Basic Mean-Field Theory for Bose-Einstein Condensates

Optics Express, 2001

We study the equilibrium dynamics of a weakly interacting Bose-Einstein condensate trapped in a box. In our approach we use a semiclassical approximation similar to the description of a multi-mode laser. In dynamical equations derived from a full N -body quantum Hamiltonian we substitute all creation (and annihilation) operators (of a particle in a given box state) by appropriate c-number amplitudes. The set of nonlinear equations obtained in this way is solved numerically. We show that on the time scale of a few miliseconds the system exhibits relaxation -reaches an equilibrium with populations of different eigenstates fluctuating around their mean values.

Journal of Physics: Conference Series, 2014

We introduce a phenomenological mean-field model to describe the growth of immiscible two-species atomic Bose-Einstein condensates towards some equilibrium. Our model is based on the coupled Gross-Pitaevskii equations with the addition of dissipative terms to account for growth. While our model may be applied generally, we take a recent Rb-Cs experiment [McCarron et al., Phys. Rev. A 84 011603(R) (2011)] as a case study. As the condensates grow, they can pass through ranging transient density structures which can be distinct from the equilibrium states, although such a model always predicts the predominance of one condensate species over longer evolution times.

2014

We study the properties of Bose-Einstein Condensates in a harmonic trap. We consider a mean-field description in terms of the Gross-Pitaevskii equation. We study properties of its solutions, especially the ground state of the system, paying a special attention to the effects of interaction. To obtain numerically the exact stationary states we use the Crank-Nicholson Scheme.

arXiv (Cornell University), 2022

The piling up of a macroscopic fraction of noninteracting bosons in the lowest energy state of a system at very low temperatures is known as Bose-Einstein condensation. It took nearly 70 years to observe the condensate after their theoretical prediction. A brief history of the relevant developments, essentials of the basic theory, physics of the steps involved in producing the condensate in a gas of alkali atoms together with the pertinent theory, and some important features of the research work carried out in the last about 25 years have been dealt with. An effort has been made to present the material in a manner that it can be easily followed by undergraduate students as well as non-specialists and may even be used for classroom teaching.

Physical Review A, 2010

International Journal of Modern Physics B, 2012

Beyond the mean-field theory, a new model of the Gross-Pitaevskii equation (GPE) that describes the dynamics of Bose-Einstein condensates (BECs) is derived using an appropriate phase-imprint on the old wavefunction. This modified version of the GPE in addition to the two-body interactions term, also takes into account effects of the threebody interactions. The three-body interactions consist of a quintic term and the delayed nonlinear response of the condensate system term. Then, the modulational instability (MI) of the new GPE confined in an attractive harmonic potential is investigated. The analytical study shows that the three-body interactions destabilize more the condensate system while the external potential alleviates the instability. Numerical results confirm the theoretical predictions. Further numerical investigations of the behavior of solitons reveal that the three-body interactions enhance the appearance of solitons, increase the number of solitons generated and deeply change the lifetime of solitons. Moreover, the external potential delays the appearance of solitons. Besides, a new initial condition is introduced which enables to increase the number of solitons created and deeply affects the trail of chains of solitons generated. Moreover, the MI of a condensate without the external potential, and in a repulsive potential is also investigated.

Journal of Physics B: Atomic, Molecular and Optical Physics, 2008

We discuss the dynamics of an open two-mode Bose-Hubbard system subject to phase noise and particle dissipation. Starting from the full many-body dynamics described by a master equation the mean-field limit is derived resulting in an effective non-hermitian (discrete) Gross-Pitaevskii equation which has been introduced only phenomenologically up to now. The familiar mean-field phase space structure is substantially altered by the dissipation. Especially the character of the fixed points shows an abrupt transition from elliptic or hyperbolic to attractiv or repulsive, respectively. This reflects the metastable behaviour of the corresponding many-body system which surprisingly also leads to a significant increase of the purity of the condensate. A comparison of the mean-field approximation to simulations of the full master equation using the Monte Carlo wave function method shows an excellent agreement for wide parameter ranges.

Journal of Optics B-quantum and Semiclassical Optics, 2004

In this tutorial we present an introduction to some theoretical methods of quantum field theory applied to the description of a trapped Bose-Einstein condensate. First of all, we give a brief account of the main characteristics of the phenomenon of condensation and present the many-body Hamiltonian of the system. We outline some of the most important approaches used in the characterization of a condensed Bose gas, including the mean-field theory and the Hartree-Fock-Bogoliubov method. Finally we illustrate the use of these techniques addressing some important issues in quantum atom optics. We characterize the quantum state of a Bose-Einstein condensate (BEC) at zero temperature. We also describe a process of Beliaev coupling between quasiparticles using a method that includes terms beyond the usual Bogoliubov approach.

For the dynamics of Bose-Einstein condensates (BECs), differences between mean-field (Gross-Pitaevskii) physics and N-particle quantum physics often disappear if the BEC becomes larger and larger. In particular, the time scale for which both dynamics agree should thus become larger if the particle number increases. For BECs in a double-well potential, we find both examples for which this is the case and examples for which differences remain even for huge BECs on experimentally realistic short time scales. By using a combination of numerical and analytical methods, we show that the differences remain visible on the level of expectation values even beyond the largest possible numbers realized experimentally for BECs with ultracold atoms.

Journal of Physics B: Atomic, Molecular and Optical Physics, 2000

We present a topical review of the development of finite-temperature field theories of Bose-Einstein condensation in weakly interacting atomic gases. We highlight the difficulties in obtaining a consistent finite-temperature theory that has a gapless excitation spectrum in accordance with Goldstone's theorem and which is free from both ultraviolet and infrared divergences. We present results from the two consistent theories developed so far. These are the Hartree-Fock-Bogoliubov theory within the Popov approximation and a many-body T-matrix approach which we have termed gapless-Hartree-Fock-Bogoliubov (GHFB). Comparison with the available experimental results is made and the remaining difficulties are highlighted.

Physical Review A, 2005

A dynamical many-body theory is presented which systematically extends beyond mean-field and perturbative quantum-field theoretical procedures. It allows us to study the dynamics of strongly interacting quantumdegenerate atomic gases. The non-perturbative approximation scheme is based on a systematic expansion of the two-particle irreducible effective action in powers of the inverse number of field components. This yields dynamic equations which contain direct scattering, memory and "off-shell" effects that are not captured by the Gross-Pitaevskii equation. This is relevant to account for the dynamics of, e.g., strongly interacting quantum gases atoms near a scattering resonance, or of one-dimensional Bose gases in the Tonks-Girardeau regime. We apply the theory to a homogeneous ultracold Bose gas in one spatial dimension. Considering the time evolution of an initial state far from equilibrium we show that it quickly evolves to a non-equilibrium quasistationary state and discuss the possibility to attribute an effective temperature to it. The approach to thermal equilibrium is found to be extremely slow.

JETP Letters, 2015

We discuss a possible origin of the experimentally observed nonlinear contribution to the shift ∆Tc = Tc − T 0 c of the critical temperature Tc in an atomic Bose-Einstein condensate (BEC) with respect to the critical temperature T 0 c of an ideal gas. We found that accounting for a nonlinear (quadratic) Zeeman effect (with applied magnetic field closely matching a Feshbach resonance field B0) in the mean-field approximation results in a rather significant renormalization of the field-free nonlinear contribution b2, namely ∆Tc/T 0 c ≃ b * 2 (a/λT ) 2 (where a is the s-wave scattering length, λT is the thermal wavelength at T 0 c ) with b * 2 = γ 2 b2 and γ = γ(B0). In particular, we predict b * 2 ≃ 42.3 for the B0 ≃ 403G resonance observed in the 39 K BEC. PACS: 67.85.Hj, 67.85.Jk Studies of Bose-Einstein condensates (BECs) continue to be an important subject in modern physics (see, e.g., Refs.[1, 2, 3, 4] and further references therein). Atomic BECs are produced in the laboratory in lasercooled, magnetically-trapped ultra-cold bosonic clouds of different atomic species (including 87 Rb [5, 9], 7 Li [6], 23 N a [7], 1 H [8], 4 He [10], 41 K [11], 133 Cs [12], 174 Y b [13] and 52 Cr [14], among others). Also, a discussion of a relativistic BEC has appeared in Ref.[15] and BECs of photons are most recently under investigation [16]. In addition, BECs are successfully utilized in cosmology and astrophysics [17] as they have been shown to constrain quantum gravity models [18].

Physical Review A, 2001

This paper examines the parameter regimes in which coupled atomic and molecular Bose-Einstein condensates do not obey the Gross-Pitaevskii equation. Stochastic field equations for coupled atomic and molecular condensates are derived using the functional positive-P representation. These equations describe the full quantum state of the coupled condensates and include the commonly used Gross-Pitaevskii equation as the noiseless limit. The model includes all interactions between the particles, background gas losses, twobody losses and the numerical simulations are performed in three dimensions. It is found that it is possible to differentiate the quantum and semiclassical behaviour when the particle density is sufficiently low and the coupling is sufficiently strong.

Universitat Politècnica de Catalunya, 2019

The Gross-Pitaevskii equation (GPE) is a mean field approximation used to study Bose-Einstein condensates (BEC). Here, we present a brief derivation of the GPE without the use of second quantization, as well as some of its consequences. Next, we present the implementation in Python of some numerical methods to compute the ground state of BEC. We compare the results with well known theoretical limits and the numerical results obtained using GPELab, a MATLAB toolbox to compute stationary and dynamic solutions of the GPE.

Bose-Einstein condensation is a very general physical phenomenon which takes place not only in the systems of bosonic atoms, but also in optical wave systems. Many important features of the condensation in such diverse systems can be captured by the nonlinear Schroedinger model. Within this model we develop a statistical description in which the condensate is nonlinearly coupled to wave turbulence described by a kinetic equation. Our focus will be on the strong-condensate regime in which the three-wave interaction replaces the four-wave process operating on the preceding stages of an explosive condensate formation and its initial growth. In the strong-condensate regime, the condensate growth accelerates and becomes quadratic in time. This regime will proceed until the wave dispersion drops below a critical value and the state of dispersionless acoustic turbulence forms.