Curvature of Quantum Evolutions for Qubits in Time-Dependent Magnetic Fields

• URL: http://arxiv.org/abs/2408.14233v1
• Date: Mon, 26 Aug 2024 12:49:28 GMT
• Title: Curvature of Quantum Evolutions for Qubits in Time-Dependent Magnetic Fields
• Authors: Carlo Cafaro, Leonardo Rossetti, Paul M. Alsing,
• Abstract summary: We present an exact analytical expression of the curvature of a quantum evolution for a two-level quantum system immersed in a time-dependent magnetic field. We show that the quantum curve is nongeodesic since the geodesic efficiency of the quantum evolution is strictly less than one.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In the geometry of quantum-mechanical processes, the time-varying curvature coefficient of a quantum evolution is specified by the magnitude squared of the covariant derivative of the tangent vector to the state vector. In particular, the curvature coefficient measures the bending of the quantum curve traced out by a parallel-transported pure quantum state that evolves in a unitary fashion under a nonstationary Hamiltonian that specifies the Schr\"odinger evolution equation. In this paper, we present an exact analytical expression of the curvature of a quantum evolution for a two-level quantum system immersed in a time-dependent magnetic field. Specifically, we study the dynamics generated by a two-parameter nonstationary Hermitian Hamiltonian with unit speed efficiency. The two parameters specify the constant temporal rates of change of the polar and azimuthal angles used in the Bloch sphere representation of the evolving pure state. To better grasp the physical significance of the curvature coefficient, showing that the quantum curve is nongeodesic since the geodesic efficiency of the quantum evolution is strictly less than one and tuning the two Hamiltonian parameters, we compare the temporal behavior of the curvature coefficient with that of the speed and the acceleration of the evolution in projective Hilbert space. Furthermore, we compare the temporal profile of the curvature coefficient with that of the square of the ratio between the parallel and transverse magnetic field intensities. Finally, we discuss the challenges in extending our geometric approach to higher-dimensional quantum systems that evolve unitarily under an arbitrary time-dependent Hermitian Hamiltonian.

Related papers

• From Uncertainty Relations to Quantum Acceleration Limits [0.0]
  We provide a comparative analysis of two alternative derivations for quantum systems specified by an arbitrary finite-dimensional projective Hilbert space. We find the most general conditions needed to attain the maximal upper bounds on the acceleration of the quantum evolution.
  arXiv  Detail & Related papers  (2024-10-14T19:30:01Z)
• Unveiling the Quantum Toroidal Dipole in Nanosystems: Quantization, Interaction Energy, and Measurement [44.99833362998488]
  We investigate a quantum particle confined to a toroidal surface in the presence of a filiform current along the system's rotational axis. Our analysis reveals that the interaction between the particle and the current induces a non-zero toroidal dipole in the particle's stationary states.
  arXiv  Detail & Related papers  (2024-01-26T13:31:32Z)
• Upper limit on the acceleration of a quantum evolution in projective Hilbert space [0.0]
  We derive an upper bound for the rate of change of the speed of transportation in an arbitrary finite-dimensional projective Hilbert space. We show that the acceleration squared of a quantum evolution in projective space is upper bounded by the variance of the temporal rate of change of the Hamiltonian operator.
  arXiv  Detail & Related papers  (2023-11-30T11:22:59Z)
• From the classical Frenet-Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part II. Nonstationary Hamiltonians [0.0]
  We show how to quantify the bending and twisting of quantum curves traced by state vectors evolving under nonstationary Hamiltonians. We find that the time-varying setting exhibits a richer structure from a statistical standpoint.
  arXiv  Detail & Related papers  (2023-11-30T11:17:37Z)
• From the classical Frenet-Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part I. Stationary Hamiltonians [0.0]
  We propose a quantum version of the Frenet-Serret apparatus for a quantum trajectory in projective Hilbert space. Our proposed constant curvature coefficient is given by the magnitude squared of the covariant derivative of the tangent vector to the state vector. Our proposed constant torsion coefficient is defined in terms of the magnitude squared of the projection of the covariant derivative of the tangent vector.
  arXiv  Detail & Related papers  (2023-11-30T11:09:49Z)
• Entanglement entropy in conformal quantum mechanics [68.8204255655161]
  We consider sets of states in conformal quantum mechanics associated to generators of time evolution whose orbits cover different regions of the time domain. States labelled by a continuous global time variable define the two-point correlation functions of the theory seen as a one-dimensional conformal field theory.
  arXiv  Detail & Related papers  (2023-06-21T14:21:23Z)
• Measurement phase transitions in the no-click limit as quantum phase transitions of a non-hermitean vacuum [77.34726150561087]
  We study phase transitions occurring in the stationary state of the dynamics of integrable many-body non-Hermitian Hamiltonians. We observe that the entanglement phase transitions occurring in the stationary state have the same nature as that occurring in the vacuum of the non-hermitian Hamiltonian.
  arXiv  Detail & Related papers  (2023-01-18T09:26:02Z)
• Geometric phases along quantum trajectories [58.720142291102135]
  We study the distribution function of geometric phases in monitored quantum systems. For the single trajectory exhibiting no quantum jumps, a topological transition in the phase acquired after a cycle. For the same parameters, the density matrix does not show any interference.
  arXiv  Detail & Related papers  (2023-01-10T22:05:18Z)
• Time and Evolution in Quantum and Classical Cosmology [68.8204255655161]
  We show that it is neither necessary nor sufficient for the Poisson bracket between the time variable and the super-Hamiltonian to be equal to unity in all of the phase space. We also discuss the question of switching between different internal times as well as the Montevideo interpretation of quantum theory.
  arXiv  Detail & Related papers  (2021-07-02T09:17:55Z)
• $\mathcal{PT}$-symmetry in compact phase space for a linear Hamiltonian [0.0]
  We study the time evolution of a PT-symmetric, non-Hermitian quantum system for which the associated phase space is compact. We analyze how the non-Hermitian part of the Hamiltonian affects the time evolution of two archetypical quantum states, coherent and Dicke states.
  arXiv  Detail & Related papers  (2020-07-30T20:38:00Z)
• Unraveling the topology of dissipative quantum systems [58.720142291102135]
  We discuss topology in dissipative quantum systems from the perspective of quantum trajectories. We show for a broad family of translation-invariant collapse models that the set of dark state-inducing Hamiltonians imposes a nontrivial topological structure on the space of Hamiltonians.
  arXiv  Detail & Related papers  (2020-07-12T11:26:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.