Parabolic Trap Research Articles

Quantum degeneracy and spin entanglement in ideal quantum gases

Quantum degeneracy is the central many-body feature of ideal quantum gases stemming from quantum mechanics. In this work we address its relationship to the most fundamental form of non-classicality in many-body system, i.e. many-body entanglement. We aim at establishing a quantitative link between quantum degeneracy and entanglement in spinful ideal gases, using entanglement witness criteria based on the variance of the collective spin of the spin ensemble. We show that spin-1/2 ideal Bose gases do not possess entanglement which can be revealed from such entanglement criteria. On the contrary, ideal spin-1/2 Fermi gases exhibit spin entanglement revealed by the collective-spin variances upon entering quantum degeneracy, due to the formation of highly non-local spin singlets. We map out the regime of detectable spin entanglement for Fermi gases in free space as well as in a parabolic trap, and probe the robustness of spin entanglement to thermal effects and spin imbalance. Spin entanglement in degenerate Fermi gases is amenable to experimental observation using state-of-the-art spin detection techniques in ultracold atoms.

Dynamical self-trapping of two-dimensional binary solitons in cross-combined linear and nonlinear optical lattices.

Dynamical and self-trapping properties of two-dimensional (2D) binary mixtures of Bose-Einstein condensates in cross-combined lattices, consisting of a one-dimensional (1D) linear optical lattice (LOL) in the x direction for the first component and a 1D nonlinear optical lattice (NOL) in the y direction for the second component, are analytically and numerically investigated. The existence and stability of 2D binary matter wave solitons in these settings are demonstrated both by variational analysis and by direct numerical integration of the coupled Gross-Pitaevskii equations. We find that in the absence of the NOL, binary solitons, stabilized by the action of the 1D LOL and by the attractive intercomponent interaction, can freely move in the y direction. In the presence of the NOL, we find, quite remarkably, the existence of threshold curves in the parameter space separating regions where solitons can move from regions where the solitons become dynamically self-trapped. The mechanism underlying the dynamical self-trapping phenomenon (DSTP) is qualitatively understood in terms of a dynamical barrier induced by the NOL, similar to the Peirls-Nabarro barrier of solitons in discrete lattices. DSTP is numerically demonstrated for binary solitons that are put in motion both by phase imprinting and by the action of external potentials applied in the y direction. In the latter case, we show that the trapping action of the NOL allows one to maintain a 2D binary soliton at rest in a nonequilibrium position of a parabolic trap or to prevent it from falling under the action of gravity. Possible applications of the results are also briefly discussed.

Appropriateness of HSE06 exchange-correlation energies for post transition metal sesquioxides Ga2O3 and In2O3: Peculiarities in electronic states, momentum densities and thermoluminescence

Compton scattering measurements using 59.54 keV γ-rays have been reported for sesquioxides Ga2O3 and In2O3. To precisely compare the experimental and theoretical momentum densities, different exchange and correlation potentials within density functional theory and its hybridisation with Hartree-Fock (PBESOL0 and HSE06) have been considered. χ2 fitting analysis depicts HSE06 as a better choice for both sesquioxides. Present band structures, density of states and UV–vis measurements unambiguously support the use of HSE06 for such oxides and resolves issues related to appropriateness of exchange and correlation potentials. Higher covalent character in Ga2O3 than In2O3 using Mulliken's population is well explained by equal-valence-electron-density scaled CPs. Most prominent and firstly reported features in band structures of In2O3 are presence of parabolic trapping centres at Γ, H and N points with three direct band gaps of similar energy (3.37 eV), which may lead to its higher luminescence properties including potential applications in thermoluminescence dosimeters.

Generating soliton trains through Floquet engineering

We study a gas of interacting ultracold bosons held in a parabolic trap in the presence of an optical lattice potential. Treating the system as a discretized Gross-Pitaevskii model, we show how Floquet engineering, by rapidly ``shaking'' the lattice, allows the ground state of the system to be converted into a train of bright solitons by inverting the sign of the hopping energy. We study how the number of solitons produced depends on the system's nonlinearity and the curvature of the trap, show how the technique can be applied in both the high- and low-driving-frequency regimes, and demonstrate the phenomenon's stability against noise. We conclude that the Floquet approach is a useful and stable method of preparing solitons in cold-atom systems.

Quasi-one-dimensional harmonically trapped quantum droplets

We theoretically consider effectively one-dimensional quantum droplets in a symmetric Bose-Bose mixture confined in a parabolic trap. We systematically investigate ground and excited families of localized trapped modes which bifurcate from eigenstates of the quantum harmonic oscillator as the number of particles departs from zero. Families of nonlinear modes have nonmonotonous behavior of chemical potential on the number of particles and feature bistability regions. Excited states are unstable close to the linear limit, but become stable when the number of particles is large enough. In the limit of large density, we derive a modified Thomas-Fermi distribution. Smoothly decreasing the trapping strength down to zero, one can dynamically transform the ground-state solution to the solitonlike quantum droplet, while excited trapped states break in several moving quantum droplets.

Enhanced Efficiency at Maximum Power in a Fock-Darwin Model Quantum Dot Engine.

We study the performance of an endoreversible magnetic Otto cycle with a working substance composed of a single quantum dot described using the well-known Fock-Darwin model. We find that tuning the intensity of the parabolic trap (geometrical confinement) impacts the proposed cycle's performance, quantified by the power, work, efficiency, and parameter region where the cycle operates as an engine. We demonstrate that a parameter region exists where the efficiency at maximum output power exceeds the Curzon-Ahlborn efficiency, the efficiency at maximum power achieved by a classical working substance.

A Method for the Dynamics of Vortices in a Bose-Einstein Condensate: Analytical Equations of the Trajectories of Phase Singularities

We present a method to study the dynamics of a quasi-two dimensional Bose-Einstein condensate which initially contains several vortices at arbitrary locations. The method allows one to find the analytical solution for the dynamics of the Bose-Einstein condensate in a homogeneous medium and in a parabolic trap, for the ideal non-interacting case. Secondly, the method allows one to obtain algebraic equations for the trajectories of the position of phase singularities present in the initial condensate along with time (the vortex lines). With these equations, one can predict quantities of interest, such as the time at which a vortex and an antivortex contained in the initial condensate will merge. For the homogeneous case, this method was introduced in the context of photonics. Here, we adapt it to the context of Bose-Einstein condensates, and we extend it to the trapped case for the first time. Also, we offer numerical simulations in the non-linear case, for repulsive and attractive interactions. We use a numerical split-step simulation of the non-linear Gross-Pitaevskii equation to determine how these trajectories and quantities of interest are changed by the interactions. We illustrate the method with several simple cases of interest, both in the homogeneous and parabolically trapped systems.

Theoretical and numerical evidence for the potential realization of the Peregrine soliton in repulsive two-component Bose-Einstein condensates

The present work is motivated by the recent experimental realization of the Townes soliton in an effective two-component Bose-Einstein condensate by B. Bakkali-Hassan et al. [Phys. Rev. Lett. 127, 023603 (2021)]. Here, we use a similar multicomponent platform to exemplify theoretically and numerically, within the mean-field Gross-Pitaevskii framework, the potential toward the experimental realization of a different fundamental wave structure, namely the Peregrine soliton. Leveraging the effective attractive interaction produced within the mixture's minority species in the immiscible regime, we illustrate how initialization of the condensate with a suitable power-law decaying spatial density pattern yields the robust emergence of the Peregrine wave in the absence and in the presence of a parabolic trap. We then showcase the spontaneous emergence of the Peregrine soliton via a suitably crafted wide Gaussian initialization, again both in the homogeneous case and in the trap scenario. It is also found that narrower wave packets may result in periodic revivals of the Peregrine soliton, while broader ones give rise to a cascade of Peregrine solitons arranged in a so-called Christmas-tree structure. Strikingly, the persistence of these rogue-wave structures is demonstrated in certain temperature regimes as well as in the presence of transversal excitations through three-dimensional computations in a quasi-one-dimensional regime. This proof-of-principle illustration is expected to represent a practically feasible way to generate and observe this rogue wave in realistic current ultracold atom experimental settings.

Breaking of Huygens–Fresnel principle in inhomogeneous Tomonaga–Luttinger liquids

Tomonaga–Luttinger liquids (TLLs) can be used to effectively describe one-dimensional quantum many-body systems such as ultracold atoms, charges in nanowires, superconducting circuits, and gapless spin chains. Their properties are given by two parameters, the propagation velocity and the Luttinger parameter. Here we study inhomogeneous TLLs where these are promoted to functions of position and demonstrate that they profoundly affect the dynamics: in general, besides curving the light cone, we show that propagation is no longer ballistically localized to the light-cone trajectories, different from standard homogeneous TLLs. Specifically, if the Luttinger parameter depends on position, the dynamics features pronounced spreading into the light cone, which cannot be understood via a simple superposition of waves as in the Huygens–Fresnel principle. This is the case for ultracold atoms in a parabolic trap, which serves as our main motivation, and we discuss possible experimental observations in such systems.

Ordering and Dynamics of Interacting Colloidal Particles under Soft Confinement

Confinement can induce substantial changes in the physical properties of macromolecules in suspension. Soft confinement is a particular class of restriction where the boundaries that constraint the particles in a region of the space are not well-defined. This scenario leads to a broader structural and dynamical behavior than observed in systems enclosed between rigid walls. In this contribution, we study the ordering and diffusive properties of a two-dimensional colloidal model system subjected to a one-dimensional parabolic trap. Increasing the trap strength makes it possible to go through weak to strong confinement, allowing a dimensional transition from two- to one-dimension. The non-monotonic response of the static and dynamical properties to the gradual dimensionality change affects the system phase behavior. We find that the particle dynamics are connected to the structural transitions induced by the parabolic trap. In particular, at low and intermediate confinement regimes, complex structural and dynamical scenarios arise, where the softness of the external potential induces melting and freezing, resulting in faster and slower particle diffusion, respectively. Besides, at strong confinements, colloids move basically along one direction, and the whole system behaves structurally and dynamically similar to a one-dimensional colloidal system.

Polarization angle dependence of the breathing mode in confined one-dimensional dipolar bosons

Probing the radial collective oscillation of a trapped quantum system is an accurate experimental tool to investigate interactions and dimensionality effects. We consider a fully polarized quasi-one dimensional dipolar quantum gas of bosonic dysprosium atoms in a parabolic trap at zero temperature. We model the dipolar gas with an effective quasi-one dimensional Hamiltonian in the single-mode approximation, and derive the equation of state using a variational approximation based on the Lieb-Liniger gas Bethe Ansatz wavefunction or perturbation theory. We calculate the breathing mode frequencies while varying polarization angles by a sum-rule approach, and find them in good agreement with recent experimental findings.

Landau levels, edge states, and gauge choice in 2D quantum dots

We examine the behavior of a charged particle in a two dimensional quantum dot in the presence of a magnetic field. Emphasis is placed on the high magnetic field regime. Compared to free space geometry, confinement in a dot geometry provides a more realistic system where edge effects arise naturally. It also serves to remove the otherwise infinite degeneracy due to the magnetic field; nonetheless, as described in this paper, additional ingredients are required to produce sensible results. We treat both circular and square geometries, and in the latter, we explicitly demonstrate the gauge invariance of the energy levels and wave function amplitudes. The characteristics of bulk states closely resemble those of free space states. For edge states, with sufficiently high quantum numbers, we achieve significant differences in the square and circular geometries. Both circular and square geometries are shown to exhibit level crossing phenomena, similar to parabolic dots, where the confining potential is a parabolic trap. Confinement effects on the probability current are also analyzed; it is the edge states that contribute non-zero current to the system. The results are achieved using straightforward matrix mechanics, in a manner that is accessible to novices in the field. On a more pedagogical note, we also provide a thorough review of the theory of single electron Landau levels in free space and illustrate how the introduction of surfaces naturally leads to a more physically transparent description of a charged particle in a magnetic field.

Semiclassical dynamics of a dark soliton in a one-dimensional bosonic superfluid in an optical lattice

We study quantum dynamics of a dark soliton in a one-dimensional Bose gas in an optical lattice within the truncated Wigner approximation. A previous work has revealed that in the absence of quantum fluctuations, dynamical stability of the dark soliton significantly depends on whether its phase kink is located at a lattice site or a link of two neighboring sites. It has also shown that the dark soliton is unstable in a regime of strong quantum fluctuations regardless of the phase-kink position. To bridge the gap between the classical and strongly quantum regimes, we investigate the dynamical stability of the dark soliton in a regime of weak quantum fluctuations. We find that the position dependence of the dynamical stability gradually diminishes and eventually vanishes as the strength of quantum fluctuations increases. This classical-to-quantum crossover of the soliton stability remains even in the presence of a parabolic trapping potential. We suggest that the crossover behavior can be used for experimentally diagnosing whether the instability of a dark soliton is due to quantum fluctuations or classical dynamical instability.

Inhomogeneity of a one-dimensional Yukawa system in a trap

A system of like charges interacting with a weak repulsive Yukawa potential and confined in a one-dimensional parabolic electrostatic trap is under consideration. It is shown that inter-particle distance in this system grows from the center of the structure to its periphery. The same effect takes place for the mean square displacement of particles from their equilibrium positions, which corresponds to the amplitude of thermal oscillations, and for the value of Lindemann parameter. This leads to different degrees of order for particles located at different distances from the center of the structure. This result might be important for the study of phase transitions in dusty, colloidal and one-component plasmas.

Bifurcation analysis of stationary solutions of two-dimensional coupled Gross–Pitaevskii equations using deflated continuation

Recently, a novel bifurcation technique known as deflated continuation was applied to the single-component nonlinear Schrödinger (NLS) equation with a parabolic trap in two spatial dimensions. This bifurcation analysis revealed previously unknown solutions, shedding light on this fundamental problem in the physics of ultracold atoms. In the present work, we take this a step further by applying deflated continuation to two coupled NLS equations, which – feature a considerably more complex landscape of solutions. Upon identifying branches of solutions, we construct the relevant bifurcation diagrams and perform spectral stability analysis to identify parametric regimes of stability and instability and to understand the mechanisms by which these branches emerge. The method reveals a remarkable wealth of solutions. These include both well-known states arising from the Cartesian and polar small amplitude limits of the underlying linear problem, but also a significant number of more complex states that arise through (typically pitchfork) bifurcations.

Inhomogeneity of Structural and Dynamical Characteristics of Dusty Plasma in a Gas Discharge

Dusty plasma crystals are commonly considered as spatially homogeneous structures with equal values of their structural and thermodynamic parameters in various regions of the system. Inhomogeneity of the interparticle distance, the amplitude of thermal particle oscillations, the Lindemann parameter, and the coupling parameter is shown to arise even in a system of particles with a simplified interaction model including a screened Coulomb potential and a parabolic trap. The kinetic energy distribution for particles in an inhomogeneous dusty plasma monolayer is investigated: it is shown that the mean kinetic energy of particles in the central part of the structure can differ significantly from the mean kinetic energy of particles on its periphery due to the inhomogeneity of structural parameters. The detected effects are verified using analytical methods and numerical simulations and by comparison with a dusty plasma experiment in a dc glow discharge. The results obtained are of great importance for investigating finite systems of like charges in a trap and studying the mechanism of phase transitions in dusty plasma systems.

High-frequency Bounds for the Helmholtz Equation Under Parabolic Trapping and Applications in Numerical Analysis

This paper is concerned with resolvent estimates on the real axis for the Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet boundary conditions when the obstacle is trapping. There are two resolvent estimates for this situation currently in the literature: (i) in the case of elliptic trapping the general “worst case” bound of exponential growth applies, and examples show that this growth can be realized through some sequence of wavenumbers; (ii) in the prototypical case of hyperbolic trapping where the Helmholtz equation is posed in the exterior of two strictly convex obstacles (or several obstacles with additional constraints) the nontrapping resolvent estimate holds with a logarithmic loss. This paper proves the first resolvent estimate for parabolic trapping by obstacles, studying a class of obstacles the prototypical example of which is the exterior of two squares (in two dimensions) or two cubes (in three dimensions), whose sides are parallel. We show, via developments of the vector-field/multiplier argument of Morawetz and the first application of this methodology to trapping configurations, that a resolvent estimate holds with a polynomial loss over the nontrapping estimate. We use this bound, along with the other trapping resolvent estimates, to prove results about integral equation formulations of the boundary value problem in the case of trapping. Feeding these bounds into existing frameworks for analyzing finite and boundary element methods, we obtain the first wavenumber-explicit proofs of convergence for numerical methods for solving the Helmholtz equation in the exterior of a trapping obstacle.

Ideal gas model of Bose-Einstein condensates confined in the parabolic trap

By using the one-dimensional canonical partition function, we modelled an ideal gas-like form of a set of Bose-Einstein condensates confined by a three-dimensional anisotropic parabolic trap. The model itself was constructed by taking the eigenenergies of the one-dimensional Gross-Pitaevskii equation in the longitudinal direction and enabling the harmonic volume as the inverse cube of average geometric trapping frequency to substitute the real volume. In this paper, we showed that the condensates form an ideal gas represented by its equation of state and have similar mature to Einstein’s solid-like model with the corrections in both the low and high temperatures in the internal energy formulations.

Energies of the static solitary wave solutions of the one-dimensional Gross-Pitaevskii equation

We calculated the energies of the static solitary wave solutions of the one-dimensional Gross-Pitaevskii equation with the time-dependent parabolic trap, the time-dependent scattering wave length of s-wave, and the time-dependent external potential describing a gain or loss term. Some written solutions of the equation were used, two of which are based on the experimental results. The solutions satisfy the condition of solitary wave solution since they are localized over the space. By this argument, the energies were obtained by integrating the Hamiltonian density over the space formulated in the classical field theory. To do that, we constructed the appropriate Lagrangian density representing the equation by initially writing the ansatz Lagrangian density and then substituting into the Euler-Lagrange equation. We found that two of the solutions have the same energies and the other one should mathematically have the pure imaginary function describing the gain-loss term to achieve the real energy.

Controlled generation of dark-bright soliton complexes in two-component and spinor Bose-Einstein condensates

We report on the controlled creation of multiple soliton complexes of the dark-bright type in one-dimensional two-component, three-component and spinor Bose-Einstein condensates. The formation of these states is based on the so-called matter wave interference of separated condensate fragments. In all three cases a systematic numerical study is carried out upon considering different variations of each systems' parameters both in the absence and in the presence of a harmonic trap. It is found that the judicious selection of the initial separation or the chemical potential of the participating components can be utilized to tailor the number of nucleated states. The latter increases as the former parameters are increased. Similarities and differences of the distinct models considered herein are showcased while the robustness of the emerging states is illustrated via the numerical experiments demonstrating their long time propagation. Importantly, for the spinorial system, we unravel the existence of beating dark soliton arrays that are formed due to the spin-mixing dynamics. These states persist in the presence of a parabolic trap, often relevant for associated experimental realizations.