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Complex Langevin simulation of Bose-Einstein condensate under extreme conditions

智也 早田

Published Mar 24, 2015• 0 citations
Publisher

Abstract

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

DOI
10.15083/00008078

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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
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.
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.
Quantum field theories with a complex action suffer from a sign problem in stochastic nonperturbative treatments, making many systems of great interest – such as polarized or mass-imbalanced fermions and QCD at finite baryon density – extremely challenging to treat numerically. Another such system is that of bosons at finite angular momentum; experimentalists have successfully achieved vortex formation in supercooled bosonic atoms, and have measured quantities of interest such as the moment of inertia. However, the rotation results in a complex action, making the usual numerical treatments of the theory unusable. This thesis treats systems of nonrelativistic bosons with finite angular momentum using two approaches. One approach is to determine the virial coefficients using a semi-classical lattice approximation (SCLA). Through this approach, we are able to compute the thermodynamic equation of state of the bosons for finite trapping frequency, rotation, and inter-particle interaction. The second approach uses the complex Langevin (CL) method – a method which employs an extension of the Langevin equation to complex space and circumvents the sign problem to compute the full quantum behavior of a low energy system of interacting, trapped, and rotating bosons. We examine the density and angular momentum of the system using all three methods, but the CL method in principal allows us to compute properties unique to rotating superfluids, in particular to show the formation of density singularities (so-called vortex lattices) and compute the circulation of the fluid around those vortices. This work advances our understanding of the quantum effects of rotation on ultracold bosonic gases in two different limits.
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.
This thesis manuscript presents experimental results on Bose-Einstein condensates in optical lattices and in the regime of fast rotation. In an optical lattice an array of inde-pendent 2-dimensional Bose-Einstein condensates is produced from an initially 3-dimen-sional cloud. The number of 2D BECs in this array can be controlled by superimposing a magnetic field gradient along the axis of the lattice and then selectively evaporating the sites. When producing a small number of sites and letting them interfere, dislocations in the interference pattern were observed, which must be due to nontrivial phase defects, as for example free vortices. Such vortices and vortex-antivortex pairs play an important role in the superfluid to normal phase transition in a 2-dimensional gas, the Kosterlitz-Thouless transition. With 3-dimensional condensates the limit of fast rotation is studied in a hybrid potential contain-ing a small quartic term. In this trap rotation frequencies larger than the harmonic trapping fre-quency can be reached. These experiments are a first step to a possible observation of strongly correlated states close to the regime of the fractional quantum Hall effect.