One-dimensional Lattice Research Articles

Non-Hermitian jumps in disordered lattices

In the context of non-Hermitian Anderson localization, we study the physics of wave propagation in one-dimensional waveguide lattices, with randomly distributed gain or loss. Despite the Anderson localization of all eigenstates, the system exhibits counterintuitive propagation by quantized jumps between states located around distant sites. Such a novel effect was recently experimentally demonstrated in optical fiber loop networks. We provide a systematic way of understanding the underlying physical mechanism of such an effect. The role of eigenstates' nonorthogonality and biorthogonal projection is of central importance to our work, and is systematically examined in the symmetric non-Hermitian model, as well as the nonsymmetric Hatano-Nelson Hamiltonian. Our methodology can be applied to any non-Hermitian disordered system that contains complex elements with loss and/or gain, and thus exploits the meaning of wave transport in complex open systems.

Non-Hermitian morphing of topological modes.

Topological modes (TMs) are usually localized at defects or boundaries of a much larger topological lattice1,2. Recent studies of non-Hermitian band theories unveiled the non-Hermitian skin effect (NHSE), by which the bulk states collapse to the boundary as skin modes3-6. Here we explore the NHSE to reshape the wavefunctions of TMs by delocalizing them from the boundary. At a critical non-Hermitian parameter, the in-gap TMs even become completely extended in the entire bulk lattice, forming an 'extended mode outside of a continuum'. These extended modes are still protected by bulk-band topology, making them robust against local disorders. The morphing of TM wavefunction is experimentally realized in active mechanical lattices in both one-dimensional and two-dimensional topological lattices, as well as in a higher-order topological lattice. Furthermore, by the judicious engineering of the non-Hermiticity distribution, the TMs can deform into a diversity of shapes. Our findings not only broaden and deepen the current understanding of the TMs and the NHSE but also open new grounds for topological applications.

Growth rate of a stochastic growth process driven by an exponential Ornstein–Uhlenbeck process

We study the stochastic growth process in discrete time xi+1 = (1 + μi)xi with the growth rate μi=ρeZi−12Var(Zi) proportional to the exponential of an Ornstein–Uhlenbeck (O–U) process dZt = −γZtdt + σdWt sampled on a grid of uniformly spaced times with time step τ. Using large deviation theory, we compute the asymptotic growth rate (Lyapunov exponent) λ=limn→∞1nlogE[xn]. We show that this limit exists, under appropriate scaling of the O–U parameters, and is expressed as the solution of a variational problem. The asymptotic growth rate is equal to the thermodynamical pressure of a one-dimensional lattice gas with attractive exponential potentials. For Zt, a stationary O–U process of the lattice gas coincides with a model considered previously by Kac and Helfand. We derive upper and lower bounds on λ. In the large mean-reversion limit γnτ ≫ 1, the two bounds converge and the growth rate is given by a lattice version of the van der Waals equation of state. The predictions are tested against numerical simulations of the stochastic growth model.

High-quality resonances in quasi-periodic clusters of scatterers for flexural waves

Multiple scattering theory is applied to the study of clusters of point-like scatterers attached to a thin elastic plate and arranged in quasi-periodic distributions. Two types of structures are specifically considered: the twisted bilayer and the quasi-periodic line. The former consists in a couple of two-dimensional lattices rotated a relative angle, so that the cluster forms a moiré pattern. The latter can be seen as a periodic one-dimensional lattice where an incommensurate modulation is superimposed. Multiple scattering theory allows for the fast and efficient calculation of the resonant modes of these structures as well as for their quality factor, which is thoroughly analyzed in this work. The results show that quasi-periodic structures present a large density of states with high quality factors, being therefore a promising way for the design of high quality wave-localization devices.

Quantum phases of dipolar bosons in one-dimensional optical lattices

We theoretically analyze the phase diagram of a quantum gas of bosons that interact via repulsive dipolar interactions. The bosons are tightly confined by an optical lattice in a quasi-one-dimensional geometry. In the single-band approximation, their dynamics is described by an extended Bose-Hubbard model where the relevant contributions of the dipolar interactions consist of density-density repulsion and correlated tunneling terms. We evaluate the phase diagram for unit density using numerical techniques based on the density-matrix renormalization group algorithm. Our results predict that correlated tunneling can significantly modify the parameter range of the topological insulator phase. At vanishing values of the onsite interactions, moreover, correlated tunneling promotes the onset of a phase with a large number of low-energy metastable configurations.

Manipulations of cold atoms in a one-dimensional ring lattice

The dynamics of Bose–Einstein condensates in a one-dimensional ring optical lattice is related to the interplay of the open system conditions and the initial atomic distribution. The circulation evaluation model based on the relative atomic density on each site is established to analyze the circulation strength under low dissipation. The direction of the circulation due to the symmetry breaking can be manipulated by changing the primary atomic distribution and the number of lattice sites. The effect of the atomic interaction on the atom flow in the optical ring lattice is analyzed to maintain the current interval rotation behavior.

Bayesian learning for optimal control of quantum many-body states in optical lattices

Strongly correlated quantum many-body states provide invaluable resources for state-of-the-art quantum information science, ranging from quantum simulation to quantum metrology. In this work, we propose a supervised machine learning algorithm for optimal control of quantum many-body atomic states in optical lattices by numerical simulations based on the Bayesian method of Gaussian process regression (GPR). We combine this method with the time evolving block decimation (TEBD) algorithm for the preparation of the Heisenberg antiferromagnetic state in a system of bosonic atoms confined with a one-dimensional optical lattice. The quantum many-body ground state of 80 atoms is efficiently optimized within a few hundred machine learning iterations, reaching a state fidelity above $96%$. With a multistep learning strategy, we find the machine-learning-based optimal control method is scalable to large systems owing to its transferability. Its robustness against noise is demonstrated by considering imperfections that are typically present in optical lattice experiments. In the application of the GPR method to the preparation of the two-dimensional (2D) antiferromagnetic state, a state fidelity of $94%$ is reached for a 2D array of $6\ifmmode\times\else\texttimes\fi{}6$ spins through the time dependent variational principle (TDVP) algorithm, confirming the generalizability of our method. We further optimize the Hamiltonian ramping sequence crossing a quantum phase transition of the quantum $XXZ$ spin chain with the exact diagonalization method, from which a control protocol for generating the long-sought atomic Greenberger-Horne-Zeilinger state is obtained. We believe the proposed optimal control method for quantum Hamiltonian ground-state preparation would benefit present ultracold-atom experiments in the study of strongly correlated many-body physics.

Quantum phase diagram for two-species hardcore bosons in one-dimensional optical lattices with resonantly driven Rabi frequency

We propose an experimental realization of the time-periodically modulated Rabi frequency and suggest density-dependent hoppings of two species hardcore bosons in a one-dimensional optical lattice. Distinct from the previous work [Phys. Rev. Research {\bf 2}, 013275 (2020)], we study effects in the first resonance region. In the effective Hamiltonian, the intra-species hopping occurs only if the density discrepancy of the other species on these sites is zero, while the inter-species one is allowed once the relevant density discrepancy becomes nonzero. At integer-$1$ filling, the quantum phase diagram of the effective Hamiltonian is determined by the perturbation analysis together with numerical calculations. We find that in the limit of dominant $J_{1}$, the system becomes a double-degenerate dimerized state, with spontaneously breaking the translation symmetry. The interplay of $J_{0}$, $J_{1}$ and the fixed ${\bar U}=1$ leads to three BKT transition lines and a tricritical BKT point. Exact transition lines are obtained by the level spectroscopic technique. Besides, general physical properties, including the charge gap, neutral gap, superfluid density and dimerization strength, are investigated as well.

Competing insulating phases in a dimerized extended Bose-Hubbard model

We study the ground-state properties of the extended Bose-Hubbard model in a one-dimensional dimerized optical lattice. In the limit of strong on-site repulsion, i.e., hardcore bosons, and strong nearest-neighbor interaction, a stable density-wave (DW) phase is obtained at half-filling as a function of lattice dimerization. Interestingly, at quarter-filling we obtain the signatures of an insulating phase which has the character of both the bond order (BO) and the DW insulators, which we call a bond-order density-wave (BODW) phase. Moreover, we show that for a fixed hopping dimerization there occurs a BO-DW phase crossover as a function of the nearest-neighbor interaction and the BODW phase is more robust when the hopping dimerization is stronger. We further examine the stability of the BODW phase in the limit of finite on-site interactions.

Coexistence of extended and localized states in the one-dimensional non-Hermitian Anderson model

In one-dimensional Hermitian tight-binding models, mobility edges separating extended and localized states can appear in the presence of properly engineered quasi-periodical potentials and coupling constants. On the other hand, mobility edges don't exist in a one-dimensional Anderson lattice since localization occurs whenever a diagonal disorder through random numbers is introduced. Here, we consider a nonreciprocal non-Hermitian lattice and show that the coexistence of extended and localized states appears with or without diagonal disorder in the topologically nontrivial region. We discuss that the mobility edges appear basically due to the boundary condition sensitivity of the nonreciprocal non-Hermitian lattice.

Phonon damping in one-dimensional lattices with asymmetric interactions

Phonon damping in one-dimensional lattices with asymmetric interactions

Non-Hermitian mode-locking synthesized by parity-time and anti-parity-time symmetric modulations.

We study the pulse characteristics in a laser mode-locked by active modulators with non-Hermitian driven signals. The signal assembles a parity-time (P T) symmetric and an anti-parity-time (A P T) symmetric function with fundamental and harmonic frequencies, respectively, inducing the complex coupling between modes in the frequency domain. A one-dimensional synthetic lattice is used to analyze the spectral mode coupling. By enlarging the weight and harmonic order of the A P T part of the signal, the optical spectrum can be adjusted from redshift to blueshift. Simultaneously, the pulse duration and spectral width are shortened and broadened, respectively. The work explores the role of non-Hermitian modulation in the mode-locked laser area.

A chaotic lattice field theory in one dimension* *Dedicated to Fritz Haake 1941–2019 [

Motivated by Gutzwiller’s semiclassical quantization, in which unstable periodic orbits of low-dimensional deterministic dynamics serve as a WKB ‘skeleton’ for chaotic quantum mechanics, we construct the corresponding deterministic skeleton for infinite-dimensional lattice-discretized scalar field theories. In the field-theoretical formulation, there is no evolution in time, and there is no ‘Lyapunov horizon’; there is only an enumeration of lattice states that contribute to the theory’s partition sum, each a global spatiotemporal solution of system’s deterministic Euler–Lagrange equations. The reformulation aligns ‘chaos theory’ with the standard solid state, field theory, and statistical mechanics. In a spatiotemporal, crystallographer formulation, the time-periodic orbits of dynamical systems theory are replaced by periodic d-dimensional Bravais cell tilings of spacetime, each weighted by the inverse of its instability, its Hill determinant. Hyperbolic shadowing of large cells by smaller ones ensures that the predictions of the theory are dominated by the smallest Bravais cells. The form of the partition function of a given field theory is determined by the group of its spatiotemporal symmetries, that is, by the space group of its lattice discretization, best studied on its reciprocal lattice. Already one-dimensional lattice discretization is of sufficient interest to be the focus of this paper. In particular, from a spatiotemporal field theory perspective, ‘time’-reversal is a purely crystallographic notion, a reflection point group, leading to a novel, symmetry quotienting perspective of time-reversible theories and associated topological zeta functions.

Pairs, trimers, and BCS-BEC crossover near a flat band: Sawtooth lattice

We investigate pairing and superconductivity in the attractive Fermi Hubbard model on the one-dimensional sawtooth lattice, which exhibits a flat band by fine-tuning the hopping rates. We first solve the two-body problem, both analytically and numerically, to extract the binding energy and the effective mass of the pairs. Based on the DMRG method, we address the ground-state properties of the many-body system, assuming equal spin populations. We compare our results with those available for a linear chain, where the model is integrable by Bethe ansatz, and show that the multiband nature of the system substantially modifies the physics of the BCS-BEC crossover. Near a flat band, the chemical potential remains always close to its zero-density limit predicted by the two-body physics. In contrast, the pairing gap exhibits a remarkably strong density dependence and, differently from the pair binding energy, it is no longer peaked at the flat-band point. We show that these results can be interpreted in terms of polarization screening effects, due to an anomalous attraction between pairs in the medium and single fermions. Importantly, we unveil that three-body bound states (trimers) exist in the sawtooth lattice, in sharp contrast with the linear chain geometry, and we compute their binding energy. The nature of these states is investigated via a strong coupling variational approach, revealing that they originate from tunneling-induced exchange processes.

Single-particle excitations and metal-insulator transition of ultracold Fermi atoms in one-dimensional optical lattice with spin-orbit coupling

The dynamic structure factors reflecting the excitation spectra were investigated in a one-dimensional (1D) optical lattice with a spin-orbit coupling (SOC) effect. The results reveal that the single-particle excitations of both the density and spin dynamical structure factors are strongly reconstructed and split owing to the SOC effect, and a hat-like excitation band appears in the high-binding-energy region. The hat-like excitation band of the density dynamical structure factor exhibits an arc form, and has a pocket in the spin dynamical structure factor. In particular, only a gapless single-particle excitation point is left for both the density dynamical structure factor and the spin dynamical structure factor when the SOC strength reaches a critical point at half-filling. A stronger SOC strength causes the gapless excitation points to disappear, which indicates that the metal-insulator transition occurs. The metal-insulator transition only appears in half-filling and lightly doped regimes.

Single and multi-vertices solitons in lattices of active Morse - van der Pol units

The dynamical evolution of one-dimensional lattice of active units (otherwise said particles) endowed with anharmonic Morse interactions and active–dissipative van der Pol negative friction is here studied both analytically and numerically. One of the directions of this evolution is the formation and stable propagation of dissipative soliton-like perturbations (in short solitons) against the background of some stationary displacement of units (pedestal). It is shown that the evolution of units at leading-front of the soliton is determined by mutual repulsive forces, while that of units at trailing-tail of the soliton is determined by the active–dissipative forces. Approximate analytically solvable models of the leading-front and the trailing-tail are obtained, which make it possible to predict soliton parameters and admissible range of pedestal values with high accuracy. There is also a scenario of N-vertices soliton formation emerging from a single soliton and (N-1)-vertices soliton. The possibility of controlling the number of vertices of multi-vertices solitons is offered, a feature which can be used in the problems of energy accumulation and transport in periodic structures.

Experimental band flip and band closure in guided-mode resonant optical lattices

We demonstrate band flip in one-dimensional dielectric photonic lattices presenting numerical and experimental results. In periodic optical lattices supporting leaky Bloch modes, there exists a second stop band where one band edge experiences radiation loss resulting in guided-mode resonance (GMR), while the other band edge becomes a nonleaky bound state in the continuum (BIC). To illustrate the band flip, band structures for two different lattices are provided by calculating zero-order reflectance with respect to wavelength and incident angle. We then provide three photonic lattices, each with a different fill factor, consisting of photoresist gratings on Si3N4 sublayers with glass substrates. The designs are fabricated using laser interferometric lithography. The lattice parameters are characterized and verified with an atomic force microscope. The band transition under fill-factor variation is accomplished experimentally. The measured data are compared to simulation results and show good agreement.

ON THE EXISTENCE OF EIGENVALUES OF THE DISCRETE SCHRÖDINGER OPERATOR

We study the discrete Schrödinger operators associated to a system of two fermions with an indefinite sign interaction potential with energy on the nearest neighbooring sites on the one-dimensional lattice . We obtain the conditions on the parameter for the operator to have zero or one eigenvalue below the essential spectrum.

Nonlinear localized modes in one-dimensional nanoscale dark-state optical lattices

Abstract Optical lattices (OLs) with conventional spatial periodic λ/2, formed by interfering the counterpropagating laser beams with wavelength λ, are versatile tools to study the dynamical and static properties of ultracold atoms. OLs with subwavelength spatial structure have been realized in recent quantum-gas experiment, offering new possibility for nonlinear and quantum control of ultracold atoms at the nano scale. Herein, we study theoretically and numerically the formation, property, and dynamics of matter-wave localized gap modes of Bose–Einstein condensates loaded in a one-dimensional nanoscale dark-state OL consisted of an array of optical subwavelength barriers. The nonlinear localized modes, in the forms of on- and off-site fundamental gap solitons, and dipole ones, are demonstrated; and we uncover that, counterintuitively, these modes exhibit always a cusplike (side peaks) mode even for a deeply subwavelength adiabatic lattice, contrary to the previously reported results in conventional deep OLs where the localized gap modes are highly confined in a single lattice cell. The (in)stability features of all the predicted localized modes are verified through the linear-stability analysis and direct perturbed simulations. Our predicted results are attainable in current ultracold atoms experiments with the cutting-edge technique, pushing the nonlinear control of ultracold atoms with short-period OLs as an enabling technology into subwavelength structures.

Small mode volume topological photonic states in one-dimensional lattices with dipole-quadrupole interactions

We study the topological photonic states in one-dimensional lattices analog to the Su-Schrieffer-Heeger model beyond the dipole approximation. The electromagnetic resonances of the lattices supported by near-field interactions between the plasmonic nanoparticles are studied analytically with coupled dipole-quadrupole method. The topological phase transition in the bipartite lattices is determined by the change of Zak phase. Our results reveal the contribution of quadrupole moments to the near-field interactions and the band topology. It is found that the topological edge states in nontrivial lattices have both dipolar and quadrupolar nature. The quadrupolar edge states are not only orthogonal to the dipolar edge states, but also spatially localized at different sublattices. Furthermore, the quadrupolar topological edge states, which coexist at the same energy with the quadrupolar flat band have shorter localization length and hence smaller mode volume than the conventional dipolar edge states. The findings deepen our understanding in topological systems that involve higher-order multipoles, or in analogy to the wave functions in quantum systems with higher-orbital angular momentum, and may be useful in designing topological systems for confining light robustly and enhancing light-matter interactions.