An atomic Bose-Einstein condensate (BEC) is often described as a macroscopic object which can be approximated by a coherent state. This, on the surface, would appear to indicate that its behavior should be close to being classical. In this paper, we clarify the extent of how "classical" a BEC is by exploring the semiclassical equations for BECs under the mean field Gaussian approximation. Such equations describe the dynamics of a condensate in the classical limit in terms of the variables < x > and < p > as well as their respective variances. We compare the semiclassical solution with the full quantum solution based on the Gross-Pitaevskii Equation (GPE) and find that the interatomic interactions which generate nonlinearity make the system less "classical." On the other hand, many qualitative features are captured by the semiclassical equations, and the equations to be solved are far less computationally intensive than solving the GPE which make them ideal for providing quick diagnostics, and for obtaining new intuitive insight.
An atomic Bose-Einstein condensate (BEC) is often described as a macroscopic object which can be approximated by a coherent state. This, on the surface, would appear to indicate that its behavior should be close to being classical. In this paper, we clarify the extent of how "classical" a BEC is by exploring the semiclassical equations for BECs under the mean field Gaussian approximation. Such equations describe the dynamics of a condensate in the classical limit in terms of the variables < x > and < p > as well as their respective variances. We compare the semiclassical solution with the full quantum solution based on the Gross-Pitaevskii Equation (GPE) and find that the interatomic interactions which generate nonlinearity make the system less "classical." On the other hand, many qualitative features are captured by the semiclassical equations, and the equations to be solved are far less computationally intensive than solving the GPE which make them ideal for providing quick diagnostics, and for obtaining new intuitive insight.
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.
We present an approach for the analysis of Bose-Einstein condensates in a few mode approximation. This method has already been used to successfully analyze the vibrational modes in various molecular systems and offers a perspective on the dynamics in many particle bosonic systems. We discuss a system consisting of a Bose-Einstein condensate in a triple well potential. Such systems correspond to classical Hamiltonian systems with three degrees of freedom. The semiclassical approach allows a simple visualization of the eigenstates of the quantum system referring to the underlying classical dynamics. From this classification we can read off the dynamical properties of the eigenstates such as particle exchange between the wells and entanglement without further calculations. In addition, this approach offers insights into the validity of the mean-field description of the many particle system by the Gross-Pitaevskii equation, since we make use of exactly this correspondence in our semiclassical analysis. We choose a three mode system in order to visualize it easily and, moreover, to have a sufficiently interesting structure, although the method can also be extended to higher dimensional systems.
Bose-Einstein condensate (BEC) is a new state of matter, experimentally realized in a dilute alkali gas of weakly interacting bosons. The gas is cooled to very low temperatures, of the order of nano-Kelvin, at which point a large fraction of the atoms occupy the lowest quantum state. Under such conditions, the quantum effects become apparent on a macroscopic scale. Many fundamental ideas
of quantum mechanics have been realized and proven to be true in experiments,
by studying the Bose-Einstein condensates. Amongst them, the observation of
macroscopic matter wave interference, the creation of vortices, dark and bright
solitons etc, find special mention. The dynamics of this system in presence of an
optical lattice reveals a superfluid to Mott insulator quantum phase transition.
This has opened a new avenue for research on BEC and its applications. From
a theoretical standpoint, the dynamics of BEC can be described by means of an
effective mean-field theory. This formalism is much simpler than the many-body
Schr¨odinger equation and based on a classical nonlinear evolution equation, known
as Gross-Pitaevskii equation. The nonlinear nature of the GPE leads to several interesting phenomena like soliton propagation, vortices, four wave mixing etc.,
which are closely related to nonlinear fiber optics.
The present thesis deals with nonlinear excitations and different phases of BEC
and is organized as follows: after a brief introduction, in chapter two, we start
with BEC in a one-dimensional shallow optical lattice, experiencing both cubic
and quintic nonlinearity. A superfluid density wave, of twice the periodicity of
lattice potential, is identified. This indicates matter redistribution, where atoms
from neighbouring sites have been depleted leading to a density wave behaviour.
Interestingly, it ceases to exist when only one of these interactions is operative.
We predict the loss of superfluidity through a classical dynamical phase transition,
where modulational instability leads to the loss of phase coherence. In presence
of pure two-body interaction, the superfluid density shows a periodicity commensurate
with the lattice potential. Apart from the superfluid state, a density wave
insulating phase is also found to exist, possessing two frequency modulations commensurate
with the lattice potential. A dynamical phase transition connects the
superfluid phase with an insulating phase, which in turn, is connected with the
density wave insulating phase. The solutions are found to be marginally stable, as
per the Vakhitov-Kolokolov (VK) criterion.
Chapter three describes the dynamics of the atomic-molecular Bose-Einstein
condensate (AMBEC) by mean-field Gross-Pitaevskii equation (GPE), taking into
account the atom-molecular interaction, as well as atom-molecule inter-conversion.
The two-body atom-atom, atom-molecule and molecule-molecule interactions are
analogous to the cubic nonlinearity of the standard GPE, whereas, the quadratic
nonlinear term arising from the inter-conversion between atoms and molecules is
identical to the scenario in strongly coupled BEC in one dimension. The presence
of both weak and strong interactions adds further richness to this system. We
find the molecular BEC possesses a complex dark (grey) soliton, while the atomic condensate is a bright one. The energy and momentum of this system is computed
in order to see the effect of the both weak and strong interactions on Lieb mode
profile and compare it with the pure atomic condensate profile. Another class
of solutions exist in this system, where both atomic and molecular condensates
possess bright solitons.
In chapter four, we investigate the effect of a harmonic trap on the dynamics
of grey solitons in Bose-Einstein condensates. It is observed that, these nonlinear
excitations of the cigar-shaped Bose-Einstein condensate have strong coupling
with the trap at low momenta and hence can be effectively isolated from the
Bogoliubov sound modes, which respond weakly to harmonic confinement. This
strong coupling with the trap also makes the grey soliton amenable for control and
manipulation through trap modulation and temporal variation of the two-body interaction.
We have shown that in presence of an expulsive oscillator trap, the grey
soliton gets amplified, while having a compression in its width. Exact solutions are
also obtained in case of the two component Bose-Einstein condensates (TBECs),
where one component is a grey soliton and the other a bright one. The presence
of the trap and inter-species interaction significantly affects the grey soliton dynamics.
Similar to the single component, in the present case, we have shown that
both grey and bright soliton can be compressed simultaneously. A detailed study
of the energy and momentum of the soliton is reported, in order to find the effect
of the trap on the Lieb mode.
In chapter five, we study a BEC-defect complex in a quasi-one dimensional
scenario, where the defect atom moves in the condensate with constant velocity.
Explicit solutions of the coupled mean-field equations, describing defect-grey soliton
dynamics are obtained, demonstrating the coexistence of grey soliton and a
localized defect. Unlike the case of dark soliton, where the defect trapping center
has vanishing superfluid density, the moving grey soliton necessarily possesses a finite superfluid component at the defect location. The presence of the defect can
lower the grey soliton’s maximum velocity, as compared to the defect free case,
where the maximum velocity is the sound velocity. We have also carried out the
stability analysis of the obtained solutions using renormalized momentum formalism
and Vakhitov-Kolokolov criteria and find that the obtained solutions are stable
through both the methods. The grey soliton’s energy gets substantially modified
through its interaction with the defect, opening up the possibility of its control
through defect dynamics.
Chapter six describes a parametrically forced Bose-Einstein condensate in the
presence of a general moving optical lattice. The interaction between the atoms in
the condensate and the time dependent lattice potential leads to a novel propagating
superfluid matter wave, which can be controlled through chirp management.
This system, when placed in a trap, accelerates and undergoes rapid nonlinear
compression, controlled by the chirp. The density achieves its maximum, when
the matter wave changes direction. It is also shown that this system exhibits dynamical
superfluid insulator transition, where superfluidity breaks down and the
condensate transits to an insulating phase. The stability has been investigated
using Vakhitov-Kolokolov criterion and found that it depends on the chirp, as well
as on the nature of the atom-atom interaction. The exact expression for energy
is obtained and analyzed in detail to gain physical understanding of the chirp
management of the sinusoidal excitations and also the dynamical phase transition.
Since the first experimental observation of the Bose-Einstein condensation of a gas of alkali atoms, six years ago, there has been an increasing interest on this new, highly interdisciplinary, branch of physics. It was realized, from the very beginning that, despite of the diluteness of such systems, the nonlinearity was crucial on the understanding of both the equilibrium and dynamical properties. In this Thesis we point out that there is a further key aspect that must be taken into account when the condensates are trapped in deep optical lattices: discreteness. The interplay between discreteness and nonlinearity raises a new class of deeply non-trivial phenomena which disappear in the continuum limit of the Gross-Pitaevskii equation (GPE). Recognizing and exploiting such interplay is the main topic of this Thesis. We think that the dynamics of a Bose-Einstein condensate (BEC) in optical lattices is going to become of central interest in the next future. Optical potentials provide BEC's waveguides, which prevent the spatial spreading of the BEC wave packets and preserve their coherence over long distances. This has allowed to build high sensitive interferometers, including gyroscopes and gravitometers. From the theoretical point of view, the big challenge is to reconsider the meaning of phenomena which have been well understood in homogeneous systems: superfluidity, for instance, is in the top list. A further example is given by quantum phase transitions. The connection and the interplay with the language and insights of the nonlinear physics community is a second challenge. We believe that the results obtained in the BEC context can provide a new prospective view into old problems. The understanding of the modulational instability in terms of phase randomization and a (classical) dynamical phase transition are among this line, as well as the possibility to generalize the Landau criteria for superfluidity to discrete systems. On the other hand, it is clear that nonlinear physicists can have much to say about BEC. With this Thesis we hope we have provided, together with original results, some links between these different languages and views.
Summary form only given. With the realisation of Bose-Einstein condensation in dilute gases, it has been possible to study the dynamics of quantum fields at finite temperatures. The Gross-Pitaevskii equation (GPE) has proved to be a very useful tool for describing Bose-Einstein condensates (BECs) and has shown very good agreement with experimental results at low temperatures. In the present paper we apply the mean-field approximation to describe the quantum state of a Bose-Einstein condensate, using a phase-space representation to visualise such a state.
Bose–Einstein Condensation (BEC) of atomic gases has attracted a renewed theoretical and experimental interest in quantum many-body systems at extremely low temperatures (Pethick & Smith; 2002). This excitement stems from two favorable features: (1) by applying magnetic fields and lasers, most of the system parameters, such as the shape, dimensionality, internal states of the condensates, and even the strength of the interatomic interactions, are controllable; (2) due to the diluteness, the mean-field theory explains experiments quite well. In particular, the Gross–Pitaevskii (GP) equation demonstrates its validity as a basic equation for the condensate dynamics. The GP equation is the counterpart of the nonlinear Schrodinger (NLS) equation in nonlinear optics. Thus, a study based on nonlinear analysis is possible and important. In nonlinear physics, a soliton is remarkable object not only for the fact that exact solutions can be obtained but also for its usefulness as a communications tool due to its robustness. In general, solitons are formed under the balance between nonlinearity and dispersion. For atomic condensates, the former is attributed to the interatomic interactions, while the latter comes from the kinetic energy. Either dark or bright solitons are allowable depending on the positive or negative sign of the interatomic coupling constants g, respectively, and indeed have been observed in a quasione-dimensional (q1D) optically constructed waveguide (Strecker et al.; 2002) (Khaykovich et al.; 2002). Such matter-wave solitons are expected in atom optics for applications in atom laser, atom interferometry, and coherent atom transport (Meystre; 2002). In this chapter, we extend the analysis of the matter-wave solitons to a multicomponent case by considering the so-called spinor condensate (Stenger et al.; 1998) whose spin degrees of freedom are liberated under optical traps. Based on theoretical and experimental results, we introduce a new integrable model which describes the dynamical properties of the matter-wave soliton of spinor condensates (Ieda et al.; 2004a). We employ the inverse scattering method to solve this model exactly. As a result, we predict the occurrence of undiscovered physical phenomena such as macroscopic spin precession and spin switching. The chapter is organized as follows. In Sec. 2, the mean field theory of condensate is briefly reviewed. Section 3 introduces an effective interatomic coupling in a q1D condensate. Using these results, we consider a spinor condensate in q1D regime in Sec. 4. Then, in Sec. 5, we