Phase coherence and localization of Bose-Einstein Condensates (BEC) in optical lattices is a notable property that is changing under the superfluid-Mott insulator phase transition. For a single-component BEC, long-range phase coherence across lattice wells is lost as the lattice height is sufficiently increased. Phase coherences of two-component overlapping mixture of BECs, however, are nontrivial because of the interspecies interaction between components, affecting localization of condensates. For such cases, the BECs form two interdependent mean-field lattices made of each component in addition to optical lattices. We prepare initial BEC clouds as a collection of small Bogoliubov quasiparticles added to classical mean-fields, and obtain the dynamics under phase space representations, where the density operators are projected onto phase space. Among various phase space representations, we focus on a Truncated Wigner Approximation (TWA) and a positive P representation, and we construct a hybrid model to combine them. Since a high volume of numerical integration is required for sufficient convergence from ensembles, we construct efficient numerical methods to reduce the sample variance. We demonstrate time evolution of phase coherence of one-component and two-component BECs in state-dependent optical lattices. For one-component BECs, we show that phase coherence loss depends on lattice heights. In the cycling process of lattice height changes, we investigate nonadiabatic effects depending on initial ramp-up speeds. The hybrid model compared with the TWA exhibits temporal fluctuations in coherence and sudden phase diffusion at the end of ramp-up of a lattice. For two-component BECs, we consider that the two wavefunctions are localized at periodic potential wells whose positions differ by a half-period between the two. Dependence on population imbalance between two components becomes another factor on coherence loss. We show that under gradual loading of an optical lattice effective only to first component, loss of phase coherence of first atoms is enhanced and that of second atoms is weakened as the fraction of second atoms is increased in the mixture. We also demonstrate effects of localizing mean-field lattices on coherence loss under various lattice heights.
Phase coherence and localization of Bose-Einstein Condensates (BEC) in optical lattices is a notable property that is changing under the superfluid-Mott insulator phase transition. For a single-component BEC, long-range phase coherence across lattice wells is lost as the lattice height is sufficiently increased. Phase coherences of two-component overlapping mixture of BECs, however, are nontrivial because of the interspecies interaction between components, affecting localization of condensates. For such cases, the BECs form two interdependent mean-field lattices made of each component in addition to optical lattices. We prepare initial BEC clouds as a collection of small Bogoliubov quasiparticles added to classical mean-fields, and obtain the dynamics under phase space representations, where the density operators are projected onto phase space. Among various phase space representations, we focus on a Truncated Wigner Approximation (TWA) and a positive P representation, and we construct a hybrid model to combine them. Since a high volume of numerical integration is required for sufficient convergence from ensembles, we construct efficient numerical methods to reduce the sample variance. We demonstrate time evolution of phase coherence of one-component and two-component BECs in state-dependent optical lattices. For one-component BECs, we show that phase coherence loss depends on lattice heights. In the cycling process of lattice height changes, we investigate nonadiabatic effects depending on initial ramp-up speeds. The hybrid model compared with the TWA exhibits temporal fluctuations in coherence and sudden phase diffusion at the end of ramp-up of a lattice. For two-component BECs, we consider that the two wavefunctions are localized at periodic potential wells whose positions differ by a half-period between the two. Dependence on population imbalance between two components becomes another factor on coherence loss. We show that under gradual loading of an optical lattice effective only to first component, loss of phase coherence of first atoms is enhanced and that of second atoms is weakened as the fraction of second atoms is increased in the mixture. We also demonstrate effects of localizing mean-field lattices on coherence loss under various lattice heights.
A binary mixture of interacting Bose-Einstein condensates (BEC) in the presence of fragmentation-driving external lattice potentials forms two interdependent mean-field lattices made of each component. These effective mean-field lattices, like ordinary optical lattices, can induce additional fragmentation and phase coherence loss of BECs between lattice sites. In this study, we consider the nonequilibrium dynamics of two hyperfine states of one-dimensional Bose-Einstein condensates, subjected to state-dependent optical lattices. Our numerical calculations using the truncated Wigner approximation (TWA) show that phase coherence in a mixture of two-component BECs can be lost not just by optical lattices, but by mean-field lattices gradually formed by other components, and we reveal that such an effect of internal mean-field lattices, however, is limited, contrary to external optical lattices, in regard to phase decoherence.
Bose-Einstein condensates (BECs) in an optical lattice driven out of equilibrium and the subsequent dynamics are investigated in the limit of weak atom-atom interactions both within the classical mean-field approximation and by incorporating an approximate model for quantum fluctuations. A prediction of pulsating dynamical instability in which atoms periodically collect together and subsequently disperse back into the initial homogeneous state is made. This instability is a unique feature of this Hamiltonian system characterized by a single unstable mode and can be considered as a remnant of the integrability of the non-integrable system. A phase-space model has been employed to incorporate the quantum fluctuations within the truncated Wigner approximation. It is observed that the quasiperiodic behavior of the pulsating dynamics persists in the individual stochastic realizations that represent the typical experimental outcomes, but the effect of quantum fluctuations is manifested in the ensemble averages of various properties of the system. Quantum effects are increasingly prominent and the revivals of the ensemble averages of the wave function become progressively weaker when the effective interaction strength is increased. Motivated by these predictions, we apply similar methods to two-component BECs. Within the mean-field model we demonstrate a spontaneous emergence of a vector soliton like structure, a bound pair of a soliton and an anti-soliton, from the homogeneous imbalanced mixture of two BEC species. This is a feature that uniquely combines phase separation, pulsating instability, and formation of a dark-bright soliton pair when the parameters involved are tuned properly. ^
We study the localization of a noninteracting and weakly interacting Bose-Einstein condensate (BEC) with spin-orbit coupling loaded in a quasiperiodic bichromatic optical lattice potential using the numerical solution and variational approximation of a binary mean-field Gross-Pitaevskii equation with two pseudospin components. We confirm the existence of the stationary localized states in the presence of the spin-orbit and Rabi couplings for an equal distribution of atoms in the two components. We find that the interaction between the spin-orbit and Rabi couplings favors the localization or delocalization of the BEC depending on the phase difference between the components. We also studied the oscillation dynamics of the localized states for an initial population imbalance between the two components.
Ultracold atoms in optical lattices provide a highly controllable environment for the clean experimental realization of various model Hamiltonians from condensed matter and statistical physics. For example, the two-component Bose-Hubbard model, which reduces to an anisotropic spin-1/2 Heisenberg model in a certain limit and thus allows for the study of quantum magnetism, can be implemented by using bosons with two different internal states that couple differently to an optical lattice potential. In this thesis, I present our first experiments with two-component hyperfine-state mixtures of ultracold 87Rb atoms in a state-dependent optical lattice, both in the strongly correlated regime and in the context of nonlinear atom optics. For the production of 87Rb Bose-Einstein condensates we have developed a moving-coil transporter apparatus featuring a magnetic TOP trap which serves as a 'phase-space funnel' to load a crossed optical dipole trap. The apparatus further incorporates a three-dimensional optical lattice setup with simultaneously usable hyperfine state-dependent and state-independent lattice beams of different spacing along the vertical axis. Internal state control is performed via rf and microwave Rabi pulses and Landau-Zener sweeps. As a first step towards studying strongly correlated two-component mixtures, we have realized a state-selective superfluid-to-Mott insulator transition, where one component enters the Mott insulator regime, while the other one stays superfluid. Using the state-dependent lattice we can tune the second component's properties from highly superfluid to strongly localized. At both extremes we find a reduction of the coherence of the primary component, i.e. a shift of the Mott transition to smaller values of the ratio U/t of interaction to tunneling. We ascribe this to a polaron-like dressing on the one hand, and a 'quantum emulsion' causing a disordered atomic background potential on the other hand. Further, we have observed and studied four-wave mixing of two-component matter waves. Using state-selective Kapitza-Dirac diffraction of a two-component BEC, we prepare seed and pump modes differing both in momentum and internal state. A novel collinear four-wave mixing process then populates the initially empty output modes. While this process can complicate studies of bosonic mixtures loaded into state-dependent optical lattices, it might prove useful for possible applications in quantum atom optics. | Ultracold atoms in optical lattices provide a highly controllable environment for the clean experimental realization of various model Hamiltonians from condensed matter and statistical physics. For example, the two-component Bose-Hubbard model, which reduces to an anisotropic spin-1/2 Heisenberg model in a certain limit and thus allows for the study of quantum magnetism, can be implemented by using bosons with two different internal states that couple differently to an optical lattice potential. In this thesis, I present our first experiments with two-component hyperfine-state mixtures of ultracold 87Rb atoms in a state-dependent optical lattice, both in the strongly correlated regime and in the context of nonlinear atom optics. For the production of 87Rb Bose-Einstein condensates we have developed a moving-coil transporter apparatus featuring a magnetic TOP trap which serves as a 'phase-space funnel' to load a crossed optical dipole trap. The apparatus further incorporates a three-dimensional optical lattice setup with simultaneously usable hyperfine state-dependent and state-independent lattice beams of different spacing along the vertical axis. Internal state control is performed via rf and microwave Rabi pulses and Landau-Zener sweeps. As a first step towards studying strongly correlated two-component mixtures, we have realized a state-selective superfluid-to-Mott insulator transition, where one component enters the Mott insulator regime, while the other one stays superfluid. Using the state-dependent lattice we can tune the second component's properties from highly superfluid to strongly localized. At both extremes we find a reduction of the coherence of the primary component, i.e. a shift of the Mott transition to smaller values of the ratio U/t of interaction to tunneling. We ascribe this to a polaron-like dressing on the one hand, and a 'quantum emulsion' causing a disordered atomic background potential on the other hand. Further, we have observed and studied four-wave mixing of two-component matter waves. Using state-selective Kapitza-Dirac diffraction of a two-component BEC, we prepare seed and pump modes differing both in momentum and internal state. A novel collinear four-wave mixing process then populates the initially empty output modes. While this process can complicate studies of bosonic mixtures loaded into state-dependent optical lattices, it might prove useful for possible applications in quantum atom optics. | 110 pages
We study the phase coherence property of Bose−Einstein condensates confined in a one-dimensional optical lattice formed by a standing-wave laser field. The lattice depth is determined using a method of Kapitza–Dirac scattering between a condensate and a short pulse lattice potential. Condensates are then adiabatically loaded into the optical lattice. The phase coherence property of the confined condensates is reflected by the interference patterns of the expanded atomic cloud released from the optical lattice. For weak lattice, nearly all of the atoms stay in a superfluid state. However, as the lattice depth is increased, the phase coherence of the whole condensate sample is gradually lost, which confirms that the sub-condensates in each lattice well have evolved into number-squeezed states.
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.