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arXiv:1805.06261 · Physical Review A

Temperature-dependent density profiles of dipolar droplets

Enes Aybar, Mehmet Özgür Oktel

Published Jan 23, 2019• 35 citations
arXivHTMLPDF

Abstract

Recently, trapped dipolar gases were observed to form high-density droplets in a regime where mean-field theory predicts collapse. These droplets present a form of equilibrium where quantum fluctuations are critical for stability. So far, the effect of quantum fluctuations has only been considered at zero temperature through the local chemical potential arising from the Lee-Huang-Yang correction. Here, we extend the theory of dipolar droplets to nonzero temperatures using Hartree-Fock-Bogoliubov theory (HFBT) and show that the equilibrium is strongly affected by temperature fluctuations. HFBT, together with local density approximation for excitations, reproduces the zero-temperature results and predicts that the condensate density can change dramatically even at low temperatures where the total depletion is small. In particular, we find that typical experimental temperatures ($T\ensuremath{\sim}100$ nK) can significantly modify the transition between low-density and droplet phases.

DOI
10.1103/physreva.99.013620

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Recently, trapped dipolar gases were observed to form high-density droplets in a regime where mean-field theory predicts collapse. These droplets present a form of equilibrium where quantum fluctuations are critical for stability. So far, the effect of quantum fluctuations has only been considered at zero temperature through the local chemical potential arising from the Lee-Huang-Yang correction. Here, we extend the theory of dipolar droplets to nonzero temperatures using Hartree-Fock-Bogoliubov theory (HFBT) and show that the equilibrium is strongly affected by temperature fluctuations. HFBT, together with local density approximation for excitations, reproduces the zero-temperature results and predicts that the condensate density can change dramatically even at low temperatures where the total depletion is small. In particular, we find that typical experimental temperatures (T\ensuremath∼100T\ensuremath{\sim}100T\ensuremath∼100 nK) can significantly modify the transition between low-density and droplet phases.
Droplet states of ultracold gases which are stabilized by fluctuations have recently been observed in dipolar and two-component Bose gases. These systems present an alternate form of equilibrium where an instability at the mean-field level is arrested by higher-order correlations, making the droplet states sensitive probes of fluctuations. In a recent paper, we argued that thermal fluctuations can play an important role for droplets even at low temperatures where the noncondensed density is much smaller than the condensate density. We used the Hartree-Fock-Bogoliubov theory together with the local density approximation for fluctuations to obtain a generalized Gross-Pitaevskii (GP) equation and solved it with a Gaussian variational ansatz to show that the transition between the low density and droplet states can be significantly modified by the temperature. In this paper, we first solve the same GP equation numerically with a time-splitting spectral method to check the validity of the Gaussian variational ansatz. Our numerical results are in good agreement with the Gaussian ansatz for a large parameter regime and show that the density of the gas is most strongly modified by temperature near the abrupt transition between a pancake-shaped cloud and the droplet. For cigar-shaped condensates, as in the recent Er experiments, the dependence of the density on temperature remains quite small throughout the smooth transition. We then consider the effect of temperature on the collective oscillation frequencies of the droplet using both a time-dependent Gaussian variational ansatz and real-time numerical evolution. We find that the oscillation frequencies depend significantly on the temperature close to the transition for the experimentally relevant temperature regime ($\ensuremath{\simeq}100\phantom{\rule{4pt}{0ex}}\mathrm{nK}$).
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 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.
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.
Quantum fluctuations play a pivotal role in determining the ground state properties and phase transitions of ultra-cold bosonic systems. This study presents a comprehensive theoretical investigation of fluctuation effects beyond mean-field theory in dilute Bose gases, examining their impact on the superfluid–Mott insulator transition, quantum droplet formation, and critical phenomena. Beginning with the Gross–Pitaevskii mean-field framework , we systematically incorporate quantum corrections through Bogoliubov theory, yielding the Lee–Huang–Yang (LHY) energy correction scaling as . The quantum depletion quantifies the fraction of atoms driven from the condensate by zero-point fluctuations. For the Bose–Hubbard model , quantum Monte Carlo simulations establish the superfluid–Mott insulator critical point at , shifted 15% from the mean-field prediction. Critical exponents and confirm 3D XY universality. Quantum droplets in binary mixtures are stabilized by the repulsive LHY correction against mean-field collapse, with equilibrium densities matching experimental observations in K mixtures. Finite-size scaling analysis and the Ginzburg criterion delineate regimes where fluctuation effects dominate. All theoretical predictions are validated against precision measurements in ultracold atomic gases.
We investigate the effects of quantum correlations on dipolar quantum droplets. To this end, we derive self-consistent time-dependent Hartree-Fock-Bogoliubov equations that fairly describe the dynamics of the order parameter, the normal quantum correlations, and anomalous quantum correlations of the droplet. We analyze the density profiles, the critical number of particles, the condensate depletion, and the pair correlation function. Our predictions are compared with very recent experimental and quantum Monte Carlo simulations results and excellent agreement is found.