Semiclassical entanglement entropy for spin-field interaction

TL;DR

The paper develops a semiclassical framework to describe entanglement dynamics in a bipartite spin–boson system by formulating forward and backward propagators in a product of canonical and spin coherent states. By extending the classical phase space into the complex domain, it introduces entangled-boundary-condition trajectories and derives a semiclassical expression for the entanglement entropy $E_T^{\mathrm{sc}}$, depending on classical trajectories and their stability. The approach generalizes previous real-trajectory treatments by including multiple complex trajectories, yielding remarkable accuracy for entanglement dynamics even beyond the Ehrenfest time, as demonstrated in a representative spin–field example. The method offers a systematic route to study entanglement in hybrid quantum systems and highlights the crucial role of complex trajectories in capturing long-time quantum correlations.

Abstract

We study a general bipartite quantum system consisting of a spin interacting with a bosonic field, with the initial state prepared as the product of a spin coherent state and a canonical coherent state. Our goal is to develop a semiclassical framework to describe the entanglement dynamics between these two subsystems. Using appropriate approximations, we derive a semiclassical expression for the entanglement entropy that depends exclusively on the trajectories of the underlying classical description. By analytically extending the classical phase space into the complex domain, we identify additional complex trajectories that significantly improve the accuracy of the semiclassical description. The inclusion of these complex trajectories allows us to capture the entanglement dynamics with remarkable precision, even well beyond the Ehrenfest time. The approach is illustrated with a representative example, where the role of real and complex trajectories in reproducing the quantum entanglement entropy is explicitly demonstrated.

Figures (2)

• Figure 1: Contour lines of $f^{\text{\tiny R}}(\alpha_1)=0$ (dashed black lines) and $f^{\text{\tiny I}}(\alpha_1)=0$ (solid red lines) in the $\alpha_1$ complex plane. The thick-solid blue line represents the unit circle. The points where a black curve crosses a red one is a solution of Eq. \ref{['TranscEq']}. Panels (a) to (e) refer to different values of $\tau$: $10^{-5}$, $0.05$, $0.1$, $0.2$, and $0.5$, respectively. Panel (f) shows an amplification of the region marked with a gray square in panel (e) and identifies three structures $\mathrm{St}_1$, $\mathrm{St}_2$, and $\mathrm{St}_3$. At last, panels (g) to (i) amplify each structure seen in panel (f). We plot the curves using the following numerical values: $\lambda=|z_0|^2=|s_0|^2=1$ and $j=5$.
• Figure 2: Entanglement as function of time $\tau$. In all panels, the thick solid blue line represents the entirely quantum calculation $E^{\mathrm{q}}_T(\hat{\rho}_0)$ while the other curves show its semiclassical approximations. In panel (a), the dashed black line refers to the semiclassical entanglement \ref{['Efinal']}, including only real trajectories. The solid red line also represents $E^{\mathrm{sc}}_T(\hat{\rho}_0)$, but considering the complex trajectories generated by the roots of $f(\alpha_1)$ belonging to structure $\mathrm{St}_1$. This curve is repeated in panel (b), now as a dashed black line, to be compared with the semiclassical entanglement formula including the trajectories associated with structure $\mathrm{St}_2$ (solid red line). Panels (c)-(e) show the results of the inclusion of the complex trajectories associated with the structures $\mathrm{St}_3$, $\mathrm{St}_4$, and $\mathrm{St}_5$, following the same reasoning used from panel (a) to (b). That is, in a given panel, we always reproduce the better semiclassical result of the previous panel for comparison.