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Modern Physics Letters B

Coherent State Representation for Generalized Bosonic Bogolyubov Transformation

Hong-Yi Fan, Yue Fan

Published Nov 20, 1997• 9 citations
Publisher

Abstract

We show that the generalized bosonic Bogolyubov transformation generator [Formula: see text] where G is a n×n matrix, has the manifest coherent state representation [Formula: see text] This expression has the advantage of exhibiting the transition from classical transformation to its quantum mechanical correspondence apparent and providing us with a convenient approach for calculating some Bogolyubov transformed states.

DOI
10.1142/s0217984997001377

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We show that the generalized bosonic Bogolyubov transformation generator [Formula: see text] where G is a n×n matrix, has the manifest coherent state representation [Formula: see text] This expression has the advantage of exhibiting the transition from classical transformation to its quantum mechanical correspondence apparent and providing us with a convenient approach for calculating some Bogolyubov transformed states.
The formalism of mapping a classical canonical transformation to quantum unitary operators for a boson system is generalized to fermion systems. In terms of the ``integration within an ordered product'' technique for fermion systems, the transition of a classical transformation in Grassmann number space to a quantum image is manifestly evaluated in the fermion coherent-state representation. The normally ordered unitary operators that generate the Bogolyubov-Valatin transformation and the SU(2) rotation are derived in this way.
Using the completeness relation composed of the coherent state and of the eigenket of bosonic creation operator, we establish a one-to-one correspondence between the z -transform and the quantum-mechanical transform from the representation by number states to the representation by coherent states (Bargmann representation). In this way, the quantum-mechanical version of the various properties of z -transform are obtained and the operators for embodying these properties in the Fock space are derived, which may find applications in quantum states engineering.
Classical linear transformations in coordinate-momentum phase space are mapped to unitary quantum-mechanical operators in Hilbert space to produce a generalization of squeeze operators. The classical to quantum transition is manifestly apparent in the derivation. The unitary operators are evaluated in the coherent-state representation using the ``integration within ordered products'' technique in the one- and two-mode cases. Application of these operators results in a new generalization of coherent states.