Breaking conservation law enables steady-state entanglement out of equilibrium

Abstract

We show how entangled steady states can be prepared by purely dissipative dynamics in a system coupled to a thermal environment. While entanglement is hindered by thermalization when the system and environment exchange a conserved quantity, we demonstrate that breaking this conservation law through the system-environment interaction drives the system to a nonequilibrium steady state. Such an interaction will generate multiple competing equilibration channels, effectively mimicking baths at distinct chemical potentials. When the environment also supports long-range correlations, these channels mediate nonlocal dissipation capable of generating entanglement. We illustrate the scheme in a model of two nitrogen-vacancy (NV) centers weakly coupled to a spin-pumped magnet, where tuneable magnon excitations enable steady-state entanglement over finite distances. Our results identifies a general mechanism for dissipative entanglement generation, rooted in the conservation structure and environmental correlations rather than fine-tuned coherent control or active driving.

Figures (4)

• Figure 1: (a) The environment is prepared in a $Q$ grand-canonical ensemble at $Q$-chemical potential $\mu$ and temperature $T$. Breaking the conservation law of $Q$ via the coupling between system and environment (indicated by the crack) enables a genuine nonequilibrium steady state. For $\mu<0$, the quantity $Q$ is continuously injected into the environment, which attempts to restore equilibrium at $\mu=0$. Maintaining the environment at finite $\mu<0$ is achieved via pumping. (b) Implementation with NVs coupled via dipole-dipole interaction to a magnetic environment with temperature $T$ and spin accumulation $\mu$. To achieve steady-state entanglement, we need both local and correlated dissipation processes that are described by distinct effective temperatures, $T(0)$ and $T(r)$, see Eq. \ref{['eq:T']}.
• Figure 2: With a symmetry-breaking interaction, a pumped environment with chemical potential $\mu$ can be decomposed into multiple effective reservoirs with distinct chemical potentials $n\mu$ with $n\in \mathbb{Z}$.
• Figure 3: Steady‐state concurrence $\mathcal{C}$ of a two-qubit system as a function of (a) $T(0)$ and $T(r)$, with $\abs{\Gamma_e(r)}/\Gamma_e(0)=0.99$ exemplifying strong nonlocal dissipation, where unphysical region corresponds to temperatures violating the positivity of the Lindblad evolution sm, and (b) local temperature $k_\text{B}T(0)/\Delta$ for fixed nonlocal temperature $k_\text{B}T(r)/\Delta = 0.2$. For $k_\text{B}T(0)/\Delta$ below this value, the system approaches the unphysical region, whose precise extent depends on $\abs{\Gamma_e(r)}/\Gamma_e(0)$, and no entanglement is observed sm.
• Figure 4: (a) Dispersion relation of magnetic excitations of the inverted magnet. The state is stabilized by a spin accumulation $\mu<-b$, set by the external field. The crossings at $\omega=\pm\Delta$ indicate excitations associated with the emission and absorption processes of the system. (b) Nonlocal temperature $k_\text{B}T(r)/\Delta$ as a function of qubit separation $r/\sqrt{As/\Delta}$ for $b/\Delta=1.0$ and $1.2$. We choose the coupling ratio to be $\abs{\lambda_1/\lambda_{-1}}=0.135$, which sets local temperature at $k_{B}T(0)/\Delta=-0.25$. The zeros of $T(r)$ corresponds to the zeros of the Bessel function. (c) Steady-state concurrence $\mathcal{C}$ as a function of $b/\Delta$ and $r/\sqrt{As/\Delta}$. The same coupling ratio is used as in (b). The condition $b/\Delta \geq 1$ ensures that both emission and absorption processes are present. Contour lines of the relative strength of the nonlocal absorption, $\abs{\Gamma_a(r)}/\Gamma_a(0) = 0.999$, $0.99$, and $0.9$, are overlaid.