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Communications in Theoretical Physics

Dynamics of 2 × 2 matrix non-Hermitian quantum systems on Bloch sphere

Libin Fu

Published Mar 11, 2024• 3 citations
Publisher

Abstract

Abstract By casting evolution to the Bloch sphere, the dynamics of 2 × 2 matrix non-Hermitian systems are investigated in detail. This investigation reveals that there are four kinds of dynamical modes for such systems. The different modes are classified by different kinds of fixed points, namely, the elliptic point, spiral point, critical node, and degenerate point. The Hermitian systems and the unbroken P T non-Hermitian cases belong to the category with elliptic points. The degenerate point just corresponds to the systems with exceptional point (EP). The topological properties of the fixed point are also discussed. It is interesting that the topological charge for the degenerate point is two, while the others are one.

DOI
10.1088/1572-9494/ad3223

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Abstract By casting evolution to the Bloch sphere, the dynamics of 2 × 2 matrix non-Hermitian systems are investigated in detail. This investigation reveals that there are four kinds of dynamical modes for such systems. The different modes are classified by different kinds of fixed points, namely, the elliptic point, spiral point, critical node, and degenerate point. The Hermitian systems and the unbroken P T non-Hermitian cases belong to the category with elliptic points. The degenerate point just corresponds to the systems with exceptional point (EP). The topological properties of the fixed point are also discussed. It is interesting that the topological charge for the degenerate point is two, while the others are one.
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.
We establish a geometric generalization of the Bloch sphere for two-level quantum systems with non-Hermitian Hamiltonians using the Spacetime Algebra (STA) formulation. By lifting the state density operator from the even subalgebra to the full STA, we show that the state space expands from the unit 2-sphere to a future light cone. The non-unitary time evolution generated by a general non-Hermitian Hamiltonian corresponds to proper orthochronous Lorentz transformations on the null vectors. We classify the Hamiltonian dynamics into four distinct geometric classes: spatial rotations (corresponding to PT\mathcal{PT}PT-symmetric systems), pure boosts (anti-PT\mathcal{PT}PT-symmetric systems), null rotations (exceptional points), and general mixtures. We also use this formulation to study several results from PT-symmetric quantum mechanics, including the topological features of the exceptional points.
An attempt is made to study the quantum criticality in non-Hermitian system with topological characterization. We use the zero mode solutions to characterize the topological phases and, criticality and also to construct the phase diagram. The Hermitian counterpart of the model Hamiltonian possess quite a few interesting features such as Majorana zero modes (MZMs) at criticality, unique topological phase transition on the critical line and hence these unique features are of an interest to study in the non-Hermitian case also. We observe a unique behavior of critical lines in presence of non-Hermiticity. We study the topological phase transitions in the non-Hermitian case using parametric curves which also reveal the gap closing point through exceptional points. We study bulk and edge properties of the system where at the edge, the stability dependence behavior of MZMs at criticality is studied and at the bulk we study the effect of non-Hermiticity on the topological phases by investigating the behavior of the critical lines. The study of non-Hermiticity on the critical lines revels the rate of receding of the topological phases with respect to the increase in the value of non-Hermiticity. This work gives a new perspective on topological quantum criticality in non-Hermitian quantum system.
In recent years, two independent research fields, i.e. non-Hermitian andstrongly correlated systems have been merged, forming an important researchfield in physics. The progress of relevant theories and experiments hasreshaped our understanding of matter. In this field, the research object isnot limited to the influence of non-Hermiticity on the energy spectrum andthe eigenstate properties of many-body systems. Researchers have paid more attentionto the manipulation of quantum states. It is universally received that the exceptional point is the most significant featurethat distinguishes non-Hermitian quantum mechanics from Hermitian quantum mechanics. In addition to the recent advances in non-Hermitian topological band theory and quantum sensing around the exceptional points, this paper concentrates on the non-Hermitian critical dynamical phenomenon and its application to the quantum many-body system. When the system has an exceptional point, an arbitrary initial state belonging to the coalescent subspace will be projected on the coalescent state. Based on the directionality of the evolved quantum state, this paper reviews our several representative researches in recent years, including local-field-induced dynamical magnetization, quantum phase transition in transverse field, Ising model at non-zero temperature, quantum mold casting in the center-environment system, as well as superconducting state preparation in the non-Hermitian strongly correlated system. We also focus on the new preparation methods and detection schemes of non-equilibrium quantum states related to exception points.
Abstract Exceptional points are interesting physical phenomena in non-Hermitian physics at which the eigenvalues are degenerate and the eigenvectors coalesce. In this paper, we find that in projected non-Hermitian two-level systems (sub-systems under projecting partial Hilbert space) the singularities of exceptional points (EPs) is due to basis defectiveness rather than energy degeneracy or state coalescence. This leads to the discovery of extended exceptional points (EEPs). For EEPs, more subtle structures (e.g. the so-called Bloch peach), additional classification, and ‘hidden’ quantum phase transitions are explored. By using the topologically protected sub-space from two edge states in the non-Hermitian Su–Schrieffer–Heeger model as an example, we illustrate the physical properties of different types of EEPs.
We suggest to employ the dissipative nature of open quantum systems for the purpose of parameter estimation: the dynamics of open quantum systems is typically described by a quantum dynamical semigroup generator The eigenvalues of are complex, reflecting unitary as well as dissipative dynamics. For certain values of parameters defining non-Hermitian degeneracies emerge, i.e. exceptional points (EP). The dynamical signature of these EPs corresponds to a unique time evolution. This unique feature can be employed experimentally to locate the EPs and thereby to determine the intrinsic system parameters with a high accuracy. This way we turn the disadvantage of the dissipation into an advantage. We demonstrate this method in the open system dynamics of a two-level system described by the Bloch equation, which has become the paradigm of diverse fields in physics, from NMR to quantum information and elementary particles.