The thermodynamic stability of quantized vortex patterns in rotating Bose-Einstein condensates is assessed at finite temperature using complex Langevin sampling. We construct a temperature-rotation frequency phase diagram and find that that vortices are stabilized at lower rotation speeds by the addition of quantum and thermal fluctuations. The coherent states field theoretic representation of the imaginary time path integral enables efficient simulation of large systems at finite temperature, and the complex Langevin simulation scheme bypasses the sign problems that arise from the complex-valued coherent states fields as well as the gauge potential describing solid body rotation. Field operators allow us to generate high-resolution images of particle and momentum density of the cloud. Quantized vortices appear as dark spots on density images, and vector plots of cloud momentum detail circulation around each vortex.
The thermodynamic stability of quantized vortex patterns in rotating Bose-Einstein condensates is assessed at finite temperature using complex Langevin sampling. We construct a temperature-rotation frequency phase diagram and find that that vortices are stabilized at lower rotation speeds by the addition of quantum and thermal fluctuations. The coherent states field theoretic representation of the imaginary time path integral enables efficient simulation of large systems at finite temperature, and the complex Langevin simulation scheme bypasses the sign problems that arise from the complex-valued coherent states fields as well as the gauge potential describing solid body rotation. Field operators allow us to generate high-resolution images of particle and momentum density of the cloud. Quantized vortices appear as dark spots on density images, and vector plots of cloud momentum detail circulation around each vortex.
In this thesis, we perform ab-initio lattice simulations of the nonrelativistic and relativistic Bose-Einstein condensates under strong external fields such as the electric field, the magnetic field, and the rotation.All the lattice actions discussed in this thesis are complex, so that the conventional Monte Carlo method su↵ers from the notorious sign problem.The path integral measure is complex due to the complex action, so that the important sampling algorithm cannot be applied.To overcome this di culty, we adopt the complex Langevin method, which has been developed in the context of finite density quantum chromodynamics to attack the fermion sign problem.In particular, we perform ab-initio simulation of the quantum vortex nucleation by the magnetic field or the rotation.In dilute and low temperature system, quantum and thermal fluctuations are negligible.The system can be remarkably well described by the mean field approximation.However, when quantum or thermal fluctuation becomes large, it is highly nontrivial how the quantized vortices behave.Around the critical values of temperature, chemical potential, magnetic field, or angular velocity, the fluctuation grows and the mean-field description inevitably breaks down.To understand the quantum vortex nucleation in such situations, the ab-initio simulation is necessary as discussed in this thesis.The direct evidence of the quantum vortex nucleation is the quantization of circulation.We show that the circulation is quantized in the superconducting (superfluid) phase far from the critical chemical potential, but it is not just above the critical chemical potential.We observe that the quantized circulation is blurred by the quantum fluctuation as the chemical potential is getting close to its critical value.In our simulation, the quantum fluctuation of vortex number is observed, while the averaged circulation is clearly quantized.At first glance, these two facts seem to be incompatible.However, we show that the fluctuation of vortex number behaves as Gaussian and, as a result of cancellation, the average circulation becomes exact integer.iii
In this review, we give an overview of the experimental and theoretical advances in the physics of quantized vortices in dilute atomic-gas Bose--Einstein condensates in a trapping potential, especially focusing on experimental research activities and their theoretical interpretations. Making good use of the atom optical technique, the experiments have revealed many novel structural and dynamic properties of quantized vortices by directly visualizing vortex cores from an image of the density profiles. These results lead to a deep understanding of superfluid hydrodynamics of such systems. Typically, vortices are stabilized by a rotating potential created by a laser beam, magnetic field, and thermal gas. Finite size effects and inhomogeneity of the system, originating from the confinement by the trapping potential, yield unique vortex dynamics coupled with the collective excitations of the condensate. Measuring the frequencies of the collective modes is an accurate tool for clarifying the character of the vortex state. The topics included in this review are the mechanism of vortex formation, equilibrium properties, and dynamics of a single vortex and those of a vortex lattice in a rapidly rotating condensate.
Bose-Einstein condensation, first predicted by Bose and Einstein in 1924, was finally realized in dilute, weakly interacting gases of alkali atoms in 1995. A few years later, quantized vortices - intimately related to the phenomenon of superfluidity - were created and observed in these novel quantum many-body systems.\n\nIn this Thesis, vortices in dilute atomic Bose-Einstein condensates are studied theoretically and computationally using microscopic mean-field theory formalisms. Special emphasis is on the structure and the stability of vortices at finite temperatures. The formation of quantized circulation in Bose-Einstein condensates is investigated, as well as multiply quantized vorticity and the adiabaticity of the precessing motion of a vortex.\n\nThis research strongly relies on numerically solving the Gross-Pitaevskii equation for the macroscopic condensate wavefunction and the coupled Bogoliubov equations and their variants for the collective quasiparticle excitation eigenmodes. The computed elementary excitation spectra are used, for instance, in determining the local stability of various condensate states. The vortex formation process is studied at finite temperatures in terms of the surface modes of the condensate providing the Landau critical velocities for the breakdown of superfluidity. Free energy calculations are used for determining the thermodynamic equilibrium states.\n\nSingly quantized, axisymmetric vortex states are found to be locally energetically stable even in the zero-temperature limit within the self-consistent finite-temperature approximations. This would imply for the vortex a precession direction opposite to that observed in the experiment. Adiabatic approximation of quantum mechanics is formulated for Bose-Einstein condensed systems and is applied to a case of a moving vortex line. The experimentally observed precessing motion of the vortex is found to be nonadiabatic. A method to create stable multiply quantized vortices using an additional pinning potential is presented. The computed critical trap rotation frequencies for vortex nucleation at finite temperatures are in agreement with experiments and the predictions of zero-temperature field theories.
Since the first realizations of Bose-Einstein condensation in dilute alkali-atom vapours in 1995, these novel quantum gases have attracted wide interest. The main focus of the research has been on finding out the coherence and superfluidity properties of the condensates. From the theoretical point of view, these systems provide a rare possibility to model an interacting many-particle system exhaustively and quantitatively accurately from first-principles quantum field theories. However, the challenge of developing a correct and computationally feasible finite-temperature formalism still remains.\n\nThe coherence and superfluidity properties are intimately related to the existence and stability of quantized vortices. In this thesis, the structure and stability of vortex states in weakly interacting Bose-Einstein condensates is investigated within microscopic finite-temperature mean-field theories. An effective scheme for computationally solving the Bogoliubov-de Gennes quasiparticle eigenequations is developed, and applied to study the microscopic structure of vortex states in partially condensed vapours. In contrast to experimental observations and the predictions of the zero-temperature Bogoliubov approximation, stationary vortex states are found to be metastable within the self-consistent Popov approximation and its extensions even in the zero-temperature limit.\n\nThe reasons underlying this discrepancy are analysed in detail. A criterion for the validity of stationary mean-field approximations in modelling slowly time-dependent systems is derived, and applied to precessing off-axis vortex states. Under typical experimental conditions moving vortices are shown to be deformed due to the inability of the thermal cloud to follow the vortex core rigidly. Consequently, the validity of the commonly used stationary mean-field theories in describing such states is shown to be questionable. Finally, an analysis of uniformly precessing off-axis vortex states within a fully time-dependent Popov approximation is presented. Even dynamically deformed vortices are shown to become energetically metastable as the vortex core is partially filled with noncondensed gas in the course of thermalization. This self-stabilization mechanism of vortex states, shown to occur under very general circumstances, is an inherent property of all self-consistent formalisms having a structure similar to the Popov approximation. However, it is argued to be affected by the improper treatment of the noncondensate dynamics within these formalisms, and the need for a theory taking properly into account the coupled dynamics of the condensate and the noncondensate in modelling vortex states is emphasized. The computational scheme developed is also applied to investigate possibilities to energetically stabilize multiply quantized vortex states by piercing an externally rotated condensate with a blue-detuned laser beam.
We study the dynamics of a single vortex and a pair of vortices in quasi two-dimensional Bose-Einstein condensates at finite temperatures. To this end, we use the stochastic Gross--Pitaevskii equation, which is the Langevin equation for the Bose--Einstein condensate. For a pair of vortices, we study the dynamics of both the vortex-vortex and vortex-antivortex pairs, which are generated by rotating the trap and moving the Gaussian obstacle potential, respectively. Due to thermal fluctuations, the constituent vortices are not symmetrically generated with respect to each other at finite temperatures. This initial asymmetry coupled with the presence of random thermal fluctuations in the system can lead to different decay rates for the component vortices of the pair, especially in the case of two corotating vortices.