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Sabre Kais - One of the best experts on this subject based on the ideXlab platform.

  • Singularity of the time-energy uncertainty in adiabatic perturbation and cycloids on a Bloch Sphere
    Scientific reports, 2016
    Co-Authors: Franco Nori, Sabre Kais
    Abstract:

    Adiabatic perturbation is shown to be singular from the exact solution of a spin-1/2 particle in a uniformly rotating magnetic field. Due to a non-adiabatic effect, its quantum trajectory on a Bloch Sphere is a cycloid traced by a circle rolling along an adiabatic path. As the magnetic field rotates more and more slowly, the time-energy uncertainty, proportional to the length of the quantum trajectory, calculated by the exact solution is entirely different from the one obtained by the adiabatic path traced by the instantaneous eigenstate. However, the non-adiabatic Aharonov- Anandan geometric phase, measured by the area enclosed by the exact path, approaches smoothly the adiabatic Berry phase, proportional to the area enclosed by the adiabatic path. The singular limit of the time-energy uncertainty and the regular limit of the geometric phase are associated with the arc length and arc area of the cycloid on a Bloch Sphere, respectively. Prolate and curtate cycloids are also traced by different initial states outside and inside of the rolling circle, respectively. The axis trajectory of the rolling circle, parallel to the adiabatic path, is shown to be an example of transitionless driving. The non-adiabatic resonance is visualized by the number of cycloid arcs.

  • singularity of the time energy uncertainty in adiabatic perturbation and cycloids on a Bloch Sphere
    arXiv: Quantum Physics, 2015
    Co-Authors: Franco Nori, Sabre Kais
    Abstract:

    The adiabatic perturbation is shown to be singular from the exact solution of a spin-1/2 particle in a uniformly rotating magnetic field. Due to a non-adiabatic effect, its quantum trajectory on a Bloch Sphere is a cycloid traced by a circle rolling along an adiabatic path. As the magnetic field rotates more and more slowly, the time-energy uncertainty, proportional to the distance of the quantum trajectory, calculated by the exact solution is entirely different from the one obtained by the adiabatic path traced by the instantaneous state. However, the non-adiabatic Aharonov-Anandan geometric phase, measured by the area enclosed by the exact path, approaches smoothly the adiabatic Berry phase, proportional to the area enclosed by the adiabatic path. The singular limit of the time-energy uncertainty and the regular limit of the geometric phase are associated with the arc length and arc area of the cycloid on a Bloch Sphere, respectively. Prolate and curtate cycloids are also traced by different initial states outside and inside of the rolling circle, respectively. The axis trajectory of the rolling circle, parallel to the adiabatic path, is shown to be an example of transitionless driving. The non-adiabatic resonance is visualized by the number of complete cycloid arcs.

Tim Palmer - One of the best experts on this subject based on the ideXlab platform.

  • Discretization of the Bloch Sphere, fractal invariant sets and Bell's theorem.
    Proceedings. Mathematical physical and engineering sciences, 2020
    Co-Authors: Tim Palmer
    Abstract:

    An arbitrarily dense discretization of the Bloch Sphere of complex Hilbert states is constructed, where points correspond to bit strings of fixed finite length. Number-theoretic properties of trigonometric functions (not part of the quantum-theoretic canon) are used to show that this constructive discretized representation incorporates many of the defining characteristics of quantum systems: completementarity, uncertainty relationships and (with a simple Cartesian product of discretized Spheres) entanglement. Unlike Meyer's earlier discretization of the Bloch Sphere, there are no orthonormal triples, hence the Kocken-Specker theorem is not nullified. A physical interpretation of points on the discretized Bloch Sphere is given in terms of ensembles of trajectories on a dynamically invariant fractal set in state space, where states of physical reality correspond to points on the invariant set. This deterministic construction provides a new way to understand the violation of the Bell inequality without violating statistical independence or factorization, where these conditions are defined solely from states on the invariant set. In this finite representation, there is an upper limit to the number of qubits that can be entangled, a property with potential experimental consequences.

A. R. P. Rau - One of the best experts on this subject based on the ideXlab platform.

  • Bloch Sphere like construction of su 3 hamiltonians using unitary integration
    Journal of Physics A, 2009
    Co-Authors: Sai Vinjanampathy, A. R. P. Rau
    Abstract:

    The Bloch Sphere is a familiar and useful geometrical picture of the time evolution of a single spin or a quantal two-level system. 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 that describe the time evolution operator. The first, called the base manifold, is the counterpart of the S2 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.

  • Bloch Sphere like construction of SU(3) Hamiltonians using unitary integration
    Journal of Physics A: Mathematical and Theoretical, 2009
    Co-Authors: Sai Vinjanampathy, A. R. P. Rau
    Abstract:

    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 and Bloch-Sphere constructions for SU ( N ) groups with a complete description of the SU(4) group
    Physical Review A, 2008
    Co-Authors: Dmitry B. Uskov, A. R. P. Rau
    Abstract:

    A two-Sphere (``Bloch'' or ``Poincare'') is familiar for describing the dynamics of a spin-$1∕2$ 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 $\mathrm{SU}(N)$ 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.

Nicholas P. Bigelow - One of the best experts on this subject based on the ideXlab platform.

  • Raman fingerprints on the Bloch Sphere of a spinor Bose-Einstein condensate
    arXiv, 2016
    Co-Authors: Justin T. Schultz, Maitreyi Jayaseelan, J. D. Murphree, Azure Hansen, Nicholas P. Bigelow
    Abstract:

    We explore the geometric interpretation of a diabatic, two-photon Raman process as a rotation on the Bloch Sphere for a pseudo-spin-1/2 system. The spin state of a spin-1/2 quantum system can be described by a point on the surface of the Bloch Sphere, and its evolution during a Raman pulse is a trajectory on the Sphere determined by properties of the optical beams: the pulse area, the relative intensities and phases, and the relative frequencies. We experimentally demonstrate key features of this model with a $^{87}$Rb spinor Bose-Einstein condensate, which allows us to examine spatially dependent signatures of the Raman beams. The two-photon detuning allows us to precisely control the spin density and imprinted relative phase profiles, as we show with a coreless vortex. With this comprehensive understanding and intuitive geometric interpretation, we use the Raman process to create and tailor as well as study and characterize exotic topological spin textures in spinor BECs.

  • Creating ‘Optics’ for Singular Atom Optics with Spinor Bose–Einstein condensates
    Frontiers in Optics 2016, 2016
    Co-Authors: Justin T. Schultz, Maitreyi Jayaseelan, Azure Hansen, Joseph D Murphree, Nicholas P. Bigelow
    Abstract:

    We present methods to create waveplates and phase plates for pseudo-spin-1/2 atomic systems via two-photon Raman interactions. The interactions are geometrically represented as rotations on the Bloch Sphere.

  • Full Bloch Bose-Einstein Condensates
    Frontiers in Optics 2012 Laser Science XXVIII, 2012
    Co-Authors: Azure Hansen, Justin T. Schultz, Nicholas P. Bigelow
    Abstract:

    We create and characterize an angular momentum texture in a spinor Bose-Einstein condensate that, in analogy to the full Poincare laser beams of singular optics, fully covers the Bloch Sphere.

Roman Orus - One of the best experts on this subject based on the ideXlab platform.

  • Topological order on the Bloch Sphere
    New Journal of Physics, 2020
    Co-Authors: Rotem Liss, Tal Mor, Roman Orus
    Abstract:

    A Bloch Sphere is the geometrical representation of an arbitrary two-dimensional Hilbert space. Possible classes of entanglement and separability for the pure and mixed states on the Bloch Sphere were suggested by [M. Boyer, R. Liss, T. Mor, PRA 95, 032308 (2017)]. Here we construct a Bloch Sphere for the Hilbert space spanned by one of the ground states of Kitaev's toric code model and one of its closest product states. We prove that this Sphere contains only one separable state, thus belonging to the fourth class suggested by the said paper. We furthermore study the topological order of the pure states on its surface and conclude that, according to conventional definitions, only one state (the toric code ground state) seems to present non-trivial topological order. We conjecture that most of the states on this Bloch Sphere are neither ``trivial'' states (namely, they cannot be generated from a product state using a trivial circuit) nor topologically ordered. In addition, we show that the whole setting can be understood in terms of Grover rotations with gauge symmetry, akin to the quantum search algorithm.