So far many theoretical considerations for experiments of Bose-Einstein condensation (BEC) of alkali atoms in harmonic traps are based on the Gross-Pitaevskii (GP) equation. In this report, we attempt to formulate the BEC in the language of quantum field theory in order to go beyond the approximation of GP equation. First the formulation at zero-temperature is presented, and then it is extended to finite-temperature case by means of Thermo Field Dynamics. Numerical calculations, which are inevitable even in the unperturbative formulation of our approach, are performed. For illustration, the corrections at one-loop level to the original GP equation are given. We also calculate the effects of quantum and thermal fluctuations to the distribution of condensed atoms numerically.
So far many theoretical considerations for experiments of Bose-Einstein condensation (BEC) of alkali atoms in harmonic traps are based on the Gross-Pitaevskii (GP) equation. In this report, we attempt to formulate the BEC in the language of quantum field theory in order to go beyond the approximation of GP equation. First the formulation at zero-temperature is presented, and then it is extended to finite-temperature case by means of Thermo Field Dynamics. Numerical calculations, which are inevitable even in the unperturbative formulation of our approach, are performed. For illustration, the corrections at one-loop level to the original GP equation are given. We also calculate the effects of quantum and thermal fluctuations to the distribution of condensed atoms numerically.
So far much theoretical consideration of experiments on the Bose-Einstein condensation (BEC) of alkali-metal atoms in harmonic traps is based on the Gross-Pitaevskii (GP) equation. In this paper, we attempt to formulate the BEC in the language of quantum field theory and to estimate the quantum and thermal fluctuation effects, which are neglected in the approximation using the GP equation. First, the formulation at zero temperature is developed, and then it is extended to the finite-temperature case by means of thermofield dynamics. We treat the zero-energy mode with care, so that the canonical commutation relations hold. As a result, an infrared divergence appears, but it can be renormalized into the observed condensate number. Numerical calculations are performed. For illustration, the corrections at one-loop level to the original GP equation are given. We also calculate numerically the effects of quantum and thermal fluctuations on the distribution of condensed atoms.
The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local-density approximation. Consistency of the theory for temperatures through the Bose condensation point Tc requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length asc, increases the number of condensate atoms at each temperature, and raises the value of Tc relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near Tc. Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.
In the mean-field approximation, a trapped Bose-Einstein condensate at zero temperature is described by the Gross-Pitaevskii equation for the condensate or, equivalently, by the hydrodynamic equations for the number density and current density. These equations receive corrections from quantum field fluctuations around the mean field. We calculate the semiclassical corrections to these equations for a general time-independent state of the condensate, extending previous work to include vortex states as well as the ground state. In the Thomas-Fermi limit, the semiclassical corrections can be taken into account by adding a local correction term to the Gross-Pitaevskii equation. At second order in the Thomas-Fermi expansion, the semiclassical corrections can be taken into account by adding local correction terms to the hydrodynamic equations.
This chapter introduces the fundamental concepts of Bose–Einstein condensates (BECs), providing a comprehensive theoretical foundation for subsequent investigation on their localization and numerical simulation. It begins with a historical overview of BECs, tracing their theoretical origins and experimental realization. The chapter then focuses on the theoretical perspective of scalar BECs and how it can be modelled by a single-component Gross–Pitaevskii (GP) equation in the mean-field description, a variable-coefficient nonlinear Schrödinger (NLS) equation encompassing a trapping potential and interatomic interaction. The mean-field description is then extended to govern the dynamics of vectorial condensates described by a coupled GP equation with intra and inter-species (atomic) interactions. Subsequently, the chapter discusses the origin of fermionic condensates, highlighting their distinct pairing mechanisms. Beyond mean-field approaches are then introduced to address the limitations of the GP framework, incorporating quantum fluctuations and correlations. The chapter concludes with an outline of the textbook, setting the stage for analytical and numerical investigations of ultracold atoms in ensuing chapters.
Trapped Bose-condensed atomic gases are remarkable because, in spite of the fact that these are very dilute systems, they exhibit robust coherent dynamic behaviour when perturbed. These quantum “wisps of matter” are a new phase of highly coherent matter. While binary collisions are very infrequent, the large coherent mean field associated with the Bose condensate ensures that interactions play a crucial role in determining the collective response of these trapped superfluid gases. In our discussion of the theory of collective oscillations of atomic condensates, the macroscopic Bose wavefunction Φ( r , t ) plays a central role. This wavefunction is the BEC order parameter. As discussed in Chapter 1, the initial attempts at defining this order parameter began with the pioneering work of London (1938a), were further developed by Bogoliubov (1947) and finally extended to deal with any Bose superfluid using the general quantum field theoretic formalism developed by Beliaev (1958a). Almost all this early theoretical work was limited to T = 0 where, in a dilute weakly interacting Bose gas, all the atoms are in the condensate. The first extension of these ideas to nonuniform Bose condensates was by Pitaevskii (1961) and, independently, by Gross (1961), which led to the now famous Gross–Pitaevskii (GP) equation of motion for Φ( r , t ). Before the discovery of BEC in trapped gases, the time-dependent GP equation was mainly used to study vortices in Bose superfluids, which involve a spatially nonuniform ground state. Apart from this application, the GP equation was largely unknown. The situation changed overnight in 1995 with the creation of trapped nonuniform Bose condensates in atomic gases.