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arXiv:1206.3181 · New Journal of Physics

Universal dynamics on the way to thermalization

Boris Nowak, Jan Schole, Thomas Gasenzer

Published Sep 30, 2014• 45 citations
arXivHTMLPDF

Abstract

It is demonstrated how a many-body system far from thermal equilibrium can exhibit universal dynamics in passing a nonthermal fixed point. As an example, the process of Bose–Einstein (BE) condensation of a dilute cold gas is considered. If the particle flux into the low-energy modes, induced, for example by a cooling quench, is sufficiently strong, the Bose gas develops a characteristic power-law single-particle spectrum , and critical slowing down in time occurs. The fixed point is shown to be marked by the creation and dilution of tangled vortex lines. Alternatively, for a weak cooling quench and particle flux, the condensation process runs quasi-adiabatically, passing by the fixed point in the far distance, and the signatures of critical scaling remain absent.

DOI
10.1088/1367-2630/16/9/093052

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It is demonstrated how a many-body system far from thermal equilibrium can exhibit universal dynamics in passing a nonthermal fixed point. As an example, the process of Bose–Einstein (BE) condensation of a dilute cold gas is considered. If the particle flux into the low-energy modes, induced, for example by a cooling quench, is sufficiently strong, the Bose gas develops a characteristic power-law single-particle spectrum , and critical slowing down in time occurs. The fixed point is shown to be marked by the creation and dilution of tangled vortex lines. Alternatively, for a weak cooling quench and particle flux, the condensation process runs quasi-adiabatically, passing by the fixed point in the far distance, and the signatures of critical scaling remain absent.
Abstract Using a non-perturbative quantum kinetic framework, we develop a unified description of the far-from-equilibrium dynamics of three-dimensional Bose gases following cooling quenches across the Bose-Einstein condensation transition. By tracking the spatio-temporal evolution of the momentum distribution, we show that the equilibrium condensation threshold simultaneously acts as a dynamical critical point, separating distinct far-from-equilibrium universality classes governed by different non-equilibrium attractors. While quenches above the transition exhibit a single-timescale relaxation toward a thermal fixed point, quenches below the transition display a crossover from a transient weak-turbulence regime to a coarsening fixed point governed by the diffusive recombination of vortex lines. Quenches directly to the condensation threshold, finally, are controlled by a previously unidentified critical fixed point characterized by the superdiffusive spreading of critical fluctuations and a distinct set of dynamical exponents. Together, these dynamical scaling laws establish a far-from-equilibrium counterpart of the condensation phase transition, in which the equilibrium critical point also organizes the long-time non-equilibrium dynamics.
We outline a kinetic theory of nonthermal fixed points for the example of a dilute Bose gas, partially reviewing results obtained earlier, thereby extending, complementing, generalizing, and straightening them out. We study universal dynamics after a cooling quench, focusing on situations where the time evolution represents a pure rescaling of spatial correlations, with time defining the scale parameter. The nonequilibrium initial condition set by the quench induces a redistribution of particles in momentum space. Depending on conservation laws, this can take the form of a wave-turbulent flux or of a more general self-similar evolution, signaling the critically slowed approach to a nonthermal fixed point. We identify such fixed points using a nonperturbative kinetic theory of collective scattering between highly occupied long-wavelength modes. In contrast, a wave-turbulent flux, possible in the perturbative Boltzmann regime, builds up in a critically accelerated self-similar manner. A key result is the simple analytical universal scaling form of the nonperturbative many-body scattering matrix, for which we lay out the concrete conditions under which it applies. We derive the scaling exponents for the time evolution as well as for the power-law tail of the momentum distribution function, for a general dynamical critical exponent zzz and an anomalous scaling dimension $\ensuremath{\eta}$. The approach of the nonthermal fixed point is, in particular, found to involve a rescaling of momenta in time ttt by ${t}^{\phantom{\rule{0.16em}{0ex}}\ensuremath{\beta}}$, with $\ensuremath{\beta}=1/z$, within our kinetic approach independent of $\ensuremath{\eta}$. We confirm our analytical predictions by numerically evaluating the kinetic scattering integral as well as the nonperturbative many-body coupling function. As a side result, we obtain a possible finite-size interpretation of wave-turbulent scaling recently measured by Navon et al.
Using a non-perturbative quantum kinetic framework, we develop a unified description of the far-from-equilibrium dynamics of three-dimensional Bose gases following cooling quenches across the Bose-Einstein condensation transition. By tracking the spatio-temporal evolution of the momentum distribution, we show that the equilibrium condensation threshold simultaneously acts as a dynamical critical point, separating distinct far-from-equilibrium universality classes governed by different non-equilibrium attractors. While quenches above the transition exhibit a single-timescale relaxation toward a thermal fixed point, quenches below the transition display a crossover from a transient weak-turbulence regime to a coarsening fixed point governed by the diffusive recombination of vortex lines. Quenches directly to the condensation threshold, finally, are controlled by a previously unidentified critical fixed point characterized by the superdiffusive spreading of critical fluctuations and a distinct set of dynamical exponents. Together, these dynamical scaling laws establish a far-from-equilibrium counterpart of the condensation phase transition, in which the equilibrium critical point also organizes the long-time non-equilibrium dynamics.
Abstract This chapter considers nearly coherent Bose gases brought far out of equilibrium and their behavior in the context of connections between universal properties, (quasi-)topological field configurations, and turbulent dynamics. The isolated Bose gas, on its way back to thermal equilibrium, can approach metastable nonequilibrium configurations and spend a long time in their vicinity. In such configurations, termed nonthermal fixed points, the system shows universal long-range properties manifest through scaling, i.e. self-similar correlations. The time evolution near such fixed points undergoes critical slowing down. The spatial field pattern, at the same time, is characterized by the appearance of defects and domain formation whose geometry gives rise to the particular scaling laws seen in the correlation functions. An overall picture is obtained that connects well-known concepts for describing universal dynamics such as wave turbulence, superfluid turbulence, and (quasi-)topological excitations. The extension to fermionic and gauge fields should reveal many interesting new aspects.
In this thesis, we study numerically critical dynamics in the ultracold Bose gas in one and two spatial dimensions. We concentrate on two specific setups, both amenable for experimental realisation: Hamiltonian parameter quenches in a two-component Bose gas in one spatial dimension and a driven-dissipative single-component gas in two spatial \ndimensions. The setups are chosen to excite critical dynamics, either via quenches close to a quantum critical point or via nucleation of vortex defects. The goal is to identify critical scaling and universal scaling forms in the time evolution of the respective systems. The analysis for the two-component Bose gas reveals that short-time quench dynamics can be described by a universal crossover function, where the quench-induced energy appears as the relevant energy scale. For the single-component gas, we find a new universal phase of time evolution, characterised by an anomalously slow phase ordering process of vortex defects. We discuss our results in the light of concepts of prethermalisation, generalised Gibbs ensembles and non-thermal fixed points, for universal critical \nphenomena far from thermal equilibrium.
Nonthermal fixed points in the evolution of a quantum many-body system quenched far out of equilibrium manifest themselves in a scaling evolution of correlations in space and time. We develop a low-energy effective theory of nonthermal fixed points in a bosonic quantum many-body system by integrating out long-wavelength density fluctuations. The system consists of NNN distinguishable spatially uniform Bose gases with U(N)\text{U}(N)U(N)-symmetric interactions. The effective theory describes interacting Goldstone modes of the total and relative-phase excitations. It is similar in character to the nonlinear Luttinger-liquid description of low-energy phonons in a single dilute Bose gas, with the markable difference of a universal nonlocal coupling function depending, in the large-NNN limit, only on momentum, single-particle mass, and density of the gas. Our theory provides a perturbative description of the nonthermal fixed point, technically easy to apply to experimentally relevant cases with a small number of fields NNN. Numerical results for N=3N=3N=3 allow us to characterize the analytical form of the scaling function and confirm the analytically predicted scaling exponents. The predicted and observed exponentially suppressed coherence at short distances takes the form of that of a quasicondensate in low-dimensional equilibrium systems. The fixed point which is dominated by the relative phases is found to be Gaussian, while a non-Gaussian fixed point is anticipated to require scaling evolution with a distinctly lower power of time.