Universal and Tunable Sudden Freezing of Entanglement Volume

TL;DR

The paper addresses how entanglement freezing—a cessation of entanglement dynamics—can occur universally in N-qubit systems when total excitation is conserved under $U(1)$-symmetric dynamics. By introducing the entanglement volume $Y_s$ and detailing its dependence on one-to-others concurrences, the authors show that under generic excitation-number-conserving evolutions the total volume can exhibit abrupt freezing and thawing, with permanent freezing possible for suitable initial states. They derive an algebraic, geometry-based mechanism explaining the phenomenon, reveal how the frozen value and duration are tunable via the initial mixing angle $\theta$, the excitation number $e$, and the system size $N$, and extend the results to open-system settings. The work provides a universal, platform-agnostic explanation and suggests concrete experimental avenues in optical lattices, cavity QED, trapped ions, and related quantum simulators. Overall, it offers a geometry-grounded framework for controlling multipartite entanglement dynamics with potential applications in quantum information processing and metrology.

Abstract

In a system where two identical two-level atoms interact with their common one-mode cavity field, it is shown that entanglement can become abruptly frozen in time, remaining at a constant value for a period of time until it begins to thaw from this value from the entanglement sharing perspective [Ding et al., Phys. Rev. A 103, 032418 (2021)]. We generalize this exotic behavior of entanglement sharing dynamics to more general systems with arbitrary N qubits, instead of restricting to the atom-cavity mode interaction system. We also demonstrate methods to control the entanglement freezing time and freezing value, and we discover a nontrivial dynamics where entanglement is frozen permanently. In addition, we show that this phenomenon is not a coincidence but a universal feature in a variety of systems with a geometric explanation of the mechanisms.

Figures (8)

• Figure 1: Time evolution of the entanglement volume $Y_s = Y_{A|BC} + Y_{B|AC} + Y_{C|AB}$ for the initial state $\ket{\psi_0} = \ket{001}$. The entanglement volume (solid line) is bounded by $Y_s \le 2$ and displays plateaus where it remains constant. The dashed lines show the individual one-to-other entanglement, which continue to oscillate even when $Y_s$ is frozen. Sharp corners at the plateau boundaries indicate non-analytic transitions in the entanglement dynamics.
• Figure 2: Time evolution of the entanglement volume $Y_s = Y_{A|BC}+Y_{B|AC}+Y_{C|AB}$ for different initial states $\ket{\psi_0} = \cos(\theta)\ket{001} + \sin(\theta)\ket{111}$. For $\theta < \pi/4$, $Y_s$ exhibits periodic freezing plateaus separated by oscillations (temporary freezing). For $\theta \ge \pi/4$, $Y_s$ remains constant in time, indicating permanent freezing.
• Figure 3: Time evolution of the entanglement volume $Y_s(t)$ for different system sizes $N$ with the initial-state parameter $\theta = \pi/12$. For all values of $N$, $Y_s(t)$ initially increases from zero, passes through a shallow dip, and then enters a broad freezing region. Both the frozen value and the duration of the plateau increase with the qubit number $N$, indicating that larger systems exhibit stronger and more persistent entanglement volume freezing.
• Figure 4: Fraction of freezing time $R_f$ (top row) and frozen entanglement value $Y_s(\mathrm{Freezing})$ (bottom row) as functions of the initial-state parameter $\theta$ and the qubit number $N$. Panels (a,b) correspond to the dynamics of initial state $\ket{\psi_0}=\cos{(\theta)}\ket{10\cdots0}+\sin(\theta)\ket{11\cdots1}$, while panels (c,d) correspond to the dynamics of initial state $\ket{\psi_0}=\frac{\cos{(\theta)}}{\sqrt{3}}(\ket{10\cdots0}+\cdots+\ket{0\cdots010}+\cdots+\ket{0\cdots01})+\sin(\theta)\ket{11\cdots1}$. White regions in (b) mark parameter sets where no freezing occurs.
• Figure 5: Time evolution of the entanglement volume $Y_s$ for different initial states of the two-cavity system. For $\theta \le \pi/4$, $Y_s(t)$ increases, forms a finite-time freezing plateau, and then decays to a steady value. For $\theta > \pi/4$, $Y_s(t)$ remains constant over the entire evolution, indicating permanent entanglement freezing. Both the frozen value and the freezing duration can be continuously tuned by the initial-state parameter $\theta$.