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arXiv:math-ph/0503062 · Journal of Mathematical Physics

Generalized coherent and squeezed states based on the h(1)⊕su(2) algebra

Nibaldo Alvarez-Moraga, Véronique Hussin

Published May 1, 2002• 17 citations
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Abstract

States which minimize the Schrödinger–Robertson uncertainty relation are constructed as eigenstates of an operator which is an element of the h(1)⊕su(2) algebra. The relations with supercoherent and supersqueezed states of the supersymmetric harmonic oscillator are given. Moreover, we are able to compute general Hamiltonians which behave like the harmonic oscillator Hamiltonian or are related to the Jaynes–Cummings Hamiltonian.

DOI
10.1063/1.1462858

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States which minimize the Schrödinger–Robertson uncertainty relation are constructed as eigenstates of an operator which is an element of the h(1)⊕su(2) algebra. The relations with supercoherent and supersqueezed states of the supersymmetric harmonic oscillator are given. Moreover, we are able to compute general Hamiltonians which behave like the harmonic oscillator Hamiltonian or are related to the Jaynes–Cummings Hamiltonian.
Robertson intelligent states which minimize the Schr\" odinger-Robertson uncertainty relation are constructed as eigenstates of a linear combination of Weyl generators of the su(3)su(3)su(3) algebra. The construction is based on the analytic representations of su(3)su(3)su(3) coherent states. New classes of coherent and squeezed states are explicitly derived.
A general approach is presented for constructing coherent states for supersymmetric systems. It uses Rogers's supermanifold formulation of supergroups to extend the group-theoretic method. Supercoherent states are explicitly obtained for the supersymmetric harmonic oscillator. They are shown to be eigenstates of the supersymmetric annihilation operator and to be minimum-uncertainty states. Two more-complex situations with extended physical supersymmetries are also considered: an electron moving in a constant magnetic field, and the electron-monopole system. The supercoherent states for these systems are found using super Baker-Campbell-Hausdorff relations and their interpretation is elucidated.
This study generalizes the supersymmetric coherent states introduced by Aragone and Zypman in 1986. The Hamiltonian of the supersymmetric quantum harmonic oscillator leads to the definition of the generalized supersymmetric annihilation operators as a 3-parameter family. Their eigenstates are the generalized supercoherent states, which can be calculated explicitly for three relevant regions of parameter space. The uncertainty in position and momentum is discussed, with specific concentration on where the uncertainty is saturated, where it is bounded, and where it is unbounded.