从经典的 Frenet-Serret 装置到量子力学演化的曲率和扭转。第二部分.非稳态哈密顿

IF 2.1 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL
Paul M. Alsing, Carlo Cafaro
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引用次数: 0

摘要

在本文中,我们从几何学的角度阐述了如何量化在非稳态哈密顿力下演化的状态向量所描绘的量子曲线的弯曲和扭曲。具体地说,基于现有的静态哈密顿几何观点,我们讨论了将我们的理论构造推广到时变曲率和扭转系数都起关键作用的时变量子力学场景。具体地说,我们提出了一种量子版的 Frenet-Serret 装置,该装置适用于投影希尔伯特空间中的量子轨迹,该轨迹是由平行传输的纯量子态在指定薛定谔演化方程的时变哈密顿下单元演化出来的。时变曲率系数由切线向量|T〉到状态向量|Ψ〉的协变导数的平方的大小确定,并测量量子曲线的弯曲程度。时变扭转系数则由切线矢量|T〉到状态矢量|Ψ〉的协变导数投影的大小平方给出,与|T〉和|Ψ〉正交,此外还测量量子曲线的扭转。我们发现,从统计学的角度来看,时变设置呈现出更丰富的结构。例如,与不依赖于时间的构型不同,我们发现广义方差的概念不可逆转地进入了量子态在非稳态汉密尔顿下演化出的曲线扭转的定义中。为了从物理上说明我们的构造的意义,我们将其应用于一个由正弦振荡时变势指定的精确可解时变双态拉比问题。在这种情况下,我们证明曲率系数和扭转系数的分析表达式完全可以用两个实三维矢量来描述,即指定量子系统的布洛赫矢量和外部施加的时变磁场。尽管我们证明了扭转在任意随时间变化的单量子位哈密顿演化过程中同等于零,但我们还是研究了曲率系数在不同动力学情况下的时间行为,包括非共振和共振状态,以及强驱动和弱驱动构型。虽然我们的形式主义适用于任意维度的纯量子态,但随着维度的增加,相关曲率的分析推导和轨道模拟会变得相当复杂。因此,最后我们简要评述了将我们的几何形式主义应用于在一般非稳态哈密顿下单元演化的高维量子系统的可能性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
From the classical Frenet–Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part II. Nonstationary Hamiltonians

In this paper, we present a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by state vectors evolving under nonstationary Hamiltonians. Specifically, relying on the existing geometric viewpoint for stationary Hamiltonians, we discuss the generalization of our theoretical construct to time-dependent quantum-mechanical scenarios where both time-varying curvature and torsion coefficients play a key role. Specifically, we present a quantum version of the Frenet–Serret apparatus for a quantum trajectory in projective Hilbert space traced out by a parallel-transported pure quantum state evolving unitarily under a time-dependent Hamiltonian specifying the Schrödinger evolution equation. The time-varying curvature coefficient is specified by the magnitude squared of the covariant derivative of the tangent vector |T to the state vector |Ψ and measures the bending of the quantum curve. The time-varying torsion coefficient, instead, is given by the magnitude squared of the projection of the covariant derivative of the tangent vector |T to the state vector |Ψ, orthogonal to |T and |Ψ and, in addition, measures the twisting of the quantum curve. We find that the time-varying setting exhibits a richer structure from a statistical standpoint. For instance, unlike the time-independent configuration, we find that the notion of generalized variance enters nontrivially in the definition of the torsion of a curve traced out by a quantum state evolving under a nonstationary Hamiltonian. To physically illustrate the significance of our construct, we apply it to an exactly soluble time-dependent two-state Rabi problem specified by a sinusoidal oscillating time-dependent potential. In this context, we show that the analytical expressions for the curvature and torsion coefficients are completely described by only two real three-dimensional vectors, the Bloch vector that specifies the quantum system and the externally applied time-varying magnetic field. Although we show that the torsion is identically zero for an arbitrary time-dependent single-qubit Hamiltonian evolution, we study the temporal behavior of the curvature coefficient in different dynamical scenarios, including off-resonance and on-resonance regimes and, in addition, strong and weak driving configurations. While our formalism applies to pure quantum states in arbitrary dimensions, the analytic derivation of associated curvatures and orbit simulations can become quite involved as the dimension increases. Thus, finally we briefly comment on the possibility of applying our geometric formalism to higher-dimensional qudit systems that evolve unitarily under a general nonstationary Hamiltonian.

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来源期刊
CiteScore
3.40
自引率
22.20%
发文量
274
审稿时长
6 months
期刊介绍: This journal publishes short communications, research and review articles devoted to all applications of geometric methods (including commutative and non-commutative Differential Geometry, Riemannian Geometry, Finsler Geometry, Complex Geometry, Lie Groups and Lie Algebras, Bundle Theory, Homology an Cohomology, Algebraic Geometry, Global Analysis, Category Theory, Operator Algebra and Topology) in all fields of Mathematical and Theoretical Physics, including in particular: Classical Mechanics (Lagrangian, Hamiltonian, Poisson formulations); Quantum Mechanics (also semi-classical approximations); Hamiltonian Systems of ODE''s and PDE''s and Integrability; Variational Structures of Physics and Conservation Laws; Thermodynamics of Systems and Continua (also Quantum Thermodynamics and Statistical Physics); General Relativity and other Geometric Theories of Gravitation; geometric models for Particle Physics; Supergravity and Supersymmetric Field Theories; Classical and Quantum Field Theory (also quantization over curved backgrounds); Gauge Theories; Topological Field Theories; Strings, Branes and Extended Objects Theory; Holography; Quantum Gravity, Loop Quantum Gravity and Quantum Cosmology; applications of Quantum Groups; Quantum Computation; Control Theory; Geometry of Chaos.
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