431 research outputs found
Shannon-Wehrl entropy for cosmological and black hole squeezing
We discuss the Shannon-Wehrl entropy within the squeezing vocabulary for the
cosmological and black hole particle production.Comment: 4 pages, 2 Figures available from the author(s), LaTex, IFUG-11/94 r,
contribution to Harmonic Oscillators 2, Cocoyoc, March 23-25/1994, To be
published in NASA Conference Series (1994/1995
Korteweg-de Vries adiabatic index solitons in barotropic open FRW cosmologies
Applying standard mathematical methods, it is explicitly shown how the
Riccati equation for the Hubble parameter H(\eta) of barotropic open FRW
cosmologies is connected with a Korteweg-de Vries equation for adiabatic index
solitons. It is also shown how one can embed a discrete sequence of adiabatic
indices of the type n^2({3/2}\gamma -1)^2 (\gamma \neq 2/3) in the sech FRW
adiabatic index solitonComment: 5 pages, without figure
Quantum phase uncertainty in mutually unbiased measurements and Gauss sums
Mutually unbiased bases (MUBs), which are such that the inner product between
two vectors in different orthogonal bases is constant equal to the inverse
, with the dimension of the finite Hilbert space, are becoming
more and more studied for applications such as quantum tomography and
cryptography, and in relation to entangled states and to the Heisenberg-Weil
group of quantum optics. Complete sets of MUBs of cardinality have been
derived for prime power dimensions using the tools of abstract algebra
(Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions
the cardinality is much less. The bases can be reinterpreted as quantum phase
states, i.e. as eigenvectors of Hermitean phase operators generalizing those
introduced by Pegg & Barnett in 1989. The MUB states are related to additive
characters of Galois fields (in odd characteristic p) and of Galois rings (in
characteristic 2). Quantum Fourier transforms of the components in vectors of
the bases define a more general class of MUBs with multiplicative characters
and additive ones altogether. We investigate the complementary properties of
the above phase operator with respect to the number operator. We also study the
phase probability distribution and variance for physical states and find them
related to the Gauss sums, which are sums over all elements of the field (or of
the ring) of the product of multiplicative and additive characters. Finally we
relate the concepts of mutual unbiasedness and maximal entanglement. This
allows to use well studied algebraic concepts as efficient tools in our quest
of minimal uncertainty in quantum information primitives.Comment: 11 page
The hyperbolic, the arithmetic and the quantum phase
We develop a new approach of the quantum phase in an Hilbert space of finite
dimension which is based on the relation between the physical concept of phase
locking and mathematical concepts such as cyclotomy and the Ramanujan sums. As
a result, phase variability looks quite similar to its classical counterpart,
having peaks at dimensions equal to a power of a prime number. Squeezing of the
phase noise is allowed for specific quantum states. The concept of phase
entanglement for Kloosterman pairs of phase-locked states is introduced.Comment: accepted for publication for the special issue of J. Opt. B, in
relation to ICSSUR, Puebla (Mexico): Foundations of Quantum Optics, to be
published in June 200
Integrable Abel equations and Vein's Abel equation
We first reformulate and expand with several novel findings some of the basic
results in the integrability of Abel equations. Next, these results are applied
to Vein's Abel equation whose solutions are expressed in terms of the third
order hyperbolic functions and a phase space analysis of the corresponding
nonlinear oscillator is also providedComment: 12 pages, 4 figures, 17 references, online at Math. Meth. Appl. Sci.
since 7/28/2015, published 4/201
Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations
We emphasize two connections, one well known and another less known, between
the dissipative nonlinear second order differential equations and the Abel
equations which in its first kind form have only cubic and quadratic terms.
Then, employing an old integrability criterion due to Chiellini, we introduce
the corresponding integrable dissipative equations. For illustration, we
present the cases of some integrable dissipative Fisher, nonlinear pendulum,
and Burgers-Huxley type equations which are obtained in this way and can be of
interest in applications. We also show how to obtain Abel solutions directly
from the factorization of second-order nonlinear equationsComment: 6 pages, 7 figures, published versio
Ermakov-Lewis Invariants and Reid Systems
Reid's m'th-order generalized Ermakov systems of nonlinear coupling constant
alpha are equivalent to an integrable Emden-Fowler equation. The standard
Ermakov-Lewis invariant is discussed from this perspective, and a closed
formula for the invariant is obtained for the higher-order Reid systems (m\geq
3). We also discuss the parametric solutions of these systems of equations
through the integration of the Emden-Fowler equation and present an example of
a dynamical system for which the invariant is equivalent to the total energyComment: 8 pages, published versio
Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping
We introduce a special type of dissipative Ermakov-Pinney equations of the
form v_{\zeta \zeta}+g(v)v_{\zeta}+h(v)=0, where h(v)=h_0(v)+cv^{-3} and the
nonlinear dissipation g(v) is based on the corresponding Chiellini integrable
Abel equation. When h_0(v) is a linear function, h_0(v)=\lambda^2v, general
solutions are obtained following the Abel equation route. Based on particular
solutions, we also provide general solutions containing a factor with the phase
of the Milne type. In addition, the same kinds of general solutions are
constructed for the cases of higher-order Reid nonlinearities. The Chiellini
dissipative function is actually a dissipation-gain function because it can be
negative on some intervals. We also examine the nonlinear case
h_0(v)=\Omega_0^2(v-v^2) and show that it leads to an integrable hyperelliptic
caseComment: 15 pages, 5 figures, 1 appendix, 21 references, published versio
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