Fig. 1. Classical diffusion coefficient D_c of particle 1 versus m for ε = 0.1 (triangles), 2 (squares), and 4 (circles). The dashed line (in red) denotes the classical diffusion coefficient of unperturbed case, i.e., ε = 0. Inset: mean square of classical momentum of particle 1 versus time for m = 0.001. Dashed line (in red) denotes the fitting function of the form 〈p12(t)〉=Dct with D_c = 0.087. The parameters are K₁ = 1.8, K₂ = 0, λ = 10, and L = π.

Fig. 2. (a) Time dependence of the classical (red line) mean energy and quantum mean energy (black lines) of particle 1. From top to bottom, black solid lines correspond to ℏ_eff = 0.01, 0.02, 0.03, 0.04, and 0.05, respectively. The parameters are K₁ = 1.8, K₂ = 0, m = 0.001, ε = 2, λ = 10, and L = π. For comparison, the dashed line (in blue) denotes the quantum mean energy of the unperturbed case, i.e., ε = 0 with K = 1.8 and ℏ_eff = 0.01. (b) Momentum distribution at the time t = 500 with ℏ_eff = 0.01 (blue curve) and 0.05 (black curve). Dash-dotted line (in red) indicates the fitting function of the Gaussian form |ψ₁(p)|²∝ e^−p2/ζ. Dashed line (in red) denotes the exponential fitting |ψ₁(p)|²∝ e^−|p|/ξ.

Fig. 3. (a), (b) Time dependence of the mean energy. In (a), ℏ_eff = 0.01, from top to bottom solid lines correspond to δKmax=0.012, 0.008, 0.005, and 0. In (b), δKmax=0.012, from top to bottom solid lines correspond to ℏ_eff = 0.01, 0.02, 0.03, and 0.05. For comparison, red-dashed lines denote the classical mean energy with δKmax=0. (c) Momentum distribution at the time t = 500 with δKmax=0.012 for ℏ_eff = 0.01 and 0.05, respectively. The parameters are K = 1.8 and L = π. Black curves denote the momentum distributions for the two-particle systems with m = 0.001, K₁ = 1.8, ε = 2, λ = 10, and L = π (same as in Fig. 1(b)).

Fig. 4. (a) The ratio ℛ versus ℏ_eff with m = 0.001 (triangles), 0.01 (circles), and 1 (squares). Dashed lines (in red) denote the exponential fitting, i.e., ℛ ∝ e^−αℏ_eff. (b) Phase diagram of quantum diffusion as a function of ℏ_eff and m, where circles and triangles indicate the threshold values of ℏeffc and ℏeffd which correspond to the appearance of classically chaotic diffusion and dynamical localization, respectively. The coupling strength is ε = 2. Other parameters are the same as those in Fig. 2.

Fig. 5. Purity P versus time with K₂ = 0 (circles), 0.03 (triangles), 0.05 (diamonds), 0.09 (pentagrams), and 0.12 (squares). Dashed lines (in blue) denote the fitting function of the form P∝t−η. Dashed-dotted (in red) line indicates the exponential fitting P∝exp(−γt). The parameters are K₁ = 5, m = 0.01, ℏ = 10⁻⁴, ε = 20ℏ, λ = 10, and L = π.