We investigate the ground state and low-lying excitations of a Bose-Einstein condensate (BEC) of dipolar molecules interacting via short-range contact and long-range dipole dipole interactions (ddi). We find the mean field ground state via the variational principle using a separable bosonic wave function as the variational state. This state can be refined to include small excitations in the form of plane waves, and the results of the variational calculation are a set of dispersion relations for the low-lying excitations. We analyze the polarization and stability of the ground state as a function of the strength of the external polarizing electric field, the strength of the ddi, and the strength of the contact interactions. In a certain parameter regime of strong contact interactions that oppose the dipole -dipole interactions, we identify a possible signature of a first -order phase transition: the polarization of the gas discontinuously flips as the electric field is continuously ramped through zero field. By examining the dispersion relations, we find that the ground state in this parameter regime is unstable.
We investigate the ground state and low-lying excitations of a Bose-Einstein condensate (BEC) of dipolar molecules interacting via short-range contact and long-range dipole dipole interactions (ddi). We find the mean field ground state via the variational principle using a separable bosonic wave function as the variational state. This state can be refined to include small excitations in the form of plane waves, and the results of the variational calculation are a set of dispersion relations for the low-lying excitations. We analyze the polarization and stability of the ground state as a function of the strength of the external polarizing electric field, the strength of the ddi, and the strength of the contact interactions. In a certain parameter regime of strong contact interactions that oppose the dipole -dipole interactions, we identify a possible signature of a first -order phase transition: the polarization of the gas discontinuously flips as the electric field is continuously ramped through zero field. By examining the dispersion relations, we find that the ground state in this parameter regime is unstable.
We investigate the properties of the ground state and low-lying excitations of an oblate Bose-Einstein condensate composed of rigid rotor molecules in the presence of an external polarizing electric field. We build in a quantum model of molecular polarizability by including the full manifold of rotational states. The interplay between spatial and microscopic degrees of freedom via feedback between the molecular polarizability and inter-molecular dipole-dipole interactions leads to a rich quasi-particle spectrum. Under large applied fields, we reproduce the well-understood density-wave rotonization that appears in a fully polarized dipolar BEC, but under smaller applied fields, we predict the emergence of a spin wave instability and possible new stable ground state phases.
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 present an analytic Bogoliubov description of a Bose-Einstein condensate of polar molecules trapped in a quasi-two-dimensional geometry and interacting via internal state-dependent dipole-dipole interactions. We derive the mean-field ground-state energy functional, and we derive analytic expressions for the dispersion relations, Bogoliubov amplitudes, and static structure factors. This method can be applied to any homogeneous, two-component system with linear coupling and direct, momentum-dependent interactions. The properties of the mean-field ground state, including polarization and stability, are investigated, and we identify three distinct instabilities: a density-wave rotonization that occurs when the gas is fully polarized, a spin-wave rotonization that occurs near zero polarization, and a mixed instability at intermediate fields. The nature of these instabilities is clarified by means of the real-space density-density correlation functions, which characterize the spontaneous fluctuations of the ground state, and the momentum-space structure factors, which characterize the response of the system to external perturbations. We find that the gas is susceptible to both density-wave and spin-wave responses in the polarized limit but only a spin-wave response in the zero-polarization limit. These results are relevant for experiments with rigid rotor molecules such as RbCs, $\mathrm{\ensuremath{\Lambda}}$-doublet molecules such as ThO that have an anomalously small zero-field splitting, and doublet-$\mathrm{\ensuremath{\Sigma}}$ molecules such as SrF where two low-lying opposite-parity states can be tuned to zero splitting by an external magnetic field.
In this work several aspects of dipolar Bose-Einstein condensates are investigated. These condensates have attracted much interest since their experimental realization and offer the chance to explore the novel, fascinating, and sometimes unexpected physics originating from the dipole-dipole interaction. For the theoretical description the time-dependent variational principle with an ansatz of coupled Gaussian wave packets is applied to the analysis of this form of matter. In order to describe physical systems, where the shape of the condensate wave function cannot be approximated adequately by origin-centered Gaussian functions, translational and rotational degrees of freedom of the particular wave packets are included. This provides not only a full-fledged alternative method to full-numerical grid calculations that are demanding in computation time, used for the determination of ground states and investigation of the real-time dynamics, but also allows for the calculation of excited states and the examination of stability properties and offers the possibility for a deeper physical understanding of dipolar condensates. The variational method is applied to three different physical systems. In the first one anisotropic quasi-two-dimensional solitons are investigated. Ground states and small excitations are presented and the full flexibility of the approach is utilized in the simulation of the collision of two solitons. Thereby simulations of collisions are performed, where initially the solitons are in the repelling side-by-side configuration and move towards each other with a specific momentum. It is shown that the results are in excellent agreement with full-numerical grid calculations. Collisions with various initial parameters such as the initial momentum and impact factor are performed, allowing to classify the qualitative interactions of two solitons and illuminating the effects of the solitons' two-dimensional nature on the collision process. In the second system dipolar Bose-Einstein condensates in a triple-well potential are investigated. This constitutes a well-suited model system for periodic optical potentials with important contributions of the non-local and anisotropic dipole-dipole interaction, which show a variety of effects such as self-organization and formation of patterns. Here, a macroscopic sample of dipolar bosons in the mean-field limit is addressed. The analysis goes beyond the calculation of ground states and clarifies the role of excited and metastable states in such systems. In particular, the formation of phases is found to originate from the interplay of several states with distinct stability properties. As some of the phases are formed by metastable states, special attention is paid to the characteristics of phase transitions in real time and the dynamical stabilization of the condensate. The dynamical simulations that support these results, constitute a way for an experimental observation of these findings. The third physical application is an open quantum system. Here, an external potential models gain and loss of particles and features parity with simultaneous time-reversal symmetry. For these dipolar condensates real eigenvalues of the nonlinear Hamiltonian are found, preserving the PT-symmetry of the external potential, as well as states that break this symmetry. A detailed stability analysis of these, in general complex states, is performed. The comparison with a similar system featuring a non-dipolar condensate demonstrates the role of the dipolar interaction for novel effects. Furthermore, the real-time dynamics is investigated, which reveals, among other effects, long-living oscillations of the two wells' population.
This thesis contains a study of ultracold paramagnetic atoms or polar molecules characterized by a long-range anisotropic dipolar interaction. We particularly focus on two aspects of ultracold dipolar gases. In the first problem the ground state properties of dipolar Bose-Einstein condensates (BEC) are investigated. This problem has gained importance due to recent experimental advances in achieving a condensate of Chromium atoms and ongoing research to produce quantum degenerate polar molecules. In the second problem, we consider possible applications of ultracold polar molecules to rotation sensing and interferometry. First, we concentrate on the interplay between the trapping geometry and dipole-dipole interaction for a polarized dipolar bosonic condensate. As the dipole-dipole interaction is attractive along the polarized direction, the lowest energy state of the BEC is always a collapsed state. However by applying a trapping potential along the polarization direction it is possible to achieve a metastable dipolar BEC. By numerically solving the Gross-Pitaevskii equation, we show that below a critical interaction strength, a metastable state exists depending on the trapping geometry. We also show that a novel feature of dipolar BEC is the appearance of different structural metastable ground states for certain combinations of trapping geometry and particle number. Next, by mixing in single component fermions we show that dipolar BEC can be stabilized against collapse in pancake shaped or cylindrical traps. We also show that the excitation spectrum of the BEC may have a minimum for non-zero momentum, termed a “roton minimum”. This minimum leads to a transition to stable or metastable density-wave states depending on the density of the bosons and boson-fermion interaction strength. In the second problem, we study a proposal for a large-angle coherent beam splitter for polar molecules. By taking into account the effect of a quasi-static external electric field on the rotational levels of the polarized molecules we show that it is possible to coherently split a stationary cloud of molecules into two counter-propagating components. We then investigate the effect of longitudinal acceleration on the transverse motion of the particles, assuming that the longitudinal motion of the molecules can be approximated classically by a wave packet with some mean velocity while the transverse motion is governed by quantum mechanics. We propose a particular time-dependent shape of acceleration to minimize the excitations in the transverse motion. Our theory is also applicable to the general case of particles moving along a circular guide with time-dependent longitudinal velocity. In addition, we include the effects of velocity fluctuations due to noise in the accelerating field.
We present a mean field analysis of the effects of molecular rotation on the excitation spectrum and stability of ultracold dipolar gases. For an unpolarized homogeneous gas interacting with a pure dipole-dipole interaction, we find that for the rotational state L = 1 the dipole-dipole interaction causes a splitting of the translation-rotation energy levels into a single M = 0 and a doubly degenerate M = ±1 excitation. For all other rotational states, the dipole-dipole interaction does not lead to coupling of translations and rotations and therefore has no effect on the rotational degeneracy of the excitations. The addition of arbitrarily small electric fields is found to introduce instabilities similar to those known to arise in the fully polarized dipolar gas. As in the case of a fully polarized gas, addition of a large enough short range repulsive potential is seen to stabilize the system, with the critical value of the repulsive interaction required for stabilization being larger when rotations are included.