We perform detailed analytical and numerical studies of a recently proposed method for a spectroscopic measurement of the superfluid fraction of an ultracold atomic gas [N. R. Cooper and Z. Hadzibabic, Phys. Rev. Lett. 104, 030401 (2010)]. Previous theoretical work is extended by explicitly including the effects of nonzero temperature and interactions, and assessing the quantitative accuracy of the proposed measurement for a one-component Bose gas. We show that for suitably chosen experimental parameters the method yields an experimentally detectable signal and a sufficiently accurate measurement. This is illustrated by explicitly considering two key examples: First, for a weakly interacting three-dimensional Bose gas it reproduces the expected result that below the critical temperature the superfluid fraction closely follows the condensate fraction. Second, it allows a clear quantitative differentiation of the superfluid and the condensate density in a strongly interacting Bose gas.
We perform detailed analytical and numerical studies of a recently proposed method for a spectroscopic measurement of the superfluid fraction of an ultracold atomic gas [N. R. Cooper and Z. Hadzibabic, Phys. Rev. Lett. 104, 030401 (2010)]. Previous theoretical work is extended by explicitly including the effects of nonzero temperature and interactions, and assessing the quantitative accuracy of the proposed measurement for a one-component Bose gas. We show that for suitably chosen experimental parameters the method yields an experimentally detectable signal and a sufficiently accurate measurement. This is illustrated by explicitly considering two key examples: First, for a weakly interacting three-dimensional Bose gas it reproduces the expected result that below the critical temperature the superfluid fraction closely follows the condensate fraction. Second, it allows a clear quantitative differentiation of the superfluid and the condensate density in a strongly interacting Bose gas.
We study the condensate and superfluid fraction of a homogeneous gas of weakly interacting bosons in three spatial dimensions by adopting a self-consistent Popov approximation, comparing this approach with other theoretical schemes. Differently from the superfluid fraction, we find that at finite temperature the condensate fraction is a non-monotonic function of the interaction strength, presenting a global maximum at a characteristic value of the gas parameter, which grows as the temperature increases. This non-monotonic behavior has not yet been observed, but could be tested with the available experimental setups of ultracold bosonic atoms confined in a box potential. We clearly identify the region of parameter space that is of experimental interest to look for this behavior and provide explicit expressions for the relevant observables. Finite size effects are also discussed within a semiclassical approximation.
Quantitative numerical analyses of interacting dilute Bose-Einstein condensates are most often based on semiclassical approximations. Since the complex-valued field-theoretic action of the Bose gas does not offer itself to the direct application of standard Monte Carlo techniques, simulations beyond their scope by now almost exclusively rely on quantum-mechanical techniques. Here we explore an alternative approach based on a Langevin-type sampling in an extended state space, known as the complex Langevin (CL) algorithm. While the use of the CL technique has a long-standing history in high-energy physics, in particular in the simulation of QCD at finite baryon density, applications to ultracold atoms are still in their infancy. Here we examine the applicability of the CL approach for a one- and two-component, three-dimensional nonrelativistic Bose gas in thermal equilibrium, below and above the Bose-Einstein phase transition. By comparison with analytic descriptions at the Gaussian level, including Bogoliubov and Hartree-Fock theory, we find that the method allows computing reliably and efficiently observables in the regime of experimentally accessible parameters. Close to the transition, quantum corrections lead to a shift of the critical temperature which we reproduce within the numerical range known in the literature. With this work, we aim to provide a first test of CL as a potential out-of-the-box tool for the simulation of experimentally realistic situations, including trapping geometries and multicomponent (or multiple species) models.
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.
We report on a superfluid properties of a Bose- Einstein condensate in a rotating harmonic trap. We employ the semiclassical approximation, which is the density of state approach. Our approach provided the finite size effect. The superfluid fraction of bosons and its heat capacity at constant volume are investigated. The calculated results provide a good qualitative agreement with the measured experimental results for superfluid helium 4H. Moreover, furnish useful qualitatively theoretical results for a spectroscopic measurement of the superfluid fraction of an ultracold atomic gas [N. R. Cooper and Z. Hadzibabic, Phys. Rev. Lett. 104, 030401 (2010)]. Our results show that both The superfluid fraction of bosons and its heat capacity depend on the number of the atoms. This significant dependence should be considered when the it in situ widths are used in extracting measurable experimental quantities. Our approach can be extended to consider the rotating condensate in optical potential .
Recent experiments on ultracold Bose gases in two dimensions have provided evidence for the existence of the Berezinskii-Kosterlitz-Thouless (BKT) phase via analysis of the interference between two independent systems. In this work we study the two-dimensional quantum degenerate Bose gas at finite temperature using the projected Gross-Pitaevskii equation classical field method. Although this describes the highly occupied modes of the gas below a momentum cutoff, we have developed a method to incorporate the higher momentum states in our model. We concentrate on finite-sized homogeneous systems in order to simplify the analysis of the vortex pairing. We determine the dependence of the condensate fraction on temperature and compare this to the calculated superfluid fraction. By measuring the first order correlation function we determine the boundary of the Bose-Einstein condensate and BKT phases, and find it is consistent with the superfluid fraction decreasing to zero. We reveal the characteristic unbinding of vortex pairs above the BKT transition via a coarse-graining procedure. Finally, we model the procedure used in experiments to infer system correlations [Hadzibabic et al., Nature 441, 1118 (2006)], and quantify its level of agreement with directly calculated in situ correlation functions.