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Annual Meeting Optical Society of America

Pseudospin approach to the dynamics and squeezing of SU(2) and SU(1,1) coherent states

P. K. Aravind

Published Jan 1, 1987• 0 citations
Publisher

Abstract

The semiclassical Dicke model and the ideal parametric amplifier are described by Hamiltonians linear in the SU(2) and SU( 1,1) group generators, respectively. We use the usual pseudospin vector as well as its SU(1,1) analog1 to give a unified description of the dynamics of the coherent states in these models. The pseudospin dynamics is governed by a precession (or pseudoprecession) equation, hence simple geometrical arguments can be used to infer the evolution of the coherent state parameters. To study fluctuations and squeezing we introduce a correlation tensor which is a higher-order generalization of the pseudospin vector. The correlation tensor obeys a generalized precession equation which can be factored into scalar, vector, and quadrupole parts and solved exactly by application of the Cayley-Hamilton theorem. Our approach to the calculation of squeezing and fluctuations provides an alternative to the more traditional approach employing disentangling theorems. It also provides an attractive and closely parallel geometrical description of coherent state evolution and squeezing in the two models.

DOI
10.1364/oam.1987.thpo21

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The semiclassical Dicke model and the ideal parametric amplifier are described by Hamiltonians linear in the SU(2) and SU( 1,1) group generators, respectively. We use the usual pseudospin vector as well as its SU(1,1) analog1 to give a unified description of the dynamics of the coherent states in these models. The pseudospin dynamics is governed by a precession (or pseudoprecession) equation, hence simple geometrical arguments can be used to infer the evolution of the coherent state parameters. To study fluctuations and squeezing we introduce a correlation tensor which is a higher-order generalization of the pseudospin vector. The correlation tensor obeys a generalized precession equation which can be factored into scalar, vector, and quadrupole parts and solved exactly by application of the Cayley-Hamilton theorem. Our approach to the calculation of squeezing and fluctuations provides an alternative to the more traditional approach employing disentangling theorems. It also provides an attractive and closely parallel geometrical description of coherent state evolution and squeezing in the two models.
The semi-classical Dicke model and the ideal parametric amplifier are described by Hamiltonians linear in the SU(2) and SU(1, 1) group generators, respectively. We use the usual pseudospin vector as well as its SU(1, 1) analog to give a unified description of the dynamics of the coherent states in these models. The pseudospin vector obeys a precession (or pseudoprecession) equation; hence simple geometrical arguments can be used to relate its evolution to that of the coherent-state parameters. To study fluctuations and squeezing we introduce the pseudospin tensor, which is a higher-order generalization of the pseudospin vector. The pseudospin tensor obeys a generalized precession equation that can be factored into scalar, vector, and quadrupole parts and solved exactly by application of the Cayley–Hamilton theorem. Squeezing is related directly to the behavior of the. pseudospin tensor and can therefore be calculated without invoking the group-theoretical disentangling formulas. Our approach is thus elementary in character. It also provides an attractive and closely parallel geometrical account of coherent-state evolution and squeezing in the two models.
The atomic pseudospin operators describing a two-level atom, and obeying the SU(2) Lie algebra, can exhibit reduced quantum fluctuations (“squeezing”) in certain states of the atom. These states include both pure and mixed states of the atom. Parameterizing the atom by the spherical polar components S, θ, and ϕ of its pseudospin vector S, we determine the parameter ranges in which squeezing can occur and the magnitude of this squeezing. We illustrate our considerations by examining two models, namely, the coherent-state Jaynes–Cummings model and the optical Bloch equations, in which mixed-state SU(2) squeezing arises dynamically.
The theory of generalized coherent states associated with the SU(2) and SU(1,1) Lie algebras is discussed and applied in an investigation of possible reductions of fluctuations. In the framework of a system of N two-level atoms the squeezing of angular-momentum [SU(2)] fluctuations is exhibited for optical coherent transients involving the photon echo. The SU(1,1) fluctuations are discussed and established for general two-photon processes involving dynamical variables different from the creation and annihilation operators.