Quantum Fluctuations of the Center of Mass and Relative Parameters of Nonlinear Schrödinger Breathers.

We study quantum fluctuations of macroscopic parameters of a nonlinear Schrödinger breather-a nonlinear superposition of two solitons, which can be created by the application of a fourfold quench of the scattering length to the fundamental soliton in a self-attractive quasi-one-dimensional Bose gas. The fluctuations are analyzed in the framework of the Bogoliubov approach in the limit of a large number of atoms N, using two models of the vacuum state: white noise and correlated noise. The latter model, closer to the ab initio setting by construction, leads to a reasonable agreement, within 20% accuracy, with fluctuations of the relative velocity of constituent solitons obtained from the exact Bethe-ansatz results [Phys. Rev. Lett. 119, 220401 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.220401] in the opposite low-N limit (for N≤23). We thus confirm, for macroscopic N, the breather dissociation time to be within the limits of current cold-atom experiments. Fluctuations of soliton masses, phases, and positions are also evaluated and may have experimental implications.

Rise and Fall of a Bright Soliton in an Optical Lattice.

We study an ultracold atomic gas with attractive interactions in a one-dimensional optical lattice. We find that its excitation spectrum displays a quantum soliton band, corresponding to N-particle bound states, and a continuum band of other, mostly extended, states. For a system of a finite size, the two branches are degenerate in energy for weak interactions, while a gap opens above a threshold value of the interaction strength. We find that the interplay between degenerate extended and bound states has important consequences for both static and dynamical properties of the system. In particular, the solitonic states turn out to be protected from spatial perturbations and random disorder. We discuss how such dynamics implies that our system effectively provides an example of a quantum many-body system that, with the variation of the bosonic lattice filling, crosses over from integrable nonergodic to nonintegrable ergodic dynamics, through nonintegrable-nonergodic regimes.

Dissociation of One-Dimensional Matter-Wave Breathers due to Quantum Many-Body Effects.

We use the ab initio Bethe ansatz dynamics to predict the dissociation of one-dimensional cold-atom breathers that are created by a quench from a fundamental soliton. We find that the dissociation is a robust quantum many-body effect, while in the mean-field (MF) limit the dissociation is forbidden by the integrability of the underlying nonlinear Schrödinger equation. The analysis demonstrates the possibility to observe quantum many-body effects without leaving the MF range of experimental parameters. We find that the dissociation time is of the order of a few seconds for a typical atomic-soliton setting.

Geometry of quantum observables and thermodynamics of small systems.

We show that for classical and quantum observables, the integrability-to-ergodicity transition leaves constant the sum of (a) the ensemble variance of the temporal average and (b) the ensemble average of temporal variance. The induced Frobenius (Hilbert-Schmidt) geometry of quantum observables encodes how eigenstate thermalization appears, the inverse participation ratio decreases, and the integrals of motion disappear during the transition. We use it to optimize the set of conserved quantities entering the generalized Gibbs ensemble for integrable, near-integrable, or mesoscopic systems.

Thermalization and its mechanism for generic isolated quantum systems.

An understanding of the temporal evolution of isolated many-body quantum systems has long been elusive. Recently, meaningful experimental studies of the problem have become possible, stimulating theoretical interest. In generic isolated systems, non-equilibrium dynamics is expected to result in thermalization: a relaxation to states in which the values of macroscopic quantities are stationary, universal with respect to widely differing initial conditions, and predictable using statistical mechanics. However, it is not obvious what feature of many-body quantum mechanics makes quantum thermalization possible in a sense analogous to that in which dynamical chaos makes classical thermalization possible. For example, dynamical chaos itself cannot occur in an isolated quantum system, in which the time evolution is linear and the spectrum is discrete. Some recent studies even suggest that statistical mechanics may give incorrect predictions for the outcomes of relaxation in such systems. Here we demonstrate that a generic isolated quantum many-body system does relax to a state well described by the standard statistical-mechanical prescription. Moreover, we show that time evolution itself plays a merely auxiliary role in relaxation, and that thermalization instead happens at the level of individual eigenstates, as first proposed by Deutsch and Srednicki. A striking consequence of this eigenstate-thermalization scenario, confirmed for our system, is that knowledge of a single many-body eigenstate is sufficient to compute thermal averages-any eigenstate in the microcanonical energy window will do, because they all give the same result.

Example of a quantum anomaly in the physics of ultracold gases.

In this Letter, we propose an experimental scheme for the observation of a quantum anomaly--quantum-mechanical symmetry breaking--in a two-dimensional harmonically trapped Bose gas. The anomaly manifests itself in a shift of the monopole excitation frequency away from the value dictated by the Pitaevskii-Rosch dynamical symmetry [L. P. Pitaevskii and A. Rosch, Phys. Rev. A 55, R853 (1997)]. While the corresponding classical Gross-Pitaevskii equation and the hydrodynamic equations derived from it do exhibit this symmetry, it is--as we show in our paper--violated under quantization. The resulting frequency shift is of the order of 1% of the carrier, well in reach for modern experimental techniques. We propose using the dipole oscillations as a frequency gauge.

Typical, finite baths as a means of exact simulation of open quantum systems.

There is presently considerable interest in accurately simulating the evolution of open systems for which Markovian master equations fail. Examples are systems that are time dependent and/or strongly damped. A number of elegant methods have now been devised to do this, but all use a bath consisting of a continuum of harmonic oscillators. While this bath is clearly appropriate for, e.g., systems coupled to the electromagnetic field, it is not so clear that it is a good model for generic many-body systems. Here we explore a different approach to exactly simulating open systems: using a finite bath chosen to have certain key properties of thermalizing many-body systems. To explore the numerical resources required by this method to approximate an open system coupled to an infinite bath, we simulate a weakly damped system and compare to the evolution given by the relevant Markovian master equation. We obtain the Markovian evolution with reasonable accuracy by using an additional averaging procedure, and elucidate how the typicality of the bath generates the correct thermal steady state via the process of "eigenstate thermalization."

An exactly solvable model for the integrability-chaos transition in rough quantum billiards.

A central question of dynamics, largely open in the quantum case, is to what extent it erases a system's memory of its initial properties. Here we present a simple statistically solvable quantum model describing this memory loss across an integrability-chaos transition under a perturbation obeying no selection rules. From the perspective of quantum localization-delocalization on the lattice of quantum numbers, we are dealing with a situation where every lattice site is coupled to every other site with the same strength, on average. The model also rigorously justifies a similar set of relationships, recently proposed in the context of two short-range-interacting ultracold atoms in a harmonic waveguide. Application of our model to an ensemble of uncorrelated impurities on a rectangular lattice gives good agreement with ab initio numerics.

Supersymmetric quantum mechanics and solitons of the sine-Gordon and nonlinear Schrödinger equations.

We present a case demonstrating the connection between supersymmetric quantum mechanics (SUSYQM), reflectionless scattering, and soliton solutions of integrable partial differential equations. We show that the members of a class of reflectionless Hamiltonians, namely, Akulin's Hamiltonians, are connected via supersymmetric chains to a potential-free Hamiltonian, explaining their reflectionless nature. While the reflectionless property in question has been mentioned in the literature for over two decades, the enabling algebraic mechanism was previously unknown. Our results indicate that the multisoliton solutions of the sine-Gordon and nonlinear Schrödinger equations can be systematically generated via the supersymmetric chains connecting Akulin's Hamiltonians. Our findings also explain a well-known but little-understood effect in laser physics: when a two-level atom, initially in the ground state, is subjected to a laser pulse of the form V(t) = (nh/τ)/cosh(t/τ), with n being an integer and τ being the pulse duration, it remains in the ground state after the pulse has been applied, for any choice of the laser detuning.

Dynamics of relaxation and fluctuations of the equilibrium state in an incompletely chaotic system.

An incompletely chaotic system, with a perturbation of the integrable part that does not obey selection rules, relaxes to an equilibrium state that lies between the initial state and thermal equilibrium. We analyze here a system of two atoms in a circular transversally harmonic waveguide. The dynamics of expectation values of generic observables and their fluctuations in the long-time limit are studied for this model. The relaxation demonstrates a nonexponential behavior and slows down as the initial-state energy increases. The fluctuation amplitude has a tendency to decrease with an increase of the initial state width.

Threshold for chaos and thermalization in the one-dimensional mean-field bose-hubbard model.

We study the threshold for chaos and its relation to thermalization in the 1D mean-field Bose-Hubbard model, which, in particular, describes atoms in optical lattices. We identify the threshold for chaos, which is finite in the thermodynamic limit, and show that it is indeed a precursor of thermalization. Far above the threshold, the state of the system after relaxation is governed by the usual laws of statistical mechanics.

Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons.

In this Letter we pose the question of whether a many-body quantum system with a full set of conserved quantities can relax to an equilibrium state, and, if it can, what the properties of such a state are. We confirm the relaxation hypothesis through an ab initio numerical investigation of the dynamics of hard-core bosons on a one-dimensional lattice. Further, a natural extension of the Gibbs ensemble to integrable systems results in a theory that is able to predict the mean values of physical observables after relaxation. Finally, we show that our generalized equilibrium carries more memory of the initial conditions than the usual thermodynamic one. This effect may have many experimental consequences, some of which have already been observed in the recent experiment on the nonequilibrium dynamics of one-dimensional hard-core bosons in a harmonic potential [T. Kinoshita et al., Nature (London) 440, 900 (2006)10.1038/nature04693].

Rigorous approach to the problem of ultraviolet divergencies in dilute Bose gases.

In this Letter we consider a system of N pairwise finite-range interacting atoms and prove rigorously that in the zero-range interaction limit all the eigenstates and eigenenergies of the Hamiltonian converge to those corresponding to N atoms interacting via the Fermi-Huang regularized pseudopotential. Next, we show that the latter eigensystem (if treated exactly) is invariant under a nontrivial transformation of the interaction potential. Finally, we realize that most of the approximate schemes of many-body physics do not exhibit this invariance: We use this property to resolve all inconsistencies of the Hartree-Fock-Bogoliubov variational formalism known thus far.

Producing Bose-Einstein condensates using optical lattices.

We relate the entropies of ensembles of atoms in optical lattices to atoms in simple traps. We then determine which ensembles of lattice-bound atoms will adiabatically transform into a Bose condensate. This shows a feasible approach to Bose condensation without evaporative cooling.

Short-distance correlation properties of the Lieb-Liniger system and momentum distributions of trapped one-dimensional atomic gases.

We derive exact closed-form expressions for the first few terms of the short-distance Taylor expansion of the one-body correlation function of the Lieb-Liniger gas. As an intermediate result, we obtain the high-p asymptotics of the momentum distribution of both free and harmonically trapped atoms and show that it obeys a universal 1/p(4) law for all values of the interaction strength. We discuss the ways to observe the predicted momentum distributions experimentally, regarding them as a sensitive identifier for the Tonks-Girardeau regime of strong correlations.