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Measurement of the density profile of quantized vortices and of the equation of state in a 3D interacting Bose gas

Carmelo Mordini

Published Jan 1, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.15168/11572_368246

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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. 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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.
Bose-Einstein condensates (BECs), with their superfluid behavior, quantized vortices, and high-level of control over trap geometry and other system parameters provide a compelling environment for studies of quantum fluid dynamics. Recently there has been an influx of theoretical and numerical progress in understanding the superfluid dynamics associated with two-dimensional quantum turbulence, with expectations that complementary experiments will soon be realized. In this dissertation I present progress in the development of an experimental toolkit that will enable such experimental studies of two-dimensional quantum turbulence. My approach to developing this toolkit has been twofold: first, efforts aimed at the development of experimental techniques for generating large disordered vortex distributions within a BEC; and second, efforts directed towards the design, implementation, and characterization of a quantum vortex microscope. Quantum turbulence in a superfluid is generally regarded as a disordered tangle of quantized vortices in three dimensions, or a disordered planar distribution of quantized vortices in two dimensions. However, not all vortex distributions, even large disordered ones, are expected to exhibit robust signatures of quantum turbulence. Identification and development of techniques for controlled forcing or initialization of turbulent vortex distributions is now underway. In this dissertation, I will discuss experimental techniques that were examined during the course of my dissertation research, namely generation of large disordered distributions of vortices, and progress towards injecting clusters of vortices into a BEC. Complimentary to vortex generation is the need to image these vortex distributions. The nondeterministic nature of quantum turbulence and other far-from-equilibrium superfluid dynamics requires the development of new imaging techniques that allow one to obtain information about vortex dynamics from a single BEC. To this end, the first vortex microscope constructed as part of my dissertation research enabled the first in situ images of quantized vortices in a single-component BEC, obtained without prior expansion. I have further developed and characterized a second vortex microscope, which has enabled the acquisition of multiple in situ images of a lattice of vortex cores, as well as the acquisition of single in situ images of vortex cores in a BEC confined in a weak hybrid trap. In this dissertation, I will discuss the state-of-the-art of imaging vortices and other superfluid phenomena in the University of Arizona BEC lab, as indicated by the examined performance of the quantum vortex microscope.
Quantum gases are well isolated, highly controllable, and defect-free systems which can simulate many body quantum phenomena that have been studied in condensed matter systems.To study two-dimensional superfluid, we have developed an experimental apparatus that can produce Bose-Einstein condensates (BEC) of 23 Na atoms.The apparatus can generate a pure condensate of 10 7 atoms in an optically plugged magnetic quadrupole trap within 17 s.The Berezinskii-Kosterlitz-Thouless (BKT) theory provides a general framework of the superfluid phase transition in two dimension, which does not involve spontaneous symmetry breaking and emergence of an order parameter below a critical temperature.Instead, it is the formation of vortex pairs with opposite circulations below the critical temperature that mediates the superfluid phase transition.Recently, it has been demonstrated that degenerate Bose gases confined in highly oblate harmonic potentials undergo the BKT phase transition.This thesis focuses on our experimental research on thermal phase fluctuations (i.e., long-wavelength phonons and vortex pairs) in the superfluid i phase of trapped quasi-2D BECs.We have developed a quantitative probe to measure phase fluctuations in the 2D superfluid by free expansion, where the phase fluctuations in the sample were revealed as density modulations in the course of expansion.The power spectrum of the density fluctuations showed an oscillatory shape and the scaling behavior of the peak positions could be understood as a Talbot effect in matter waves.Employing this method, we demonstrated the thermal origin of phase fluctuations.We also investigated relaxation dynamics of nonequilibrium states of the quasi-2D system using the power spectrum.Thermally excited vortex pairs are the characteristic feature of the 2D superfluid.The quantized vortices are conventionally observed by a densitydepleted core after expanding a trapped sample.The method, however, cannot be applied to the 2D sample because the density modulations after free expansion lower the vortex core visibility.We enhanced the core visibility by radial compression of the sample before the expansion so the phonon modes in the 2D sample were relaxed in a 3D environment.Measuring vortex distributions, we revealed the pairing feature by spatial correlations of the vortex positions.We also studied BKT-BEC crossover phenomena in a finite-size sample trapped in a quasi-2D harmonic potential by investigating the vortex profiles at various temperatures.Condensates of atoms have an internal spin structure so they can host various kinds of topological excitations.The two-dimensional Skyrmion is one of the topological spin textures in the anti-ferromagnetic spinor condensate and ii we imprinted the structure using the magnetic field sweep technique.The 2D Skyrmion spin texture has a finite Berry curvature because of the non-coplanar spin configuration.We studied a geometric Hall effect, with condensates trapped in a harmonic potential with the Skyrmion spin texture.Under a linear driving of the spin texture, we observed a condensate dipole motion resonantly developed into a circular motion, which demonstrates the existence of an effective Lorentz force.
Dilute gas Bose-Einstein condensates (BECs) provide unique and powerful experimental platforms to study fluid turbulence. Some aspects of BEC hydrodynamics are specific to atomic quantum fluids such as quantized vortices and flexible trapping geometries. However, there are features of turbulence that are universal and simpler to understand in such systems. BECs are also of interest in fundamental physics in their own right and a rich synthesis of theory and experiment has yielded powerful numerical methods to simulate and study BEC dynamics. In this dissertation I present two experiments, one conducted and one proposed, describing novel aspects of the generation and detection of vorticity in Bose-Einstein condensates. Models of BEC dynamics at zero temperature are provided by the Gross-Pitaevskii equation. Simple and fast numerical solutions of this equation have yielded a wealth of literature. However, in experimental reality the atomic ensemble exists at a finite temperature and consists of a BEC coexisting with a thermal cloud of non-condensed atoms. The interaction between the condensate and the thermal fraction yields rich and complex physics that require more advanced models. We present an experiment demonstrating the relaxation dynamics of a BEC in a rotating trap perturbed by a repulsive laser barrier. The data provided by this experiment are valuable to further development of theoretical models that incorporate interactions between the BEC and noncondensed atoms. In the field of quantum fluid dynamics, an experimental method to determine the position and circulation of vortices is a highly sought-after capability. Onsager's point-vortex model of turbulence completely determines the kinetic energy spectrum of an incompressible fluid by these degrees of freedom. We present proof-of-principle simulations that describe a new method of spatially sampling the velocity field of a two-dimensional BEC by using an optical lattice analogously to a Shack-Hartmann Wavefront Sensor. Extracting vortex information from the appropriately sampled velocity field can be accomplished either qualitatively or with detection algorithms. This method requires minimal experimental infrastructure and is generally applicable across atomic species. The implications of measuring a condensate velocity field are broad and these initial results provide the first step towards realizing a valuable tool in BEC physics.
Atom chips are a great tool for creation of low dimensional magnetic trapping geometries via micro-structures on the chip surface. Such structures allow the creation of time-dependent magnetic and electric potentials with highly accurate spatial and temporal dependency. As part of this thesis we have investigated the coherence dynamics in one-dimensional Bose-Einstein condensate while creating a sudden change in the atomic trapping potential. Such sudden changes create phase perturbations of the wave function, which leads to density perturbations. Analyzing these changes enables studies of the evolution of the coherence in a one-dimensional Bose gas with dynamically changing boundary conditions. Of particular interest is the study of prethermalization which can be understood in an integrable systems as so-called generalized Gibbs state. This state does not decay, but in case that there are perturbations that break integrability, this state relaxes further to a thermal state. To get a good understanding of such 1D systems we first investigated the transition from 3D to 1D Bose gas by observing both in situ and time of flight density profiles and analyzing the spatial variations in atom number as a function of temperature, geometry, and atomic density. High quality imaging is essential in these types of atomic physics experiments, and therefore a whole chapter is devoted to a new optimization method of absorption imaging. In this method we have taken into account the quantum nature of both the atomic medium and imaging light. Last, we have outlined an experiment that utilizes one-dimensional Bose-Einstein condensate as an analogue model of quantum field theory, in particular the dynamical Casimir effect and Hawking radiation. We do so by dynamically splitting a condensate along its long axis to a Y-like shape and measure the differential phase between the branches.