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arXiv:1304.8099 · Physical Review A

Few interacting fermions in a one-dimensional harmonic trap

Tomasz Sowiński, Tobias Graß, Omjyoti Dutta, Maciej Lewenstein

Published Sep 6, 2013• 86 citations
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Abstract

We study spin-1/2 fermions, interacting via a two-body contact potential, in a one-dimensional harmonic trap. Applying exact diagonalization, we investigate their behavior at finite interaction strength and discuss the role of the ground-state degeneracy which occurs for sufficiently strong repulsive interaction. Even low temperature or a completely depolarizing channel may then dramatically influence the system's behavior. We calculate level occupation numbers as signatures of thermalization, and we discuss the mechanisms to break the degeneracy.

DOI
10.1103/physreva.88.033607

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We study spin-1/2 fermions, interacting via a two-body contact potential, in a one-dimensional harmonic trap. Applying exact diagonalization, we investigate their behavior at finite interaction strength and discuss the role of the ground-state degeneracy which occurs for sufficiently strong repulsive interaction. Even low temperature or a completely depolarizing channel may then dramatically influence the system's behavior. We calculate level occupation numbers as signatures of thermalization, and we discuss the mechanisms to break the degeneracy.
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We provide a detailed study of the properties of a few interacting spin 1 / 2 fermions trapped in a one-dimensional harmonic oscillator potential. The interaction is assumed to be well represented by a contact delta potential. Numerical results obtained by means of direct diagonalization techniques are combined with analytical expressions for both the non-interacting and strongly interacting regime. The N = 2 case is used to benchmark our numerical techniques with the known exact solution of the problem. After a detailed description of the numerical methods, in a tutorial-like manner, we present the static properties of the system for N = 2 , 3 , 4 and 5 particles, e.g., low-energy spectrum, one-body density matrix, ground-state densities. Then, we consider dynamical properties of the system exploring first the excitation of the breathing mode, using the dynamical structure function and corresponding sum-rules, and then a sudden quench of the interaction strength.
We employ ab initio methods of quantum chemistry to investigate spin-1/2 fermions interacting via a two-body contact potential in a one-dimensional harmonic trap. The convergence of the total energy with the size of the one-particle basis set is analytically investigated for the two-body problem and the same form of the convergence formula is numerically confirmed to be valid for the many-body case. Benchmark calculations for two to six fermions with the full configuration interaction method equivalent to the exact diagonalization approach, and the coupled cluster (CC) method including single, double, triple, and quadruple excitations are presented. The convergence of the correlation energy with the level of excitations included in the CC model is analyzed. The range of the interaction strength for which single-reference CC methods work is examined. Next, the CC method restricted to single, double, and noniterative triple excitations, CCSD(T), is employed to study a two-component Fermi gas composed of 6–80 atoms in a one-dimensional harmonic trap. The density profiles of trapped atomic clouds are also reported. Finally, a comparison with experimental results for few-fermion systems is presented. Upcoming possible applications and extensions of the presented approach are discussed.
The ground-state properties of a few spin-1/2 fermions with different masses and interacting via short-range contact forces are studied within an exact diagonalization approach. It is shown that, depending on the shape of the external confinement, different scenarios of the spatial separation between components manifested by specific shapes of the density profiles can be obtained in the strong interaction limit. We find that the ground-state of the system undergoes a specific transition between orderings when the confinement is changed adiabatically from a uniform box to a harmonic oscillator shape. We study the properties of this transition in the framework of the finite-size scaling method adopted to few-body systems.
We study a system of few fermions in a one-dimensional harmonic trap and focus on the case of dipolar majority particles in contact with a single impurity. The impurity is used both for quenching the system and for tracking the system evolution after the quench. Employing exact diagonalization, we investigate relaxation and thermalization properties. In the absence of dipolar interactions, the system dynamics remains oscillatory even on long time scales. On the other hand, repulsive as well as attractive dipolar interactions lead to quick relaxation to the diagonal ensemble average, which is significantly different from corresponding thermal averages. A Wigner-shaped level spacing distribution indicates level repulsion and thus chaotic dynamical behavior due to the presence of dipolar interactions.
An asymptotically exact many-body theory for spin-polarized interacting fermions in a one-dimensional harmonic atom trap is developed using the bosonization method and including backward scattering. In contrast to the Luttinger model, backscattering in the trap generates one-particle potentials which must be diagonalized simultaneously with the two-body interactions. Inclusion of backscattering becomes necessary because backscattering is a dominant interaction process between confined identical one-dimensional fermions. The bosonization method is applied to the calculation of one-particle matrix elements at zero temperature. A detailed discussion of the validity of the results from bosonization is given, including a comparison with direct numerical diagonalization in fermionic Hilbert space. A model for the interaction coefficients is developed along the lines of the Luttinger model with only one coupling constant K. With these results, particle densities, the Wigner function, and the central pair-correlation function are calculated and displayed for large fermion numbers. It is shown how interactions modify these quantities. The anomalous dimension of the pair-correlation function in the center of the trap is also discussed and found to be in accord with the Luttinger model.