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Physical Review A

Phase transitions and dynamic currents in a pseudo-Hermitian system of coupled sites with tunable gain and loss

Yan-Zi Jing, Kewen Xiao, Wen-Yuan Wang

Published Aug 7, 2024• 4 citations
Publisher

Abstract

The investigation of pseudo-Hermitian systems has garnered widespread attention due to their distinctive property of purely real eigenvalues splitting into complex conjugate pairs. This transition profoundly modifies the system's behavior, specifically giving rise to remarkable dynamical effects. Here we explore the novel characteristics of phase transitions and dynamic currents in a pseudo-Hermitian system of coupled sites with tunable gain and loss. With tunable gain and loss, we verify that the system uniquely supports two pseudo-Hermitian configurations, which exhibit distinct symmetries. The phase transitions from the $\mathcal{PT}$-symmetric phase to the spontaneous broken $\mathcal{PT}$-symmetric phase, along with the associated high-order exceptional points (EPs), are unambiguously demonstrated in both types of symmetry configurations. We measure the dynamic current within the four coupled sites, which allows us to probe the EPs and phase transitions between distinct phases. Our results offer insights into the phase transitions and dynamics of pseudo-Hermitian systems, with potential applications in the design and manipulation of novel quantum phenomena across various fields including photonics and other related quantum systems.

DOI
10.1103/physreva.110.023711

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The investigation of pseudo-Hermitian systems has garnered widespread attention due to their distinctive property of purely real eigenvalues splitting into complex conjugate pairs. This transition profoundly modifies the system's behavior, specifically giving rise to remarkable dynamical effects. Here we explore the novel characteristics of phase transitions and dynamic currents in a pseudo-Hermitian system of coupled sites with tunable gain and loss. With tunable gain and loss, we verify that the system uniquely supports two pseudo-Hermitian configurations, which exhibit distinct symmetries. The phase transitions from the PT\mathcal{PT}PT-symmetric phase to the spontaneous broken PT\mathcal{PT}PT-symmetric phase, along with the associated high-order exceptional points (EPs), are unambiguously demonstrated in both types of symmetry configurations. We measure the dynamic current within the four coupled sites, which allows us to probe the EPs and phase transitions between distinct phases. Our results offer insights into the phase transitions and dynamics of pseudo-Hermitian systems, with potential applications in the design and manipulation of novel quantum phenomena across various fields including photonics and other related quantum systems.
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 study the exceptional-point (EP) structure and the associated quantum dynamics in a system consisting of a non-Hermitian qubit and a Hermitian qubit. We find that the system possesses two sets of EPs, which divide the system-parameter space into P T -symmetry unbroken, partially broken and fully broken regimes, each with distinct quantum-dynamics characteristics. Particularly, in the partially broken regime, while the P T -symmetry is generally broken in the whole four-dimensional Hilbert space, it is preserved in a two-dimensional subspace such that the quantum dynamics in the subspace are similar to those in the P T -symmetry unbroken regime. In addition, we reveal that the competition between the inter-qubit coupling and the intra-qubit driving gives rise to a complex pattern in the EP variation with system parameters.
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PT\mathcal{PT}PT-like phase transitions from square roots of supersymmetric Hamiltonians
We introduce a general framework for realizing PT\mathcal{PT}PT-like phase transitions in non-Hermitian systems without imposing explicit parity--time (PT\mathcal{PT}PT) symmetry. The approach is based on constructing a Hamiltonian as the square root of a supersymmetric partner energy-shifted by a constant. This formulation naturally leads to bipartite dynamics with balanced gain and loss and can incorporate non-reciprocal couplings. The resulting systems exhibit entirely real spectra over a finite parameter range precisely when the corresponding passive Hamiltonian lacks a zero mode. As the non-Hermitian parameter representing gain and loss increases, the spectrum undergoes controlled real-to-complex transitions at second-order exceptional points. We demonstrate the versatility of this framework through several examples, including well-known models such as the Hatano--Nelson (HN) and complex Su--Schrieffer--Heeger (cSSH) lattices. Extending the formalism to qqq-commuting matrices further enables the systematic realization of higher-order exceptional points in systems with unidirectional couplings. Overall, this work uncovers new links between non-Hermitian physics and supersymmetry, offering a practical route to engineer photonic arrays with tunable spectral properties beyond what is achievable with explicit PT\mathcal{PT}PT-symmetry.
Parity-time (PT)(\mathcal{PT})(PT) symmetric systems are classical, gain-loss systems whose dynamics are governed by non-Hermitian Hamiltonians with exceptional-point (EP) degeneracies. The eigenvalues of a PT\mathcal{PT}PT-symmetric Hamiltonian change from real to complex conjugates at a critical value of gain-loss strength that is called the PT\mathcal{PT}PT breaking threshold. Here, we obtain the PT\mathcal{PT}PT threshold for a one-dimensional, finite Kitaev chain---a prototype for a ppp-wave superconductor---in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, on-site potential, and the distance between the gain and loss sites. In addition to a robust, nonlocal threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band structure of the Hermitian Kitaev model. In particular, for an even chain with zero on-site potential, we find a re-entrant PT\mathcal{PT}PT-symmetric phase bounded by second-order EP contours. Our numerical results are supplemented by analytical calculations for small system sizes.
The exploration of the effect of non-Hermitian (NH) in quantum systems is gaining renewed momentum as a result of the recent progress in experiments. It has been indicated that the ability to engineer the orders of exceptional points (EPs) can be fundamental to its applications in quantum control, such as accelerating entanglement generation; however, an initial survey of the interactions between different orders of EPs is still missing. In this work, we try to partially address this issue by employing a non-reciprocally coupled cavity system which can be experimentally realizable with a two-dimensional reconfigurable NH gauged laser array. With deliberate design, our systems display both second and third orders of EPs. The sign of a novel, purely quantum continuous entanglement phase transition is observed, which reveals how multi-EP interactions alter entanglement dynamics across different regimes while maintaining accelerated generation. Our findings reveil the genuinely quantum NH physics of higher-order EPs, which has opened up a new avenue for future investigations of the higher-order degeneracy application and the dynamic phase transitions of quantum systems.