Quantum States of Light by Akira Furusawa

Author:Akira Furusawa
Language: eng
Format: epub
Publisher: Springer Japan, Tokyo

1.11 Squeezing Operation and Squeezed States

We will describe a squeezed vacuum and general squeezed states in a different way from the previous section in order to explain them more extensively. We will do it in historical order.

A coherent state is an eigenstate of annihilation operator as mentioned in Sect. 1.2. We make a Bogoliubov transformation of operator and get operator as follows:

(1.154)

Squeezed states will then be the eigenstates of operator . Here the Bogoliubov transformation is called a squeezing operation. Note that the Bogoliubov transformation was introduced to explain the process of creation of Cooper pairs of electrons in superconducting materials in the BCS theory. Therefore the squeezing process makes pairwise photons, where a photon is a Boson and an electron is a Fermion and we have to be careful about the difference between the commutation relationship and the anti-commutation relationship.

The Bogoliubov transformation can be regarded as time-evolution of an operator in the Heisenberg picture. In the Schrödinger picture we treat time-evolution of a quantum state with the Schrödinger equation (Eq. (1.41)) as follows:

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