Nested-sphere description of the N-level Chern number and the generalized Bloch hypersphere

• URL: http://arxiv.org/abs/2110.06934v2
• Date: Sun, 24 Apr 2022 18:15:22 GMT
• Title: Nested-sphere description of the N-level Chern number and the generalized Bloch hypersphere
• Authors: Cameron J.D. Kemp, Nigel R. Cooper, F. Nur \"Unal
• Abstract summary: The geometric interpretation of (pseudo)spin 1/2 systems on the Bloch sphere has been appreciated across different areas ranging from condensed matter to quantum information and high energy physics. We here employ a coherence vector description to theoretically characterize a general N-level system on the higher dimensional generalized Bloch (hyper)sphere. We demonstrate that for the N-level case, there is an exterior two-sphere that provides a useful characterization of the system, notably by playing a primary role in determining the Chern number.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The geometric interpretation of (pseudo)spin 1/2 systems on the Bloch sphere has been appreciated across different areas ranging from condensed matter to quantum information and high energy physics. Although similar notions for larger Hilbert spaces are established in mathematics, they have been so far less explored beyond the two-level case for practical usage in condensed matter settings, or have involved restrictions to sub manifolds within the full Hilbert space. We here employ a coherence vector description to theoretically characterize a general N-level system on the higher dimensional generalized Bloch (hyper)sphere by respecting the structure of the underlying SU(N) algebra and construct physically intuitive geometric pictures for topological concepts. Focusing on two spatial dimensions, we reveal a geometric interpretation for the Chern number in larger Hilbert spaces in terms of a nested structure comprising N-1 two-spheres. We demonstrate that for the N-level case, there is an exterior two-sphere that provides a useful characterization of the system, notably by playing a primary role in determining the Chern number. The external sphere can be directly measured in ultracold atoms via well-established band mapping techniques, thereby imparting knowledge of the topological nature of state. We also investigate how the time evolution of the coherence vector defined on the generalized Bloch hypersphere can be utilized to extract the full state vector in experiments, allowing us to develop a tomography scheme involving quenches for three-level systems. Our geometric description opens up a new avenue for the interpretation of the topological classification and the dynamical illustration of multi-level systems, which in turn is anticipated to help in the design of new experimental probes.

Related papers

• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Perfectly Spherical Bloch Hyper-spheres from Quantum Matrix Geometry [0.0]
  Exploiting analogies between the precessing quantum spin system and the charge-monopole system, we construct Bloch hyper-spheres with $itexact$ spherical symmetries in arbitrary dimensions. The geometry of Bloch hyper-spheres is exactly equal to the quantum Nambu geometry of higher dimensional fuzzy spheres.
  arXiv  Detail & Related papers  (2024-02-11T10:33:33Z)
• Gaussian Entanglement Measure: Applications to Multipartite Entanglement of Graph States and Bosonic Field Theory [50.24983453990065]
  An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers. We present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states. By providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory.
  arXiv  Detail & Related papers  (2024-01-31T15:50:50Z)
• Gravitational quantum switch on a superposition of spherical shells [0.0]
  In a gravitational quantum switch, the order of operations applied by two agents on a target system is entangled with the state of the geometry. We consider a model describing the superposition of geometries produced by distinct arrangements of spherical mass shells. We show that a protocol for the implementation of a gravitational quantum switch can be formulated in such a system.
  arXiv  Detail & Related papers  (2023-06-19T14:52:09Z)
• Complementarity between quantum entanglement, geometrical and dynamical appearances in N spin-$1/2$ system under all-range Ising model [0.0]
  Modern geometry studies the interrelations between elements such as distance and curvature. We explore these structures in a physical system of $N$ interaction spin-$1/2$ under all-range Ising model.
  arXiv  Detail & Related papers  (2023-04-11T15:26:19Z)
• Continuous percolation in a Hilbert space for a large system of qubits [58.720142291102135]
  The percolation transition is defined through the appearance of the infinite cluster. We show that the exponentially increasing dimensionality of the Hilbert space makes its covering by finite-size hyperspheres inefficient. Our approach to the percolation transition in compact metric spaces may prove useful for its rigorous treatment in other contexts.
  arXiv  Detail & Related papers  (2022-10-15T13:53:21Z)
• Pointillisme \`a la Signac and Construction of a Pseudo Quantum Phase Space [0.0]
  We construct a quantum-mechanical substitute for the symplectic phase space. The total space of this fiber bundle consists of geometric quantum states. We show that the set of equivalence classes of unitarily related geometric quantum states is in a one-to-one correspondence with the set of all Gaussian wavepackets.
  arXiv  Detail & Related papers  (2022-07-31T16:43:06Z)
• Geometry Interaction Knowledge Graph Embeddings [153.69745042757066]
  We propose Geometry Interaction knowledge graph Embeddings (GIE), which learns spatial structures interactively between the Euclidean, hyperbolic and hyperspherical spaces. Our proposed GIE can capture a richer set of relational information, model key inference patterns, and enable expressive semantic matching across entities.
  arXiv  Detail & Related papers  (2022-06-24T08:33:43Z)
• Neural-Network Quantum States for Periodic Systems in Continuous Space [66.03977113919439]
  We introduce a family of neural quantum states for the simulation of strongly interacting systems in the presence of periodicity. For one-dimensional systems we find very precise estimations of the ground-state energies and the radial distribution functions of the particles. In two dimensions we obtain good estimations of the ground-state energies, comparable to results obtained from more conventional methods.
  arXiv  Detail & Related papers  (2021-12-22T15:27:30Z)
• The shape of higher-dimensional state space: Bloch-ball analog for a qutrit [0.0]
  We show that it is possible to construct a three dimensional model for the qutrit state space. Besides being of indisputable theoretical value, this opens the door to a new type of representation.
  arXiv  Detail & Related papers  (2020-12-01T15:57:13Z)
• Probing chiral edge dynamics and bulk topology of a synthetic Hall system [52.77024349608834]
  Quantum Hall systems are characterized by the quantization of the Hall conductance -- a bulk property rooted in the topological structure of the underlying quantum states. Here, we realize a quantum Hall system using ultracold dysprosium atoms, in a two-dimensional geometry formed by one spatial dimension. We demonstrate that the large number of magnetic sublevels leads to distinct bulk and edge behaviors.
  arXiv  Detail & Related papers  (2020-01-06T16:59:08Z)

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.