Turbulent scaling phenomena are studied in an ultracold Bose gas away from thermal equilibrium. Fixed points of the dynamical evolution are characterized in terms of universal scaling exponents of correlation functions. The scaling behavior is determined analytically in the framework of quantum field theory, using a nonperturbative approximation of the two-particle irreducible effective action. While perturbative Kolmogorov scaling is recovered at higher energies, scaling solutions with anomalously large exponents arise in the infrared regime of the turbulence spectrum. The extraordinary enhancement in the momentum dependence of long-range correlations could be experimentally accessible in dilute ultracold atomic gases. Such experiments have the potential to provide insight into dynamical phenomena directly relevant also in other present-day focus areas like heavy-ion collisions and early-universe cosmology.
Turbulent scaling phenomena are studied in an ultracold Bose gas away from thermal equilibrium. Fixed points of the dynamical evolution are characterized in terms of universal scaling exponents of correlation functions. The scaling behavior is determined analytically in the framework of quantum field theory, using a nonperturbative approximation of the two-particle irreducible effective action. While perturbative Kolmogorov scaling is recovered at higher energies, scaling solutions with anomalously large exponents arise in the infrared regime of the turbulence spectrum. The extraordinary enhancement in the momentum dependence of long-range correlations could be experimentally accessible in dilute ultracold atomic gases. Such experiments have the potential to provide insight into dynamical phenomena directly relevant also in other present-day focus areas like heavy-ion collisions and early-universe cosmology.
Universal dynamics of a dilute Bose gas is studied within the kinetic regime where the time evolution of the mode occupation number is governed by a wave-Boltzmann equation. The universality manifests itself in the form of the dynamical evolution which can be a cascade in the context of Kolmogorov-Zakharov wave turbulence or a self-similar shift in time and space. Which is the case depends on the relevant global conservation laws and on the particular range of scales in which the respective dynamics takes place. A nonperturbative kinetic equation is derived by applying the Schwinger-Keldysh closed-time functional integral and the 2-particle-irreducible formalism to an action of complex scalar Bose fields with quartic interaction. The resulting dynamic equation for two-point correlations are reduced to a wave-Boltzmann-type kinetic equation with an effective T-matrix which depends explicitly on the mode occupation numbers. This nonperturbative dependence on the solution occurs in the infrared regime where wave numbers are large and collective scattering of many particles prevails. Thus, the scaling analysis of wave turbulence theory can be applied. We explicitly calculate the effective T-matrix analytically taking into account an infrared cutoff for dealing with the infrared divergences. Our results show that the scaling behaviour of the T-matrix departs from the one predicted by naive dimensional counting due to the presence of the infrared cutoff. The Kolmogorov-Zakharov and the self-similar exponents are evaluated by power counting using our T-matrix and the results are confirmed by numerical integration of the wave-Boltzmann scattering integral. The scaling exponents governing the time evolution are determined by means of the global conservation laws and the kinetic equation. Depending on the scaling properties of the quasiparticle spectrum, the momentum cutoff scale in the infrared evolves critically slowed in time. The respective scaling exponent is universal in the nonperturbative regime regardless of whether the process is an inverse cascade or a self-similar shift towards the infrared. Thus, our results provide a general framework for classifying nonthermal fixed points in dilute ultracold Bose gases. They pave the way to a straightforward generalisation and application to trapped systems in different dimensionalities and to systems with more than one internal degree of freedom. Our results furthermore provide a possible interpretation of recent experimental results on wave turbulence in an ultracold Bose gas. %We studied the universal dynamics of diluted Bose gas within the kinetic regime where time evolution of occupation number is governed by the Boltzmann's scattering integral. %There are three possible scenarios, depending on the scaling behaviour of occupation number. They are the particle and energy cascade which are the steady state of turbulence transport and self-similar evolution where local occupation evolves in time. %We evaluated the scaling exponents in both momentum and time scales in each case under the constraints of global number and energy conservation. Furthermore, we can go beyond the perturbative regime by employing the closed-time-path contour and the 2-particle-irreducible formalism to the Gross-Pitaevskii action. The kinetic equation is modified by an effective T-matrix as a result of the resummation scheme. However, the structure of kinetic equation remains the same as in the perturbative case. Thus, all exponents in the nonperturbative regime depart from those in the perturbative regime because of the scaling of the effective T-matrix.
Nonthermal fixed points of the dynamics of a dilute degenerate Bose gas far from thermal equilibrium are analyzed in two and three spatial dimensions. Universal power-law distributions, previously found within a nonperturbative quantum-field theoretical approach and recently shown to be related to vortical dynamics and superfluid turbulence [Phys. Rev. B 84, 020506(R) (2011)], are studied in detail. The results imply an interpretation of the scaling behavior in terms of independent vortex excitations of the superfluid and show that the statistics of topological excitations can be described in the framework of wave turbulence. The particular scaling exponents observed in the single-particle momentum distributions are found to be consistent with irreversibility as well as conservation laws obeyed by the wave interactions. Moreover, long-wavelength acoustic excitations of the vortex-bearing condensate, driven by vortex annihilations, are found to follow a nonthermal power law. Considering vortex correlations in a statistical model, the long-time departure from the nonthermal fixed point is related to vortex-antivortex pairing. The studied nonthermal fixed points are accessible in cold-gas experiments. The results shed light on fundamental aspects of superfluid turbulence and have strong potential implications for related phenomena, for example, in early universe inflation or quark-gluon plasma 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 z 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 t 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.
In these notes we discuss recent developments in the field of non-equilibrium quantum dynamics. Specifically, we consider nearly coherent Bose gases brought far out of equilibrium and study their behaviour in view of connections between universal properties, (quasi-)topological field configurations and turbulent dynamics. We demonstrate that the isolated Bose gas, on its way back to thermal equilibrium, can approach metastable non-equilibrium configurations and spend a long time in their vicinity. In such configurations, which have been termed non-thermal fixed points, the system shows universal long-range properties manifest through scaling, i.e., self-similar correlations. 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. We obtain an overall picture which connects well-known concepts for describing universal dynamics such as wave-turbulence, superfluid turbulence, and (quasi-)topological excitations. This allows to bring together a wide range of concepts and methods with a large spectrum of applications.
Abstract Closed quantum systems far from thermal equilibrium can show universal dynamics near attractor solutions, known as non-thermal fixed points, generically in the form of scaling behaviour in space and time. A systematic classification and comprehensive understanding of such scaling solutions are tasks of future developments in non-equilibrium quantum many-body theory. In this tutorial review, we outline several analytical approaches to non-thermal fixed points and summarise corresponding numerical and experimental results. The analytic methods include a non-perturbative kinetic theory derived within the two-particle irreducible effective action formalism, as well as a low-energy effective field theory framework. As one of the driving forces of this research field are numerical simulations, we summarise the main results of exemplary cases of universal dynamics in ultracold Bose gases. This encompasses quantum vortex ensembles in turbulent superfluids as well as recently observed real-time instanton solutions in one-dimensional spinor condensates.
Nonthermal fixed points of the dynamics of a dilute degenerate Bose gas are analyzed in two and three spatial dimensions. For such systems, universal power-law distributions, previously found within a nonperturbative quantum-field theoretic approach, are shown to be related to vortical dynamics and superfluid turbulence. The results imply an interpretation of the momentum scaling at the nonthermal fixed points in terms of independent vortex excitations of the superfluid. Long-wavelength acoustic excitations on the top of these are found to follow a nonthermal power law. The results shed light on fundamental aspects of superfluid turbulence and have strong potential implications for related phenomena, e.g., in early-universe inflation or quark-gluon plasma dynamics.