Layered chaos in mean-field and quantum many-body dynamics

We investigate the dimension of the phase-space attractor of a quantum chaotic many-body ratchet in the mean-field limit. Specifically, we explore a driven Bose-Einstein condensate in three distinct dynamical regimes—Rabi oscillations, chaos, and self-trapping regimes—and for each of them we calculate the correlation dimension. For the ground state of the ratchet formed by a system of field-free noninteracting particles, we find four distinct pockets of chaotic dynamics throughout these regimes. We show that a measurement of local density in each of the dynamical regimes has an attractor characterized by a higher fractal dimension, $D_R=2.59\pm 0.01$, $D_C=3.93\pm 0.04$, and $D_S=3.05\pm 0.05$, compared to the global measure of current, $D_R=2.07\pm 0.02$, $D_C=2.96\pm 0.05$, and $D_S=2.30\pm 0.02$. The deviation between local and global measurements of the attractor's dimension corresponds to an increase towards higher condensate depletion, which remains constant for long time scales in both Rabi and chaotic regimes. The depletion is found to scale polynomially with particle number $N$, namely, as $N^{\beta }$ with ${\beta }_R=0.51\pm 0.004$ and ${\beta }_C=0.18\pm 0.004$ for the two regimes. Thus, we find a strong deviation from the mean-field results, especially in the chaotic regime of the quantum ratchet. The ratchet also reveals quantum revivals in the Rabi and self-trapping regimes but not in the chaotic regime, with revival times scaling linearly in particle number for Rabi dynamics. Based on the obtained results, we outline pathways for the identification and characterization of emergent phenomena in driven many-body systems. This includes the identification of many-body localization from the many-body measures of the system, the influence of entanglement on the rate of the convergence to the mean-field limit, and the establishment of a polynomial scaling of the Ehrenfest time at which the mean-field description fails to describe the dynamics of the system.

Figure 1

Quantum ratchet via toroidal condensate. (a) Off center rotation of a toroidal Bose-Einstein condensate [14, 15, 16, 17, 18] generates the effect of driving by two counterpropagating waves in one spatial dimension [19]. Only when the amplitudes of the two waves are unequal can such a system present a ratchet effect due to asymmetric coupling to positive and negative angular momentum modes. (b) Space-time plot of the drive given in (a) with equal amplitude fields. (c) Once the drive amplitudes differ, the BEC becomes coupled to positive and negative modes, easy visible in the symmetries of the space-time plot for this case.

Figure 2

$0–1$ test coordinates. (a)–(i) Examples of the mapped coordinates in the $0–1$ test for chaos for the 3GP and six site DNLS for all dynamical regimes. As expected in the regular regimes, the Rabi (left) and self-trapping (right) dynamics are mapped to tori in the $\left(pq\right)$ plane. The center column, corresponding to the chaotic regime, displays random-walk-like behavior with the points spreading over the effective phase space. This diffusive behavior is characteristic of chaotic dynamics under the map used in the $0–1$ test, making use of linear scaling in the mean-squared displacement to identify chaos.

Figure 3

$0–1$ test for chaos. The output of the $0–1$ test for chaos, $K$, shows distinct pockets of chaos (gray shading) that are the same for each model and measure, with no chaos observed after the transition to self-trapping. This is in contrast to the quantum many-body dynamics and level statistics in [21], which predicted one bulk region given by the gray hashed region. The current and local density in DNLS agree exceptionally well with the 3GP, indicating that the inclusion of higher angular momentum modes in the DNLS does not affect the presence of chaos.

Figure 4

Delay reconstructed attractors. (a)–(c), (d)–(f) The current in both the 3GP and DNLS, respectively, have qualitatively similar attractors for each dynamical regime. The differences in coordinate ranges are due to the exclusion of the scaling factors for the angular momentum modes in the 3GP, which has no effect on the attractor dimension. (g)–(i) The local density in the DNLS has attractors that are visibly higher dimensional when compared to the current. In the Rabi and self-trapping regimes we see densely filled surfaces with no holes, unlike the currents, which appear as tori. The attractors for the chaotic regime all show signs of self-similarity, and have deviated from regular torus shapes to a higher-dimensional region in phase space. Coloring is meant only to provide contrast to aid the eye.

Figure 5

Correlation sums. (a)–(i) Gray dashed lines indicate the start and end of the linear scaling regions used to calculate the correlation dimension, with the embedding dimension decreasing down and to the right. Each model and measure approaches a constant slope in the indicated region, with the range of $\epsilon$ checked for being within the size of delay reconstructed attractors.

Figure 6

Converging correlation dimension. Each panel corresponds to the slope of the correlation sum given in Fig. 5 with the same letter. Each model and interaction strength shows a plateau corresponding to the convergence of the correlation dimension in the embedding dimension. The final correlation dimension is given by the average over all $m$ after the plateau is reached (see Table 1 for numerical values). The slow increase in dimension as $m$ increases after the plateau, as seen in (b) and (i), is a typical artifact of noise in the time series [27].

Figure 7

Mean-field fractal dimension. The correlation dimension, $D_2$, shows sharp increases for the regimes where chaos was identified by the $0–1$ test for chaos (vertical gray regions). The dimensions of the current in the 3GP and the DNLS follow the same trend with increasing interaction strength, further reaffirming the effectiveness of the three mode model. However, the local density in the DNLS is seen to be greater than the particle current regardless of interaction. This is due to the inclusion of rapid oscillations with the drive potential that get averaged out when the whole lattice is considered. Lines are meant only to guide the eye.

Figure 8

Dynamics beyond convergence time: (a)–(d) share the horizontal axis given by (d). (a) The normalized particle current, $I/I_{\text{max}}$, for the mean-field 3GP. (b) The 3LS for particle number $N=6$ has an envelope that decays to a nearly constant value beyond the initial time it is converged to the mean-field result. This is then followed by a quantum revival of nearly Rabi-like oscillations, then the pattern repeats. (c), (d) Normalized particle currents $N=12$ and 16, respectively, show the same qualitative behavior as $N=6$, with envelopes and regions of near constant current increasing with $N$.

Figure 9

Condensate depletion. (a)–(c) Depletion of the original condensate for each dynamical regime in the 3LS. The depletion is defined as $D=1-{\lambda }_1/N$, where ${\lambda }_1/N$ is the ratio of the largest eigenvalue of the single-particle density matrix $\langle {\widehat{a}}_i{\widehat{a}}_j\rangle$ to a total number of particles, $N$, that form the BEC. Physically, depletion measures the portion of the BEC that remains in a single-particle mode. The large values of depletion throughout the dynamical regimes reveal significant deviations from the mean-field description of the BEC. Thus, persistent depletion indicates many-body dynamics of the condensate that cannot be captured within the mean-field approximation. (d)–(f), (g)–(i) The same mean-field interaction strength with increased particle number, i.e., $N{U}_{\text{3LS}}=\text{const}$. The Rabi and self-trapping regimes show clear quantum revivals of a condensate, while the chaotic regime remains depleted over multiple macroscopic modes.

Figure 10

Onset of depletion. The black points are data for $C$ and the red curves are the fitting functions. (a), (b) Onset of depletion in the Rabi and chaotic regimes, respectively, as measured by the turning point of the $tanh$ fit. Both scale polynomially with particle number $N$. For Rabi dynamics the onset of depletion scales with $N^{0.51\pm 0.004}$, while for chaotic dynamics it scales as $N^{0.18\pm 0.004}$.

Figure 11

Fidelity. (a)–(c) Overlap of the system with its initial state, $|\langle \psi (0\left)\right|\psi \left(t\right)\rangle |$, for Rabi, chaotic, and self-trapping dynamics, respectively. (c)–(f), (g)–(i) The same measure for increased system size, with columns corresponding to regimes. The Rabi and self-trapping regimes have large overlaps with the initial condensate, with the revivals in the Rabi regime corresponding to the decrease in depletion (see Fig. 9). Chaotic dynamics have little overlap with the initial condensate, with the fidelity decreasing as the particle number is increased. (j) Mean time for $|\langle \psi (0\left)\right|\psi \left(t\right)\rangle |\ge 0.75$ in the Rabi regime increases linearly in particle number, following $T_{\text{revive}}=5.79N-4.41$.