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Acta Physica Sinica

SUPERCOHERENT STATES AND BERRY'S PHASE

C Chen

Published Jan 1, 1992• 0 citations
OpenAlex

Abstract

Supercoherent states as the result of acting of displacement operator on a supersymmetric hamiltonian are introduced and discussed. Berry's phase is calculated for coherent states as their parameters are taken to be varying adiabatically.

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SUPERCOHERENT STATES AND BERRY'S PHASE

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We define the Berry phase for the Heisenberg operators. This definition is motivated by the calculation of the phase shifts by different techniques. These techniques are: the solution of the Heisenberg equations of motion, the solution of the Schrodinger equation in coherent-state representation, and the direct computation of the evolution operator. Our definition of the Berry phase in the Heisenberg representation is consistent with the underlying supersymmetry of the model in the following sense. The structural blocks of the Hamiltonians of supersymmetrical quantum mechanics ('superpairs') are connected by transformations which conserve the similarity in structure of the energy levels of superpairs. These transformations include transformation of phase of the creation-annihilation operators, which are generated by adiabatic cyclic evolution of the parameters of the system.
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.
The authors present the supercoherent states. They are introduced as the eigenstates of the supersymmetric annihilation operator of the quantum mechanical supersymmetric harmonic oscillator. They have a compact expression in terms of the standard (bosonic) coherent states. For each value of the complex parameter z there are two linearly independent orthogonal supercoherent states. One of them has pure fermionic character and is the more classical, while the other is fully supersymmetric (i.e. the mean value of the Klein operator when the system is set in this state vanishes). There are no purely bosonic supercoherent states. The supercoherent fermionic states saturate both the Heisenberg uncertainty relation and the new entropic uncertainty recently introduced by Deutsch (1983), while the supersymmetric ones almost (but not exactly) make it. They give a classical (or almost classical) mean value for the energy of the system and do not spread along their time evolution.