Entanglement and Spin Squeezing in Non-Hermitian Phase Transitions

We show that non-Hermitian dynamics generate substantial entanglement in many-body systems. We consider the non-Hermitian Lipkin-Meshkov-Glick model and show that its phase transition occurs with maximum multiparticle entanglement: There is full $N$-particle entanglement at the transition, in contrast to the Hermitian case. The non-Hermitian model also exhibits more spin squeezing than the Hermitian model, showing that non-Hermitian dynamics are useful for quantum metrology. Experimental implementations with trapped ions and cavity QED are discussed.

Figure 1

Entanglement properties for $N=20$ spins. (a) Averaged quantum Fisher information $\overline{F}/N^2$ (solid line) indicates multiparticle entanglement, while rescaled concurrence $C_R$ (dashed line) indicates two-particle entanglement. (b) Wigner function on the Bloch sphere for $V=V_c$.

Figure 2

Eigenvalues of $H$ for $N=20$, showing (a) imaginary parts and (b) the gap between the two largest imaginary parts. (c) $⟨{\sigma }_z⟩$ of the steady state.

Figure 3

Scaling of various quantities with $N$. (a) $V_c$ (blue circles) and $V^{*}$ (red asterisks). (b) The gap at $V^{*}$. (c) Averaged quantum Fisher information $\overline{F}/N^2$ at $V_c$ (blue circles) and $V^{*}$ (red asterisks). (d) $⟨{\sigma }_z⟩$ at $V^{*}$.

Figure 4

(a) Spin squeezing for $N=20$ as a function of $V$. (b) Minimum ${\xi }^2$ for different $N$.

Figure 5

Non-Hermitian evolution of $N=1000$ spins with $V=V^{*}$, starting from $|\downarrow \downarrow \dots \downarrow ⟩$. (a) Spin squeezing at current time (solid line) and steady state (dotted line). (b) Probability of no decay event.