1 condensates.
The spin − 1/2 BH Hamiltonians in the Floquet unitaries are characterized by the fixed parameters J↑ = u↑ = u↓ = u↑↓ = u = J, while the remaining parameters alternate according to
with the constraint μ↑,r = − μ↓,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 \(\left\vert \downarrow \right\rangle\) 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↑↓ not to inadvertently polarize the time-dependent state.
Evolution times for the BH Hamiltonians, e.g., τ, are expressed in units of inverse tunneling 1/J, and to compute \({\hat{U}}^{{t}^{{\prime} }}\) we apply trotterization. When τ ≫ 0.1 holds, the application of the two-step scheme reproduces COE spectral statistics for the spin-1/2 bosonic system. In contrast, by following the three-step periodic scheme the system transitions from exhibiting COE to CUE spectral statistics as τ1 and τ2 increase, and has persistent CUE statistics for τ1, τ2 ≫ 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, presented above, 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 Supplementary Note 3 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↑↓,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↑ = 20 J. 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↑↓. The underlying reason can be traced back to the fact that the spin-mixing tunneling term breaks the SU(2) symmetry of the system149,167. 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) as shown in Supplementary Note 3.
The periodically driven spinless BH model exhibits chaotic behavior when the Hamiltonians of the Floquet unitary are given by \({\hat{H}}_{1}={\hat{H}}_{{{{{{{{\rm{BH}}}}}}}}}\) with {J = 1, u = J, μr ∈ [ − J, J]}, and \({\hat{H}}_{2}={\hat{H}}_{{{{{{{{\rm{BH}}}}}}}}}\) with {J = 0, u = 1, μr = 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.
The spinor Hamiltonian described in Eq. (4) can also be expressed in the Fock basis \(\left\vert {n}_{-1},{n}_{0},{n}_{1}\right\rangle\), where ni is the population in each hyperfine state. By utilizing the SU(3) Lie algebra operators168,
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 − ν hyperfine level, respectively. Note the presence of spin-mixing collision interactions130 in the first two terms of Eq. (14). Similarly the TACT Hamiltonian defined in Role of locality in the bump regime and Thouless time Section, can be written in the Fock basis,
We construct the Floquet unitaries of both the BH models and the spinor condensates in the Fock basis.
Let us emphasize that without \({p}_{z}{\hat{F}}_{z}\) term in Eq. (14), the condensate preserves the inversion symmetry \(\hat{{{{{{{{\mathcal{P}}}}}}}}}={e}^{i\pi {\hat{F}}_{x}}\) which separates the Hilbert space into two independent symmetry sectors. For a Fock basis representation \(\vert {N}_{0},{{{{{{{\mathcal{M}}}}}}}}\rangle\) where \({{{{{{{\mathcal{M}}}}}}}}={N}_{1}-{N}_{-1}\) is the magnetization, the parity operator acts on the states as \(\hat{{{{{{{{\mathcal{P}}}}}}}}}\vert {N}_{0},{{{{{{{\mathcal{M}}}}}}}}\rangle =\vert {N}_{0},-{{{{{{{\mathcal{M}}}}}}}}\rangle\). Hence a more convenient basis choice is in terms of even and odd parity sectors, where the even parity sector consists of states that are even under the application of \(\hat{{{{{{{{\mathcal{P}}}}}}}}}\), e.g., \(\vert {N}_{0},0\rangle\) and \((\vert {N}_{0},{{{{{{{\mathcal{M}}}}}}}}\rangle +\vert {N}_{0},-{{{{{{{\mathcal{M}}}}}}}}\rangle )/\sqrt{2}\), and the odd parity sector consists of states \((\vert {N}_{0},{{{{{{{\mathcal{M}}}}}}}}\rangle -\vert {N}_{0},-{{{{{{{\mathcal{M}}}}}}}}\rangle )/\sqrt{2}\). Therefore, in order to capture the bump-ramp-plateau behavior in SFF, we either perform projection onto the symmetry subspace with even parity \(\vert {N}_{0},{{{{{{{\mathcal{M}}}}}}}}\rangle \to (\vert {N}_{0},{{{{{{{\mathcal{M}}}}}}}}\rangle +\vert {N}_{0},-{{{{{{{\mathcal{M}}}}}}}}\rangle )/\sqrt{2}\), or we add linear Zeeman field in the z − direction, as in Eq. (14), and compute the unprojected SFF. The latter approach is experimentally feasible.
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,
For concreteness throughout the text, we assume a spinor condensate in the Thomas-Fermi limit and confined in a 1D trap169, such that c1/N ∝ N−1/3 holds.
VE-protocol
We utilize an alternative protocol, which we refer to as the VE-protocol, based on refs. 160,170 to test whether, and if so how, tTh depends on the choice of the driving protocol. The VE protocol is defined by the Floquet unitary, \({\hat{U}}_{{{{{{{{\rm{V\; E}}}}}}}}}=\mathop{\prod }\nolimits_{m}^{M}{e}^{-i{\hat{H}}_{m}\tau }\) where \({\hat{H}}_{m}\) is a spinless BH Hamiltonian with a different random potential landscape at each Floquet step μr,m ∈ [ − 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 (Supplementary Note 1).
Survival probability and spectral form factor
In this section, we discuss the relationship between survival probability and spectral form factor, and investigate the survival probability of randomly sampled initial states. The survival probability for the system of interest with initial state \(\left\vert \psi \right\rangle\) and time evolution operator \(\hat{U}\) is given by Eq. (8). When the state \(\left\vert \psi \right\rangle\) is chosen to be the equal-amplitude superposition of all eigenstates of \({\hat{U}}\), then
Next, we consider the survival probability of two randomly sampled initial states, namely,
-
1.
Globally randomly rotated states, i.e.
$$\left\vert \psi \right\rangle =\hat{W}\left\vert \Psi \right\rangle ,\quad \hat{V}\in {{{{{{{\rm{CUE}}}}}}}}(D)={{{{{{{\rm{CUE}}}}}}}}({q}^{L})$$(17)where \(\hat{W}\) is a random unitary drawn from the D-by-D CUE, D and q are the Hilbert space dimension and the on-site Hilbert space dimension respectively. \(\left\vert \Psi \right\rangle\) is any pure state including product states.
-
2.
Locally randomly rotated states, i.e.
$$\left\vert \psi \right\rangle =\left(\mathop{\bigotimes}\limits_{i}^{L} {\hat{v}}_{i}\right)\left\vert \Psi \right\rangle ,\quad {\hat{v}}_{i}\in {{{{{{{\rm{CUE}}}}}}}}(q).$$(18)
Using the two-design formula for m-by-m random unitaries \(\hat{W}\) from CUE,
one can compute Eq. (11) for both cases in Eqs. (17) and (18).
Data availability
All relevant data presented in the Manuscript are available from the corresponding author upon reasonable request.
Code availability
All relevant code used in the Manuscript are available from the corresponding author upon reasonable request.
References
Bohigas, O., Giannoni, M.-J. & Schmit, C. Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52, 1 (1984).
Brody, T. A. et al. Random-matrix physics: spectrum and strength fluctuations. Rev. Mod. Phys. 53, 385–479 (1981).
Haq, R. U., Pandey, A. & Bohigas, O. Fluctuation properties of nuclear energy levels: Do theory and experiment agree? Phys. Rev. Lett. 48, 1086–1089 (1982).
Beenakker, C. W. J. Random-matrix theory of quantum transport. Rev. Mod. Phys. 69, 731–808 (1997).
Beenakker, C. W. J. Random-matrix theory of majorana fermions and topological superconductors. Rev. Mod. Phys. 87, 1037–1066 (2015).
Roushan, P. et al. Spectroscopic signatures of localization with interacting photons in superconducting qubits. Science 358, 1175–1179 (2017).
D’Alessio, L., Kafri, Y., Polkovnikov, A. & Rigol, M. From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics. Adv. Phys. 65, 239–362 (2016).
Chan, A., De Luca, A. & Chalker, J. T. Spectral statistics in spatially extended chaotic quantum many-body systems. Phys. Rev. Lett. 121, 060601 (2018).
Kos, P., Ljubotina, M. & Prosen, T. Many-body quantum chaos: Analytic connection to random matrix theory. Phys. Rev. X 8, 021062 (2018).
García-García, A. M. & Verbaarschot, J. J. M. Spectral and thermodynamic properties of the sachdev-ye-kitaev model. Phys. Rev. D 94, 126010 (2016).
Cotler, J. S. et al. Black holes and random matrices. J. High Energy Phys. 2017 https://doi.org/10.1007/JHEP05(2017)118 (2017).
Cotler, J., Hunter-Jones, N., Liu, J. & Yoshida, B. Chaos, complexity, and random matrices. J. High Energy Phys. 2017, 48 (2017).
Smilga, A. V.Continuous Advances in QCD (WORLD SCIENTIFIC, 1995).
Akemann, G. Random matrix theory and quantum chromodynamics. Oxford Scholarship Online https://doi.org/10.1093/oso/9780198797319.003.0005 (2018).
Bloch, I., Dalibard, J. & Nascimbene, S. Quantum simulations with ultracold quantum gases. Nat. Phys. 8, 267–276 (2012).
Browaeys, A. & Lahaye, T. Many-body physics with individually controlled rydberg atoms. Nat. Phys. 16, 132–142 (2020).
Davis, K. B. et al. Bose-einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75, 3969–3973 (1995).
Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E. & Cornell, E. A. Observation of bose-einstein condensation in a dilute atomic vapor. Science 269, 198–201 (1995).
Chu, S., Hollberg, L., Bjorkholm, J. E., Cable, A. & Ashkin, A. Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure. Phys. Rev. Lett. 55, 48–51 (1985).
Pethick, C. J. & Smith, H.Bose-Einstein Condensation in Dilute Gases: (Cambridge University Press, Cambridge, 2008), 2 edn. https://www.cambridge.org/core/books/boseeinstein-condensation-in-dilute-gases/CC439EAD70D78E47E9AF536DA7B203EC.
Greiner, M., Bloch, I., Mandel, O., Hänsch, T. W. & Esslinger, T. Exploring phase coherence in a 2d lattice of bose-einstein condensates. Phys. Rev. Lett. 87, 160405 (2001).
Greiner, M., Mandel, O., Hänsch, T. W. & Bloch, I. Collapse and revival of the matter wave field of a bose-einstein condensate. Nature 419, 51–54 (2002).
Bakr, W. S., Gillen, J. I., Peng, A., Fölling, S. & Greiner, M. A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice. Nature (London) 462, 74–77 (2009).
Jaksch, D., Bruder, C., Cirac, J. I., Gardiner, C. W. & Zoller, P. Cold bosonic atoms in optical lattices. Phys. Rev. Lett. 81, 3108–3111 (1998).
Weimer, H., Müller, M., Lesanovsky, I., Zoller, P. & Büchler, H. P. A rydberg quantum simulator. Nat. Phys. 6, 382–388 (2010).
Greiner, M., Mandel, O., Esslinger, T., Hänsch, T. W. & Bloch, I. Quantum phase transition from a superfluid to a mott insulator in a gas of ultracold atoms. Nature 415, 39–44 (2002).
Kinoshita, T., Wenger, T. & Weiss, D. S. A quantum newton’s cradle. Nature 440, 900–903 (2006).
Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008).
Cheneau, M. et al. Light-cone-like spreading of correlations in a quantum many-body system. Nature 481, 484–487 (2012).
Tarruell, L., Greif, D., Uehlinger, T., Jotzu, G. & Esslinger, T. Creating, moving and merging dirac points with a fermi gas in a tunable honeycomb lattice. Nature 483, 302–305 (2012).
Islam, R. et al. Measuring entanglement entropy in a quantum many-body system. Nature 528, 77–83 (2015).
Schreiber, M. et al. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science 349, 842–845 (2015).
Kaufman, A. M. et al. Quantum thermalization through entanglement in an isolated many-body system. Science 353, 794–800 (2016).
Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (2017).
Mazurenko, A. et al. A cold-atom fermi–hubbard antiferromagnet. Nature 545, 462–466 (2017).
de Léséleuc, S. et al. Observation of a symmetry-protected topological phase of interacting bosons with rydberg atoms. Science 365, 775–780 (2019).
Rispoli, M. et al. Quantum critical behaviour at the many-body localization transition. Nature 573, 385–389 (2019).
Semeghini, G. et al. Probing topological spin liquids on a programmable quantum simulator. Science 374, 1242–1247 (2021).
Deutsch, J. M. Quantum statistical mechanics in a closed system. Phys. Rev. A 43, 2046–2049 (1991).
Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E 50, 888–901 (1994).
Gornyi, I., Mirlin, A. & Polyakov, D. Interacting electrons in disordered wires: Anderson localization and low-t transport. Phys. Rev. lett. 95, 206603 (2005).
Basko, D., Aleiner, I. & Altshuler, B. Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states. Annals Phys.321, 1126 – 1205 (2006).
Nandkishore, R. & Huse, D. A. Many-body localization and thermalization in quantum statistical mechanics. Annual Rev. Condensed Matter Phys. 6, 15–38 (2015).
Turner, C. J., Michailidis, A. A., Abanin, D. A., Serbyn, M. & Papić, Z. Weak ergodicity breaking from quantum many-body scars. Nat. Phys. 14, 745 (2018).
Moudgalya, S., Rachel, S., Bernevig, B. A. & Regnault, N. Exact excited states of nonintegrable models. Phys. Rev. B 98, 235155 (2018).
Sala, P., Rakovszky, T., Verresen, R., Knap, M. & Pollmann, F. Ergodicity breaking arising from hilbert space fragmentation in dipole-conserving hamiltonians. Phys. Rev. X 10, 011047 (2020).
Khemani, V., Hermele, M. & Nandkishore, R. Localization from hilbert space shattering: From theory to physical realizations. Phys. Rev. B 101, 174204 (2020).
Wang, X., Ghose, S., Sanders, B. C. & Hu, B. Entanglement as a signature of quantum chaos. Phys. Rev. E 70, 016217 (2004).
Kim, H. & Huse, D. A. Ballistic spreading of entanglement in a diffusive nonintegrable system. Phys. Rev. Lett. 111, 127205 (2013).
Parra-Murillo, C. A., Madroñero, J. & Wimberger, S. Quantum diffusion and thermalization at resonant tunneling. Phys. Rev. A 89, 053610 (2014).
Hosur, P., Qi, X.-L., Roberts, D. A. & Yoshida, B. Chaos in quantum channels. J. High Energy Phys. 2016, 1–49 (2016).
Maldacena, J., Shenker, S. H. & Stanford, D. A bound on chaos. J. High Energy Phys. 2016, 106 (2016).
Patel, A. A., Chowdhury, D., Sachdev, S. & Swingle, B. Quantum butterfly effect in weakly interacting diffusive metals. Phys. Rev. X 7, 031047 (2017).
Li, J. et al. Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator. Phys. Rev. X 7, 031011 (2017).
Chen, X., Zhou, T., Huse, D. A. & Fradkin, E. Out-of-time-order correlations in many-body localized and thermal phases. Annalen der Physik 529, 1600332 (2016).
Luitz, D. J. & Bar Lev, Y. Information propagation in isolated quantum systems. Phys. Rev. B 96, 020406 (2017).
Parker, D. E., Cao, X., Avdoshkin, A., Scaffidi, T. & Altman, E. A universal operator growth hypothesis. Phys. Rev. X 9, 041017 (2019).
Dağ, C. B. & Duan, L.-M. Detection of out-of-time-order correlators and information scrambling in cold atoms: Ladder-XX model. Phys. Rev. A 99, 052322 (2019).
Murthy, C. & Srednicki, M. Bounds on chaos from the eigenstate thermalization hypothesis. Phys. Rev. Lett. 123, 230606 (2019).
Xu, S. & Swingle, B. Accessing scrambling using matrix product operators. Nat. Phys. 16, 199–204 (2020).
Neill, C. et al. Ergodic dynamics and thermalization in an isolated quantum system. Nat. Phys. 12, 1037–1041 (2016).
Gärttner, M. et al. Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet. Nat. Phys. 13, 781–786 (2017).
Wei, K. X., Ramanathan, C. & Cappellaro, P. Exploring localization in nuclear spin chains. Phys. Rev. Lett. 120, 070501 (2018).
Landsman, K. A. et al. Verified quantum information scrambling. Nature 567, 61–65 (2019).
Meier, E. J., Ang’ong’a, J., An, F. A. & Gadway, B. Exploring quantum signatures of chaos on a floquet synthetic lattice. Phys. Rev. A 100, 013623 (2019).
Braumüller, J. et al. Probing quantum information propagation with out-of-time-ordered correlators. Nat. Phys. 18, 172–178 (2022).
Shen, H., Zhang, P., Fan, R. & Zhai, H. Out-of-time-order correlation at a quantum phase transition. Phys. Rev. B 96, 054503 (2017).
Heyl, M., Pollmann, F. & Dóra, B. Detecting equilibrium and dynamical quantum phase transitions in ising chains via out-of-time-ordered correlators. Phys. Rev. Lett. 121, 016801 (2018).
Dağ, C. B., Sun, K. & Duan, L.-M. Detection of quantum phases via out-of-time-order correlators. Phys. Rev. Lett. 123, 140602 (2019).
Sun, Z.-H., Cai, J.-Q., Tang, Q.-C., Hu, Y. & Fan, H. Out-of-time-order correlators and quantum phase transitions in the rabi and dicke models. Annalen der Physik 532, 1900270 (2020).
Dağ, C. B., Duan, L.-M. & Sun, K. Topologically induced prescrambling and dynamical detection of topological phase transitions at infinite temperature. Phys. Rev. B 101, 104415 (2020).
Pilatowsky-Cameo, S. et al. Positive quantum lyapunov exponents in experimental systems with a regular classical limit. Phys. Rev. E 101, 010202 (2020).
Xu, T., Scaffidi, T. & Cao, X. Does scrambling equal chaos? Phys. Rev. Lett. 124, 140602 (2020).
Friedrich, H. & Wintgen, H. The hydrogen atom in a uniform magnetic field – an example of chaos. Phys. Rep. 183, 37–79 (1989).
Eckhardt, B. Quantum mechanics of classically non-integrable systems. Phys. Rep. 163, 205–297 (1988).
Oganesyan, V. & Huse, D. A. Localization of interacting fermions at high temperature. Phys. Rev. B 75, 155111 (2007).
Lombardi, M. & Seligman, T. H. Universal and nonuniversal statistical properties of levels and intensities for chaotic rydberg molecules. Phys. Rev. A 47, 3571–3586 (1993).
Mehta, M. L.Random Matrices (Academic Press, 2004).
Haake, F.Quantum Signatures of Chaos (Springer, 2010).
Dyson, F. J. Statistical theory of the energy levels of complex systems. i. J. Mathematical Phys. 3, 140–156 (1962).
Bertini, B., Kos, P. & Prosen, T. Exact spectral form factor in a minimal model of many-body quantum chaos. Phys. Rev. lett. 121, 264101 (2018).
Chan, A., De Luca, A. & Chalker, J. T. Solution of a minimal model for many-body quantum chaos. Phys. Rev. X 8, 041019 (2018).
Friedman, A. J., Chan, A., De Luca, A. & Chalker, J. T. Spectral statistics and many-body quantum chaos with conserved charge. Phys. Rev. Lett. 123, 210603 (2019).
Flack, A., Bertini, B. & Prosen, T. Statistics of the spectral form factor in the self-dual kicked ising model. Phys. Rev. Res. 2 https://doi.org/10.1103/physrevresearch.2.043403 (2020).
Šuntajs, J., Bonča, J., Prosen, Tcv & Vidmar, L. Quantum chaos challenges many-body localization. Phys. Rev. E 102, 062144 (2020).
Sierant, P., Delande, D. & Zakrzewski, J. Thouless time analysis of anderson and many-body localization transitions. Phys. Rev. Lett. 124, 186601 (2020).
Sierant, P., Lewenstein, M. & Zakrzewski, J. Polynomially filtered exact diagonalization approach to many-body localization. Phys. Rev. Lett. 125, 156601 (2020).
Liao, Y., Vikram, A. & Galitski, V. Many-body level statistics of single-particle quantum chaos. Phys. Rev. Lett. 125 https://doi.org/10.1103/physrevlett.125.250601 (2020).
Roy, D. & Prosen, Tcv Random matrix spectral form factor in kicked interacting fermionic chains. Phys. Rev. E 102, 060202 (2020).
Chan, A., De Luca, A. & Chalker, J. T. Spectral lyapunov exponents in chaotic and localized many-body quantum systems. Phys. Rev. Res. 3, 023118 (2021).
Moudgalya, S., Prem, A., Huse, D. A. & Chan, A. Spectral statistics in constrained many-body quantum chaotic systems. Phys. Rev. Res. 3, 023176 (2021).
Bertini, B., Kos, P. & Prosen, T. Random matrix spectral form factor of dual-unitary quantum circuits. Commun. Mathematical Phys. 387, 597–620 (2021).
Garratt, S. J. & Chalker, J. T. Local pairing of feynman histories in many-body floquet models. Phys. Rev. X 11, 021051 (2021).
Šuntajs, J., Prosen, T. & Vidmar, L. Spectral properties of three-dimensional anderson model. Annals Phys. 435, 168469 (2021).
Garratt, S. & Chalker, J. Many-body delocalization as symmetry breaking. Phys. Rev. Lett. 127 https://doi.org/10.1103/physrevlett.127.026802 (2021).
Li, J., Prosen, T. & Chan, A. Spectral statistics of non-hermitian matrices and dissipative quantum chaos. Phys. Rev. Lett. 127, 170602 (2021).
Prakash, A., Pixley, J. H. & Kulkarni, M. Universal spectral form factor for many-body localization. Phys. Rev. Res. 3, L012019 (2021).
Liao, Y. & Galitski, V. Emergence of many-body quantum chaos via spontaneous breaking of unitarity. Phys. Rev. B 105, L140202 (2022).
Chan, A., Shivam, S., Huse, D. A. & Luca, A. D. Many-body quantum chaos and space-time translational invariance. Nat. Commun. 13 https://doi.org/10.1038/s41467-022-34318-1 (2022).
Cornelius, J., Xu, Z., Saxena, A., Chenu, A. & del Campo, A. Spectral filtering induced by non-hermitian evolution with balanced gain and loss: Enhancing quantum chaos. Phys. Rev. Lett. 128, 190402 (2022).
Šuntajs, J. & Vidmar, L. Ergodicity breaking transition in zero dimensions. Phys. Rev. Lett. 129, 060602 (2022).
Roy, D., Mishra, D. & Prosen, T. Spectral form factor in a minimal bosonic model of many-body quantum chaos. Phys. Rev.E 106 https://doi.org/10.1103/physreve.106.024208 (2022).
Shivam, S., De Luca, A., Huse, D. A. & Chan, A. Many-body quantum chaos and emergence of ginibre ensemble. Phys. Rev. Lett. 130, 140403 (2023).
Bunin, G., Foini, L. & Kurchan, J. Fisher zeroes and the fluctuations of the spectral form factor of chaotic systemshttps://arxiv.org/abs/2207.02473 (2022).
Gharibyan, H., Hanada, M., Shenker, S. H. & Tezuka, M. Onset of random matrix behavior in scrambling systems. J. High Energy Phys. 2018 https://doi.org/10.1007/JHEP07(2018)124 (2018).
Altland, A. & Bagrets, D. Quantum ergodicity in the syk model. Nuclear Physics B 930, 45–68 (2018).
Hunter-Jones, N. & Liu, J. Chaos and random matrices in supersymmetric SYK. J. High Energy Phys. 2018 https://doi.org/10.1007/jhep05(2018)202 (2018).
Winer, M., Jian, S.-K. & Swingle, B. Exponential ramp in the quadratic sachdev-ye-kitaev model. Phys. Rev. Lett. 125, 250602 (2020).
Khramtsov, M. & Lanina, E. Spectral form factor in the double-scaled syk model. J. High Energy Phys. 2021, 1–38 (2021).
Saad, P., Shenker, S. H. & Stanford, D. A semiclassical ramp in syk and in gravity (2019). 1806.06840.
Bousso, R. et al. Snowmass white paper: Quantum aspects of black holes and the emergence of spacetime https://arxiv.org/abs/2201.03096 (2022).
Duan, L.-M., Demler, E. & Lukin, M. D. Controlling spin exchange interactions of ultracold atoms in optical lattices. Phys. Rev. Lett. 91, 090402 (2003).
de la Cruz, J., Lerma-Hernández, S. & Hirsch, J. G. Quantum chaos in a system with high degree of symmetries. Phys. Rev. E 102, 032208 (2020).
Wittmann W, K., Castro, E. R., Foerster, A. & Santos, L. F. Interacting bosons in a triple well: Preface of many-body quantum chaos. Phys. Rev. E 105, 034204 (2022).
Łydżba, P. & Sowiński, T. Signatures of quantum chaos in low-energy mixtures of few fermions. Phys. Rev. A 106, 013301 (2022).
Vasilyev, D. V., Grankin, A., Baranov, M. A., Sieberer, L. M. & Zoller, P. Monitoring quantum simulators via quantum nondemolition couplings to atomic clock qubits. PRX Quantum 1 https://doi.org/10.1103/PRXQuantum.1.020302 (2020).
Joshi, L. K. et al. Probing many-body quantum chaos with quantum simulators. Phys. Rev. X 12 https://doi.org/10.1103/physrevx.12.011018 (2022).
Schlunk, S. et al. Signatures of quantum stability in a classically chaotic system. Phys. Rev. Lett. 90, 054101 (2003).
Wimberger, S. & Buchleitner, A. Saturation of fidelity in the atom-optics kicked rotor. J. Phys. B: Atomic, Mol. Optical Phys. 39, L145 (2006).
Wintgen, D. & Friedrich, H. Classical and quantum-mechanical transition between regularity and irregularity in a hamiltonian system. Phys. Rev. A 35, 1464–1466 (1987).
Buchleitner, A. & Kolovsky, A. R. Interaction-induced decoherence of atomic bloch oscillations. Phys. Rev. Lett. 91, 253002 (2003).
Le, A.-T., Morishita, T., Tong, X.-M. & Lin, C. D. Signature of chaos in high-lying doubly excited states of the helium atom. Phys. Rev. A 72, 032511 (2005).
Tomadin, A., Mannella, R. & Wimberger, S. Many-body interband tunneling as a witness of complex dynamics in the bose-hubbard model. Phys. Rev. Lett. 98, 130402 (2007).
Santos, L. F. & Rigol, M. Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization. Phys. Rev. E 81, 036206 (2010).
Parra-Murillo, C. A., Madroñero, J. & Wimberger, S. Two-band bose-hubbard model for many-body resonant tunneling in the wannier-stark system. Phys. Rev. A 88, 032119 (2013).
Lazarides, A., Das, A. & Moessner, R. Equilibrium states of generic quantum systems subject to periodic driving. Phys. Rev. E 90, 012110 (2014).
Rautenberg, M. & Gärttner, M. Classical and quantum chaos in a three-mode bosonic system. Phys. Rev. A 101, 053604 (2020).
Anh-Tai, T. D., Mikkelsen, M., Busch, T. & Fogarty, T. Quantum chaos in interacting bose-bose mixtures (2023). 2301.04818.
Fisher, M. P. A., Weichman, P. B., Grinstein, G. & Fisher, D. S. Boson localization and the superfluid-insulator transition. Phys. Rev. B 40, 546–570 (1989).
Law, C. K., Pu, H. & Bigelow, N. P. Quantum spins mixing in spinor bose-einstein condensates. Phys. Rev. Lett. 81, 5257–5261 (1998).
Sachdev, S. & Ye, J. Gapless spin-fluid ground state in a random quantum heisenberg magnet. Phys. Rev. Lett. 70, 3339 (1993).
Kitaev, A. A simple model of quantum holography. In KITP strings seminar and Entanglement (2015). http://online.kitp.ucsb.edu/online/entangled15/kitaev/; http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
Swingle, B., Bentsen, G., Schleier-Smith, M. & Hayden, P. Measuring the scrambling of quantum information. Phys. Rev. A94 https://doi.org/10.1103/physreva.94.040302 (2016).
Mandel, O. et al. Controlled collisions for multi-particle entanglement of optically trapped atoms. Nature 425, 937–940 (2003).
Gadway, B., Pertot, D., Reimann, R. & Schneble, D. Superfluidity of interacting bosonic mixtures in optical lattices. Phys. Rev. Lett. 105, 045303 (2010).
Stamper-Kurn, D. M. & Ueda, M. Spinor bose gases: Symmetries, magnetism, and quantum dynamics. Rev. Mod. Phys. 85, 1191–1244 (2013).
Anquez, M. et al. Quantum kibble-zurek mechanism in a spin-1 bose-einstein condensate. Phys. Rev. Lett. 116, 155301 (2016).
Yang, H.-X. et al. Observation of dynamical quantum phase transitions in a spinor condensate. Phys. Rev. A 100, 013622 (2019).
Evrard, B., Qu, A., Dalibard, J. & Gerbier, F. From many-body oscillations to thermalization in an isolated spinor gas. Phys. Rev. Lett. 126, 063401 (2021).
Guhr, T., Mueller-Groeling, A. & Weidenmueller, H. A. Random Matrix Theories in Quantum Physics: Common Concepts. Phys. Reports. 299, 189–425 (1998). ArXiv: cond-mat/9707301.
Wintersperger, K. et al. Realization of an anomalous floquet topological system with ultracold atoms. Nat. Phys. 16, 1058–1063 (2020).
Lellouch, S., Bukov, M., Demler, E. & Goldman, N. Parametric instability rates in periodically driven band systems. Phys. Rev. X 7, 021015 (2017).
Goldman, N. & Dalibard, J. Periodically driven quantum systems: effective hamiltonians and engineered gauge fields. Phys. Rev. X 4, 031027 (2014).
Schiulaz, M., Torres-Herrera, E. J. & Santos, L. F. Thouless and relaxation time scales in many-body quantum systems. Phys. Rev. B 99, 174313 (2019).
Pique, J. P., Chen, Y., Field, R. W. & Kinsey, J. L. Chaos and dynamics on 0.5–300 ps time scales in vibrationally excited acetylene: Fourier transform of stimulated-emission pumping spectrum. Phys. Rev. Lett. 58, 475–478 (1987).
Zewail, A. H. Femtochemistry. J. Phys. Chem. 97, 12427–12446 (1993).
Alekseyev, A. B. et al. Spin-orbit effects in photodissociation of sodium iodide. J. Chem. Phys. 113, 1514–1523 (2000).
Daley, A. J., Pichler, H., Schachenmayer, J. & Zoller, P. Measuring entanglement growth in quench dynamics of bosons in an optical lattice. Phys. Rev. Lett. 109, 020505 (2012).
Lavoine, L., Hammond, A., Recati, A., Petrov, D. S. & Bourdel, T. Beyond-mean-field effects in rabi-coupled two-component bose-einstein condensate. Phys. Rev. Lett. 127, 203402 (2021).
Lukin, A. et al. Probing entanglement in a many-body–localized system. Science 364, 256–260 (2019).
Bordia, P. et al. Probing slow relaxation and many-body localization in two-dimensional quasiperiodic systems. Phys. Rev. X 7, 041047 (2017).
Ho, T.-L. Spinor bose condensates in optical traps. Phys. Rev. Lett. 81, 742 (1998).
Yi, S., Müstecaplioğlu, Ö., Sun, C.-P. & You, L. Single-mode approximation in a spinor-1 atomic condensate. Phys. Rev. A 66, 011601 (2002).
Mistakidis, S. I. et al. Cold atoms in low dimensions–a laboratory for quantum dynamics. arXiv preprint arXiv:2202.11071 (2022).
Bookjans, E. M., Vinit, A. & Raman, C. Quantum phase transition in an antiferromagnetic spinor bose-einstein condensate. Phys. Rev. Lett. 107, 195306 (2011).
Zhang, W., Zhou, D. L., Chang, M.-S., Chapman, M. S. & You, L. Coherent spin mixing dynamics in a spin-1 atomic condensate. Phys. Rev. A 72, 013602 (2005).
Dağ, C. B., Wang, S.-T. & Duan, L.-M. Classification of quench-dynamical behaviors in spinor condensates. Phys. Rev. A 97, 023603 (2018).
Yi, S. & Pu, H. Magnetization, squeezing, and entanglement in dipolar spin-1 condensates. Phys. Rev. A 73, 023602 (2006).
Kajtoch, D. & Witkowska, E. Spin squeezing in dipolar spinor condensates. Phys. Rev. A 93, 023627 (2016).
Vermersch, B., Elben, A., Dalmonte, M., Cirac, J. I. & Zoller, P. Unitary n-designs via random quenches in atomic hubbard and spin models: Application to the measurement of rényi entropies. Phys. Rev. A 97, 023604 (2018).
Poulin, D., Laflamme, R., Milburn, G. J. & Paz, J. P. Testing integrability with a single bit of quantum information. Phys. Rev. A 68, 022302 (2003).
Müller, M., Lesanovsky, I., Weimer, H., Büchler, H. P. & Zoller, P. Mesoscopic rydberg gate based on electromagnetically induced transparency. Phys. Rev. Lett. 102, 170502 (2009).
Jiang, L. et al. Anyonic interferometry and protected memories in atomic spin lattices. Nat. Phys. 4, 482–488 (2008).
Sherson, J. F. et al. Single-atom-resolved fluorescence imaging of an atomic mott insulator. Nature 467, 68–72 (2010).
Daley, A. J., Kollath, C., Schollwöck, U. & Vidal, G. Time-dependent density-matrix renormalization-group using adaptive effective hilbert spaces. J. Statistical Mechanics: Theory Exp. 2004, P04005 (2004).
Cao, L. et al. A unified ab initio approach to the correlated quantum dynamics of ultracold fermionic and bosonic mixtures. J. Chem. Phys. 147, 044106 (2017).
Mistakidis, S. I., Koutentakis, G. M., Grusdt, F., Schmelcher, P. & Sadeghpour, H. R. Inducing spin-order with an impurity: phase diagram of the magnetic bose polaron. New J. Phys. 24, 083030 (2022).
Yukawa, E., Ueda, M. & Nemoto, K. Classification of spin-nematic squeezing in spin-1 collective atomic systems. Phys. Rev. A 88, 033629 (2013).
Pu, H., Law, C. K., Raghavan, S., Eberly, J. H. & Bigelow, N. P. Spin-mixing dynamics of a spinor bose-einstein condensate. Phys. Rev. A 60, 1463–1470 (1999).
Elben, A., Vermersch, B., Dalmonte, M., Cirac, J. I. & Zoller, P. Rényi entropies from random quenches in atomic hubbard and spin models. Phys. Rev. Lett. 120, 050406 (2018).
Acknowledgements
The authors 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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C.B.D., S.I.M, A.C. and H.R.S. wrote and revised the Manuscript, discussed and interpreted the results. C.B.D. conceived and led the project, and developed the code behind the data. S.I.M. and C.B.D. performed simulations and analyzed the results. A.C. and C.B.D. developed the experimental protocol.
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Dağ, C.B., Mistakidis, S.I., Chan, A. et al. Many-body quantum chaos in stroboscopically-driven cold atoms. Commun Phys 6, 136 (2023). https://doi.org/10.1038/s42005-023-01258-1
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DOI: https://doi.org/10.1038/s42005-023-01258-1
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