Cubic-quintic nonlinearity in superfluid Bose-Bose mixtures in optical lattices: Heavy solitary waves, barrier-induced criticality, and current-phase relations
We study superfluid (SF) states of strongly interacting Bose-Bose mixtures with equal mass and intracomponent interaction in optical lattices both in the presence and absence of a barrier potential. We show that the SF order parameters obey the two-component nonlinear Schr\"odinger equation (NLSE) with not only cubic but also quintic nonlinearity in the vicinity of the first-order transitions to the Mott insulators with even fillings. In the case of no barrier potential, we analyze solitary-wave solutions of the cubic-quintic NLSE. When the SF state changes from a ground state to a metastable one, a standard dark solitary wave turns into a bubblelike dark solitary wave, which has a nonvanishing density dip and no $\ensuremath{\pi}$-phase kink even in the case of a standing solitary wave. It is shown that the former and latter solitary waves are dynamically unstable against an out-of-phase fluctuation and an in-phase fluctuation, respectively, and the dynamical instabilities are weakened when one approaches the transition point. We find that the size and the inertial mass of the solitary waves diverge at the first-order transition point. We suggest that the divergence of the inertial mass may be detected through measurement of the relation between the velocity and the phase jump of the solitary wave. In the presence of a barrier potential, we reveal that when the barrier strength exceeds a certain critical value, the SF state that was metastable without the barrier is destabilized towards complete disjunction of the SF. The presence of the critical barrier strength indicates that the strong barrier potential qualitatively changes the criticality near the metastability limit of the SF state. We derive critical behaviors of the density, the compressibility, and the critical current near the metastability limit induced by the barrier. It is also found that the relation between the supercurrent and the phase jump across the barrier exhibits a peculiar behavior, owing to the nontopological nature of the bubblelike solitary wave.
We study superfluid (SF) states of strongly interacting Bose-Bose mixtures with equal mass and intracomponent interaction in optical lattices both in the presence and absence of a barrier potential. We show that the SF order parameters obey the two-component nonlinear Schr"odinger equation (NLSE) with not only cubic but also quintic nonlinearity in the vicinity of the first-order transitions to the Mott insulators with even fillings. In the case of no barrier potential, we analyze solitary-wave solutions of the cubic-quintic NLSE. When the SF state changes from a ground state to a metastable one, a standard dark solitary wave turns into a bubblelike dark solitary wave, which has a nonvanishing density dip and no \ensuremathπ-phase kink even in the case of a standing solitary wave. It is shown that the former and latter solitary waves are dynamically unstable against an out-of-phase fluctuation and an in-phase fluctuation, respectively, and the dynamical instabilities are weakened when one approaches the transition point. We find that the size and the inertial mass of the solitary waves diverge at the first-order transition point. We suggest that the divergence of the inertial mass may be detected through measurement of the relation between the velocity and the phase jump of the solitary wave. In the presence of a barrier potential, we reveal that when the barrier strength exceeds a certain critical value, the SF state that was metastable without the barrier is destabilized towards complete disjunction of the SF. The presence of the critical barrier strength indicates that the strong barrier potential qualitatively changes the criticality near the metastability limit of the SF state. We derive critical behaviors of the density, the compressibility, and the critical current near the metastability limit induced by the barrier. It is also found that the relation between the supercurrent and the phase jump across the barrier exhibits a peculiar behavior, owing to the nontopological nature of the bubblelike solitary wave.
In a one-dimensional shallow optical lattice, in the presence of both cubic and quintic nonlinearity, a superfluid density wave is identified in a Bose-Einstein condensate. 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 a certain parameter domain, the competition between lattice potential and the interactions is shown to give rise to a stripe phase, where atoms are confined in finite domains. In a pure two-body case, apart from the known superfluid and insulating phases, a density wave insulating phase is found to exist, possessing two frequency modulations commensurate with the lattice potential.
We study ground-state phase diagrams and excitation spectra of Bose-Bose mixtures in an optical lattice by applying the Gutzwiller approximation to the two-component Bose-Hubbard model. A case of equal hoppings and equal intra-component interactions for both components is considered. Due to the existence of inter-component interaction, there appear several quantum phases, such as the superfluid, paired superfluid, and counterflow superfluid phases. We find that the transition from superfluid (SF) to Mott insulator (MI) with even filling factors can be of the first order for a wide range of the chemical potential. We calculate the excitation spectra as a useful probe to identify the quantum phases and the SF-to-MI transitions. In the excitation spectra of the SF phase, there are two gapless modes and a few gapful modes, which respectively correspond to phase and amplitude fluctuations of the order parameters. At the SF-to-MI transition point, we show that the energy gaps of certain amplitude modes reach zero for the second-order transition while they remain finite for the first-order one. Since the excitation spectrum can be measured by the Bragg scattering, we calculate the dynamical structure factor by using the linear response theory. We consider three types of density fluctuations, and show that the density fluctuations are coupled to different excitation branches in different quantum phases.
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.
In a one-dimensional shallow optical lattice, in the presence of both cubic and quintic nonlinearity, a superfluid density wave is identified in a Bose-Einstein condensate. 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 a certain parameter domain, the competition between lattice potential and the interactions is shown to give rise to a stripe phase, where atoms are confined in finite domains. In a pure two-body case, apart from the known superfluid and insulating phases, a density wave insulating phase is found to exist, possessing two frequency modulations commensurate with the lattice potential.
We examine the quench dynamics across quantum phase transitions from a Mott insulator (MI) to a superfluid (SF) phase in a two-component bosonic mixture in an optical lattice. We show that two-component Bose mixtures exhibit qualitatively different quantum dynamics than one-component Bose gas. In addition to second-order MI-SF transitions, we investigate quench dynamics across a first-order MI-SF phase boundary. The Bose mixtures show the critical slowing down of dynamics near the critical transition point, as proposed by the Kibble-Zurek mechanism. For MI-SF transitions with homogeneous lattice-site distributions in the MI phase, the dynamical critical exponents extracted by the power-law scaling of the proposed quantities obtained via numerical simulations are in very close agreement with the mean-field predictions.
We demonstrate the existence of two types of dark gap solitary waves-the dark gap solitons and the dark gap soliton clusters-in Bose-Einstein condensates trapped in a bichromatic optical superlattice with cubic-quintic nonlinearities. The background of these dark soliton families is different from the one in a common monochromatic linear lattice; namely, the background in our model is composed of two types of Gaussian-like pulses, whereas in the monochromatic linear lattice, it is composed of only one type of Gaussian-like pulses. Such a special background of dark soliton families is convenient for the manipulation of solitons by the parameters of bichromatic and chemical potentials. The dark soliton families in the first, second, and third bandgap in our model are studied. Their stability is assessed by the linear-stability analysis, and stable as well as unstable propagation of these gap solitons are displayed. The profiles, stability, and perturbed evolution of both types of dark soliton families are distinctly presented in this work.