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Contributions to Plasma Physics

Quantum Breathing Mode of Trapped Particles: From Nanoplasmas to Ultracold Gases

Jan Willem Abraham, M. Bönitz

Published Jan 1, 2014• 35 citations
Publisher

Abstract

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)

DOI
10.1002/ctpp.201300066

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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)
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.
Ultracold atoms are exceptional tools to explore the physics of quantum matter. In fact, the high degree of tunability of ultracold Bose and Fermi gases makes them ideal systems for quantum simulation and for investigating macroscopic manifestations of quantum effects, such as superfluidity. In ultracold gas research, a central role is played by collective oscillations. They can be used to study different dynamical regimes, such as superfluid, collisional, or collisionless limits or to test the equation of state of the system. In this thesis, we present a unified description of collective oscillations in low dimensions covering both Bose and Fermi statistics, different trap geometries and zero as well as finite temperature, based on the formalism of hydrodynamics and sum rules. We discuss the different behaviour exhibited by the second excited breathing mode in the collisional regime at low temperature and in the collisionless limit at high temperature in a 1D trapped Bose gas with repulsive contact interaction. We show how this mode exhibits a single-valued excitation spectrum in the collisional regime and two different frequencies in the collisionless limit. Our predictions could be important for future research related to the thermalization and damping phenomena in this low-dimensional system. We show that 1D uniform Bose gases exhibit a non-monotonic temperature dependence of the chemical potential characterized by an increasing-with-temperature behaviour at low temperature. This is due to the thermal excitation of phonons and reveals an interesting analogy with the behaviour of superfluids. Finally, we investigate a gas with a finite number N of atoms in a ring geometry at T = 0. We discuss explicitly the deviations of the thermodynamic behaviour in the ring from the one in the large N limit.
Interacting particles in a harmonic trap are known to possess a radial collective oscillation---the breathing mode (BM). We show that a quantum system has two BMs and analyze their properties by exactly solving the time-dependent Schr\"odinger equation. We report that the frequency of one BM changes with system dimensionality, the particle spin and the strength of the pair interaction and propose a scheme that gives direct access to key properties of trapped particles, including their many-body effects.
We theoretically investigate the ground-state properties and breathing-mode collective excitations of three-dimensional dipolar Bose gases in anisotropic harmonic traps incorporating quantum fluctuations. Combining a Gaussian variational ansatz with a non-perturbative sum-rule analysis, we derive explicit analytical expressions for both axial and radial breathing-mode frequencies, which are validated by numerical solutions of the time-dependent extended Gross-Pitaevskii equation. Our theoretical predictions show excellent agreement with existing experimental data for 166^{166}166Er and 162^{162}162Dy gases. By constructing comprehensive phase diagrams across the parameter space of the sss-wave scattering length, atom number, and trap aspect ratio, we reveal both discontinuous first-order phase transitions and smooth crossovers between the dilute Bose-Einstein condensate and dense quantum droplet phases. We confirm that the enhanced incompressibility induced by quantum fluctuations significantly elevates the breathing-mode frequencies in the droplet phase compared to conventional weakly interacting Bose gases. Furthermore, the system undergoes a phase transition and a crossover over the scattering length under the quasi-two-dimensional and quasi-one-dimensional confinements, characterized by discontinuous jumps and continuous crossovers in peak density and atomic cloud sizes, respectively. Our work offers a rigorous and highly accurate framework to characterize collective excitations in dipolar quantum gases, providing quantitative insights for forthcoming ultracold atom experiments in lanthanide atoms and polar molecules.
Advances in trapping and cooling of ultracold gases over the last several decades have made it possible to test many formerly outstanding predictions from disparate branches of physics. This thesis touches on three historical problems that have found new life recently in the context of ultracold Bose gases of alkali atoms. The first problem revolves around an outstanding prediction from Boltzmann over a century and half old that the breathing mode of a isotropically trapped classical gas should oscillate indefinitely. I analyze recent experimental results [Nat. Phys. 11, 1009 (2015)], and attribute observed damping sources to trap imperfections. The second question is about the analogue of first and second sound modes from liquid helium in trapped dilute gases. I present the results of a joint theoretical/experimental investigation of the breathing mode of a finite temperature Bose-Einstein condensate (BEC), attributing a striking collapse revival behavior of the resultant oscillation to in-phase and out-of-phase normal modes of the thermal cloud and condensate. The third problem is that of the formation of Borromean ring-like three-body bound states, referred to as Efimov trimers, in strongly-interacting few-body systems. I extend the predicted spectrum of Efimov states into the realm of many degenerate internal levels, and investigate the difficult three-body elastic scattering problem.\nThese questions are part of the broader theme of this thesis: How can our understanding of few-body physics in the ultracold limit be translated into statements about the bulk behavior of an ultracold gas? For weakly-interacting Bose gases, this translation is well-known: the many-body properties of the gas are well-described by the tracking just the one and two particle correlations. I analyze a generalization of this procedure to higher order correlations, the general connection between few-body physics and correlations in a dilute gas, and results for the emergence of Efimov physics in the magnetic phase of the strongly-interacting Bose gas.
Examples of resonant phenomena in varieties of macroscopic and microscopic systems are abundant in nature. These features are observable in a range of systems from ocean waves to Fermi and Bose gases in very sophisticated sate-of-the-art experiments. For past few decades, there have been a lot of developments in investigations of collective oscillations in semiconductor heterostructures, cold fermion and boson gases. In some cases theoretical predictions on the behavior of such oscillations predate the experimental evidences by decades, and in other cases there are still unexplained experimental results. In real experiments the system is always finite and confined by external forces which gives rise to many interesting behaviors and lately numerous theoretical works have also addressed the issue of confinement in such systems. This work addresses an important factor in theoretical considerations, namely the interparticle interactions. In the first part of this thesis, I present a quantum-phenomenological calculation of the collective mode frequencies of a 1D interacting electron gas laterally confined by a parabolic potential under the influence of an external magnetic field, which will also be complemented by a perturbative solution. In the second part, I talk about the effect on finite non-parabolicity in the confinement potential. It will be showed that the collective oscillations of this nature are detuned by such irregularity and also that they have finite lifetime as opposed to the parabolic case. In the final part I discuss the collective modes in an interacting Bose-Einstein Condensate (BEC) coupled with its thermal cloud in a pancake-shaped confinement. The geometry is taken to be infinite in the radial direction and confined laterally with a parabolic potential. I will show how the coupled system behaves behaves in presence of interaction, and also how to incorporate non-parabolicty to the solution.