Symmetry-Enforced Non-Hermitian Jarzynski Equality in an SU(2)-Rotated Family of Hybrid $\mathcal{PT}$--$\mathcal{APT}$ Systems

Abstract

The Jarzynski equality is a cornerstone of nonequilibrium thermodynamics, linking work statistics to equilibrium free-energy differences. Although it has been extensively verified in classical and quantum Hermitian settings, its status in non-Hermitian dynamics remains under debate. Here we show that, in a postselected no-quantum-jump framework, a conditional non-Hermitian Jarzynski equality holds when the transition probabilities obey a parity-exchange symmetry. We study a constructed family of two-level hybrid Hamiltonians formed as linear combinations of parity-time ($\mathcal{PT}$) and anti-parity-time ($\mathcal{APT}$) symmetric terms, and demonstrate using complementary geometric and algebraic arguments that the parity-exchange symmetry persists throughout the corresponding $\mathrm{SU}(2)$-rotated orbit. Relative to previous $\mathcal{PT}$-focused conditional Jarzynski equality results, the advance here is an extension of the symmetry criterion from the isolated $\mathcal{PT}$ endpoint to a broader $\mathcal{PT}$--$\mathcal{APT}$ hybrid family. Experimentally, we implement three representative points, $θ_k = 0, π/4, π/2$, in a single trapped $^{171}\mathrm{Yb}^+$ ion and measure the resulting work distributions under cyclic protocols with $ΔF = 0$, confirming the predicted symmetry criterion at those points. Our results establish a symmetry-based extension of the conditional non-Hermitian Jarzynski relation within this restricted two-level setting.