This work considers the semi-classical description of two applications involving cold atoms. This is, on one hand, the behavior of a BOSE-EINSTEIN condensate in hybrid systems, i.e. in contact with a microscopic object (carbon nanotubes, fullerenes, etc.). On the other, the evolution of phase space distributions in matter wave interferometers utilizing ray tracing methods was discussed. For describing condensates in hybrid systems, one can map the GROSS-PITAEVSKII equation, a differential equation in the complex-valued macroscopic wave function, onto a system of two differential equations in density and phase. Neglecting quantum dispersion, one obtains a semiclassical description which is easily modified to incorporate interactions between condensate and microscopical object. In our model, these interactions comprise attractive forces (CASIMIR-POLDER forces) and loss of condensed atoms due to inelastic collisions at the surface of the object. Our model exhibited the excitation of sound waves that are triggered by the object’s rapid immersion, and spread across the condensate thereafter. Moreover, local particle loss leads to a shrinking of the bulk condensate. We showed that the total number of condensed particles is decreasing potentially in the beginning (large condensate, strong mean field interaction), while it decays exponentially in the long-time limit (small condensate, mean field inetraction negligible). For representing the physics of matter wave interferometers in phase space, we utilized the WIGNER function. In semi-classical approximation, which again consists in ignoring the quantum dispersion, this representation is subject to the same equation of motion as classical phase space distributions, i.e. the LIOUVILLE equation. This implies that time evolution of theWIGNER function follows a phase space flow that consists of classical trajectories (classical transport). This means, for calculating a time-evolved distribution, one has know the initial distribution and one has to solve the classical equations of motion. Concerning the initial distribution, we have studied a stationary solution of the nonlinear LIOUVILLE equation, the LAMBERT density. We saw that it agrees very well with results from singleparticle quantum mechanics as well as the MAXWELL-BOLTZMANN distribution in the weakly interacting limit. Likewise, in the strongly interacting limit, familiar results of the THOMAS-FERMI approximation are recovered. A distribution that is first prepared in a trap and then released can be described quite conveniently in terms of WIGNER functions. However, propagation in optical potentials associated to the interferometer elements (beam splitter, Pi-half-pulse) do not satisfy the condition of the semiclassical approximation. Nevertheless, one finds discrete before-after mappings that describe the effect of these elements on incident distributions. This leads to several channels of phase space propagation which relate to the interferometer paths and interferences between them. The formalism for WIGNER functions in an interferometer can be translated straightforwardly into a ray tracing algorithm. As mentioned above, this algorithm solves the classical equations of motion and computes time-evolved distributions, using values of the initial distribution. This procedure, in contrast to most analytical solutions, does not require the HAMILTON function to be quadratic (e.g. free propagation, const. acceleration, harmonic oscillator). We compared simulation results to analytic expressions in case of freely propagating GAUSS distribution. They showed perfect agreement, especially for the functional dependence of wave length and contrast of interference fringes on the laser pulse timings.
This work considers the semi-classical description of two applications involving cold atoms. This is, on one hand, the behavior of a BOSE-EINSTEIN condensate in hybrid systems, i.e. in contact with a microscopic object (carbon nanotubes, fullerenes, etc.). On the other, the evolution of phase space distributions in matter wave interferometers utilizing ray tracing methods was discussed. For describing condensates in hybrid systems, one can map the GROSS-PITAEVSKII equation, a differential equation in the complex-valued macroscopic wave function, onto a system of two differential equations in density and phase. Neglecting quantum dispersion, one obtains a semiclassical description which is easily modified to incorporate interactions between condensate and microscopical object. In our model, these interactions comprise attractive forces (CASIMIR-POLDER forces) and loss of condensed atoms due to inelastic collisions at the surface of the object. Our model exhibited the excitation of sound waves that are triggered by the object’s rapid immersion, and spread across the condensate thereafter. Moreover, local particle loss leads to a shrinking of the bulk condensate. We showed that the total number of condensed particles is decreasing potentially in the beginning (large condensate, strong mean field interaction), while it decays exponentially in the long-time limit (small condensate, mean field inetraction negligible). For representing the physics of matter wave interferometers in phase space, we utilized the WIGNER function. In semi-classical approximation, which again consists in ignoring the quantum dispersion, this representation is subject to the same equation of motion as classical phase space distributions, i.e. the LIOUVILLE equation. This implies that time evolution of theWIGNER function follows a phase space flow that consists of classical trajectories (classical transport). This means, for calculating a time-evolved distribution, one has know the initial distribution and one has to solve the classical equations of motion. Concerning the initial distribution, we have studied a stationary solution of the nonlinear LIOUVILLE equation, the LAMBERT density. We saw that it agrees very well with results from singleparticle quantum mechanics as well as the MAXWELL-BOLTZMANN distribution in the weakly interacting limit. Likewise, in the strongly interacting limit, familiar results of the THOMAS-FERMI approximation are recovered. A distribution that is first prepared in a trap and then released can be described quite conveniently in terms of WIGNER functions. However, propagation in optical potentials associated to the interferometer elements (beam splitter, Pi-half-pulse) do not satisfy the condition of the semiclassical approximation. Nevertheless, one finds discrete before-after mappings that describe the effect of these elements on incident distributions. This leads to several channels of phase space propagation which relate to the interferometer paths and interferences between them. The formalism for WIGNER functions in an interferometer can be translated straightforwardly into a ray tracing algorithm. As mentioned above, this algorithm solves the classical equations of motion and computes time-evolved distributions, using values of the initial distribution. This procedure, in contrast to most analytical solutions, does not require the HAMILTON function to be quadratic (e.g. free propagation, const. acceleration, harmonic oscillator). We compared simulation results to analytic expressions in case of freely propagating GAUSS distribution. They showed perfect agreement, especially for the functional dependence of wave length and contrast of interference fringes on the laser pulse timings.
After the first experimental realization of a Bose-Einstein condensate (BEC) in 1995, BECs have become a subject of intense experimental and theoretical study. In this dissertation, I present our results on the classical and quantum dynamics of BECs at zero temperature under different scenarios. First, I consider the analog of slow light in the collision of two BECs near a Feshbach resonance. The scattering length then becomes a function of the collision energy. I derive a generalization of the Gross-Pitaevskii equation for incorporating this energy dependence. In certain parameter regimes, the group velocity of a BEC traveling through another BEC decreases. I also study the feasibility of an experimental realization of this phenomena. Second, I analyze an experiment in which a BEC in a ring-shaped trap is stirred by a rotating barrier. The phase drop across and current flow through the barrier is measured from spiral-shaped density profiles created by interfering the BEC in the ring-shaped trap and a concentric reference BEC after release from all trapping potentials. I show that a free-particle expansion is sufficient to explain the origin of the spiral pattern and relate the phase drop to the geometry of a spiral. I also bound the expansion times for which the phase drop can be accurately determined and study the effect of inter-atomic interactions on the expansion time scales. Third, I study the dynamics of few-mode BECs when they become dynamically unstable after preparing an initial state at a saddle point of the Hamiltonian. I study the dynamics within the truncated Wigner approximation (TWA) and find that, due to phase-space mixing, the expectation value of an observable relaxes to a steady-state value. Using the action-angle formalism, we derive analytical expressions for the steady-state value and the time evolution towards this value. I apply these general results to two systems: a condensate in a double-well potential and a spin-1 (spinor) condensate. Finally, I study quantum corrections beyond the TWA in the semiclassical limit. I derive general expressions for the dynamics of an observable by using the van Vleck-Gutzwiller propagator and find that the interference of classical paths leads to non-perturbative corrections. As a case study, I consider a single-mode nonlinear oscillator; this system displays collapse and revival of observables. I find that the interference of classical paths, which is absent in the TWA, leads to revivals.
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.
A major goal of the semiclassical theory is to understand and treat a quantum system in the regime of large quantum numbers using information from its classical limit. In this framework, quantum phenomena are addressed through a path integral perspective, i.e. the weighted interference of phases accumulated along classical trajectories. While this formalism has been successful in explaining interference phenomena, for instance, in mesoscopic single-particle quantum systems, the recent extension of the semiclassical theory allows now also the treatment of interacting quantum many-body systems. This thesis aims to contribute to these ongoing efforts to apply semiclassical techniques to quantum many-body systems. It is focused on bosonic quantum many-body systems, where the thermodynamic limit of a large number of particles represents an alternative version of a semiclassical limit. Within the derivation of the corresponding semiclassical theory, mean-field equations, i.e. effective nonlinear wave equations describing the matter wave, re-emerge as Hamilton's equations of motion of an abstract Hamilton function, defining the above thermodynamic limit as the classical limit. Within this thesis, two different topics are studied. The first one deals with the so-called “Out-of-Time-Order Correlator” (OTOC), the expectation value of the squared commutator of two local operators at different times. The OTOC provides a direct probe for the presence of chaos in the classical limit (if any) of a quantum many-body system. This is based on that for short times the OTOC directly relates to the stability of classical solutions upon changes in their initial conditions. One of the defining properties of chaos is that this stability displays an exponential growth, with the rate called the classical Lyapunov exponent. Immediately, this implies an exponential growth of the OTOC for short times, from which the Lyapunov exponent can be extracted. For later times, one observes a saturation of the OTOC as a consequence of unitary quantum time evolution. The Ehrenfest time, the time scale which marks the onset of quantum interference, separates these different dynamical behaviors. In this thesis a thorough understanding of the concept of chaos in the classical limit of quantum many-body systems is provided, and with that the underlying interference mechanisms involved in the early exponential growth and the later saturation of OTOCs are identified. It is found that the pre-Ehrenfest time exponential growth is given through the interference of multiple contributions, all of them essentially following only a single solution of mean-field equations. Conversely the post-Ehrenfest time behavior stems from the contributions of fundamentally different mean-field solutions, which display correlated dynamics only for a limited amount of time, of the order of Ehrenfest time. The second topic covered in this thesis deals with the coherent transport of cold bosonic atoms through an Aharonov-Bohm ring structure. This setup consists of two semi-infinite wave guides attached on the opposite sides of a wave guide ring which is pierced by a synthetic gauge field. Within the ring, the atoms are further subject to both a weak disorder potential and particle-particle interaction. In the non-interacting case, the disorder-averaged transmission probability as a function of the encircled flux displays the well-known Al'tshuler-Aronov-Spivak (AAS) oscillations, i.e. oscillations with a frequency twice as large as Aharonov-Bohm oscillations. This thesis provides further insight into the influence of a weak particle-particle interaction on AAS oscillations, for which numerical results predict an interaction-based inversion of peaks. Here, using semiclassical techniques to solve the corresponding mean-field problem, this inversion is successfully reproduced, and a first indicator of the relevant mechanism leading to the inversion is found through contributions of self-averaging scattering path constellations.
We present a generalized model for the evolution of atomic wave-packets in matter-wave interferometers. This method provides an efficient tool for analyzing the performance of atomic interferometers using atom clouds prepared in a trap as a Bose-Einstein condensate (BEC) or as a thermal ensemble. Predictions of the model for dynamic properties such as wave-packet size and phase are in excellent agreement with explicit numerical solutions of the non-linear Gross-Pitaevskii equations and enable fast calculations of interferometric performance in regimes where full numerical solutions become impractical. As a starting point, the static Thomas-Fermi (TF) approximation for a BEC in a harmonic trap is generalized to the whole range of atom-atom interaction strengths: from non-interacting atoms (low densities) to the standard TF limit (high atomic densities, as long as the condensate approximation still holds). In particular, this generalization allows a good estimation of atomic cloud properties along the transition from a three-dimensional to a quasi-one-dimensional BEC in an elongated trap. We then develop a theoretical model of wave-packet evolution in time-dependent conditions. The model is applicable for a wide range of dynamical problems involving evolution in time-dependent potentials and in a changing mean-field atomic repulsion due to splitting and separation of wave-packets. We use the model for studying two effects that influence interferometric coherence: imperfect spatial recombination in a two-state interferometer (the so-called "Humpty-Dumpty effect") and phase diffusion due to number uncertainty in the two interferometer arms, which was previously studied thoroughly only for interferometric schemes where the BECs in the two arms stay trapped (for example, in a double-well potential).
We present a general method for calculating wave-packet evolution in the context of ultracold atoms in matter-wave interferometers. This method provides an efficient tool for analyzing the performance of atomic interferometers based on atom clouds initially prepared in a trap in a Bose-Einstein condensate (BEC) state or in a thermal distribution. Without having to solve explicitely the Gross-Pitaevskii or the Schrodinger equation, this provides a good estimation of dynamic wave-packet properties such as size and phase across the wave-packet, and slows to predict the coherence of the interferometer. We develop a generalized Thomas-Fermi approximation for a BEC in a harmonic trap, which is valid for any atom-atom interaction strength. In particular, this approximation also provides a good estimation of the transition from a three-dimensional to a one-dimensional BEC in an elongated trap. The approximation is followed by a theory of dynamic wave-packet evolution when the trap is fully or partially switched off and the atoms are split and propagate, for example, in free space or in a matter waveguide. The method is applied for studying two examples of interferometric effects: reduction of coherence of two-state interferometers due to imperfect spatial recombination of the two interferometer arms (the Humpty-Dumpty effect) and phase diffusion due to number uncertainty in the two interferometer arms, which was previously studied thoroughly only for interferometric schemes where the BECs in the two arms stay trapped (for example, in a double-well potential). For both effects we extend the applicability of the theory to a wide range of interferometric scenarios that were not included in previous theories.
The central theme of this thesis is that of coherence in bosonic matter-wave fields. Analogous to optical coherence; the measure of the capacity of a photonic beam to produce fringes upon interference; matter-wave coherence is in many ways the defining characteristic of the BEC phase of identical massive bosons. Its study therefore plays a critical role in the quest to better understand the physics of ultra-cold quantum gases, which has been the goal of this research. This work approaches coherence from a number of different angles and is for this reason presented in two parts. The first of these delves into the physics of perfectly coherent matterwave fields which are adequately described by ‘mean field’ equations, while the second is devoted to origins and repercussions of the diminished coherence evident in actual atom laser experiments. The presentation of the original work of this project begins with a detailed study of the interaction between an atomic and a molecular condensate, coherently coupled at zero temperature via one of several possible mechanisms (a Feshbach resonance, for example) in three spatial dimensions. Despite the presence of repulsive scattering interaction between the atoms, the mean-field equations of motion are known to support bright solitary waves — time-independent solutions which remain localised even in the absence of a trapping potential. (These solutions are the matter-wave analogue of spatio-temporal solitons in quadratic nonlinear media, but can also represent stationary states of the Schr¨odinger-Newton equations describing bosons interacting via a gravitational field.) Ignoring all interactions save the repulsive inter-atomic scattering mentioned, a variational ansatz approach is employed to analytically approximate these stationary solutions using Gaussian density distributions of the atomic and molecular clouds. By evolving these approximate solutions numerically, a comprehensive map of the accessible parameter space is constructed, showing primarily stable localised propagation. Parameter space regions where the approximate Gaussian solutions display strong ‘breathing’ oscillations are identified, and a modified ansatz having exponentially decaying tails is found to dramatically reduce the amplitude of these oscillations. Finally, by seeding a numerical relaxation algorithm with our Gaussian solutions, we find that the only parameter space region in which the approximate solutions diffuse under propagation is also devoid of exact localised stationary solutions. The discussion then proceeds beyond the scope of mean field theory, and turns to a detailed treatment of the quantum mechanical observable corresponding to the centre of mass of systems of identical particles. Firstly, careful consideration is given to its operational definition in the presence of fluctuating particle numbers. Possible pitfalls due to naive approximations when the number fluctuations are large are also addressed. Attention is then given to the fundamental quantum noise associated with this variable, which provides a physical mechanism for diminished first order coherence in degenerate Bose gases, even at zero temperature. Characteristic lower bounds (standard quantum limits) to the resulting mean position uncertainty are identified both for systems of identical bosons and identical fermions with comparable density profiles. Interestingly, fermions are found to have an intrinsically lower mean-position uncertainty than bosons, a fact which is attributed to the action of Pauli repulsion between atoms confined to a fixed density distribution. The results of numerical simulations of the exact quantum evolution of mean position variance are then presented. These demonstrate firstly the quantum diffusion of the mean position of one dimensional matter-wave solitons and secondly the evolution of variance in the centre of mass of a Bose gas undergoing forced evaporative cooling. Finally, attention is paid to the fact that the Penrose-Onsager criterion, which essentially equates Bose condensation with perfect first-order matter-wave coherence, fails to correctly detect the presence of BEC when a priori knowledge of the exact condensate mode is imperfect or unavailable. Insight is gained by drawing analogies between this criterion and entropic measures of entanglement between particular system subspaces. This motivates employment of the concept of entanglement of formation as a means to providing a more accurate measure of condensation in the general case of mixed system states. Unfortunately, this new measure remains adversely sensitive to superpositions of condensate modes. For this reason, alternative criteria based on higher order particle number correlation functions are also proposed.