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arXiv:2110.06934 · Physical Review Research

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

Cameron J. D. Kemp, Nigel R. Cooper, F. Nur Ünal

Published May 13, 2022• 19 citations
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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 $\mathrm{SU}(N)$ 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 $N\ensuremath{-}1$ 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 another avenue for the interpretation of the topological classification and the dynamical illustration of multilevel systems, which in turn is anticipated to help in the design of new experimental probes.

DOI
10.1103/physrevresearch.4.023120

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The geometric interpretation of (pseudo)spin-1/21/21/2 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 NNN-level system on the higher dimensional generalized Bloch (hyper)sphere by respecting the structure of the underlying SU(N)\mathrm{SU}(N)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\ensuremath−1N\ensuremath{-}1N\ensuremath−1 two-spheres. We demonstrate that for the NNN-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 another avenue for the interpretation of the topological classification and the dynamical illustration of multilevel systems, which in turn is anticipated to help in the design of new experimental probes.
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. We here theoretically characterize a general N-level system by its coherence vector described on the higher dimensional generalized Bloch (hyper)sphere, where topological properties such as the Berry curvature and the quantum metric take simple forms set by the SU(N) algebra. Focusing on two spatial dimensions, we present 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.
A two-sphere (``Bloch'' or ``Poincare'') is familiar for describing the dynamics of a spin-1∕21∕21∕2 particle or light polarization. Analogous objects are derived for unitary groups larger than SU(2) through an iterative procedure that constructs evolution operators for higher-dimensional SU(N)\mathrm{SU}(N)SU(N) in terms of lower-dimensional ones. We focus, in particular, on the SU(4) of two qubits which describes all possible logic gates in quantum computation and entangled states in quantum-information sciences. For a general Hamiltonian of SU(4) with 15 parameters, and for Hamiltonians of its various subgroups so that fewer parameters suffice, we derive Bloch-like rotation of unit vectors analogous to the one familiar for a single spin in a magnetic field. The unitary evolution of a quantal spin pair is thereby expressed as rotations of real, many-dimensional vectors. Correspondingly, the manifolds involved are Bloch two-spheres along with higher dimensional manifolds such as a four-sphere for the SO(5) subgroup and an eight-dimensional Grassmannian manifold for the general SU(4). The latter may also be viewed as two, mutually orthogonal, real six-dimensional unit vectors moving on a five-sphere with an additional phase constraint. This geometrical picture for two spins provides the extension and generalization of the Bloch sphere that has proved invaluable for the understanding of the dynamics of a single spin.
The Bloch sphere is a familiar and useful geometrical picture of the dynamics of a single spin or two-level system's quantum evolution. The analogous geometrical picture for three-level systems is presented, with several applications. The relevant SU(3) group and su(3) algebra are eight-dimensional objects and are realized in our picture as two four-dimensional manifolds describing the time evolution operator. The first, called the base manifold, is the counterpart of the S^2 Bloch sphere, whereas the second, called the fiber, generalizes the single U(1) phase of a single spin. Now four-dimensional, it breaks down further into smaller objects depending on alternative representations that we discuss. Geometrical phases are also developed and presented for specific applications. Arbitrary time-dependent couplings between three levels or between two spins (qubits) with SU(3) Hamiltonians can be conveniently handled through these geometrical objects.
Geometric phases are important in quantum physics and are now central to fault-tolerant quantum computation. For spin 1∕21∕21∕2, the Bloch sphere S2{S}^{2}S2, together with a U(1) phase, provides a complete SU(2) description. We generalize to NNN-level systems and SU(N)\mathrm{SU}(N)SU(N) in terms of a $2(N\ensuremath{-}1)$-dimensional base space and reduction to a $(N\ensuremath{-}1)$-level problem, paralleling closely the two-dimensional case. This iteratively solves the time evolution of an NNN-level system and gives $(N\ensuremath{-}1)$ geometric phases explicitly. A complete analytical construction of an S4{S}^{4}S4 Bloch-like sphere for two qubits is given for the Spin(5) or SO(5) subgroup of SU(4).
Abstract Observations in quantum weak measurements are determined by complex numbers called weak values. We present a geometrical interpretation of the argument of weak values of general Hermitian observables in N-dimensional quantum systems in terms of geometric phases. We formulate an arbitrary weak value in terms of three real vectors on the unit sphere in N 2 − 1 dimensions, S N 2 − 2 . These vectors are linked to the initial and final states, and to the weakly measured observable, respectively. We express pure states in the complex projective space of N − 1 dimensions, C P N − 1 , which has a non-trivial representation as a 2N − 2 dimensional submanifold of S N 2 − 2 (a generalization of the Bloch sphere for qudits). The argument of the weak value of a projector on a pure state of an N-level quantum system describes a geometric phase associated to the symplectic area of the geodesic triangle spanned by the vectors representing the pre-selected state, the projector and the post-selected state in C P N − 1 . We then proceed to show that the argument of the weak value of a general observable is equivalent to the argument of an effective Bargmann invariant. Hence, we extend the geometrical interpretation of projector weak values to weak values of general observables. In particular, we consider the generators of SU(N) given by the generalized Gell–Mann matrices. Finally, we study in detail the case of the argument of weak values of general observables in two-level systems and we illustrate weak measurements in larger dimensional systems by considering projectors on degenerate subspaces, as well as Hermitian quantum gates. To conclude, we discuss the interpretation and usefulness of geometric phases in weak values in connection to weak measurements.
We show evidence which suggests that the quench dynamics of general N -level Chern insulators can be topologically characterised by a linking number that is related to the equilibrium topology of the insulator. Recently, it was demonstrated that the intuitive picture of the Chern number in terms of the wrapping of a sphere, that famously exists in two-level systems, can be generalised to N-level systems, where it corresponds to the wrapping of N − 1 spheres that are nested in one another. The exterior sphere in this nested structure plays a dominant role in determining the Chern number. Inspired by a previous work which focuses on the quench dynamics of two-level systems, we make use of the dominant role played by the exterior sphere and use its surface vector as a Hopf map. For a topologically trivial initial Hamiltonian, the homotopy classes of such maps give rise to the homotopy group π3(S2) = Z, and is used in the characterisation of the quenched topological phase. The topological index is the Hopf invariant and is equal to the linking number of the preimages of the Hopf map.