Quantum Scars and Regular Eigenstates in a Chaotic Spinor Condensate

Quantum many-body scars consist of a few low-entropy eigenstates in an otherwise chaotic many-body spectrum, and can weakly break ergodicity resulting in robust oscillatory dynamics. The notion of quantum many-body scars follows the original single-particle scars introduced within the context of quantum billiards, where scarring manifests in the form of a quantum eigenstate concentrating around an underlying classical unstable periodic orbit. A direct connection between these notions remains an outstanding problem. Here, we study a many-body spinor condensate that, owing to its collective interactions, is amenable to the diagnostics of scars. We characterize the system’s rich dynamics, spectrum, and phase space, consisting of both regular and chaotic states. The former are low in entropy, violate the eigenstate thermalization hypothesis, and can be traced back to integrable effective Hamiltonians, whereas most of the latter are scarred by the underlying semiclassical unstable periodic orbits, while satisfying the eigenstate thermalization hypothesis. We outline an experimental proposal to probe our theory in trapped spin-1 Bose-Einstein condensates.

Figure 1

(a) The entanglement entropy ${\mathcal{S}}_n^{(1)}$ of the eigenvalues of one-body reduced density matrix with respect to energy density $E_n/N$. ${\mathcal{S}}_n^{(1)}$ distinguishes thermal eigenstates, which nearly saturate the upper bound ${\mathcal{S}}_n^{(1)}=\ln 3\approx 1.1$, from an array of towers of regular eigenstates with lower entropy. (b)–(d) Phase space distributions ${\mathcal{P}}_n(n_0,\theta )$ of selected eigenstates, corresponding to the purple circle, green triangle, and red square markers in panel (a). Regular states (b) lie within a restricted region of the phase space, whereas the scarred states (c),(d) concentrate around the classical UPOs (white and red dashed lines) to some variable degree [strong for (c), scarness $D_n\approx 2.1$, and weak for (d), $D_n\approx 1.2$]. Here, we consider $N=200$ particles.

Figure 2

(a) Poincaré section of the classical equations of motion Eq. (3) at energy $E/N=0.24$, showing a mixed phase space. The colors represent the Lyapunov exponent $\lambda$ of trajectories in phase space. The UPO at the same energy is represented by a solid black line (${\lambda }_{\mathrm{UPO}}=0.12$). The dark blue dots at $n_0\ll 1$ correspond to stable periodic orbits ($\lambda \approx 0$). The density of states vanishes in the gray background. (b) Distribution of the scarness $D_n$ for ${10}^3$ eigenstates at energy $E=0.24N$.

Figure 3

(a) Eigenstate expectation values $⟨{\widehat{n}}_0⟩$, showing a narrow branch of thermal states with small fluctuations of $⟨{\widehat{n}}_0⟩$, in agreement with ETH, and towers of regular states violating ETH. The red circles are obtained from a numerical diagonalization of the effective Hamiltonian Eq. (6). Level spacing statistics for the regular (b) and thermal (c) states. The solid lines depict the Poisson and the Wigner-Dyson distributions, respectively.

Figure 4

Time evolution of (a) $⟨{\widehat{n}}_0⟩$ and (b) the fidelity for an initial Fock state overlapping with regular states (green lines), a coherent state initiated on a UPO (red lines) and a coherent state in the chaotic sea (blue lines). The line intensity denotes increasing atom number (from 64 to $N=1024$). In (a), the solid black line is the microcanonical ensemble prediction with an energy window that excludes the regular states, and it is identical for all initial states that are at the same energy $E=0.24N$. In (c), we show the fidelity achieved at the first revival as a function of the atom number.