Generation of concurrence in a generalized central spin model with a three-spin interacting environment

TL;DR

This work analyzes how a three-spin interaction in an Ising environment influences entanglement generation in a generalized central-spin model. By mapping the spin chain to fermions and solving with exact diagonalization, the authors identify critical lines $h=J_3\pm1$ and $h=-J_3$, where bipartite entanglement is enhanced in the chain, yet suppressed in the three-spin-dominated region. They then couple two central spins locally to the chain and study both equilibrium and non-equilibrium dynamics: equilibrium dynamics exhibit dip-and- revival patterns governed by quasi-particle velocities, while non-equilibrium dynamics show a growth of concurrence followed by a two-stage decay, with maximal entanglement near a multicritical point and sensitivity to the sign of the final field due to the non-invariance of the Hamiltonian when $J_3 \neq 0$. The results reveal that three-spin interactions can actively generate, tune, and sustain bi-partite entanglement between central spins through environmental criticality and decoherence-channel dynamics, with potential experimental realization in optical lattices or trapped ions. Overall, the paper highlights a mechanism by which higher-order spin interactions enhance quantum correlations in open quantum systems and provides a framework for exploring entanglement generation in more complex frustrated environments.

Abstract

We consider the three-spin Ising model to study the effect of three-spin interacting term on bi-partitie entanglement between adjacent spins. The three-dominated disordered region has tri-partite entanglement causing a vanishingly small concurrence, while it acquires maximum value around the critical points. Considering the above model as an environment, we construct a generalized central spin model where two central spins, initially in an unentangled pure state, are coupled locally to two distinct sites of the environmental spin chain. We study the generation of mixed state entanglement between the central spins when the transverse field of the environment is kept fixed, and suddenly quenched, referring to equilibrium and non-equilibrium dynamics of the central spins, respectively. For the critical environment in the equilibrium, the concurrence shows a dip-revival structure governed by quasi-particle movement. In the non-equilibrium study, we find an initial growth of concurrence followed by a two-stage fall for the inter-phase quench which is governed by dynamic decoherence channels. The central spins are maximally entangled for a quench in the vicinity of a multicritical point, which arises due to three-spin interaction only. The concurrence becomes long-lived for an intra-phase quench, and this sustainability depends on the strength of the three-spin interaction. Therefore, the three-spin interaction indeed helps in generating bi-partite entanglement in the central spins.

Figures (10)

• Figure 1: Equilibrium phase diagram of the three-spin interacting Ising model Eq. (\ref{['eq:tsim-Ham']}). Solid lines show the Ising (three-spin dominated) phase boundaries $h=J_3\pm 1$ ($h=-J_3$), and the dotted line marks the boundary between the incommensurate and commensurate phases.
• Figure 2: Plot for concurrence $C(\rho)$ between nearest neighbour spins, computed using Eq. (\ref{['eq:cnc']}), as a function of transverse field $h$ in the three-spin interacting Ising model Eq. (\ref{['eq:tsim-Ham']}) for $J_3 = 0$ to $1.7$. We consider $N=20$ for exact diagonalization calculation.
• Figure 3: Illustration explaining the schematic GCSM where the central yellow part represents the two-qubit spin system with spins $A$ and $B$. Spin $A (B)$ is coupled locally to site $p (q)$, identified by blue bonds, of the environmental three-spin Ising model. The central as well as the environmental spins are represented by brown arrows.
• Figure 4: Equilibrium plot shows the behavior of system's concurrence $C(\rho_S^I)$ with time on the $h=J_3+1$ critical line for ($J_3=0, h=1.0$, $d=0$ and $J_3=0.2, h=1.2$, $d=0$) and at the multi-critical point ($J_3=0.5, h=-0.5$, $d=0$). We consider $N=20$ spins for exact diagonalization calculation.
• Figure 5: Time evolution of concurrence $C(\rho_S^I)$ as a function of time at critical points corresponding to $h = 1 + J_3$. (a) For $J_3 = 0.5$, $h_I = h_F = 1.5$, a sharp dip is observed at $t = N/2$, consistent with coherent quasiparticle interference and linear dispersion with group velocity $v_{\text{max}} = 2$. (b) For $J_3 = 0.165$, $h_I = h_F = 1.165$, the concurrence dip occurs near $t \approx 15$, corresponding to $v_{\text{max}} \approx 1.33$. We consider $d=1$ and $N=20$ spins for exact diagonalization calculation.