Particles In A One-dimensional Lattice Research Articles

Bootstrapping the Kronig-Penney model

Recently, bootstrap methods from conformal field theory have been adapted for studying the energy spectrum of various quantum mechanical systems. In this paper, we consider the application of these methods in obtaining the spectrum from the Schr\"odinger equation with periodic potentials, paying particular attention to the Kronig-Penney model of a particle in a one-dimensional lattice. With an appropriate choice of operator basis involving position and momenta, we find that the bootstrap approach efficiently computes the band gaps of the energy spectrum but has trouble effectively constraining the minimum energy. We show how applying more complex constraints involving higher powers of momenta can potentially remedy such a problem. We also propose an approach for analytically constructing the dispersion relation associated with the Bloch momentum of the system.

Tunable exciton-polaritons emerging from WS2 monolayer excitons in a photonic lattice at room temperature

Engineering non-linear hybrid light-matter states in tailored lattices is a central research strategy for the simulation of complex Hamiltonians. Excitons in atomically thin crystals are an ideal active medium for such purposes, since they couple strongly with light and bear the potential to harness giant non-linearities and interactions while presenting a simple sample-processing and room temperature operability. We demonstrate lattice polaritons, based on an open, high-quality optical cavity, with an imprinted photonic lattice strongly coupled to excitons in a WS2 monolayer. We experimentally observe the emergence of the canonical band-structure of particles in a one-dimensional lattice at room temperature, and demonstrate frequency reconfigurability over a spectral window exceeding 85 meV, as well as the systematic variation of the nearest-neighbour coupling, reflected by a tunability in the bandwidth of the p-band polaritons by 7 meV. The technology presented in this work is a critical demonstration towards reconfigurable photonic emulators operated with non-linear photonic fluids, offering a simple experimental implementation and working at ambient conditions.

Scattering of two particles in a one-dimensional lattice

This study concerns the two-body scattering of particles in a one-dimensional periodic potential. A convenient ansatz allows for the separation of center-of-mass and relative motion, leading to a discrete Schr\"odinger equation in the relative motion that resembles a tight-binding model. A lattice Green's function is used to develop the Lippmann-Schwinger equation, and ultimately derive a multiband scattering $K$ matrix which is described in detail in the two-band approximation. Two distinct scattering lengths are defined according to the limits of zero relative quasimomentum at the top and bottom edges of the two-body collision band. Scattering resonances occur in the collision band when the energy is coincident with a bound state attached to another higher or lower band. Notably, repulsive on-site interactions in an energetically closed lower band lead to collision resonances in an excited band.

Convergence rate from systems of balance laws to isotropic parabolic systems, a periodic case

It is proved that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish the global-in-time error estimates between the smooth solutions to the partially dissipative hyperbolic systems and those to the isotropic parabolic limiting systems in a three dimensional torus, rather than in the one dimensional whole space (Appl. Anal. 100(5) (2021) 1079–1095). This avoids the condition raised for the strong connection between the flux and the source term and make the result obtained more generalized. In the proof, we provide a similar stream function technique which is valid for the three dimensional periodic case. Similar method is provided for the one-dimensional periodic case. As applications of the results, we give several examples arising from physical models at the end of the paper.

Two-particles bounded states as a mechanism to weaken the Anderson localization in systems with structural disorder

Abstract We study numerically the effect of the on-site Hubbard interaction U on the Anderson localization of two-interacting particles in a one-dimensional lattice with structural disorder. By solving the time-dependent Schrodinger equation to follow the time evolution of an initially localized two-electron state, we show the inter-particle interaction employing a non monotonic behavior on the Anderson localization. This non monotonic behavior is consistent with the results depicted by several many-body models in random potentials, which suggests an universal phenomenology for disordered systems. By studying the numerical solution of the time-independent Schrodinger equation, we reveal the origin of this non monotonic behavior: competitive aspects of the bounded and unbounded states of two-particle eigenstates. Ultimately, we show the disorder destroying the coherent Bloch oscillations induced by an electric field, even while particles have the proper interaction strength to correlate their motions.

Typical and extreme entropies of long-lived isolated quantum systems

In this paper, we investigate and compare two well-developed definitions of entropy relevant for describing the dynamics of isolated quantum systems: bipartite entanglement entropy and observational entropy. In a model system of interacting particles in a one-dimensional lattice, we numerically solve for the full quantum behavior of the system. We characterize the fluctuations, and find the maximal, minimal, and typical entropy of each type that the system can eventually attain through its evolution. While both entropies are low for some "special" configurations and high for more "generic" ones, there are several fundamental differences in their behavior. Observational entropy behaves in accord with classical Boltzmann entropy (e.g. equilibrium is a condition of near-maximal entropy and uniformly distributed particles, and minimal entropy is a very compact configuration). Entanglement entropy is rather different: minimal entropy "empties out" one partition while maximal entropy apportions the particles between the partitions, and neither is typical. Beyond these qualitative results, we characterize both entropies and their fluctuations in some detail as they depend on temperature, particle number, and box size.

Interpolation inequalities in <inline-formula><tex-math id="M1">\begin{document}$ \mathrm W^{1,p}( {\mathbb S}^1) $\end{document}</tex-math></inline-formula> and <i>carré du champ</i> methods

This paper is devoted to an extension of rigidity results for nonlinear differential equations, based on carre du champ methods, in the one-dimensional periodic case. The main result is an interpolation inequality with non-trivial explicit estimates of the constants in W1,p(S1) with p ≥ 2. Mostly for numerical reasons, we relate our estimates with issues concerning periodic dynamical systems. Our interpolation inequalities have a dual formulation in terms of generalized spectral estimates of Keller-Lieb-Thirring type, where the differential operator is now a p-Laplacian type operator. It is remarkable that the carre du champ method adapts to such a nonlinear framework, but significant changes have to be done and, for instance, the underlying parabolic equation has a nonlocal term whenever p≠2.

Asymptotically perfect efficient quantum state transfer across uniform chains with two impurities

The ability to transfer quantum information from one location to another with high fidelity is of central importance to quantum information science. Unfortunately for the simplest system of a uniform chain (a spin chain or a particle in a one-dimensional lattice), the state transfer time grows exponentially in the chain length $N$ at fixed fidelity. In this work we show that the addition of an impurity near each endpoint, coupled to the uniform chain with strength $w$, is sufficient to ensure efficient and high-fidelity state transfer. An eigenstate localized in the vicinity of the impurity can be tuned into resonance with chain extended states by tuning $w(N)\propto N^{1/2}$; the resulting avoided crossing yields resonant eigenstates with large amplitudes on the chain endpoints and approximately equidistant eigenvalues. The state transfer time scales as $t\propto N^{3/2}$ and its fidelity $F$ approaches unity in the thermodynamic limit $N\to\infty$; the error scales as $1-F\propto N^{-1}$. Thus, with the addition of two impurities, asymptotically perfect state transfer with a uniform chain is possible even in the absence of external control.

Response of a particle in a one-dimensional lattice to an applied force: Dynamics of the effective mass

We study the behavior of the expectation value of the acceleration of a particle in a one-dimensional periodic potential when an external homogeneous force is suddenly applied. The theory is formulated in terms of modified Bloch states that include the interband mixing induced by the force. This approach allows us to understand the behavior of the wave packet, which responds with a mass that is initially the bare mass, and subsequently oscillates around the value predicted by the effective mass. If Zener tunneling can be neglected, the expression obtained for the acceleration of the particle is valid over time scales of the order of a Bloch oscillation, which are of interest for experiments with cold atoms in optical lattices. We discuss how these oscillations can be tuned in an optical lattice for experimental detection.

Lieb–Thirring inequalities on some manifolds

We prove Lieb–Thirring inequalities with improved constants on the two-dimensional $\mathbb S^2$ sphere and the two-dimensional torus $\mathbb T^2$. In the one-dimensional periodic case we obtain a simultaneous bound for the negative trace and the number of negative eigenvalues.

Lattice-Induced Resonances in One-Dimensional Bosonic Systems

We study the resonant effects produced when a Feshbach dimer crosses a scattering continuum band of atoms in an optical lattice. We numerically obtain the exact spectrum of two particles in a one-dimensional lattice and develop an effective atom-dimer Hamiltonian that accurately captures resonant effects. The lattice-induced resonances lead to the formation of bound states simultaneously above and below the scattering continuum and significantly modify the curvature of the dimer dispersion relation. The nature of the atom-dimer coupling depends strongly on the parity of the dimer state leading to a novel coupling in the case of negative parity dimers.

Combined effect of periodic gates and external fields on the diffusion coefficient of a single particle.

A general analytical expression to describe the diffusion of a single particle in a one-dimensional lattice with periodically distributed gates of lifetime (tau) and while under the influence of a constant external field is calculated. A formulation based on a microscopic model and a diffusion relaxation condition is used to derive an equation for the diffusion coefficient as a function of the concentration of gates (c), the lifetime (tau) of such gates, and the strength of the external field (p). The theory is compared against Monte Carlo simulations, and limiting cases are used to reproduce previously published results on a variety of phenomena.

Diffusion of two repulsive particles in a one-dimensional lattice

The problem of the lattice diffusion of two particles coupled by a contact repulsive interaction is solved by finding analytical expressions of the two-body probability characteristic function. The interaction induces anomalous drift with a vanishing velocity, the average coordinate of each particle growing at large times as $t^{1/2}$. The leading term of the mean square dispersions displays normal diffusion, with a diffusion constant made smaller by the interaction by the non-trivial factor $1-1/\pi$. Space continuous limit taken from the lattice calculations allows to establish connection with the standard problem of diffusion of a single fictitious particle constrained by a totally reflecting wall. Comparison between lattice and continuous results display marked differences for transient regimes, relevant with regards to high time resolution experiments, and in addition show that, due to slowly decreasing subdominant terms, lattice effects persist even at very large times.

Annihilation and Coagulation Reactions in Low-Dimensional Substrata: Effects of Probability of Reaction and Short Range Interactions

We study the bimolecular coagulation A + A → A and annihilation A + A → 0 reactions with diffusion and probability p of reaction in one dimension and annihilation reaction with diffusion and short range interactions in two-dimensional fractal and nonfractal percolation clusters. In these cases there is no input of particles and the particle density Ρ decays as a function of time t. The effect of p and interactions become relevant at short times. In one dimension analytical approximations of the density for coagulation and annihilation reactions are found which agree very well with Monte Carlo results of Ρ(t) for all times and for small values of p. For percolation clusters and large values of the nearest-neighbor repulsive interaction Ubetween particles, we find that a simple mean-field approximation works at short times. The length of the interval where this approximation holds increases as U increases. For the case of repulsive nearest-neighbor with attractive next nearest-neighbor interactions we find an exponential decay of the density at short times. We also analyze the steady-state regime of annihilation and coagulation reactions ofimmobile reactants with reaction probability and input of particles in a one-dimensional lattice. Analytical approximations of the particle density and of the n-particle correlations are obtained. The limit of zero input rate is solved exactly. The analytical results were confirmed by Monte Carlo simulations.

Quantum dynamics of a nonintegrable system

The quantum motion of a periodically kicked rotator is shown to be related to Anderson's problem of motion of a quantum particle in a one-dimensional lattice in the presence of a static-random potential. Classically, the first problem is nonintegrable and, for certain values of the parameters, exhibits chaos and diffusion in phase space; in the second problem, diffusion takes place in configuration space. Quantum phase interference, however, is known to suppress diffusion in Anderson's problem and to produce quasiperiodic motion. By establishing a mapping between the two systems we show that a similar effect determines the dynamics of the quantum rotator. As a result its wave functions are localized in phase space and their time evolution is quasiperiodic. This result explains the quantum recurrences and boundedness of the energy found in recent numerical work.

ON LINEAR WIDTHS OF SOBOLEV CLASSES AND CHAINS OF EXTREMAL SUBSPACES

The linear and trigonometric -diameters of the class in are calculated in this paper. For the linear diameter it is proved that, when and , This formula, together with the known results for other , finishes the solution of the problem of asymptotic computation of the linear diameters for the Sobolev classes in the one-dimensional periodic case when .Bibliography: 28 titles.