Pseudo-Hermitian operators generalize the concept of Hermiticity. This class of operators includes the quasi-Hermitian operators, which reformulate quantum theory while retaining real-valued measurement outcomes and unitary time evolution. This thesis is devoted to the study of locality in quasi-Hermitian theory, the symmetries and conserved quantities associated with non-Hermitian operators, and the perturbative features of pseudo-Hermitian matrices. In addition to the presented original research, scholars will appreciate the lengthy introduction to non-Hermitian physics. Local quasi-Hermitian observable algebras are examined. Expectation values of local quasi-Hermitian observables equal expectation values of local Hermitian observables. Thus, quasi-Hermitian theories do not increase the values of nonlocal games set by Hermitian theories. Furthermore, Bell's inequality violations in quasi-Hermitian theories never exceed the Tsirelson bound of Hermitian quantum theory. Exceptional points, which are branch points in the spectrum, are a perturbative feature unique to non-Hermitian operators. Cusp singularities of algebraic curves are related to higher-order exceptional points. To exemplify novelties of non-Hermiticity, one-dimensional lattice models with a pair of non-Hermitian defect potentials with balanced loss and gain, $Δ\pm i γ$ , are explored. When the defects are nearest neighbour, the entire spectrum becomes complex when $γ$ is tuned past a second-order exceptional point. When the defects are at the edges of the chain and the hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the $\mathcal{PT}$ -phase transition is dictated by the topological phase. Chiral symmetry and representation theory are used to derive large classes of pseudo-Hermitian operators with closed-form intertwining operators.Pseudo-Hermitian operators generalize the concept of Hermiticity. This class of operators includes the quasi-Hermitian operators, which reformulate quantum theory while retaining real-valued measurement outcomes and unitary time evolution. This thesis is devoted to the study of locality in quasi-Hermitian theory, the symmetries and conserved quantities associated with non-Hermitian operators, and the perturbative features of pseudo-Hermitian matrices. In addition to the presented original research, scholars will appreciate the lengthy introduction to non-Hermitian physics. Local quasi-Hermitian observable algebras are examined. Expectation values of local quasi-Hermitian observables equal expectation values of local Hermitian observables. Thus, quasi-Hermitian theories do not increase the values of nonlocal games set by Hermitian theories. Furthermore, Bell's inequality violations in quasi-Hermitian theories never exceed the Tsirelson bound of Hermitian quantum theory. Exceptional points, which are branch points in the spectrum, are a perturbative feature unique to non-Hermitian operators. Cusp singularities of algebraic curves are related to higher-order exceptional points. To exemplify novelties of non-Hermiticity, one-dimensional lattice models with a pair of non-Hermitian defect potentials with balanced loss and gain, $Δ\pm i γ$, are explored. When the defects are nearest neighbour, the entire spectrum becomes complex when $γ$ is tuned past a second-order exceptional point. When the defects are at the edges of the chain and the hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the $\mathcal{PT}$-phase transition is dictated by the topological phase. Chiral symmetry and representation theory are used to derive large classes of pseudo-Hermitian operators with closed-form intertwining operators.

arXiv:2306.04044Jun 6, 20230 citations

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

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.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.

Physical Review AAug 7, 20244 citations

Algebraic method for pseudo-Hermitian Hamiltonians

Junqing Li, Yan-Gang Miao, Zhao Xue

Jul 25, 20115 citations

Higher-order exceptional point in a pseudo-Hermitian cavity optomechanical system

Wei Xiong, Zhuanxia Li, Yiling Song +3 more

Higher-order exceptional points (EPs), resulting from non-Hermitian degeneracies, have shown greater advantages in sensitive enhancement than second-order EPs (EP2s). Therefore, seeking higher-order EPs in various quantum systems is important for quantum information science. Here we propose a benchmark cavity optomechanical (COM) system consisting of a mechanical resonator coupled to two cavities via radiation pressure for predicting the third-order exceptional point (EP3). We first give the pseudo-Hermitian condition for the non-Hermitian COM system by taking the bath effects into account. Then we consider the mechanical gain effect, and we find that the pseudo-Hermitian COM system without $\mathcal{PT}$ symmetry can host both EP3 and EP2 for symmetric and asymmetric cavities. In the symmetric case, only EP3 or EP2 can be predicted in the parameter space, but EP3 and EP2 can be transformed into each other by tuning the optomechanical coupling strength in the asymmetric case. We further consider the case of one cavity with gain. For this case, the pseudo-Hermitian COM system is $\mathcal{PT}$ symmetric and can also host EP3 or EP2. The influence of system parameters on them is discussed. Our proposal provides a potential way to realize sensitive detection and study other physical phenomena around higher-order EPs in non-Hermitian COM systems.Higher-order exceptional points (EPs), resulting from non-Hermitian degeneracies, have shown greater advantages in sensitive enhancement than second-order EPs (EP2s). Therefore, seeking higher-order EPs in various quantum systems is important for quantum information science. Here we propose a benchmark cavity optomechanical (COM) system consisting of a mechanical resonator coupled to two cavities via radiation pressure for predicting the third-order exceptional point (EP3). We first give the pseudo-Hermitian condition for the non-Hermitian COM system by taking the bath effects into account. Then we consider the mechanical gain effect, and we find that the pseudo-Hermitian COM system without $\mathcal{PT}$ symmetry can host both EP3 and EP2 for symmetric and asymmetric cavities. In the symmetric case, only EP3 or EP2 can be predicted in the parameter space, but EP3 and EP2 can be transformed into each other by tuning the optomechanical coupling strength in the asymmetric case. We further consider the case of one cavity with gain. For this case, the pseudo-Hermitian COM system is $\mathcal{PT}$ symmetric and can also host EP3 or EP2. The influence of system parameters on them is discussed. Our proposal provides a potential way to realize sensitive detection and study other physical phenomena around higher-order EPs in non-Hermitian COM systems.

arXiv:2109.12232Physical Review ADec 13, 202156 citations

Pseudo-Hermitian Systems, Involutive Symmetries and Pseudofermions

O. Cherbal, D. Trifonov, M. Zenad

arXiv:1605.02064International Journal of Theoretical PhysicsSep 13, 20161 citation

Δ\pm i γ

, are explored. When the defects are nearest neighbour, the entire spectrum becomes complex when

γ

is tuned past a second-order exceptional point. When the defects are at the edges of the chain and the hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the

\mathcal{PT}

-phase transition is dictated by the topological phase. Chiral symmetry and representation theory are used to derive large classes of pseudo-Hermitian operators with closed-form intertwining operators.

\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.

\mathcal{PT}

symmetry can host both EP3 and EP2 for symmetric and asymmetric cavities. In the symmetric case, only EP3 or EP2 can be predicted in the parameter space, but EP3 and EP2 can be transformed into each other by tuning the optomechanical coupling strength in the asymmetric case. We further consider the case of one cavity with gain. For this case, the pseudo-Hermitian COM system is

\mathcal{PT}

symmetric and can also host EP3 or EP2. The influence of system parameters on them is discussed. Our proposal provides a potential way to realize sensitive detection and study other physical phenomena around higher-order EPs in non-Hermitian COM systems.