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Physical Review Letters

Squeezing and quantum groups

E. Celeghini, Mario Rasetti, Giuseppe Vitiello

Published Apr 22, 1991• 111 citations
Publisher

Abstract

Generalized quasicoherent states for the Weyl-Heisenberg quantum group have been defined by Biedenharn and MacFarlane. In this Letter other quantum Weyl-Heisenberg coherent states are defined for complex q in the usual Fock space. Such states are shown to exhibit interesting squeezing properties, in particular when \ensuremath{\Vert}q\ensuremath{\Vert}\ensuremath{\approxeq}1, for the q analog to the harmonic oscillator.

DOI
10.1103/physrevlett.66.2056

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Generalized quasicoherent states for the Weyl-Heisenberg quantum group have been defined by Biedenharn and MacFarlane. In this Letter other quantum Weyl-Heisenberg coherent states are defined for complex q in the usual Fock space. Such states are shown to exhibit interesting squeezing properties, in particular when \ensuremath{\Vert}q\ensuremath{\Vert}\ensuremath{\approxeq}1, for the q analog to the harmonic oscillator.
Generalized q-boson operators associated with the simultaneous creation of several q-bosons are introduced. These operators give rise to a realization of the quantum Weyl-Heisenberg algebra and are applied to the construction of corresponding Holstein-Primakoff realizations of the quantum algebras SU q (2) and SU q (1,1). Coherent states of these algebras are defined in the various ways suggested by the equivalent definitions of the harmonic oscillator coherent states, and some of their properties are studied. Particular attention is devoted to the squeezing properties of the quadrature of the electromagnetic field in these states.
The multiphoton squeezed states defined in this paper are generalizations of the conventional coherent (Glauber) and squeezed (Yuen) states previously discussed by many authors. We define multiphoton generalizations of the latter by a unified class of states that includes the Holstein–Primakoff realizations of SU(2) and SU(1, 1) as well as the standard harmonic oscillator coherent states (Weyl–Heisenberg group) and squeezed states in a general framework that allows also non-Hermitian realizations. We determine the squeezing properties of these states in a unified formalism and study numerically their dependence on the parameter classifying the states.
The noise (variance squared) of a component of the electromagnetic field - considered as a quantum oscillator - in the vacuum is equal to one half, in appropriate units (taking Planck's constant and the mass and frequency of the oscillator all equal to 1). A practical definition of a squeezed state is one for which the noise is less than the vacuum value - and the amount of squeezing is determined by the appropriate ratio. Thus the usual coherent (Glauber) states are not squeezed, as they produce the same variance as the vacuum. However, it is not difficult to define states analogous to coherent states which do have this noise-reducing effect. In fact, they are coherent states in the more general group sense but with respect to groups other than the Heisenberg-Weyl Group which defines the Glauber states. The original, conventional squeezed state in quantum optics is that associated with the group SU(1,1). Just as the annihilation operator a of a single photon mode (and its hermitian conjugate a, the creation operator) generates the Heisenberg Weyl algebra, so the pair-photon operator a(sup 2) and its conjugate generates the algebra of the group SU(1,1). Another viewpoint, more productive from the calculational stance, is to note that the automorphism group of the Heisenberg-Weyl algebra is SU(1,1). Needless to say, each of these viewpoints generalizes differently to the quantum group context. Both are discussed. The following topics are addressed: conventional coherent and squeezed states; eigenstate definitions; exponential definitions; algebra (group) definitions; automorphism group definition; example: signal-to-noise ratio; q-coherent and q-squeezed states; M and P q-bosons; eigenstate definitions; exponential definitions; algebra (q-group) definitions; and automorphism q-group definition.
A set of deformed k-boson operators are defined and it is shown that the nonlinear transformations from the usual deformed oscillator operators to the deformed k-boson operators form an Abelian group. It is shown that the multiboson operators satisfy quantum Weyl–Heisenberg group. The generalized Holstein–Primakoff realizations in terms of these multibosons operators are used to define the quantum group-theoretic coherent states of SUq(1,1) and SUq(2). The uncertainties ΔX(k′,q) and ΔP(k′,q) with respect to those group-theoretic and generalized coherent states are computed explicitly and the results show that these coherent states are squeezed.
Using a generalization of the q-commutation relations, we develop a formalism in which we define generalized q-bosonic operators. This formalism includes both types of the usual q-deformed bosons as special cases. The coherent states of these operators show interesting and novel noise reduction properties including simultaneous squeezing in both field components, unlike the conventional case in which squeezing is permitted in only one component. This also contrasts with the usual quantum group deformation which also only permits one-component squeezing.