Two-dimensional Harmonic Potential Research Articles

Stochastic currents and efficiency in an autonomous heat engine.

We experimentally demonstrate that a Brownian gyrator of a colloidal particle confined in a two-dimensional harmonic potential with different effective temperatures on orthogonal axes can work as an autonomous heat engine capable of extracting work from the heat bath, generated by an optical feedback trap. The results confirm the theoretically predicted thermodynamic currents and validate the attainability of Carnot efficiency as well as the trade-off relation between power and efficiency. We further show that current fluctuations and the entropy production rate are time independent in the steady state and their product near the Carnot efficiency is close to the lower bound of the thermodynamic uncertainty relation.

Bose—Einstein condensates with tunable spin—orbit coupling in the two-dimensional harmonic potential: The ground-state phases, stability phase diagram and collapse dynamics

Bose—Einstein condensates with tunable spin—orbit coupling in the two-dimensional harmonic potential: The ground-state phases, stability phase diagram and collapse dynamics

Perturbative expansion of irreversible work in symmetric and asymmetric processes.

The systematic expansion method of the solution of the Fokker-Planck equationis developed by generalizing the formulation proposed in [J. Phys. A: Math. Theor. 50, 325001 (2017)10.1088/1751-8121/aa7af4]. Using this method, we obtain an alternative formula to calculate the mean work perturbatively which is applicable to systems with degeneracy in the eigenvalues of the Fokker-Planck operator. This method enables us to study how the geometrical symmetry affects thermodynamic description of a Brownian particle. To illustrate the application of the derived theory, we consider the Fokker-Planck equationwith a two-dimensional harmonic potential. To investigate the effect of symmetry of the potential, we study thermodynamic properties in symmetric and asymmetric deformation processes of the potential: the rotational symmetry of the harmonic potential is held in the former, but it is broken in the latter. Optimized deformations in these processes are defined by minimizing mean work. Comparing these optimized processes, we find that the difference between the symmetric and asymmetric processes is maximized when the deformation time of the potential is given by a critical time which is characterized by the relaxation time of the Fokker-Planck equation. This critical time in the mean work is smaller than that of the change of the mean energy because of the hysteresis effect in the irreversible processes.

Dirac particles in the presence of a constant magnetic field and harmonic potential with spin symmetry

In this paper, we study the effect of the constant magnetic field on energy levels of the Dirac particles such as electron, proton and heavy ions. We calculate the energy eigenvalues of the Dirac equation in the presence of the magnetic field and two-dimensional harmonic oscillator potential with spin symmetry by using the supersymmetric quantum mechanics and asymptotic iteration methods.

Stationary properties of a non-Markovian Brownian gyrator

We investigate the stochastic behavior of a non-Markovian version of an elementary Brownian gyrator. The model is defined by overdamped Langevin-like dynamics with a two-dimensional harmonic potential that presents distinct principal axes and is coupled to heat baths at different temperatures. The thermal noises are assumed to be Gaussian, and are related to friction forces through a dissipation memory kernel. The stationary states present rotational motion with non-trivial average torques due to harmonic, friction and fluctuating thermal forces. However, the Markovian limit of the system exhibits a zero average torque produced by fluctuating thermal forces. For the case of stochastic torque exerted by harmonic force, the cumulant-generating function is calculated exactly. We also study the average heat fluxes in the steady-state regime, where a memory-dependent behavior is observed.

Observing the emergence of a quantum phase transition shell by shell.

Many-body physics describes phenomena that cannot be understood by looking only at the constituents of a system1. Striking examples are broken symmetry, phase transitions and collective excitations2. To understand how such collective behaviour emerges as a system is gradually assembled from individual particles has been a goal in atomic, nuclear and solid-state physics for decades3-6. Here we observe the few-body precursor of a quantum phase transition from a normal to a superfluid phase. The transition is signalled by the softening of the mode associated with amplitude vibrations of the order parameter, usually referred to as a Higgs mode7. We achieve fine control over ultracold fermions confined to two-dimensional harmonic potentials and prepare closed-shell configurations of 2, 6 and 12 fermionic atoms in the ground state with high fidelity. Spectroscopy is then performed on our mesoscopic system while tuning the pair energy from zero to a value larger than the shell spacing. Using full atom counting statistics, we find the lowest resonance to consist of coherently excited pairs only. The distinct non-monotonic interaction dependence of this many-body excitation, combined with comparison with numerical calculations allows us to identify it as the precursor of the Higgs mode. Our atomic simulator provides a way to study the emergence of collective phenomena and the thermodynamic limit, particle by particle.

Engineered swift equilibration of a Brownian gyrator.

In the context of stochastic thermodynamics, a minimal model for nonequilibrium steady states has been recently proposed: the Brownian gyrator (BG). It describes the stochastic overdamped motion of a particle in a two-dimensional harmonic potential, as in the classic Ornstein-Uhlenbeck process, but considering the simultaneous presence of two independent thermal baths. When the two baths have different temperatures, the steady BG exhibits a rotating current, a clear signature of nonequilibrium dynamics. Here, we consider a time-dependent potential, and we apply a reverse-engineering approach to derive exactly the required protocol to switch from an initial steady state to a final steady state in a finite time τ. The protocol can be built by first choosing an arbitrary quasistatic counterpart, with few constraints, and then adding a finite-time contribution which only depends upon the chosen quasistatic form and which is of order 1/τ. We also get a condition for transformations which, in finite time, conserve internal energy, useful for applications such as the design of microscopic thermal engines. Our study extends finite-time stochastic thermodynamics to transformations connecting nonequilibrium steady states.

Critical potentials and fluctuations phenomena with quartic, sextic, and octic anharmonic oscillator potentials

This is an investigation of the energy levels of three classes of two-dimensional anharmonic oscillator potentials characterized by free-, one-, and two- parameters with quartic, sextic, and octic anharmonicity. The wave functions and energy eigenvalues are found through the discretization using the nine-point finite-difference method. It is proved that the free-parameter potentials provide a “bridge” between the two-dimensional harmonic oscillator potential and two-dimensional infinite square well potential. However, the one- and two- parameter potentials show a transition from tunneling splitting mode to the spectra of free-parameter potentials crossing to the spectrum of infinite square well potential and in the same time present new features. One can distinguish cases of fluctuations phenomena in the ground state and excited states. We obtained, for the first time, a ground state with a four-peak structure. The quantum tunneling between the four minima of one- and two- parameter potentials depends on the characteristics of the separating barrier. These phase transitions are not of the ordinary thermodynamic category, but truly they are phase transitions in the structure of the low-lying energy spectra. The concept of critical potentials is associated with the potentials at the phase-transition points. The coherent quadrupole-octupole motion model, using these potentials, is applied to describe yrast and non-yrast energy sequences with alternating parity in several even-even nuclei from different regions, namely, in 100Mo, 146Ba, 226Ra, and 226Th.

Two-dimensional non-Hermitian harmonic oscillator: coherent states

In this study, we introduce a two-dimensional complex harmonic oscillator potential with space and time reflection symmetries. The corresponding time-independent Schrödinger equation yields real eigenvalues with complex eigenfunctions. We also construct the coherent state of the system by using a superposition of 12 eigenfunctions. Using the complex correspondence principle for the probability density we investigate the possible modifications in the probability densities due to the non-Hermitian aspect of the Hamiltonian.

A solvable model of Landau quantization breakdown

Physics of two-dimensional electron gases under perpendicular magnetic field often displays three distinct stages when increasing the field amplitude: a low field regime with classical magnetotransport, followed at intermediate field by a Shubnikov-de Haas phase where the transport coefficients present quantum oscillations, and, ultimately, the emergence at high field of the quantum Hall effect with perfect quantization of the Hall resistance. A rigorous demonstration of this general paradigm is still limited by the difficulty in solving models of quantum Hall bars with macroscopic lateral dimensions and smooth disorder. We propose here the exact solution of a simple model exhibiting similarly two sharp transitions that are triggered by the competition of cyclotron motion and potential-induced drift. As a function of increasing magnetic field, one observes indeed three distinct phases showing respectively fully broken, partially smeared, or perfect Landau level quantization. This model is based on a non-rotationally invariant, inverted two-dimensional harmonic potential, from which a full quantum solution is obtained using 4D phase space quantization. The developed formalism unifies all three possible regimes under a single analytical theory, as well as arbitrary quadratic potentials, for all magnetic field values.

Magnetic Shell Structure of 2D-Trapped Fermi Gases in the Flat-Band Lieb Lattices

We investigate the magnetic shell structure of repulsively interacting two-component Fermi gases trapped in a two-dimensional harmonic potential and loaded on the optical Lieb lattices. We employ the real-space dynamical mean-field theory (R-DMFT) to explicitly consider the trap potential in a self-consistent way. Computing the profiles of particle density and local magnetization across the lattice sites in the trap, we find that the incompressible core with ferrimagnetic ordering appears with the density plateau at the trap center, which is surrounded by the shell of the normal metallic phase. We examine the incompressibility of the core by adding more particles and creating the higher spin-population imbalance. While the core area expands from the outer shell with added particles and increased polarization, the excess particles are prohibited from going inside the core, and thus the density plateau is unchanged at the half-filling with the same magnetic ordering. In addition, we find that the feature of the phase separation differs with the sublattices, where the interstitial sites causing the flat band dispersion shows the signature of the abrupt transition in the density and magnetization at the boundary between the core and surrounding shells.

Assessment of local density approximation based exchange–correlation functional for a two-dimensional spin polarized dipolar Fermi gas

In this paper we assess the accuracy of the local density approximation (LDA) based expressions for exchange and correlation functionals associated with a two-dimensional, spin-polarized, dipolar Fermi gas. For this purpose we consider and study an exactly solvable system of two interacting spin-polarized dipolar fermions confined in a two-dimensional harmonic oscillator potential. We determine the exact triplet ground state wavefunction, energy and the density by separating the time-independent Schrodinger equation in center-of-mass and relative coordinates and numerically solving the eigenvalue equation in relative coordinate. The exact Kohn–Sham orbitals, corresponding energies and exact Kohn–Sham potential are obtained from the exact density by inverting the Kohn–Sham equation. These results serve as benchmark data for assessing the accuracies of the LDA based expressions for a two-dimensional, spin-polarized, dipolar Fermi gas. We find that LDA results for total energy, interaction energy, and Kohn–Sham eigenvalues match well with the exact results for smaller values of dipole–dipole interaction strength. For higher values of interaction strength, LDA results for the total energy are underestimated. On the other hand, the LDA based interaction potentials overestimate the repulsion between the dipoles and their spatial profiles differ considerably from their exact counterparts and the discrepancy between them is higher for stronger interaction.

Bose–Einstein condensation temperature of finite systems

In studies of the Bose–Einstein condensation of ideal gases in finite systems, the divergence problem usually arises in the equation of state. In this paper, we present a technique based on the heat kernel expansion and zeta function regularization to solve the divergence problem, and obtain the analytical expression of the Bose–Einstein condensation temperature for general finite systems. The result is represented by the heat kernel coefficients, where the asymptotic energy spectrum of the system is used. Besides the general case, for systems with exact spectra, e.g. ideal gases in an infinite slab or in a three-sphere, the sums of the spectra can be obtained exactly and the calculation of corrections to the critical temperatures is more direct. For a system confined in a bounded potential, the form of the heat kernel is different from the usual heat kernel expansion. We show that as long as the asymptotic form of the global heat kernel can be found, our method works. For Bose gases confined in three- and two-dimensional isotropic harmonic potentials, we obtain the higher-order corrections to the usual results of the critical temperatures. Our method can also be applied to the problem of generalized condensation, and we give the correction of the boundary on the second critical temperature in a highly anisotropic slab.

Stable Vortex Clusters in Two-Dimensional Bose-Einstein Condensates with Spatially Modulated Defocusing Nonlinearity

We investigate the vortex dynamic behavior of Bose-Einstein condensate in a spatially two-dimensional harmonic potential with spatially modulated defocusing nonlinearity. Kinds of exact vortex solutions of the Gross-Pitaevskii equation are obtained by designing various laser potentials which can be experimentally realized. In particular, we demonstrate that the spatially modulated defocusing nonlinearity supports stable vortex clusters. For the stationary vortex solutions, the vortex clusters are dynamically stable, and the circulation direction of each vortex and vortex core positions remain unchanged. However, for the non-stationary vortex solutions, stable and unstable vortex clusters exist. For the stable vortex clusters, the vortex core positions are constant during dynamical evolution, while the phase increase directions vary periodically. Additionally, corresponding to the unstable vortex clusters, vortices disappear and appear periodically in the condensate.

Two-photon absorption of the spatially confined LiH molecule.

In the present contribution we study the influence of spatial restriction on the two-photon dipole transitions between the X1Σ+ and A1Σ+ states of lithium hydride. The bond-length dependence of the two-photon absorption strength is also analyzed for the first time in the literature. The highly accurate multiconfiguration self-consistent field (MCSCF) method and response theory are used to characterize the electronic structure of the studied molecule. In order to render the effect of orbital compression we apply a two-dimensional harmonic oscillator potential, mimicking the topology of cylindrical confining environments (e.g. carbon nanotubes, quantum wires). Among others, the obtained results provide evidence that at large internuclear distances the TPA response of lithium hydride may be significantly enhanced and this effect is much more pronounced upon embedding of the LiH molecule in an external confining potential. To understand the origin of the observed variation in the two-photon absorption response a two-level approximation is employed.

Ground-state dipole moment of the spatially confined carbon monoxide and boron fluoride molecules

In this contribution the influence of spatial confinement on the dipole moment of carbon monoxide and boron fluoride was investigated. The spatial confinement was modeled by applying the two-dimensional harmonic oscillator potential. The dipole moment calculations were performed for both molecules using their experimental geometries and also including the effect of geometry relaxation in the presence of confining potential. Our results demonstrate that the dipole moment is noticeably affected by the confinement. Moreover, it was found that the changes in the dipole moment, caused by the presence of harmonic oscillator potential, are substantially different for each of the studied molecules.

Scattering-induced quantum correlation in electronic waveguides with static magnetic impurities

Entanglement generation due to low-energy scattering of the transporting electrons in an electronic waveguide by a quantum dot magnetic impurity is theoretically investigated. The transverse confining potential of the waveguide is considered as a two-dimensional harmonic potential, and the interaction of the electron with the impurity is described by a zero-range pseudopotential modulated by an Ising or a Heisenberg spin interaction. Our calculation shows that the scattering process leads to creation of a considerable amount of entanglement in the state of the reflected and transmitted electrons. The situation is extended to the scattering of the electrons by two well-separated magnetic impurities localized on the nanowire axis. It is shown that the scattering process causes the magnetic impurities embedded in the nanowire to share their quantum information; subsequently, they can be entangled by spin interaction with the injected electron. The created entanglement between the impurities is calculated and discussed. It is shown that the exact three-dimensional problem can be approximated as a one-dimensional problem under certain circumstances. The approximate results are compared to exact calculations and discussed.

Analytical study of parameter regions of dynamical instability for two-component Bose-Einstein condensates with coaxial quantized vortices

The dynamical instability of weakly interacting two-component Bose--Einstein condensates with coaxial quantized vortices is analytically investigated in a two-dimensional isotopic harmonic potential. We examine whether complex eigenvalues appear on the Bogoliubov--de Gennes equation, implying dynamical instability. Rather than solving the Bogoliubov--de Gennes equation numerically, we rely on a perturbative expansion with respect to the coupling constant which enables a simple, analytic approach. For each pair of winding numbers and for each magnetic quantum number, the ranges of inter-component coupling constant where the system is dynamically unstable are exhaustively obtained. Co-rotating and counter-rotating systems show distinctive behaviors. The latter is much more complicated than the former with respect to dynamical instability, particularly because radial excitations contribute to complex eigenvalues in counter-rotating systems.

Influence of variations in the electron–electron interaction on the ground state metric space “band structure” of a two-electron magnetic system

We consider a model system of two electrons confined in a two-dimensional harmonic oscillator potential, with the electrons interacting via an α/r2 potential, and subject to a magnetic field applied perpendicular to the plane of confinement. Our results show that variations in the strength of the electron–electron interaction generate a “band structure” in ground state metric spaces, which shares many characteristics with those generated as a result of varying the confinement potential. In particular, the metric spaces for wavefunctions, particle densities, and paramagnetic current densities all exhibit distinct “bands” and “gaps”. The behavior of the polar angle of the bands also shares traits with that obtained by varying the confinement potential, but the behavior of the arc lengths of the bands on the metric space spheres can be seen to be different for the two cases and opposite for a large range of angular momentum values. The findings here and in Refs. [1,2] demonstrate that the “band structure” that arises in ground state metric spaces when a magnetic field is applied is a robust feature.

About diverse behavior of the molecular electric properties upon spatial confinement

We report on the influence of spatial restriction on the static electric properties of two π-electron molecules, namely carbonyl sulfide and chloroacetylene. A two-dimensional harmonic oscillator potential has been applied in order to mimic the effect of orbital compression. All components of the studied tensorial quantities were obtained using the finite-field method. Among others, the nonmonotonic changes of the first hyperpolarizability, observed for the spatially limited OCS molecule, are described for the first time in the literature. Moreover, it has been found that upon embedding in cylindrical confining potential behavior of dipole moment is different for each of the investigated molecular systems. In order to explain the obtained diverse trends, the Hirshfeld technique as well as the concept of hyperpolarizability density has been adopted.