We work out two different analytical methods for calculating the boundary of the Mott-insulatorsuperfluid (MI-SF) quantum phase transition for scalar bosons in cubic optical lattices of arbitrary dimension at zero temperature which improve upon the seminal mean-field result. The first one is a variational method, which is inspired by variational perturbation theory, whereas the second one is based on the field-theoretic concept of effective potential. Within both analytical approaches we achieve a considerable improvement of the location of the MI-SF quantum phase transition for the first Mott lobe in excellent agreement with recent numerical results from Quantum Monte-Carlo simulations in two and three dimensions. Thus, our analytical results for the whole quantum phase diagram can be regarded as being essentially exact for all practical purposes.

Turbulence, the complicated fluid behavior of nonlinear and statistical nature, arises in many physical systems across various disciplines, from tiny laboratory scales to geophysical and astrophysical ones. The notion of turbulence in the quantum world was conceived long ago by Onsager and Feynman, but the occurrence of turbulence in ultracold gases has been studied in the laboratory only very recently. Albeit new as a field, it already offers new paths and perspectives on the problem of turbulence. Herein we review the general properties of quantum gases at ultralow temperatures paying particular attention to vortices, their dynamics and turbulent behavior. We review the recent advances both from theory and experiment. We highlight, moreover, the difficulties of identifying and characterizing turbulence in gaseous Bose-Einstein condensates compared to ordinary turbulence and turbulence in superfluid liquid helium and spotlight future possible directions.

Based on standard field-theoretic considerations, we develop an effective action approach for investigating quantum phase transitions in lattice Bose systems at arbitrary temperature. We begin by adding to the Hamiltonian of interest a symmetry breaking source term. Using time-dependent perturbation theory, we then expand the grand-canonical free energy as a double power series in both the tunneling and the source term. From here, an order parameter field is introduced in the standard way and the underlying effective action is derived via a Legendre transformation. Determining the Ginzburg-Landau expansion to first order in the tunneling term, expressions for the Mott insulator-superfluid phase boundary, condensate density, average particle number, and compressibility are derived and analyzed in detail. Additionally, excitation spectra in the ordered phase are found by considering both longitudinal and transverse variations of the order parameter. Finally, these results are applied to the concrete case of the Bose-Hubbard Hamiltonian on a three-dimensional cubic lattice, and compared with the corresponding results from mean-field theory. Although both approaches yield the same Mott insulator-superfluid phase boundary to first order in the tunneling, the predictions of our effective action theory turn out to be superior to the mean-field results deeper into the superfluid phase.

As a consequence of the quantization of its vorticity, superfluid systems may present the simplest form of turbulence and the exploration of turbulence in Bose condensed gases may create a gateway to better understand various turbulent phenomena. In this work, a magnetically trapped atomic condensate of 87 Rb atoms is used to investigate the emergence of quantum turbulence. Vortices and anti-vortices are generated by applying an off axis, sinusoidal, magnetic field gradient and agitating the condensate to inject kinetic energy. Vortices are created on the periphery and propagate through the cloud, setting up experimental conditions favorable for turbulence. Once a turbulent regime has been produced, the condensate is released from its trapping potential and allowed to freely expand. Measurements of the atomic density profile after a time of flight are used to gain insight into the in situ momentum distribution of the system. These images show clear deviations between the non-turbulent and turbulent density profiles, in both the distribution of momentum and its average magnitude.

Quantized vortices have been observed in a variety of superfluid systems, from 4 He to condensates of alkali-metal bosons and ultracold Fermi gases along the BEC-BCS crossover. In this article we study the stability of singly quantized vortex lines in dilute dipolar self-bound droplets. We first discuss the energetic stability region of dipolar vortex excitations within a variational ansatz in the generalized nonlocal Gross-Pitaevskii functional that includes quantum fluctuation corrections. We find a wide region where stationary solutions corresponding to axially-symmetric vortex states exist. However, these singly-charged vortex states are shown to be unstable, either by splitting the droplet in two fragments or by vortex-line instabilities developed from Kelvin-wave excitations. These observations are the results of large-scale fully three-dimensional simulations in real time. We conclude with some experimental considerations for the observation of such states and suggest possible extensions of this work.

We consider a superfluid cloud composed of a Bose-Einstein condensate oscillating within a magnetic trap (dipole mode) where, due to the existence of a Feshbach resonance, an effective periodic time-dependent modulation in the scattering length is introduced. Under this condition, collective excitations such as the quadrupole mode can take place. We approach this problem by employing both the Gaussian and the Thomas-Fermi variational Ansätze. The resulting dynamic equations are analyzed by considering both linear approximations and numerical solutions, where we observe coupling between dipole and quadrupole modes. Aspects of this coupling related to the variation of the dipole oscillation amplitude are analyzed. This may be a relevant effect in situations where oscillation in a magnetic field in the presence of a bias field B takes place, and should be considered in the interpretation of experimental results.

Turbulence is characterized by a large number of degrees of freedom, distributed over several length scales, that result into a disordered state of a fluid. The field of quantum turbulence deals with the manifestation of turbulence in quantum fluids, such as liquid helium and ultracold gases. We review, from both experimental and theoretical points of view, advances in quantum turbulence focusing on atomic Bose-Einstein condensates. We also explore the similarities and differences between quantum and classical turbulence. Lastly, we present challenges and possible directions for the field. We summarize questions that are being asked in recent works, which need to be answered in order to understand fundamental properties of quantum turbulence, and we provide some possible ways of investigating them. arXiv:1903.12215v1 [cond-mat.quant-gas]

We propose a scheme for generating two-dimensional turbulence in harmonically trapped atomic condensates with the novelty of controlling the polarization (net rotation) of the turbulence. Our scheme is based on an initial giant (multicharged) vortex which induces a large-scale circular flow. Two thin obstacles, created by blue-detuned laser beams, speed up the decay of the giant vortex into many singly-quantized vortices of the same circulation; at the same time, vortex-antivortex pairs are created by the decaying circular flow past the obstacles. Rotation of the obstacles against the circular flow controls the relative proportion of positive and negative vortices, from the limit of strongly anisotropic turbulence (almost all vortices having the same sign) to that of isotropic turbulence (equal number of vortices and antivortices). Using the new scheme, we numerically study quantum turbulence and report on its decay as a function of the polarization. We finally present a phenomenological model for the decay rate of vortex number which fits our numerical experiment curves, with the novelty of taking into account polarization time-dependence.