Spectral form factor and power spectrum for trapped rotating interacting bosons: An exact diagonalization study

TL;DR

This work investigates quantum-chaos signatures in trapped rotating interacting bosons by computing the spectral form factor (SFF) and power spectrum from exact-diagonalization of the lowest 100 energy levels in a quasi-2D harmonic trap with Gaussian repulsion. By varying interaction strength and angular momentum, it identifies integrable (Poisson-like) behavior in the non-rotating moderate regime, a pseudo-integrable regime upon moderate vortex formation, and a strongly chaotic regime when both strong interactions and multiple vortices are present, as reflected in dip–ramp–plateau SFF structures and $P_k$ exponents near $\alpha=1$. The analysis highlights condensate depletion due to vortex formation and strong interactions as the key mechanism driving the crossover to chaos, with consistent signatures across non-rotating, single-vortex, and multi-vortex configurations. These results connect Bose-Einstein condensate vortex physics to quantum-chaos diagnostics and demonstrate how exact diagonalization can reveal regime-dependent spectral correlations in many-body bosonic systems.

Abstract

We present an exact diagonalization study of the spectral form factor and the power spectrum for externally impressed rotating bosons in a quasi-two-dimensional harmonic trap interacting via repulsive Gaussian potential. Our focus is on the spectral correlations arising from the condensate depletion as a result of the formation of quantized vortices and the strong interaction, which leads to chaos in the system. We consider two distinct interaction regimes: moderate, where the interaction energy is small compared to the trap energy, and strong, where it is comparable. For the non-rotating case, the spectral form factor (SFF) in the moderate interaction regime exhibits a dip-plateau structure characteristic of integrable systems, while in the strong interaction regime it develops a weak ramp, signalling a transition to pseudo-integrable behavior. For the rotating single-vortex state, the SFF in the moderate interaction regime shows the onset of a weak ramp consistent with pseudo-integrability, whereas in the strong interaction regime the ramp becomes significantly more pronounced, indicating strongly chaotic behavior. For the multi-vortex states in the strong interacting regime, the SFF exhibits a clear dip-ramp-plateau structure, indicative of chaotic behavior. The corresponding power spectrum results further corroborate these findings: integrable system exhibits the characteristic $1/f^α$ (with $α\approx2$) noise, chaotic system shows the ubiquitous $1/f^α$ (with $α\approx 1$) noise, and pseudo-integrable system yields intermediate exponents with $1 < α< 2$.

Figures (6)

• Figure 1: (Color online) Spectral form factor $K(\tau)$ (with moving time average in a logarithmic window) on the log-log scale in the moderate interaction regime with $g_{2}=0.3669$ (upper panel) and the strong interaction regime with $g_{2}=3.669$ (lower panel) for $N=12, 16$ and $20$ bosons with total angular momentum $L_{z}=0$. The horizontal dashed line corresponds to the asymptotic limit of the SFF, $\langle K(\tau) \rangle = 1/L$, with $L$ denoting the total number of energy levels considered. The orange dotted vertical lines mark the dip time $\tau_{dip}$ and the blue dotted vertical lines mark the Heisenberg time $\tau_{H}$.
• Figure 2: (Color online) Plot of the power spectrum $P_{k}$ vs $k$ on the log-log scale in the moderate (left) and the strong (right) interaction regimes for $N=12, 16$ and $20$ bosons with total angular momentum $L_{z}=0$. The dots are our numerical results of $N=12, 16$ and $20$ bosons. The blue dashed, the red dashed and the green dashed lines are the straight line fits for $N=12, 16$ and $20$ bosons, respectively. The black square line and the black solid line are the reference lines for the GOE and the Poisson statistics, respectively.
• Figure 3: (Color online) Spectral form factor $K(\tau)$ (with moving time average in a logarithmic window) on the log-log scale in the moderate interaction regime with $g_{2}=0.3669$ (upper panel) and the strong interaction regime with $g_{2}=3.669$ (lower panel) for $N=12, 16$ and $20$ bosons with total angular momentum $L_{z}=N$. The horizontal dashed line corresponds to the asymptotic limit of the SFF, $\langle K(\tau) \rangle = 1/L$, with $L$ denoting the total number of energy levels considered. The orange dotted vertical lines mark the dip time $\tau_{dip}$ and the blue dotted vertical lines mark the Heisenberg time $\tau_{H}$.
• Figure 4: (Color online) Plot of the power spectrum $P_{k}$ vs $k$ on the log-log scale in the moderate (left) and the strong (right) interaction regimes for $N=12, 16$ and $20$ bosons with total angular momentum $L_{z}=N$. The dots are our numerical results of $N=12, 16$ and $20$ bosons. The blue dashed, the red dashed and the green dashed lines are the straight line fits for $N=12, 16$ and $20$ bosons, respectively. The black square line and the black solid line are the reference lines for the GOE and the Poisson statistics, respectively.
• Figure 5: (Color online) Spectral form factor $K(\tau)$ (with moving time average in a logarithmic window) on the log-log scale in the strong interaction regime with $g_{2}=3.669$ for $N=12, 16$ and $20$ bosons with total angular momentum $L_{z}=2N$ (upper panel) and $L_{z}=3N$ (lower panel). The horizontal dashed line corresponds to the asymptotic limit of the SFF, $\langle K(\tau) \rangle = 1/L$, with $L$ denoting the total number of energy levels considered. The orange dotted vertical lines mark the dip time $\tau_{dip}$ and the blue dotted vertical lines mark the Heisenberg time $\tau_{H}$.