Single-well Potential Research Articles

Theory of inelastic confinement-induced resonances due to the coupling of center-of-mass and relative motion

A detailed study of the anharmonicity-induced resonances caused by the coupling of center-of-mass and relative motion is presented for a system of two ultracold atoms in single-well potentials. As has been confirmed experimentally, these inelastic confinement-induced resonances are of interest, since they can lead to coherent molecule formation, losses, and heating in ultracold atomic gases. A perturbative model is introduced to describe the resonance positions and the coupling strengths. The validity of the model and the behavior of the resonances for different confinement geometries are analyzed in comparison with exact numerical ab initio calculations. While such resonances have so far only been detected for large positive values of the $s$-wave scattering length, it is found that they are present also for negative $s$-wave scattering lengths, i. e. for attractive interactions. The possibility to coherently tune the resonances by a variation of the external confinement geometry might pave the way for coherent molecule association where magnetic Feshbach resonances are inaccessible.

Proton transfer in the molecular complexes of phosphorus acids with DMSO

Proton transfer processes in various H-bonded complexes of phosphoric (H3PO4), phosphorous (H3PO3) and methylphosphonic (H2MePO3) acids with dimethyl sulfoxide (DMSO), i.e., (acid)2–DMSO, acid–(DMSO) n for n = 1, 2 and H3PO4–(DMSO)3 have been studied. The potential energy surface (PES) for proton transfer was investigated using the B3LYP level of theory, and the solvent effect on the PES was included using the conductor polarized continuum model. The energy curves for proton transfer from acid to DMSO oxygen atom in all investigated complexes represent single-well potentials, if no constraints are imposed on the system. The PES obtained by optimizing the geometry of each complex at fixed O···O distances has two nonsymmetric minima with respect to the energy barrier. In these cases, the distance at which a barrier starts to rise is slightly different. The energy barrier for proton transfer in all considered complexes increases in the series of acids: H3PO4 < H3PO3 < H2MePO3. The energies associated with proton transfer in the complexes of all investigated acid with DMSO become higher with increasing number of DMSO molecules. For all complexes, the effect of DMSO environment favors a proton transfer.

Mechanical manifestations of bursting oscillations in slowly rotating systems

This study is concerned with certain mechanical systems that comprise discrete masses moving along slowly rotating objects. The corresponding equation of relative motion is derived, with the rotating motion creating slowly varying external excitation. Depending on the system parameters, two cases are distinguished: two-well and single-well potential, i.e. the Duffing bistable oscillator and a pure cubic oscillator. It is illustrated that both systems can exhibit bursting oscillations, consisting of fast oscillations around the slow flow. Their mechanisms are explained in terms of bifurcation theory: the first one with respect to the existence of certain saddle-node bifurcation points, and the second one by creation of a certain hysteresis loop. The exact expressions for the slow flow are derived, in the first case as a discontinuous curve, and in the second one as a continuous curve. The influence of the excitation magnitude, which is a potential control parameter, on the characteristics of bursting oscillations is numerically illustrated.

Effects of confinement for single-well potentials

We study bound states generated by a unique potential minimum in the situation where the system is strongly confined to a bounded region containing the minimum (by imposing Dirichlet boundary conditions). In this case, the eigenvalues of the confined system differ from those of the unconfined system by an exponentially small quantity in the semiclassical limit. An asymptotic expansion for this shift is established. The formulas are evaluated explicitly for the harmonic oscillator and an application to the Coulomb potential at a fixed angular momentum is given.

Solitons and vortices in nonlinear potential wells

We consider self-trapping of topological modes governed by the one- and two-dimensional (1D and 2D) nonlinear-Schrödinger/Gross–Pitaevskii equation with effective single- and double-well (DW) nonlinear potentials induced by spatial modulation of the local strength of the self-defocusing nonlinearity. This setting, which may be implemented in optics and Bose–Einstein condensates, aims to extend previous studies, which dealt with single-well nonlinear potentials. In the 1D setting, we find several types of symmetric, asymmetric and antisymmetric states, paying attention to scenarios of the spontaneous symmetry breaking. The single-well model is extended by including rocking motion of the well, which gives rise to Rabi oscillations between fundamental and dipole modes. Analysis of the 2D single-well setting gives rise to stable modes in the form of ordinary dipoles, vortex–antivortex dipoles (VADs), and vortex triangles (VTs), which may be considered as produced by spontaneous breaking of the axial symmetry. The consideration of the DW configuration in 2D reveals diverse types of modes built of components trapped in the two wells, which may be fundamental states and vortices with topological charges m = 1 and 2, as well as VADs (with m = 0) and VTs (with m = 2).

Origin of Dissimilar Single-Molecule Magnet Behavior of Three MnII(2)MoIII Complexes Based on [MoIII(CN)7]4- Heptacyanomolybdate: Interplay of MoIII-CN-MnII Anisotropic Exchange Interactions.

The origin of contrasting single-molecule magnet (SMM) behavior of three MnII2MoIII complexes based on [MoIII(CN)7]4– heptacyanomolybdate is analyzed; only the apical Mn2Mo isomer exhibits SMM properties with Ueff = 40.5 cm(-1) and TB = 3.2 K, while the two equatorial isomers are simple paramagnets [Qian, K.; J. Am. Chem. Soc. 2013, 135, 13302]. A microscopic theory of anisotropic spin coupling between orbitally degenerate [MoIII(CN)7](4-) complexes (pentagonal bipyramid) and bound MnII ions is developed. It is shown that the [MoIII(CN)7](4-) complex has a unique property of uniaxial anisotropic spin coupling in the apical and equatorial MoIII-CN-MnII pairs, H̑eff = -Jxy(SMoxSMnx + SMoySMny) - JzSMozSMnz, regardless of their actual low symmetry. The difference in the SMM behavior originates from a different ratio between the anisotropic exchange parameters Jz and Jxy for the apical and equatorial Mo-CN-Mn groups. In the apical Mn2Mo isomer, an Ising-type anisotropic spin coupling (Jz = -34, Jxy = -11 cm(-1)) produces a double-well potential of spin states resulting in SMM behavior. Exchange anisotropy of an xy-type (|Jz| < |Jxy|) in the equatorial Mn2Mo isomers results in a single-well potential with no SMM properties. The prospects of anisotropic uniaxial spin coupling in engineering of high Ueff and TB values are discussed.

Specific features of the properties of lattice vibrations in the Cd1–y Hg y Te alloy formed by the CdTe semiconductor and the HgTe semimetal

The specific features of the properties of Hg-Te lattice vibrations have been considered within the percolation model of a mixed Cd1–yHgyTe crystal (alloy), in which the composite medium is formed by the alloy regions enriched in HgTe and CdTe with different Hg-Te vibrational states and, hence, with different characteristics of vibrational modes (frequency and damping parameter). In the HgTe-enriched alloy, the properties of Hg-Te lattice vibrations are determined by a double-well lattice potential for the Hg atom with the off-center localization at low temperatures. In the CdTe-enriched alloys, the properties of Hg-Te lattice vibrations are determined by a single-well lattice potential for the Hg atom in the center of the anion tetrahedron, and the Hg-Te vibrational states are localized. For the HgTe-enriched alloy, the Hg-Te vibrational states are extended. In the percolation scheme of the transformation of the vibrational spectrum of the Cd1 - yHgyTe alloy with the composition y, the “137 cm−1” mode is a split-off mode of Hg-Te vibrations, or one of the modes of the percolation doublet of HgTe-like vibrations.

Solitons in a nonlinear Schrödinger equation withPT-symmetric potentials and inhomogeneous nonlinearity: Stability and excitation of nonlinear modes

We report branches of explicit expressions for nonlinear modes in parity-time $(\mathcal{PT})$-symmetric potentials of several types. For the single-well and double-well potentials the found solutions are two-parametric and appear to be stable even when the $\mathcal{PT}$ symmetry of respective underlying linear models is broken. Based on the examples of these solutions we describe an algorithm of excitation of a stable nonlinear mode in a model whose linear limit is unstable. The method is based on the adiabatic change of the control parameter driving the mode along a branch bifurcating from a stable linear mode. The suggested algorithm is confirmed by extensive numerical simulations.

The gap between the first two eigenvalues of Schrödinger operators with single-well potential

In this paper the Schrödinger operators endowed with Neumann or Dirichlet boundary conditions, where the potential q ∈ L1[0, π] is single-well with transition point a ∈ (0, π), are considered. Suppose {νn}n ≥ 0 and {λn}n ≥ 1 are the set of eigenvalues for Neumann or Dirichlet problem, respectively, and the potential q(x) possesses an additional condition, we present estimates of lower bound for the differences ν1−ν0 and λ2−λ1, respectively. The results show that the energy transition for quantum particles jumping to the ground state is related to transition point a.

Nonlinear analysis of energy harvesting systems with fractional order physical properties

An electromechanical energy harvesting system with a fractional order current–voltage relationship for the electrical circuit and fractional power law in the restoring force of its mechanical part is considered to act as an energy harvester. Our results show that, under a single-well potential configuration, for a small amplitude of the perturbation, as the order of derivative increases, the resonant amplitude of mechanical vibration decreases while the bending degree (hardening case) remains fairly constant. For a large amplitude of the perturbation, the output power is increased due to the hardening effects. Under a double-well configuration, the fractional power stiffness \(\alpha \) strongly affects the crossing well dynamics (large amplitude motion) and consequently the output electrical power. The harvested electric power appears to be maximal for deterministic and random excitation for small \(\alpha \). High-level noise intensity is found to reduce the output power in the region of resonance and surprisingly increases the output in other regions of \(\alpha \). For sufficiently large amplitude of harmonic excitation, this effect is realized in a stochastic resonance.

Stationary states in two-dimensional systems driven by bivariate Lévy noises.

Systems driven by α-stable noises could be very different from their Gaussian counterparts. Stationary states in single-well potentials can be multimodal. Moreover, a potential well needs to be steep enough in order to produce stationary states. Here it is demonstrated that two-dimensional (2D) systems driven by bivariate α-stable noises are even more surprising than their 1D analogs. In 2D systems, intriguing properties of stationary states originate not only due to heavy tails of noise pulses, which are distributed according to α-stable densities, but also because of properties of spectral measures. Consequently, 2D systems are described by a whole family of Langevin and fractional diffusion equations. Solutions of these equations bear some common properties, but also can be very different. It is demonstrated that also for 2D systems potential wells need to be steep enough in order to produce bounded states. Moreover, stationary states can have local minima at the origin. The shape of stationary states reflects symmetries of the underlying noise, i.e., its spectral measure. Finally, marginal densities in power-law potentials also have power-law asymptotics.

Intricate internal rotation surface and fundamental infrared transitions of the n-propyl radical.

The potential energy surface for methylene hindered internal rotation is examined for the n-propyl radical, a molecule fundamental to combustion chemistry. Six stationary points are identified, and four of them are unique: 1, 2, TS1, and TS2. The remaining two structures 1' and TS1' are mirror images with respect to 1 and TS1. Focal point analysis, converged to the complete basis set limit of coupled-cluster theory with single, double, triple, and perturbative quadruple excitations [CCSDT(Q)], is employed to obtain the relative energies of these structures. A one-dimensional potential energy surface (PES) is constructed by explicitly mapping out a distinguished reaction path via constrained geometry optimizations. A "double-well" feature is observed on the electronic PES, but under the adiabatic approximation, the enthalpic (0 K) PES becomes a regular single-well potential with the expected 180° periodicity. The corresponding one-dimensional vibrational Schrödinger equation is solved using the Cooley-Numerov approach to obtain vibrational states of the methylene torsional motion. The predicted barrier for internal rotation is 105.5 and 137.2 cm(-1) for the electronic and enthalpic surfaces, respectively. Anharmonic (fundamental) vibrational frequencies are predicted for structure 1 using second-order vibrational perturbation theory, and the band origins for 11 modes are reported. Comparison with previous electron spin resonance and infrared spectroscopic work, in addition to other theoretical investigations, is made where possible.

Symmetry Breaking and a Dynamical Property of a Dipolar Bose–Einstein Condensate in a Triple-Well Potential

We investigate the properties of a three-dimensional (3D) dipolar Bose–Einstein condensate (BEC) in a triple-well potential. Symmetry breaking and self-trapping (SBST) phenomena are demonstrated for 52Cr atoms in the 3D triple-well potential using numerical solutions of the Gross–Pitaevskii (GP) equation. The results show that the ground-state density distributions are affected markedly by the long-range nature and anisotropy of the dipolar interaction and the isotropic s-wave contact interaction. In addition, the dynamical picture of the SBST induced by a gradual transformation of the single-well potential into a triple-well one is also illustrated. SBST phenomena are also presented by adjusting the height of the middle potential well when the dipole orientation is aligned in the dipole axis.

Comparison Between Maxwell–Schrödinger and Maxwell–Newton Hybrid Simulations for Multi-Well Electrostatic Potential

A novel hybrid approach to the dynamics of electron interacting with time-dependent electromagnetic fields, namely, the Maxwell-Schrodinger approach, has been employed to study a system of electron confined in single- and multi-well electrostatic potentials subjected to pulsed laser fields. A comparison of the results of simulation to those obtained by the conventional Maxwell-Newton approach has been made by calculating the time response of the current density and the time-evolution of the electric field at an observation point. The results obtained by these two distinct approaches agree very well for the single-well potential while disagree qualitatively for the multi-well potential. This clearly demonstrates limitation of applicability of the conventional Maxwell-Newton scheme to study electron dynamics in electromagnetic fields.

Tunnel splitting of the spectrum and bilocalization of eigenfunctions in an asymmetric double well

We consider the one-dimensional stationary Schrodinger equation with a smooth double-well potential. We obtain a criterion for the double localization of wave functions, exponential splitting of energy levels, and the tunneling transport of a particle in an asymmetric potential and also obtain asymptotic formulas for the energy splitting that generalize the formulas known in the case of a mirror-symmetric potential. We consider the case of higher energy levels and the case of energies close to the potential minimums. We present an example of tunneling transport in an asymmetric double well and also consider the problem of tunnel perturbation of the discrete spectrum of the Schrodinger operator with a single-well potential. Exponentially small perturbations of the energies occur in the case of local potential deformations concentrated only in the classically forbidden region. We also calculate the leading term of the asymptotic expansion of the tunnel perturbation of the spectrum.

Mechanical control of heat conductivity in molecular chains

We discuss a possibility to control heat conductivity in molecular chains by means of external mechanical loads. To illustrate such possibilities we consider first well-studied one-dimensional chain with degenerate double-well potential of the nearest-neighbor interaction. We consider varying lengths of the chain with fixed number of particles. Number of possible energetically degenerate ground states strongly depends on the overall length of the chain, or, in other terms, on average length of the link between neighboring particles. These degenerate states correspond to mechanical equilibria; therefore, one can say that formation of such structures mimics a process of plastic deformation. We demonstrate that such modification of the chain length can lead to quite profound (almost fivefold) reduction of the heat conduction coefficient. Even more profound effect is revealed for a model with a single-well nonconvex potential. It is demonstrated that in a certain range of constant external forcing, this model becomes effectively double-well and has a multitude of possible states of equilibrium for fixed value of the external load. Due to this degeneracy, the heat-conduction coefficient can be reduced by two orders of magnitude. We suggest a mechanical model of a chain with periodic double-well potential, which allows control of the heat transport. The models considered may be useful for description of heat transfer in biological macromolecules and for control of the heat transport in microsystems. The possibility of the heat transport control in more realistic three-dimensional systems is illustrated by simulation of a three-dimensional model of polymer α-helix. In this model, the mechanical stretching also brings about the structural inhomogeneity and, in turn, to essential reduction of the heat conductivity.

Design of a surface electrode trap for parallel ion strings

We report a linear surface-electrode trap that can be used to form parallel ion strings. By adjusting the balance of the radio-frequency (RF) voltages applied to central and RF electrodes, the RF pseudopotential can be varied from single-well to double-well in the radial direction. Ions located on two parallel lines of the RF potential null are in principle free from excess micromotion if appropriate static voltages are applied. Calcium ions were trapped for the evaluation of the designed electrode. An ion string in the single-well potential and two ion strings in the double-well potential were observed by changing the RF voltages. Such traps could be used for quantum simulation of coupled spin systems.

Pitchfork bifurcation and vibrational resonance in a fractional-order Duffing oscillator

The pitchfork bifurcation and vibrational resonance are studied in a fractional-order Duffing oscillator with delayed feedback and excited by two harmonic signals. Using an approximation method, the bifurcation behaviours and resonance patterns are predicted. Supercritical and subcritical pitchfork bifurcations can be induced by the fractional-order damping, the exciting high-frequency signal and the delayed time. The fractional-order damping mainly determines the pattern of the vibrational resonance. There is a bifurcation point of the fractional order which, in the case of double-well potential, transforms vibrational resonance pattern from a single resonance to a double resonance, while in the case of single-well potential, transforms vibrational resonance from no resonance to a single resonance. The delayed time influences the location of the vibrational resonance and the bifurcation point of the fractional order. Pitchfork bifurcation is the necessary condition for the double resonance. The theoretical predictions are in good agreement with the numerical simulations.

Symmetry breaking and a dynamical property of a dipolar Bose–Einstein condensate in a double-well potential

Abstract We investigate the properties of a three-dimensional (3D) dipolar Bose–Einstein condensate (BEC) in a double-well potential (DWP). The symmetry breaking and self-trapping (SBST) phenomena that the original symmetric ground state is replaced by a new asymmetric one and localized in one of the wells are demonstrated for Dy 164 atoms in the 3D DWP by means of numerical solutions of the Gross–Pitaevskii (GP) equation. The results show that the SBST properties are affected dramatically by the magnetization direction for purely dipolar BEC. The SBST under various scattering lengths are also studied. In addition, the dynamical picture of the SBST induced by a gradual transformation of the single-well potential into a double-well is also illustrated.

Specific features of the properties of Hg-Te lattice vibrations in Cd1 − y Hg y Te alloys enriched with CdTe

In alloys (solid solutions) Cd1 − yHgyTe enriched with CdTe, the properties of Hg-Te lattice vibrations are determined by a single-well lattice potential for the Hg atom located at the center of the anion tetrahedron. The HgTe-enriched alloys retain the anomalous properties of Hg-Te lattice vibrations, which are determined by a double-well lattice potential for the Hg atom located in the off-center position at low temperatures; a shallower well of the potential for the Hg atom located at the center of the anion tetrahedron is filled upon thermal activation. The polarizability of the ion pair in the TO mode of Hg-Te lattice vibrations with the Hg atom located at the center of the anion tetrahedron for the CdTe-enriched alloys considerably exceeds the polarizability of the pair with the off-center Hg atom for the HgTe-enriched alloys.