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

TL;DR

This paper extends the non-Hermitian Jarzynski equality to a broader class of $ ext{SU}(2)$-rotated hybrid $ ext{PT}$--$ ext{APT}$ systems, demonstrating symmetry conditions and experimental verification in a trapped ion setup.

Contribution

It introduces a symmetry-based extension of the non-Hermitian Jarzynski equality from $ ext{PT}$-symmetric to hybrid $ ext{PT}$--$ ext{APT}$ systems, supported by experimental results.

Findings

Conditional Jarzynski equality holds under parity-exchange symmetry.

Symmetry persists throughout the $ ext{SU}(2)$-rotated orbit.

Experimental measurements confirm the symmetry criterion at selected points.

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, $\theta_k = 0, \pi/4, \pi/2$, in a single trapped $^{171}\mathrm{Yb}^+$ ion and measure the resulting work distributions under cyclic protocols with $\Delta F = 0$, 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.