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Squeezed states: A geometric framework

Ali, S. T., James A. Brooke, Jean‐Pierre Gazeau

Published Jan 1, 1992• 0 citations
OpenAlex

Abstract

A general definition of squeezed states is proposed and its main features are illustrated through a discussion of the standard optical coherent states represented by 'Gaussian pure states'. The set-up involves representations of groups on Hilbert spaces over homogeneous spaces of the group, and relies on the construction of a square integrable (coherent state) group representation modulo a subgroup. This construction depends upon a choice of a Borel section which has a certain permissible arbitrariness in its selection; this freedom is attributable to a squeezing of the defining coherent states of the representation, and corresponds in this way to a sort of gauging.

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A general definition of squeezed states is proposed and its main features are illustrated through a discussion of the standard optical coherent states represented by 'Gaussian pure states'. The set-up involves representations of groups on Hilbert spaces over homogeneous spaces of the group, and relies on the construction of a square integrable (coherent state) group representation modulo a subgroup. This construction depends upon a choice of a Borel section which has a certain permissible arbitrariness in its selection; this freedom is attributable to a squeezing of the defining coherent states of the representation, and corresponds in this way to a sort of gauging.
We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.
Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary — or, in general, projective unitary — representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.
We derive a generalization of the well-known theorem for the square integrability of a unitary irreducible representation of a locally compact group. The generalization covers the case of representations admitting vector coherent states. The result is illustrated by an example drawn from the isochronous Galilei group. The construction yields a wide variety of coherent states, labeled by phase space points, which satisfy a resolution of the identity condition, and incorporate spin degrees of freedom.
Abstract The paper presents a new method of geometric solution of a Schrödinger equation by constructing an equivalent first-order partial differential equation with a bigger number of variables. The equivalent equation shall be restricted to a specific subspace with auxiliary conditions which are obtained from a coherent state transform. The method is applied to the fundamental case of the harmonic oscillator and coherent state transform generated by the minimal nilpotent step three Lie group—the group (also known under many names, e.g. quartic group). We obtain a geometric solution for an arbitrary minimal uncertainty state used as a fiducial vector. In contrast, it is shown that the well-known Fock–Segal–Bargmann transform and the Heisenberg group require a specific fiducial vector to produce a geometric solution. A technical aspect considered in this paper is that a certain modification of a coherent state transform is required: although the irreducible representation of the group is square-integrable modulo a subgroup , the obtained dynamic is transverse to the homogeneous space .