Violent relaxation is a process that occurs in systems with long-range interactions. It has the peculiar feature of dramatically amplifying small perturbations, and rather than driving the system to equilibrium, it instead leads to slowly evolving configurations known as quasistationary states that fall outside the standard paradigm of statistical mechanics. Violent relaxation was originally identified in gravity-driven stellar dynamics; here, we extend the theory into the quantum regime by developing a quantum version of the Hamiltonian mean field (HMF) model which exemplifies many of the generic properties of long-range interacting systems. The HMF model can either be viewed as describing particles interacting via a cosine potential, or equivalently as the kinetic XY model with infinite-range interactions, and its quantum fluid dynamics can be obtained from a generalized Gross-Pitaevskii equation. We show that singular caustics that form during violent relaxation are regulated by interference effects in a universal way described by Thom's catastrophe theory applied to waves and this leads to emergent length scales and timescales not present in the classical problem. In the deep quantum regime we find that violent relaxation is suppressed altogether by quantum zero-point motion. Our results are relevant to laboratory studies of self-organization in cold atomic gases with long-range interactions.
Violent relaxation is a process that occurs in systems with long-range interactions. It has the peculiar feature of dramatically amplifying small perturbations, and rather than driving the system to equilibrium, it instead leads to slowly evolving configurations known as quasistationary states that fall outside the standard paradigm of statistical mechanics. Violent relaxation was originally identified in gravity-driven stellar dynamics; here, we extend the theory into the quantum regime by developing a quantum version of the Hamiltonian mean field (HMF) model which exemplifies many of the generic properties of long-range interacting systems. The HMF model can either be viewed as describing particles interacting via a cosine potential, or equivalently as the kinetic XY model with infinite-range interactions, and its quantum fluid dynamics can be obtained from a generalized Gross-Pitaevskii equation. We show that singular caustics that form during violent relaxation are regulated by interference effects in a universal way described by Thom's catastrophe theory applied to waves and this leads to emergent length scales and timescales not present in the classical problem. In the deep quantum regime we find that violent relaxation is suppressed altogether by quantum zero-point motion. Our results are relevant to laboratory studies of self-organization in cold atomic gases with long-range interactions.
In general, classical fully-connected systems are known to undergo violent relaxation. This phenomenon refers to the relaxation of observables to stationary, non-thermal, values on a finite timescale, despite their long-time dynamics being dominated by mean-field effects in the thermodynamic limit. Here, we analyze the ``quantum" violent relaxation by studying the dynamics of generic many-body systems with two-body, all-to-all, interactions in the thermodynamic limit. We show that, in order for violent relaxation to occur very specific conditions on the spectrum of the mean-field effective Hamiltonian have to be met. These conditions are hardly met and ``quantum" violent relaxation is observed rarely with respect to its classical counterpart. Our predictions are validated by the study of a spin model which, depending on the value of the coupling, shows a transition between violent-relaxation and a generic prethermal phase. We also analyze a spin version of the quantum Hamiltonian-Mean-Field model, which is shown not to exhibit violent-relaxation. Finally, we discuss how the violent-relaxation picture emerges back in the classical limit. Our results demonstrate how, even in the mean-field regime, quantum effects have a rather dramatic impact on the dynamics, paving the way to a better understanding of light-matter coupled systems.
Systems with long-range interactions, while relaxing towards equilibrium, sometimes get trapped in long-lived non-Boltzmann quasistationary states (QSS) which have lifetimes that grow algebraically with the system size. Such states have been observed in models of globally coupled particles that move under Hamiltonian dynamics either on a unit circle or on a unit spherical surface. Here, we address the ubiquity of QSS in long-range systems by considering a different dynamical setting. Thus, we consider an anisotropic Heisenberg model consisting of classical Heisenberg spins with mean-field interactions and evolving under classical spin dynamics. Our analysis of the corresponding Vlasov equation for time evolution of the phase space distribution shows that in a certain energy interval, relaxation of a class of initial states occurs over a timescale which grows algebraically with the system size. We support these findings by extensive numerical simulations. This work further supports the generality of occurrence of QSS in long-range systems evolving under Hamiltonian dynamics.
While the dynamics of fully connected systems is dominated by mean-field effect, in the classical limit the single-particle observables are observed to relax toward a nonthermal stationary value. This phenomenon is known as violent relaxation and it is generally absent at the quantum-mechanical level, where single-particle observables exhibits long-lived oscillations. In this paper we explain this discrepancy by determining some very restrictive conditions that quantum single-site Hamiltonians should meet in order for the system to undergo relaxation. We thus check them by introducing a new model (the so-called w model) which can exhibit dynamical phase transition among thermal behavior, persistent excitations, and violent relaxation. We also propose a way to implement it within light-matter coupling. We finally explain how the classical limit restore this behavior, thus showing that, even if the mean-field dynamics of quantum models is usually thought to be classical, quantum effects still play an important role in it.
Abstract We study the non-equilibrium dynamics of a one-dimensional Bose gas with long-range interactions that decay as ( 1 r α ) ( 0.5 < α < 4.0 ). We investigate exotic dynamics when the interactions are suddenly switched from strongly repulsive to strongly attractive, a procedure known to generate super-Tonks-Girardeau gases in systems with contact interactions. We find that relaxation is achieved through a complex intermediate dynamics demonstrated by violent fragmentation and chaotic delocalization. We establish that the relaxed state exhibits classical gaseous characteristics and an asymptotic state associated with unbounded entropy production. The phase diagram shows an exponential boundary between the coherent (quantum) gas and the chaotic (classical) gas. We show the universality of the dynamics by also presenting analogous results for spinless fermions. Weaker quench protocols give a certain degree of control over the relaxation process and induce a slower initial entropy growth. Our study showcases the complex relaxation behavior of tunable long-range interacting systems that could be engineered in state-of-the-art experiments, e.g. in trapped ions or Rydberg atoms.
We describe a new class of nonequilibrium quantum many-body phenomena in the form of networks of caustics that dominate the many-body wave function in the semiclassical regime following a sudden quench. It includes the light cone-like propagation of correlations as a particular case. Caustics are singularities formed by the birth and death of waves and form a hierarchy of universal patterns whose natural mathematical description is via catastrophe theory. Examples in classical waves range from rainbows and gravitational lensing in optics to tidal bores and rogue waves in hydrodynamics. Quantum many-body caustics are discretized by second quantization (``quantum catastrophes'') and live in Fock space, which can potentially have many dimensions. We illustrate these ideas using the Bose Hubbard dimer and trimer models, which are simple enough that the caustic structure can be elucidated from first principles and yet run the full range from integrable to nonintegrable dynamics. The dimer gives rise to discretized versions of fold and cusp catastrophes whereas the trimer allows for higher catastrophes including the codimension-3 hyperbolic and elliptic umbilics, which are organized by, and projections of, an 8-dimensional corank-2 catastrophe known as X9. These results describe a hitherto unrecognized form of universality in quantum dynamics organized by singularities that manifest as strong fluctuations in mode population probabilities.
Equations of the Schrödinger-Newton type appear naturally in different domains of research. In cosmology, they are used to model the formation of large scale structures in the universe, while, in astrophysics, they are employed as a semi-classical description for the evolution of elementary particles of dark matter. The Schrödinger-Newton equation is also used in optics, to describe the propagation of a beam of light through a medium with a thermo-optical nonlinearity. All of these systems, despite being very different from each other, are characterized by long range interactions. Long range interacting systems, relax generically to out-of-equilibrium quasi-stationary states, with significant astrophysical examples such as galaxies and globular clusters. These states cannot be described by standard statistical mechanics, but are formed through a process of very different nature, called violent relaxation. However, astrophysical time-scales are so large that it is not possible to directly observe the relaxation dynamics. In this thesis, we develop a table-top experimental model that captures the dynamics of long range interacting systems and allows to directly observe violent relaxation mechanism, leading to the formation of a table-top optical analogue of a galaxy. The experiment allows to control a range of parameters, including the nonlocal (gravitational) interaction strength and the strength of quantum gravity coupling ħ/m, thus providing an effective test-bed for gravitational models that cannot otherwise be directly studied in experimental settings. In addition, a central part of the thesis is devoted to the numerical study of the Schrödinger-Newton equation. Here, we present a new efficient numerical method which can be employed not only to solve the Schrödinger-Newton equation, but any Schrödinger-like equations, with different kind of interactions. The numerical method developed, which exploits the freedom provided by the gauge condition of the potential, is an improvement of the integrating factor technique. Optimal gauge conditions are derived considering the equation and the temporal numerical resolution with an adaptive embedded scheme of arbitrary order. We show that this optimization increases significantly the overall computational speed, sometimes by a factor five or more. Moreover, we make an extensive comparison of the new method developed, together with other popular integrators, commonly use to numerically solve this kind of equations. We focus in particular on splitting algorithms and methods belonging to the integrating factor family. Comparisons are done in one and two spatial dimensions, with different boundary conditions, both for the Schrödinger--Newton equation and the non-linear Schrödinger equation. We conclude that for the short range potential of the non-linear Schrödinger equation, integrating factor methods perform better than the Split-Step algorithm, while for the long range potential of the Schrödinger--Newton equation it depends on the particular system considered.