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arXiv:1311.5371 · New Journal of Physics

Quantum breathing mode of trapped systems in one and two dimensions

Jan Willem Abraham, M. Bönitz, Chris McDonald, Gianfranco Orlando, Thomas Brabec

Published Jan 2, 2014• 18 citations
arXivHTMLPDF

Abstract

We investigate the quantum breathing mode (monopole oscillation) of trapped fermionic particles with Coulomb and dipole interaction in one and two dimensions. This collective oscillation has been shown to reveal detailed information on the many-particle state of interacting trapped systems and is thus a sensitive diagnostics for a variety of finite systems, including cold atomic and molecular gases in traps and optical lattices, electrons in metal clusters and in quantum confined semiconductor structures or nanoplasmas. An improved sum rule formalism allows us to accurately determine the breathing frequencies from the ground state of the system, avoiding complicated time-dependent simulations. In combination with the Hartree–Fock and the Thomas–Fermi approximations this enables us to extend the calculations to large particle numbers N on the order of several million. Tracing the breathing frequency to large N as a function of the coupling parameter of the system reveals a surprising difference of the asymptotic behavior of one-dimensional and two-dimensional harmonically trapped Coulomb systems.

DOI
10.1088/1367-2630/16/1/013001

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We investigate the quantum breathing mode (monopole oscillation) of trapped fermionic particles with Coulomb and dipole interaction in one and two dimensions. This collective oscillation has been shown to reveal detailed information on the many-particle state of interacting trapped systems and is thus a sensitive diagnostics for a variety of finite systems, including cold atomic and molecular gases in traps and optical lattices, electrons in metal clusters and in quantum confined semiconductor structures or nanoplasmas. An improved sum rule formalism allows us to accurately determine the breathing frequencies from the ground state of the system, avoiding complicated time-dependent simulations. In combination with the Hartree–Fock and the Thomas–Fermi approximations this enables us to extend the calculations to large particle numbers N on the order of several million. Tracing the breathing frequency to large N as a function of the coupling parameter of the system reveals a surprising difference of the asymptotic behavior of one-dimensional and two-dimensional harmonically trapped Coulomb systems.
Abstract Particles spatially confined in trapping potentials have attracted increasing interest over the recent decade. Of particular importance are systems of charged particles, such as non‐neutral plasmas, nanoplasmas, electrons in metal clusters, electrons in quantum‐confined semiconductor structures (“artificial atoms”), electrons on the surface of liquid helium, ions in traps or highly charged particles (grains) in dusty plasmas. A second example of recent interest are systems with other kinds of pair interactions, including dipole interaction, which is important for excitons in quantum wells or ultracold Fermi and Bose gases in traps and optical lattices. Trapped systems are fundamentally different from macroscopic systems since they are dominated by strong spatial inhomogeneity and finite size effects (the properties depend on the exact particle number). Furthermore, by changing the strength of the confinement potential, the many‐particle state of the system can be externally controlled—from weak coupling (gas‐like) to strong coupling (crystal‐like). While trapped classical particles are meanwhile well understood and accessible to first‐principle computer simulations, their quantum counterparts still pose big challenges, both for experiment and theory. Therefore, collective properties that can be easily measured or computed and allow to diagnose the many‐particle state of the system are of prime importance. It has been found that the quantum breathing mode (monopole oscillation) is one of the most important such properties. In recent years a number of theoretical studies has demonstrated that the quantum breathing mode is ideally suited to measure the coupling strength (the degree of nonideality) of a trapped system, its kinetic and interaction energy and other key observables. This give rise to a novel kind of “spectroscopy” of trapped systems. In this review these developments are summarized. The quantum breathing mode is studied for trapped fermions and bosons with Coulomb and dipole interaction, respectively. A systematic description of collective oscillations and especially the breathing mode is provided. Making use of time‐dependent perturbation theory, it is shown how the corresponding breathing frequencies are connected to the properties of the initial equilibrium system. This gives rise to the application of the quantum mechanical sum rules. It is demonstrated how an improved version of the conventional sum rule formulas is suitable for an accurate description of the breathing mode in small systems. Finally, the dependence of the breathing mode on the particle number N is analyzed and the limit of large N is studied for one‐dimensional and two‐dimensional systems. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Extending our previous work, we explore the breathing mode---the (uniform) radial expansion and contraction of a spatially confined system. We study the breathing mode across the transition from weak to moderate couplings and confirm that its frequency is not independent of the pair interaction strength (coupling parameter). We present the results of time-dependent Hartree-Fock simulations for 2 to 20 fermions with Coulomb interaction and show that the breathing frequency has a minimum for a small particle number. We validate the accuracy of our results, comparing them to exact configuration interaction results for up to eight particles.
The breathing mode collective-excitation frequency of a Bose-Einstein condensate is studied in a highly asymmetric (two- or one-dimensional) trap by using a sum-rule method and a time-dependent variational method. The collective-excitation frequency of the breathing mode in the Thomas-Fermi limit for a purely two- or one-dimensional trap is reproduced by a general result for a three-dimensional trap. In the case of a two- or one-dimensional trap, we obtain the lowest-order correction for the excitation frequency due to the finiteness of the trap frequency in the tightly trapped direction or the finiteness of the number of atoms.
We theoretically investigate breathing oscillations of weakly-interacting degenerate Fermi gases in highly-anisotropic harmonic oscillator traps. If the traps are not highly anisotropic, the fermions behave as three-dimensional (3D) gases and exhibit the coupled breathing oscillations as studied in a previous paper [T. Maruyama and T. Nishimura, Phys. Rev. A 75 (2007) 033611]; Otherwise the fermions exhibit quasi-low-dimensional (QLD) properties derived from specific structures in their single-particle spectrum, called QLD structures. In the present paper, we focus on effects of the QLD structures on the breathing oscillations of the two-component fermions with symmetric population densities. Here we develop the semi-classical Thomas-Fermi approximation extended to the highly-anisotropic systems and obtain the collective frequencies in the sum-rule-scaling method and perturbation theory. As a result, we reveal that the effects of the QLD structures can not be seen in the transverse modes in the first-order perturbation and appear only in the longitudinal modes with hierarchies reflecting the QLD structures. We also demonstrate time-evolution of the oscillations in the present framework.
For two-dimensional (2D) atomic Fermi gases in harmonic traps, the SO(2,1) symmetry is broken by the interatomic interaction explicitly via the contact correlation operator. Consequently, the frequency of the breathing mode ${\ensuremath{\omega}}_{B}$ of the 2D Fermi gas can be different from $2{\ensuremath{\omega}}_{0}$, with ${\ensuremath{\omega}}_{0}$ the trapping frequency of harmonic potentials. At zero temperature, we use the sum rules of density correlation functions to yield upper bounds for ${\ensuremath{\omega}}_{B}$. We further calculate ${\ensuremath{\omega}}_{B}$ through the Euler equations in the hydrodynamic regime. The obtained value of ${\ensuremath{\omega}}_{B}$ satisfies the upper bounds and shows deviation from $2{\ensuremath{\omega}}_{0}$, which can be as large as about 88%8.