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arXiv:1807.07514 · Physical Review Letters

Prescaling in a Far-from-Equilibrium Bose Gas

Christian-Marcel Schmied, A. N. Mikheev, Thomas Gasenzer

Published May 3, 2019• 32 citations
arXivHTMLPDF

Abstract

Nonequilibrium conditions give rise to classes of universally evolving configurations of quantum many-body systems at nonthermal fixed points. While the fixed point and thus full scaling in space and time is generically reached at very long evolution times, we propose that systems can show prescaling much earlier in time, in particular, on experimentally accessible timescales. During the prescaling evolution, some well-measurable properties of spatial correlations already scale with the universal exponents of the fixed point while others still show scaling violations. Prescaling is characterized by the evolution obeying conservation laws associated with the remaining symmetry which also defines the universality class of the asymptotically reached nonthermal fixed point. Here, we consider N=3 species of spatially uniform three-dimensional Bose gases, with identical inter- and intraspecies interactions. During prescaling, the full U(N) symmetry of the model is broken to U(N-1) while the conserved transport, reflecting explicit and emerging symmetries, leads to the buildup of rescaling quasicondensate distributions.

DOI
10.1103/physrevlett.122.170404

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Nonequilibrium conditions give rise to classes of universally evolving configurations of quantum many-body systems at nonthermal fixed points. While the fixed point and thus full scaling in space and time is generically reached at very long evolution times, we propose that systems can show prescaling much earlier in time, in particular, on experimentally accessible timescales. During the prescaling evolution, some well-measurable properties of spatial correlations already scale with the universal exponents of the fixed point while others still show scaling violations. Prescaling is characterized by the evolution obeying conservation laws associated with the remaining symmetry which also defines the universality class of the asymptotically reached nonthermal fixed point. Here, we consider N=3 species of spatially uniform three-dimensional Bose gases, with identical inter- and intraspecies interactions. During prescaling, the full U(N) symmetry of the model is broken to U(N-1) while the conserved transport, reflecting explicit and emerging symmetries, leads to the buildup of rescaling quasicondensate distributions.
Abstract Predicting the dynamics of quantum systems far from equilibrium represents one of the most challenging problems in theoretical many-body physics1, 2. While the evolution of a many-body system is in general intractable in all its details, relevant observables can become insensitive to microscopic system parameters and initial conditions. This is the basis of the phenomenon of universality. Far from equilibrium, universality is identified through the scaling of the spatio-temporal evolution of the system, captured by universal exponents and functions. Theoretically, this has been studied in examples as different as the reheating process in inflationary Universe cosmology3, 4, the dynamics of nuclear collision experiments described by quantum chromodynamics5, 6, and the post-quench dynamics in dilute quantum gases in non-relativistic quantum field theory7–11. However, an experimental demonstration of such scaling evolution in space and time in a quantum many-body system has been lacking. Here we observe the emergence of universal dynamics by evaluating spatially resolved spin correlations in a quasi-one-dimensional spinor Bose–Einstein condensate12–16. For long evolution times we extract the scaling properties from the spatial correlations of the spin excitations. From this we find the dynamics to be governed by an emergent conserved quantity and the transport of spin excitations towards low momentum scales. Our results establish an important class of non-stationary systems whose dynamics is encoded in time-independent scaling exponents and functions, signalling the existence of non-thermal fixed points10, 17, 18. We confirm that the non-thermal scaling phenomenon involves no fine-tuning of parameters, by preparing different initial conditions and observing the same scaling behaviour. Our analogue quantum simulation approach provides the basis with which to reveal the underlying mechanisms and characteristics of non-thermal universality classes. One may use this universality to learn, from experiments with ultracold gases, about fundamental aspects of dynamics studied in cosmology and quantum chromodynamics.
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.
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Coarsening of an isolated far-from-equilibrium quantum system is a paradigmatic many-body phenomenon, relevant from subnuclear to cosmological length scales and predicted to feature universal dynamic scaling. Here, we observed universal scaling in the coarsening of a homogeneous two-dimensional Bose gas, with exponents that match analytical predictions. For different initial states, we reveal universal scaling in the experimentally accessible finite-time dynamics by elucidating and accounting for the initial-state-dependent prescaling effects. The methods we introduce allow direct comparison between cold-atom experiments and nonequilibrium field theory and are applicable to any study of universality far from equilibrium.
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.
We explore the far-from-equilibrium dynamics of Bose gases in a universal regime associated to nonthermal fixed points. While previous investigations concentrated on scaling functions and exponents describing equal-time correlations, we compute the additional scaling functions and dynamic exponent zzz characterizing the frequency dependence or dispersion from unequal-time correlations. This allows us to compare the characteristic condensation and correlation times from a finite-size scaling analysis depending on the system's volume.