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arXiv:1108.0847 · Journal of Physics B Atomic Molecular and Optical Physics

Quantum Bose liquids with logarithmic nonlinearity: self-sustainability and emergence of spatial extent

А. V. Avdeenkov, Konstantin G. Zloshchastiev

Published Sep 13, 2011• 115 citations
arXivHTMLPDF

Abstract

The Gross-Pitaevskii (GP) equation is a long-wavelength approach widely used to describe the dilute Bose-Einstein condensates (BEC). However, in many physical situations, such as higher densities, this approximation unlikely suffices hence one might need models which would account for long-range correlations and multi-body interactions. We show that the Bose liquid described by the logarithmic wave equation has a number of drastic differences from the GP one. It possesses the self-sustainability property: while the free GP condensate tends to spill all over the available volume the logarithmic one tends to form a Gaussian-type droplet - even in the absence of an external trapping potential. The quasi-particle modes of the logarithmic BEC are shown to acquire a finite size despite the bare particles being assumed point-like, i.e., the spatial extent emerges here as a result of quantum many-body correlations. Finally, we study the elementary excitations and demonstrate that the background density changes the topological structure of their momentum space which, in turn, affects their dispersion relations. Depending on the density the latter can be of the massive relativistic, massless relativistic, tachyonic and quaternionic type.

DOI
10.1088/0953-4075/44/19/195303

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The Gross-Pitaevskii (GP) equation is a long-wavelength approach widely used to describe the dilute Bose-Einstein condensates (BEC). However, in many physical situations, such as higher densities, this approximation unlikely suffices hence one might need models which would account for long-range correlations and multi-body interactions. We show that the Bose liquid described by the logarithmic wave equation has a number of drastic differences from the GP one. It possesses the self-sustainability property: while the free GP condensate tends to spill all over the available volume the logarithmic one tends to form a Gaussian-type droplet - even in the absence of an external trapping potential. The quasi-particle modes of the logarithmic BEC are shown to acquire a finite size despite the bare particles being assumed point-like, i.e., the spatial extent emerges here as a result of quantum many-body correlations. Finally, we study the elementary excitations and demonstrate that the background density changes the topological structure of their momentum space which, in turn, affects their dispersion relations. Depending on the density the latter can be of the massive relativistic, massless relativistic, tachyonic and quaternionic type.
We study a relativistic scalar field model for self-bound Bose-Einstein condensates (BECs) by analyzing a nonlinear Klein-Gordon equation with cubic and logarithmic interactions. This framework captures essential features of quantum droplets, such as self-trapping and finite energy configurations, which emerge from the interplay between attractive and repulsive terms. By performing the non-relativistic limit, we derive a generalized Gross-Pitaevskii equation with a logarithmic correction, consistent with recent models used to describe ultra-cold atomic gasses beyond mean-field theory. We construct the corresponding Lagrangian density, identify conserved quantities via Noether's theorem, and compute the energy-momentum tensor. Numerical solutions of the BEC parameters are shown, establishing the foundations for a field theoretical description of relativistic condensates with a logarithmic interaction. This model provides a unified approach to investigate relativistic effects in quantum droplets and enriches the theoretical landscape of Bose-Einstein condensates with non-standard interactions. The resulting dynamics exhibit stable oscillatory regimes consistent with self-bound condensate configurations.
A unified model of a dilute Bose–Einstein condensate is proposed, combining the logarithmic and Gross–Pitaevskii (GP) nonlinear terms in a wave equation, where the GP term describes two-body interactions, as suggested by the standard perturbation theory; while the logarithmic term is essentially nonperturbative, and takes into account quantum vacuum effects. The model is shown to have excellent agreement with sound propagation data in the condensate of cold sodium atoms known since the now classic works by Andrews and collaborators. The data also allowed us to place constraints on two of the unified model’s parameters, which describe the strengths of the logarithmic and GP terms. Additionally, we suggest an experiment constraining the value of the third parameter (the characteristic density scale of the logarithmic part of the model), using the conjectured attraction–repulsion transition of many-body interaction inside the condensate.
We describe recent development of quantum hydrodynamics for ultracold Bose particle studying and consider different kinds of interactions. The method of derivation of equations describing the evolution of the neutral Bose particle system at low temperatures is described. Despite the fact that we consider the neutral particles we account the short-range interaction between particles. We consider the particles in the Bose-Einstein condensate (BEC) state. This method is called the method of quantum hydrodynamics, because natural for of the quantum mechanics rewritten in terms of material fields of observable quantities in three dimensional space is the set of equations, which look like the hydrodynamics equations. It can be shown that from the quantum hydrodynamics equations can be derived macroscopic non-linear Schrodinger equation. Most famous non-linear Schrodinger equation is the Gross-Pitaevskii (GP) equation, which contains nonlinearity of the third degree. There are generalizations of the GP equation. New term appears in the GP equation at account of the three-particle interaction. This term contains nonlinearity of the fifth degree. At more detailed account of the two particle interaction we come to the nonlocal non-linear Schrodinger equation. This equation contains spatial derivatives of the order parameter in the non-linear terms caused by the interaction. In this terminology the GP equation corresponds to the first order by the interaction radius. For the BEC of the neutral particles with anisotropic long-range dipole-dipole interaction the generalization of the GP equation was also suggested. Detailed analyses of the applicability conditions shows that this equation valid for the system of dipoles parallel to each other, which do not change their direction, and where the dipole-dipole interaction interferences translational motion of particles.
A one-dimensional (1D) Gross-Pitaevskii (GP) equation which governs the dynamics of Bose-Einstein condensates (BECs) with time-dependent scattering length in a harmonic potential is considered. Through a modified phase imprint transformation into the macroscopic order parameter, we suitably generate a derivative cubic-quintic Gross-Pitaevskii (DCQ-GP) equation that describes the dynamics of condensates in the harmonic potential for both twoand three-body interatomic interactions in the presence of nonlinear dispersion. Employing a modified lens-type transformation, we reduce the DCQ-GP equation into a derivative cubic-quintic Ginzburg-Landau (DCQ-GL) equation that allows us to investigate the modulational instability (MI) of the system and study, through the analytical exact solutions of the considered 1D GP equation, the effects of the three-body interactions and nonlinear dispersion on the dynamics of the BECs. We observe that for BECs with repulsive two-body interatomic interactions, the attractive three-body interactions and the nonlinear dispersion have opposite effects on the condensate: (i) The attractive three-body interactions enhance the instability, while the nonlinear dispersion softens the instability of the system; (ii) the attractive three-body interactions contributes to the ejections of atoms from the condensates while the nonlinear dispersion contributes to the accumulation of atoms in the condensates; (iii) the attractive three-body interactions contributes to the broadening of the matter wave solitons, while the nonlinear dispersion contributes to the compression of the matter wave solitons.
Sound propagation in quantum droplets differs from that in conventional Bose-Einstein condensates (BECs) because of their self-bound nature and the role of quantum fluctuations. We investigate sound propagation in one-dimensional quantum droplets formed by a symmetric Bose-Bose mixture, focusing on finite-size and confinement effects. Using the extended Gross-Pitaevskii equation, we extract the sound velocity from the real-time propagation of localized density perturbations and compare it with the low-energy excitation spectrum. We find that, unlike in a conventional BEC, the sound velocity of a finite droplet is strongly affected by its density profile and quantum-pressure contribution. It decreases with increasing particle number as the droplet evolves from a Gaussian-like to a flat-top profile, approaching the bulk quantum-droplet value. In contrast, external harmonic confinement compresses the droplet and enhances the sound velocity, driving the system toward the acoustic behavior of a trapped BEC. Our results establish sound propagation as a sensitive probe of finite-size effects and the crossover between self-bound quantum droplets and conventional Bose gases, and suggest a feasible route for experimental observation in ultracold 39^{39}39K droplets.