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Journal of Mathematical Physics

Vectorlike coherent states with noncompact stability group

Loucas C. Papaloucas, Jakub Rembieliński, W. Tybor

Published Oct 1, 1989• 4 citations
Publisher

Abstract

An investigation is made of coherent states that differ from the usual ones in two ways: (a) they are connected with the coset space G/H, where the stability subgroup H may be noncompact; and (b) the notion of an H-invariant ray is replaced by the more general notion of an H-invariant subspace. A general framework is given for vectorlike coherent states with the help of the nonlinear realization technique as well as with the rigged Hilbert space theory. The vectorlike coherent states are found for the Poincaré group and the hyperbolic coherent states are found for the SU(1,1) group.

DOI
10.1063/1.528517

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An investigation is made of coherent states that differ from the usual ones in two ways: (a) they are connected with the coset space G/H, where the stability subgroup H may be noncompact; and (b) the notion of an H-invariant ray is replaced by the more general notion of an H-invariant subspace. A general framework is given for vectorlike coherent states with the help of the nonlinear realization technique as well as with the rigged Hilbert space theory. The vectorlike coherent states are found for the Poincaré group and the hyperbolic coherent states are found for the SU(1,1) group.
Abstract. Starting with the canonical coherent states, we demonstrate that all the so-called nonlinear coherent states, used in the physical literature, as well as large classes of other generalized coherent states, can be obtained by changes of bases in the underlying Hilbert space. This observation leads to an interesting duality between pairs of generalized coherent states, bringing into play a Gelfand triple of (rigged) Hilbert spaces. Moreover, it is shown that in each dual pair of families of nonlinear coherent states, at least one family is related to a (generally) non-unitary projective representation of the Weyl-Heisenberg group, which can then be thought of as characterizing the dual pair. 1.
Starting with the canonical coherent states, we demonstrate that all the so-called nonlinear coherent states, used in the physical literature, as well as large classes of other generalized coherent states, can be obtained by changes of bases in the underlying Hilbert space. This observation leads to an interesting duality between pairs of generalized coherent states, bringing into play a Gelfand triple of (rigged) Hilbert spaces. Moreover, it is shown that in each dual pair of families of nonlinear coherent states, at least one family is related to a (generally) non-unitary projective representation of the Weyl–Heisenberg group, which can then be thought of as characterizing the dual pair.
Within the generalized definition of coherent states as group orbits we study the orbit spaces and the orbit manifolds in the projective spaces constructed from linear representations. Invariant functions are suggested for arbitrary groups. The group SU (2) is studied in particular and the orbit spaces of its j = 1/2 and 1 representations completely determined. The orbits of SU (2) in CP N can be either two- or three-dimensional, the former being either isomorphic to S 2 or to RP 2 and the latter being isomorphic to quotient spaces of RP 3 . We end with a look from the same perspective to the quantum mechanical space of states in particle mechanics.
The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.
We investigate dynamics of the SU (1,1) coherent states with the use of the group transformations diagonalizing the coherence preserving Hamiltonian driving the physical system. The model physical system we consider may be viewed as a particular case of the generalized time-dependent harmonic oscillator, or as a generalization of the degenerate parametric amplifier, with the pumping field having modulated amplitude and a nonresonant phase. A Hamiltonian of such a system is given as a linear combination of the SU (1,1) generators with time-dependent coefficients, and the group transformations, mentioned above, transform this Hamiltonian to an expression containing only one generator, i.e. diagonalize the Hamiltonian. Trajectories of the complex coherent state parameter in the phase space (Lobachevskii plane) can be divided into two classes: compact trajectories never approaching the unit circle (boundary of the phase space) and noncompact trajectories approaching the unit circle from inside asymptotically, after sufficiently long time. The character of the dynamics is reflected by the time behaviour of the parameters of the group transformation diagonalizing the Hamiltonian. The main observation is that in the case of noncompact dynamics absolute values of the group parameters increase indefinitely, although in some cases they may exhibit a singular behaviour with regions of rapid variations, sudden changes of their values or cusplike singularities. For compact dynamics the group transformation parameters remain bounded and exhibit an oscillatory behaviour as functions of time.