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Tunneling Theory for Few-Body Systems in One-Dimensional Traps

Rikard Lundmark

Published Jan 1, 2014• 1 citation
OpenAlex

Abstract

In recent experiments, carried out at the University of Heidelberg, tunneling rates of ultracold distinguishable fermions out of an optomagnetical trap have been measured. The fermions interact by a tunable short-ranged interaction,\nand the trap is asymmetric making the trapped quantum system\neffectively one-dimensional. In this thesis, a method for calculating the energy levels and tunneling rates of one and two interacting particles out of a very general onedimensional potential well is devised. The method is based on expanding the Schrodinger equation of the system in a complex-momentum basis. This is done utilizing the so-called Berggren completeness relation. Ultimately,\nthe basis expansion leads to a complex symmetric non-Hermitian eigenvalue problem for a large, dense matrix.\nThe general method is applied to a system of trapped, ultracold fermionic atoms in a setup that closely resembles the Heidelberg experiments. The short-ranged interaction is modeled as a point-interaction, and the trap potential\nis regularized at large distances from the interesting region. The obtained energies and decay rates are contrasted to results obtained using the Wentzel-Kramers-Brillouin (WKB) approximation. Notable differences\ncan be observed, and these may be due to insufficiency of the WKB approximation to accurately describe the system in question.

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In recent experiments, carried out at the University of Heidelberg, tunneling rates of ultracold distinguishable fermions out of an optomagnetical trap have been measured. The fermions interact by a tunable short-ranged interaction,\nand the trap is asymmetric making the trapped quantum system\neffectively one-dimensional. In this thesis, a method for calculating the energy levels and tunneling rates of one and two interacting particles out of a very general onedimensional potential well is devised. The method is based on expanding the Schrodinger equation of the system in a complex-momentum basis. This is done utilizing the so-called Berggren completeness relation. Ultimately,\nthe basis expansion leads to a complex symmetric non-Hermitian eigenvalue problem for a large, dense matrix.\nThe general method is applied to a system of trapped, ultracold fermionic atoms in a setup that closely resembles the Heidelberg experiments. The short-ranged interaction is modeled as a point-interaction, and the trap potential\nis regularized at large distances from the interesting region. The obtained energies and decay rates are contrasted to results obtained using the Wentzel-Kramers-Brillouin (WKB) approximation. Notable differences\ncan be observed, and these may be due to insufficiency of the WKB approximation to accurately describe the system in question.
We present one-dimensional simulation results for the cold-atom tunneling experiments by the Heidelberg group [Z\"urn et al., Phys. Rev. Lett. 108, 075303 (2012); Z\"urn et al., Phys. Rev. Lett. 111, 175302 (2013)] on one or two 6Li^{6}\mathrm{Li}6Li atoms confined by a potential that consists of an approximately harmonic optical trap plus a linear magnetic-field gradient. At the noninteracting particle level, we find that the Wentzel-Kramers-Brillouin approximation may not be used as a reliable tool to extract the trapping potential parameters from the experimentally measured tunneling data. We use our numerical calculations along with the experimental tunneling rates for the noninteracting system to reparametrize the trapping potential. The reparametrized trapping potentials serve as input for our simulations of two interacting particles. For two interacting (distinguishable) atoms on the upper branch, we reproduce the experimentally measured tunneling rates, which vary over several orders of magnitude, fairly well. For infinitely strong interaction strength, we compare the time dynamics with that of two identical fermions and discuss the implications of fermionization on the dynamics. For two attractively interacting atoms on the molecular branch, we find that single-particle tunneling dominates for weakly attractive interactions, while pair tunneling dominates for strongly attractive interactions. Our first set of calculations yields qualitative but not quantitative agreement with the experimentally measured tunneling rates. We obtain quantitative agreement with the experimentally measured tunneling rates if we allow for a weakened radial confinement.
Few- and many-fermion systems on the verge of stability, and consisting of strongly interacting particles, appear in many areas of physics. The theoretical modeling of such systems is a very difficult problem. In this work we present a theoretical framework that is based on the rigged Hilbert space formulation. The few-body problem is solved by exact diagonalization using a basis in which bound, resonant, and non-resonant scattering states are included on an equal footing. Current experiments with ultracold atoms offer a fascinating opportunity to study universal properties of few-body systems with a high degree of control over parameters such as the external trap geometry, the number of particles, and even the interaction strength. In particular, particles can be allowed to tunnel out of the trap by applying a magnetic-field gradient that effectively lowers the potential barrier. The result is a tunable open quantum system that allows detailed studies of the tunneling mechanism. In this Paper we introduce our method and present results for the decay rate of two distinguishable fermions in a one-dimensional trap as a function of the interaction strength. In particular, we present for the first time several technical and numerical details of our approach, recently published in Lundmark et al. (Phys Rev A 91:041601, 2015). We also show results from a careful analysis of the numerical convergence.
In this work numerical methods for solving the problem of two ultracold correlated fermionic 6Li atoms in a one-dimensional optical dipole trap are described. The underlying quantum mechanical model contains the typical approximation of a pointlike interaction for ultracold gases and the exact shape of the trap potential, which is known from the experiment. Using these methods, in which solving the time-dependent Schrödinger equation plays a central role, it was possible to obtain the characteristical energy spectrum dependent on the interaction strength and the significant eigenstates. On the basis of these results typical features of a transition called “Fermionization“ have been observed and reference values could be determined. In addition latest tunneling experiments were simulated, which are used to prove the Fermionization. For this purpose the decay characteristics of the prepared two-particle states in the trap potential with a potenial barrier were observed.
The system of interacting, trapped fermions in one dimension has been of interest in both the theoretical and experimental communities. This system is realizable experimentally using ultracold atoms in traps, where the interactions can be tuned to simulate a number of important situations in nuclear theory, condensed matter, quantum information, and QCD. Theoretically, however, this system remains a challenge to treat, and no known benchmarks exist for the ground state energy, Tan's contact, or density profiles for the few- to many-body regime. This project implements a lattice Monte Carlo (LMC) method to solve for these quantities. The method blends hybrid Monte Carlo (HMC) - a pillar of lattice quantum chromodynamics (lattice QCD) - with a non-uniform lattice defined using Gauss-Hermite quadrature points and weights. This coordinate basis is the natural one for the harmonic oscillator trapping potential, and can be generalized to traps of other shapes. Using this method, we determine the ground-state energy and Tan's contact of attractively interacting few-fermion systems in a one-dimensional harmonic trap, for a range of couplings and particle numbers. Complementing those results, we show the corresponding density profiles. We present results for N=4,...,20N = 4,...,20N=4,...,20 particles - and the method is capable of extending beyond that. The method is the first lattice calculation of its kind, and is exact up to statistical and systematic uncertainties, which we account for. Our results are therefore a benchmark for other methods and a prediction for ultracold-atom experiments.
Ultracold atomic systems provide a versatile platform for exploring quantum phenomena, offering tunable interactions and diverse trapping geometries. In this study, we investigate a one-dimensional system of trapped fermionic atoms using the composite boson formalism, which describes pairs of opposite-spin fermions as cobosons (short for composite bosons). We solve the Schr\"odinger equation for a $2\ensuremath{\uparrow}+2\ensuremath{\downarrow}$ fermionic system across a broad range of attractive and repulsive interactions by constructing the system's wave function as a superposition of two-coboson states. We determine key observables such as particle density profiles and two-body correlations. The density profiles calculated agree with both the Tonks-Girardeau and Lieb-Liniger limits and provide a portrait of the transition between those regimes. In the strong repulsive regime, the ansatz captures the Fridel-Wigner transition characterized by a doubling of the peaks in the density profiles. Additionally, we compute the low-lying energy spectrum and estimate the pairing gap. Our results highlight the usefulness of the coboson superposition ansatz for exploring quantum phenomena in strongly correlated few-body systems.
We study quasi-one-dimensional few-particle systems consisting of one to six ultracold fermionic atoms in two different spin states with attractive interactions. We probe the system by deforming the trapping potential and by observing the tunneling of particles out of the trap. For even particle numbers, we observe a tunneling behavior that deviates from uncorrelated single-particle tunneling indicating the existence of pair correlations in the system. From the tunneling time scales, we infer the differences in interaction energies of systems with different number of particles, which show a strong odd-even effect, similar to the one observed for neutron separation experiments in nuclei.