We theoretically investigate various beyond mean-field effects on Bose gases at zero temperature featuring the anisotropic and long-range dipole-dipole interaction in addition to the isotropic and short-range contact interaction. Within the realm of the Bogoliubov--de Gennes theory, we consider static properties and low-lying excitations of both homogeneous and harmonically trapped dipolar bosonic gases. For the homogeneous system, the condensate depletion, the ground-state energy, the equation of state, and the speed of sound are discussed in detail. Making use of the local density approximation, we extend these results in order to study the properties of a dipolar Bose gas in a harmonic trap and in the regime of large particle numbers. After deriving the equations of motion for the general case of a triaxial trap, we analyze the influence of quantum fluctuations on important properties of the gas, such as the equilibrium configuration and the low-lying excitations in the case of a cylinder-symmetric trap. In addition to the monopole and quadrupole oscillation modes, we also discuss the radial quadrupole mode. We find that the latter acquires a quantum correction exclusively due to the dipole-dipole interaction. As a result, we identify the radial quadrupole as a reasonably accessible source for the signature of dipolar many-body effects and stress the enhancing character that dipolar interactions have for quantum fluctuations in the other oscillation modes.
We theoretically investigate various beyond mean-field effects on Bose gases at zero temperature featuring the anisotropic and long-range dipole-dipole interaction in addition to the isotropic and short-range contact interaction. Within the realm of the Bogoliubov--de Gennes theory, we consider static properties and low-lying excitations of both homogeneous and harmonically trapped dipolar bosonic gases. For the homogeneous system, the condensate depletion, the ground-state energy, the equation of state, and the speed of sound are discussed in detail. Making use of the local density approximation, we extend these results in order to study the properties of a dipolar Bose gas in a harmonic trap and in the regime of large particle numbers. After deriving the equations of motion for the general case of a triaxial trap, we analyze the influence of quantum fluctuations on important properties of the gas, such as the equilibrium configuration and the low-lying excitations in the case of a cylinder-symmetric trap. In addition to the monopole and quadrupole oscillation modes, we also discuss the radial quadrupole mode. We find that the latter acquires a quantum correction exclusively due to the dipole-dipole interaction. As a result, we identify the radial quadrupole as a reasonably accessible source for the signature of dipolar many-body effects and stress the enhancing character that dipolar interactions have for quantum fluctuations in the other oscillation modes.
We investigate the influence of quantum fluctuations upon dipolar Bose gases by means of the Bogoliubov--de Gennes theory. Thereby, we make use of the local density approximation to evaluate the dipolar exchange interaction between the condensate and the excited particles. This allows to obtain the Bogoliubov spectrum analytically in the limit of large particle numbers. After discussing the condensate depletion and the ground-state energy correction, we derive quantum-corrected equations of motion for harmonically trapped dipolar Bose gases by using superfluid hydrodynamics. These equations are subsequently applied to analyze the equilibrium configuration, the low-lying oscillation frequencies, and the time-of-flight dynamics. We find that both atomic magnetic and molecular electric dipolar systems offer promising scenarios for detecting beyond mean-field effects.
The work of this thesis concerns the properties of a Bose gas of polarised dipoles that have long-range and anisotropic interactions. Our work is divided into two parts. First, the stability of a dipolar Bose gas at finite temperature (both above and below the critical Bose-Einstein condensation (BEC) temperature Tc). Second, the fluctuations of a dipolar BEC at zero and small finite temperature (T ≪ Tc) in a regime where rotonic excitations emerge. \n \nPart I: Stability of a Trapped Finite Temperature Dipolar Bose Gas: Above Tc we implement a semiclassical Hartree-Fock theory and characterise the dependence of the stability boundary on temperature, trap geometry and the strength of the dipole-dipole interaction and contact interaction. We find that stability is greatly enhanced above Tc and that trap geometry continues to play a key role. Furthermore, we find that for oblate traps a novel double instability feature emerges. \nTo extend our stability analysis to the low temperature regime, T < Tc, we develop a beyond semiclassical Hartree theory. We use this to characterise the stability boundary as a function of geometry. Interestingly, we find large beyond semiclassical effects above Tc for prolate trapping geometries. We characterise thermal effects on biconcave condensate states. \n \nPart II: Rotons and Fluctuations in a Trapped Dipolar Condensate: To study density fluctuations we implement a numerical scheme to solve the Gross-Pitaevskii equation and the Bogoliubov de Gennes equations. We find that the phonon and roton gases spatially separate and we characterise the role of the anomalous density on the density fluctuations of the thermally activated rotons. We develop a numerical scheme that calculates number fluctuations within cells of various shapes and sizes, and find a strong peak in the fluctuations when the cell size is around half the roton wavelength, which should be detectable by current experiments. By tailoring the cell shape we predict that experiments should be able to detect the effects of individual roton modes. \nFor the study of zero temperature fluctuations we deploy the Gross-Pitaevskii and Bogoliubov de Gennes equations to calculate the dynamic and static structure factors for a highly oblate BEC. We find a clear signature of the roton gas dispersion relation within the structure factors. This signature should be detectible in current experiments using Bragg spectroscopy.
The quest for quantum degenerate Fermi gases interacting through the anisotropic and long-range dipole-dipole interaction is an exciting and fast developing branch within the cold-atom research program. Recent experimental progress in trapping, cooling, and controlling polar molecules with large electric dipole moments has, therefore, motivated much theoretical effort. In a recent letter, we have briefly discussed the application of a variational time-dependent Hartree-Fock approach to study theoretically both the static and the dynamic properties of such a system in a cylinder-symmetric harmonic trap. We focused on the hydrodynamic regime, where collisions ensure the equilibrium locally. Here we present a detailed theory extended to encompass the general case of a harmonic trap geometry without any symmetry. After deriving the equations of motion for the gas, we explore their static solutions to investigate key properties like the aspect ratios in both real and momentum space as well as the stability diagram. We find that, despite the lack of symmetry of the trap, the momentum distribution remains cylinder symmetric. The equations of motion are then used to study the low-lying hydrodynamic excitations, where, apart from the quadrupole and monopole modes, the radial quadrupole mode is also investigated. Furthermore, we study the time-of-flight dynamics as it represents an important diagnostic tool for quantum gases. We find that the real-space aspect ratios are inverted during the expansion, while that in momentum space becomes asymptotically unity. In addition, anisotropic features of the dipole-dipole interaction are discussed in detail. These results could be particularly useful for future investigations of strongly dipolar heteronuclear polar molecules deep in the quantum degenerate regime.
Abstract The properties of ultracold quantum gases of bosons with dipole–dipole interaction are investigated at finite temperature in the frame of representative ensembles theory. Self-consistent coupled equations of motion are derived for the condensate and the non-condensate components. Corrections due to the dipolar interaction to condensate depletion, the anomalous density and thermodynamic quantities such as the ground state energy, the equation of state, the compressibility and the presure are calculated in the homogeneous case at both zero and finite temperatures. Effects of interaction and temperature on the structure factor are also discussed. Within the realm of the local density approximation, we generalize our results to the case of a trapped dipolar gas.
In this thesis we study the effects of the dipole-dipole interaction on the physics of ultracold quantum gases, both bosonic and fermionic, within the theoretical framework provided by the mean-field regime. This kind of interaction takes place in ultracold atomic gases (for instance 52Cr or 164Dy) due to their atomic magnetic dipole moment, and in ultracold molecular gases due to the magnetic or electric dipole moment. \n \nIn the case of quantum gases of bosonic atoms, or Bose-Einstein condensates, the dipole-dipole interaction can be studied within mean-field approximation using the Gross-Pitaevskii equation, which now contains a new non-linear term due to the dipole-dipole interaction. We investigate, on the one hand dipolar condensates confined in harmonic traps, and on the other dipolar condensates confined in toroidal traps. In the harmonic geometry, our focus is on the study of the ground state and the quantized vortex state, where the density profile is characterized as well as some properties leading to the process of vortex formation, such as the critical frequency and the energy barrier that has to be overcome to bring the vortex from the surface to the centre of the gas. We finish the study of dipolar condensates in harmonic traps by dynamically simulating the precession frequency of an off-center vortex in a non-rotating condensate. \n \nIn the toroidal geometry the dipolar effects are strongly magnified when the polarization axis of the dipoles is perpendicular to the trap symmetry axis. In this case, the anisotropic structure of the density can be understood as the response of the system to the double-well effective potential along the ring. We have studied the dynamics of this system when the initial number of atoms in the left and right wells is imbalanced, predicting Josephson and self-trapping oscillations depending on the initial condition. This has led us to name this new system as Self-induced Josephson Junction. We have studied in detail the self-trapping regime and we have seen that the particle flux inversion is closely related to the crossing of vortices across the Josephson junctions. This result opens the door to establishing a more direct connection between the phase-slip regime, widely addressed in superfluid helium, and the self-trapping regime of condensates. \n \nIn the case of quantum gases of fermionic dipolar particles, we have studied how the radial quadrupole mode allows one to distinguish between hydrodynamic and collisionless regimes. We have analytically calculated the frequency of this mode in the mean-field approximation, generalizing the results from the Thomas-Fermi approximation for trapped ideal Fermi gases. On the one hand, we observe that the frequency in the hydrodynamic regime is smaller than in non-dipolar Fermi gases, while in the collisionless regime the frequency is larger or smaller than that corresponding to the non-interacting system depending on the geometry of the harmonic trap. On the other hand, we predict that reducing the trap deformation (aspect ratio) an observable jump in the frequency of the radial quadrupole mode would take place, which would correspond to the transition between the collisionless and hydrodynamic regimes, for instance when the gas undergoes the transition to the superfluid state.
A microscopic description of the zero-energy two-body ground state and many-body static properties of anisotropic homogeneous gases of bosonic dipoles in two dimensions at low densities is presented and discussed. By changing the polarization angle with respect to the plane, we study the impact of the anisotropy, present in the dipole-dipole interaction, on the energy per particle, comparing the results with mean-field predictions. We restrict the analysis to the regime where the interaction is always repulsive, although the strength of the repulsion depends on the orientation with respect to the polarization field. We present a series expansion of the solution of the zero-energy two-body problem, which allows us to find the scattering length of the interaction and to build a suitable Jastrow factor that we use as a trial wave function for both a variational and diffusion Monte Carlo simulation of the infinite system. We find that the anisotropy has an almost negligible impact on the ground-state properties of the many-body system in the universal regime where the scattering length governs the physics of the system. We also show that scaling in the gas parameter persists in the dipolar case up to values where other isotropic interactions with the same scattering length yield different predictions.