We study numerically and analytically isolated interacting quantum systems that are taken out of equilibrium instantaneously (quenched). The probability of finding the initial state in time, the so-called fidelity, decays fastest for systems described by full random matrices, where simultaneous many-body interactions are implied. In the realm of realistic systems with two-body interactions, the dynamics is slower and depends on the interplay between the initial state and the Hamiltonian characterizing the system. The fastest fidelity decay in this case is Gaussian and can persist until saturation. A simple general picture, in which the fidelity plays a central role, is also achieved for the short-time dynamics of few-body observables. It holds for initial states that are eigenstates of the observables. We also discuss the need to reassess analytical expressions that were previously proposed to describe the evolution of the Shannon entropy. Our analyses are mainly developed for initial states that can be prepared in experiments with cold atoms in optical lattices. INTRODUCTIONDespite the ubiquity of many-body quantum systems out of equilibrium, they are much less understood than quantum systems in equilibrium. To advance our understanding and to construct a general picture, it is necessary to identify the elements that lead to similar dynamics. Determining how fast these systems evolve in time [1-5] is also essential for the development of algorithms for quantum optimal control [6]. In these two contexts, the unitary evolution of isolated many-body quantum systems is of particular interest due, in part, to the connection with current experiments in optical lattices [7][8][9][10][11][12][13][14][15][16][17]. The latter are quasi-isolated systems, where coherent evolutions can be studied for very long times.The evolution of an isolated system can be initiated by changing instantaneously the parameters of a certain initial Hamiltonian which is brought into a new final Hamiltonian. This abrupt perturbation is referred to as a quench. The system starts off in an eigenstate of the initial Hamiltonian. The fidelity (return probability) [18,19], which is defined as the overlap between the initial state and its evolved counterpart, is a way to characterize the system evolution. This quantity is related to the Loschmidt echo. It is also analogous to the characteristic function of the probability distribution of work [20][21][22][23] and is therefore likely to find applications in quantum thermodynamics, particularly in studies related with the quantification of the work done to take quantum systems out of equilibrium. The fidelity decays exponentially when the final Hamiltonian is chaotic [24][25][26][27][28][29][30][31][32]. In fact, this behavior is expected to hold even in integrable Hamiltonians provided the initial state be sufficiently delocalized in the energy eigenbasis [33][34][35].Here, we extend the results obtained in Ref. [36] and show that the fidelity can have a faster than exponential behavior. The fidelity ...

Finite interacting Fermi systems with a mean-field and a chaos generating two-body interaction are modeled by one plus two-body embedded Gaussian orthogonal ensemble of random matrices with spin degree of freedom [called EGOE(1+2)-s]. Numerical calculations are used to demonstrate that, as lambda , the strength of the interaction (measured in the units of the average spacing of the single-particle levels defining the mean-field), increases, generically there is Poisson to GOE transition in level fluctuations, Breit-Wigner to Gaussian transition in strength functions (also called local density of states) and also a duality region where information entropy will be the same in both the mean-field and interaction defined basis. Spin dependence of the transition points lambda_{c} , lambdaF, and lambdad , respectively, is described using the propagator for the spectral variances and the formula for the propagator is derived. We further establish that the duality region corresponds to a region of thermalization. For this purpose we compared the single-particle entropy defined by the occupancies of the single-particle orbitals with thermodynamic entropy and information entropy for various lambda values and they are very close to each other at lambda=lambdad.

In a m particle quantum system, the rank of interactions and the nature of particles (fermions or bosons) can strongly affect the dynamics of the system. To explore this, we study non-equilibrium dynamics with the particles in a one-body mean-field and quenched by an interaction of bodyrank k = 2, 3, . . ., m. Using Fermionic Embedded Gaussian Orthogonal Ensembles (FEGOE) andBosonic Embedded Gaussian Orthogonal Ensembles (BEGOE) of one plus k-body interactions (also the Unitary variants FEGUE and BEGUE), it is seen that the short time decay of the survival probability of many-particle systems is given by the Fourier transform of the generating function v(E|q) of q-Hermite polynomials. Deriving formulas for q for both fermion and boson systems as a function of m, k and number of single particle states N , we have verified that the Fourier transform of v(E|q) agrees very well with numerical ensemble calculations for both fermion and boson systems. These results bridge the gap between the known results for k = 2 and k = m.

For a two-species boson system, it is possible to introduce a fictitious (F) spin for the bosons such that the two projections of F represent the two species. Then, for m bosons the total fictitious spin F takes values m/2, m/2 − 1,…, 0 or 1/2. For such a system with m number of bosons in Ω number of single-particle levels, each doubly degenerate, we introduce and analyze an embedded Gaussian orthogonal ensemble (GOE) of random matrices generated by random two-body interactions that conserve F-spin (BEGOE(1+2)-F); with degenerate single-particle levels, we have BEGOE(2)-F. Embedding algebra for BEGOE(1+2)-F ensemble is U(2Ω)⊃U(Ω)⊗SU(2) with SU(2) generating F-spin. A method for constructing the ensembles in fixed-(m, F) spaces has been developed. Numerical calculations show that for BEGOE(1+2)-F, the fixed-(m, F) density of states is close to Gaussian and level fluctuations follow the GOE in the dense limit. Similarly, generically there is Poisson to GOE transition in level fluctuations as the interaction strength (measured in the units of the average spacing of the single-particle levels defining the mean field) is increased. The interaction strength needed for the onset of the transition is found to decrease with increasing F. Formulas for the fixed-(m, F) space eigenvalue centroids and spectral variances are derived for a given member of the ensemble and also for the variance propagator for the fixed-(m, F) ensemble-averaged spectral variances. Using these, covariances in eigenvalue centroids and spectral variances are analyzed. The variance propagator clearly shows that the BEGOE(2)-F ensemble generates ground states with spin F = Fmax = m/2. Natural F-spin ordering (Fmax, Fmax − 1, Fmax − 2, …, 0 or 1/2) is also observed with random interactions. Going beyond these, we also introduce pairing symmetry in the space defined by BEGOE(1+2)-F. Expectation values of the pairing Hamiltonian show that random interactions generate ground states with a maximum value for the expectation value for a given F and in these it is largest for F = Fmax = m/2.