Quantum—Classical Correspondence for the Electromagnetic Field II: P, Q, and Wigner Representations

Abstract

The definition of the P representation as the Fourier transform of the normal-ordered characteristic function can be generalized by simply taking different characteristic functions — characteristic functions that give operator averages in other than normal order. Here we will look at two new representations: the Q representation, which is defined in terms of the characteristic function that gives operator averages in antinormal order, and the Wigner representation, defined in terms of the characteristic function that gives operator averages in symmetric, or Weyl, order. This is not a comprehensive list. Cahill and Glauber [4.1], and Agarwal and Wolf [4.2] have introduced formalisms in which whole classes of different representations are defined. In particular, Agarwal and Wolf take the possibilities to their ultimate extreme and develop a very general and elegant formalism which they call the phase-space calculus. These general formalisms are not of much interest, however, when it comes to applications. The P, Q, and Wigner representations are the only examples that have traditionally seen any use in quantum optics. They are special cases within the classes defined by Cahill and Glauber, and Agarwal and Wolf. In Volume 2 we will meet one recent addition to the list which has been used quite extensively, particularly in the treatment of squeezing and related nonclassical effects. This is the positive P representation introduced by Drummond and Gardiner [4.3]. As the name suggests, the positive P representation is closely related to the Glauber—Sudarshan P representation. We postpone its discussion, however, until we have acquired the background needed to appreciate its special purpose and application. Certain properties of the positive P representation are still only partly understood; this representation therefore belongs with the modern research topics that are taken up in Volume 2.