310 research outputs found
The Mutually Unbiased Bases Revisited
The study of Mutually Unbiased Bases continues to be developed vigorously,
and presents several challenges in the Quantum Information Theory. Two
orthonormal bases in are said mutually unbiased if
the scalar product has modulus
. In particular this property has been introduced in order to allow
an optimization of the measurement-driven quantum evolution process of any
state when measured in the mutually unbiased bases
. At present it is an open problem to find the maximal
umber of mutually Unbiased Bases when is not a power of a prime number.
\noindent In this article, we revisit the problem of finding Mutually Unbiased
Bases (MUB's) in any dimension . The method is very elementary, using the
simple unitary matrices introduced by Schwinger in 1960, together with their
diagonalizations. The Vandermonde matrix based on the -th roots of unity
plays a major role. This allows us to show the existence of a set of 3 MUB's in
any dimension, to give conditions for existence of more than 3 MUB's for
even or odd number, and to recover the known result of existence of MUB's
for a prime number. Furthermore the construction of these MUB's is very
explicit. As a by-product, we recover results about Gauss Sums, known in number
theory, but which have apparently not been previously derived from MUB
properties.Comment: International Conference on Transport and Spectral Problems in
Quantum Mechanics held in Honor of Jean-Michel Combes, Cergy Pontoise :
France (2006
A proof of the Gutzwiller Semiclassical Trace Formula using Coherent States Decomposition
The Gutzwiller semiclassical trace formula links the eigenvalues of the
Scrodinger operator ^H with the closed orbits of the corresponding classical
mechanical system, associated with the Hamiltonian H, when the Planck constant
is small ("semiclassical regime"). Gutzwiller gave a heuristic proof, using the
Feynman integral representation for the propagator of ^H. Later on
mathematicians gave rigorous proofs of this trace formula, under different
settings, using the theory of Fourier Integral Operators and Lagrangian
manifolds. Here we want to show how the use of coherent states (or gaussian
beams) allows us to give a simple and direct proof.Comment: 17 pages, LaTeX, available on http://qcd.th.u-psud.f
Phase-space semiclassical analysis - Around semiclassical trace formulae
ThéorieInternational audienc
The quantum fidelity for the time-dependent singular quantum oscillator
In this paper we perform an exact study of ``Quantum Fidelity'' (also called
Loschmidt Echo) for the time-periodic quantum Harmonic Oscillator of
Hamiltonian : when compared with the quantum evolution
induced by (), in the case where is a -periodic
function and a real constant. The reference (initial) state is taken to be
an arbitrary ``generalized coherent state'' in the sense of Perelomov. We show
that, starting with a quadratic decrease in time in the neighborhood of ,
this quantum fidelity may recur to its initial value 1 at an infinite sequence
of times {}. We discuss the result when the classical motion induced by
Hamiltonian is assumed to be stable versus unstable. A
beautiful relationship between the quantum and the classical fidelity is also
demonstrated
Circulant matrices, gauss sums and mutually unbiased I. The prime number case
In this paper, we consider the problem of Mutually Unbiased Bases in prime
dimension . It is known to provide exactly mutually unbiased bases. We
revisit this problem using a class of circulant matrices. The
constructive proof of a set of mutually unbiased bases follows, together
with a set of properties of Gauss sums, and of bi-unimodular sequences
A Trace Formula for Products of Diagonal Matrix Elements in Chaotic Systems
We derive a trace formula for , where
is the diagonal matrix element of the operator in the energy basis
of a chaotic system. The result takes the form of a smooth term plus
periodic-orbit corrections; each orbit is weighted by the usual Gutzwiller
factor times , where is the average of the classical
observable along the periodic orbit . This structure for the orbit
corrections was previously proposed by Main and Wunner (chao-dyn/9904040) on
the basis of numerical evidence.Comment: 8 pages; analysis made more rigorous in the revised versio
A constant of quantum motion in two dimensions in crossed magnetic and electric fields
We consider the quantum dynamics of a single particle in the plane under the
influence of a constant perpendicular magnetic and a crossed electric potential
field. For a class of smooth and small potentials we construct a non-trivial
invariant of motion. Do to so we proof that the Hamiltonian is unitarily
equivalent to an effective Hamiltonian which commutes with the observable of
kinetic energy.Comment: 18 pages, 2 figures; the title was changed and several typos
corrected; to appear in J. Phys. A: Math. Theor. 43 (2010
How do wave packets spread? Time evolution on Ehrenfest time scales
We derive an extension of the standard time dependent WKB theory which can be
applied to propagate coherent states and other strongly localised states for
long times. It allows in particular to give a uniform description of the
transformation from a localised coherent state to a delocalised Lagrangian
state which takes place at the Ehrenfest time. The main new ingredient is a
metaplectic operator which is used to modify the initial state in a way that
standard time dependent WKB can then be applied for the propagation.
We give a detailed analysis of the phase space geometry underlying this
construction and use this to determine the range of validity of the new method.
Several examples are used to illustrate and test the scheme and two
applications are discussed: (i) For scattering of a wave packet on a barrier
near the critical energy we can derive uniform approximations for the
transition from reflection to transmission. (ii) A wave packet propagated along
a hyperbolic trajectory becomes a Lagrangian state associated with the unstable
manifold at the Ehrenfest time, this is illustrated with the kicked harmonic
oscillator.Comment: 30 pages, 3 figure
Semiclassical wave packet dynamics for Hartree equations
We study the propagation of wave packets for nonlinear nonlocal Schrodinger
equations in the semi-classical limit. When the kernel is smooth, we construct
approximate solutions for the wave functions in subcritical, critical and
supercritical cases (in terms of the size of the initial data). The validity of
the approximation is proved up to Ehrenfest time. For homogeneous kernels, we
establish similar results in subcritical and critical cases. Nonlinear
superposition principle for two nonlinear wave packets is also considered.Comment: 28 pages. Some errors fixed in Section 2.
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