Chaos and localized phases in a two-body linear kicked rotor system

Anjali Nambudiripad, J. Bharathi Kannan, and M. S. Santhanam

Phys. Rev. E 109, 034206 – Published 13 March 2024

Abstract

Despite the periodic kicks, a linear kicked rotor (LKR) is an integrable and exactly solvable model in which the kinetic energy term is linear in momentum. It was recently shown that spatially interacting LKRs are also integrable, and results in dynamical localization in the corresponding quantum regime. Similar localized phases exist in other nonintegrable models such as the coupled relativistic kicked rotors. This work, using a two-body LKR, demonstrates two main results; first, it is shown that chaos can be induced in the integrable linear kicked rotor through interactions between the momenta of rotors. An analytical estimate of its Lyapunov exponent is obtained. Second, the quantum dynamics of this chaotic model, upon variation of kicking and interaction strengths, is shown to exhibit a variety of phases: classically induced localization, dynamical localization, subdiffusive, and diffusive phases. We point out the signatures of these phases from the perspective of entanglement production in this system. By defining an effective Hilbert space dimension, the entanglement growth rate can be understood using appropriate random matrix averages.

Received 24 April 2023
Accepted 22 February 2024

DOI:https://doi.org/10.1103/PhysRevE.109.034206

Physics Subject Headings (PhySH)

Dynamical localization Quantum chaos Quantum chaotic transport

Nonlinear Dynamics

Authors & Affiliations

Anjali Nambudiripad ^*, J. Bharathi Kannan^*, and M. S. Santhanam

Indian Institute of Science Education and Research, Pune 411 008, India

^*These two authors contributed equally.

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Vol. 109, Iss. 3 — March 2024

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Images

Figure 1
Left column shows the $(x_{1}, p_{1})$ projection of the stroboscopic section for a selection of parameters $(K, k_{p})$ . The right column shows the time evolution of mean classical energy ${〈 E 〉}_{c} = 〈 p_{1}^{2} + p_{2}^{2} 〉$ (corresponding quantum energy growth is shown in Fig. 3). The numerically computed Lyapunov exponent $λ$ is also shown. Other parameters are $α_{1} = \sqrt{3}, α_{2} = \sqrt{5}$ . The system is evolved for $t = 10^{4}$ kicks, and the classical energy is averaged over $10^{4}$ initial conditions. The red curve is the analytical estimate of classical energy growth obtained through quasi-linear approximation.
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Figure 2
Largest Lyapunov exponent $λ_{\max}$ shown as a function of $K_{s}$ in semilog axis. Red symbols are computed from the map (6), and the black line is obtained by evolving using the Jacobian in Eq. (9). For $K_{s} ≫ 1$ , a good agreement is seen between the largest Lyapunov exponent computed by both methods. The dashed (blue) line has slope 1 and is a guide to the eye. Each value of the Lyapunov exponent is averaged with respect to $θ$ .
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Figure 3
(Left panel) The evolution of quantum mean energy ${〈 E 〉}_{q}$ . (Right panel) shows the corresponding classical (black dashed line) and quantum momentum distribution (blue line) $f (p_{1})$ after $10^{4}$ kicks. The parameters $(K, k_{p})$ (shown on top of each figure) correspond to that shown in Fig. 1 for classical dynamics. Since the energy growth follows ${〈 E 〉}_{q} \sim t^{β}$ , the numerically estimated $β$ is also shown in left panel. The other parameters are $α_{1} = \sqrt{3}, α_{2} = \sqrt{5}, ℏ = 1$ .
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Figure 4
Image map of exponent $β$ is plotted as a function of $k_{p}$ and $K$ . The other parameters are $α_{1} = \sqrt{3}, α_{2} = \sqrt{5}$ and $ℏ = 1$ . This plot is obtained by evolving an initial wave packet for $t = 10^{4}$ kicks. The observed quantum mean energy growth is regressed against ${〈 E 〉}_{q} \propto t^{β}$ to numerically estimate $β$ . The four highlighted points correspond to the parameters chosen for illustration in Figs. 1 and 3. Classical dynamics in region I is regular, and is chaotic in regions II–IV. See text for a detailed explanation of quantum dynamics in regions I–IV. The dashed lines demarcating regions II, III, and IV are approximate indicators of boundaries.
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Figure 5
(a) Diffusive mean energy growth for classical and quantum model. Both display similar growth $\sim t^{β}$ , with $β \approx 1.0$ . (b) The classical (black dashed line) and quantum (blue line) momentum distributions (at $t = 5000$ ) have a Gaussian profile. Other parameters are $α_{1} = \sqrt{3}, α_{2} = \sqrt{5}, ℏ = 1$ .
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Figure 6
Numerically computed entanglement $S (t)$ for $m$ -LKR (solid blue line). (a) $(K, k_{p}) = (0.6, 2.0)$ corresponds to dynamical two-body localized phase and saturated entanglement, (b) (2.0,2.0) corresponds to quantum subdiffusive energy growth regime and logarithmic entanglement production, (c) (7.5,7.5) corresponds to quantum diffusive energy growth regime and logarithmic entanglement production. The black dashed line is an analytical estimate $S_{RMT} (t)$ obtained by using Eq. (15) in Eq. (16). The other parameters of $m$ -LKR are $α_{1} = \sqrt{3}, α_{2} = \sqrt{5}, ℏ = 1$ .
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