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arXiv:2403.08875 · Physical review. B./Physical review. B

Quantum fluctuating theory for one-dimensional shock waves

Andrew Urilyon, Stefano Scopa, Giuseppe Del Vecchio Del Vecchio, Jacopo De Nardis

Published Jan 2, 2025• 3 citations
arXivHTMLPDF

Abstract

We study the formation and the subsequent dynamics of shock waves in repulsive one-dimensional Bose gases during the free expansion of a density hump. By building coherent Fermi states for interacting Bethe fermions, we define a quantum fluctuating initial state expressed in terms of universal quantities, namely the density and the Luttinger parameter. In the integrable case, this fluctuating state is then evolved by generalized hydrodynamics (GHD) and, differently from nonfluctuating initial states, it develops density ripples on top of the hydrodynamic mean value. Our analysis gives a general theory of quantum ripples and wave breaking in integrable and quasi-integrable one-dimensional liquids and clarifies the role of the interaction strength. In particular, for strongly/intermediately interacting bosons, we find quantum ripples originating from low-energy modes at the Fermi surface interfering when transported by GHD. In the low-coupling limit, near the quasicondensate regime, we find instead that density ripples have a semiclassical nature, and their description requires information on the curvature of the Fermi surface.

DOI
10.1103/physrevb.111.045401

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We study the formation and the subsequent dynamics of shock waves in repulsive one-dimensional Bose gases during the free expansion of a density hump. By building coherent Fermi states for interacting Bethe fermions, we define a quantum fluctuating initial state expressed in terms of universal quantities, namely the density and the Luttinger parameter. In the integrable case, this fluctuating state is then evolved by generalized hydrodynamics (GHD) and, differently from nonfluctuating initial states, it develops density ripples on top of the hydrodynamic mean value. Our analysis gives a general theory of quantum ripples and wave breaking in integrable and quasi-integrable one-dimensional liquids and clarifies the role of the interaction strength. In particular, for strongly/intermediately interacting bosons, we find quantum ripples originating from low-energy modes at the Fermi surface interfering when transported by GHD. In the low-coupling limit, near the quasicondensate regime, we find instead that density ripples have a semiclassical nature, and their description requires information on the curvature of the Fermi surface.
The theory of generalized hydrodynamics (GHD) was recently developed as a new tool for the study of inhomogeneous time evolution in many-body interacting systems with infinitely many conserved charges. It offers a full and efficient framework to capture the long-wavelengths nonequilibrium dynamics in experimentally realizable one-dimensional systems, like Bose gases with delta repulsion. In this letter, we study the nonlinear sound waves emanating from initial density accumulations in the zero-temperature Bose gas. We numerically confirm the emergence of GHD by comparing its predictions with a full quantum simulation performed using the NRG-TSA-ABACUS algorithm. We prove that at zero temperature, GHD is equivalent to conventional Galilean hydrodynamics with nontrivial equations of state. We further predict, at long wavelengths and large densities, the appearance of sharp profiles in finite times, that immediately dissolve (no sustained shocks). The analysis is performed in the full interaction range, and is not restricted to either weak- or strong-repulsion regimes.
The theory of generalized hydrodynamics (GHD) was recently developed as a new tool for the study of inhomogeneous time evolution in many-body interacting systems with infinitely many conserved charges. In this Letter, we show that it supersedes the widely used conventional hydrodynamics (CHD) of one-dimensional Bose gases. We illustrate this by studying "nonlinear sound waves" emanating from initial density accumulations in the Lieb-Liniger model. We show that, at zero temperature and in the absence of shocks, GHD reduces to CHD, thus for the first time justifying its use from purely hydrodynamic principles. We show that sharp profiles, which appear in finite times in CHD, immediately dissolve into a higher hierarchy of reductions of GHD, with no sustained shock. CHD thereon fails to capture the correct hydrodynamics. We establish the correct hydrodynamic equations, which are finite-dimensional reductions of GHD characterized by multiple, disjoint Fermi seas. We further verify that at nonzero temperature, CHD fails at all nonzero times. Finally, we numerically confirm the emergence of hydrodynamics at zero temperature by comparing its predictions with a full quantum simulation performed using the NRG-TSA-abacus algorithm. The analysis is performed in the full interaction range, and is not restricted to either weak- or strong-repulsion regimes.
Physical systems made of many interacting quantum particles can often be described by Euler hydrodynamic equations in the limit of long wavelengths and low frequencies. Recently such a classical hydrodynamic framework, now dubbed generalized hydrodynamics (GHD), was found for quantum integrable models in one spatial dimension. Despite its great predictive power, GHD, like any Euler hydrodynamic equation, misses important quantum effects, such as quantum fluctuations leading to nonzero equal-time correlations between fluid cells at different positions. Focusing on the one-dimensional gas of bosons with delta repulsion, and on states of zero entropy, for which quantum fluctuations are larger, we reconstruct such quantum effects by quantizing GHD. The resulting theory of quantum GHD can be viewed as a multicomponent Luttinger liquid theory, with a small set of effective parameters that are fixed by the thermodynamic Bethe ansatz. It describes quantum fluctuations of truly nonequilibrium systems where conventional Luttinger liquid theory fails.
The evolution of an initially smooth spatial inhomogeneity in the density of a one-dimensional Fermi gas is well described by classical mechanics. The classical evolution leads to the formation of a shock wave: the density develops kinks in its coordinate dependence. We show that quantum corrections to the shock wave produce density ripples which run off the kinks. Despite their quantum origin, the amplitude and period of the ripples are expressed only in terms of classical objects derived from a smooth density profile.
The dynamics of strongly interacting many-body quantum systems are notoriously complex and difficult to simulate. A recently proposed theory called generalized hydrodynamics (GHD) promises to efficiently accomplish such simulations for nearly integrable systems. We test GHD with bundles of ultracold one-dimensional (1D) Bose gases by performing large trap quenches in both the strong and intermediate coupling regimes. We find that theory and experiment agree well over dozens of trap oscillations, for average dimensionless coupling strengths that range from 0.3 to 9.3. Our results show that GHD can accurately describe the quantum dynamics of a 1D nearly integrable experimental system even when particle numbers are low and density changes are large and fast.
The formation of dispersive shock waves in one-dimensional Bose gas represents a limitation of Generalized Hydrodynamics (GHD) due to the coarse-grained nature of the theory. Nevertheless, GHD accurately captures long-wavelength behavior, thus indicating an implicit knowledge of the underlying microscopic physics. Such representations are already known through the Whitham modulation theory, where dispersionless equations describe the evolution of the slowly varying shock wave parameters. Here we study the correspondence between Whitham's approach to the Gross-Pitaevskii equation and GHD in the semiclassical limit and beyond. Our findings enable the recovery of the shock wave solution directly from GHD simulations, which we demonstrate for both zero and finite temperature. Additionally, we study how free expansion protocols affect the shock wave density and their implications for experimental detection. The combined picture of Whitham and GHD lends itself to additional physical interpretation regarding the formation of shock waves. Published by the American Physical Society 2024