Entanglement Transfer Dynamics in a Two-Leg Spin Ladder Under a Selective Magnetic Field

Abstract

We investigate the dynamical transfer of bipartite entanglement through a two-leg spin-1/2 ladder governed by the anisotropic Heisenberg (XXZ-type) model with a selective magnetic field applied exclusively to the mediating rungs. Starting from a maximally entangled initial rung pair, we demonstrate high-fidelity entanglement transfer to the terminal pair (F_max = 0.9998 for N = 3 rung pairs), with the intermediate rungs remaining effectively disentangled throughout. The dynamics is governed by two independent timescales: a fast carrier oscillation at frequency omega_fast = 2*sqrt(1 + 4d^2) J (set by local rung physics, field-independent) and a slow transfer envelope with period T_slow = 2.37 h/J^2 (set by virtual inter-rung coupling, field-dependent). The effective inter-rung coupling J_eff = alpha(d,g) J^2/h is derived via second-order perturbation theory through two parallel virtual paths. We systematically study the effects of magnetic field strength, Hamiltonian anisotropy, and initial state on transfer quality, establish a global parameter space map of the fidelity, and demonstrate robustness under uncorrelated coupling disorder (mean F_max > 0.998 for delta <= 10%). All results are obtained by exact diagonalisation for systems of up to N = 5 rung pairs; extension to larger systems requires tensor-network methods such as DMRG. Compared to one-dimensional chain proposals, the ladder geometry enables a spatially selective control mechanism that suppresses intermediate entanglement while preserving coherent transfer, providing a distinct route to engineered quantum channels.

Figures (6)

• Figure 1: Schematic of the two-leg spin-$\frac{1}{2}$ ladder for $N=3$ rung pairs (six sites). Filled circles are spin-$\frac{1}{2}$ sites. Solid lines indicate Heisenberg interactions along the rungs (vertical, coupling $J_{\perp}$) and legs (horizontal, coupling $J_{\parallel}$). The magnetic field $h$ is applied selectively to the mediating rung (sites 3,4) only. $C_{12}$, $C_{34}$, $C_{56}$ denote the concurrences of the initial, mediating, and terminal pairs respectively.
• Figure 2: Time evolution of the concurrences $C_{12}(t)$, $C_{34}(t)$, and $C_{56}(t)$ for the reference parameter set ($J_\perp = J_\parallel = 1$, $g = 1$, $d = 0.5$, $h = 100$) with initial state $|\Phi^+\rangle^{1,2} \otimes |0\rangle^{\otimes 4}$. (a) The initial pair $C_{12}$ oscillates periodically between 0 and 1. (b) The mediating pair $C_{34}$ remains close to zero throughout the dynamics. (c) The terminal pair $C_{56}$ oscillates in perfect antiphase with $C_{12}$, reaching values close to unity. This behaviour demonstrates coherent entanglement transfer through the ladder: the mediating rung acts as a transparent quantum channel.
• Figure 3: The two independent timescales governing entanglement transfer ($J_\perp = J_\parallel = 1$, $g = 1$, $d = 0.5$, $h = 100$). (a) Slow transfer period $T_{\rm slow} = 2t^*$ vs magnetic field strength $h$. The linear scaling (dashed line) confirms the virtual-process origin: $J_{\rm eff} = \alpha(d,g)\,J^2/h$. (b) Concurrence $C_{56}(t)$ showing the fast carrier oscillation. Vertical dark red lines mark successive fast periods at $t = kT_{\rm fast}$, where $T_{\rm fast} = \pi/\omega_{\rm fast} \approx 1.11$ with $\omega_{\rm fast} = 2\sqrt{2}\,J$ (independent of $h$). The two timescales differ by a factor of $\sim\!100$ for $h = 100$: the fast carrier frequency is set by local rung physics while the slow transfer envelope is governed by virtual inter-rung coupling.
• Figure 4: Transfer fidelity for reference parameters ($J_\perp=J_\parallel=1$, $g=1$, $d=0.5$, $h=100$). (a) $F(t)$ (purple) and $C_{56}(t)$ (green dashed). Classical limit $F=0.5$ shown dotted. $F_{\max}=0.9998$ at $t^*\approx1.11$. (b) Parametric $F$ vs $C_{56}$, colour-coded by $t$. Curve lies above $F=C_{56}$ (dashed red).
• Figure 5: Peak fidelity $F_{\max}(g,d)$ for $J_\perp=J_\parallel=1$, $h=100$, $t\in[0,10]$. Black contours: $F_{\max}=0.99$. Cyan dashed: $F_{\max}=0.9$. Red star: reference parameters ($g=1$, $d=0.5$). Both axes at $g=0$ and $d=0$ show low fidelity; the diagonal band shows that both anisotropy parameters must be non-zero.