Quantum chaotic interacting N -particle systems are assumed to show fast and irreversible spreading of quantum information on short (Ehrenfest) time scales ∼ log N . Here we show that, near criticality, certain many-body systems exhibit fast initial scrambling, followed subsequently by oscillatory behavior between reentrant localization and delocalization of information in Hilbert space. We consider both integrable and nonintegrable quantum critical bosonic systems with attractive contact interaction that exhibit locally unstable dynamics in the corresponding many-body phase space of the large-N limit. Semiclassical quantization of the latter accounts for many-body correlations in excellent agreement with simulations. Most notably, it predicts an asymptotically constant local level spacing /τ , again given by τ ∼ log N . This unique timescale governs the long-time behavior of out-of-time-order correlators that feature quasi-periodic recurrences indicating reversibility.The dynamics of quantum information in complex many-body (MB) systems presently attracts a lot of attention [1, 2] ranging from atomic and condensed quantum matter to high energy physics. The evolution of an (excited) quantum MB system towards a state of thermal equilibrium usually goes along with the scrambling of quantum correlations, encoded in the initial state, across the system's many degrees of freedom. Such dynamics requires an improved understanding of MB quantum chaos and the link with thermalization [3-6] and its suppression [1,7,8].Echo protocols, measuring how a perturbation affects successive forward and backward propagations in time, sensitively probe the stability of complex quantum dynamics. Here, out-of-time-order correlators (OTOCs) [9-11]play a central role, with first experimental implementations [12-14], allowing to distinguish various classes of MB systems by their operator growth. On the one side there are slow scramblers, such as systems in the MB localized phase exhibiting logarithmically slow operator spreading [15-18] or, e.g., Luttinger liquids [19] showing only quadratic increase. On the other side, an exponentially fast initial growth of OTOCs is commonly viewed as a quantum signature of MB chaos. Examples comprise systems with holographic duals to black holes [10,20], the SYK-model [11,[21][22][23], and condensed matter systems close to a quantum phase transition (QPT) [24][25][26][27] or exhibiting chaos in the classical limit of large particle number N . In such large-N systems, the exponential growth rate for OTOCs is given by the Lyapunov exponent of their classical counterpart [10,[27][28][29][30][31][32][33][34] and prevails up to the Ehrenfest log N time where MB quantum interference sets in [32,35]. Subsequent OTOC time evolution towards an ergodic limit is then often governed by slow classical modes [36].Here we show that exponentially fast scrambling need not necessarily lead to quantum information loss: There exist systems exhibiting initial growth of complexity without relaxation, i.e., after a quench to ...

The interplay between single-particle interference and quantum indistinguishability leads to signature correlations in many-body scattering. We uncover these with a semiclassical calculation of the transmission probabilities through mesoscopic cavities for systems of noninteracting particles. For chaotic cavities we provide the universal form of the first two moments of the transmission probabilities over ensembles of random unitary matrices, including weak localization and dephasing effects. If the incoming many-body state consists of two macroscopically occupied wave packets, their time delay drives a quantum-classical transition along a boundary determined by the bosonic birthday paradox. Mesoscopic chaotic scattering of Bose-Einstein condensates is, then, a realistic candidate to build a boson sampler and to observe the macroscopic Hong-Ou-Mandel effect. DOI: 10.1103/PhysRevLett.116.100401 In quantum mechanics, identical particles are indistinguishable and their very identity is, then, affected by quantum fluctuations and interference effects. A prominent type of many-body (MB) correlations is exemplified by the celebrated Hong-Ou-Mandel (HOM) effect [1], by now the standard indicator of MB coherence in quantum optics. There, the probability of observing two photons leaving in different arms of a beam splitter is measured. As a function of the delay between the arrival times of the incoming pulses, the coincidence probability shows a characteristic dip that can be seen as an effective quantum-classical transition (QCT), where the difference in arrival times dephases the MB interference due to quantum indistinguishability [2]. In recent years, a wealth of hallmark experimental studies of MB scattering has gone beyond this scenario [3][4][5][6][7][8][9]. The aim is to reach a regime where for a random single-particle (SP) scattering matrix σ, and due to MB interference, the complexity in the calculation of MB scattering probabilities as a function of σ beats classical computers; this is called the boson sampling (BS) problem [10]. However, while current optical devices [5,9] reach photon occupations (below 6) far from the required regime of large number of particles, it is not clear how to sample σ uniformly on platforms based on trapped ions [11], cold atoms [12], and spin chains [13].Here we study mesoscopic MB scattering of massive particles depicted in Fig. 1(a). While formally identical to the optical situation in that it relates SP scattering matrices with MB scattering probabilities, it allows for large occupations through, e.g., Bose-Einstein condensation. Moreover, a standard result from quantum chaos [14] says that complex SP interference due to classical chaos inside such a mesoscopic scattering cavity Ω transforms averages over small changes of the incoming energies into averages over an appropriate ensemble of unitary matrices, thus providing a genuine sampling over random scattering matrices. With experimental techniques for preparation

Abstract. We present a novel analytical approach for the calculation of the mean density of states in many-body systems made of confined indistinguishable and noninteracting particles. Our method makes explicit the intrinsic geometry inherent in the symmetrization postulate and, in the spirit of the usual Weyl expansion for the smooth part of the density of states in single-particle confined systems, our results take the form of a sum over clusters of particles moving freely around manifolds in configuration space invariant under elements of the group of permutations. Being asymptotic, our approximation gives increasingly better results for large excitation energies and we formally confirm that it coincides with the celebrated Bethe estimate in the appropriate region. Moreover, our construction gives the correct high energy asymptotics expected from general considerations, and shows that the emergence of the fermionic ground state is actually a consequence of an extremely delicate large cancellation effect. Remarkably, our expansion in cluster zones is naturally incorporated for systems of interacting particles, opening the road to address the fundamental problem about the interplay between confinement and interactions in many-body systems of identical particles.PACS numbers: 74.20.Fg, 75.10.Jm, 71.10.Li, 73.21.La A geometrical approach to the mean density of states in many-body quantum systems 2

We show that a theory of complex scattering between many-body (Fock) states can be constructed such that its classical limit is a canonical transformation thus encoding quantum interference in the semiclassical form of the associated unitary operator. Based on this idea, we study the different coherent effects expected under different choices of the many-body states and provide different representations of the associated transition probabilities. In this way, we derive exact relations and representations of the scattering process that can be used to attack timely problems related with Boson Sampling. Linear scattering of photonsWe consider a typical scattering scenario, where a highly coherent many photon state of light is injected through waveguides into a complex array of optical elements, such as e.g. in [1][2][3][4][5]. We further assume that decoherence and dephasing due to losses and /or coupling with uncontrolled degrees of freedom can be neglected. The simulation of such a scattering process between multiparticle photonic (or in general, bosonic) input and output states is a computationally hard problem because it involves, as shown bellow, the calculation of permanents of large matrices. The complexity of this problem is expected to render the problem of sampling the space of matrices with a distribution given by their permanents, the Boson Sampling (BS) problem, also hard. Thus, a quantum optical device that samples for us scattering probabilities between many-body states constitutes a quantum computer that eventually beats any classical computer [6] in the BS task, an observation that has attracted enormous attention during the last years [7][8][9][10][11].The physical operation of our scattering device consists of mapping the incoming many-photon states |in〉 into the output states |out〉. By injecting the same incoming state several times and counting the number of times we get |out〉 as output, we will eventually obtain the transition probabilityand our goal is to study this quantity.As any other quantum state of the field, the |in〉, |out〉 states belong to the Hilbert (Fock) space H of the system, which consists of all possible linear combinations of Fock states [12] |n〉 := |n 1 , n 2 , . . . , n M 〉specifying the set of integer occupation numbers n 1 , . . . , n M . An occupation number n i specifies how many photons (bosons) occupy the i th single-particle state. The choice of these channels (or orbitals) is a matter of convenience, depending on the particular features of the system. In the scattering problem there are two preferred options to construct the Fock space, namely, by defining occupation numbers specifying how many photons occupy a given single-particle state with either incoming or outgoing boundary conditions in the asymptotic region far away from the scatterer. The operators that create a particle in the case of given incoming boundary conditions are denoted byb † , and their action on the vacuum state |0〉 produces Fock states in the incoming modes [12]:

Due to the vast growth of the many-body level density with excitation energy, its smoothed form is of central relevance for spectral and thermodynamic properties of interacting quantum systems. We compute the cumulative of this level density for confined one-dimensional continuous systems with repulsive short-range interactions. We show that the crossover from an ideal Bose gas to the strongly correlated, fermionized gas, i.e., partial fermionization, exhibits universal behaviour: Systems with very few up to many particles share the same underlying spectral features. In our derivation we supplement quantum cluster expansions with short-time dynamical information. Our nonperturbative analytical results are in excellent agreement with numerics for systems of experimental relevance in cold atom physics, such as interacting bosons on a ring (Lieb-Liniger model) or subject to harmonic confinement. Our method provides predictions for excitation spectra that enable access to finite-temperature thermodynamics in large parameter ranges.