In this dissertation we make use of the many analogies between quantum optical and ultra-cold atomic and molecular systems in order to study the coherence properties of the latter with methods of non-linear optics. We adapt the XFROG method that has first been developed for the characterization of ultra-short laser pulses, to the problem of reconstructing both amplitude and phase of the condensate wavefunction of a Bose-Einstein-condensate (BEC). Using the example of a vortex state we study the dependence of the reconstruction quality on the number of measurements and different sources of noise and we find that the method is feasible with available experimental technology. Exploiting the similarity between the coherent formation of ultra-cold molecules and optical sum frequency generation we devise a scheme for measuring second-order correlations of atoms through density measurements of molecules. We use perturbation theory in the cases of weak and strong coupling between atoms and molecules to calculate the momentum distribution of the molecules for the cases where the molecules are formed from a BEC, a normal Fermi gas and a Fermi gas with superfluidity in a Bardeen-Cooper-Schrieffer (BCS) state. These calculations are supplemented by exact integrations of Schroedinger's equation in the single mode approximation for the molecules. Atoms in a BEC are collectively transformed into molecules with a narrow momentum distribution reflecting the long coherence length of atoms in the BEC. For the normal Fermi gas molecules are formed non-collectively and their momentum distribution is much wider. The momentum distribution of molecules from a BCS state looks similar to the BEC case: The superfluid component leads to collectively formed molecules with a very narrow momentum distribution and the unpaired fraction gives rise to non-collectively formed molecules with a much wider momentum distribution similar to the normal Fermi gas case. The counting statistics of the molecules from a BEC is that of a coherent state, from a normal Fermi gas it is that of a thermal state and the BCS case interpolates between the two.
In this dissertation we make use of the many analogies between quantum optical and ultra-cold atomic and molecular systems in order to study the coherence properties of the latter with methods of non-linear optics. We adapt the XFROG method that has first been developed for the characterization of ultra-short laser pulses, to the problem of reconstructing both amplitude and phase of the condensate wavefunction of a Bose-Einstein-condensate (BEC). Using the example of a vortex state we study the dependence of the reconstruction quality on the number of measurements and different sources of noise and we find that the method is feasible with available experimental technology. Exploiting the similarity between the coherent formation of ultra-cold molecules and optical sum frequency generation we devise a scheme for measuring second-order correlations of atoms through density measurements of molecules. We use perturbation theory in the cases of weak and strong coupling between atoms and molecules to calculate the momentum distribution of the molecules for the cases where the molecules are formed from a BEC, a normal Fermi gas and a Fermi gas with superfluidity in a Bardeen-Cooper-Schrieffer (BCS) state. These calculations are supplemented by exact integrations of Schroedinger's equation in the single mode approximation for the molecules. Atoms in a BEC are collectively transformed into molecules with a narrow momentum distribution reflecting the long coherence length of atoms in the BEC. For the normal Fermi gas molecules are formed non-collectively and their momentum distribution is much wider. The momentum distribution of molecules from a BCS state looks similar to the BEC case: The superfluid component leads to collectively formed molecules with a very narrow momentum distribution and the unpaired fraction gives rise to non-collectively formed molecules with a much wider momentum distribution similar to the normal Fermi gas case. The counting statistics of the molecules from a BEC is that of a coherent state, from a normal Fermi gas it is that of a thermal state and the BCS case interpolates between the two.
Physics of ultra cold atoms and molecules is one of the most rapidly developing areas of the modern atomic, molecular, optical physics and quantum optics.This area covers several sub-areas of diverse importance, but definitely two trends can be easily identifies: i) fundamental research on properties of ultra cold atoms and molecules, and in particular atomic and molecular condensates trapped and lattice gases; here the fundamental questions of nature and form of phase transitions in degenerate systems are being posed and studied; ii) more applications oriented research on ultracold atoms and molecules, involving in particular applications in precision measurements, atomic clocks, frequency standards, atom and molecule interferometry, quantum information, quantum engineering, "super" chemistry etc.This thesis concerns both of these major trends.The applications, or more specifically quantum engineering aspect are discussed in the first four chapters of the thesis, which deal with optical manipulations of ultra cold gases, in particular optical generation of vortices in Bose condensates, quasi-solitons in Fermi gases, and retarded dipole-dipole interactions in Bose, or Fermi gases.This thesis consists of four main chapters.Chapter 2, "Optical generation of vortices in Bose Einstein Condensates".In this chapter we study theoretically the generation of vortices by phase imprinting in trapped Bose-Einstein condensates.Phase imprinting is achieved by passing a short off-resonance laser pulse through an appropriately designed absorption plate, and impinging it on a condensate.We answer the fundamental question of how the circulation and the vortex are introduced into the system, and discuss the creation of vortex arrays.We discuss vortex dynamics and interactions in such configurations.Various methods of vortex detection based on interference combined with Bragg scattering are also analyzed.In Chapter 3, "Fermionic 'Solitons'" we investigate in detail the concept of low energy excitations of ultra cold Fermi gases.This excitations are generated using the same idea of phase imprinting as that employed in the generation of vortices in BEC.We address in this chapter some questions concerning the generation (via phase imprinting) of soliton-like solutions in a single-component Fermi gas of neutral atoms at zero and finite temperatures.By using both numerical and analytical calculations, we find the conditions under which it is possible to find in non-interacting Fermi gases, quasisolitons which resemble the properties of solitons in non-linear integrable equations.We present the results for both spatially homogeneous and trapped cases, and emphasize the importance of Fermi statistics and the absence of interactions for the existence of such solutions.In Chapter 4, "Generating Dipole Condensate" we keep studying the possibilities of optical manipulations of ultracold gases.In particular, we study two different ways to laser-induce dipole-dipole interactions in trapped Bose-Einstein condensates.The first one employs the fact that those atoms in high Rydberg states posses a large dipolar moment.We first analyze the idea of imposing the dipole moment as a time average over short stroboscopic pulses.These pulses are coherently transporting the atoms from the ground atomic state to some high Rydberg state and back.The second idea is based on the observation that in strong static or oscillatory electric fields, atoms polarize and acquire dipole moment.We analyze the possibilities and limitations of inducing such a high electric field using lasers of low frequency.In Chapter 5, "Dipolar BCS", we continue our study of trapped fermionic gases, with the analysis of the BCS transition in trapped dipolar fermionic gases.We find that the critical strength of the dipole-dipole interaction to achieve BCS pairing, depends on the trap geometry.We find the particular form of this dependence, and present the solutions for the order parameter.This first chapter is meant as a general introduction to the thesis.This introduction is by no means exhaustive, and should be rather treated as a collection of basic facts concerning ultra cold gases, without rigorous derivations.The last subsection of the introductory chapter, discusses very briefly the most important achievements and trends in the field and the current state of the art.
Since the achievement of Bose-Einstein condensation (BEC), the progress in matterwave \nphysics has been immense. Among the many recent achievements there is \nthe miniaturization of atom traps, demonstration of the superfluid-Mott insulator \nquantum-phase transition in optical lattices and the experimental demonstration of \nthe BEC-BCS crossover in ultra-cold gases. \nMiniaturization of atom traps using micro-structured wires on a chip is one important \nstep towards an on-chip cold-atom device. These so-called \\atom chips" provide \nhigh control and versatility for trapping and guiding the ultra-cold atomic clouds. \nParticularly interesting is the use of these microchips to build mesoscopic devices \nfor cold atomic clouds as, for instance, in the case of an atom-cloud interferometer. \nHowever, these mesoscopic devices require coherent transport of the atom cloud. A \ngeneral method to treat decoherence due to current fluctuations in multi-wire atomchip \ntraps is presented in the rst part of this thesis. The decoherence rate Γ shows a \nstrong dependence on the distance between the wire and the atom cloud, r0, scaling \nas Γ ~ r-4 0 for a single atom waveguide. Considering an interferometer device, a \nstrong dependence of the decoherence rate on the trap geometry is found. \n \nStudying many-body eects in ultra-cold quantum gases is another important \nresearch eld. Experiments using ultra-cold quantum gases in optical lattices have \ndemonstrated the superuid-Mott insulator quantum phase transition and manybody \nentanglement. Optical lattices are based on a periodic modulation of the light \nintensity, generated by retro-reected laser beams. Correlations of the atomic cloud \nbetween dierent lattice sites of the optical lattice play a central role in these manybody \nexperiments. The dierent phases of the superuid-Mott insulator system can \nbe characterized by the dierent behavior of the inter-lattice site correlations. There \nare several numerical methods such as Quantum Monte Carlo (QMC) simulations, \nDensity Matrix Renormalization Group (DMRG) simulations, exact-diagonalization, \nor the Gutzwiller ansatz, to investigate the dynamics of an ultra-cold gas in an optical \nlattice theoretically. The Gutzwiller method, corresponding to the mean-eld solution, \nallows for the treatment of large lattice sizes. Mean-eld approaches have proven \nto be very useful to describe many-body physics. However, diculties arise in the correct \ndescription of the behavior of the decay of inter-lattice site correlations. Based on \nthe Gutzwiller approach, we have developed a method which allows the successive inclusion \nof inter-lattice site correlations. Comparing the results for the particle-number \nuctuations and the correlation function obtained from pure Gutzwiller calculations, \nto calculations which perturbatively include short-range correlations and calculations \n \nusing \\quasi-exact methods", showed a considerable improvement relative to the pure \nGutzwiller results due to the inclusion of short-range correlations. \nMany-body eects do not only arise in periodic potentials, but become increasingly \nimportant at ultra-low temperatures. The formation of Bose-Einstein condensates \nrequires an overlap of the atom wavefunctions and, hence, the formation of \na single condensate wavefunction. Another example of a many-body state is the \nsuperuid-BCS state, commonly used as a description of superconductivity. Here, \nfermions in dierent hyperne states form Cooper pairs. Experiments with ultracold \nquantum gases enable a variation of the interparticle interaction, e.g. , by using \na Feshbach resonance. Using Feshbach resonances to tune the interaction strength \nhas enabled the experimental observation of the crossover from a superuid-BCS \nstate to a Bose-Einstein condensate of molecules. A useful way to characterize the \ndierent states of ultra-cold quantum gases is to investigate the particle-number uctuations. \nIn this thesis we suggest to divide the atomic cloud into bins and consider \nthe atom-number uctuations in these bins. We calculate the full counting statistics \nfor dierent physical systems of ultra-cold gases (e.g. bosonic gases, fermionic gases, \nand spin mixtures). In particular, we consider the BCS-state as a rst trial example \nto show that there is a strong variation in the particle-number statistics at the \ncrossover from a superuid-BCS state to a Bose-Einstein condensate of molecules.
We calculate the number statistics of a single-mode molecular field excited by photo-association or via a Feshbach resonance from an atomic Bose-Einstein condensate (BEC), a normal atomic Fermi gas, and a Fermi system with pair correlations (BCS state). We find that the molecule formation from a BEC leads for short times to a coherent molecular state in the quantum optical sense. Atoms in a normal Fermi gas, on the other hand, result for short times in a molecular field analog of a classical chaotic light source. The BCS situation is intermediate between the two and goes from producing an incoherent to a coherent molecular field with an increasing gap parameter. This distinct signature of the initial atomic state in the resulting molecular field makes single molecule counting into a powerful diagnostic tool.
We calculate the momentum distribution and the second-order correlation function in momentum space, ${g}^{(2)}(\mathbf{p},{\mathbf{p}}^{\ensuremath{'}},t)$ for molecular dimers that are coherently formed from an ultracold atomic gas by photoassociation or a Feshbach resonance. We investigate using perturbation theory how the quantum statistics of the molecules depend on the initial state of the atoms by considering three different initial states: a Bose-Einstein condensate (BEC), a normal Fermi gas of ultracold atoms, and a BCS-type superfluid Fermi gas. The cases of strong and weak coupling to the molecular field are discussed. It is found that BEC and BCS states give rise to an essentially coherent molecular field with a momentum distribution determined by the zero-point motion in the confining potential. On the other hand, a normal Fermi gas and the unpaired atoms in the BCS state give rise to a molecular field with a broad momentum distribution and thermal number statistics. It is shown that the first-order correlations of the molecules can be used to measure second-order correlations of the initial atomic state.
In this habilitation-thesis a collection of 14 experiments is presented. In these experiments we have developed and used novel techniques to control ultracold gases on the quantum level. In brief, Bose-Einstein condensates (BEC) and degenerate Fermi gases of laser-cooled alkali atoms are subjected to specially designed optical and magnetic fields. The fields influence in a coherent way the behavior, properties and dynamics of the gases. On this basis we developed tools to prepare, coherently manipulate, and analyze pre-defined quantum states. With this high level of control, it was possible to investigate interesting physics phenomena, and we were able to achieve several breakthroughs in the field of ultracold atoms. Highlights of our experimental results include the demonstration of phase engineering of a BEC wavefunction where we created solitons in a BEC and studied their propagation. In another experiment we constructed a spatially resolved matter wave interferometer with which we mapped out the phase distribution on a condensate. Illuminating the BEC with a periodic pattern, i.e. an optical lattice, allowed us to study solid states physics phenomena. In a second set of experiments we demonstrated optically tuning of the interaction between atoms with optical Feshbach resonances. Employing either magnetically tunable Feshbach resonances or laser radiation, we were able to produce ground state molecules in well defined quantum states. This led to the production of the first molecular BEC. By tuning the coupling of atom pairs in a Fermi gas of atoms, we could for the first time investigate the so-called BEC-BCS crossover which describes the continuous change from a BEC superfluid to a Bardeen-Cooper Schrieffer (BCS) superfluid. Due to the excellent control on the quantum level, the developed cold atom techniques open up intriguing prospects for future applications and experiments. They represent a general tool box to investigate fundamental physical phenomena in a pure and undisturbed environment. In the future, cold atoms might serve as a testing ground for fundamental theories or as quantum simulators for complex 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.