We consider a harmonically trapped few-boson system under rotation and investigate the ground-state properties beyond the usual ``lowest-Landau-level'' approximation by using exact diagonalizations in a restricted Hilbert subspace. We find that both the effective interaction energy and density distribution are strongly affected by the two-body interaction strength.
We consider a harmonically trapped few-boson system under rotation and investigate the ground-state properties beyond the usual ``lowest-Landau-level'' approximation by using exact diagonalizations in a restricted Hilbert subspace. We find that both the effective interaction energy and density distribution are strongly affected by the two-body interaction strength.
Abstract We use exact diagonalization to study an interacting system of N spinless bosons with finite-range Gaussian repulsion, confined in a quasi-two-dimensional harmonic trap with and without an introduced rotation. The diagonalization of the Hamiltonian matrix using Davidson algorithm in subspaces of quantized total angular momentum L z is carried out to obtain the N -body lowest eigenenergy and eigenstate. To bring out the effect of quantum (Bose) statistics and consequent phase stiffness (rigidity) of the variationally obtained many-body wavefunction on various physical quantities, our study spans from few-body ( N = 2) to many-body ( N = 16) systems. Further, to examine the finite-range effect of the repulsive Gaussian potential on many-body ground state properties of the Bose-condensate, we obtain the lowest eigenstate, the critical angular velocity of single vortex state and the quantum correlation (measured) in terms of von Neumann entanglement entropy and degree of condensation. It is found that for small values of the range (measured by the parameter σ ) of Gaussian potential, the ground state energy increases for few-boson (2 ⩽ N ⩽ 8) systems but decreases for many-boson ( N > 8) systems. On the other hand for relatively large values of the range of Gaussian potential, the ground state energy exhibits a monotonic decrease, regardless of the number of bosons N . For a given N , there is found an optimal value of the range of Gaussian potential for which the first vortex (with L z = N ) nucleates at a lower value of the rotational angular velocity Ω c1 compared to the zero-range ( δ -function) potential. Further, we observe that the inter-particle interaction and the introduced rotation are competing effects with latter being dominant over the former. With increase in the range of Gaussian potential, the value of von Neumann entropy decreases and the degree of condensation increases implying an enhanced quantum correlation and phase rigidity.
We present an exact diagonalization study of the spectral properties of bosons harmonically confined in a quasi-2D plane and interacting via repulsive Gaussian potential. We consider the lowest 100 energy levels for systems of N=12,16, and 20 bosons in two distinct regimes: (a) when the interaction energy is small compared to the trap energy (moderate interaction) and (b) when the interaction energy is comparable to the trap energy (strong interaction), for the nonrotating (L_{z}=0) as well as the rotating single-vortex state (L_{z}=N). For higher angular momenta, L_{z}=2N and L_{z}=3N, only the strong interaction regime is considered. While the nearest-neighbor spacing distribution (NNSD) P(s) and the ratios of consecutive level spacings distribution P(r) are used to study the short-range correlations, the Dyson-Mehta Δ_{3} statistic and the level number variance Σ^{2}(L) are used to examine the long-range correlations. In the moderate interaction regime, the nonrotating system exhibits Poisson distribution, a characteristic of the regular energy spectra. In the strong interaction regime, the nonrotating system exhibits chaotic behavior signified by GOE distribution. Furthermore, in the rotating case for the single-vortex state (L_{z}=N) in the moderate interaction regime, the system exhibits signatures of weak chaos with some degree of regularity in the energy-level spectra. However, in the strong interaction regime for the rotating case with L_{z}=N, 2N and 3N, the system exhibits strong chaotic behavior. The rotation is found to contribute to enhancement of chaotic behavior in the system for both the moderate and the strong interaction regimes. Our results of NNSD analysis are supported by the analysis of the ratios of consecutive level spacings distribution P(r), which does not involve unfolding.
We present results for the ground states of a system of spin-1 bosons in a rotating trap. We focus on the dilute, weakly interacting regime, and restrict the bosons to the quantum states in the lowest Landau level (LLL) in the plane (disk), sphere, or torus geometries. We map out parts of the zero-temperature phase diagram, using both exact quantum ground states and LLL mean-field configurations. For the case of a spin-independent interaction we present exact quantum ground states at angular momentum L<N. For general values of the interaction parameters, we present mean-field studies of general ground states at slow rotation and of lattices of vortices and skyrmions at higher rotation rates. Finally, we discuss quantum Hall liquid states at ultrahigh rotation.
We consider systems of a small number of interacting bosons confined to harmonic potentials in one and two dimensions. By exact numerical diagonalization of the many-body Hamiltonian we determine the low-lying excitation energies and the ground-state energy and density profile. We discuss the dependence of these quantities on both interaction strength g and particle number N. The ground-state properties are compared to the predictions of the Gross-Pitaevskii equation, and the agreement is surprisingly good even for relatively low particle numbers. We also calculate the specific heat based on the obtained energy spectra.
We investigate a rotating system of N spinless bosons confined in quasi-two-dimensional harmonic trap with repulsive finite-range Gaussian interaction potential of large s-wave scattering length. Exact diagonalization of the Hamiltonian matrix in subspaces of quantized total angular momenta Lz is carried out to obtain the N-body ground states in slow rotating regime of 0≤Lz≤2N. We study the finite-range effects of the Gaussian repulsion potential on the many-body ground state properties in different subspaces of quantized Lz. The study ranges from the few-body (N=2) system to the many-body (N=16), system where quantum (Bose) statistics becomes perceptible. In particular, we analyze the effect of interaction range of the Gaussian potential on the ground state energy of different Lz states as well on the critical angular velocity Ωc1 of first vortex (Lz=N) state. For an optimal value of the range of the Gaussian interaction potential, the nucleation of the first centered vortex may begin at a lower value of the rotational angular velocity as compared to the zero-range interaction potential. Moreover, we explore the role of interaction potential range on the quantum correlation (measured in terms of the degree of condensation and the von Neumann entropy), for small systems (N=2 to 8) where an understanding of few-body effects provide valuable insight into large systems (N=8 to 16) which admit the possibility of many-body correlation. The results obtained indicate that the finite-range Gaussian interaction potential enhances the degree of condensation compared to the zero-range interaction potential.
In this thesis, we study the physical properties of several ultracold few-boson systems depending on the interactions between their constituents. Nowadays, experimentally, it is possible to have great control with high precision over the geometry and the interactions between the particles, making them an excellent setup to test directly the principles of quantum mechanics. A very interesting point is to study the evolution of their properties with the number of particles. The theoretical study of these systems pretends to microscopically understand the current experimental results and give support to new experimental developments.\n\n\t\t\t\t The method that will be used is the exact diagonalization of the Hamiltonian of the system. As we will see, in spite of the attempts to improve it, the method is limited by the fact that, in practice, it is only useful to study few-particle systems. The method has several advantages. First of all, one has access to both the ground and the excited states. In second place, the method is variational and converges to the exact solution as long as the Hilbert space in which we diagonalize is enlarged.\n\t\t\t\t Moreover, since we have access to the states of the system, it is possible to calculate any observable quantity of interest.\n\n\t\t\t\t First, we will study a system of spinless bosons trapped in a two-dimensional harmonic potential. The effect of the trap is to keep the system bound. It will be seen how the presence of a repulsive interaction changes the energy spectrum and other properties of the system. For instance, the density profile, which is usually measurable, and also the two-body distribution function, which is intimately related to the existence of correlations.\n\n\t\t\t\t Afterwards, the focus will be on the particular case of having only two bosons in the system interacting through a strong repulsive force. Inspired by the one-dimensional case where the fermionization phenomenon takes place in the strongly-interacting limit, we will study whether in two dimensions there is a resembling reminiscent effect. In other words, we will analyze if there are properties of the two strongly-interacting bosons in two dimensions that are like the ones of two noninteracting fermions.\n\n\t\t\t\t After that, we will tackle the localization phenomenon in a one-dimensional system that is caused by an external speckle potential that introduces disorder in the system. We will show that the localization is a robust phenomenon against repulsive contact interactions.\n\n\t\t\t\t Finally, we will study the influence of the spin-orbit coupling in a system of bosons with two possible pseudospin components, associated, for instance, to two hyperfine levels, confined in a two-dimensional harmonic trap. We will present an exhaustive analysis of the combined effects of the interaction and the spin-orbit coupling in the spectrum and the properties of the system. In particular we show the existence of a crossover in the ground state of the system susceptible to be experimentally identified.