Rich dynamical behavior in periodically-driven systems has attracted extensive attention in recent years.^[1–3] It is found that periodical driving induces quantized transport,^[4,5] topological phase transition,^[6–8] and thermalization of quantum dynamics,^[9,10] just to name a few. Among these, the quantum transport in momentum space is particularly important, as it provides an insight on the mapping of transport phenomenon from position to momentum space. As an example, the dynamical localization (DL) in momentum space^[11–13] is an analog of the Anderson localization in position space,^[14] both of which result from the disordered feature of the system.^[15] Nowadays, the transport phenomenon in momentum-space lattice has been a fruitful subject in physics, where the intriguing phenomena, such as topologically-protected quantum walk^[16] and exponentially-fast diffusion^[17–21] have been reported.

Theoretical investigations have stimulated experimental studies on the transport phenomenon in periodically-driven quantum systems. Recently, the DL was observed in the atom-optics experiments on the periodically-driven ultracold atoms^[22] and molecules.^[23] Even more remarkable, experimental advances in atom-optics have made it possible to actively tune the driven laser to be a quasiperiodic^[24–26] or random sequence,^[27] and thus to investigate the diffusion behavior in a controlled manner, e.g., the Anderson transition for the quasiperiodical kicking systems, and the decoherence-induced subdiffusion for the random kicking systems. These investigations greatly broaden our understanding on the exotic transport phenomenon induced by quantum coherence.^[28–30]

It is known that the unavoidable coupling between system and environment destroys quantum coherence,^[31] and consequently leads to the appearance of the classically chaotic diffusion from the underlying quantum dynamics.^[32–41] Interestingly, the influence from a single chaotic particle can effectively produce the decoherence effects.^[42–46] The quantum–classical correspondence (QCC) which is induced by the interaction with a few degrees of freedom^[47–50] has been recently observed in a ultracold atoms experiment.^[51] In the past decades, the influence of interatomic interactions on the quantum diffusion has received considerable investigations.^[52–54] Rich diffusion behaviors, such as subdiffusion^[55–61] and exponentially-fast diffusion,^[18,19,21] have been observed. This topic regains attention as it is closely associated with the issue of dynamical many-body localization,^[62–65] since the system of coupled periodically-kicking particles is an ideal system to explore the diffusion behavior in discrete momentum-space lattice.^[66–69] At present, the fate of dynamical many-body localization under the interaction of particles is still an open question.^[70]

Motivated by these studies, we investigate the quantum diffusion in a system involving two-coupled particles. The system is periodically driven by impulsive fields. We concentrate on both the classical and quantum dynamics of one particle (say particle 1) under the interaction with the other one (say particle 2). Interestingly, the effects of particle 2 with mass m on the classical diffusion of particle 1 with unity mass decrease as m decreases, and it is negligibly small when m ≪ 1. The reason is that particle 2 of very small mass possesses little energy to affect the classical motion of particle 1. More importantly, the classically vanishing small influence is able to destroy the DL of particle 1. It even leads to the appearance of classically chaotic diffusion in quantum dynamics when the effective Planck constant ℏ_eff is small enough. To characterize the quantum entanglement, we numerically investigate the purity of the quantum state. The issue we address here is the sensitivity of entanglement to the classical chaoticity of the external degrees of freedom. We find that the time dependence of purity exhibits the power law decay for the regular motion of the classical dynamics of particle 2, and the exponential decay for the chaotic motion.

The paper is organized as follows. In Section 2, we describe the system and show the QCC of particle 1 subjected to the influence from a very small particle. In Section 3, we give a purity description of quantum entanglement. Summary is present in Section 4.

We consider two interacting particles which are trapped in an infinitely square well and periodically kicked by optical lattices. The Hamiltonian reads

It is useful to introduce a set of scaled dimensionless units. Time is scaled by T, i.e., t′ = t/T, which means that t′ denotes the number of kicks. The canonical coordinate and momentum variables are redefined as and , which satisfy the commutation relation , with the effective Planck constant . The masses of the two particles are rescalled by m₁, thereby a dimensionless mass is m = m₂/m₁. The kicking strength in dimensionless units takes the form . For the interaction term, the scaled coupling strength is , and . The dimensionless width of the infinite square well has the expression L′ = 2k_l L, For clarity, from here on, we omit the superscript of prime in these variables. After rescalling, the dimensionless Hamiltonian is described as

We concentrate on both the classical and quantum dynamics of particle 1. Numerically, we use the split-operator method to simulate the time evolution of the quantum dynamics. The initial state is a product state of the ground state of each particle, i.e., ψ(0) = ϕ₁(0)ϕ₂(0) with . The classical evolution of a trajectory is addressed by the fourth order Runge–Kutta Method. The initial condition of a trajectory (p₁,x₁,p₂,x₂) is such that the momentum of each particle is zero, p₁ = p₂ = 0, the coordinates x₁ and x₂ are uniformly distributed in the region [0,L].

The typical character of classically chaotic diffusion is the linear growth of the mean energy with time, i.e., 〈 p²(t) 〉_c = D_c t, where 〈 ⋅ 〉_c stands for the ensemble average over many trajectories and D_c is usually termed as the classical diffusion coefficient (CDC).^[71–74] This phenomenon is traditionally referred as normal diffusion. Our numerical investigations show that the mean energy of particle 1 linearly increases with time if the kick strength is strong enough (see the inset in Fig. 1). In our numerical simulations, the number of the classical trajectories is 10⁴ and the evolution over one period is separated into 10⁵ numbers of steps. In order to investigate the effects of particle 2 on the classical diffusion of particle 1, we numerically calculate the CDC of this particle, i.e., for different m. Here, t_f is the total time during which one may numerically track the time evolution. In our numerical experiments, t_f is on the scale of several hundreds of kicking periods, which can ensure the high precision of the numerical results due to the linear growth of mean energy with time. Interestingly, with the decrease of m, D_c decreases to the saturation level which almost equals with the CDC of the unperturbed case (see Fig. 1 for ε = 2 and 4). This demonstrates that the influence from particle 2 decreases as m decreases and it is even negligible if m ≪ 1. For weak coupling (e.g., ε = 0.1), the D_c of particle 1 has slight difference from that of the unperturbed case as m varies, since the two particles have little effects on each other for very weak coupling.

Fig. 1.

Figure Option ViewDownloadNew Window

Fig. 1. Classical diffusion coefficient Dc 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 with Dc = 0.087. The parameters are K1 = 1.8, K2 = 0, λ = 10, and L = π.

We numerically investigate the quantum diffusion of particle 1 when its classical counterpart is almost unaffected which is ensured by the condition m ≪ 1. The quantum mean energy of this subsystem exhibits DL for large ℏ_eff [e.g., ℏ_eff = 0.05 in Fig. 2(a)]. Interestingly, the quantum diffusion of particle 1 undergoes the transition from DL to classically-chaotic diffusion as ℏ_eff decreases. For small enough ℏ_eff (e.g., for ℏ_eff = 0.01), the quantum diffusion is in good agreement with its classical counterpart, which is a clear evidence of QCC. At first glance, such QCC happening for very small ℏ_eff is trivial. However, by comparing with the quantum mean energy with ε = 0, one can find that the quantum diffusion of the unperturbed case is eventually suppressed by quantum coherence after long time evolution [see Fig. 2(a) for ℏ_eff = 0.01]. Therefore, the interaction plays a key role for the appearance of QCC. To reveal rich features of quantum diffusion, we numerically investigate the probability density distribution in momentum space. Our numerical results show that for large ℏ_eff [e.g., ℏ_eff = 0.05 in Fig. 2(b)], the wave packet is exponentially localized in momentum space, which is a feature of DL. For small enough ℏ_eff [e.g., ℏ_eff = 0.01 in Fig. 2(b)], the momentum distribution is in a good agreement with the Gaussian function, which is a signature of the appearance classically chaotic diffusion. The change from the exponentially-localized profile of wavepackets to the Gaussian form reveals the quantum to classical transition.

Fig. 2.

Figure Option ViewDownloadNew Window

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 ℏeff = 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 ℏeff = 0.01. (b) Momentum distribution at the time t = 500 with ℏeff = 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.

In chaotic situation, the motion of particle 2 of very small mass behaves like random noise. It is known that external noises can destroy quantum interference and thus leads to the classically chaotic diffusion.^[75–77] As a further step, we consider the case that the kick strength contains random noises, for which the Hamiltonian reads

Fig. 3.

Figure Option ViewDownloadNew Window

Fig. 3. (a), (b) Time dependence of the mean energy. In (a), ℏeff = 0.01, from top to bottom solid lines correspond to , 0.008, 0.005, and 0. In (b), , from top to bottom solid lines correspond to ℏeff = 0.01, 0.02, 0.03, and 0.05. For comparison, red-dashed lines denote the classical mean energy with . (c) Momentum distribution at the time t = 500 with for ℏeff = 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)).

We quantify the difference between quantum and diffusion coefficients of particle 1 by using the ratio ℛ = D_q/D_c, where the quantum diffusion coefficient is defined as . In the ideal case, the DL (D_q = 0) and QCC (D_q = D_c) correspond to ℛ = 0 and 1, respectively. In numerical simulations, we evolve the system for 500 periods to calculate both D_c and D_q, and thus obtain the value ℛ. Figure 4(a) shows the dependence of ℛ on ℏ_eff for various m. It is seen that with increasing ℏ_eff, the ℛ exponentially decays as ℛ ∝ e^−αℏ_eff from almost unity to the saturation level which is around ℛ = 0.1. This clearly demonstrates the exponentially-fast departure of the quantum dynamics from its classical limit. Note that the saturation region of ℛ ≈ 0.1 indicates the appearance of DL. Due to the finite time evolution in numerical simulations, the value of D_q for DL is not exactly zero, which results in the saturation of ℛ ≈ 0.1. Detailed observations show that the decay rate α of ℛ increases with the decrease of m, which reveals that the quantum dynamics of particle 1 is much easier to deviate from its classical counterpart if the mass of particle 2 gets smaller. This is due to the fact that the influence from particle 2 on particle 1 decreases as m decreases.

Fig. 4.

Figure Option ViewDownloadNew Window

Fig. 4. (a) The ratio ℛ versus ℏeff with m = 0.001 (triangles), 0.01 (circles), and 1 (squares). Dashed lines (in red) denote the exponential fitting, i.e., ℛ ∝ e−αℏeff. (b) Phase diagram of quantum diffusion as a function of ℏeff and m, where circles and triangles indicate the threshold values of and 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.

Furthermore, we define the criteria of the appearance of the classically chaotic diffusion and DL as ℛ = 0.9 and 0.1, respectively. We numerically obtain the threshold values of ℏ_eff corresponding to ℛ = 0.9 and 0.1, which are separately termed as and . The dependence of and on m is depicted in Fig. 4(b). One can find three different regions in the parameter space (ℏ, m). Blow the curve of , there is the zone of QCC for which the quantum system exhibits chaotic diffusion. Above the curve , the quantum system exhibits the DL since ℏ_eff is so large that the the quantum coherence is not destroyed by the interaction. There is a transition region between the classically chaotic diffusion zone and the DL zone. Our numerical result is useful for guiding the investigation on the diffusion behavior in cold-atom experiments.

A commonly used measure of quantum entanglement is purity , where ρ₁ is the reduced density matrix of particle 1 by tracing out the degree of particle 2 from the density matrix of the two-particle system.^[78,79] The sensitivity of the quantum entanglement to external perturbations has been extensively studied in the context of quantum chaos.^[80–82] In the present work, we numerically investigate the effects of the classically chaotic dynamics of particle 2 on the generation of quantum entanglement. We consider the case that the coupling strength is in the form , for which the coupling is almost negligible in the semiclassical limit ℏ_eff ≪ 1. Our numerical results show that for small K₂, the purity decays in the power law of time, i.e., (see Fig. 5). For strong kick strength (e.g. K₂ = 0.09 and 0.12), the purity decays exponentially fast with time, i.e., . Moreover, the time-dependence of purity is almost unchanged as long as the K₂ is sufficiently strong. Our numerical result demonstrates that there is a regime where the exponentially-fast growth of entanglement is stable provided that the chaoticity of the external degree of freedom is strong enough.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. Purity 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 . Dashed-dotted (in red) line indicates the exponential fitting . The parameters are K1 = 5, m = 0.01, ℏ = 10−4, ε = 20ℏ, λ = 10, and L = π.

We numerically investigate the entanglement involving a few degrees of freedom via two coupled particles. We show that, the effect of particle 2 on the classical behavior of particle 1 decreases as its mass (m) decreases. Under the classically weak perturbation (m ≪ 1), the quantum diffusion behavior of particle 1 undergoes a transition from DL to chaotic diffusion with the decrease of ℏ_eff. We numerically investigate the difference between quantum and classical diffusions for a wide regime of ℏ_eff and m, and find the exponential decay of ℛ with ℏ_eff. By using this quantity, we define the boundary for the appearance of DL and classically-chaotic diffusion. Numerically, we obtain a “phase” diagram of the quantum diffusion in the parameters space (ℏ_eff, m). The quantum to classical transition is accompanied by the growth of entanglement. For the vanishingly small interaction ε ∝ ℏ with ℏ_eff ≪ 1, the time decay of purity is exponentially fast in condition that the classical dynamics of particle 2 is strongly chaotic. Such exponentially-fast entanglement is stable in the sense that the time dependence of purity is almost unchanged as K₂ varies. Our investigation has important implication for the fundamental problem of the quantum to classical transition.