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.
It is a well-established fact that statistical properties of energy-level spectra are the most efficient tool to characterize nonintegrable quantum systems. The statistical behavior of different systems such as complex atoms, atomic nuclei, two-dimensional Hamiltonians, quantum billiards, and noninteracting many bosons has been studied. The study of statistical properties and spectral fluctuations in interacting many-boson systems has developed interest in this direction. We are especially interested in weakly interacting trapped bosons in the context of Bose-Einstein condensation (BEC) as the energy spectrum shows a transition from a collective nature to a single-particle nature with an increase in the number of levels. However this has received less attention as it is believed that the system may exhibit Poisson-like fluctuations due to the existence of an external harmonic trap. Here we compute numerically the energy levels of the zero-temperature many-boson systems which are weakly interacting through the van der Waals potential and are confined in the three-dimensional harmonic potential. We study the nearest-neighbor spacing distribution and the spectral rigidity by unfolding the spectrum. It is found that an increase in the number of energy levels for repulsive BEC induces a transition from a Wigner-like form displaying level repulsion to the Poisson distribution for
P(s). It does not follow the Gaussian orthogonal ensemble prediction. For repulsive interaction, the lower levels are correlated and manifest level-repulsion. For intermediate levels
P(s) shows mixed statistics, which clearly signifies the existence of two energy scales: external trap and interatomic interaction, whereas for very high levels the trapping potential dominates, generating a Poisson distribution. Comparison with mean-field results for lower levels are also presented. For attractive BEC near the critical point we observe the Shnirelman-like peak near
s=0, which signifies the presence of a large number of quasidegenerate states.
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.
It is a well-known fact that the statistical behaviors of level fluctuation and level correlation in the energy-level spectra are the most efficient tool to characterize quantum chaos in nonintegrable quantum systems. The system of interacting trapped bosons is a complex system where the low-lying energy levels are highly influenced by the level repulsion. In this case, interatomic interaction is a dominating fact with strong level correlation between distant levels. Here we numerically calculate the correlation function, number variance, and Dyson-Mehta least-square deviation for the low-lying levels for a few thousand interacting trapped bosons, and our data show good analogy with the Gaussian orthogonal ensemble (GOE) results with a signature of chaos. In the next part of our study, the energy spectrum of these low-lying levels is considered as a discrete signal and the fluctuation of the excitation energy is considered as discrete time series. Then we calculate numerically the height-height correlation function for different order of momentum. In our study logarithmic correlation structure is found instead of multiscaling structure, and we observe that spectral statistics are compatible with those of GOE.
The emergence of quantum chaos in a system of trapped interacting bosons with externally impressed rotation is studied through spectral form factor (SFF) and power spectrum using exact diagonalization. Two distinct interaction regimes are considered: the moderate, when the interaction energy is small compared to the trap energy and the strong, when the interaction energy is comparable to the trap energy. In the moderate interaction regime, the SFF for the non-rotating case exhibits a dip-plateau structure with absence of linear ramp, indicating integrable behavior, while for the single-vortex state the SFF exhibits a discernible linear ramp consistent with pseudo-integrable behavior. In the strong interaction regime, the non-rotating case exhibits emergence of a linear ramp with small time span in SFF, indicating that the system has moved further towards chaotic regime but continues to be pseudo-integrable. For the single-vortex and the multi-vortex states in strong interaction regime, the span of the linear ramp in SFF increases progressively with rotation, indicating the system has moved into strong chaotic regime consistent with Gaussian orthogonal ensemble. The power spectrum results with exponent lying in the interval
1≲α≲2 are consistent with the findings of SFF. An understanding of the observed crossover from integrable to quantum chaos is presented in terms of the macroscopic occupation of a single-particle quantum state---the Bose-Einstein condensation---and its depletion driven by interaction and rotation.
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.
We present exact diagonalization study on a system of
10≤N≤24 spinless bosons interacting via repulsive Gaussian potential, harmonically confined in
xy-plane with an externally impressed rotation about the
z-axis. The two-body interaction strength in the Gaussian potential is taken in the strongly interacting regime with values of interaction range in the regime
0≤σ≤1. The diagonalization of the
N-body Hamiltonian matrix, in subspaces of total angular momentum in the regime
0≤Lz≤4N corresponds to the filling fraction
ν≲3.2 is carried out to obtain the variationally exact ground-state wavefunction and the corresponding eigenvalue. It is found that an increase in interaction range
σ leads to (a) a systematic decrease in energy, (b) an increase in the critical angular velocity
Ωci of the
i{th} vortex state and (c) an increase in the largest eigenvalue
λ1, (condensate fraction), of one-particle reduced density matrix (OPRDM). The von Neumann entropy
S1(Lz,σ), quantifying the quantum entanglement between the particles in the many-body ground state, is largely found to decrease with increase in
σ. Crossings in von Neumann entropy for several of the angular momentum states are observed with variation in
σ. The response of the Bose-Einstein condensate to rotation is examined through
Lz(σ)−Ω(σ) stability graph for several values of
σ. A vortex state with larger plateau length on the
Lz−Ω graph is considered to be more stable. The internal structure of the condensate, as depicted by the conditional probability distribution (CPD) in the body-fixed frame, exhibits characteristic features with interaction range
σ. One such feature is the merging of the cores of the two-vortex state.