We investigate continuous-time quantum walks of two fermionic atoms loaded in one-dimensional optical lattices with on-site interaction and subjected to a Zeeman field. The quantum walks are accompanied by spin-flipping processes. We calculate the time-dependent density distributions of the two fermions with opposite spins which are initially positioned at the center site by means of an exact numerical method. Besides the usual fast linear expansion behavior, we find an interesting spin-flipping-induced localization in the time evolution of density distributions. We show that the fast linear expansion behavior can be restored by simply ramping up the Zeeman field or further increasing the spin-flipping strength. The intrinsic origin of this exotic phenomenon is attributed to the emergence of a flat band in the single-particle spectrum of the system. Furthermore, we investigate the effect of on-site interaction on the dynamics of the quantum walkers. The two-particle correlations are calculated and the signal of localization is also shown therein. A simple potential experimental application of this interesting phenomenon is proposed.
We investigate continuous-time quantum walks of two fermionic atoms loaded in one-dimensional optical lattices with on-site interaction and subjected to a Zeeman field. The quantum walks are accompanied by spin-flipping processes. We calculate the time-dependent density distributions of the two fermions with opposite spins which are initially positioned at the center site by means of an exact numerical method. Besides the usual fast linear expansion behavior, we find an interesting spin-flipping-induced localization in the time evolution of density distributions. We show that the fast linear expansion behavior can be restored by simply ramping up the Zeeman field or further increasing the spin-flipping strength. The intrinsic origin of this exotic phenomenon is attributed to the emergence of a flat band in the single-particle spectrum of the system. Furthermore, we investigate the effect of on-site interaction on the dynamics of the quantum walkers. The two-particle correlations are calculated and the signal of localization is also shown therein. A simple potential experimental application of this interesting phenomenon is proposed.
We investigate continuous-time quantum walks of two indistinguishable anyons in one-dimensional lattices with both on-site and nearest-neighbor interactions based on the fractional Jordan-Wigner transformation. It is shown that the two-body correlations in position space are symmetric about the initial sites of two quantum walkers in the Bose limit ($\ensuremath{\chi}=0$) and Fermi limit ($\ensuremath{\chi}=1$), while in momentum space this happens only in the Bose limit. An interesting asymmetry arises in the correlation for most cases with the statistical parameter $\ensuremath{\chi}$ varying in between. It turns out that the origin of this asymmetry comes from the fractional statistics that anyons obey. On the other hand, the two-body correlations of hard-core anyons in position space show uniform behaviors from antibunching to cowalking regardless of the statistical parameter. The momentum correlations in the case of strong interaction undergo a smooth process of two stripes smoothly merging into a single one, i.e. the evolution of fermions into hard-core bosons.
We consider the nonequilibrium dynamics of an interacting spin- fermion gas in a one-dimensional optical lattice after switching off the confining potential. In particular, we study the creation and the time evolution of spatially separated, spin-entangled fermionic pairs. The time-dependent density-matrix renormalization group is used to simulate the time evolution and evaluate the two-site spin correlation functions, from which the concurrence is calculated. We find that the typical distance between entangled fermions depends crucially on the onsite interaction strength, and that a time-dependent modulation of the tunnelling amplitude can enhance the production of spin entanglement. Moreover, we discuss the prospects of experimentally observing these phenomena using spin-dependent single-site detection.
We theoretically explore the tunneling dynamics for the tight-binding model of a single spin-orbit-coupled atom trapped in an optical lattice subjected to lattice shaking and to time-periodic Zeeman field. By means of analytical and numerical methods, we demonstrate that the spin-orbit (SO) coupling adds some results to the tunneling dynamics in both multiphoton resonance and far-off-resonance parameter regimes. When the driving frequency is resonant with the static Zeeman field (multiphoton resonances), we obtain an unexpected dynamical localization (DL) phenomenon where the single SO-coupled atom is restricted to making perfect two-site Rabi oscillation accompanied by spin flipping. By using the unconventional DL phenomenon, we are able to generate a ratchetlike effect which enables directed atomic motion towards different directions and accompanies periodic spin flipping under the action of SO coupling. For the far-off-resonance case, we show that by suppressing the usual intersite tunneling alone, it is possible to realize a type of spin-conserving second-order tunneling between next-nearest-neighboring sites, which is not accessible in the conventional lattice system without SO coupling. We also show that simultaneous controls of the usual intersite tunneling and the SO-coupling-related second-order tunneling are necessary for quasienergies flatness (collapse) and completely frozen dynamics to exist. These results may be relevant to potential applications such as spin-based quantum information processing and design of spintronics devices.
We investigate continuous-time quantum walks of two indistinguishable particles (bosons, fermions or hard-core bosons) in one-dimensional lattices with nearest-neighbour interactions. The two interacting particles can undergo independent- and/or co-walking dependent on both quantum statistics and interaction strength. We find that two strongly interacting particles may form a bound state and then co-walk like a single composite particle with statistics-dependent propagation speed. Such an effective single-particle picture of co-walking is analytically derived in the context of degenerate perturbation and the analytical results are well consistent with direct numerical simulation. In addition to implementing universal quantum computation and observing bound states, two-particle quantum walks offer a novel route to detecting quantum statistics. Our theoretical results can be examined in experiments of light propagations in two-dimensional waveguide arrays or spin-impurity dynamics of ultracold atoms in one-dimensional optical lattices.
In this thesis, we discuss quantum phase transitions in low-dimensional optical lattices, namely one- and two-dimensional lattices. The dimensional confinement is realized in experiments by suppressing the hopping in the extra dimensions through a deep potential barrier that prevents the atoms to tunnel across dimensions. The dimensionality considerably affects the physical behavior of the system. For instance, in one dimension, the celebrated Fermi-liquid theory does not describe the many-body behavior of a fermionic system. There are no single particle excitations, but instead collective behavior, and spin and charge degrees of freedom are decoupled . In the first part of this thesis, we consider one-dimensional fermionic gases under the effect of time-dependent driving forces. First, we describe a proposal to realize non-standard Hubbard models that display correlated-hopping terms by modulating interactions through a time-dependent magnetic field. The extra terms describe hopping processes that depend on the particle-density difference in neighboring sites. The effective model is obtained in arbitrary spatial dimensions, but only in one-dimension it is known to be exactly solvable for specific values of the parameters. Moreover, an SU(2) symmetry in the charge sector allows to construct non-trivial superconducting eigenstates (eta-states), discovered by Yang as excited states of the Hubbard model. Then, we study a one-dimensional bipartite optical lattice with fermionic atoms at half-filling. The lattice potential barriers have heights alternating in magnitude. As a consequence, the hopping amplitudes also alternate in magnitude and the elementary unit cell has two sites. This system is known to display Peierls dimerization at half-filling: the fermions form dimers in neighboring sites that inhibit transport and open a gap in the spectrum. After applying shaking on this system we discover that one can dynamically induce a phase transition to an unconventional metallic state with four Fermi-points. The second part of this thesis describes bosonic systems in two-dimensional optical lattices. First, we consider a square optical lattice with alternating deep-shallow wells arranged in a checkerboard pattern. This lattice is realized in the experimental laboratory of Prof. Hemmerich in Hamburg. In the experiment, the visibility is extracted from the time-of-flight imaging to quantify the phase coherence of the bosonic gas. We develop a mean-field theory and determine the phase diagram of this model. We interpret the experimental data by assuming that at large detuning, only the deeper wells are populated and form an imbalanced Mott insulator with empty shallow wells and integer density on the deeper wells. We support this hypothesis by large-scale Gutzwiller calculations. Then, we investigate a system of bosons in the Lieb lattice, which is the lattice geometry of the Cu-O plane in cuprates . We consider configurations of the optical potential where s and p orbitals sitting on neighboring wells become resonant and hybridize. The interest in this problem comes from the possibility to realize time-reversal broken phases, usually known as Varma phases for electronic systems, which have been proposed to explain the pseudo-gap regime of cuprate superconductors. We show the conditions under which the symmetry breaking in a four-band model can lead to states with loop currents. Moreover, the lowest branch of the excitation spectrum has a non-vanishing Chern number and therefore supports topological excitations.
Coherent transport of atoms trapped in an optical lattice can be controlled by microwave-induced spin flips that correlate with site-to-site hopping. We study the controllability of homogeneous one-dimensional systems of noninteracting atoms in the absence of site addressability. Given these restrictions, we construct a deterministic protocol to map an initially localized Wannier state to a wave packet that is coherently distributed over n sites. As an example, we consider a one dimensional quantum walk in the presence of both realistic photon scattering and inhomogeneous broadening of the microwave transition due to the optical lattice. Using composite pulses to suppress errors, fidelities of over 95 can be achieved for a 25-step walk. We extend the protocol for state preparation to analytic solutions for arbitrary unitary maps given homogeneous systems and in the presence of time-dependent uniform forces. Such control is important for applications in quantum information processing, such as quantum computing and quantum simulations of condensed matter phenomena.