Quantization of the Optical Phase Space S^2 = {phi mod 2pi, I > 0} in Terms of the Group SO(1,2)

Abstract

The problem of quantizing the canonical pair angle and action variables phi and I is almost as old as quantum mechanics itself and since decades a strongly debated but still unresolved issue in quantum optics. The present paper proposes a new approach to the problem, namely quantization in terms of the group SO(1,2): The crucial point is that the phase space S^2 = {phi mod 2pi, I>0} has the global structure S^1 x R^+ (a simple cone) and cannot be quantized in the conventional manner. As the group SO(1,2) acts appropriately on that space its unitary representations of the positive discrete series provide the correct quantum theoretical framework. The space S^2 has the conic structure of an orbifold R^2/Z_2. That structure is closely related to the center Z_2 of the symplectic group Sp(2,R). The basic variables on S^2 are the functions h_0 = I, h_1 = I cosphi and h_2 = -I sinphi, the Poisson brackets of which obey the Lie algebra so(1,2). In the quantum theory they are represented by self-adjoint generators K_0, K_1 and K_2 of a unitary representation. A crucial prediction is that the classical Pythagorean relation h_1^2+h_2^2 = h_0^2 may be violated in the quantum theory. For each representation one can define 3 different types of coherent states the complex phases of which can be "measured" by means of K_1 and K_2 alone without introducing any new phase operators! The SO(1,2) structure of optical squeezing and interference properties as well as that of the harmonic oscillator are analyzed in detail. The new coherent states can be used for the introduction of (Husimi type) Q and (Sudarshan-Glauber type) P representations of the density operator. The 3 operators K_0, K_1 and K_2 are fundamental in the sense that one can construct (composite) position and momentum operators out of them!