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.
Highlights
Trapped systems are of major interest in many fields of research
To work out how the equilibrium approach is connected to existing theories for the collective motions of many-body systems, we further extend the theoretical foundations of the breathing mode, providing a systematic description in terms of time-dependent perturbation theory
We show that this unexpected behavior is indicated by the lowest frequency of the breathing oscillation as well as a localization parameter which relates the average extension of the system to that of an ideal quantum system
Summary
Trapped systems are of major interest in many fields of research. Prominent examples are correlated electrons in metal clusters, e.g. [1], confined plasmas [2], ultracold quantum gases in traps or optical lattices [3, 4, 5], electrons in quantum dots [6, 7, 8, 9, 10] (“artificial atoms”), excitons in bilayers and quantum wells, e.g. [11, 12, 13, 14, 15] trapped ions [16], and colloidal particles [17]. To work out how the equilibrium approach is connected to existing theories for the collective motions of many-body systems, we further extend the theoretical foundations of the breathing mode, providing a systematic description in terms of time-dependent perturbation theory. This allows one to conclude that the breathing mode is characterized by the spectrum of the initial Hamiltonian. We briefly recapitulate the most important sum rule formulas and show how their accuracy can be improved Using this formalism, we analyze how the breathing frequency depends on the particle number, the coupling parameter and the dimensionality of the system
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