Abstract Due to the nonunitary time evolution and possibly complex energy eigenvalues in non-Hermitian systems, it is still under debate how to properly deal with the dynamics of time-dependent non-Hermitian Hamiltonian. Recently a quantum metric framework has been proposed to study the dynamics of generated defects of a non-Hermitian system under linear quench. Here, we provide an explicit expression for the endowed Hamiltonian under quantum metric for a general two-level non-Hermitian system. Then we propose two exactly solvable models for the study of nonadiabatic dynamics of non-Hermitian systems, and analyze the defect production using the metric method. We find that, in contrast to the direct normalization method, the metric method can reproduce the symmetry of generated defects. The power-law scaling of generated defects with respect to quench time is also obtained.
Abstract Due to the nonunitary time evolution and possibly complex energy eigenvalues in non-Hermitian systems, it is still under debate how to properly deal with the dynamics of time-dependent non-Hermitian Hamiltonian. Recently a quantum metric framework has been proposed to study the dynamics of generated defects of a non-Hermitian system under linear quench. Here, we provide an explicit expression for the endowed Hamiltonian under quantum metric for a general two-level non-Hermitian system. Then we propose two exactly solvable models for the study of nonadiabatic dynamics of non-Hermitian systems, and analyze the defect production using the metric method. We find that, in contrast to the direct normalization method, the metric method can reproduce the symmetry of generated defects. The power-law scaling of generated defects with respect to quench time is also obtained.
Non-Hermiticity in quantum Hamiltonians leads to nonunitary time evolution and possibly complex energy eigenvalues, which can lead to a rich phenomenology with no Hermitian counterpart. In this work, we study the dynamics of an exactly solvable non-Hermitian system, hosting both PT-symmetric and PT-broken modes subject to a linear quench. Employing a fully consistent framework, in which the Hilbert space is endowed with a nontrivial dynamical metric, we analyze the dynamics of the generated defects. In contrast to Hermitian systems, our study reveals that PT -broken time evolution leads to defect freezing and hence the violation of adiabaticity. This physics necessitates the so-called metric framework, as it is missed by the oft used approach of normalizing quantities by the time-dependent norm of the state. Our results are relevant for a wide class of experimental systems.
Non-Hermiticity in quantum Hamiltonians leads to nonunitary time evolution and possibly complex energy eigenvalues, which can lead to a rich phenomenology with no Hermitian counterpart. In this work, we study the dynamics of an exactly solvable non-Hermitian system, hosting both PT-symmetric and PT-broken modes subject to a linear quench. Employing a fully consistent framework, in which the Hilbert space is endowed with a nontrivial dynamical metric, we analyze the dynamics of the generated defects. In contrast to Hermitian systems, our study reveals that PT-broken time evolution leads to defect freezing and hence the violation of adiabaticity. This physics necessitates the so-called metric framework, as it is missed by the oft used approach of normalizing quantities by the time-dependent norm of the state. Our results are relevant for a wide class of experimental systems.
Non-unitary dynamics in quantum mechanics arise in open quantum systems, where the interaction between the system and its environment is not negligible. One way this coupling can be realised is through measurement, where the backaction from the measurement apparatus induces non-unitary dynamics in the system. An effective non-Hermitian description arises within the no-jump approximation of a measured system, in which the time-evolved states are explicitly normalised at all times. Non-Hermiticity in quantum Hamiltonians gives rise to non-unitary time evolution and possibly complex energy eigenvalues, which can lead to a rich phenomenology with no Hermitian counterpart. In this thesis, we explore these two interconnected concepts related to non-unitary dynamics: non-Hermiticity and measurements. However, instead of effective non-Hermiticity, we consider the cases where non-Hermiticity is realised exactly. This setup necessitates a treatment under the self-consistent frameworks of non-Hermitian quantum mechanics in order to restore its probabilistic interpretation. Concretely, the self-consistent metric formalism restores unitarity of the evolution through a modified Hilbert space endowed with a dynamical metric. Through comparative studies using exactly solvable models, we find that the physics of the two approaches of non-Hermitian quantum mechanics is fundamentally different. This has a profound physical impact in terms of defect production in quenches across an exceptional point (EP). We demonstrate this through a non-Hermitian system hosting both PT-symmetric and PT-broken modes subject to a linear quench. Our results highlight the role played by the EP and the spontaneous breaking of the PT-symmetry in contributing to a non-zero defect density, even in the limit of an infinitely slow drive. Further, we uncover generalised transformations underpinning the results, which are incorporated in the dynamical metric. Going beyond purely theoretical considerations, we implement two novel dilation schemes to emulate the metric dynamics in a digital quantum simulator. Our protocols enable controlled access to all PT-symmetry regimes and to a direct tomography of the dynamical metric, opening the possibility of studying a new kind of non-Hermitian physics in engineered matter. To explore the second theme of the thesis, we investigate the dynamics of entanglement entropy in the continuously monitored Bose-Hubbard model. Our numerical data reveals a counterintuitive enhancement of superfluid correlations in a parameter regime, in which the entanglement entropy increases with the onsite repulsion strength in the presence of measurement. This phenomenon arising from the interplay between hopping, interaction, and measurement, is also present in the effective central charge extracted from a fit of the logarithmic scaling form of the entropy. In summary, this thesis reveals the richness of non-unitary quantum dynamics. Through an in-depth study of the different aspects of non-Hermitian systems, we illustrate the non-trivial physics uncovered only with the consideration of the dynamical metric. By bridging theoretical insights and physical realisation through novel dilation protocols realisable in engineered matter, our work lays the groundwork for further exploration of dynamical aspects in non-Hermitian and monitored quantum systems.
Abstract In this series of works, we study exactly solvable non-unitary time evolutions in one-dimensional quantum critical systems ranging from quantum quenches to time-dependent drivings. In this part I, we are motivated by the recent works of Kontsevich and Segal (2021 arXiv: 2105.10161 ) and Witten (2021 arXiv: 2111.06514 ) on allowable complex spacetime metrics in quantum field theories. In general, such complex spacetime metrics will lead to non-unitary time evolutions. In this work, we study the universal features of such non-unitary time evolutions based on exactly solvable setups. Various physical quantities including the entanglement Hamiltonian and entanglement spectrum, entanglement entropy, and energy density at an arbitrary time can be exactly solved. Due to the damping effect introduced by the complex time, the excitations in the initial state are gradually damped out in time. The non-equilibrium dynamics exhibit universal features that are qualitatively different from the case of real-time evolutions. For instance, for an infinite system after a global quench, the entanglement entropy of the semi-infinite subsystem will grow logarithmically in time, in contrast to the linear growth in a real-time evolution. Moreover, we study numerically the time-dependent driven quantum critical systems with allowable complex spacetime metrics. It is found that the competition between driving and damping leads to a steady state with an interesting entanglement structure.
Despite acute interest in the dynamics of non-Hermitian systems, there is a lack of consensus in the mathematical formulation of non-Hermitian quantum mechanics in the community. Different methodologies are used in the literature to study non-Hermitian dynamics. This ranges from consistent frameworks like biorthogonal quantum mechanics and metric approach characterized by modified inner products, to normalization by time-dependent norms inspired by open quantum systems. In this work, we systematically explore the similarities and differences among these various methods. Utilizing illustrative models with exact solutions, we demonstrate that these methods produce not only quantitatively different results but also distinct physical interpretations. For dissipative systems where non-Hermiticity arises as an approximation, we find that simply dividing by the norm in the PT -broken regime closely aligns with the full master equation solutions. In contrast, for quantum systems where non-Hermiticity can be engineered exactly, incorporating metric dynamics is crucial for the probabilistic interpretation of quantum mechanics, necessitating the generalizations of similarity transformations and unitarity to non-Hermitian systems. This study lays the groundwork for further exploration of non-Hermitian Hamiltonians, potentially leveraging generalized transformations for novel physical phenomena.
Non-Hermitian physics provides an effective description of open and nonequilibrium systems and hosts many novel and intriguing phenomena such as exceptional points and non-Hermitian skin effect. Despite extensive theoretical and experimental studies, however, how to properly deal with the nonadiabatic dynamics in driven non-Hermitian quantum system is still under debate. Here, we develop a theoretical framework based on time-dependent biorthogonal quantum formalism by redefining the associated state to obtain the gauge-independent transition probability, and study the nonadiabatic dynamics of a linearly driven non-Hermitian system. In contrast to the normalization method that leads to a modified Kibble-Zurek scaling behavior, our approach predicts that the defect production at exceptional points exhibits power-law scaling behaviors conforming to the Kibble-Zurek mechanism. In the fast quench regime, universal scaling behaviors are also found with respect to the initial quenching parameter, which can be explained by the impulse-adiabatic approximation. Moreover, as trespassing the PT -broken region, the phenomenon of defect freezing, i.e., violation of adiabaticity, is observed.