Nonlinear Classical Dynamics described by a Density Matrix in the Classical Limit

TL;DR

This work studies the classical limit of nonlinear quantum-classical hybrids within a MaxEnt framework, showing that a closed Lie algebra of quantum observables and smooth classical potentials yield a classical limit described by a pure-density projector. The key analytic result demonstrates that in the limit \\mathcal{I} \ ightarrow 0 (equivalently \\sqrt{I} \ ightarrow 0 with fixed energy), decoherence reduces the MaxEnt density operator to a single pure state whose evolution matches the classical analogue of the Hamiltonian. The authors derive autonomous evolution equations for Lagrange multipliers and classical variables, employ a unitary transformation to simplify the density operator, and relate the invariant parameters via \\mathcal{I}_{\\lambda} to \\mathcal{I}. The findings generalize prior numerical and analytic results to a broad class of quantum–classical systems and offer a principled route to analyze semiclassical chaos, dissipation, and interface dynamics across quantum information, condensed matter, and quantum optics.

Abstract

We examine the classical limit of a fairly general nonlinear semiclassical hybrid system within a MaxEnt framework. The consistency of the hybrid dynamics requires algebraic constraints on quantum operators and smoothness conditions for the classical variables. Analytically, we demonstrate that the classical limit is characterized by a pure density matrix representing a single state, which reproduces the dynamics of its classical analogue. To illustrate the methodology, we revisit and synthesize two previously studied examples.

Figures (6)

• Figure 1: Poincaré sections for the case $A = 0$, corresponding to the specific Hamiltonian $\hat{H}_1 = \frac{1}{2}\left(\frac{\hat{p}^{2}}{m_q} + \frac{P_A^{2}}{m_{cl}} + m_q \omega^2 \hat{x}^{2}\right)$ from Eq. (\ref{['Hpaper']}), where $\omega^2 = \omega_q^2 + e^2 A^2$. The energy is set to $E = 0.6$. Figures (a) and (b) correspond to $I = 0.17361$ and $I = 2.341311 \times 10^{-5}$, respectively, both within the transition (mesoscopic) regime. These sections are bounded by the curves $\frac{\langle \hat{p}^{2} \rangle}{m_q} + m_q \omega_q^2 \langle \hat{x}^{2} \rangle = 2E$, associated with the total energy, and $\langle \hat{x}^2 \rangle \langle \hat{p}^2 \rangle = I$, which reflects the Uncertainty Principle (see Eq. (\ref{['eqInv']})). These plots demonstrate the coexistence of chaotic dynamics with the Heisenberg Principle, illustrating semiclassical chaos. Figure (c) corresponds to $I = 3.6 \times 10^{-23}$. Figure (d) shows $x^2$ versus $p^2$, representing the limits $\lim\limits_{\hbar \rightarrow 0} \lim\limits_{I \rightarrow \hbar^2/4} \langle \hat{x}^2 \rangle$ and $\lim\limits_{\hbar \rightarrow 0} \lim\limits_{I \rightarrow \hbar^2/4} \langle \hat{p}^2 \rangle$ (see Section \ref{['currentres']}), which coincide with the classical case $I = 0$. This plot is bounded solely by $\frac{p^2}{m_q} + m_q \omega_q^2 x^2 = 2E$, and illustrates classical chaos. All results were validated by confirming the time invariance of the dynamical quantities $E$ and $I$, with a numerical precision of $10^{-80}$.
• Figure 2: 3D Poincaré sections with $A = 0$ and their projections, corresponding to the specific case $\hat{H}_2 = \frac{1}{2} \left( \omega_q (\hat{x}^2 + \hat{p}^2) + \omega_{cl}(A^2 + P_A^2)\hat{I} + e_q^{cl} A^2 \hat{x}^2 \right)$ from Eq. (\ref{['Hpaper']}). This system is highly relevant in condensed matter physics. We present an example in the dissipative regime, where energy is not conserved. Consequently, only one dynamical invariant remains, enabling the construction of 3D Poincaré sections. The initial energy is $E(0) = 0.6$. Figures (a), (b), and (c) display the Poincaré sections in the space of $\langle \hat{x}^2 \rangle$, $\langle \hat{p}^2 \rangle$, and $\langle \hat{L} \rangle$, along with their respective projections onto the $\langle \hat{x}^2 \rangle$–$\langle \hat{p}^2 \rangle$ plane (see details in Ref. disipativo). The corresponding values of $I$ are $I = 0.17361$, $2.34131 \times 10^{-5}$, and $3.6 \times 10^{-23}$, respectively. Figure (d) corresponds to $I = 0$, which represents the classical limit: $\lim\limits_{\hbar \rightarrow 0} \lim\limits_{I \rightarrow \hbar^2/4} \langle \hat{x}^2 \rangle$, $\lim\limits_{\hbar \rightarrow 0} \lim\limits_{I \rightarrow \hbar^2/4} \langle \hat{p}^2 \rangle$, and $\lim\limits_{\hbar \rightarrow 0} \lim\limits_{I \rightarrow \hbar^2/4} \langle \hat{L} \rangle$. The dissipative parameter is $\eta = 0.05$. Note the curve $\langle \hat{x}^2 \rangle \langle \hat{p}^2 \rangle = I$ (see Eq. (\ref{['eqInv']})) in the projections of Figs. (a), (b), and (c).
• Figure 3: Energy as a function of time in the dissipative regime of the Hamiltonian shown in Fig. \ref{['dis_case']}, for the specific initial condition $E(0) = 0.6$. The final energy remains non-zero, consistent with the Heisenberg Uncertainty Principle disipativo.
• Figure 4: Poincaré sections for $A = 0$, corresponding to the Lagrange multipliers Eqs. (\ref{['lamb+s,p']}), in the particular case $\hat{H}_1 = \frac{1}{2}\left(\frac{\hat{p}^{2}}{m_q} + \frac{P_A^{2}}{m_{cl}} + m_q \omega^2 \hat{x}^{2}\right)$. We take $\hbar = 1.0 \times 10^{-40}$ and fix the energy at $E = 0.6$. Fig. (a) corresponds to $I = 0.17361$, Fig. (b) to $I = 2.341311 \times 10^{-5}$, and Fig. (c) to $I = 3.6 \times 10^{-23}$.
• Figure 5: 3D Poincaré sections with $A = 0$ and their projections, calculated using Eqs. (\ref{['lamb+s,p']}). The Hamiltonian considered is $\hat{H}_2 = \frac{1}{2} \left(\omega_q(\hat{x}^2 + \hat{p}^2) + \omega_{cl}(A^2 + P_A^2)\hat{I} + e_q^{cl} A^2 \hat{x}^2\right)$. We take $\hbar = 1.0 \times 10^{-40}$ and set the initial energy value to $E(0) = 0.6$. This example corresponds to the dissipative case with $\eta = 0.05$. Figures (a), (b), and (c) show the Poincaré sections in the space of $\lambda_1$, $\lambda_2$, and $\lambda_3$, along with their respective projections onto the $\lambda_1$–$\lambda_2$ plane. The corresponding values of $I$ are $0.17361$, $2.34131 \times 10^{-5}$, and $3.6 \times 10^{-23}$, respectively.