One-dimensional Periodic Potential Research Articles

Regularized Scarf potentials: energy band structure and supersymmetry

The singular one-dimensional periodic Scarf potential is regularized by means of one-parameter square well counter-terms. It is shown that the regularized spectrum converges formally to the conventional Scarf energy bands for specific values of the parameter. The behaviour of the regularizations under supersymmetric transformations is also investigated; this is a key point for the algebraic solvability of the regularized Scarf potential.

Pinning of vortices in a Bose-Einstein condensate by an optical lattice.

We consider the ground state of vortices in a Bose-Einstein condensate. We show that turning on a weak optical periodic potential leads to a transition from the triangular Abrikosov vortex lattice to phases where the vortices are pinned by the optical potential. We discuss the phase diagram of the system for a two-dimensional optical periodic potential with one vortex per optical lattice cell. We also discuss the influence of a one-dimensional optical periodic potential on the vortex ground state. The latter situation has no analog in other condensed-matter systems.

Umklapp collisions and center-of-mass oscillations of a trapped Fermi gas.

Starting from the Boltzmann equation, we study the center-of-mass oscillation of a harmonically trapped normal Fermi gas in the presence of a one-dimensional periodic potential. We show that for values of the Fermi energy above the first Bloch band the center of mass motion is overdamped in the collisional regime due to umklapp processes. This should be contrasted with the behavior of a superfluid where one instead expects the occurrence of persistent Josephson-like oscillations.

Theory of minigap plasmons in a two-dimensional electron gas subjected to a one-dimensional periodic potential

We investigate the plasmon excitations in a two-dimensional (2D) electron gas subjected to a one-dimensional (1D) weak periodic potential. We derive and discuss the dispersion relations for both intrasubband and intersubband excitations within the framework of Bohm-Pines' random-phase approximation. For such an anisotropic system with spatially modulated charge density, we observe a splitting of the 2D plasmon dispersion. The splitting is caused by the superlattice effect of the charge-density modulation on the collective excitation spectrum. Several illustrative examples are presented on the computed plasmon excitation energy as a function of the propagation vector as well as the Bloch vector. We also discuss how the tunneling and the potential amplitude influence the minigaps and the plasmon excitations.

Minigap plasmons in a two-dimensional electron gas subjected to a one-dimensional periodic potential

We investigate the plasmon excitations in a two-dimensional electron gas subjected to a one-dimensional weak periodic potential. We derive and discuss the dispersion relations for both intrasubband and intersubband excitations within the framework of Bohm–Pines' random-phase approximation. For such an anisotropic system with spatially modulated charge density, we observe a splitting of the 2D plasmon dispersion. The splitting is caused by the superlattice effect of the charge-density modulation on the collective excitation spectrum. We also discuss how the tunneling and the potential amplitude affect the plasmon excitations.

Dynamics of Bloch oscillations

We study the dynamics of Bloch oscillations in a one-dimensional periodic potential plus a (relatively weak) static force. The tight-binding and single-band approximations are analysed in detail, and also in a classicalized version. A number of numerically exact results obtained from wavepacket propagation are analysed and interpreted in terms of the tight-binding and single-band model, both in co-ordinate and momentum space.

Quantum and classical aspects of activated surface diffusion

The specific features of classical and quantum activated diffusion of a particle over a surface, modeled by a one-dimensional periodic potential, are analyzed in the low-to-moderate friction limit, in which the kinetics of the process is determined by the energy relaxation. Different models for the energy transition probability are considered with special emphasis on the exponential model which leads to significant simplification of the problem. New expressions are presented for the escape rate, mean squared path length and diffusion coefficient of an activated particle whose energy exchange dynamics is described by an exponential kernel. A universal behavior pj∼j−3/2 exp(−Δj) (where Δ depends only on the friction strength) is found for the distribution pj of diffusive hopping lengths j. It is identical for classical and quantum activated diffusion, does not depend on the details of the model used or on the characteristic energy loss of the particle to the bath. Quantum effects (tunneling) demonstrate themselves only in the absolute values of hopping rates, which for the weak damping regime considered in this paper, lead to a decrease of rates and, thus, the diffusion coefficient. This quantum suppression of diffusion is shown to be equivalent to an effective increase in the activation barrier, caused by quantum above barrier-reflection.

Formula omitted]-symmetry in one-dimensional quantum periodic potentials

A complete band structure is shown to exist for one-dimensional periodic non-Hermitian potentials exhibiting PT -symmetry. The full band spectrum is exactly given and some of its properties discussed, specially those concerning the role of the imaginary parameters of the couplings. Infinite periodic arrays constructed with finite chains each one made of N different ultralocal couplings are used as models of this one-dimensional complex quantum wire.

Band structure, elementary excitations, and stability of a Bose-Einstein condensate in a periodic potential

We investigate the band structure of a Bose-Einstein condensate in a one-dimensional periodic potential by calculating stationary solutions of the Gross-Pitaevskii equation which have the form of Bloch waves. We demonstrate that loops ("swallow tails") in the band structure occur both at the Brillouin zone boundary and at the center of the zone, and they are therefore a generic feature. A physical interpretation of the swallow tails in terms of periodic solitons is given. The linear stability of the solutions is investigated as a function of the strength of the mean-field interaction, the magnitude of the periodic potential, and the wave vector of the condensate. The regions of energetic and dynamical stability are identified by considering the behavior of the Gross-Pitaevskii energy functional for small deviations of the condensate wave function from a stationary state. It is also shown how for long-wavelength disturbances the stability criteria may be obtained within a hydrodynamic approach.

Dynamic structure factor of a Bose-Einstein condensate in a one-dimensional optical lattice

We study the effect of a one-dimensional periodic potential on the dynamic structure factor of an interacting Bose-Einstein condensate at zero temperature. We show that, due to phononic correlations, the excitation strength toward the first band develops a typical oscillating behavior as a function of the momentum transfer, and vanishes at even multiples of the Bragg momentum. The effects of interactions on the static structure factor are found to be significantly amplified by the presence of the optical potential. Our predictions can be tested in stimulated photon scattering experiments.

Characteristic times and current inversion in inertial ratchets with colored thermal noise

We perform a systematical numerical study of the statistical behaviour of a particle immersed in a colored thermal bath moving with friction in a one-dimensional asymmetric and periodic potential and subjected to an external sinusoidal force. To this end, we characterize the problem in terms of a set of three characteristic times and two strength variables that determine the evolution of the system. We solve the corresponding equations of motion by using an iterative approach. We analyze different bath temperature regimes and study the appearance of a current. We also determine whether the evolution is diffusive or not. We find that current inversion is a very robust phenomenon appearing in an oscillatory fashion as a function of the frequency of the time-dependent external force.

Supersymmetry and solvable periodic potentials

We use the formalism of supersymmetric quantum mechanics to enlarge considerably the limited class of analytically solvable one-dimensional periodic potentials. In particular, we derive and discuss the energy-band structure of the Lame potentials pm sn2 (x, m) and associated Lame potentials pm sn2 (x, m) + qm cn2(x, m)/dn2 (x, m), both of which involve Jacobi elliptic functions with modulus parameter m. We find several new analytic expressions for band-edge energies and wave functions. The supersymmetric partners of Lame and associated Lame potentials constitute even more new solvable potentials with exactly the same energy-band structure.

Phase behavior of colloidal molecular crystals on triangular light lattices.

We investigate the melting process of a two-dimensional charge-stabilized colloidal system in the presence of a triangular substrate potential. This potential is formed by an optical interference pattern that allows the substrate strength to be varied continuously. By means of an additional scanned optical tweezer the particle density can be adjusted to different numbers m of colloidal particles per substrate minima; here we concentrate on the case of trimers, i.e., m = 3. Because trimers exhibit additional internal degrees of rotational freedom, the phase behavior of such a system is very different from homogeneous or one-dimensional periodic substrate potentials.

Optical properties of remotely doped AlAs/GaAs coupled quantum wire arrays. I. Periodic quantum confinement and localization of minority carriers

We perform low-temperature optical experiments on remotely doped lateral superlattices epitaxially grown on vicinal GaAs substrates. This quantum wire system provides a degenerate electron gas of density n s ≃ 8 X 10 1 1 cm - 2 subjected to a tunable one-dimensional periodic potential of amplitude ≃20 meV and period ≃30 nm, i.e., at the scale of its Fermi energy (E F ≃25 meV) and Fermi wavelength (λ F ≃30 nm). We show that the one-dimensional quantum confinement strongly affects both photoluminescence and photoluminescence excitation properties of the degenerate electron system. Through a careful analysis of the optical emission and absorption linear polarization spectra together with their temperature dependence, we distinguish between features due to (i) the periodic one-dimensional confinement and (ii) the localization anisotropy of photocreated valence-band carriers.

Magnetoresistance in a GaAs–AlGaAs two-dimensional electron gas with one-dimensional periodic potential fabricated by AFM local anodization

We report the application of the local anodization technique by using an atomic force microscope to GaAs-AlGaAs two-dimensional electron system. We have measured the magnetoresistance in samples with a one-dimensional periodic potential induced by this method, to investigate the properties of the introduced potential. The observed Weiss oscillations and the negative magnetoresistance are analyzed in samples with different ID potential strength controlled by the anodization conditions. The electron localization observed in the samples may be attributed to the inhomogeneity of the potential induced by the local anodization.

Some exact results for mid-band and zero band-gap states of associated Lamé potentials

Applying certain known theorems about one-dimensional periodic potentials, we show that the energy spectrum of the associated Lamé potentials, a(a+1)msn2(x,m)+b(b+1)m cn2(x,m)/dn2(x,m), consists of a finite number of bound bands followed by a continuum band when both a and b take integer values. Further, if a and b are unequal integers, we show that there must exist some zero band-gap states, i.e., doubly degenerate states with the same number of nodes. More generally, in case a and b are not integers, but either a+b or a−b is an integer (a≠b), we again show that several of the band-gaps vanish due to degeneracy of states with the same number of nodes. Finally, when either a or b is an integer and the other takes a half-integral value, we obtain exact analytic solutions for several mid-band states.

New exactly solvable supersymmetric periodic potentials

Using the formalism of supersymmetric quantum mechanics, we give an exact solution for a family of one-dimensional periodic potentials, which are the supersymmetric partners of the potential proportional to the trigonometric function cos(2x) such that the Schrodinger equation for this potential is named the Mathieu equation mathematically. We show that the new potentials are distinctly different from their original ones. However, both have the same energy band structure. All the potentials obtained in this paper are free of singularities.

Low-energy electron transmission experiments on graphite

Low-energy-electron transmission (LEET) spectra were measured for graphite using electron energies below 30 eV. The observed LEET spectra have broadened square-wave-like features, and comparison with the conduction-band density of states above the vacuum level measured by ultraviolet photoemission and inverse photoemission spectroscopies indicated that the conduction-band density of states was not observed in the LEET spectra except band gaps. It is concluded that band gaps are more strongly reflected in LEET spectra than other features in the density-of-states. It is expected that electron-interference effects along the surface normal dominate LEET features, even for a very thick sample, where the energy dependence of electron-transmission probability through a one-dimensional periodic potential along the surface normal does not reflect the variation of the density of states (except band gaps).

THE ROTATION NUMBER APPROACH TO EIGENVALUES OF THE ONE-DIMENSIONAL p-LAPLACIAN WITH PERIODIC POTENTIALS

The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function ρ(λ) has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval ρ−1(n/2) in ℝ yield the corresponding periodic or anti-periodic eigenvalues. However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained is a partial generalization of the spectrum theory of the one-dimensional Schrödinger operators with periodic potentials.

Novel phases and reentrant melting of two-dimensional colloidal crystals

We investigate two-dimensional (2D) melting in the presence of a one-dimensional (1D) periodic potential as, for example, realized in recent experiments on 2D colloids subjected to two interfering laser beams. The topology of the phase diagram is found to depend primarily on two factors: the relative orientation of the 2D crystal and the periodic potential troughs, which selects a set of Bragg planes running parallel to the troughs, and the commensurability ratio p=a(')/d of the spacing a(') between these Bragg planes to the period d of the periodic potential. The complexity of the phase diagram increases with the magnitude of the commensurabilty ratio p. Rich phase diagrams, with "modulated liquid," "floating," and "locked floating" solid and smectic phases are found. Phase transitions between these phases fall into two broad universality classes, roughening and melting, driven by the proliferation of discommensuration walls and dislocations, respectively. We discuss correlation functions and the static structure factor in these phases, and make detailed predictions about the universal features close to the phase boundaries. We predict that for charged systems with highly screened short-range interactions, these melting transitions are generically reentrant as a function of the strength of the periodic potential, a prediction that is in accord with recent 2D colloid experiments. Implications of our results for future experiments are also discussed.