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.

• The paper demonstrates that parity-exchange symmetry enforces the non-Hermitian Jarzynski equality across an SU(2)-rotated family of hybrid Hamiltonians.
• It employs a two-point measurement protocol and trapped-ion experiments to verify theoretical predictions over both static and cyclic driving protocols.
• Results reveal that restoring mirror symmetry in transition probabilities sustains the fluctuation theorem even when conventional PT symmetry is broken.

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

Overview

This paper extends the conceptual and experimental foundation of non-equilibrium fluctuation theorems—specifically, the Jarzynski equality (JE)—into a broad class of non-Hermitian quantum systems beyond conventional $\mathcal{PT}$-symmetric models. The authors demonstrate both theoretically and experimentally that the validity of a non-Hermitian version of the JE is enforced not by $\mathcal{PT}$ symmetry per se but by an operator-level parity-exchange symmetry of the transition probabilities under a specific class of SU(2)-rotated hybrid Hamiltonians interpolating between $\mathcal{PT}$ and $\mathcal{APT}$ symmetry.

A single trapped ${}^{171}$Yb$^+$ ion is employed to implement non-Hermitian evolution within a two-level subspace, with energies defined via the Hermitian part of the hybrid Hamiltonian. Using the two-point measurement (TPM) protocol and postselected no-jump trajectories, the experiments investigate the JE across a continuous family of SU(2)-rotated hybrid models. Key results indicate that mirror and parity-exchange symmetry suffice for the non-Hermitian JE to hold, even when $\mathcal{PT}$ is broken, provided the appropriate operational and measurement prescriptions are adopted.

Theoretical Foundations

Non-Hermitian Quantum Thermodynamics and Jarzynski Equality

For systems subject to non-equilibrium driving, the JE provides the exact relation $\langle e^{-\beta W} \rangle = e^{-\beta \Delta F}$ for work fluctuations, connecting the ensemble-averaged exponentiated work $\mathcal{APT}$0 to the free energy difference $\mathcal{APT}$1, regardless of how far the process is from equilibrium. In non-Hermitian quantum systems, intrinsic ambiguity arises in defining trajectories and work due to the lack of a Hermitian generator and the possibility of complex spectra.

The paper implements a TPM framework, initializing in a thermal distribution defined by the eigenstates of the Hermitian part of the Hamiltonian, applying a non-Hermitian evolution (with no-jump postselection), and redefining the work as the difference between the eigenvalues obtained in the initial and final projective measurements.

Generalized SU(2) Family and Parity-Exchange Symmetry

The central theoretical advance is the construction of a hybrid Hamiltonian family parameterized by angle $\mathcal{APT}$2:

$\mathcal{APT}$3

This Hamiltonian interpolates via SU(2) rotations between the $\mathcal{APT}$4 ($\mathcal{APT}$5) and $\mathcal{APT}$6 ($\mathcal{APT}$7) endpoints, maintaining a real Hermitian spectrum in the TPM scheme.

The main analytical result is that the validity of the (conditional) non-Hermitian JE is enforced by the symmetry condition

$\mathcal{APT}$8

where $\mathcal{APT}$9 is the normalized probability for TPM measurement transitions, corresponding to a generalized parity-exchange symmetry with respect to the energy eigenbasis of the Hermitian part. This operator-level symmetry, originating from an involutive $\mathcal{PT}$0 operator, is preserved along the entire SU(2)-rotated family.

Time-dependent driving and detunings that move the Hamiltonian outside the SU(2)-protected subspace generally break this symmetry, leading to violations of the JE except at special evolution times where the Floquet Hamiltonian re-enters the canonical hybrid form.

Experimental Realization

Trapped Ion Implementation

A single $\mathcal{PT}$1Yb$\mathcal{PT}$2 ion in a Paul trap is manipulated via microwave coupling and engineered dissipation to realize the non-Hermitian dynamics. The dissipative component is controlled optically and projects population outside the two-level system, creating an effective non-Hermitian Hamiltonian within the qubit subspace after postselection.

The experiment utilizes deterministic eigenstate preparation, engineered SU(2) basis rotations, and time-dependent protocols to realize representative points ($\mathcal{PT}$3) along the hybrid orbit. State preparation fidelity and system drifts are carefully controlled and benchmarked by quantum state tomography.

Protocol and Measurements

Each run comprises:

• Preparation of eigenstates $\mathcal{PT}$4 corresponding to the Hermitian part
• Non-Hermitian evolution with piecewise postselection to mitigate decay-induced detection loss
• Final projection in the rotated energy eigenbasis
• Evaluation of transition probabilities and computation of $\mathcal{PT}$5

Three classes of driving protocols are explored:

1. Constant $\mathcal{PT}$6 (static hybrid Hamiltonian)
2. Cyclically ramped $\mathcal{PT}$7
3. Sinusoidal time-dependent detuning $\mathcal{PT}$8 (driving system beyond canonical hybrid subspace)

Results

Symmetry-Enforced Validity of the Non-Hermitian JE

• For both static and cyclic protocols confined to the SU(2)-protected subspace, experimental and theoretical results show $\mathcal{PT}$9, confirming that the JE holds across the hybrid family, not just at the $\mathcal{PT}$0- or $\mathcal{PT}$1-symmetric endpoints.
• Analysis shows that parity-exchange symmetry of transition probabilities is robustly preserved and is the operational mechanism enforcing the JE.

Breakdown and Revival under Generalized Driving

• When time-dependent detuning is introduced, the parity-exchange symmetry and thus the JE fails for generic evolution intervals; only at discrete times, where the stroboscopic Floquet Hamiltonian regains the hybrid symmetry, does the JE revive.
• High-resolution temporal scanning in the experiment captures these symmetry revival points, directly correlating zero-crossings in symmetry-breaking observables with restoration of the JE.

Intermediate Angle Verification

• Experimental confirmation at the maximally hybridized $\mathcal{PT}$2 underscores that the physical symmetry, not the algebraic subclass, governs the fluctuation relations, completing the SU(2) family test.

Implications and Future Directions

The identification of parity-exchange symmetry as the operative criterion for non-Hermitian JE validity generalizes non-Hermitian statistical mechanics frameworks and decouples the fluctuation theorem from strict $\mathcal{PT}$3 symmetry. This insight could guide the control and design of open quantum systems for quantum thermodynamics and information processing, expanding the range of acceptable non-Hermitian generators.

Experimentally, the passive no-jump scheme faces a tradeoff between the exactness of postselected dynamics and unfavorable signal decay. Transitioning to active $\mathcal{PT}$4-symmetric platforms with gain/loss balancing may circumvent this cost, though fundamental questions about fluctuation relations under such conditions remain open.

The symmetry-based perspective presented here paves the way to investigate higher-dimensional non-Hermitian systems, topological phenomena, and integrated non-equilibrium quantum technologies leveraging precisely engineered dissipative or non-Hermitian resources.

Conclusion

This work demonstrates that the conditional non-Hermitian Jarzynski equality is not exclusive to the unbroken $\mathcal{PT}$5 regime but is instead enforced by a generic parity-exchange symmetry of transition probabilities within SU(2)-rotated two-level non-Hermitian systems. Experimental verification using a trapped-ion platform corroborates the theoretical predictions and highlights the primacy of operational symmetry over algebraic class in determining the thermodynamic behavior of open quantum systems. This framework forms a scalable basis for exploring non-equilibrium fluctuation theorems in complex, engineered non-Hermitian quantum platforms (2605.10099).