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arXiv:2101.09727 · Physica Scripta

Characterization of the energy level-structure of a trapped dipolar Bose gas via mean-field parametric resonances

Asaad R Sakhel, Roger R Sakhel

Published Jun 28, 2021• 1 citation
arXivHTMLPDF

Abstract

Abstract Analyzing the energy levels of a trapped Bose–Einstein condensate (BEC) can be difficult when dipole-dipole interactions (DDIs) are present. To address this issue, this study focuses on the parametric resonances (PRs) in the mean-field dynamics of a one-dimensional dipolar BEC (DBEC) over widely varying trapping geometries, with the primary objective of characterizing the energy levels of this system via analytical methods. This is achieved by matching the PR energies to the energy levels of the confining trap using perturbative methods. Further, this research reveals the role of the interplay between DDIs and the trapping geometry in defining the energies and amplitudes of the PRs. The PRs are induced by a negative Gaussian potential with a depth that oscillates with respect to time; DDIs also play a role in this induction. The dynamics of this system are modeled using the time-dependent Gross–Pitaevskii equation (TDGPE), which is numerically solved via the Crank–Nicolson method. The PRs are discussed based on analytical methods. First, we show that PRs similar to the ones obtained from the TDGPE can be reproduced via the Lagrangian variational method. Second, the energies at which the PRs occur are closely matched with the energy levels of the corresponding trap, calculated using the time-independent perturbation theory. Third, the most probable transitions between the trap energy levels yielding PRs are determined based on the time-dependent perturbation theory. The primary contribution of this research is that the energy levels of a DBEC within a complex trapping potential could be characterized.

DOI
10.1088/1402-4896/ac0f3a

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Abstract Analyzing the energy levels of a trapped Bose–Einstein condensate (BEC) can be difficult when dipole-dipole interactions (DDIs) are present. To address this issue, this study focuses on the parametric resonances (PRs) in the mean-field dynamics of a one-dimensional dipolar BEC (DBEC) over widely varying trapping geometries, with the primary objective of characterizing the energy levels of this system via analytical methods. This is achieved by matching the PR energies to the energy levels of the confining trap using perturbative methods. Further, this research reveals the role of the interplay between DDIs and the trapping geometry in defining the energies and amplitudes of the PRs. The PRs are induced by a negative Gaussian potential with a depth that oscillates with respect to time; DDIs also play a role in this induction. The dynamics of this system are modeled using the time-dependent Gross–Pitaevskii equation (TDGPE), which is numerically solved via the Crank–Nicolson method. The PRs are discussed based on analytical methods. First, we show that PRs similar to the ones obtained from the TDGPE can be reproduced via the Lagrangian variational method. Second, the energies at which the PRs occur are closely matched with the energy levels of the corresponding trap, calculated using the time-independent perturbation theory. Third, the most probable transitions between the trap energy levels yielding PRs are determined based on the time-dependent perturbation theory. The primary contribution of this research is that the energy levels of a DBEC within a complex trapping potential could be characterized.
Abstract We report parametric resonances (PRs) in a numerical investigation of a driven one-dimensional, interacting, and disordered Bose–Einstein condensate (BEC) confined in different traps. The BEC is excited by an oscillating Gaussian obstacle along a broad range of driving frequencies Ω. The PRs are detected via a quantity that is closely related to the time-average of the kinetic energy. The significant result of this work is that the trapping geometry plays a major role in defining the values of Ω at which PRs arise and controls their response to disorder. As such, it reveals the interplay of trapping geometry and disorder in these resonances. The dynamics of the modal coefficient C 0 ( t ) as well as that of the phase-mismatch δ ( t ) between the C 0 ( t ) and C 1 ( t ) are examined at and away from PR. At PR, | C 0 ( t )| is generally found to be lower in magnitude than away from it, demonstrating that the atoms leave the n = 0 ground state towards higher states. In the harmonic oscillator trap, the dynamic pattern of δ ( t ) is found to be quite robust against changes in the disorder strength contrary to the box potential. This is because in the box the ratio of the random-potential and kinetic energies is higher than in the harmonic trap signaling that the influence of disorder is weaker in the latter.
We study a system of ultra-cold dipolar Bose gas atoms confined in a two-dimensional (2D) harmonic trap with a dipolar impurity implanted at the center of the trap. Due to recent experimental progress in dipolar condensates, we focused on calculating properties of dipolar impurity systems that might guide experimentalists if they choose to study impurities in dipolar gases. We used the Gross–Pitaevskii formalism solved numerically via the split-step Crank–Nicolson method. We chose parameters of the background gas to be consistent with dysprosium (Dy), one of the strongest magnetic dipoles and of current experimental interest, and used chromium (Cr), erbium (Er), terbium (Tb), and Dy for the impurity. The dipole moments were aligned by an external field along what was chosen to be the z-axis, and we studied 2D confinements that were perpendicular or parallel to the external field. We show density contour plots for the two confinements, 1D cross-sections of the densities, calculated self-energies of the impurities while varying both number of atoms in the condensate and the symmetry of the trap. We also calculated the time evolution of the density of an initially pure system where an impurity is introduced. Our results show that while the self-energy increases in magnitude with increasing number of particles, it is reduced when the trap anisotropy follows the natural anisotropy of the gas, i.e., elongated along the z-axis in the case of parallel confinement. This work builds upon work conducted in Bose gases with zero-range interactions and demonstrates some of the features that could be found when exploring dipolar impurities in 2D Bose gases.
We analytically and numerically study nonlinear dynamics in Bose–Einstein condensates (BECs) induced either by a harmonic modulation of the interaction or by the geometry of the trapping potential. To analytically describe BEC dynamics, we use a perturbative expansion based on the Poincaré–Lindstedt analysis of a Gaussian variational ansatz, whereas in the numerical approach we use numerical solutions of both a variational system of equations and the full time-dependent Gross–Pitaevskii equation. The harmonic modulation of the atomic s-wave scattering length of a BEC of 7 Li was achieved recently via Feshbach resonance, and such a modulation leads to a number of nonlinear effects, which we describe within our approach: mode coupling, higher harmonics generation and significant shifts in the frequencies of collective modes. In addition to the strength of atomic interactions, the geometry of the trapping potential is another key factor for the dynamics of the condensate, as well as for its collective modes. The asymmetry of the confining potential leads to important nonlinear effects, including resonances in the frequencies of collective modes of the condensate. We study in detail such geometric resonances and derive explicit analytic results for frequency shifts for the case of an axially symmetric condensate with two- and three-body interactions. Analytically obtained results are verified by extensive numerical simulations.
This thesis investigates the geometric properties of dipolar Bose-Einstein condensates confined in harmonic and bubble-shaped traps.We develop a set of analytical and variational methods to describe the geometric shape of the condensate, the corresponding dipole-induced deformation, as well as the transition between filled and hollow clouds.Within the Thomas-Fermi regime in the Gross-Pitaevskii mean field theory, we present a convolution-based approximation and a generalized analytic solution to the Thomas-Fermi equation in the context of polynomial trapping potentials.Beyond mean-field, we explore the emergence of angular density modulations driven by quantum fluctuations and provide a variational model to estimate the number of droplets forming in the bubble trap.
Dipolar Bose--Einstein condensates (BECs), i.e., BECs with long-range dipole-dipole interactions (DDI), exhibit extraordinarily rich physics and serve as a versatile experimental platform for exploring exotic quantum states of matter. Within the mean-field framework, a dipolar BEC can be described by the Gross--Pitaevskii (GP) model with a DDI term. Its stationary states consist of the ground state, which minimizes the GP energy functional under the normalization constraint, and excited states, which are higher-energy critical points that typically correspond to constrained saddle points. Excited states encode collective excitation modes and play a key role in the dynamical stability and quantum phase transitions of cold-atom systems. However, in contrast to the extensively studied ground state, they remain far less explored numerically, especially in dipolar systems. In this work, we develop a numerical method based on the constrained gentlest ascent dynamics (CGAD) that systematically computes excited states of dipolar BECs by stably targeting constrained saddle points with any prescribed Morse index. Specifically, a semi-implicit sine-pseudospectral discretization with an orthonormalization correction step is adopted, and the nonlocal DDI terms are evaluated efficiently and accurately via the nonuniform fast Fourier transform (NUFFT). The resulting scheme is applied to one-, two-, and three-dimensional dipolar BECs, enabling a detailed numerical investigation of their excited states. Remarkably, our simulations reveal that excited states can persist even in regimes where the ground state of the condensate becomes mean-field unstable. This work provides a computational framework for studying excited states in dipolar quantum gases and offers insights for future theoretical and experimental investigations.