Many-Body Quantum Chaos and Entanglement in a Quantum Ratchet

We uncover signatures of quantum chaos in the many-body dynamics of a Bose-Einstein condensate-based quantum ratchet in a toroidal trap. We propose measures including entanglement, condensate depletion, and spreading over a fixed basis in many-body Hilbert space, which quantitatively identify the region in which quantum chaotic many-body dynamics occurs, where random matrix theory is limited or inaccessible. With these tools, we show that many-body quantum chaos is neither highly entangled nor delocalized in the Hilbert space, contrary to conventionally expected signatures of quantum chaos.

Figure 1

Quantum ratchet in a ring trap. (a) BEC (blue torus) trapped in a ring geometry (yellow), being rotated off center (black arrow). The driving breaks generalized $P$ and $T$ symmetries, giving rise to the ratchet effect. (b) The system can be treated as a one-dimensional condensate, driven by the symmetry breaking potential $V(t)$, generated by the rotation of the trap. (c) The effective three-level system, derived from the Floquet-inspired $(t,t^{\prime })$ formalism [13], provides a simplified treatment of the quantum ratchet. The system exhibits three dynamical regimes in particle current, Rabi oscillations, chaos, and self-trapping, for increasing particle interactions.

Figure 2

Random Matrix Theory Analysis. (a) The quasienergy level statistics of the driven Bose-Hubbard model obtained from the Floquet operator. The dynamical regimes are Rabi oscillations, chaos, and self-trapping for interaction strengths of $\tilde{U}=0.011$, 0.021, and 0.034, respectively, and the black line indicates Poisson statistics. The quasienergy level spacings have increasing probability as the spacing tends to zero for both the regular and chaotic regimes, contrary to the prediction of Random Matrix Theory. (b) Level repulsion is recovered in the truncated three-level system, shown for particle number $N=50$. (c) The Brody distribution (solid curves in (b)), which interpolates between Poisson statistics ($\eta =0$) and Wigner-Dyson statistics ($\eta =1$, black curve), has turning points which identify the chaotic regime. The fits ignore the initial fluctuations for $\tilde{U}<0.008$, since they correspond to Rabi dynamics. (d) Critical $\tilde{U}$ as a function of $N$, which asymptotically define the chaotic regime [gray shading in (c)], where the colored regions give the error in $g$.

Figure 3

Quantum many-body measures. (a)–(c) The chaotic regime is indicated by vertical shading spanning from ${\tilde{U}}_s$ to ${\tilde{U}}_e$, with the solid black line giving the maximally chaotic point ${\tilde{U}}_m$, according to the level statistics in the large $N$ limit. (a) Entropy and (b) depletion show characteristic increases in the chaotic regime in $x$ space, $k$ space, and the truncated Floquet space representations of the ratchet [shared key in (a)]. (c) Inverse participation ratio shows a characteristic increase in the chaotic regime only in $k$ space and the truncated Floquet space [shared key in (a)]. Each measure allows a fit (solid lines), which can be used to extract the onset of chaos, maximal chaos, and end point of chaos. (d) Entropy and depletion exhibit similar trends regardless of lattice length in $x$ space. The lack of increasing entropy indicates the system can be simulated using matrix product states methods.