We investigate the effects of the finite-range interactions of six rotating ultracold bosonic atoms using a Gaussian-type interatomic interaction model. The model is analyzed numerically by exact diagonalization within the Lowest Landau Level (LLL) approximation and semiclassical approximation. The result of exact diagonalization shows that the ground-state angular momentum changes discretely with increasing angular velocity. For the short-range limit, the ground-state angular momentum and wavefunctions agree with those of the delta interaction evaluated by Bertsch and Papenbrock [Phys. Rev. A 63, 023616 (2001)]. Different from the delta interaction, the ground-state angular momenta higher than 30, i.e., N(N − 1), are observed at a high angular frequency as a result of the finite-range two-body interactions. For the intermediate-range interaction, the sequence of ground-state angular momenta increases in steps of five, which was not found in previous works on the Gaussian interaction. For the long-range limit of Gaussian interaction, we find that the ground-state angular momenta increase in steps of six. These steps of the ground-state angular momentum according to the width of the Gaussian interactions are explained by semiclassical and classical analysis based on the rovibrating molecule picture. The increments of the ground-state angular momentum of five and six are explained by the semiclassical quantization condition of the rotational and vibrational modes of fivefold and sixfold molecules, respectively. Our analysis based on the classical model also confirms that the fivefold molecule picture is more stable than the sixfold molecule picture in the intermediate range of the Gaussian interaction. These results suggest that the Gaussian interaction model can be used to emulate and characterize interactions by their width as the model can reproduce various rotational states including the ground states of rotating systems interacting with delta-type and Coulomb type interactions.
We investigate the effects of the finite-range interactions of six rotating ultracold bosonic atoms using a Gaussian-type interatomic interaction model. The model is analyzed numerically by exact diagonalization within the Lowest Landau Level (LLL) approximation and semiclassical approximation. The result of exact diagonalization shows that the ground-state angular momentum changes discretely with increasing angular velocity. For the short-range limit, the ground-state angular momentum and wavefunctions agree with those of the delta interaction evaluated by Bertsch and Papenbrock [Phys. Rev. A 63, 023616 (2001)]. Different from the delta interaction, the ground-state angular momenta higher than 30, i.e., N(N − 1), are observed at a high angular frequency as a result of the finite-range two-body interactions. For the intermediate-range interaction, the sequence of ground-state angular momenta increases in steps of five, which was not found in previous works on the Gaussian interaction. For the long-range limit of Gaussian interaction, we find that the ground-state angular momenta increase in steps of six. These steps of the ground-state angular momentum according to the width of the Gaussian interactions are explained by semiclassical and classical analysis based on the rovibrating molecule picture. The increments of the ground-state angular momentum of five and six are explained by the semiclassical quantization condition of the rotational and vibrational modes of fivefold and sixfold molecules, respectively. Our analysis based on the classical model also confirms that the fivefold molecule picture is more stable than the sixfold molecule picture in the intermediate range of the Gaussian interaction. These results suggest that the Gaussian interaction model can be used to emulate and characterize interactions by their width as the model can reproduce various rotational states including the ground states of rotating systems interacting with delta-type and Coulomb type interactions.
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.
The Hamiltonian for a small number $N\ensuremath{\leqslant}11$ of bosons in a rapidly rotating harmonic trap, interacting via a short range (contact potential) or a long range (Coulomb) interaction, is studied via an exact diagonalization in the lowest Landau level. Our analysis shows that, for both low and high fractional fillings, the bosons localize and form rotating boson molecules (RBMs) consisting of concentric polygonal rings. Focusing on systems with the number of trapped atoms sufficiently large to form multiring bosonic molecules, we find that, as a function of the rotational frequency and regardless of the type of repulsive interaction, the ground-state angular momenta grow in specific steps that coincide with the number of localized bosons on each concentric ring. Comparison of the conditional probability distributions (CPDs) for both interactions suggests that the degree of crystalline correlations appears to depend more on the fractional filling $\ensuremath{\nu}$ than on the range of the interaction. The RBMs behave as nonrigid rotors, i.e., the concentric rings rotate independently of each other. At filling fractions $\ensuremath{\nu}<1∕2$, we observe well developed crystallinity in the CPDs (two-point correlation functions). For larger filling fractions $\ensuremath{\nu}>1∕2$, observation of similar molecular patterns requires consideration of even higher-order correlation functions.
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.
Abstract The ground and low-lying collective states of a rotating system of N = 3 bosons harmonically confined in quasi-two-dimension and interacting via repulsive finite-range Gaussian potential is studied in weakly to moderately interacting regime. The N-body Hamiltonian matrix is diagonalized in subspaces of quantized total angular momenta 0 ≥ L ≥ 4N to obtain the ground and low-lying eigenstates. Our numerical results show that breathing modes with N-body eigenenergy spacing of 2ħω⊥, known to exist in strictly 2D system with zero-range (δ-function) interaction potential, may as well exist in quasi-2D system with finite-range Gaussian interaction potential. To gain an insight into the many-body states, the von Neumann entropy is calculated as a measure of quantum correlation and the conditional probability distribution is analyzed for the internal structure of the eigenstates. In the rapidly rotating regime the ground state in angular momentum subspaces L = (q/2)N (N − 1) with q = 2, 4 is found to exhibit the anticorrelation structure suggesting that it may variationally be described by a Bose-Laughlin like state. We further observe that the first breathing mode exhibits features similar to the Bose-Laughlin state in having eigenenergy, von Neumann entropy and internal structure independent of interaction for the three-boson system considered here. On the contrary, for eigenstates lying between the Bose-Laughlin like ground state and the first breathing mode, values of eigenenergy, von Neumann entropy and internal structure are found to vary with interaction.
We investigate a system of N spinless bosons confined in quasi-two-dimensional harmonic trap with repulsive two-body finite-range Gaussian interaction potential of large s-wave scattering length. Exact diagonalization of the Hamiltonian matrix is carried out to obtain the N-body ground state as well as low-lying excited states, using Davidson algorithm in beyond lowest-Landau-level approximation. We examine the finite-range effects of the interaction potential on the many-body ground state energy as also the degree of condensation of the Bose-condensed gas. The results obtained indicate that the finite-range Gaussian interaction potential enhances the degree of condensation compared to the zero-range interaction potential. We further analyze the effect of finite-range interaction potential on the breathing mode collective excitation. Our theoretical results may be relevant for experiments currently conducted on quasi-two-dimensional Bose gas with more realistic interaction potential.
We present an exact diagonalization study of the breathing mode collective excitations for a rotating Bose-Einstein condensate of N=10 spinless bosons interacting via repulsive finiterange Gaussian potential and harmonically confined in quasi-two-dimension. The yrast state and the low-lying excited states are variationally obtained in given subspaces of the quantized total angular momentum L employing the beyond lowest Landau level approximation in slowly rotating regime with 0≤L<2N. For a given L, the low-energy eigenspectra (bands) are obtained in weakly to moderately interacting regime. Further, for a given interaction, the split in low-lying eigenenergies with increasing L is the precursor to spontaneous symmetry breaking of the axisymmetry associated with the entry of the first vortex. With increase in repulsive interaction, the value of the first breathing mode increases for stable total angular momentum states L=0 and N, but decreases for intermediate 0<L<N metastable states. The position of the observed first breathing modes in the eigenspectrum remains unchanged as the interaction is varied over several orders of magnitude.