The emergence of quantum chaos in a system of trapped interacting bosons with external rotation is studied through spectral form factor (SFF) and power spectrum using exact diagonalization. Two distinct interaction regimes are considered: the moderate and the strong. In the moderate interaction regime, the SFF for the non-rotating case exhibits a dip-plateau structure with absence of linear ramp, reflecting integrable behavior arising from the macroscopic occupation of a single-particle state due to Bose-Einstein condensation. For the single-vortex state, the SFF exhibits a discernible linear ramp, consistent with pseudo-integrable behavior as a consequence of small depletion of the macroscopically occupied single-particle state with one unit of angular momentum. In the strong interaction regime for the non-rotating case, the transfer of bosons from the macroscopically occupied single-particle state to other incoherent states leads to emergence of a linear ramp with small span in SFF, indicating crossover to pseudo-integrable behavior, whereas for the single-vortex and the multi-vortex states, the span of the linear ramp increases, arising from the combined effects of interaction and rotation, leading to significant transfer of bosons out of the condensate and indicating strong chaotic behavior consistent with Gaussian orthogonal ensemble. The corresponding power spectrum results are consistent with the findings of SFF with power exponent lying in the interval $1 < α< 2$.
The emergence of quantum chaos in a system of trapped interacting bosons with external rotation is studied through spectral form factor (SFF) and power spectrum using exact diagonalization. Two distinct interaction regimes are considered: the moderate and the strong. In the moderate interaction regime, the SFF for the non-rotating case exhibits a dip-plateau structure with absence of linear ramp, reflecting integrable behavior arising from the macroscopic occupation of a single-particle state due to Bose-Einstein condensation. For the single-vortex state, the SFF exhibits a discernible linear ramp, consistent with pseudo-integrable behavior as a consequence of small depletion of the macroscopically occupied single-particle state with one unit of angular momentum. In the strong interaction regime for the non-rotating case, the transfer of bosons from the macroscopically occupied single-particle state to other incoherent states leads to emergence of a linear ramp with small span in SFF, indicating crossover to pseudo-integrable behavior, whereas for the single-vortex and the multi-vortex states, the span of the linear ramp increases, arising from the combined effects of interaction and rotation, leading to significant transfer of bosons out of the condensate and indicating strong chaotic behavior consistent with Gaussian orthogonal ensemble. The corresponding power spectrum results are consistent with the findings of SFF with power exponent lying in the interval 1<α<2.
We present an exact diagonalization study of the spectral properties of bosons harmonically confined in a quasi-2D plane and interacting via repulsive Gaussian potential. We consider the lowest 100 energy levels for systems of N=12,16, and 20 bosons in two distinct regimes: (a) when the interaction energy is small compared to the trap energy (moderate interaction) and (b) when the interaction energy is comparable to the trap energy (strong interaction), for the nonrotating (L_{z}=0) as well as the rotating single-vortex state (L_{z}=N). For higher angular momenta, L_{z}=2N and L_{z}=3N, only the strong interaction regime is considered. While the nearest-neighbor spacing distribution (NNSD) P(s) and the ratios of consecutive level spacings distribution P(r) are used to study the short-range correlations, the Dyson-Mehta Δ_{3} statistic and the level number variance Σ^{2}(L) are used to examine the long-range correlations. In the moderate interaction regime, the nonrotating system exhibits Poisson distribution, a characteristic of the regular energy spectra. In the strong interaction regime, the nonrotating system exhibits chaotic behavior signified by GOE distribution. Furthermore, in the rotating case for the single-vortex state (L_{z}=N) in the moderate interaction regime, the system exhibits signatures of weak chaos with some degree of regularity in the energy-level spectra. However, in the strong interaction regime for the rotating case with L_{z}=N, 2N and 3N, the system exhibits strong chaotic behavior. The rotation is found to contribute to enhancement of chaotic behavior in the system for both the moderate and the strong interaction regimes. Our results of NNSD analysis are supported by the analysis of the ratios of consecutive level spacings distribution P(r), which does not involve unfolding.
We numerically study the spectral statistics of open quantum many-body systems (OQMBS) as signatures of quantum chaos (or the lack thereof), using the dissipative spectral form factor (DSFF), a generalization of the spectral form factor to complex spectra. We show that the DSFF of chaotic OQMBS generically displays the quadratic ramp-plateau behaviour of the Ginibre ensemble from random matrix theory, in contrast to the linear ramp-plateau behaviour of the Gaussian ensemble in closed quantum systems. Furthermore, in the presence of many-body interactions, such RMT behaviour emerges only after a time scale τdev, which generally increases with system size for sufficiently large system size, and can be identified as the non-Hermitian analogue of the many-body Thouless time. The universality of the random matrix theory behavior is demonstrated by surveying twelve models of OQMBS, including random Kraus circuits (quantum channels) and random Lindbladians (Liouvillians) in several symmetry classes, as well as Lindbladians of paradigmatic models such as the Sachdev-Ye-Kitaev (SYK), XXZ, and the transverse field Ising models. We devise an unfolding and filtering procedure to remove variations of the averaged density of states which would otherwise hide the universal RMT-like signatures in the DSFF for chaotic OQMBS. Beyond chaotic OQMBS, we study the spectral statistics of non-chaotic OQMBS, specifically the integrable XX model and a system in the many-body localized (MBL) regime in the presence of dissipation, which exhibit DSFF behaviours distinct from the ramp-plateau behaviour of random matrix theory. Lastly, we study the DSFF of Lindbladians with the Hamiltonian term set to zero, i.e. only the jump operators are present, and demonstrate that the results of RMT universality and scaling of many-body Thouless time survive even without coherent evolution.
Ergodicity is a fundamental principle of statistical mechanics underlying the behavior of generic quantum many-body systems.However, how this universal many-body quantum chaotic regime emerges due to interactions remains largely a puzzle.This paper demonstrates using both heuristic arguments and a microscopic calculation that a dephasing mechanism, similar to Altshuler-Aronov-Khmelnitskii dephasing in the theory of localization, underlies this transition to chaos.We focus on the behavior of the spectral form factor (SFF) as a function of "time" t, which characterizes level correlations in the many-body spectrum.The SFF can be expressed as a sum over periodic classical orbits and its behavior hinges on the interference of trajectories related to each other by a time translation.In the absence of interactions, time-translation symmetry is present for each individual particle, which leads to a fast exponential growth of the SFF and correspondingly loss of correlations between many-body levels.Interactions lead to dephasing, which disrupts interference, and breaks the massive time-translation symmetry down to a global time-translation/energy conservation.This in turn gives rise to the hallmark linear-in-t ramp in the SFF reflecting Wigner-Dyson level repulsion.This general picture is supported by a microscopic analysis of an interacting many-body model.Specifically, we study the complex SYK 2 + SYK 2 2 model, which allows to tune between an integrable and chaotic regime.It is shown that the dephasing mass vanishes in the former case, which maps to the noninteracting complex SYK 2 model via a time reparameterization.In contrast, the chaotic regime gives rise to dephasing, which suppresses the exponential ramp of the noninteracting theory and induces correlations between many-body levels.
The emergence of random matrix spectral correlations in interacting quantum systems is a defining feature of quantum chaos. We study such correlations in terms of the spectral form factor and its moments in interacting chaotic few- and many-body systems, modeled by suitable random-matrix ensembles. We obtain the spectral form factor exactly for large Hilbert space dimension. Extrapolating those results to finite Hilbert space dimension we find a universal transition from the non-interacting to the strongly interacting case, which can be described as a simple combination of these two limits. This transition is governed by a single scaling parameter. In the bipartite case we derive similar results also for all moments of the spectral form factor. We confirm our results by extensive numerical studies and demonstrate that they apply to more realistic systems given by a pair of quantized kicked rotors as well. Ultimately we complement our analysis by a perturbative approach covering the small coupling regime.
Correlations between the energies of a system's spectrum are one of the defining features of quantum chaos. They can be probed using the spectral form factor (SFF). We investigate how each spectral distance contributes in building this two-point correlation function. Specifically, starting from the spectral distribution of kth-neighbor-level spacing (knLS), we provide analytical expressions for the kth-neighbor spectral form factor (knSFF). We do so for the three Gaussian random matrix ensembles and the ``Poisson'' ensemble of uncorrelated energy levels. We study the properties of the knSFF, namely its minimum value and the time at which this minimum is reached, as well as the energy spacing with the deepest knSFF. This allows us to quantify the contribution of each individual knLS to the SFF ramp, which is a characteristic feature of quantum chaos. In particular, we show how the onset of the ramp, characterized either by the dip or the Thouless time, shifts to shorter times as contributions from longer-range spectral distance are included. Interestingly, the even and odd neighbors contribute quite distinctively, the first being the most important to built the ramp. They respectively yield a resonance or antiresonance in the ramp. All of our analytical results are tested against numerical realizations of random matrices. We complete our analysis and show how the introduced tools help characterize the spectral properties of a physical many-body system by looking at the interacting XXZ Heisenberg model with local on-site disorder that allows transitioning between the chaotic and integrable regimes.
We consider systems of fermions evolved by noninteracting unitary circuits with correlated on-site potentials. When these potentials are drawn from the eigenvalue distribution of a circular random matrix ensemble, the single-particle sector exhibits chaotic dynamics. The corresponding many-body spectral form factors (SFFs) can be computed exactly, revealing signatures of single-particle chaos in the many-body spectral statistics. Due to the absence of interactions the SFF grows exponentially in time, a result which we demonstrate through simple arguments, scaling collapses, and closed-form evaluation of the SFF. Upon introducing interactions, the SFF crosses over to a linear growth regime consistent with many-body random matrix universality. Our exact results for the SFF provide a baseline for future studies of the crossover between single-particle and many-body random matrix behavior.