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arXiv:1508.00365

Trapped electrons in the quantum degenerate regime

Ronen M. Kroeze, Jom Luiten, S. J. J. M. F. Kokkelmans

Published Aug 3, 2015• 0 citations
arXivHTMLPDF

Abstract

A full strength Coulomb interaction between trapped electrons can be felt only in absence of a neutralizing background. In order to study quantum degenerate electrons without such a background, an external trap is needed to compensate for the strong electronic repulsion. As a basic model for such a system, we study a trapped electron pair in a harmonic trap with an explicit inclusion of its Coulomb interaction. We find the eigenenergy of the ground state, confirming earlier work in the context of harmonium. We extend this to a complete set of properly scaled energies for any value of the trapping strength, including the excited states. The problem is solved either numerically or by making harmonic approximations to the potential. As function of the trapping strength a crossover can be made from the strongly to the weakly-coupled regime, and we show that in both regimes perturbative methods based on a pair-wise electron description would be effective for a many-particle trapped electron system, which resembles a Wigner crystal in the ground state of the strongly coupled limit.

DOI
10.48550/arxiv.1508.00365

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A full strength Coulomb interaction between trapped electrons can be felt only in absence of a neutralizing background. In order to study quantum degenerate electrons without such a background, an external trap is needed to compensate for the strong electronic repulsion. As a basic model for such a system, we study a trapped electron pair in a harmonic trap with an explicit inclusion of its Coulomb interaction. We find the eigenenergy of the ground state, confirming earlier work in the context of harmonium. We extend this to a complete set of properly scaled energies for any value of the trapping strength, including the excited states. The problem is solved either numerically or by making harmonic approximations to the potential. As function of the trapping strength a crossover can be made from the strongly to the weakly-coupled regime, and we show that in both regimes perturbative methods based on a pair-wise electron description would be effective for a many-particle trapped electron system, which resembles a Wigner crystal in the ground state of the strongly coupled limit.
The three-dimensional Schrodinger equation for three electrons in a parabolic confinement potential (with strength measured by the frequency ω) can be decoupled into three pair problems, provided the expectation value of the center of mass vector R is small compared with the average distance between the electrons. This should be fulfilled at the strong correlation limit (small ω), where the electron system tends to crystallize. The remaining part of the Hamiltonian, which is not included in the independent pair model, is taken into account in perturbation theory. The complementary treatment of the weak correlation limit (large ω) considers noninteracting electrons as a zeroth-order approximation and includes the electron–electron interaction in perturbation theory. It turns out that both approaches match satisfactorily for intermediate ω. Our results are compared with those obtained with the Hartree–Fock, configuration interaction, multiconfigurational complete active space, and stochastic variational method approaches and the data from a Wigner crystal treatment in a harmonic approximation.
A system of bosons in a two-dimensional harmonic trap in the presence of Rashba-type spin-orbit coupling is investigated. An analytic treatment of the ground state of a single atom in the weak-coupling regime is presented and used as a basis for a perturbation theory in the interacting two-boson system. The numerical diagonalization of both the single-particle and the two-boson Hamiltonian matrices allows us to go beyond those approximations and obtain not only the ground state, but also the low-energy spectra and the different energy contributions separately. We show that the expectation value of the spin-orbit term is related to the expectation value of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{z}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{L}}_{z}$ for the eigenstates of the system, regardless of the trapping potential. The low-energy states of the repulsively interacting two-boson system are characterized. In the presence of a sufficiently strong interaction and spin-orbit coupling strength, there is a direct energy-level crossing in the ground state of the system between states of different Jz{J}_{z}Jz​, the third component of the total angular momentum, that changes its structure. This is reflected in a discontinuity in the different energy terms and it is signaled in the spatial density of the system.
Simple few-body systems often serve as theoretical laboratories across various branches of theoretical physics. A prominent example is the two-electron Harmonium model, which has been widely studied over the past three decades to gain insights into the nature of the electron-electron correlations in many-electron quantum systems. Building on our previous work [Riyahi et al., Phys. Rev. B 108, 245155 (2023)], we introduce an analogous model consisting of an electron and a positively charged particle (PCP) with variable mass, interacting via Coulomb forces while confined by external harmonic potentials. Termed the exotic Harmonium model, this provides insights into the electron-PCP correlations, a cornerstone of the emerging field of the ab initio study of multi-component many-body quantum systems. Through a systematic exploration of the parameter space and numerical solutions of the corresponding Schrödinger equation, we identify two extreme regimes: the atom-like and the particle-in-trap-like behavior. The electron-PCP correlation dominates in the atom-like regime, significantly influencing physical observables, while its role diminishes in the particle-in-trap-like limit. Between these two extremes lies a complex intermediate regime that challenges qualitative interpretation. Overall, the exotic Harmonium model offers a powerful framework to unravel the electron-PCP correlations across diverse systems, spanning particles of varying masses and conditions, from ambient to high-pressure environments.
We conduct a theoretical study of SU(N\mathrm{SU}(NSU(N) fermions confined by a one-dimensional harmonic potential. First, we introduce a numerical approach for solving the trapped interacting few-body problem, by which one may obtain accurate energy spectra across the full range of interaction strengths. In the strong-coupling limit, we map the SU(N\mathrm{SU}(NSU(N) Hamiltonian to a spin-chain model. We then show that an existing, extremely accurate ansatz---derived for a Heisenberg SU(2) spin chain---is extendable to these NNN-component systems. Lastly, we consider balanced SU(N\mathrm{SU}(NSU(N) Fermi gases that have an equal number of particles in each spin state for N=2,3,4N=2,\phantom{\rule{0.16em}{0ex}}3,\phantom{\rule{0.16em}{0ex}}4N=2,3,4. In the weak- and strong-coupling regimes, we find that the ground-state energies rapidly converge to their expected values in the thermodynamic limit with increasing atom number. This suggests that the many-body energetics of NNN-component fermions may be accurately inferred from the corresponding few-body systems of NNN distinguishable particles.
Abstract Many quantum phenomena responsible for key applications in material science and quantum chemistry arise in the strongly correlated regime. This is at the same time, a costly regime for computer simulations. In the limit of strong correlation analytic solutions exist, but as we move away from this limit numerical simulation are needed, and accurate quantum solutions do not scale well with the number of interacting particles. In this work we propose to use few-particle harmonic traps in combination with twisted light as a quantum emulator to investigate the transition into a strongly-correlated regime. Using both analytic derivations and numerical simulations we generalize previous findings on 2 Coulomb interacting fermions trapped in a one-dimensional harmonic trap to the case of 3 fermions. The 4 signatures of strong correlation we have identified in the one-dimensional harmonic trap are: (i) the ground state density is highly localized around N equilibrium positions, where N is the number of particles, (ii) the symmetric and antisymmetric ground state wavefunctions become degenerate, (iii) the von Neumann entropy grows, (iv) the energy spectrum is fully characterized by N normal modes or less. Our findings describe the low-energy behavior of electrons in quantum wires and ions in Paul traps. Similar features have also been reported for cold atoms in optical lattices.
We study, by means of exact-diagonalization techniques, the ground state of a few-fermion system with strong short-range repulsive interactions trapped by a harmonic potential in one spatial dimension. Even when the ground-state density profile displays, at strong coupling, very well pronounced Wigner oscillations with a 4kF4{k}_{\mathrm{F}}4kF​ periodicity, the pair-correlation function does not show any signature of Wigner-molecule-type correlations. For the sake of comparison, we present also numerical results for few-electron systems with Coulomb interactions, demonstrating that their ground state at strong coupling is, on the contrary, a Wigner molecule.