Antiscarring from eigenstate stacking in a chaotic spinor condensate

TL;DR

This work demonstrates that quantum scars in a chaotic many-body system are complemented by antiscarred states to maintain phase-space uniformity within an energy window; by extending the eigenstate stacking theorem to a spin-1 chaotic spinor condensate with a semiclassical SU(3) coherent-state limit, the authors link early-time dynamics and the shortest periodic orbit period $t^*$ to the required window $\Delta E > 2\pi/t^*$. They show GOE-like spectral statistics and ETH across the spectrum while observing robust scar signatures, and they explicitly reveal antiscarring via projected stacking and Husimi projections, with approximate uniformity improving in the thermodynamic limit. The study offers a concrete path to experimental observation of antiscarring and deepens the connection between quantum chaos, scarring, and ergodicity in many-body quantum systems.

Abstract

We reveal a feature of quantum scarring in systems with many particles: Quantum scars, living densely near an unstable periodic orbit, must be compensated by corresponding antiscarred states suppressed there to establish the uniformity of the whole. The uniformity of the underlying phase space is linked to early-time dynamics -- a regime beyond the predictions of random matrix theory and encapsulated in the eigenstate stacking theorem. By extending the domain of the stacking theorem, we apply our theory to a chaotic spinor Bose-Einstein condensate, whose quantum scar dynamics have recently been observed in the laboratory. Our work uncovers how scarring of some eigenstates affects the rest of the chaotic and thermal spectrum in quantum systems with many particles.

Figures (8)

• Figure 1: (a) The Poincare section at energy density $E=0.24$ with colorbar denoting the Lyapunov exponents of the trajectories. Almost entire phase space is chaotic, with a small regular island at $n_0\sim 0.7$. The periodic orbit highlighted in red is unstable with Lyapunov exponent $\lambda=0.31$. (b) Periods of the UPO family $T_{\text{UPO}}$ are found to be continuous. At $p=0.5$, $T_{\text{UPO}}$ spans from $\sqrt{2}\pi$ to infinity. (c) Spectral rigidity $\Delta_3$ as a function of energy window $\Delta E$ with increasing system size $N$ (light to dark green). The saturation is observed after the energy width $\sim 2\pi/T_{\text{UPO}}^*$, marked by the dashed gray line. (d) Normalized connected spectral form factor (cSFF) computed for $N=100$ (blue) and $N=150$ (orange). The cSFF follows the GOE prediction marked with the dashed black curve, whereas at early times it peaks at the period of the shortest UPO $T_{\text{UPO}}^*$ (inset).
• Figure 2: (a,b) One-body entanglement entropy and eigenstate expectation value $\langle n_0 \rangle$ with respect to energy density $E/N$ for $N=200$ atoms and $p=0.5$. (c) Distribution of ratio of nearest-neighbor energy levels (gap ratio) matches well with the Wigner-Dyson statistics of GOE (red). (d) Distribution of scaled eigenstate element $\eta$ within energy range $0.18 < E_n /N <0.3$ agrees with Porter-Thomas distribution of GOE (i.e., $\chi^2$ distribution with one degree of freedom), marked by the red line.
• Figure 3: (a) The scarmometer $\mathcal{F}_n$ of \ref{['eq:scarmometer']}, plotted with respect to energy density. (b,c) Projected Husimi-Q distribution of cumulative scar and antiscarring around UPO at $E_0/N=0.24$ with $N=100$, \ref{['eq:proj_stack']}. Here the colormap is normalized by the maximum value. The gray area indicates no density of states. The cumulative scar is obtained by stacking scar eigenstates with $\mathcal{F}_n > 0.03$ indicated by the dashed red line in (a).
• Figure 4: (a) The survival probability of an initial state on UPO (red) and off UPO, i.e., on a chaotic trajectory (blue). (b) The first three revival amplitudes for dynamics starting on UPO (red) and off UPO (blue). The dynamics started on UPO has revivals robust to increasing the system size, whereas the dynamics off UPO does not show revivals in the thermodynamic limit. (c) The spin$-0$ population dynamics for two different initial states at the same energy where one is chosen on the UPO (red) $\ket{\zeta_s}\equiv\ket{n_0,m,\theta,\eta}=\ket{0.4,0,\pi,0}$ and the other off UPO on a chaotic trajectory (blue) $\ket{\zeta_c}=\ket{0.4,0,0,\pi}$. The saturation values denote the system size given in the legend. The black solid line is the microcanonical ensemble prediction.
• Figure 5: (a) Projected eigenstate stacking at energy density center $E_0/N=0.24$ with energy window fixed $\Delta E/N=0.6$ with increased uniformity observed with increasing $N$. (b) The deviation from perfect uniformity $\sigma$ with respect to energy window for different atom numbers and for different system sizes. (c) Projected eigenstate stacking at energy density center $E_0/N=0.24$ with atom number fixed to $N=100$ and for smaller energy windows than $\Delta E/N=0.6$, showing more inhomogeneity. (d) Projected eigenstate stacking with atom number fixed to $N=100$ and energy window $\Delta E/N=0.6$ with different energy density center $E_0/N$. Approximate uniformity at finite-sizes persists. All phase space diagrams in (c) and (d) share the same colorbar as in (a).