Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.28028

Sound propagation in one-dimensional quantum droplets

Yuan Zizhou, Jiarui Xiao, Xiao-Long Chen

Published Aug 28, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $^{39}$K droplets.

DOI
10.48550/arxiv.2608.28028

Related papers

Ground-state properties and Bogoliubov modes of a harmonically trapped one-dimensional quantum droplet

X. C. Du, Yifan Fei, Xiao-Long Chen +1 more

We study the stationary and excitation properties of a one-dimensional quantum droplet in a two-component Bose mixture trapped in a harmonic potential. By constructing the energy functional for the inhomogeneous mixture, we elaborate the extended Gross-Pitaevskii equation applicable to both symmetric and asymmetric mixtures into a universal form, and the equations in two different dimensionless schemes are in a duality relation; that is, the unique parameters that are left are the inverse of each other. The Bogoliubov equations for the trapped droplet are obtained by linearizing the small density fluctuation around the ground state, and the low-lying excitation modes are calculated numerically. It is found that the confinement trap easily changes the flat-top structure for large droplets and intensively alters the mean-square radius and the chemical potential. The breathing mode of the confined droplet connects the self-bound and ideal-gas limits, with the excitation in the weakly interacting Bose condensate for large particle numbers lying in between. We explicitly show how the continuum spectrum of the excitation is split into discrete modes and finally taken over by the harmonic trap. Two critical particle numbers are identified by the minimum size of the trapped droplet and the maximum breathing-mode energy, both of which are found to decrease exponentially with the trapping parameter.

arXiv:2309.05245•Physical Review A•Sep 18, 2023•22 citations

Many-body and temperature effects in two-dimensional quantum droplets in Bose–Bose mixtures

Abdelâali Boudjemâa

We study the equilibrium properties of self-bound droplets in two-dimensional Bose mixtures employing the time-dependent Hartree-Fock-Bogoliubov theory. This theory allows one to understand both the many-body and temperature effects beyond the Lee-Huang-Yang description. We calculate higher-order corrections to the excitations, the sound velocity, and the energy of the droplet. Our results for the ground-state energy are compared with the diffusion Monte Carlo data and good agreement is found. The behavior of the depletion and anomalous density of the droplet is also discussed. At finite temperature, we show that the droplet emerges at temperatures well below the Berezinskii-Kosterlitz-Thouless transition temperature. The critical temperature strongly depends on the interspecies interactions. Our study is extended to the finite size droplet by numerically solving the generalized finite-temperature Gross-Pitaevskii equation which is obtained self-consistently from our formalism in the framework of the local density approximation.

arXiv:2111.04025•Scientific Reports•Nov 5, 2021•23 citations

One-dimensional asymmetrically interacting quantum droplets in Bose-Bose mixtures

Huiyun Xiao, Xinran Zhang, Junli Liu +3 more

We theoretically investigate ground-state properties and collective excitations of one-dimensional quantum droplets in asymmetric Bose-Bose mixtures with unequal intraspin interactions. Using the extended Gross-Pitaevskii equation supported by variational, sum-rule, and linearization methods, we show that the intraspin interaction ratio substantially alters the droplet's density profile, driving a transition from Gaussian-like to flat-top shapes. By examining two experimentally relevant parameter regions, we analyze density profiles, radii, peak densities, and excitation spectra to distinguish quantum phases and to depict phase diagrams in the space of asymmetric interaction ratio and total atom number. We carefully study the frequencies of both well-known dipole and breathing modes and less-explored spin-dipole and spin-breathing modes. The breathing-mode frequency decreases monotonically with interaction ratio, approaching asymptotically the result of a conventional weakly interacting Bose gas. It varies nonmonotonically with total atom number, peaking at a critical point that highlights the crucial role of quantum fluctuations. In contrast, spin modes display distinct temporal spin density distributions and reveal in-phase and out-of-phase relative dynamics between components. Their frequencies depend instead monotonically on the interaction ratio and atom number. Our results provide a comprehensive understanding of asymmetric quantum droplets and link to experimentally accessible regimes in ultracold $^{39}$K atomic gases.

arXiv:2601.16808•Physical Review A•May 12, 2026•3 citations

Quantum droplets in one-dimensional Bose-Bose mixtures: beyond the Lee-Huang-Yang description

Abdelâali Boudjemâa, Karima Abbas

Abstract The static and dynamic properties of self-bound quantum droplets in a one-dimensional Bose-Bose mixture are discussed in the spirit of the Hartree–Fock-Bogoliubov theory. This latter enables us to provide beyond the Lee-Huang-Yang (LHY) quantum corrections to the equation of state at both zero and finite temperatures. In the uniform case our results for the ground-state energy and the critical temperature are confirmed through comparison with Quantum Monte-Carlo simulation and with available theoretical results. The density profiles are supported by numerical simulations of the generalized Gross-Pitaevskii equation which selfconsistently includes higher-order terms originating from the normal and anomalous fluctuations under the local density approximation. We show that the density exhibits a dip near its center in the flat-top plateau region for large interspecies interactions. We exemplify the impact of the beyond LHY corrections on the spatiotemporal evolution of the self-bound droplet in the presence of excitation induced by periodic density modulation. It is found that higher-order corrections may lead to the formation of a train of small droplets. We then extend our study for the case of inhomogeneous droplets in quasi one-dimensional Bose mixtures.

New Journal of Physics•Sep 1, 2023•10 citations

Quantum self-bound droplets in Bose-Bose mixtures: Effects of higher-order quantum and thermal fluctuations

Nadia Guebli, Abdelâali Boudjemâa

We systematically study the effects of higher-order quantum and thermal fluctuations on the stabilization of self-bound droplets in Bose mixtures employing the time-dependent Hartree-Fock-Bogoliubov theory. We calculate the ground-state energy, the droplet equilibrium density, the depletion and anomalous density of the droplets as well as the critical temperature as a function of the relevant parameters. Our findings are compared with previous analytical predictions and diffusion Monte Carlo simulations. We employ our theory together with the local density approximation for quantum and thermal fluctuations to obtain an extended finite-temperature Gross-Pitaevskii equation. The density profiles and breathing modes of the droplet are deeply examined in terms of the interaction strength and the temperature by numerically solving the developed generalized Gross-Pitaevskii equation.

arXiv:2108.08632•Physical Review A•Aug 11, 2021•39 citations

Phonon Stability of Quantum Droplets in Dipolar Bose Gases

Fan Zhang, Lan Yin

Stabilized by quantum fluctuations, dipolar Bose–Einstein condensates can form self-bound liquid-like droplets. However in the Bogoliubov theory, there are imaginary phonon energies in the long-wavelength limit, implying dynamical instability of this system. A similar instability appears in the Bogoliubov theory of a binary quantum droplet, and is removed due to higher-order quantum fluctuations as shown recently [Gu Q and Yin L 2020 Phys. Rev. B 102 220503(R)]. We study the excitation energy of a dipolar quantum droplet in the Beliaev formalism, and find that quantum fluctuations significantly enhance the phonon stability. We adopt a self-consistent approach without the problem of complex excitation energy in the Bogoliubov theory, and obtain a stable anisotropic sound velocity which is consistent with the superfluid hydrodynamic theory, but slightly different from the result of the extended Gross–Pitaevskii equation due to quantum depletion. A modified Gross–Pitaevskii equation in agreement with the Beliaev theory is proposed, which takes the effect of quantum fluctuations into account more completely.

arXiv:2109.03599•Chinese Physics Letters•Jun 1, 2022•18 citations
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.
We study the stationary and excitation properties of a one-dimensional quantum droplet in a two-component Bose mixture trapped in a harmonic potential. By constructing the energy functional for the inhomogeneous mixture, we elaborate the extended Gross-Pitaevskii equation applicable to both symmetric and asymmetric mixtures into a universal form, and the equations in two different dimensionless schemes are in a duality relation; that is, the unique parameters that are left are the inverse of each other. The Bogoliubov equations for the trapped droplet are obtained by linearizing the small density fluctuation around the ground state, and the low-lying excitation modes are calculated numerically. It is found that the confinement trap easily changes the flat-top structure for large droplets and intensively alters the mean-square radius and the chemical potential. The breathing mode of the confined droplet connects the self-bound and ideal-gas limits, with the excitation in the weakly interacting Bose condensate for large particle numbers lying in between. We explicitly show how the continuum spectrum of the excitation is split into discrete modes and finally taken over by the harmonic trap. Two critical particle numbers are identified by the minimum size of the trapped droplet and the maximum breathing-mode energy, both of which are found to decrease exponentially with the trapping parameter.
We study the equilibrium properties of self-bound droplets in two-dimensional Bose mixtures employing the time-dependent Hartree-Fock-Bogoliubov theory. This theory allows one to understand both the many-body and temperature effects beyond the Lee-Huang-Yang description. We calculate higher-order corrections to the excitations, the sound velocity, and the energy of the droplet. Our results for the ground-state energy are compared with the diffusion Monte Carlo data and good agreement is found. The behavior of the depletion and anomalous density of the droplet is also discussed. At finite temperature, we show that the droplet emerges at temperatures well below the Berezinskii-Kosterlitz-Thouless transition temperature. The critical temperature strongly depends on the interspecies interactions. Our study is extended to the finite size droplet by numerically solving the generalized finite-temperature Gross-Pitaevskii equation which is obtained self-consistently from our formalism in the framework of the local density approximation.
We theoretically investigate ground-state properties and collective excitations of one-dimensional quantum droplets in asymmetric Bose-Bose mixtures with unequal intraspin interactions. Using the extended Gross-Pitaevskii equation supported by variational, sum-rule, and linearization methods, we show that the intraspin interaction ratio substantially alters the droplet's density profile, driving a transition from Gaussian-like to flat-top shapes. By examining two experimentally relevant parameter regions, we analyze density profiles, radii, peak densities, and excitation spectra to distinguish quantum phases and to depict phase diagrams in the space of asymmetric interaction ratio and total atom number. We carefully study the frequencies of both well-known dipole and breathing modes and less-explored spin-dipole and spin-breathing modes. The breathing-mode frequency decreases monotonically with interaction ratio, approaching asymptotically the result of a conventional weakly interacting Bose gas. It varies nonmonotonically with total atom number, peaking at a critical point that highlights the crucial role of quantum fluctuations. In contrast, spin modes display distinct temporal spin density distributions and reveal in-phase and out-of-phase relative dynamics between components. Their frequencies depend instead monotonically on the interaction ratio and atom number. Our results provide a comprehensive understanding of asymmetric quantum droplets and link to experimentally accessible regimes in ultracold 39^{39}39K atomic gases.
Abstract The static and dynamic properties of self-bound quantum droplets in a one-dimensional Bose-Bose mixture are discussed in the spirit of the Hartree–Fock-Bogoliubov theory. This latter enables us to provide beyond the Lee-Huang-Yang (LHY) quantum corrections to the equation of state at both zero and finite temperatures. In the uniform case our results for the ground-state energy and the critical temperature are confirmed through comparison with Quantum Monte-Carlo simulation and with available theoretical results. The density profiles are supported by numerical simulations of the generalized Gross-Pitaevskii equation which selfconsistently includes higher-order terms originating from the normal and anomalous fluctuations under the local density approximation. We show that the density exhibits a dip near its center in the flat-top plateau region for large interspecies interactions. We exemplify the impact of the beyond LHY corrections on the spatiotemporal evolution of the self-bound droplet in the presence of excitation induced by periodic density modulation. It is found that higher-order corrections may lead to the formation of a train of small droplets. We then extend our study for the case of inhomogeneous droplets in quasi one-dimensional Bose mixtures.
We systematically study the effects of higher-order quantum and thermal fluctuations on the stabilization of self-bound droplets in Bose mixtures employing the time-dependent Hartree-Fock-Bogoliubov theory. We calculate the ground-state energy, the droplet equilibrium density, the depletion and anomalous density of the droplets as well as the critical temperature as a function of the relevant parameters. Our findings are compared with previous analytical predictions and diffusion Monte Carlo simulations. We employ our theory together with the local density approximation for quantum and thermal fluctuations to obtain an extended finite-temperature Gross-Pitaevskii equation. The density profiles and breathing modes of the droplet are deeply examined in terms of the interaction strength and the temperature by numerically solving the developed generalized Gross-Pitaevskii equation.
Stabilized by quantum fluctuations, dipolar Bose–Einstein condensates can form self-bound liquid-like droplets. However in the Bogoliubov theory, there are imaginary phonon energies in the long-wavelength limit, implying dynamical instability of this system. A similar instability appears in the Bogoliubov theory of a binary quantum droplet, and is removed due to higher-order quantum fluctuations as shown recently [Gu Q and Yin L 2020 Phys. Rev. B 102 220503(R)]. We study the excitation energy of a dipolar quantum droplet in the Beliaev formalism, and find that quantum fluctuations significantly enhance the phonon stability. We adopt a self-consistent approach without the problem of complex excitation energy in the Bogoliubov theory, and obtain a stable anisotropic sound velocity which is consistent with the superfluid hydrodynamic theory, but slightly different from the result of the extended Gross–Pitaevskii equation due to quantum depletion. A modified Gross–Pitaevskii equation in agreement with the Beliaev theory is proposed, which takes the effect of quantum fluctuations into account more completely.