The relative-density amplitude profile of a quantized vortex line in He II is calculated from a generalized Gross-Pitaevskii (GP) equation. The ordinary GP equation is a Hartree nonlinear integrodifferential equation for the order parameter of an interacting system of bosons. The generalization is made by replacing the two-body potential in the average field by a local $T$ matrix, which is calculated from a realistic interatomic potential for helium. In configuration space the $T$ matrix is similar to the potential, except in the repulsive region where it is much softer. The only adjustable parameter in the generalized GP equation is the condensate fraction for which a theoretical value of 0.10 and an experimental value of 0.024 are used. In all cases the density profile is a monotonically increasing function with an effective core radius of from 3.7 to 4.7 \AA{}.
The relative-density amplitude profile of a quantized vortex line in He II is calculated from a generalized Gross-Pitaevskii (GP) equation. The ordinary GP equation is a Hartree nonlinear integrodifferential equation for the order parameter of an interacting system of bosons. The generalization is made by replacing the two-body potential in the average field by a local T matrix, which is calculated from a realistic interatomic potential for helium. In configuration space the T matrix is similar to the potential, except in the repulsive region where it is much softer. The only adjustable parameter in the generalized GP equation is the condensate fraction for which a theoretical value of 0.10 and an experimental value of 0.024 are used. In all cases the density profile is a monotonically increasing function with an effective core radius of from 3.7 to 4.7 \AA{}.
The Gross-Pitaevskii equation for the order parameter of a superfluid boson system is derived in a form which is applicable to a system in which the two-body potential has a strong repulsive core, as is the case for liquid helium. The equation of motion for the order parameter, defined as the matrix element of the field annihilation operator between the states of an ($N\ensuremath{-}1$)-particle and an N-particle system, is obtained. The term describing the average potential due to the other particles is expanded in a perturbation series, in which the coherentstate representation is used. The resulting expansion is the same as using the Bogoliubov approximation and Wick's theorem, and shows that the Bogoliubov expansion is exact in the limit of an infinite number of particles. The perturbation expansion is partially resummed to give a Hartree field in which the potential has been replaced by the T matrix, which gives a finite average effective field.
Quantum vortex dynamics in Bose-Einstein condensates or superfluid helium can be informatively described by the Gross-Pitaevskii (GP) equation. Various approximate analytical formulas for a single stationary vortex are recalled and their shortcomings demonstrated. Significantly more accurate two-point [2/2] and [3/3] Pad\'e approximants for stationary vortex profiles are presented. Two straight, singly quantized, antiparallel vortices, located at a distance d0 apart, form a vortex dipole, which, in the GP model, can either annihilate or propagate indefinitely as a ``solitary wave.'' We show, through calculations performed in a periodic domain, that the details and types of behavior displayed by vortex dipoles depend strongly on the initial conditions rather than only on the separation distance d0 (as has been previously claimed). It is found, indeed, that the choice of the initial two-vortex profile (i.e., the modulus of the ``effective wave function''), strongly affects the vortex trajectories and the time scale of the process: annihilation proceeds more rapidly when low-energy (or ``relaxed'') initial profiles are imposed. The initial ``circular'' phase distribution contours, customarily obtained by multiplying an effective wave function for each individual vortex, can be generalized to explicit elliptical forms specified by two parameters; then by ``tuning'' the elliptical shape at fixed d0, a sharp transition between solitary-wave propagation and annihilation is captured. Thereby, a ``phase diagram'' for this ``AnSol'' transition is constructed in the space of ellipticity and separation and various limiting forms of the boundary are discussed.
We introduce a nonlinear Schr\"odinger equation to describe the dynamics of a superfluid Bose gas in the crossover from the weak-coupling regime, where $a{n}^{1/3}\ensuremath{\ll}1$ with a the interatomic s-wave scattering length and n the bosonic density, to the unitarity limit, where $a\ensuremath{\rightarrow}+\ensuremath{\infty}$. We call this equation the unitarity Schr\"odinger equation (USE). The zero-temperature bulk equation of state of this USE is parametrized by the Lee-Yang-Huang low-density expansion and Jastrow calculations at unitarity. With the help of the USE we study the profiles of quantized vortices and vortex-core radius in a uniform Bose gas. We also consider quantized vortices in a Bose gas under cylindrically symmetric harmonic confinement and study their profile and chemical potential using the USE and compare the results with those obtained from the Gross-Pitaevskii-type equations valid in the weak-coupling limit. Finally, the USE is applied to calculate the breathing modes of the confined Bose gas as a function of the scattering length.
In this work we studied two sistems of bosonic superfluids at zero temperature,
ultra-cold Bose gases and superfluid helium, within a simple classical field formalism and we
underlined differences and similarities between them. We also analyzed how the presence
of strong interactions, as in the case of helium, modifies the quasiparticle excitation spectrum and
subsequently the vortex line density profile. After a general review about these bosonic systems properties
we started in general from the Hartree equation of a bosonic system and,
with different hypotesis on the interaction potential between bosons, we obtained the equations
describing the two above mentioned systems, mainly focusing on elementary
quasiparticle excitations and quantized vortices.
Furthermore throughout this work the results were compared with experimental
data and aftermaths of the main theories in literature.
In this thesis I present two different research topics investigated during the course of my PhD, regarding the analysis of spatial structures in a Bose Einstein condensate. Ultracold atomic gases offer a privileged platform for such kind of experiments, thanks to the fine control that can be achieved on the system’s parameters and to the availability of advanced imaging schemes allowing for a great measurement accuracy. \nThe first topic is about the shape of quantized vortices in an elongated condensate, with the goal of providing a quantitative analysis of the density structure of a quantized vortex filament hosted in a bulk 3D superfluid. We analyzed the shape of the vortex and studied its dynamics during a free expansion, or time of flight (TOF), of the hosting BEC, with the goal of making a quantitative comparison between theory and experiment for the structure of the core of a quantized vortex in three-dimensional (3D) condensates. Simultaneously imaging the sample along orthogonal directions after a long TOF allowed to map the complete 3D shape of the vortex at the end of the free flight, while the full expansion dynamics has been simulated with numerical solutions of the Gross-Pitaevskii equation. The same data analysis procedure has been applied to both the experimental images and to the density profiles computed with the simulations to ensure a faithful comparison. \nWe were able to detail the evolution of the vortex parameters at all times combining a simple analytic scaling-law model valid at early times, experimental data for the width and the depth of the core at long expansion times, and the numerics that were used to bridge between the two. Additionally, we could check the validity of the predictions on the scaling of vortex parameters with the size of the BEC using the experimental data to interpolate between theoretical limiting models. We concluded that quantized vortex filaments can be optically imaged with standard techniques in 3D atomic BECs, at a level of accuracy which indeed is enough to show good quantitative agreement with the predictions of the GP theory for the width, depth, and overall shape of the vortex core. \nThe second topic is a measurement of the equation of state of a single component BEC. The goal of this project is to verify the non-monotonic behaviour of the chemical potential of a homogeneous Bose gas of weakly interacting particles as a function of temperature, where one expects to find a maximum across the critical point of transition to the superfluid phase. This effect is believed to be a general feature of the normal-to-superfluid phase transition: it has been already experimentally demonstrated in unitary Fermi gases, and although the same is predicted to happen also in a gas of weakly interacting bosons, no experimental evidence has been reported so far. The measurement relies on the local density approximation, which allows to extract information about the thermodynamics of a homogeneous system from accurate measurements of the local properties of a trapped one. My work has focused on developing a series of imaging and data analysis techniques to measure the 3D density profile of a harmonically trapped gas, even in regimes of extreme density such as inside a Bose condensate. With a new high-dynamic-range method we were able to image the 3D density distribution of a trapped sample, leading to a low-noise measurement of the density distribution. \nWe confirmed the existence of the non-monotonic behaviour of the chemicial potential across, and set the basis for further measurements of the thermodynamics of the system across the transition.In this thesis I present two different research topics investigated during the course of my PhD, regarding the analysis of spatial structures in a Bose Einstein condensate. Ultracold atomic gases offer a privileged platform for such kind of experiments, thanks to the fine control that can be achieved on the system’s parameters and to the availability of advanced imaging schemes allowing for a great measurement accuracy. \nThe first topic is about the shape of quantized vortices in an elongated condensate, with the goal of providing a quantitative analysis of the density structure of a quantized vortex filament hosted in a bulk 3D superfluid. We analyzed the shape of the vortex and studied its dynamics during a free expansion, or time of flight (TOF), of the hosting BEC, with the goal of making a quantitative comparison between theory and experiment for the structure of the core of a quantized vortex in three-dimensional (3D) condensates. \nSimultaneously imaging the sample along orthogonal directions after a long TOF allowed to map the complete 3D shape of the vortex at the end of the free flight, while the full expansion dynamics has been simulated with numerical solutions of the Gross-Pitaevskii equation. The same data analysis procedure has been applied to both the experimental images and to the density profiles computed with the simulations to ensure a faithful comparison. \nWe were able to detail the evolution of the vortex parameters at all times combining a simple analytic scaling-law model valid at early times, experimental data for the width and the depth of the core at long expansion times, and the numerics that were used to bridge between the two. Additionally, we could check the validity of the predictions on the scaling of vortex parameters with the size of the BEC using the experimental data to interpolate between theoretical limiting models. We concluded that quantized vortex filaments can be optically imaged with standard techniques in 3D atomic BECs, at a level of accuracy which indeed is enough to show good quantitative agreement with the predictions of the GP theory for the width, depth, and overall shape of the vortex core. \nThe second topic is a measurement of the equation of state of a single component BEC. The goal of this project is to verify the non-monotonic behaviour of the chemical potential of a homogeneous Bose gas of weakly interacting particles as a function of temperature, where one expects to find a maximum across the critical point of transition to the superfluid phase. This effect is believed to be a general feature of the normal-to-superfluid phase transition: it has been already experimentally demonstrated in unitary Fermi gases, and although the same is predicted to happen also in a gas of weakly interacting bosons, no experimental evidence has been reported so far. The measurement relies on the local density approximation, which allows to extract information about the thermodynamics of a homogeneous system from accurate measurements of the local properties of a trapped one. My work has focused on developing a series of imaging and data analysis techniques to measure the 3D density profile of a harmonically trapped gas, even in regimes of extreme density such as inside a Bose condensate. With a new high-dynamic-range method we were able to image the 3D density distribution of a trapped sample, leading to a low-noise measurement of the density distribution. \nWe confirmed the existence of the non-monotonic behaviour of the chemicial potential across, and set the basis for further measurements of the thermodynamics of the system across the transition.
The dynamics of interacting quantized vortex filaments in a rotating Bose–Einstein condensate existing in the Thomas–Fermi regime at zero temperature and obeying the Gross–Pitaevskii equation has been considered in the hydrodynamic “nonelastic” approximation. A noncanonical Hamilton equation of motion for the macroscopically averaged vorticity has been derived for a smoothly inhomogeneous array of filaments (vortex lattice) taking into account spatial nonuniformity of the equilibrium density of the condensate, which is determined by the trap potential. The minimum of the corresponding Hamiltonian describes the static configuration of the deformed vortex lattice against the preset density background. The condition of minimum can be reduced to a nonlinear second-order partial differential vector equation for which some exact and approximate solutions are obtained. It has been shown that if the condensate density has an anisotropic Gaussian profile, the equation of motion for the averaged vorticity has solutions in the form of a vector exhibiting a nontrivial time dependence, but homogeneous in space. An integral representation has also been obtained for the matrix Green function that determines the nonlocal Hamiltonian of a system of several quantized vortices of an arbitrary shape in a Bose–Einstein condensate with the Gaussian density. In particular, if all filaments are straight and oriented along one of the principal axes of the ellipsoid, we have a finitedimensional reduction that can describe the dynamics of the system of pointlike vortices against an inhomogeneous background. A simple approximate expression is proposed for the 2D Green function with an arbitrary density profile and is compared numerically with the exact result in the Gaussian case. The corresponding approximate equations of motion, describing the long-wavelength dynamics of interacting vortex filaments in condensates with a density depending only on transverse coordinates, have been derived.