Symmetric Resourceful Steady States via Non-Markovian Dissipation

Abstract

We prove a no-go theorem for symmetry-based dissipative engineering of collective-spin steady states: in spin-only Lindblad dynamics with jump operators linear in the collective-spin operators, any unique steady state exhibiting at least $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry is necessarily the maximally mixed state. We then show that bath memory lifts this obstruction, enabling unique entangled steady states with a prescribed symmetry and a metrological gain, and providing a steady-state witness of non-Markovianity. Notably, this framework is largely insensitive to the microscopic details of the bath.

Figures (4)

• Figure 1: A collective spin $\mathbf{S}$ couples linearly to its environment through linear spin operators $L_\mu$ (three shown), chosen so that the set ${L_\mu}$ is closed under $\mathbb{G}$. (a) Spin-only Markovian dynamics: if the Lindbladian has weak symmetry containing $\mathbb{Z}_2\times\mathbb{Z}_2$, our no-go theorem implies that the MMS is the only possible unique steady state. (b) Non-Markovian case: the spin couples to damped auxiliary systems that provide bath memory.
• Figure 2: (a): Hilbert-Schmidt distance between the non-Markovian steady state and the MMS reached in the Lindblad limit, as a function of $\kappa/\omega$ for fermionic, bosonic and two-level AS. The inset shows that for fermionic AS the Lindblad limit is approached with a different scaling from bosonic and two-level AS. (b): Negativity for the $2|3$ bipartition as a function of $\kappa/\omega$. Here, $H_S = (h/N)S_z^2$, with $\omega_1 = \omega_2 = \omega$, $h/\omega=10$, $\gamma/\omega = 2.5$, $g_1 = g_2 =\sqrt{\gamma \kappa/{2N}}$, $N=5$ and $\kappa = \kappa_1 = \kappa_2$.
• Figure 3: Equatorial QFI for $H_S = (h/N) S_z^2$, $L_1 = S_-$ and $L_2 = S_+$ and with bosonic AS. The region of metrological gain ($F_\perp/N > 1$) is observed for $\kappa/\omega$ small enough; the white region corresponds to $F_\perp/N < 1$. The insets display the populations in the Dicke basis for the parameters indicated by the symbols (triangle and dot), showing a concentration around the state $m=0$ as the metrological gain increases. Other parameters: $\omega_1 = \omega_2 = \omega$, $g_1 = g_2 = \sqrt{\gamma \kappa/2N}$ with $\gamma = \omega$ and $N =6$.
• Figure 4: (a) Wigner functions of spin steady states for bosonic AS, showing polyhedral symmetries: tetrahedral ($N=4$), octahedral ($N=6$), and icosahedral ($N=6$). All rotational symmetry axes are shown. Colors from green to blue correspond to values from minimum to maximum. (b) Quantum Fisher information per spin-$\tfrac{1}{2}$, equal for all collective spin components $S_{\mathbf n}$ due to steady-state isotropy, as a function of the Hamiltonian strength $h/\omega$ (SMfor full models). The hatched region marks metrologically useful states. (c) Symmetry-deviation parameter versus relative rate mismatch $\Delta\kappa/\overline{\kappa}$ (in log–log scale), showing quadratic scaling $\Delta_\mathbb{G}\propto (\Delta\kappa/\overline{\kappa})^{2}$. Parameters: $\omega_k = \omega$ and $g_k = \sqrt{\gamma \kappa/2N}$ for all modes $k$, with $\gamma=\kappa = \omega/2$, and, for panel (c), $h=\omega$ for $\mathbb{T}$ and $\mathbb{O}$, and $h=\omega/6$ for $\mathbb{I}$.