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Time invariants across a fourth-order exceptional point in a parity-time-symmetric qudit

Zhihao Bian, Lei Xiao, Kunkun Wang, Xiang Zhan, Franck Assogba Onanga, František Růžička, Wei Yi, Yogesh N. Joglekar, Peng Xue

Published Mar 23, 2019• 2 citations
OpenAlex

Abstract

Constants of motion of a closed system, such as its energy or charge, are determined by symmetries of the system. They offer global insights into the system dynamics and were instrumental to advances such as the prediction of neutrinos. In contrast, little is known about time invariants in open systems. Recently, a special class of open systems with parity-time (PT) symmetry has been intensely explored for their remarkable properties. However, a complete characterization and experimental observation of time invariants therein are both still lacking. Here we present an analytical solution for all time invariants of a broad class of PT-symmetric Hamiltonians. Using a single-photon interferometry setup, we confirm our results by simulating the quantum dynamics of a PT-symmetric qudit across a fourth-order exceptional point. We further observe the information flow in the system via the dynamics of qudit entropies. Our results demonstrate conserved quantities in a non-unitary time evolution, and point towards the rich dynamics and enhanced sensitivity of systems with higher-order exceptional points.

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Constants of motion of a closed system, such as its energy or charge, are determined by symmetries of the system. They offer global insights into the system dynamics and were instrumental to advances such as the prediction of neutrinos. In contrast, little is known about time invariants in open systems. Recently, a special class of open systems with parity-time (PT) symmetry has been intensely explored for their remarkable properties. However, a complete characterization and experimental observation of time invariants therein are both still lacking. Here we present an analytical solution for all time invariants of a broad class of PT-symmetric Hamiltonians. Using a single-photon interferometry setup, we confirm our results by simulating the quantum dynamics of a PT-symmetric qudit across a fourth-order exceptional point. We further observe the information flow in the system via the dynamics of qudit entropies. Our results demonstrate conserved quantities in a non-unitary time evolution, and point towards the rich dynamics and enhanced sensitivity of systems with higher-order exceptional points.
Non-Hermitian systems with parity-time (PT\mathcal{PT}PT) symmetry give rise to exceptional points (EPs) with exceptional properties that arise due to the coalescence of eigenvectors. Such systems have been extensively explored in the classical domain, where second- or higher-order EPs have been proposed or realized. In contrast, quantum information studies of PT\mathcal{PT}PT-symmetric systems have been confined to systems with a two-dimensional Hilbert space. Here, by using a single-photon interferometry setup, we simulate the quantum dynamics of a four-dimensional PT\mathcal{PT}PT-symmetric system across a fourth-order exceptional point. By tracking the coherent, nonunitary evolution of the density matrix of the system in PT\mathcal{PT}PT-symmetry unbroken and broken regions, we observe the entropy dynamics for both the entire system, and the gain and loss subsystems. Our setup is scalable to the higher-dimensional PT\mathcal{PT}PT-symmetric systems, and our results point towards the rich dynamics and critical properties.
We experimentally simulate nonunitary quantum dynamics using a single-photon interferometric network and study the information flow between a parity-time- (PT-)symmetric non-Hermitian system and its environment. We observe oscillations of quantum-state distinguishability and complete information retrieval in the PT-symmetry-unbroken regime. We then characterize in detail critical phenomena of the information flow near the exceptional point separating the PT-unbroken and PT-broken regimes, and demonstrate power-law behavior in key quantities such as the distinguishability and the recurrence time. We also reveal how the critical phenomena are affected by symmetry and initial conditions. Finally, introducing an ancilla as an environment and probing quantum entanglement between the system and the environment, we confirm that the observed information retrieval is induced by a finite-dimensional entanglement partner in the environment. Our work constitutes the first experimental characterization of critical phenomena in PT-symmetric nonunitary quantum dynamics.
Exceptional points (EPs) of a non-Hermitian Hamiltonian with parity-time-reversal (PT) symmetry have the potential to drastically enhance the capabilities of metrology and sensing through their power-law growing sensitivity to external perturbation. With the ability of generating and tuning dissipation in a single trapped ion system, we observe rich dynamics and detailed quantum phase transitions from the PT-symmetric phase to the symmetry-breaking phase. In this single qubit full quantum system, we develop a method to precisely determine the location of EP without any fitting parameter, and extract the eigenvalues in a unified way through all parameter regions. We can also obtain the full density matrix by quantum state tomography. Finally, we suggest from theoretical analysis that the periodically driving PT-symmetric non-Hermitian system can be used to measure the magnitude, frequency, and phase of time-dependent perturbation with EP enhancement.
The concept of parity-time (PT) symmetry originates from the framework of quantum mechanics, where if the Hamiltonian operator satisfies the commutation relation with the parity and time operators, it shows real energy spectrum. Recently, PT symmetry was introduced into various systems, such as optics, electronic circuits, acoustics, and other classical fields to further study the dynamical behaviors of the Hamiltonian and the energies. To focus on the dynamical evolution of the quantum state, here we experimentally studied the dynamical evolution of a two-level quantum system under the PT symmetric Hamiltonian using single-photon system. By enlarging the system using ancillary qubit and encoding the subsystem under the non-Hermitian Hamiltonian with post-selection, the evolution of the state can be characterized with a high fidelity. Owing to the effectively operation of the dilation method, our work provides a route for further exploiting the exotic properties of PT symmetric Hamiltonian for quantum simulation and quantum information processing.
Abstract Non-Hermitian parity-time ( PT{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}PT P T ) and anti-parity-time ( APT{{{{{{{\mathcal{APT}}}}}}}}APT APT )-symmetric systems exhibit novel quantum properties and have attracted increasing interest. Although many counterintuitive phenomena in PT{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}PT P T - and APT{{{{{{{\mathcal{APT}}}}}}}}APT APT -symmetric systems were previously studied, coherence flow has been rarely investigated. Here, we experimentally demonstrate single-qubit coherence flow in PT{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}PT P T - and APT{{{{{{{\mathcal{APT}}}}}}}}APT APT -symmetric systems using an optical setup. In the symmetry unbroken regime, we observe different periodic oscillations of coherence. Particularly, we observe two complete coherence backflows in one period in the PT{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}PT P T -symmetric system, while only one backflow in the APT{{{{{{{\mathcal{APT}}}}}}}}APT APT -symmetric system. Moreover, in the symmetry broken regime, we observe the phenomenon of stable value of coherence flow. We derive the analytic proofs of these phenomena and show that most experimental data agree with theoretical results within one standard deviation. This work opens avenues for future study on the dynamics of coherence in PT{{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}PT P T - and APT{{{{{{{\mathcal{APT}}}}}}}}APT APT -symmetric systems.