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arXiv:2311.04597 · Physical Review A

Quantum-Monte-Carlo–based functional for dysprosium dipolar systems

Raúl Bombín, Viktor Cikojević, Ferran Mazzanti, Jordi Boronat

Published Mar 19, 2024• 5 citations
arXivHTMLPDF

Abstract

We present a quantum-Monte-Carlo--based density functional to describe droplet formation and supersolidity in dipolar systems. The usual Lee-Huang-Yang term, accounting for quantum correlations in the conventional extended Gross-Pitaevskii equation (eGPE), has been substituted by the correlation energy evaluated with quantum Monte Carlo. We demonstrate the ability of our functional to reproduce existing experimental data for the minimum critical number of atoms ${N}_{\mathrm{c}}$ required for droplet formation. ${N}_{\mathrm{c}}$ is a challenging quantity for theoretical predictions, and the eGPE provides only a qualitative description of it, mainly when it is applied to dysprosium. We also use our approach to characterize the BEC-supersolid transition. The quantum-Monte-Carlo--based functional can be easily implemented in any existing eGPE code, improving the description of dipolar systems without increasing the computational cost.

DOI
10.1103/physreva.109.033312

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Abstract In quantum droplets, the mean-field energy is comparable to the Lee-Huang-Yang (LHY) energy. In the Bogoliubov theory, the LHY energy of a quantum droplet has an imaginary part, which has been neglected in most studies for practical purposes. Thus far, most theoretical studies on quantum droplets have been based on the extended Gross-Pitaevskii (GP) equation, which includes the contribution of the LHY energy to the chemical potential. In this article, we present the density-functional theory of quantum droplets. In our approach, the quantum fluctuations in quantum droplets, as described by an effective action, generate a real correlation energy that can be determined consistently. Using density-functional theory, we calculated the energy, quantum depletion fraction, and excitations of the droplet. Our results for the ground-state energy and quantum depletion fractions were consistent with the Monte Carlo results. We also discuss the implications of our theory.
Abstract The properties of a supersolid state (SS) in quasi-one-dimensional dipolar Bose–Einstein condensate is studied, considering two possible mechanisms of realization—due to repulsive three-body atomic interactions and quantum fluctuations in the framework of the Lee–Huang–Yang theory. The role of both mechanisms in the formation of SS properties has been emphasized. The proposed theoretical model, based on minimization of the energy functional, allows evaluating the amplitude of the SS for an arbitrary set of parameters in the extended Gross–Pitaevskii equation (eGPE). To explore the dynamics of the SS first we numerically construct its ground state in different settings, including periodic boundary conditions, box-like trap and parabolic potential, then impose a perturbation. In oscillations of the perturbed supersolid we observe the key manifestation of SS, namely the free flow of the superfluid fraction through the crystalline component of the system. Two distinct oscillation frequencies of the supersolid associated with the superfluid fraction and crystalline components of the wave function are identified from numerical simulations of the eGPE.
By combining first-principles path integral Monte Carlo methods and mean-field techniques, we explore the properties of cylindrically trapped doubly dipolar Bose gases. We first verify the emergence of a pancake quantum droplet at low temperatures, validating previous mean-field calculations. In a regime of small doubly dipolar interactions, first-principles calculations agree with the generalized Gross-Pitaevskii equation. Such an accordance disappears in a large interaction limit. Here the path integral Monte Carlo method estimates the strong doubly dipolar regime with accuracy. In contrast, the Gross-Pitaevskii equation does not seize quantum fluctuations in full. We also provide a complete description of the system's quantum behavior in a wide range of parameters. When the system forms a droplet, the superfluid fraction exhibits an anisotropic behavior if compared to the usual Bose gas regime. Interestingly, we observe that the transition temperature from thermal gas to droplet is higher than that of the thermal gas to a Bose-Einstein condensate, indicating the robustness of the droplet against thermal fluctuations. Further, we investigate the anisotropic behavior of the superfluid fraction during the structural transition from a pancake to a cigar-shaped droplet by varying the ratio between electric and magnetic dipole interaction strengths. Our findings furnish evidence that the stability of doubly dipolar Bose-Einstein condensates can be detected in experiments by means of dysprosium atoms.
In quantum droplets, the mean-field energy is comparable to the Lee-Huang-Yang (LHY) energy. In the Bogoliubov theory, the LHY energy of the quantum droplet has an imaginary part, but it is neglected for practical purposes. So far, most theoretical studies of quantum droplets have been based on the extended Gross-Pitaevskii (GP) equation obtained by adding the LHY energy to the GP equation. In this article, we present the density-functional theory of quantum droplets. In our approach, the quantum fluctuations in quantum droplets, as described by an effective action, generate the correlation energy which is real and can be determined self-consistently. Using the density-functional theory, we calculate higher-order corrections to the energy, the quantum depletion fraction, and the excitations of the droplet. Our results for the ground-state energy and the quantum depletion fraction are compared with the Monte Carlo results and good agreement is found. The implications of our theory are discussed.