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arXiv:1711.00586 · Physical review. E

Quantum work relations and response theory in parity-time-symmetric quantum systems

Bo-Bo Wei

Published Jan 11, 2018• 12 citations
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Abstract

In this work, we show that a universal quantum work relation for a quantum system driven arbitrarily far from equilibrium extends to a parity-time- (PT-) symmetric quantum system with unbroken PT symmetry, which is a consequence of microscopic reversibility. The quantum Jarzynski equality, linear response theory, and Onsager reciprocal relations for the PT-symmetric quantum system are recovered as special cases of the universal quantum work relation in a PT-symmetric quantum system. In the regime of broken PT symmetry, the universal quantum work relation does not hold because the norm is not preserved during the dynamics.

DOI
10.1103/physreve.97.012114

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In this work, we show that a universal quantum work relation for a quantum system driven arbitrarily far from equilibrium extends to a parity-time- (PT-) symmetric quantum system with unbroken PT symmetry, which is a consequence of microscopic reversibility. The quantum Jarzynski equality, linear response theory, and Onsager reciprocal relations for the PT-symmetric quantum system are recovered as special cases of the universal quantum work relation in a PT-symmetric quantum system. In the regime of broken PT symmetry, the universal quantum work relation does not hold because the norm is not preserved during the dynamics.
We show that the quantum Jarzynski equality generalizes to PT\mathcal{P}\mathcal{T}PT-symmetric quantum mechanics with unbroken PT\mathcal{P}\mathcal{T}PT symmetry. In the regime of broken PT\mathcal{P}\mathcal{T}PT symmetry, the Jarzynski equality does not hold as also the CPT\mathcal{C}\mathcal{P}\mathcal{T}CPT norm is not preserved during the dynamics. These findings are illustrated for an experimentally relevant system---two coupled optical waveguides. It turns out that for these systems the phase transition between the regimes of unbroken and broken PT\mathcal{P}\mathcal{T}PT symmetry is thermodynamically inhibited as the irreversible work diverges at the critical point.
Thermodynamics and information theory have been intimately related since the times of Maxwell and Boltzmann. Recently it was shown that the dissipated work in an arbitrary nonequilibrium process is related to the R\'enyi divergences between two states along the forward and reversed dynamics. Here we show that the relation between dissipated work and Renyi divergences generalizes to PT\mathcal{PT}PT-symmetric quantum mechanics with unbroken PT\mathcal{PT}PT symmetry. In the regime of broken PT\mathcal{PT}PT symmetry, the relation between dissipated work and Renyi divergences does not hold as the norm is not preserved during the dynamics. This finding is illustrated for an experimentally relevant system of two-coupled cavities.
For classical systems driven out of equilibrium, Crooks derived a relation (the Crooks-Jarzynski relation), whose special cases include a relation (the Crooks relation) equivalent to the Kawasaki non-linear response relation. We derive a quantum extension of the Crooks-Jarzynski relation without explicitly using the principle of microreversibility. Its special cases lead to the Jarzynski equality and the standard linear response theory with a Green-Kubo formula with a canonical correlation function. We also derive a quantum extension of the Crooks relation using the principle of microreversibility. Its special cases lead to the Jarzynski equality, the Crooks transient fluctuation theorem, and the fluctuation theorem for current or shear stress, which leads to a Green-Kubo formula with a symmetrized correlation function. For each quantum Crooks relation, there exists a corresponding quantum Crooks-Jarzynski relation. Using either relation, we can derive the Jarzynski equality, the fluctuation theorems mentioned above, and the standard linear response theory.
The Jarzynski equality is a cornerstone of nonequilibrium thermodynamics, linking work statistics to equilibrium freeenergy 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 post-selected 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 (PT) and anti-parity-time (APT) symmetric terms, and demonstrate using complementary geometric and algebraic arguments that the parity-exchange symmetry persists throughout the corresponding SU(2)-rotated orbit. Relative to previous PT-focused conditional Jarzynski equality results, the advance here is an extension of the symmetry criterion from the isolated PT endpoint to a broader PT−APT hybrid family. Experimentally, we implement three representative points, θk = 0, π/4, π/2, in a single trapped 171Yb+ 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.
Comprehensive study on parity-time (PT) symmetric systems demonstrates the novel properties and innovative application of non-Hermitian physics in recent years. In the quantum regime, PT symmetric physics exhibits unique quantum dynamical behaviors such as spontaneous state-distinguishability oscillation. However, the construction and control of a PT symmetric quantum system are still challenging, that and restrict the experimental investigation of PT symmetric quantum nature and application. In this Letter, we propose and construct a recycling-structure PT symmetric quantum simulator for the first time, which can effectively simulate the discrete-time dynamical process of a PT symmetric quantum system in both unbroken and broken phases, to be different from our previous work [J.-S. Tang, et al., Nat. Photonics 10, 642 (2016)]. We investigate the dynamical features of quantum state distinguishability based on the PT symmetric simulator. Our results demonstrate the novel PT symmetric quantum dynamics characterized by the periodical oscillation of state distinguishability in the unbroken phase, and the monotonic decay of that in the broken phase. This work also provides a practical experimental platform for the future intensive study of PT symmetric quantum dynamics.