14,683 research outputs found
Simplifying the spectral analysis of the volume operator
The volume operator plays a central role in both the kinematics and dynamics
of canonical approaches to quantum gravity which are based on algebras of
generalized Wilson loops. We introduce a method for simplifying its spectral
analysis, for quantum states that can be realized on a cubic three-dimensional
lattice. This involves a decomposition of Hilbert space into sectors
transforming according to the irreducible representations of a subgroup of the
cubic group. As an application, we determine the complete spectrum for a class
of states with six-valent intersections.Comment: 19 pages, TeX, to be published in Nucl. Phys.
Coherent spin manipulation in an exchange-only qubit
Initialization, manipulation, and measurement of a three-spin qubit are
demonstrated using a few-electron triple quantum dot, where all operations can
be driven by tuning the nearest-neighbor exchange interaction. Multiplexed
reflectometry, applied to two nearby charge sensors, allows for qubit readout.
Decoherence is found to be consistent with predictions based on gate voltage
noise with a uniform power spectrum. The theory of the exchange-only qubit is
developed and it is shown that initialization of only two spins suffices for
operation. Requirements for full multi-qubit control using only exchange and
electrostatic interactions are outlined.Comment: related work at http://marcuslab.harvard.ed
Uniform Semiclassical Approximation for the Wigner Symbol in Terms of Rotation Matrices
A new uniform asymptotic approximation for the Wigner symbol is given in
terms of Wigner rotation matrices (-matrices). The approximation is uniform
in the sense that it applies for all values of the quantum numbers, even those
near caustics. The derivation of the new approximation is not given, but the
geometrical ideas supporting it are discussed and numerical tests are
presented, including comparisons with the exact -symbol and with the
Ponzano-Regge approximation.Comment: 44 pages plus 20 figure
The Quantum Group Structure of 2D Gravity and Minimal Models II: The Genus-Zero Chiral Bootstrap
The F and B matrices associated with Virasoro null vectors are derived in
closed form by making use of the operator-approach suggested by the Liouville
theory, where the quantum-group symmetry is explicit. It is found that the
entries of the fusing and braiding matrices are not simply equal to
quantum-group symbols, but involve additional coupling constants whose
derivation is one aim of the present work. Our explicit formulae are new, to
our knowledge, in spite of the numerous studies of this problem. The
relationship between the quantum-group-invariant (of IRF type) and
quantum-group-covariant (of vertex type) chiral operator-algebras is fully
clarified, and connected with the transition to the shadow world for
quantum-group symbols. The corresponding 3-j-symbol dressing is shown to reduce
to the simpler transformation of Babelon and one of the author (J.-L. G.) in a
suitable infinite limit defined by analytic continuation. The above two types
of operators are found to coincide when applied to states with Liouville
momenta going to in a suitable way. The introduction of
quantum-group-covariant operators in the three dimensional picture gives a
generalisation of the quantum-group version of discrete three-dimensional
gravity that includes tetrahedra associated with 3-j symbols and universal
R-matrix elements. Altogether the present work gives the concrete realization
of Moore and Seiberg's scheme that describes the chiral operator-algebra of
two-dimensional gravity and minimal models.Comment: 56 pages, 22 figures. Technical problem only, due to the use of an
old version of uuencode that produces blank characters some times suppressed
by the mailer. Same content
Racah Sum Rule and Biedenharn-Elliott Identity for the Super-Rotation symbols
It is shown that the well known Racah sum rule and Biedenharn-Elliott
identity satisfied by the recoupling coefficients or by the symbols of
the usual rotation algebra can be extended to the corresponding
features of the super-rotation superalgebra. The structure of the
sum rules is completely similar in both cases, the only difference concerns the
signs which are more involved in the super-rotation case.Comment: 9 pages. Two misprints correcte
Uniform Approximation from Symbol Calculus on a Spherical Phase Space
We use symbol correspondence and quantum normal form theory to develop a more
general method for finding uniform asymptotic approximations. We then apply
this method to derive a result we announced in an earlier paper, namely, the
uniform approximation of the -symbol in terms of the rotation matrices. The
derivation is based on the Stratonovich-Weyl symbol correspondence between
matrix operators and functions on a spherical phase space. The resulting
approximation depends on a canonical, or area preserving, map between two pairs
of intersecting level sets on the spherical phase space.Comment: 18 pages, 5 figure
Trivalent graphs, volume conjectures and character varieties
The generalized volume conjecture and the AJ conjecture (a.k.a. the quantum
volume conjecture) are extended to U_q(\fraksl_2) colored quantum invariants
of the theta and tetrahedron graph. The \SL(2,\bC) character variety of the
fundamental group of the complement of a trivalent graph with edges in
is a Lagrangian subvariety of the Hitchin moduli space over the Riemann
surface of genus . For the theta and tetrahedron graph, we conjecture
that the configuration of the character variety is locally determined by large
color asymptotics of the quantum invariants of the trivalent graph in terms of
complex Fenchel-Nielsen coordinates. Moreover, the -holonomic difference
equation of the quantum invariants provides the quantization of the character
variety.Comment: 11 pages, 2 figure
Simple operation sequences to couple and interchange quantum information between spin qubits of different kinds
Efficient operation sequences to couple and interchange quantum information
between quantum dot spin qubits of different kinds are derived using exchange
interactions. In the qubit encoding of a single-spin qubit, a singlet-triplet
qubit, and an exchange-only (triple-dot) qubit, some of the single-qubit
interactions remain on during the entangling operation; this greatly simplifies
the operation sequences that construct entangling operations. In the ideal
setup, the gate operations use the intra-qubit exchange interactions only once.
The limitations of the entangling sequences are discussed, and it is shown how
quantum information can be converted between different kinds of quantum dot
spin qubits.Comment: 9 pages, 4 figure
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