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arXiv:2605.28606

Fractional short-time dynamics in driven quantum gases

Uri Sharell, Tilman Enss

Published May 27, 2026• 0 citations
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Abstract

Quantum gases with short-range attractive interaction tend to form pairs. For time-dependent interaction we find that the pairing amplitude at small separation satisfies a fractional differential equation (FDE). We derive analytic solutions of the pairing evolution for sudden interaction quenches and power-law drives toward resonant scattering. We observe universal short-time dynamics governed by a nonrelativistic conformal fixed point at which the momentum distribution exhibits self-similar dynamic scaling, in quantitative agreement with experiment. At longer times, many-body effects induce relaxation toward an equilibrium state. In this limit, the FDE turns into a Müller-Israel-Stewart type equation that describes a hydrodynamic attractor approaching equilibrium.

DOI
10.48550/arxiv.2605.28606

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Quantum gases with short-range attractive interaction tend to form pairs. For time-dependent interaction we find that the pairing amplitude at small separation satisfies a fractional differential equation (FDE). We derive analytic solutions of the pairing evolution for sudden interaction quenches and power-law drives toward resonant scattering. We observe universal short-time dynamics governed by a nonrelativistic conformal fixed point at which the momentum distribution exhibits self-similar dynamic scaling, in quantitative agreement with experiment. At longer times, many-body effects induce relaxation toward an equilibrium state. In this limit, the FDE turns into a Müller-Israel-Stewart type equation that describes a hydrodynamic attractor approaching equilibrium.
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The evolution of quantum gases, released from traps, are studied through hydrodynamics, both analytically and numerically, in one and two dimensions. In particular, we demonstrate the existence of long time self-similar solutions of the Euler equations, for the density and velocity fields, and derive the scaling exponents as well as the scaling functions. We find that the expanding gas develops a shock front and the size of the cloud grows in time as a powerlaw. We relate the associated exponent to that appearing in the corresponding equation of state of the quantum gas. Furthermore, we study the relaxation dynamics of a trapped quantum gas and show that the resulting steady state is in excellent agreement with that derived analytically. Our hydrodynamic approach is versatile and can be used to unravel several other far-from-equilibrium collective phenomenon of extreme nature, relevant to the growing experimental interests in quantum gases.
The hydrodynamic attractor is a concept that describes universal equilibration behavior in which systems lose microscopic details before hydrodynamics becomes applicable. We propose a setup to observe hydrodynamic attractors in ultracold atomic gases, taking advantage of the fact that driving the two-body s-wave scattering length causes phenomena equivalent to isotropic fluid expansions. We specifically consider two-component fermions with contact interactions in three dimensions and discuss their dynamics under a power-law drive of the scattering length in a uniform system. By explicit computation, we derive a hydrodynamic relaxation model. We analytically solve their dynamics and find the hydrodynamic attractor solution. Our proposed method using the scattering length drive is applicable to a wide range of ultracold atomic systems, and our results establish these as a new platform for exploring hydrodynamic attractors.
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In this Letter, we investigate the effects of a time-dependent, short-ranged interaction on the long-time expansion dynamics of Fermi gases. We show that the effects of the interaction on the dynamics is dictated by how it changes under a conformal transformation, and derive an explicit criterion for the relevancy of time-dependent interactions near both the strongly and noninteracting scale invariant limits. In addition, we show that it is possible to engineer interactions that give rise to nonexponential thermalization dynamics in trapped Fermi gases. To supplement the symmetry analysis, we perform hydrodynamic simulations to show that the moment of inertia of the trapped gas indeed follows a universal time dependence that is determined jointly by the conformal symmetry and time-dependent scattering length a(t). Our results should also be relevant to the dynamics of other systems that are nearly scale invariant and that are governed by a nonrelativistic conformal symmetry.
Recently universal dynamic scaling is observed in several systems, which exhibit a spatiotemporal self-similar scaling behavior, analogous to the spatial scaling near phase transition. The latter one arises from the emergent continuous scaling symmetry. Motivated by this, we investigate the possible relation between the scaling dynamics and the continuous scaling symmetry in this paper. We derive a theorem that the scaling invariance of the quenched Hamiltonian and the initial density matrix can lead to the universal dynamic scaling. It is further demonstrated both in a two-body system analytically and in a many-body system numerically. For the latter one, we calculate the dynamics of quantum gases quenched from the zero interaction to a finite interaction via the non-equilibrium high-temperature virial expansion. A dynamic scaling of the momentum distribution appears in certain momentum-time windows at unitarity as well as in the weak interacting limit. Remarkably, this universal scaling dynamics persists approximately with smaller scaling exponents even if the scaling symmetry is fairly broken. Our findings may offer a new perspective to interpret the related experiments. We also study the Contact dynamics in the BEC–BCS crossover. Surprisingly, the half-way time displays a maximum near unitarity while some damping oscillations occur on the BEC side due to the dimer state, which can be used to detect possible two-body bound states in experiments.