Quantization of the optical phase space 𝒮² = {ϕ mod 2π, I > 0} in terms of the group SO^↑(1, 2)

Abstract

The problem of quantizing properly the canonical pair "angle and action variables ", ϕ and I, is almost as old as quantum mechanics itself and since decades an intensively 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 = {ϕ mod 2π, I > 0} has the global structure S 1 × R + (a simple cone) and cannot be quantized in the conventional manner. As the group SO(1, 2) acts transitively, effectively and Hamilton-like on that space its irreducible unitary representations of the positive discrete series provide the appropriate quantum theoretical framework. The phase space S 2 has the conic structure of an orbifold R 2 /Z 2 . That structure is closely related to a Z 2 gauge symmetry which corresponds to the center of a 2-fold covering of SO(1, 2), the symplectic group Sp(2, R). The basic variables on the phase space are the functions h 0 = I , h 1 = I cos ϕ and h 2 = −I sin ϕ the Poisson brackets of which obey the Lie algebra so(1, 2). In the quantum theory they are represented by the self-adjoint Lie algebra generators K 0 , K 1 and K 2 of a unitary representation, where K 0 has the spectrum {k + n, n = 0, 1, . . . ; k > 0}. A crucial prediction is that the classical Pythagorean relation h 2 1 + h 2 2 = h 2 0 can be violated in the quantum theory. For each representation one can define three different types of coherent states the complex phases of which may be "measured" by means of the operators 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 additional coherent states can be used for the introduction of (Husimi type) "Q" distributions and (Sudarshan-Glauber type) "P" representations of the density operator. The three 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! The new framework poses quite a number of fascinating experimental and theoretical challenges.

Entangled photon–electron states and the number-phase minimum uncertainty states of the photon field

“…We mention, that for large coherent excitations of the mode, the moments of C and S have a similar form of the moments of the ordinary c-number cosine and sine functions. We have to note here that Kastrup (2006a) has recently raised serious objections against the use of the Susskind and Glogower cosine and sine operators in the description of quantal phase properties of the linear oscillator. On the basis of the analysis presented in Chapter 5 of his paper, he concludes that "the London-Susskind-Glogower operators k C ~ and k S ~ are not appropriate for measuring angle properties of a state!".…”

“…See also the critical review by Lynch (1995) and the book by Peřinová et al (1999) on the description of phase in optics. Concerning the recent developments of the concept of quantum phase of a linear oscillator, see the thorough group theoretical studies by Kastrup (2003Kastrup ( , 2006aKastrup ( and 2007, in which a genuinely new approach to this problem has been worked out.…”

Playing with Numbers, with Fermions and Bosons

“…In the unitarity range n ≥ 1 of the infinite register, the operator F 0 = an − 1 2 , from (11), can be identified as the fermionic realization of the quantum phase operatorê, a =ê √n , i.e., e ≡ F 0 , with F † 0 = F −1 0 : for the phase-modulus decomposition of the bosonic annihilation operator and the strictly related problem of the number-phase uncertainty, here we cite [6], for obvious reasons, and [7] for a recent review on both topics. Since F 0 is not unitary on the whole range n ≥ 0, one retrieves the well known resultsêê…”

Phase state and related nonlinear coherent states

“…First, there is an effort in theoretical physics to bring forward bosons as multimode coherent states of the universal covering group of SU(1, 1) [6][7][8][9]. Such an approach may not simplify the problems found in quantum optics, which are well developed through the Heisenberg-Weyl group provided by the number, creation and annihilation operators, but we will show in the following that it is possible to cast the phase state as a generalized SU(1, 1) coherent state based on the Lie algebraic representation of quantum phase and number operators [10][11][12]. We do not pretend to touch upon the phase problem, but our approach may provide further support to the SU(1, 1) formalism and may open a new avenue to approach some quantum optics problems.…”