1,253 research outputs found
Squeezing: the ups and downs
We present an operator theoretic side of the story of squeezed states
regardless the order of squeezing. For low order, that is for displacement
(order 1) and squeeze (order 2) operators, we bring back to consciousness what
is know or rather what has to be known by making the exposition as exhaustive
as possible. For the order 2 (squeeze) we propose an interesting model of the
Segal-Bargmann type. For higher order the impossibility of squeezing in the
traditional sense is proved rigorously. Nevertheless what we offer is the
state-of-the-art concerning the topic.Comment: 21 pages; improved presentation; it has been published by Proceedings
of the Royal Society
Holomorphic Hermite polynomials in two variables
Generalizations of the Hermite polynomials to many variables and/or to the
complex domain have been located in mathematical and physical literature for
some decades. Polynomials traditionally called complex Hermite ones are mostly
understood as polynomials in and which in fact makes them
polynomials in two real variables with complex coefficients. The present paper
proposes to investigate for the first time holomorphic Hermite polynomials in
two variables. Their algebraic and analytic properties are developed here.
While the algebraic properties do not differ too much for those considered so
far, their analytic features are based on a kind of non-rotational
orthogonality invented by van Eijndhoven and Meyers. Inspired by their
invention we merely follow the idea of Bargmann's seminal paper (1961) giving
explicit construction of reproducing kernel Hilbert spaces based on those
polynomials. "Homotopic" behavior of our new formation culminates in comparing
it to the very classical Bargmann space of two variables on one edge and the
aforementioned Hermite polynomials in and on the other. Unlike in
the case of Bargmann's basis our Hermite polynomials are not product ones but
factorize to it when bonded together with the first case of limit properties
leading both to the Bargmann basis and suitable form of the reproducing kernel.
Also in the second limit we recover standard results obeyed by Hermite
polynomials in and
Combinatorial Physics, Normal Order and Model Feynman Graphs
The general normal ordering problem for boson strings is a combinatorial
problem. In this note we restrict ourselves to single-mode boson monomials.
This problem leads to elegant generalisations of well-known combinatorial
numbers, such as Bell and Stirling numbers. We explicitly give the generating
functions for some classes of these numbers. Finally we show that a graphical
representation of these combinatorial numbers leads to sets of model field
theories, for which the graphs may be interpreted as Feynman diagrams
corresponding to the bosons of the theory. The generating functions are the
generators of the classes of Feynman diagrams.Comment: 9 pages, 4 figures. 12 references. Presented at the Symposium
'Symmetries in Science XIII', Bregenz, Austria, 200
Squeezed States and Hermite polynomials in a Complex Variable
Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec
[J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of
coherent states, related to the Hermite polynomials in a complex variable which
are orthogonal with respect to a non-rotationally invariant measure. We
investigate relations between these coherent states and obtain the relationship
between them and the squeezed states of quantum optics. We also obtain a second
realization of the canonical coherent states in the Bargmann space of analytic
functions, in terms of a squeezed basis. All this is done in the flavor of the
classical approach of V. Bargmann [Commun. Pur. Appl. Math. 14, 187 (1961)].Comment: 15 page
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