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Chinese Physics B

New 3-mode bosonic operator realization of SU(2) Lie algebra: From the point of view of squeezing

Cheng Da, Qian-Fan Chen, Hong-Yi Fan

Published Sep 1, 2014• 1 citation
Publisher

Abstract

We consider the quantum mechanical SU(2) transformation e 2λ J z J ± e −2λ J z = e ±2λ J ± as if the meaning of squeezing with e ±2λ being squeezing parameter. By studying SU(2) operators ( J ± , J z ) from the point of view of squeezing we find that ( J ± , J z ) can also be realized in terms of 3-mode bosonic operators. Employing this realization, we find the natural representation (the eigenvectors of J + or J − ) of the 3-mode squeezing operator e 2λ J z . The idea of considering quantum SU(2) transformation as if squeezing is liable for us to obtain the new bosonic operator realization of SU(2) and new squeezing operators.

DOI
10.1088/1674-1056/23/9/090302

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We consider the quantum mechanical SU(2) transformation e 2λ J z J ± e −2λ J z = e ±2λ J ± as if the meaning of squeezing with e ±2λ being squeezing parameter. By studying SU(2) operators ( J ± , J z ) from the point of view of squeezing we find that ( J ± , J z ) can also be realized in terms of 3-mode bosonic operators. Employing this realization, we find the natural representation (the eigenvectors of J + or J − ) of the 3-mode squeezing operator e 2λ J z . The idea of considering quantum SU(2) transformation as if squeezing is liable for us to obtain the new bosonic operator realization of SU(2) and new squeezing operators.
We discuss the unitary operator corresponding to the general two-mode coordinate-momentum mixed transformation , where and are arbitrary real numbers. Suitably selecting the parameters and , we obtain a new two-mode bosonic realization of the Lie algebra. We also study the squeezing effects of the squeezed vacuum associated with the new two-mode bosonic realization of the Lie algebra. The results show that the new squeezed vacuum does not possess second-order squeezing, but exhibits higher-order squeezing.
By extending the Einstein-Podolsky-Rosen bipartite entanglement to the tripartite case,we construct the common eigenvector of the tripartite centre-of -mass coordinate and two mass-weighted relative momenta,which is a new entangled state of continuum variables.The classical dilation transform of varibles in such a state induces new entangled state of continum variables,The classical dilation transform of variables in such a state induces a new three-mode squeezing operator related to a three-mode bosonic operator realization of SU(1,1) Lie algebra.a new three-mode squeezing operator related to a three-mode bosonic operator realization of SU(1,1) Lie algebra.
We find that a three-mode boson realization of the SU(1,1) Lie algebra is involved in solving the three coupled harmonic oscillators' problem. The unitary operator U, which is found to be able to transform the Fock space of a three–dimensional isotropic harmonic oscillator into the space in which the Hamiltonian of three coupled oscillators is diagonized, is further decomposed as a quantum–mechanical rotation in three–mode Hilbert space followed by an SU(1,1) squeezing transformation. The coordinate representation of this SU(1,1) unitary operator is obtained.